R?ssler U. Solid State Theory: An Introduction

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146 5 Electrons in a Periodic Potential

in the diamond structure. Likewise, the other orbitals point to the direction of

the other nearest neighbors. These directed orbitals, called sp

hybrid orbitals,

are favorable for establishing a network with tetrahedral coordination based on

co valent bonds as realized in the diamond structure.

In the same way sp

hybrid orbitals can be used to establish planar networks

based on covalent bonds as in graphite. In this case the p

orbitals sticking out

of the plane form π b onding and anti-bonding orbitals, which lead to a peculiar

band structure with vanishing gap (Problem 5.11). The band structure of this

planar network of carbon atoms (graphene) is the basis also for the electronic

structure of carbon nanotubes, which result from rolling up the carbon sheets

into cylinders.

For the diamond structure, which we pursue here, there are two atoms in the

Wigner–Seitz cell with each four directed sp

hybrid orbitals, which can be

superimposed, as depicted in the right part of Fig.

5.16, such that the positive

lobes overlap in the nearest neighbor direction or with the opposite signs to

form bonding and anti-bonding orbitals, respectively,

(r)=

+ν



r +



± φ

−ν



r −

'

. (5.85)

Here ±τ /2 are the positions of the two atoms in the Wigner–Seitz cell and ±ν

refers to the directions of the positive lobes of the sp

hybrid orbitals, which

change sign between the nearest neighbor sites. In order to calculate the band

structure, Bloch functions have to be composed of these bonding and anti-

bonding orbitals and in general, an 8 × 8 secular problem has to be solved.

The bonding orbitals yield the valence bands, the anti-b onding orbitals the

conduction bands. Both groups are separated by an energy gap and we obtain

the characteristic band structure of a semiconductor.

As an example, the valence bands of Ge calculated by diagonalising the 4 × 4

matrix for the four b onding orbitals with nearest and next nearest neighbor

interaction are shown in Fig.

5.17. These bands exhibit a strong similarity with

those of Fig.

5.11 obtained from a pseudo-potential calculation. It is interesting

also to look at the spatial electron distribution

ΧΓ

–8

–6

–4

–2

–12

–14

–10

3’

25’

energy [eV]

Fig. 5.17. Valence band dispersion (Ge) from an LCAO calculation with nearest

and next nearest neighbor coupling after [

144]

5.5 Eﬀective-Mass Approximation 147

Fig. 5.18. Contour plots of the valence electron charge density of Si showing the

bond charge: experiment (left), theory (right)after[

145]

ρ(r)=−e



n,k,occ.

|ψ

(r)|

(5.86)

determined by all occupied (valence band) states. A calculated contour plot

is shown in Fig.

5.18 for Si together with experimental data obtained from

X-ray scattering. The dominant feature is the accumulation of charge between

the neighboring lattice sites. It is the bond charge characteristic of covalent

binding.

The density of states and the energy band dispersion can be determined

experimentally using light–matter interaction (see Chap.

10). Photo-electron

spectroscopy (PES) is used to analyse the energy distribution of photo-emitted

electrons and yields approximately the density of states from which the excita-

tion takes place, while angular resolved photo-electron spectroscopy (ARPES),

which analyses the geometry of the photo-emission process, in addition, allows

to map out the energy bands.

5.5 Eﬀective-Mass Approximation

Semiconductors diﬀer from metals by the fact that the Fermi energy is in the

gap between valence an d conduction bands, which is a region with vanishing

density of states. At low temperature, there are no free carriers, which could

react on a weak p erturbation by an applied electric ﬁeld. At elevated temp er-

atures, the Fermi distribution function is smeare d out and thermal population

of the lowest conduction band states takes place together with depopulation of

the topmost valence band states. The empty valence band states correspond

to missing electrons or holes which can be understood as particles with pos-

itive charge. In doped semiconductors, the Fermi energy is shifted from the

middle of the gap to the impurity levels, which can be close to the conduction

band minimum (n doping) or to the valence band maximum (p doping) and

the thermal population/depopulation takes place between the impurity states

and the nearby band edge. The number of the thermally excited carriers will be

small compared with the number of valence electrons and only states close to

148 5 Electrons in a Periodic Potential

the conduction band minimum or valence ban d maximum become occupied

with electrons and holes, respectively. These carriers can follow an applied

electric ﬁeld and carry an electric current. Similarly, optical excitation with

photon energies exceeding the gap energy creates electron–hole pairs by lift-

ing electrons from the valence band (leaving holes behind) to the conduction

band (see Chap.

10). Thus, the near band edge states determine some of the

characteristic properties of semiconductors and deserve special attention. In

fact, most of the technological applications with semiconductors (transistors,

sensors, lasers) are based on these states.

The lowest minimum of the conduction bands is found in Si along the axis

from Γ to X (see Fig.

5.11), in Ge at the L point, but it is at the Γ point

for most of the compound semiconductors A

) in the zinc blende

structure, with one fcc lattice occupied by atoms from the third (second) and

the other by atoms from the ﬁfth (sixth) column of the periodic table. For all

these tetrahedrally coordinated semiconductors, the valence band maximum

is at the Γ point. Si and Ge with valence band maximum and conduction

band minimum at diﬀerent points of the Brillouin zone are called indire ct gap

semiconductors, while those with the band extrema at the same point of the

BZ (here the Γ point) are called dire ct gap semiconductors.

The method to describe the dispersion o f energy bands around a given

point k

in the BZ has been developed in the early days of semiconductor

physics [

146, 147]. It is an expansion around this point, which in principle

can be extended throughout the Brillouin zone but is used mainly under the

condition |k − k

|≪2π/a. Let us assume the Schroedinger equation (

5.37)

to be solved for k

Hψ

(r)=E

)ψ

(r). (5.87)

Theperiodicpartsu

(r)(withﬁxedk

and all n ) of the Bloch functions

form a complete set of lattice periodic functions. With the Bloch functions at

k written as

(r)=e

i(k−k

)·r

·r

(r) (5.88)

the Schro edinger equation reads



−

¯h

∆ + V

eﬀ

(r)+

¯h

(k − k

)

¯h

(k − k

) · p



·r

(r)=E

(k)e

·r

(r). (5.89)

Now we make use of the complete set in the expansion

(r)=



′

(k − k

′

(r), (5.90)

and insert in (

5.89) to ﬁnd

5.5 Eﬀective-Mass Approximation 149



′

(k − k

)

′

¯h

(k − k

)

− E

¯h

(k − k

) · p

·r

′

(r)=0. (5.91)

Here the eigenvalue equation at k

was applied and the energy E

(k)weare

looking for is now called E. The last term in the curly brackets containing the

momentum operator and the diﬀerence in k vectors can be treated as a per-

turbation and that is why this concept is called k ·p perturbation theory. The

expansion coeﬃcients can be found from the set of coupled linear equations



′

(k − k

)

&

′

¯h

(k − k

)

− E



′

¯h

(k − k

) · p

′

)

= 0 (5.92)

which are obtained from (

5.91) by multiplying ψ

∗

(r)fromtheleftandinte-

grating over the crystal volume. Here the matrix elements of the momentum

operator

′



∗

(r)p u

′

(r)d

r (5.93)

establish a coupling between the diﬀerent bands. Solving the secular problem



¯h

(k − k

)

− E



′

¯h

(k − k

) · p

′

)

= 0 (5.94)

yields the dispersion relations E

(k) around k

for given E

)andp

′

Formally, (

5.94) is the matrix of a ﬁrst order perturbation calculation for

degenerate states. The oﬀ-diagonal coupling by the momentum matrix ele-

ments can be eliminated to any desired order, by matrix perturbation theory

as will be outlined below.

From Sect.

5.4 we identify the states at the conduction band minimum at

=0ass anti-bonding states, while those of the valence band maximum are

p bonding states. As seen in Fig.

5.17, the dispersion around the valence band

maximum is more complex than that around the conduction band miminum.

Let us start, therefore, with describing the states at the conduction band

edge.

Conduction band edge at k

=0:

We choose the band index n = c for the lowest conduction band with the Bloch

function u

(r)=r|S deriving from the s anti-bonding orbital. This state

is coupled with the p bonding (or valence band) and with the p anti-bonding

(or higher conduction band) states (n

′

). Denoting these states |X, |Y , |Z

and |X

′

, |Y

′

, |Z

′

, respectively, we have the momentum matrix elements

150 5 Electrons in a Periodic Potential

P =

¯h

S|p

|X =

¯h

S|p

|Y  =

¯h

S|p

|Z (5.95)

and similar ones for the primed states. These matrix elements are equal due

to the symmetry of the diamond or zinc blende structure according to which

the cubic axes are equivalent. By eliminating to lowest order the coupling of

the s like state at the conduction band minimum to the p like states n

′

= c,

one arrives at the second order p erturbation expression

(k)=E

(0) +

¯h



′

=c

S|k · p|n

′



(0) − E

′

(0)

(5.96)

and by making use of (

5.95) one arrives with



′

=X,Y,Z

S|k · p|n

′



S|p

|X



+ k



(5.97)

at the approximate parabolic dispersion around the conduction band mini-

mum

(k)=E

(0) +

¯h





′

(p)

S|p

′



(0) − E

′

(0)



. (5.98)

The summation here is over all p like states. This dispersion tells us, that

the electrons occupying these states behave like free electrons but with an

eﬀective mass m

∗

given by

∗

=1+



′

(p)

S|p

′



(0) − E

′

(0)

. (5.99)

Depending on the energy denominators, only a few terms need to be considered

in the sum. For narrow gap semiconductors the contribution of the topmost

valence band dominates, while in general the coupling to the p anti-bonding

states needs to be considered. Characteristic values are m

∗

/m =0.067 for

GaAs with a band gap E

gap

= E

(0) − E

(0) = 1.52 eV and m

∗

/m =0.0135

for InSb with E

gap

=0.25 eV. By applying a magnetic ﬁeld, a fan chart of

Landau levels evolves out of this parabolic disper sion with typical spacings

of ¯hω

∗

= eB/m

∗

and the detection of the cyclotron resonance frequency (see

Sect.

4.2) provides the information about the eﬀective ma ss. We note in pass-

ing that higher order terms in the dispersion relation account for the ﬂattening

and anisotropy of the energy band away from the Γ point [

148] as can be seen

in e.g., Fig.

5.12. This nonparabolicity and warping would translate into an

energy dependence and anisotropy of the eﬀective mass.

Conduction band edge of Si at k

=(0, 0,k

As mentioned already, Si i s an indirect gap semiconductor, the minimum of

the conduction band is along the ∆ axis close to the p o int X (see Fig.

5.11).

5.5 Eﬀective-Mass Approximation 151

The lowest conduction band derives from the sp

anti-bonding states with

a k dependent hybridization. Let us denote them by |

S.Alongthe∆ axis,

the threefold valence band maximum splits into a rather ﬂat twofold band

connected with the states |X and |Y  and a band, which evolves from a

mixing of |Z with the s b onding state, is denoted here as |

Z. Instead of

(

5.95) we now have for the momentum matrix elements

¯h



S|p

|X =

¯h



S|p

|Y  =

¯h



S|p

Z = P

, (5.100)

and the perturbation series reads with k

′

= k − k

′

)=E

¯h

′2

¯h



′

=c



S|k

′

· p|n

′



) − E

′

)

. (5.101)

The inequality in (

5.100), caused by the r edu ced symmetry of the group of

the wave vector k

, leads to the anisotropic dispersion relation (with k

′

− k

))

′

)=E

¯h



+ k

∗

⊥

′

∗





(5.102)

with m

∗



and m

∗

⊥

bei n g the masses parallel and perpendicular to the ∆ axis. For

Si, the conduction band minimum is at k

=0.85 in units of 2π/a. Due to the

cubic symmetry there are minima also alon g the other equivalent directions

in the Brillouin zone, thus the conduction band of Si has six minima, which

(thermally or by doping) become equally populated. In the presence of a

magnetic ﬁeld, fan charts of Landau levels evolve from these minima, which

depend on the orientation of the magnetic ﬁeld with respect to the inverse

mass ellipsoid and so does the cyclotron mass (Problem 5 .12).

Valence band maximum at k

=0:

The valence band maximum of semiconductors with diamond or zinc blende

structure derives from the p bonding orbitals and is (without spin) threefold

degenerate. The energy dispersion is obtained from the 3 × 3 determinant

(ν, ν

′

= X, Y , Z)



(0) +

¯h

− E



νν

′

¯h



n=ν,ν

′

ν|k · p|nn|k · p|ν

′



(0) − E

(0)

=0. (5.103)

Symmetry arguments, for which we refer to the literature [

146], allow to write

the terms bilinear in the components of the wave vector as a 3 × 3matrix

(known as the Shockley

matrix [147])

William B. Shockley 1910–1989, received the Noble prize in physics 1956 jointly

with J. Bardeen and W.H. Brattain.

152 5 Electrons in a Periodic Potential

M =

⎛

⎝

+ M(k

+ k

) Nk

+ M(k

+ k

) Nk

+ M(k

+ k

)

⎞

⎠

(5.104)

with

L, M, N ∼



n=ν,ν

′

ν|p

|nn|p

|ν

′



(0) − E

(0)

, (5.105)

where diﬀerent intermediate levels contribute to L, M, and N. The secular

problem M−Eδ

ν,ν

′

 = 0 is easily solved for special directions of k.Forthe

∆ axis or k =(0, 0,k) one ﬁnds

(k)=Lk

and E

(k)=Mk

, (5.106)

where the second solution is twofold, which by comparison with Fig.

5.17 can

be identiﬁed with the upper branch of the valence band. For the Λ axis with

k =(k, k, k)/

√

3thesolutionsare

(k)=

L +2M − 2N

and E

(k)=

L +2M + N

. (5.107)

Again, the second solution is twofold and can be identiﬁed with the upper

branch for this direction. A characteristic feature is the diﬀerent curvatures of

the bands for the two solutions: the twofold band with the small curvature, cor-

responding to a large mass, is the heavy hole band, while the non-degenerate

band with the larger curvature is the light hole band. The other feature is

that these curvatures depend on the direction of k and the surfaces of con-

stant energy are warped. We note in passing that for all the other directions

the secular problem has three diﬀerent solutions. These degeneracies and the

anisotropy of the valence bands are the same as those of the acoustic phonon

branches discussed in Sect.

3.4. This is a consequence of the group of the wave

vector, which contains at least threefold rotations only for the ∆ and Λ axes.

In general, energy bands can be calculated numerically by diagonalizing

(

5.94) for a ﬁnite set of states, provided, the separation of their energy levels

at k

and the momentum matrix elements are known. For near band-edge

states in direct gap semiconductors around the Γ point, a multi-band k · p

model is frequently in use, which comprises ﬁve bands or (including spin) 14

states [

148–150].

5.6 Subbands in Semiconductor Quantum Structures

While in the previous sections of this chapter, we have looked at the elec-

tron states in three-dimensional solids, we shall now consider systems with

reduced dimensionality, which are in the focus of interest since the develop-

ment of the planar semiconductor technology of Si based ﬁeld eﬀect transistors

5.6 Subbands in Semiconductor Quantum Structures 153

(MOSFET or metal oxide Si ﬁeld eﬀect transistor)[

151–153]. Here, we would

like to describe a semiconductor heterostructure, which can be obtained by

epitaxial growth of one semiconductor (say AlAs) onto another one (GaAs).

Both systems have the same crystal structure (zinc blende) and almost the

same lattice constant, i.e., across the interface it is only the chemical natur e

of the atoms that changes. Let the growth direction be along z and the inter-

face at z = 0 extend in the x − y plane. The electronic structure of this

system is determined by the bulk band structure of the two materials with

their respective energy g aps and eﬀective masses. In particular, the b and

edges (the minima of the conduction band and the maxima of the valence

band) are shifted against each other and the interface appears as a step-like

potential for carriers at these edges. Correspondingly, a double heterostructure

AlAs/GaAs/AlAs would represent a conﬁning square well potential which for

electrons can be described by the conduction band proﬁle

(z)=

⎧

⎨

⎩

0 z<0

−V

0 <z<L

0 z>L

(5.108)

where the minimum of the energy band in AlAs is the zero point of the

energy scale and V

is the conduction ban d oﬀset between Al As and GaAs.

As it is only a few 100 meV, the conﬁned states, being close to the conduction

band minimum, can be described in the eﬀective-mass approximation with

the Hamiltonian

EMA

= −

¯h

∗

∆ + E

(z). (5.109)

The eigenfunctions of this Hamiltonian



r)=e



·r

(z) (5.110)

and the eigenvalues



)=E

¯h



∗

(5.111)

indicate the quantization with energies E

due to the conﬁnement potential

in growth direction and the free motion parallel to the interface: electrons

occupying these states would be free carriers in two dimensions. The quantum

number n denotes the subbands evolving from the conduction band of the

three-dimensional band structure.

In both, the single heterostructure and in the quantum well, free carriers

can be introduced by doping the AlAs barrier with Si. The Si atoms create

additional states close to the conduction band minimum (donor states) with

a small binding energy. Due to the negative band oﬀset, the donor electrons

cross the interface and reside in the GaAs layer, while leaving a positively

charged center behind in AlAs. This charge separation gives rise to an electro-

static potential, which is superimposed on the band edge proﬁle. For a single

154 5 Electrons in a Periodic Potential

AlGaAs:Si

AlGaAs

GaAs

Fig. 5.19. Semiconductor heterostructure (left) and conduction band proﬁle across

the interface with subband levels and Fermi energy (right)

heterostructure this leads to the picture shown in Fig.

5.19 with an almost tri-

angular potential at the interface, which now conﬁnes the carriers. The energ y

spectrum is similar to (

5.111) but the subband energies are obtained here from

a self-consistent solution of the one-dimensional Schroedinger equation

−

¯h

∗

∂

+ V (z)

(z)=E

(z) (5.112)

with the potential V (z)=V

Θ(−z)+V

(z)+V

(z) and Poisson’s equation

∂

(z)=

εε

̺(z). (5.113)

Here, V

(z) is the Hartree potential, determined by the charge distribution

̺(z) of ionized donors in the barrier (AlAs) and of the electrons occupy-

ing subband states. We recognize (

5.112) as the one-dimensional analogue to

the Schroedinger equation (

5.5) for the three-dimensional case with an eﬀec-

tive single-particle potential, which contains also an exchange–correlation part

(z). We note in passing that a more accurate description would take into

account the change of the eﬀective mass a cross the interface. The eﬀective

mass determines also the density of states, which (without non p arabolicity

eﬀects) is a constant (see Problem 4.3 and Fig.

5.20, left part). The num-

ber of subbands with occupied states depend on the areal density, which

can be controlled by the external gate voltage. The situation, when only

states in the lowest subband are occupied, is called the electric quantum limit.

A more detailed description shows, that the spin-degeneracy of the subbands

is removed due to a spin–orbit coupling caused by the asymmetric conﬁne-

ment potential at the interface [

153] (Problem 5.13). It can be tuned by an

external gate voltage and is important in manipulating the spin dynamics in

these structures (Rashba eﬀect).

The eﬀective-mass Hamiltonian (

5.109) is obtained from the conduction

band dispersion (

5.98) by replacing k

→−∆, which leads to the operator of

5.6 Subbands in Semiconductor Quantum Structures 155

N = 4

k ,D(E)

Fig. 5.20. Subband dispersion and density of states for the lowest two subbands

(left) and fan chart of Landau levels evolving from the lowest subband (right), spin

splitting is suppressed

kinetic energy for carriers close to the conduction band minimum. The corre-

sponding description of conﬁned holes is more complex because o f the va l ence

band degeneracy and would lead (without spin) to three coupled Schroedinger

equations, which for k



= 0, decouple and yield subband states for heavy

and light holes. Quantitative subband calculations including spin and non-

parabolicity eﬀects are performed within the multiband envelope function

approximation, which is based on the k · p theory [

152–154]. Due to the con-

ﬁnement potential, the energy gap, i.e., the energy diﬀerence between the

lowe st electron and the highest hole subband state, is changed as compared

with the bulk material. The possible choices of well and barrier material with

their r espective electron and hole masses and of the width L of the quantum

well allow one to design materials with deﬁned gap energies. This band gap

engineering is exploited in optical devices (see Chap.

10).

In heterostr uctures, the spatial separation between the free carriers in the

conﬁning potential and the ionized donors in the barriers (usually enlarged

by growing a spacer layer at the interface) leads to improved mobilities of the

carriers at low temperatures (when scattering with ionized impurities is dom-

inating, see Chap.

9). Under these conditions, characteristic transport lengths

become comparable or even larger than the lateral system size and phenom-

ena of mesoscopic physics can be investigated [

155–160]. Magnetotransport

experiments in these systems have led to the discovery of the quantum Hall

eﬀects and composite fermions (see Chap.

7).

A magnetic ﬁeld applied perpendicular to the plane of the two-dimensional

electron system leads to a complete discretization of the energy spectrum.

The subbands split into highly degenerate Landau levels (see Sect.

4.2)and

the ﬁlling fac t or, i.e., th e number of occupied Landau levels changes with the

magnetic ﬁeld. The ultimate case, when at suﬃciently high magnetic ﬁeld

all electrons can be accommodated in the lowest Landau level of the lowest