Semiconductor photonics. Principles and Applications [Kindle Edition]

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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)

‘