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Chapter 2 Electron Energy Bands This is a brief survey of important terms and theories related to the energy bands in semiconductors. First, the fundamental concepts of electron wave vectors k, energy dispersion E( k), and effective masses are introduced. Section 2.2 is mathematically more involved as it outlines the k · p method, which is most popular in optoelectronics for calculating the band structure. Semiconductor alloys, interfaces of different semiconductor materials, and quantum wells are covered at the end of this chapter. 2.1 Fundamentals 2.1.1 Electron Waves In the classical picture, electrons are particles that follow Newton’s laws of mechanics. They are characterized by their mass m 0 , their position r = (x,y,z), and their velocity v. However, this intuitive picture is not sufﬁcient for describing the behavior of electrons within solid crystals, where it is more appropriate to consider electrons as waves. The wave–particle duality is one of the fundamental features of quantum mechanics. Using complex numbers, the wave function for a free electron can be written as ψ( k, r) ∝ exp(i k r) = cos( k r) + i sin( k r) (2.1) with the wave vector k = (k x ,k y ,k z ). The wave vector is parallel to the electron momentum p k = m 0 v ¯ h = p ¯ h , (2.2) and it relates to the electron energy E as E = m 0 2 v 2 = p 2 2m 0 = ¯ h 2 k 2 2m 0 , (2.3) with k 2 = k 2 x + k 2 y + k 2 z . Hence, in all three directions, E( p) and E( k) are des- cribed by a parabola with the free electron mass m 0 as parameter. 13

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Chapter 2

Electron Energy Bands

This is a brief survey of important terms and theories related to the energybands in semiconductors. First, the fundamental concepts of electron wavevectors k, energy dispersion E(k), and effective masses are introduced.Section 2.2 is mathematically more involved as it outlines the k p method,which is most popular in optoelectronics for calculating the band structure.Semiconductor alloys, interfaces of different semiconductor materials, andquantum wells are covered at the end of this chapter.

2.1 Fundamentals

2.1.1 Electron WavesIn the classical picture, electrons are particles that follow Newtons laws ofmechanics. They are characterized by their mass m0, their position r = (x, y, z),and their velocity v. However, this intuitive picture is not sufficient for describingthe behavior of electrons within solid crystals, where it is more appropriate toconsider electrons as waves. The waveparticle duality is one of the fundamentalfeatures of quantum mechanics. Using complex numbers, the wave function for afree electron can be written as

(k, r) exp(i kr) = cos(kr) + i sin(kr) (2.1)

with the wave vector k = (kx, ky, kz). The wave vector is parallel to the electronmomentum p

k = m0vh

= ph, (2.2)

and it relates to the electron energy E as

E = m02

v2 = p2

2m0= h

2k2

2m0, (2.3)

with k2 = k2x + k2y + k2z . Hence, in all three directions, E( p) and E(k) are des-cribed by a parabola with the free electron mass m0 as parameter.

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