Basics of Quantum Mechanics and Solid State Physics

8/6/2019 Basics of Quantum Mechanics and Solid State Physics

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energy W with ω being the frequency of the electromagnetic radiation, Evidence camefrom numerous diffraction experiments with material particles (electrons, neutrons etc.),where interference patterns can only be explained by attributing to a moving particle such asan electron, a neutron etc. a wave with frequency

ω = h

2 212 2

W mv p

mω = = =

h h h(1)

(Achtung: Energie hier immer mit “W” bezeichnet, anstelle von “E” ) and a wavelength

p

hπ λ

2= , (2)

the de Broglie wavelength ( mv p = ).

In this quantum mechanical description a free particle (moving in a spatially constant potential) can no longer be described by a classical trajectory r (t ) but rather by a plane wave

)(e),( t ict ω−⋅=ψ r k r (3)

with 21

2W mv ω = = h being the energy of the particle and m= = h p k its momentum

vector. Particle propagation is wave-like whereas the detection process reveals the particlecharacter by transferring a well defined quantum of energy to the detector at a particular site

in space. The only reasonable interpretation of the wave function (3) in connection with the particle picture is in terms of a probability amplitude, where |ψ (r ,t )|2 = ψ * ψ is the probability(density) of detecting a particle in r at a time t . Since the wave function (3) has non-vanishingvalues everywhere in space, the propagation of a particle has to be described by a wave

packet, a superposition of many plane waves (in its simplest form a gaussian packet), i.e. inone dimension:

)()(

2

2 e)(ee),(0

t kxit kxik

k k

k adk dk t x ω−ω−

∆−

−

∫ ∫ ==ψ (4)

This wave packet is centered in k -space around one center wave vector k 0 and in real space itsmaximum amplitude propagates with the group velocity

0

0

k k v

∂ω∂

= . (5)

This group velocity v0 is identified with the classical velocity of the particle.

The description of particle propagation as the propagation of a wave packet includes alreadythe uncertainty principle of quantum mechanics. Assuming a gaussian distribution a(k ) in (4)

for the different plane waves contributing to the wave packet ψ ( x,t ) the spatial half width ∆ x of the peaked function ψ ( x,t ) is inversely related to the half width ∆k of the k -distribution by



1≅∆∆ k x . (6a)

Assuming the particle picture this is identical to

h≅∆∆ p x , (6b)

which means that a well defined space coordinate x of a particle implies a highly undefinedmomentum p. Location x and momentum p of an atomic particle, so-called complementaryvariables, cannot be determined as sharp values simultaneously. The product of their marginsof error equals Planck’s constant h .π= 2/h

This directly leads to the fundamental fact in quantum mechanics, that given a general wavefunction ψ (r ,t ) the important quantities, location r , momentum p and energy W of a particlecan no longer be simple sharp numbers as in classical mechanics, but rather they must resultas statistical mean values from corresponding measurements of space, momentum, energy etc.

The adequate mathematical description is that of observables which are attributed to themeasured quantities x, p, W etc. From the wave function (3) of a freely moving particle andfrom the particle-wave-relations (1) and (2) we easily find out, that momentum and

energy W can be regained from the ψ -function by applying the operations

k p h=

ω = h xi

∂∂h

and

t ∂i

∂−h

on the wave function. The observables momentum and energy of a particle are

therefore in general defined by the operators

xi

p

∂

∂=

hˆ , (7)

t i H

∂∂−

=hˆ , (8)

where H ˆ is called the Hamilton-operator for the energy observable. In this description thespace operator is a simple multiplication by the coordinate x. x

We can see that operators, in their function on a wave function, can not be commuted ingeneral. The operation of ˆ ˆ xp on ψ (x,t)is not identical with ˆ ˆ px on ψ (x,t), but rather

ˆˆ ˆ ˆ( ) xp px− ψ (x,t)=iħψ (x,t) . (9a)

This is valid for all kinds of ψ and we define

[ ]ˆ ˆ ˆˆ ˆ ˆ, p xp px i= − = h . (9b)

as the commutator (operator) of the space and momentum operators, which does not vanish.In general, in quantum mechanics, operators which do not commute, belong to observableswhich obey an uncertainty relation as do ˆ and ˆ p (6b); their measured values can not be

determined simultaneously as sharp numbers but rather within certain limits of uncertainty.





basis for the one-particle description of metals and semiconductors which is treated in theforthcoming chapters.

With )()2(ˆ 2r V m H +∆−= h as the Hamilton operator for the energy observable (13a) is

written

),(ˆ),( t H t t

i r r ψ =ψ ∂∂

h , or in general Dirac notation (13b)

Hit

ψ ψ ∂

=∂

h

Since for stationary problems H ˆ does not depend on time we can separate in space and timewith an ansatz

)()(),( r r ϕ=ψ t f t (14)

and find

1 1 ˆ( ) ( )( ) ( )

i f t H W const f t t

ϕ ϕ

∂= = =

∂r

rh , (15)

where the left side is only dependent on time, while the right side only depends on space.From (15) follows the solution

( ) exp W f t i = − h

t (16)

for the time dependence of all stationary solutions, while the spatial part of the wave functionϕ(r ) obeys an eigenvalue equation for the Hamilton operator

ˆ ( ) ( ) H W ϕ ϕ =r r , or in Dirac notation H W ϕ ϕ = (17)

All operators describing observables in quantum mechanics have real eigenvalues (in equ.(17) the energy eigenvalues W=Wn) and a complete, orthogonal set of eigenfunctions ϕn(r )(12) which are obtained as the result of a measurement of that particular observable. When thewave function of a particle is a particular eigenfunction ϕn(r ) of an observable, measurementof that observable yields the single sharp eigenvalue (Wn in (17)) instead of the statisticalmean values or expectation values (11), when the initial wave function is not eigenfunction of the operator ascribed to the observable (particular type of measurement).

Operators andˆ A ˆ B , which have the same system of eigenfunctions commute, i.e.ˆ ˆ AB ˆˆ BAψ ψ = , which is shortly expressed as ˆ ˆ ˆˆ ˆ ˆ, 0 AB BA A B − = = .





( ) [ (2

1 22 1 cosc cψ ψ ψ ∗ ∗= = + ⋅ − r )k r r (28)

with k ⋅(r1-r2) the phase difference between the two partial waves. The cosinus interference term isinherent to the superposition state in the double slit experiment. It is just what makes quantummechanical particle (wave-like) propagation different from classical behaviour.

Fig. 1: Double slit (gedanken) experiment with electrons (a) or atoms (b).The electron particle waves originating from two screen openings r1 and r2 produce aninterference pattern on a screen at large distance from r1 and r2 (k 1≈k 2≈k ).

e

3. Electronic Properties of Solids

Since nanoelectronic devices rely, in their performance, on the electronic properties of solids, it isnecessary to remind some fundamental facts, about electronic states and electronic transport incrystals. In particular the distinction between metals and semiconductors is of central importance.

3.1 Electronic Band Structure: Metals and Semiconductors

A crystal, such as an ideal Si-crystal has translational symmetry, its atoms are ordered along lines inthe crystal such that the potential in which the electrons of the crystal move, can be written as

V(r)=V(r+n1a1+n2a2+n3a3) , (32)

with ni as integer numbers and ai the elemental translation vectors (spanning the elementary cell of thecrystal). A crystal volume of 1cm3 contains about 1023 electrons. A simple but very powerfulapproximation is the single-particle approximation, where the dynamics of only one single electron isconsidered in the periodic crystal potential (32). But then the potential contains not only the Coulombfields of the positive atomic cores but also in an indirect way the screening action of all other electronsexcept the one we consider. With this drastic approximation we can use the one-electron Schrödinger

equation (13) or when we consider stationary states the time independent equation (17), wherecontains the periodic potential (32).

ˆ H

Since V(r) and also have crystal periodicity, one can evaluate all interesting quantities includingV(r) in a Fourier series

ˆ H

( ) iV V ⋅= ∑G r

G G r , (33)



rather than in a Fourier integral, where G=hg1+k g2+lg3 are discrete points of the so-called reciprocalspace (h, k, l integer) of wave vectors, k, with gi the spanning vectors of the reciprocal elementary cell.

Fig. 2: A plane oblique lattice (left) and its corresponding reciprocal lattice (right). Thereciprocal lattice vectors g1 and g2 lie perpendicular to a2 and a1 in real space.

For every Bravais lattice in real space one defines a reciprocal lattice (Fig. 2) with primitive vectors gi by

(2 , , 1,2,i j ij i jπδ ⋅ = ∈ g a )3 . (34a)

This is satisfied by

g1=2π(a2 x a3)/Vz, g2=2π(a3 x a1)/Vz, g3=2π(a1 x a2)/Vz (34b)

where Vz=a1.(a2 x a3) is the volume of the elementary cell in real space. For a cubic lattice the

reciprocal lattice is again cubic with g i=2π/ai. In general, vectors in real and reciprocal space (of wave

vectors) are related to each other by

Ghkl.(n1a1 + n2a2 + n3a3) = 2πm, m integer. (35)

For the solution of the Schrödinger equation (13) with periodic potential (32) one evaluates both wave function ϕ(r ) and potential V (r ) in plane waves exp(i k·r ). The periodic potentialthen is the Fourier series (33) and the whole problem shows up to be periodic in the reciprocalspace of wave vectors k . The general time independent wave function of the electron has thegeneral form

r k

k

r r ⋅=ϕ i

u e)()( (36)



with uk (r ) being periodic in real space, i.e. a plane wave with amplitude modulation uk (r ); thisis called a Bloch wave, and similarly as for simple plane waves being solutions of (13) in aconstant potential, they can be characterized by their wave vector k . The translationalsymmetry of real space transfers into reciprocal space such that the energy eigenvalues of theelectron W(k ) exhibit translational symmetry in reciprocal space:

( ) ( )hkl W W = +k k G . (37)

This symmetry property allows a qualitative insight into the electronic propertiescharacteristic of electrons moving in a periodic crystal lattice.

We consider a one-dimensional crystal of macroscopic dimension (length L) with latticeconstant a and assume at first that the periodic potential V ( x) is negligibly small, i.e. V ≅ const (= 0). Then the energy eigenvalues W(k ) are those of a free electron, 2 (2 )m= hW k , a

parabola in k -space. Now, in a gedanken experiment, the periodic potential is “switched on”

gradually, such that the energetic effect remains negligible, but the symmetry property of translational symmetry starts to be effective, i.e. the energy eigenvalues now have the property (37)

( ) ( )hW k W k G= + , (38)

In reciprocal space the energy parabola of the free electron has to be repeated on the k -axis,each time originating at multiples of the reciprocal lattice vector G (Fig. 3a). The

periodicity volume in k -space centered around the k (Γ ) point is called the 1a/2π=

G /−0=

G /+

st Brillouinzone. At the edges of the Brillouin zone, i.e. at and twoneighbouring parabolae intersect and there is a degeneracy of the energy eigenvalues. The twocorresponding solutions of the Schrödinger equation belonging to the two parabolae areequivalent. Since the Schrödinger equation is a linear differential equation the most generalsolution at this k -value is a superposition of both particular solutions. For negligible potentialvariations the corresponding plane waves are

a/2 π= a/2 π−=

2/eiGx and e (39)[ ] 2/)2/( e iGxGGi −− =

The most general solutions at G/2 are therefore the two linear superpositions

( ) )cos(ee 2/2/

a

xiGxiGx π∝+∝ψ −

+, (40a)

( ) )sin(ee 2/2/

a

xiGxiGx π∝−∝ψ −− . (40b)



Fig. 3a) The parabolic energy curves of a free electron in one dimension, periodicallycontinued in reciprocal space. The periodicity in real space is a. This W(k)dependence corresponds to a periodic lattice with a vanishing potential (“empty”lattice).

b) Energy dispersion curves W(k) for a one-dimensional lattice (lattice constant a) inthe extended zone scheme. As can be seen, the quasi-free electron approximationgives rise to forbidden and allowed energy regions due to the opening of band gaps.The parts of the bands corresponding to the free-electron parabola are indicated by

the thick lines.



Fig. 4a) Qualitative form of the potential energy V(x) of an electron in a one-dimensionallattice. The positions of the ion cores are indicated by the points with separation a

(lattice constant).

b) Probability density ρ ψ ψ ∗+ += + for the standing wave produced by Bragg reflection

at k=+π/a, the upper edge of band (1) in d).

c) Probability density ρ ψ ψ ∗− −= − for the standing wave at the lower edge of band (2)

at k=+π/a in d).

d) Splitting of the energy parabola of the free electron (- - -) at the edges of the firstBrillouin zone (k=+π/a in the one-dimensional case). To a first approximation the

gap is given by twice the corresponding Fourier coefficient VG of the potential.Periodic continuation over the whole of k-space gives rise to continuous bands (1)and (2), shown here only in the vicinity of the original energy parabola.

According to Fig. 4 the probability density accumulates negative electronic charge at

the location of the positive atomic core (minimum of V ( x)) while accumulates negative

charge in between the positive atomic cores. In comparison with the state of a travelling waveexp(ikx), where the charge density is homogenous all over the crystal, the ψ

+∗+ψ ψ

−∗−ψ ψ

– thus means anincrease of the energy of the electron, whereas ψ + is related to a lower electronic energy. Theoriginally degenerated states at ±G/2 split off and a forbidden gap in the spectrum of one-

electron states is opened at the Brillouin zone boundaries (Fig. 3). As a consequence theelectronic states in a periodic crystal, unlike for the parabola W(k ) of the free electron, nowform allowed and forbidden bands on the energy scale. Within a band of allowed states W(k )shows an oscillatory behaviour as a function of the wave vector k of the Bloch states (20).Because of periodicity in k -space the W(k ) dependence can be restricted for practical reasonsto the 1st Brillouin zone. Near the lower and upper energetic edges of the bands a parabolicdescription of the W(k ) curves is adequate. The origin of allowed electronic bands in a crystalcan also be traced back to the interaction of neighbouring atoms by overlap of the bondingatomic orbitals. Electronic bands can thus be characterized by the atomic orbitals from whichthey originate, e.g. sp-bands or d-bands etc.



Fig. 5a) Construction of the first Brillouin zone for a plane oblique lattice. The zone boundary consists of perpendicular bisectors of the shortest reciprocal latticevectors.

b) The Brillouin zones of the face centered cubic, body-.centered cubic and hexagonallattices. Points of high symmetry are denoted by Γ , L, X etc. The surfaces enclosingthe Brillouin zones are parts of the planes that perpendicularly bisect the smallestreciprocal lattice vectors. The polyhedra that are produced by these rules of

construction can be drawn about every point of the reciprocal lattice. They then fillthe entire reciprocal space. The cell produced by the equivalent construction in realspace is known as the Wigner-Seitz cell. It can be used to describe the volume thatone may assign to each point of the real crystal lattice.

The extension to three dimensions (3D) is straightforward, but much more complex W (k )functions in the 3D reciprocal k -space are obtained. Even for the most common 3D bcc andfcc lattices Brillouin zones as periodicity volumes in k -space are complex polyhedra (Fig. 5).W(k ) dependencies are then plotted as curves along symmetry directions, e.g. Γ –X, X–W etc.,to give a rough overview over the Bloch state energies as a function of k -vector. As anexample the calculated band structure of Si along symmetry lines L– Γ –X is shown in Fig. 6

[4]. A totally forbidden band (shaddowed) is seen which is characteristic for semiconductors.



The important distinction between a semiconductor and a metal is made on the basis of theoccupancy of electronic states W(k ). So far only the quantum mechanically possibleelectronic states were calculated. Their occupancy with electrons is controlled by the Pauli-

principle, which is valid for all kinds of Fermions, i.e. elementary particles with spin 1/2.Electrons of course are Fermions, they obey the so-called Fermi-statistics. According to the

Pauli-principle non-interacting Fermions can occupy a quantum mechanical state such as ψ + or ψ – only one times. Since in our consideration so far the two spin orientations of theelectrons were not taken into account, the electronic states described by the band structureW(k ) can be occupied by two electrons (opposite spins) at maximum. Up to a certain energythe states of the band structure W(k ) are thus filled, each with two electrons. When the highestoccupied state is identical with the upper energetic edge of an electronic band which isseparated from higher lying bands by an absolute gap as for Si in Fig. 6 then this material is asemiconductor . In order to excite electrons, they have to overcome the gap in the bandstructure, since no continuum of empty states is available in direct energetic neighbourhood of the highest occupied states.

Fig. 6 Calculated bandstructures of silicon and germanium. For germanium the spin-orbitsplitting is also taken into account. (After [4]). Both semiconductors are so-called



indirect semiconductors, i.e. the maximum of the valence band and the minimum of the conduction band are at different positions in the Brillouin zone. The minimumof the conduction band of silicon lies along the Γ X=[100] direction and that of germanium along the Γ L=[111] direction. The calculation for Ge is relativistic. Thespin-orbit splitting in the upper valence band near Γ is therefore resolved in contrast

to Si.

When the highest occupied energy level falls into the continuum of an allowed energy band,this band is partially occupied and partially empty. Electrons can be excited with infinitelysmall energy quanta and the material is a metal. As an example the band structure of Cu isshown in Fig. 7 along some symmetry lines [5]. The highest occupied energetic state in ametal (at zero temperature T ) is called Fermi-level E F. For Cu the Fermi energy E F crosses theenergetically wide continuum of the W(k ) band which is derived from the atomic Cu s-states.Characteristic for transition metals are the energetically sharp, relatively flat electronic bands,which are derived from the atomic d -states. In Cu these bands are all occupied, i.e. the Fermilevel WF lies approximately 2 eV higher than the upper d -band edge. For other transitionmetals as Fe, Ni, Pt the d -bands are partially empty and the Fermi energy WF lies within thed -bands.

Fig. 7 Bandstructure W(k) for copper along directions of high crystal symmetry (right).The experimental data were measured by various authors and were presentedcollectively by Courths and Hüfner [5]. The full lines showing the calculatedenergy bands and the density of states are from [5]. The experimental data agreevery well, not only among themselves, but also with the calculation.



For many purposes it is not necessary to know the complete W(k ) dependence of theelectronic band structure but rather the number of states dZ lying between W and W+dW , i.e.the density of states, is interesting (Fig. 7b).

This is easily calculated by an integration in k -space

3

3( )

(2 )

W dW

W

V dZ d k D W dW

π

+

= =∫ (41)

Here we have to take into account, that our crystal does not have infinite dimensions, butrather a macroscopic length of L or a volume V = L3 with L of typically 108 Å = 1 cm. Thismeans, that the electronic waves, the Bloch states, can only have wave vectors k or wavelengths which fit as multiples into the length L. This leads to the restrictions k i=2π ni/ L for the 3 coordinates (i=1,2,3) with ni as integers and discrete possible k i values havingdistances 2π/ L from each other. Since however the 1st Brillouin zone has a diameter of

G=2π/a, where a, the atomic distance is about 10 –8 L, k can nevertheless be assumed as quasicontinuous in k -space. But in k -space one has to attribute a volume of (2π)3/ L3=(2π)3/V toeach quantum state; the inverse value shows up in (41). If the k -volume element d 3k isseparated into an area element dS E on the energy surface and a component dk ⊥ normal to thissurface, i.e. d 3k =dS F dk ⊥, then with dW =gradk W dk ⊥ one has for the density of states(related to the crystal volume V )

3( )

1( )

(2 ) grad ( )W

W const

dS D W dW dW

W π =

=

∫

k k k

(42)

This density of states D(W) is high in regions of flat bands (e.g. d -bands in Fig. 7) sincegradk W (k ) is small. Near maxima or minima of the band structure W(k ) has a parabolic k -dependence W ∝ k

2, thus gradk W –1 ∝ k

–1, but the integration over the energy surfacerestores a k -dependence of D(W ). The density of states near a maximum or minimum thus

behaves as D(W ) ∝ k ∝ W , just similar to a free electron gas.

3.2 Fermi-Statistics in Metals and Semiconductors

According to the Pauli-Principle each electronic state can only be occupied by one electron or

by two if spin orientation is not taken into account. If we plot a probability function1<ƒ(W)<1 for the occupancy of states with energy W for vanishing temperature T , it must bea step function, which switches from the value 1 (occupied) to 0 (non occupied) at the Fermienergy WF. Here WF is given in a metal with continuously lying states by the maximumnumber of electrons which can occupy the states in the band. For increasing temperatureselectrons deep below E F can not be excited, they find no empty states in their energeticneighbourhood. Only electrons slightly below E F can be excited to empty states slightly aboveWF. For higher temperatures ƒ(W) therefore must exhibit a “smeared out” step near WF (Fig.8).



Fig. 8 The Fermi distribution function at various temperatures. The Fermi temperatureTF= /W has been taken as 5x10O

F k 4 K. The tangent at the point of inflection (- .. -)

intersects the energy axis at 2kT above W at all temperatures.O

F

A detailed thermodynamical derivation of ƒ(W) being based on Pauli-principle and theundistinguishability of electrons yields the mathematical expression.

( ) /

1( , )

e 1 F BW W k T f W T −=

+. (43)

In order to get an impression of the shape of the Fermi function near W F, one has to consider the typical energetic width of an electronic band. For Cu (Fig. 7) EF lies about 8 eV above the

s-band minimum. In contrast, at room temperature k B T amounts to (1/25) eV, i.e. even atroom temperature ƒ( E ) has a relatively sharp Fermi edge: The region, over which ƒ( E )deviates significantly from the step function is of the order of 2 k B T to each side of W F (Fig.8).

When in a metal the Fermi-energy WF cuts the band structure W(k ) in the lower part of the

band, where W(k ) is essentially parabolic2 2

( )2

k

m∗≅k h

W (m*, the so-called effective mass,

formally describes the reciprocal curvature of the energy parabola), the Fermi surface2 2

( )2

F F

k W

m∗=k h

is essentially a sphere with radius k F, the so-called Fermi wave vector. The

Fermi surface is fully contained in the 1. Brillouin zone. This case is given for monovalentalkali metals. For noble metals as Cu, Ag and Au the Fermi surface touches the boundary of the Brillouin zone. A purely parabolic energy dependence of WF(k ) is no longer given and the

Fermi-surface becomes a deformed sphere with extensions into the neighbouring Brillouinzones (Fig. 9a) [6]. For transition metals, where WF crosses the band structure in regions of complex d -band structures, the Fermi surfaces get very complicated (“monsters”) shapes in k -space (Fig. 9b) [7].

3.3 Dissipative Electronic Transport: The Electrical Resistance

Electrical conduction arises from electric field induced propagation of free electrons in a partially filled band of a metal or in semiconductors of free electrons in the conduction bandand/or free holes (empty electronic states) in the valence band. In all these cases electrons andholes are described quantum mechanically by wave packages of Bloch waves (36) and the

particle velocity v(k ) is the group velocity of the wave package. Given the(1/ ) W ∇k h



electronic band structure W (k ) the electron takes up a certain amount of energy δW in theexternal electric field E in order to enhance its energy within the band by

( )W W δ δ = ∇ = ⋅k k k v k h δ

t δ

E

(48a)

In this semiclassical description

(48b)W eδ = − ⋅E v

and . (48c)e= −k &h

From (48) the simple semiclassical dynamic equation for crystal electrons follows as

21 1( )i i

j i j

d v W

dt k k

∂= ∇ =

∂ ∂∑k &&

h hj

W k (49a)

2

2

1(i

j i j

W v

k k

∂=

∂ ∂∑&h

) jeE − (49b)

In (48b) the inverse curvature of the energy bands can be interpreted as a so-called effectivemass tensor

2

* 2

1 1 (

ij i j

W

m k

∂ = ∂ ∂

k

h

)

k , (50)

which replaces in a crystal lattice the free electron mass; all effects of the solid crystalinteraction (chemical bonding) are included. In a simple cubic lattice the effective mass is

2*

2 /m

d W dk =

h

2(51)

These effective masses, in a semiconductor for electrons in the conduction band and for

holes in the valence band describe, as essential band structure parameters, the curvatures

of W (k ) near the band edges and thus also the densities of states (44) near the conduction andvalence band edges, respectively.

*nm

* pm

Since Bloch waves (36) are stationary solutions of the Schrödinger equation in a crystal with

perfect translational symmetry, such a state, once excited in an electrical field, never woulddecay. An infinite electrical conductance would result. A finite electrical resistance is due toscattering processes of the free electrons on perturbations of the ideal translational symmetryof the crystal lattice. Such perturbations are defects, e.g. impurity atoms, dislocations etc. andthermally excited lattice vibrations (phonons). Electron-electron scattering usually plays anegligible role; because of the Pauli principle only electrons within a shell of about 2 k B T around W F can contribute. When two electrons scatter from each other, one typically becomes

faster and the other slower. But states for slower electrons are already occupied. The processis thus forbidden. Since defects can be assumed in first approximation as immobile, defectscattering is essentially elastic. Lattice vibrations are time depending oscillations of the lattice



atoms. Kinetic energy can be exchanged between these vibrations and the carriers; electronscattering on lattice vibrations is thus inelastic.

The accelerating action of an external electric field E is in stationary condition of DC currentflow balanced by scattering processes such that the carriers assume an average, so-called, drift

velocity vD on top of their random, not directed thermal velocity, which is determined by W F in a metal. In the simplest Drude-type description scattering is described phenomenologically by a friction term (∝ vD), such that the semiclassical equation of motion for a carrier followsas:

**

D D

mm v v eE

τ + = −&

x (52)

For stationary conditions the drift velocity v)0( = Dv& D is obtained as

* D xevm

τ = − E , (53)

where τ is the relaxation time which describes the decay time, after which the non-equilibriumcarrier distribution relaxes into the equilibrium distribution after switching off the externalfield. For the calculation of the current density j Drude originally (about 1900) assumed the

participation of all free carriers in a metal. Here now quantum mechanics comes into play.According to the Pauli-principle only a tiny amount of the whole free carrier density n cancontribute to the current, namely carriers in the direct energetic neighbourhood of the Fermi-energy WF. The situation is best described in reciprocal k -space, where in a simple metal theFermi-sphere is shifted on the average by a tiny k -displacement δk x due to the superimposeddrift velocity (Fig. 12).

Fig. 12) Schematic representation of the action of a constant electric field Ex on thedistribution of quasi-free electrons in reciprocal k-space. The Fermi sphere of theequilibrium distribution (dashed line, centered around (0,0,0)) is shifted by δk x=-eτEx/ħ in the state of stationary current flow (full line).

From (53) and

x D k vm δ= h* (54)

one obtains

( ) F x

e W k

τ δ = −

hx E . (55)



Only electrons near W F have to be taken into account [τ=τ(W F)] and only the electrons onstates in the shaded area (shell) of Fig. 12 contribute to the current. The current density in x-direction therefore is obtained as

∫ π−=

Shell

3

3)(

82 k x x vk d e j (56)

With the shell volume in k -space and by replacement of v x F k k δπ 24 x (on the Fermi surface

only) by v(WF), the mean value on the whole Fermi sphere, one obtains

2 22

2

( ) ( )( )

3 3 F

x F F x

e W e W k 3

2 * F F

x j v W k E E m

τ τ

π π = = (57)

Since the volume of the Fermi sphere is given by the total electron density (41)yields the old Drude expression for the current density )3(

23

nk F π=

2

*

( ) F x x

e W n x j E

m

τ σ = = E , (58)

eventhough only electrons near WF rather than all electrons contribute. The total density nenters (58) via the integration over the Fermi-sphere. Ohm’s law expressed in (58) for a metalalso holds formally for semiconductors, be it for free electrons in the conduction band, or for free holes in the valence band. The relaxation time τ as well as the effective mass m* then isascribed to electrons or holes near the band edges W

Cand W

V, respectively. According to

(47) electrons and holes are distributed according to Boltzmann statistics within theconduction and valence band and, in contrast to a metal, the whole densities of free electronsn and/or holes p near WC and/or WV contribute to the conductivity σ in (58).

In order to discuss the temperature dependence of σ(T ) for metals and semiconductors, it isconvenient to introduce the carrier mobility

*m

eτ=µ (59)

and to write (42) as

x x x j e nE E µ σ = = (60)

For metals the electron density n being given by the volume of the Fermi-sphere, is essentiallyindependent on temperature and the temperature effect on σ enters through temperaturedependent scattering processes described by µ(T ) and τ(T ), respectively. Elastic scattering onimpurities gives rise to a temperature independent so-called residual resistance ρR , whileinelastic scattering on phonons is enhanced with increasing temperature, since the scatteringcross section increases in first approximation with the increasing vibrational amplitude of the

atoms (∝ k B T in first approximation). The superposition of both scattering mechanisms yieldsthe characteristic temperature dependence of ρ(T ) for metals in Fig. 13 [8].



Fig. 13) Electrical resistance of sodium compared to the value at 290K as a function of temperature. The data points (o, ●, □) were measured for three different sampleswith differing defect concentrations. (After [8]).

In semiconductors both factors µ and n (or p for holes) in (60) depend on temperature. Thecarrier concentrations n and/or p, however, depend exponentially on temperature according to(47) and also for dopant induced conductivity (only with smaller activation energy). Themobility µ(T ), on the other hand, being determined by impurity and phonon scatteringdepends only slightly on temperature, ∝ T 3/2 for impurity scattering (dominant at lowtemperature) and ∝ T

–3/2 for phonon scattering (dominant at high temperature). For

semiconductors the conductivity σ(T ) therefore reflects essentially the exponentialdependencies n(T ) (or p(T ) for holes) of the carrier densities (Fig. 11a). Only slightmodifications due to µ(T ), essentially in the saturation regime of constant n(T ), appear.

The location of the Fermi-level WF with respect to the band edges is determined deep in the bulk of both semiconductors by the doping level. Furthermore, in thermal equilibrium WF must be equal on both sides of the semiconductor heterojunction. In order to fulfill bothrequirements, material specific well defined band offsets ∆WV, ∆WC and fixed Fermi-level

positions deep in the bulk of both semiconductors, particular band bendings with certain types

of space charge layers have to occur on both sides of the semiconductor heterointerface. For aspecial situation, n-doped high gap semiconductor on nearly intrinsic low gap material, theinterface electronic band scheme is shown in Fig. 16c.

Due to the upwards band bending within the wide gap semiconductor, bulk donor states areemptied within the space charge depletion layer, electrons originating from these donor levelsare collected in a potential well at the interface within the narrow gap semiconductor. Theseelectrons are squeezed together within a layer of thickness 1 to 2 nm, they form a quasi-twodimensional electron gas (2 DEG) with free electron movement parallel to the interface. Thenarrow gap material is not intentionally doped; the electrons which form the 2 DEG originatefrom ionized donor atoms which are spatially separated from the 2 DEG carriers. Impurityscattering even at high dopant concentrations plays a negligible role. At low temperatures theelectrons in the 2 DEG reach mobilities which are higher by orders of magnitude than in bulk semiconductors, e.g. up to 10×106 cm2/Vs in AlGaAs/GaAs heterostructures at low



temperature. These heterostructures, preferentially based on AlGaAs/GaAs, InGaAs/InP andAlGaN/GaN material systems are the basis for the fabrication of the fastest field-effect-transistor (FETs), so called HEMTs (high electron mobility transistors). Their application inwireless telecommunication and radar systems is also favoured by the extremely low noisefigures being due to the two-dimensionality of the channel together with the lack of impurity

scattering centres. Furthermore the high mobility 2 DEGs are basic systems for studyingquantum effects such as (fractional) Quantum Hall effect and transport in low dimensionalstructures.

( )(1) 2 1

2 2

m D W dW dk dW

mW π π = =

h, (67)

i.e. a strongly peaked density for each subband (Fig. 18b). This peaking is fully developed for a quantum dot in 0D, where sharp quantized energy levels result as in an artificial atom (Fig.18c).

These different shapes of the densities of states have dramatic effects on the electronic properties of confined states. While in 3D structures energetic continua of electrons and holescan recombine during optical emission in lasers, e.g., in 1D or 0D structures electrons andholes are arranged in more or less sharp energy levels. The emitted photon energy is muchsharper, i.e. centered around a mean value.

6. Conclusions

The present short overview over some basic concepts of quantum mechanics, with particular

emphasis on nanoelectronics and quantum information, electronic properties of solids and lowdimensional structures is far from being complete. It shall merely be of some help for a deeper understanding of the following chapters of the book which are closer to present researchtopics.

References

[ 1] O. Scully, B.-G. Englert and H. Walther: Nature 351, 111 (1991)

[ 2] S. Dürr, T. Nonn and G. Rempe: Nature 395, 33 (1998)

[ 3] H. Ibach and H. Lüth:“Festkörperphysik-Einführung in die Grundlagen”, 6. Auflage (Springer, Berlin,Heidelberg, New York 2002),and “Solid State Physics”, 2nd edn. (Springer, Berlin, Heidelberg, New York 1996)

[ 4] J.R. Chelikowsky, M.L. Cohen:Phys. Rev. B14, 556 (1976)



[ 5] R. Courths, S. Hüfner:Phys. Rep. 112, 55 (1984)and H. Eckhardt, L. Fritsche, J. Noffke:J. Phys. F14, 97 (1984)

[ 6] J.R. Klauder and J.E. Kunzler:“The Fermi-Surface”, Harrison and Webb, eds.Wiley, New York 1960

[ 7] D. Schoenberg:“The Physics of Metals-1, Electrons”,J.M. Ziman ed.; Cambridge 1969, p. 112

[ 8] D.K.C. McDonald, K. Mendelssohn:Proc. R. Soc. Edinburgh, Sect. A202, 103 (1950)

[ 9] H. Lüth:“Solid Surfaces, Interfaces and Thin Films”, 4th edn. (Springer, Berlin, Heidelberg,

New York 2001)

Basics of Quantum Mechanics and Solid State Physics

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