Semiconductor Opto-Electronics 9780408703260

Table of contents :
01a_Front-Matter_1973_Semiconductor-Opto-Electronics
01b_Copyright_1973_Semiconductor-Opto-Electronics
01c_PREFACE_1973_Semiconductor-Opto-Electronics
01d_UNITS_1973_Semiconductor-Opto-Electronics
02_moss1973 (1)
03_moss1973 (2)
04_moss1973 (3)
05_moss1973 (4)
06_moss1973 (5)
07_moss1973 (6)
08_moss1973 (7)
09_moss1973 (8)
10_moss1973 (9)
11_moss1973 (10)
12_moss1973 (11)
13_moss1973 (12)
14_absorption-due-to-direct-transitions-between-the-valence-band-an-1973
15_reference-list-and-author-index-1973
16_subject-index-1973

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SEMICONDUCTOR OPTO-ELECTRONICS T.S. M O S S , P h . D . , F.Inst.P. G.J. B U R R E L L , P h . D . , M.Inst.P. a n d B. E L L I S , P h . D . , A . K . C .

LONDON

BUTTERWORTHS

THE BUTTERWORTH GROUP ENGLAND Butterworth & Co (Publishers) Ltd London: 88 Kingsway, WC2B 6AB AUSTRALIA Butterworths Pty Ltd Sydney: 586 Pacific Highway, NSW 2067 Melbourne: 343 Little Collins Street, 3000 Brisbane: 240 Queen Street, 4000 CANADA Butterworth & Co (Canada) Ltd Toronto: 14 Curity Avenue, 374 NEW ZEALAND Butterworths of New Zealand Ltd Wellington: 26-28 Waring Taylor Street, 1 SOUTH AFRICA Butterworth & Co (South Africa) (Pty) Ltd Durban: 152-154 Gale Street

First published in 1973 © Butterworth & Co. (Publishers) Ltd, 1973 ISBN 0 408 70326 1

Printed in Hungary and bound in England by Chapel River Press

PREFACE

The term opto-electronics covers the basic physical phenomena a n d device behaviour which arise from the interaction between electromagnetic radiation a n d the electrons in a solid. Generally these phenomena result either from the absorption of radiation, with consequent electronic effects or, conversely, from the generation of radiation by electron flow within a semiconductor. The subject is of expanding interest b o t h in the fundamental a n d applied fields. On the academic side, the twin approach afforded by the use of optical and electrical measurements gives an invaluable insight into the fundamental properties of semiconductors, many of which are now better understood than either metals or insulators. F r o m the point of view of commercial research and device development, there is continual progress in the production of radiation detectors of wide spectral range and high sensitivity, while electroluminescent lamps and semiconductor lasers are now moving rapidly into the applications phase. The first nine chapters of this b o o k are devoted to theoretical topics, covering the interaction of electromagnetic waves with solids, dispersion theory and absorption processes, magneto-optical effects, and non-linear phenomena. Theories of photo-effects a n d photo-detectors are treated in detail, as are the theories of radiation generation and the behaviour of semiconductor lasers and lamps. F o r a small p a r t of the general theory we have drawn on an earlier b o o k by one of us (Moss, T. S., Optical Properties of Semiconductors, Butterworths, London, 1959) now out of print. In dealing with the wide range of materials which exhibit o p t o electronic effects, we have limited ourselves to three chapters covering the group IV elements, I I I - V compounds, and a selection of the most important chalcogenides. T h u s we have included virtually all materials of current applications interest as well as the m o r e important semiconductors from the point of view of opto-electronic V

vi

Preface

phenomena a n d basic understanding. The b o o k is intended primarily for physicists engaged in academic research or commercial device development a n d for honours students specialising in solid-state physics. It contains an extensive reference list/author index a n d subject index—very largely due to the efforts of Audrey M o s s — a n d we t h a n k all three wives for their assistance in the creation of the book. Farnborough, H a n t s

T.S.M. G.J.B. B.E.

UNITS

All equations in this b o o k are expressed in terms of rationalised m.k.s. units. Numerical values are quoted in SI units and, in accord with recommended practice, are expressed in general in terms of 10 multiples or 1 0 " submultiples of the basic SI units. T o assist comparison with some of the c o m m o n quantities which have been discarded, it may be noted that 3

3

1 n m = 10 A 1 W b / m (i.e. 1 T) = 10 G 2

4

1 bar (i.e. 10 N / m ) = 1-0197 kgf/cm 5

2

2

However, there are certain quantities which have well-established units and which would be unfamiliar if quoted in terms of the metre. Exceptions have therefore been made of the following Carrier concentrations are quoted in units of c m " Absorption coefficients K are in units of c m Wavenumbers are in units of c m " Detectivities D* of photodetectors are in units of cm W 3

- 1

1

vii

- 1

Hz

1 / 2

Chapter 1

OPTICAL CONSTANTS OF SOLIDS

1.1

E L E C T R O M A G N E T I C WAVES

Transmission of visible a n d infra-red radiation through conducting media and the behaviour of such radiation at the interfaces between different media are aspects of the general behaviour of electromagnetic waves which are governed by Maxwell's field equations. These equations may be written curl

(1.1a)

curl

(1.1b)

div

(l.ld)

d.lc)

where the symbols have their usual meanings. In general, homogeneous isotropic media will be considered*, so

since the relative permeability ju is unity for all materials at optical wavelengths. Also (1.2a) t For cases outside these approximations, see Born and Wolf (1964). l

2

Optical Constants of Solids

where F = Nes

(1.2b)

is the polarisation produced by displacing a charge Ne a distance s in the field direction. I t is also assumed here that Q = 0. which is valid in any medium of high conductivity a n d is usually an adequate approximation in a semiconductor. T h e relevant equations therefore become (1.3a) (1.3b) div H = 0

(1.3c)

div E = 0

(1.3d)

where e a n d /no are the permittivity a n d permeability of free space, e is the relative permittivity of the medium, and o its conductivity. Hence 0

t

which simplifies t o

with a similar equation for H. Write as a solution for one of the components of E E

= Eo exp ico(t—Olz/c)

x

(1.4)

This solution satisfies the equation provided that Ol

2

= c (eeofJ>o—i0fiolcD) 2

(1.5)

Equation (1.4) represents a wave travelling in the z direction with a propagation constant Ol which is in general complex. When the conductivity of the medium is zero, Oft is real a n d cjQl is the velocity of propagation of the wave, so that 01 is identified with n, the refractive index of the medium. t This is the preferred term for e, but 'dielectric constant' is frequently used especially when it is complex.

Optical Constants of Solids

3

The case of non-zero conductivity leads to the definition of a complex refractive index given by (ft} = c fioe (e-ia/(oeo)

(1.6a)

2

0

N o w for free space e = 1, a — 0, a n d 01=1, relation c //o«o = 1 so that

giving the well-known

2

CM} = e-iojcoeo

(1.6b)

If 01 is written in terms of real components as OL = n-ik then equation (1.4) becomes E

(1.7)

= E exp (—cokz/c) exp [ico(f—wz/c)]

x

0

This expression is seen t o represent a wave of frequency co/2n travelling with velocity c/w, a n d suffering attenuation or absorption a t a rate determined by the absorption index k. T h e equivalent expression for the magnetic field can be obtained by substituting the above solution in equation (1.3a) to give 9

H

y

=

e x

P (—tokzjc) exp [ico(t—nz/c)]

F r o m equations (1.6b) and (1.7) one can obtain n -k 2

= s

(1.8a)

Ink = a/coeo

(1.8b)

2

giving 2n = e(l +&*lafi*t®u*+e 2

2k = £ ( 1 + &*/GfiePe$)iU* -e 2

(1.9a) (1.9b)

It should be noted that a is the conductivity at the optical frequency concerned, a n d is not generally equal to the d.c. or low-frequency conductivity cr . Clearly, as a 0, k 0 a n d n — e. It has been verified experimentally that the latter relation holds provided the measurements of n a n d s are m a d e at the same frequency or, if made a t different frequencies, provided there are n o absorption bands at any intermediate frequencies. 0

2

4

Optical Constants of Solids

The absorption coefficient K for the medium is defined by the condition that the energy in the wave falls to 1 /e in a distance 1 /K. A s the energy flow is given by the Poynting vector, it is thus proportional to the product of the amplitudes of the electric and magnetic vectors. As b o t h of these contain the term exp (—cofcz/c), the attenuation is exp (—2o)kz/c) and the absorption coefficient K = 2cok/c = Aizhjl

(1.10)

where X is the wavelength in free space. Measurements of transmission through samples of the material of different thicknesses may thus be used to determine K and k directly. As equations (1.8) show that in general n ^ e, it should be pointed out that the velocity of the wave is given by c/«, not c/e . Hence measurements of properties determined by the velocity of the wave alone will give a direct value for n. Such a method would be the measurement of interference fringes produced by reflection at, or transmission through, plane parallel samples. In general, experiments of the types described above to determine n a n d k separately can only be performed on material where the absorption is n o t intense, i.e. where specimens can be prepared of thickness only a few times 1/K. F o r highly absorbing materials, where the optical properties more nearly resemble those of metals than dielectrics, it is necessary to rely on the analysis of reflection measurements using polarised radiation, which usually yields two simultaneous equations involving b o t h n a n d k. 2

1/2

When calculating n a n d k in terms of the material parameters, it is frequently convenient to calculate the displacement of the charge under the influence of the field a n d thus obtain the polarisation from equation (1.2b) ". In the case of zero conductivity this is clearly related to the relative permittivity by equation (1.2a); it may further be shown that, if (n—ik) is regarded as a complex dielectric constant in equation (1.2a), then the generalisation of this equation t o the complex case is straightforward. T h e real a n d imaginary parts yield two equations for n and k [equations (1.8)]. 1

2

t Various approaches to this equation exist in the literature; some authors define D simply as the lattice contribution to the displacement (including other contributions in the conductivity term) while others absorb the conductivity into an extended dielectric constant. These different procedures give the same end-result and are really no more than differences in notation (see, for example, Seitz, 1940).

Optical Constants of Solids

5

Contributions to the polarisation may come from several sources, from b o u n d electrons a n d from free carriers for example, a n d their effects are additive.

1.2 B E H A V I O U R A T A N

INTERFACE

The problem of the behaviour of a plane wave at the interface between two media becomes considerably more difficult when one of the media is conducting. SnelPs a n d Fresnel's laws still hold, b u t the interpretation is complicated by the fact that the angle of refraction is complex, and in the refracted wave the planes of constant phase a n d of constant amplitude n o longer coincide. Consider a plane wave travelling in a dielectric medium of refractive index GL and incident on the interface of a conducting medium where the complex refractive index is GL". T h e permeabilities fx\ a n d /Lt are taken as unity. Let the b o u n d a r y of the two media be a t z = 0, with the incident wave in the xz plane. Resolve the electric intensity Eo of this wave into components E normal t o , a n d E parallel t o , the plane of incidence. E thus lies in the y direction 2

n

p

n

m at"

Z

y

Figure 1.1. Reflection and refraction at an interface

whilst E has components E cos 0 and — E sin 0 in the x a n d z directions respectively, where is the angle between the incident wave a n d the surface n o r m a l (see Figure 1.1). p

2

p

p

6

Optical Constants of Solids

The incident wave may be written E = E exp ico[t—0l(x sin $ + z cos = —E cosec (j> = E exp ico[t—Ol(x sin 0-f z cos $)/c] x

z

p

(1.11b) The magnetic components may be found from equation (1.3a) which gives (1.12) Similarly for the reflected wave E' = E^ exp ico[t—Ol(x sin y

Ex sec ' = —E' cosec 0' =

z cos ')/c]

exp ico[t—Ol(x sin

z

(1.13a) z cos ')/c] (1.13b)

and for the refracted wave E' ' = ^ ' exp i(D[t-Gt'\x y

sin

cos ' = E p

p

p

E +E Q

n

= E' n

cos 4>" = (E -E ) p

p

cos

(1.16a) (1.16b)

Optical Constants of Solids

7

Similarly the condition for the magnetic field to be continuous through z = 0 is found t o give

cos 4> =

(E -E' )Ol n

n

01" cos k a n d n » sin 4>. It follows that a n a n d /? ^ k. It may thus be seen that R is a slowly varying function of 0, rising from a typical value ~ 3 0 % at normal incidence to 100% a t glancing incidence, as in Figure 1.2. R has the same value as jR at 0 = 0, b u t passes through a minimum near a = sin tan 0, i.e. when tan 0 ^ w, and then rises rapidly to 100% at 0 = 90°. F o r values of n between 3 a n d 6, this angle of minimum reflectivity lies between 71° a n d 81°. 2

2

2

2

n

n

p

If is measured, there are only two unknowns (a and /?) in equations (1.27) a n d (1.28), a n d so their values (and hence n a n d k) may be determined by any two measurements of reflectivity. Equation (1.27) is the simpler to analyse but, as it varies only slowly with n a n d k, it is not conducive to good accuracy. By contrast, R has the virtue that it varies rapidly with n a n d k in the neighbourhood of = t a n « , a n d measurements near this angle are potentially capable of giving good accuracy for these parameters. By measuring the ratio R /R , the computation is considerably simplified and, furthermore, the accuracy of the experiment is improved as this ratio varies more rapidly with angle of incidence than R alone and the equipment is no longer called upon to measure absolute reflectivities. This method has been used by Avery (1952, 1953) who solved the computational problem by the use of sets of curves of p

_ 1

p

n

p

12

Optical Constants of Solids

values of R /R computed for feasible values of n a n d k a t two o r three specific angles of incidence. These angles should be chosen so that one is just below a n d one just above the angle of minimum reflectivity—for example 65° a n d 83°—with perhaps a third intermediate angle which must inevitably b e fairly near the actual minimum. Measurements a t two angles give only one estimate of n a n d k, measurements a t three angles may be averaged for n a n dfc,a n d the consistency of the three values of n a n d k obtained is a useful indication of the accuracy of the experiment. If k is small, R /R has a sharp minimum a t the pseudo-Brewster angle given by sin 4> t a n $ = a - f / 5 , which approximates closely t o p

n

p

n

2

2

2

2

t a n 4> = n +k 2

2

(1.29)

2

Also (Rv/RnUn - i8 /4a 2

fcW-4)

2

(1.29a)

for high-refractive-index materials. F o r the absolute reflectivity with the electric vector in the plane of incidence, Miller a n d Johnson (1954) give a minimum value (JlpXnin = k (l-2/n )/4n 2

2

(1.29b)

2

As is clear from Figure 1.2, this expression must be smaller than that given b y equation (1.29a). Measurements using equation (1.29b), or preferably equation (1.29a), are probably as good as any polarised reflection experiments for the determination of low values of k. Even so, if (R /Rn)min = 1 % is taken as a reasonable limit of accurate measurement, this corresponds t o k = 0-6 for n = 3. These figures clearly emphasize the fact that methods of measurement based on the analysis of reflected p o larised light cannot be used for the accurate determination of values of k much less than unity. Another limitation t o the method of reflection measurement is imposed by the state of the surface of the material. In a reflection measurement on a dielectric, it is estimated that a surface layer only ~ A/50 thick is involved, while the presence of a surface film only A/200 thick was found t o raise the reflectivity of glass a t the Brewster angle from its ideal value of zero t o R ~ 1 0 " (Wood, 1933). It is thus of great importance t o ensure that the surface layers of the specimen a r e characteristic of the bulk material, a n d a r e n o t contaminated by oxide films o r work damaged d u e to polishing. Conversely, measurements of ellipticity provide a sensitive method for measuring very thin surface films, this being a valuable p

4

p

Optical Constants of Solids

13

technique for studying surface states, recombination, and oxide growth in silicon planar technology. T h e behaviour t o be expected may be seen from equations (1.17a) and (1.17b). If b o t h media are non-absorbing, the phase shift on reflection from the interface is either 0 or 180°; when either one or b o t h media are absorbing, the phase shift can be any value between 0 and 180°. Also, of course, the reflected amplitudes r and r will differ, so that an incident beam of plane-polarised radiation is in general elliptically polarised after reflection. Ellipsometric measurements determine the state of polarisation of the reflected radiation in terms of two parameters, namely A, the phase change, which is determined directly by the setting of an optical compensator necessary to restore linear polarisation, and tp the inclination of this plane of polarisation which is given by p

n

tan tp =

E /E p

=

n

r /r p

n

if E a n d E are made equal by setting the azimuth of the incident beam at 45° t o the plane of incidence. The detailed theory has been given by Archer (1962) who used pre-computed curves to interpret his results. Simple approximations for very thin films have been given by Archer (1957) w h o showed that b o t h A and rp change linearly with film thickness for films u p to 5 n m thick. Using modern techniques, film thicknesses equivalent to 0-01 nm —i.e. fractions of a mono-layer—are measurable (Meyer and Bootsma, 1969). Results for oxide and silicate films on Si have been given by Frensel (1969), for chemisorption of various gases on Si by Becker and Gobeli (1963), a n d for InSb with various surface treatments by Saxena (1965). Another technique which is useful for studying surface films is that utilising total internal reflection at an interface (Fahrenfort, 1961; and Harrick, 1960). If the rarer medium at the interface is a b sorbing, the reflection is n o longer total, since the penetration of the evanescent wave into the second material results in some loss of energy. As the penetration depth for attenuation to 1 /e is given by p

n

z = A/2TU(«| sin 0 - n ) 2

2

1 / 2

(1.30)

where «i and n are the indices for the dense and rare media, it can be seen that z < A/10 for most angles of incidence 0, so that one can study materials which absorb so strongly that it is not possible to m a k e specimens thin enough to do transmission measurements. 2

14

Optical Constants of Solids

1.4 R E F R A C T I O N A T A N A I R / C O N D U C T O R SURFACE As stated above, the behaviour of the refracted ray is still described by t h e generalised form of Snell's law Qtsm = GL" sin e it follows that A' will increase as X . F o r a semiconductor with ju = 0-1 m / V s a n d m* = 1 0 " kg (i.e. approximately the free-electron mass), o}/Ltm*/e = 1 for a wavelength of a b o u t 1000 (Jim, so that the quadratic relation can be expected to hold in general for wavelengths u p to a few hundreds of micrometres. Hence equation (2.33) becomes 2

2

3 0

(2.34) for an n-type specimen. F o r near-intrinsic semiconductors, a b sorption by b o t h types of carrier is important. T h e original D r u d e (1900) theory of absorption, which postulated two kinds of carrier, 4

38

Dispersion Theory

has been shown by Roberts (1955) to be applicable to our modern concepts of conduction by electrons a n d holes (or to light a n d heavy holes). The contributions add linearly giving (2.35) This equation has been used by Gibson (1956) to interpret absorption in G e a t m m wavelengths. It is generally found that for wavelengths ~ 10 \im or longer the mobility ^ has approximately its zero-frequency value, a n d thus for such wavelengths equation (2.33) may be used to determine absolute magnitudes; in particular it represents a method of calculating the carrier effective mass if the other parameters are measured, see, for example, results for InSb (Moss, 1954, a n d Spitzer and F a n , 1957a). Equation (2.33) may be generalised to deal with anisotropic effective masses. F o r one type of carrier in a cubic crystal where the effective masses along the three main axes are m i , m , a n d m , Brooks (1955) shows that the appropriate mass to be inserted in equation (2.33) is the conductivity mass m , given by 2

3

c

3/m = l / m i + l / m + l / m c

2

3

(2.36)

Detailed treatment of the free-carrier absorption process shows that equation (2.33) corresponds to the assumption of an energyindependent collision time. If this condition does not hold, b u t alternatively the collision time can be written as depending on the energy by the power law r = r ( e n e r g y ) ~ , then it is found (Brooks, 1955) that the absorption given by equation (2.33) is increased by a factor which may be expressed in terms of the well-known g a m m a functions, namely p

0

y(p) = I X 5 / 2 + / 0 T(5/2-p)

[I\5/2)]-*

(2.37)

F o r p = 0, y is clearly unity; for lattice scattering where p = \ , y = 1-13, and for ionised-impurity scattering where p = —3/2, y = 3-4. Little error is thus introduced by ignoring this term (i.e. taking y = 1) in intrinsic specimens where lattice scattering will predominate, b u t in impure specimens the neglect of this factor may be serious. The classical results have been largely confirmed by subsequent quantum-mechanical approaches. According to F a n (1956) when

Dispersion Theory

39

acoustic-mode scattering is predominant, the absorption given by equation (2.33) is multiplied by a factor (4/9tu) (hcojkTfi

((l+2E/fko)

2

(1 + £ / / k o )

1 / 2

>

(2.38)

where the averaging, expressed by the angular brackets, is done over all the electron energies E. This energy is of the order of kT for non-degenerate cases, so that if hco ^> kT the frequency dependence becomes c o * . A t the other extreme of long wavelengths a n d / o r high temperatures, where hco ~ . F o r scattering by optical lattice modes Visvanathan (1960) shows that the free-carrier absorption varies as co~ ' if h(co—co )/kT^> 1, where co is the frequency of the longitudinal optical mode. Both of the above authors also derive expressions for the case when ionised-impurity scattering is dominant. F o r hco kT the frequency dependence is co~ ' a n d the absorption varies as ( m * ) ~ . Broad confirmation of these results was obtained by H a g a a n d K i m u r a (1963, 1964) in the course of detailed calculations for InSb and InAs. A quantum-mechanical treatment by Demidenko (1970) taking account of non-parabolicity in the E-k relationship shows that for acoustic-mode or optical-mode scattering the co' ' a n d co~ ' dependencies should be multiplied by -1

5

3

2 5

LO

LO

3 5

3/2

1 5

2 5

[l + (hco/E r ] 2

G

(l+2fo/£ )-i G

F o r hcojE mm-> d oj /a) can be evaluated a n d curves plotted relating R to these other parameters. Sets of curves for the usual case of measurements at an air interface (i.e. n± = 1) are shown in Figures 2.6 and 2.7*. F o r large values of co r, the following approximate relations may be derived from the condition of minimum reflection 0

p

0

a n

xco

p

min

min

p

(2.42a) (2.42b) (2.42c) (2.42d) The accuracy with which co /co is given by equation (2.42c) is better than 2 % for 10(XR < 1-5H and the accuracy of equation (2.42d) is better than 5 % for 100jR < nl F r o m equations (2.42) the following may be noted: p

min

2

min

0

min

1. X is n o t identical with the wavelength of the minimum reflection; there is a correction factor which becomes considerable if rco is small. With this correction, given by Figure 2.6, precise values of N/m* are therefore easily obtainable. 2. r may be evaluated directly, more directly than from mobility measurements for example, by use of Figure 2.7. p

min

t At low values of cor, the exact expression rather than the approximate one given in equation (2.41) has been used.

45

Figure 2.6. Dependence of plasma frequency on minimum reflectivity

1

2

3

4

5

6

7

8

9

10

15

20

*min C M

Figure 2.7. Dependence of minimum reflectivity on scattering time

46

Dispersion Theory

3. The minimum value of reflectivity varies roughly as r hence as (mobility)" when the mobility is large.

2

and

2

The value of the conductivity can be obtained, since o = Ne(j, = Ne t/m* 2

(2.43)

= nle co r 2

0

p

assuming the usual relation jn = et/m* for the mobility. Measurement of conductivity in this manner should be particularly useful in studying the diffused surface layers used in the fabrication of transistors or integrated circuits, where it is often i m p o r t a n t to know the conductivity rather than carrier concentration. It is interesting to note that at a wavelength a b o u t 2 5 % greater than the plasma wavelength the refractive index goes through a b r o a d minimum (where n ^n /co r) a n d thereafter rises slowly with increasing wavelength. When the distribution of scattering times is taken into account, the equations for the real and imaginary parts of the dielectric constant become 0

p

(2.44a) and (2.44b) where the angular brackets signify averages over the distribution. These averages have been evaluated for various approximations by Schumann a n d Phillips (1967). However, for most conditions of interest, cor is considerably greater than unity a n d the c o " r " term is only a small correction. Hence, irrespective of the detailed scattering mechanism, the determination of N o r m* from Figure 2.6 will be of good accuracy. T h e effect on Ink a n d hence on ico is significant however. Equation (2.42d) shows that r c o is inversely proportional to Ink, so t h a t what is actually determined by the use of Figure 2.7 is the reciprocal of the average < r ~ / ( l + c o ' " r ~ ) ) . As most of the cox values are large, this expression tends to 1 / ( T ) , whereas the average involved in the expression for the mobility is of course (%). Hence, unless the relevant r values lie in a fairly n a r r o w bracket, precise agreement between r values determined from Figure 2.7 and from Hall-effect mobility would not be expected. 2

2

mia

min

1

2

2

- 1

Dispersion Theory

47

When the energy bands are such t h a t the carrier masses are anisotropic, the mass determined from either dispersion or plasma reflection measurements is t h e conductivity mass 3/m

=

c

l/mi+l/m +l/m 2

3

If the energy surfaces are ellipsoids of revolution with longitudinal a n d transverse masses w a n d m this becomes 7

3/m* =

t

2\m -\-\\mi x

If the E-k curves are non-parabolic, then the effective mass obtained is somewhat greater than that for the b a n d edge. It is w o r t h pointing o u t that plasma-edge measurements are usually m a d e at sufficiently long wavelengths for surface effects not t o b e troublesome a n d for the results obtained t o agree with low-frequency determinations. T h e method is particularly useful for the study of highly doped or relatively impure materials. A n extension of the treatment t o cover layers buried beneath oxide or n-layers under p-layers has been given by Moss, Hawkins, a n d Burrell (1968a). A comprehensive review of the advantages a n d limitations of these techniques for studying semiconductor properties has been given recently by Black, Lanning, a n d Perkowitz (1970) a n d measurements of the hole mass in C d S b have been given by Rheinlander (1970). Results for b o t h hole a n d electron masses in grey tin have been given by Wagner a n d Ewald(1971), while T h o m a s a n d Woolley (1971) have studied G a / I n A s a n d I n A s / S b alloys.

Chapter 3

ABSORPTION PROCESSES IN SEMICONDUCTORS

3.1 A B S O R P T I O N A N D R E F L E C T I O N S P E C T R A As indicated in Chapter 2, b o t h b o u n d a n d free electrons produce significant absorption in semiconductors. Electrons in four types of state must be considered: 1. Valence-band electrons, 2. Inner-shell electrons, 3. Free carriers—including, of course, holes as well as electrons, and 4. Electrons b o u n d to localised impurity centres or defects of some type. The absorption process of greatest importance in the study of semiconductors involves transitions from states of the first type a n d arises from the optical excitation of electrons across a forbidden g a p ^ o into the conduction band. In an ideal semiconductor at zero temperature the valence b a n d would be completely full, so that an electron could not be excited to a higher state within the band. The only possible absorption process is that due to photons having sufficient energy to excite the electrons across the forbidden gap, leaving holes in the valence band. If an electric field is applied, the electrons a n d holes will move through the crystal a n d p h o t o conductivity will be observed. In practice the resulting absorption spectrum is a continuum of intense absorption at short wavelengths, bounded by a more or less steep absorption edge (at hv = E ) beyond which the material is relatively transparent. F o r most semiconductors this edge occurs in the infra-red p a r t of the specG

48

Absorption Processes in Semiconductors

49

t r u m and the absorption can result from'direct' or 'indirect' optical transitions, depending on the energy b a n d structure of the semiconductor. In order to illustrate the distinction between direct a n d indirect transitions, consider the form of a typical b a n d structure diagram for a semiconductor; Figure 3.1a shows the b a n d structure of G a A s . This diagram, which is applicable to electron motion along the symmetry directions (111) a n d (100), shows the energy levels available t o the outer shell (valence) electrons. F o r the sake of completeness the notation normally applied to the symmetry points r, X , and L has been inserted. In this material the minimum of the conduction b a n d (Ti) a n d the m a x i m u m of the valence b a n d ( r ) b o t h occur at k = 0 as seen more clearly in Figure 3.1b, which shows an enlarged diagram of the E-k curves in the vicinity of the energy gap. i5

k=0 A- —>-

(a) (c) Figure 3.1. (a) Band structure of GaAs in the (111) and (100) directions (After Cardona, 1967), (b) Enlarged diagram of the E-IL curves for GaAs, showing only the bands in the region of the energy gap, (c) E-k curves for GaP. (Low-temperature energy values have been inserted) k

F o r a semiconductor in which the limits of the energy b a n d s occur at the same value of the wave vector k, the onset of absorption will occur at hv = E as the result of direct transitions. The G

50

Absorption Processes in Semiconductors

absorption coefficient K then rises rapidly to a b o u t 10 c m " as shown, for example, by the logarithmic plot of Figure 3.4. As a consequence of an optically induced transition, an electron is transferred between the two bands without a change in moment u m fik since the m o m e n t u m of the p h o t o n involved is negligible. In a quantum-mechanical calculation of the transition probability, this condition for allowed transitions is apparent as a k-conservation rule in which electrons with a given wave vector in the valence b a n d can make only 'vertical' transitions to states in a higher b a n d having the same wave vector. Non-vertical transitions are nominally forbidden. In some semiconductors the minimum of the conduction b a n d occurs in a different region of k space from the maximum of the valence band. G a P is an example of such a material in which the b a n d structure has the general form of Figure 3.1a but, as shown in Figure 3.1c, the relative positions of the bands are altered such that the lowest conduction-band energy occurs at points along the (100) directions. T h e intense absorption due to direct transitions will cease at the wavelength corresponding to the minimum vertical energy gap, i.e. at hv = 2-9 eV. Optical transitions of lower energy require the participation of phonons in order to conserve moment u m due to the change in electron wave vector. T w o possible p h o non interactions can occur in the optical absorption process, dependent u p o n whether the p h o n o n is absorbed or emitted. T h u s the photon energy required for excitation of an electron across the energy gap is 4

hv

for emission of a p h o n o n of energy E

E +E G

1

P

P

and hv

E

G

—E

P

for p h o n o n absorption.

These 'indirect' transitions occur with a lower probability, as calculated by second-order perturbation theory, and give rise to an absorption edge which is less steep than for direct transitions. Absorption-coefficient data can be derived readily from transmission measurements for photon energies u p to and just in excess of a direct energy gap. Typically with a sample 5 [im thick one can measure absorption coefficients u p to K ^ 2 X 1 0 c m " , b u t the measurement of higher absorption levels becomes increasingly difficult due to the requirement for thinner specimens. It is for this 4

1

Absorption Processes in Semiconductors

51

reason that most optical data in the ultra-violet p a r t of the spectrum have been derived from the measurement of normal-incidence reflectivity, which is interpreted for n a n d k as described in Section 2.4.2. T h e typical form of a reflectivity curve is shown in Figure 3.2

0-6

^0-4 >

% 0-2

en

0

5

10 15 Photon energy hv (eV)

20

25

Figure 3.2. Reflectivity of InSb in the ultra-violet (After Ehrenreich and Philipp, 1962; courtesy The Institute of Physics)

where it is apparent that several peaks are present a t p h o t o n energies u p t o a b o u t 10 eV. These peaks correspond t o transitions, vertical in k space, between valence- a n d conduction-band critical points, usually referred to as 'van Hove singularities' (van Hove, 1953). This behaviour has been observed n o t only in reflection b u t also in absorption (on evaporated films) in many semiconductors of the zinc blende type (Cardona a n d Harbeke, 1962, 1963). T h e mechanisms responsible for this structure will now be discussed; detailed reviews have been given by Phillips (1966) a n d by Greenaway a n d Harbeke (1968). A n important parameter which determines the magnitude of the absorption coefficient for a particular p h o t o n energy hv, a n d which thus in turn influences the reflectivity, is the 'joint density of states'. This is, in effect, the density of energy-level pairs which are available for optical transitions, each pair having one level in each of the two bands with an energy separation hv ^ E —E . T h e joint density of states may be shown to vary rapidly with energy near to the Van H o v e singularities which occur a t symmetry points where c

v

(3.1)

52

Absorption Processes in Semiconductors

and a t points of lower symmetry where V E (k) k

= V E (k)

c

k

v

* 0

(3.2)

The general shape of the absorption curve which arises from transitions between b a n d states in the vicinity of these critical points is dependent on the signs of the functions d (E —E )/dk% d (E -E )/dk% a n d d\E -E )/dk . F o u r possible situations can occur: 2

c

2

v

2

c

w

c

y

1. All three terms are positive a n d a so-called 'Mo-type parabolic edge' is obtained for which the absorption coefficient K increases with photon energy hv. 2. T w o terms are positive a n d one negative, corresponding t o an 'Mi-type saddle-point edge' for which K increases with hv. 3. O n e term is positive a n d two negative, giving a n ' M - t y p e saddle-point edge' for which K decreases with increasing hv. 4. All three terms are negative, giving a n ' M - t y p e parabolic edge' for which K decreases with hv. 2

3

Saddle-point edges ( M i or M ) may satisfy either equation (3.1) or (3.2), b u t parabolic edges (Mo or M ) can occur only when equation (3.1) is satisfied. In a direct-gap semiconductor, the absorption edge which occurs a t hv ^ E is always of type Mo, b u t edges of any of the four types can b e observed a t critical points of higher energy. A mathematical description of the joint density of states for the general case is given by Harbeke (1968); the joint density of states appropriate to the main absorption edge of a direct-gap semiconductor is given in Section 3.2.2. Normally, a singularity a t one point in k space would be expected to give rise to a n absorption edge (either increasing or decreasing with photon energy), whereas to explain the existence of an absorption peak a pair of singularities close in energy are required. Exciton states (as described in Section 3.4) are also thought to play a significant role in the formation of peaks a t the critical points. It was suggested by Phillips (1966) that exciton states should b e associated not only with parabolic edges b u t also with saddle points; in the latter case the excitons would be expected to form metastable states. Exciton transitions a t parabolic edges are described well by Elliott's (1957) theoretical analysis, while theoretical treatments of metastable exciton states a n d their influence on critical-point spectra have been given by Phillips (1966) a n d Velicky a n d Sak (1966). 2

3

G

Absorption Processes in Semiconductors

53

The interpretation of the'reflectivity curves shows that the interb a n d transitions are sufficiently intense (K^ 10 c m ) to give rise to structure in b o t h the real and imaginary parts of the refractive index, n and k. This main absorption band, illustrated in Figure 3.3a, is several eV wide a n d corresponds to peak values of Ink between 20 a n d 30. Integration over the absorption spectrum in accordance with equations (2.30) shows that most of the refractive index arises from this main band. F r o m the point of view of most photocon6

- 1

10 15 20 Photon e n e r g y (eV) (a)

25

10'

^ R e s i s t ra hie n l\ absorption

c10'

Absorption edge

! 10' c o

'a10

2

< ' 10

, / \ Multi-phonon / absorption^/ - Free-carrier / h— absorption

0

10

l

i

20

30

!

AO

50

Wavelength (/xm) (b) Figure 3.3. A typical absorption spectrum for a semiconductor: (a) In the ultraviolet, (b) In the infra-red 5

54

Absorption Processes in Semiconductors

ductive effects, however, it is absorption in the tail of the main band which is all important, corresponding to energies just in excess of E Q . In this tail region, which does not cover a large range of energy b u t which may extend over a wavelength range of several micrometres, Ink is commonly of the order of unity a n d the absorption constant K usually lies in the range 10 to 10 c m . As the photon energy is increased further into the ultra-violet spectrum, ultimately transitions between inner-shell electrons a n d the conduction b a n d are observed. In G e and the zinc blende semiconductors, the first excitation to be observed is from atomic d states, apparent as a small increase in reflectivity and a large increase in absorption constant in the region of 20 eV (Figures 3.2 and 3.3). Reflectivity measurements by Philipp and Ehrenreich (1962) have identified J-band transitions at 18 eV in InSb a n d InAs, in which the excitations are from the atomic d levels of I n ; it is the G a d b a n d at 22 eV which is responsible for transitions in G a A s a n d G a P . Si is a notable exception which does not exhibit rf-band excitation; the inner-shell transition of lowest energy occurs at a b o u t 100 eV due to transitions from p bands (Tomboulian and Bedo, 1956). All of these inner-core transitions and others at even shorter wavelengths are usually referred to as x-ray spectra. 3

4

- 1

It is significant that, although the inner-core transitions give rise to large absorption coefficients (K 10 c m " ) , these do not cover a large wavelength range and, therefore, according to equation (2.30b) they do n o t contribute greatly to the long-wavelength refractive index. A n intermediate range of optical frequencies exists which is sufficiently displaced from the absorption bands described above for n and k to be influenced b u t little by these bands. In this range the optical behaviour is determined mainly by the valence-band electrons (four per atom) which behave like a plasma of free particles (see Section 2.5). A theoretical treatment of this optical p r o p erties of semiconductors in this plasma dispersion region of the ultra-violet has been given by Ehrenreich and Philipp (1962). A s is characteristic of many metals in the ultra-violet, the reflectivity decreases to around 0*01 over this range, as shown in Figure 3.2. A feature of Figure 3.3b which has not been mentioned is the gradual increase in absorption which occurs on the long-wavelength side of the absorption edge. This is attributable to free-carrier a b sorption (see Section 2.5.) and, of course, is of a magnitude which 5

1

Absorption Processes in Semiconductors

55

is proportional to the concentration of free electrons or holes present. Another absorption mechanism, mentioned previously in Section 2.2.2, occurs in the far-infra-red spectrum as a result of the excitation of lattice vibrations, giving rise to the well-known Reststrahlen bands. These absorption bands are very intense in ionic materials a n d are most easily studied in reflection. The covalent semiconductors G e and Si do not exhibit this intense absorption. A t shorter wavelengths weaker absorption bands may be observed due to the excitation of two or more phonons, the intensity of t h e bands decreasing with the number of phonons involved. The general features of lattice absorption spectra are included in Figure 3.3b.

3.2 D I R E C T O P T I C A L

TRANSITIONS

3.2.1 T H E T R A N S I T I O N PROBABILITY

The absorption coefficient due to direct transitions at the fundamental edge is determined by the transition probability between pairs of energy levels in the two bands and the number of such levels which interact with the radiation. The transition probability P (f) from a filled state 0 to an empty state m in the presence of electromagnetic radiation is derived in standard textbooks using, first-order perturbation theory (see, for example, Heitler, 1944 a n d M o t t and Sneddon, 1948). In terms of the intensity I(v) of the radiation the probability that a transition has occurred after a time t is m0

(3.3) where the matrix element of the perturbation is given by (3.4) hto is the energy separation of the two levels, m is the freeelectron mass, and co is 2nv, the angular frequency of the radiation. Here n is the refractive index and I p ^ J is the square of the m o m0

0

2

5'

56

Absorption Processes in Semiconductors

mentum matrix element ", given by 1

(3.5) where (3.6) It should be noted that in this expression the wave vector q of the electromagnetic radiation has been neglected. T h e integration is performed with respect to the volume element dr, and tp and ipo are the wave functions appropriate to the two states. F o r a semiconductor these are the Bloch functions (Bloch, 1928) of the conduction and valence bands. These functions are responsible for the existence of the k-conservation law for direct interband transitions a n d take the form m

= « ( r , kv) exp i k - r , v

v

\p = w (r, kc) exp i k -r m

c

(3.7)

c

where k and k are the wave vectors appropriate to the two states in the valence a n d conduction bands, respectively, a n d w (r, k ) , « ( r , kc) are functions which have the periodicity of the lattice. Since a single pair of energy levels is being considered, these wave functions have to be normalised over the volume