1,537 270 170MB
English Pages 254 [247] Year 2016
Mauro Nisoli
Semiconductor Photonics Principles and applications
git su|:|ET.& EDITRICE I
=—'§ ESCULAPIO
Contents
Preface Band structure of semiconductors Crystals. lattices and cells
1.1 1.1.1
The Wigner-Seltz cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
The reciprocal lattice
1.3
Electrons in a periodic crystal The concept ot eitective mass Energy bands Electrons and holes in a semiconductor . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 1.5 1.5.1 1.6 1.6.1 1.6.2 1.6.3 1.6.4 1.6.5 1.7 1.8 1.8.1 1.8.2 1.8.3
2.1
~1610 13 14
. 14
Calculation oi the band structure
15 15 18 19 21 22
Tight-binding method . . . . . . . . . . . . . . . . . . . . . . . .
Crystal with one-atom basis and single atomic orbital Linear lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slmple cubic lattice . . . . . . . . . . . . . . . . . . . . . . . . . .
Band structure of semiconductors calculated by TBM The k - p method Bandstructures of a few semiconductors
25 28
Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Gallium Arsenlde . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gallium Nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electrons in semiconductors introduction
- - - . . . . - - i i ' - - - -
29 30
33
5.7
Competition between radiative and nonradiative recombination QLlCJl‘iii.ll*.“ ‘1‘.’e-ll=1
6.1 6.2 6.2.1 6.3 6.4 6.5 6.5.1 6.6 6.7 6.8 6.9 6.9.1
introduction Electronic states Electronic states in the valence band . . . . . . . . . . . . . . . . . . . . . . . . . .
Density of states Electron density Transition selection rules lntersubband transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Absorption and gain in a quantum well lntersubband absorption Strained quantum wells Transparency density and differential gain Differential gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Excitons 6.10.1 Absorption spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.2 Excltons in quantum wells . . .
Light i:r‘r\-iitin:.; -T.=iP»-
Indeed:
f(r + R) = Z fa a@~< 4 secular problem. The calculations are rather complex. To get some physical insight into the problem we will solve a simple problem of one atom basis with only an s—function.
Crystal with one-atom basis and single atomic orbital Since we have to consider only one atomic level, the coefficients {bm} are zero except for the s—leve1, where b, = 1. A single equation results in this case. If 8,, is the energy of the atomic s-level and we write ¢,(r) = ¢(r), the following functions can be introduced: a(R) = /¢"(r)¢(r — R)dr
(1.69)
.6 = - / ¢*|11..»6>|2 gm _ gm)
l (1.97)
Note that the linear terms in k are not present since we are assuming that Sno represents a maximum or a minimum of the band structure. In 1.96 and 1.97 the matrix elements (u.,,0|k ~ p|u,,10) can be written as: (u,,0|k - p|u,,/0) = luaok ~ punro dr.
(1.98)
Since k is a vector of real numbers, these matrix elements can be rewritten as:
(walk-PI116/6) = k - ("n0|P|'"»n'0)-
(1-99)
l.7 The k - p method
27
For small values of k: it is very useful to write €,,k as:
62162
51111 = 5110 + T; TTL
(1-100)
where m‘ is the effective mass. By comparing Eqs. 1.100 and 1.97 we obtain the following expression for the effective mass: 2 m,1 __- mo1 + mgk,
||2 = l1< b3) __ 21¢ (a2 "b2)(as a1_(a2
(2.17)
‘