R?ssler U. Solid State Theory: An Introduction

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156 5 Electrons in a Periodic Potential

B=18.9T

T=1.5 K

h/4e

h/6e

h/8e

h/3e

h/2e

/kΩ

10 20

n=2

n=1

Fig. 5.21. Longitudinal and Hall resistance vs. magnetic ﬁeld of a Si-MOSFET

showing the quantum Hall eﬀect after [

161]

subband, is known as the magnetic quantum limit (see Problem 4.4). Upon

reducing the magnetic ﬁeld starting from this limit, a point will be reached

when all states in the lowest Landau level are occupied, which corresponds

to ﬁlling factor ν = 2 (due to spin degeneracy). With further reduction of

the magnetic ﬁeld, the Fermi energy moves into the next higher Landau level.

Thus, by sweeping the magnetic ﬁeld, the Fermi energy will jump at ﬁeld

values corresponding to (even) integer ﬁlling factors (Fig.

5.20, right). When at

suﬃciently high magnetic ﬁelds (and low temperatures) the Zeeman splitting

can be resolved, these jumps take place at all (even and odd) integer ﬁlling

factors.

In a MOSFET or a gated heterostructure, it is possible to change the

carrier density by varying the gate voltage. Thus, for a ﬁxed magnetic ﬁeld,

the Fermi energy can be shifted through the Lan dau levels by changing the

ﬁlling factor. At integer ﬁlling factors and suﬃciently low temperature, mag-

netotransport data show striking deviations from the classical behavior of the

Drude model (Fig.

5.21): the longitudinal resistance (or conductan ce) vanishes

(as typical for an insulator) and the Hall resistance (o r conductance) exhibits

plateaus, which to extremely high accuracy are inverse integer multiples of

/h [

161]. This is known as the integer quantum Hall eﬀect,discoveredby

von Klitzing

in 1980. The high accuracy of the plateau values gave rise to a

new deﬁnition of the resistance, norma l ly deﬁned by precision measurements

in a quantum Hall experiment of h/e

, which correspo nds to the Klitzing con-

stant R

K−90

= 25812, 807 Ω intro duced by the Meter Convention in 1990. An

introduction to the theory of the integer Quantum Hall eﬀect is [

162].

Klaus von Klitzing *1943, Noble prize in physics 1985.

Problems 157

In 1983, Tsui and St¨ormer

discovered signatures of the quantum Hall

eﬀect at very low temperatures and high magnetic ﬁelds at the fractional ﬁlling

ν =1/3. The conﬁrmation of this signature complemented by the discovery

of a whole family of so-called fractio nal quantum Hall states (see Sect.

7.6)in

the following years has stimulated much experimental and theoretical work on

electron systems under such extreme conditions. The fractional quantum Hall

eﬀect is now understood to be dominated by electron–electron interaction (see

Chap.

7) while the integer quantum Hall eﬀect is essentially a single-par ticle

eﬀect caused by disorder (see Chap.

9).

Problems

5.1 Derive the Hartree–Fock equations as Euler–Lagrange equations of the

variational problem (

5.7) with the Slater determinant as ground state

wave function. Note that the single-particle wave functions depend on

space and spin coordinates.

5.2 Show that the exchange potential

(r)=−e



¯n

(r, r

′

)

|r − r

′

(5.114)

with the averaged nonlocal HF or exchange density (

5.15) reduces for

free electrons to the so-called Slater exchange potential

x,Slater

(r)=−

2π

(n(r)). (5.115)

Compare it with the exchange potential in the LDA

LDA

(r)=



nǫ

(n)



n=n(r)

, (5.116)

where ǫ

(n) is the exchange energy per electron for the jellium model

with density n.

5.3 Verify that each energy band can accommodate one electron of either

spin p e r unit cell. Remember the corr e sponding statement concerning

phonon modes.

5.4 A point at the Brillouin zone boundary can be characterized by wave

vectors k and k

′

drawn from diﬀerent reciprocal lattice vectors. Give the

relation between k and k

′

and verify that the condition of degeneracy of

the corresponding free electron states, i.e., of having the same energy, is

consistent with the condition for Bragg reﬂection.

Daniel C. Tsui *1939, Horst L. St¨ormer *1949, shared the Noble prize in Physics

1998 with R.B. Laughlin.

158 5 Electrons in a Periodic Potential

5.5 Calculate and plot the free electron energy dispersion for the square

lattice along the lines Γ → M, M → X,andX → Γforthelowest

three bands (the reciprocal lattice vectors ca n be used as band index).

Indicate the position of the Fermi energy, if each atom in the unit cell

contributes one, two, or three electrons. Calculate the disper sion for the

two lowest bands in the presence o f a per i odic potential and discuss the

modiﬁcation of the Fermi circle with radius k

= π/a by this potential.

5.6 Calculate the free electron energy bands of a fcc lattice as shown in

Fig.

8.6. Find out the degeneracies of these bands.

5.7 Compare the band structures of Al, Si, and GaAs in the context of the

almost free electron or pseudo-potential approximation. What is com-

mon (diﬀerent) in the free electron picture and which of the Fourier

coeﬃcients of the periodic potential are responsible for opening of

gaps? Which details of the band structure can be understood from this

comparison?

5.8 The atomic d states are ﬁvefold degenerate (without spin). Calculate the

crystal ﬁeld splitting of these states in a cubic lattice. Is there a diﬀerence

between the sc, bcc, and fcc crystal s tru ctures?

5.9 Show that the secular problem (

5.77) can be reduced to the standard

form

′

(k) − E

′

(k)

= 0 (5.117)

by making use of the properties of the overlap matrix S

′

(k). Give the

explicit expression of

′

(k)intermsofH

′

(k)andS

′

(k).

5.10 Calculate the dispersion of an energy band deriving from an s orbital in

a sc and bcc lattice for the lines Γ − X and Γ − L.Comparewitheach

other and with the result for the fcc lattice. How does the width of the

energy band depend on the nearest neighbor conﬁguration?

5.11 Graphene is a single sheet of carbon atoms arranged in a (two-

dimensional) hexagonal lattice with primitive lattice vectors in the (x, y)

plane

= d



√



, a

= d



−

√



(5.118)

and two carbon atoms in each cell at τ

=(0,d)andτ

=(0, 2d)

with th e nearest neighbor distance d.Thes, p

, and p

orbitals can

be combined to sp

hybrid orbitals directed towards the nearest neigh-

bors, while the p

orbitals stick out of the lattice plane and form π

bonding and anti-bonding orbitals. Formulate and solve the eigenvalue

problem for graphene using LCAO for the π

orbitals in the tight-binding

approximation and discuss the resulting energy bands.

5.12 Find the expression for the cyclotron mass of a spheroidal energy surface

E(k)=



+ k



= const. (5.119)

Problems 159

in dependence on the orientation of the magnetic ﬁeld in the xz plane.

Make use of the Peierls substitution and use a vector potential with

= A

= 0 . Set up and solve the equations of motion in the xz plane.

5.13 Solve the eigenvalue problem of free two-dimensional electrons in the xy

plane in the presence of a spin-orb it coupling

= α(k ×∇V ) · σ (5.120)

due to the conﬁnement potential V in z direction, i.e., ∇V (001).

Calculate and visualize the expectation values of the spin operator σ

in dependence on the direction of k. Identify the Kramers pairs.

Spin Waves: Magnons

The electron spi n , which does not explicitly appear in the N-particle

Hamiltonian of the solid (except for the spin-orbit coupling and Zeeman

terms), will be in the focus of this chapter. In Chaps.

4 and 5 we have addressed

already the relevance of spin in connection with magnetic properties, which

shall be studied now in more detail. The issue here will be to consider the inter-

acting electron system with dominating exchange interaction, which leads to

a spin-ordered ground state, and to describe elementary excitations out of

this ground state: spin waves or (in quantized form) magnons. In several

aspects, spin dynamics is similar to lattice dynamics (see Chap.

3)withthe

masses coupled by springs, now being replaced with spins (or their magnetic

moments) coupled with exchange interaction. Depending on the complexity

of the crystal structures on which these spin systems are realized, their spin

or magnetic order can be ferromagnetic, anti-ferromagnetic, ferrimagnetic, or

anti-ferrimagnetic [

163]. Most standard textbooks on Solid State Theory con-

tain a chapter on spin waves or magnons and magnetic properties, but there

are also special review articles [

164, 165] and monographs [116, 166–172]on

these topics. In solids with disorder, e.g. d u e to alloying, the magnetic order

takes the form of spin glasses [

16]. Theoretical concepts developed for spin

glasses have turned out to be useful also for neural networks [

173]. A variant

of magnetic order in disordered solids are diluted magnetic semicondu ctors of

which A

with Mn ions randomly replacing the A

atoms show ferromag-

netic order (see [

174–176]). The latter materials have gained much interest

in connection with spintronics [

177, 178], a concept of electronics using the

electron spin rather than its charge. Magnetic order exists as a ground state

property only below some critical temperature at which a phase transition

takes place. It can be described using the molecular or mean ﬁeld approxima-

tion. For spin excitations at the surface of magnetic solids (surface magnons)

we refer to [

179, 180].

U. R¨ossler, Solid State Theory,

DOI 10.1007/978-3-540-92762-4

 Springer-Verlag Berlin Heidelberg 2009

162 6 Spin Waves: Magnons

6.1 Preliminaries

In Chap.

4 we have seen how magnetic properties of free electrons can be

understood in terms of the single-particle picture and the Pauli principle. The

Zeeman term causes a shift of the density of states of up and down spin elec-

trons against each other (see Fig.

4.7). Occupying these states according to the

Fermi–Dirac distribution function, leads to a magnetic moment determined

by the excess of majority s pins: this is the Pauli spin paramagnetism. On the

other hand, for the case without external magnetic ﬁeld, we have found a

lowering of the single-particle and ground state energies due to the exchange

term in the HF approximation, which becomes eﬀective for electrons with

parallel spins. This eﬀect is expected to get stronger with an increase of the

number N

↑,↓

of aligned spins, i.e. with a spin polarization N

↑

− N

↓

of the

system accompanied by a magnetization M = gμ

↑

−N

↓

). Eventually, one

may ask for a condition under which a state with all spins aligned could be

the system ground state.

In this consideration, we have completely neglected the inﬂuence of the

periodic potential. Let us look, therefore, for a moment at the band structure

of ferromagnetic Fe in Fig.

6.1 as an example. We note the characteristic fea-

tures resulting from s-d hybridization discussed in Sect.

5.4, but ﬁnd that for

the ferromagnetic state the energy bands of up and down spin electrons diﬀer

from each other, ǫ

nk↑

= ǫ

nk↓

, as is typical for Bloch electrons with incomplete

d shells. A closer inspection shows that the bands of spin-up and spin-down

electrons are very similar but shifted against each other, as can b e seen in

the density of states (Fig.

6.2) by comparing the position of the pronounced

structures, caused by van Hove singularities, for majority and minority spins.

This reminds us of the free electron case with external magnetic ﬁeld, but

minority spin

majority spin

0.8

0.6

0.4

0.2

HNP Γ N

wave vector k

energy [Ry]

Fig. 6.1. Energy bands of ferromagnetic Fe after [181]

6.1 Preliminaries 163

–10 –8 –6 –4 –2 0 2

3.5

3.0

2.5

2.0

1.5

1.0

0.5

1.0

1.5

2.0

2.5

3.0

majority spin

minority spin

Energy E - E [eV]

Integrated DOS

DOS [states / eV - atom]

Fig. 6.2. Density of states (solid lines) and integrated density of states (dotted lines)

for the energy bands of ferromagnetic Fe of Fig.

6.1 (after [181])

here, the shift is determined by the spin–polarization that exists even without

an external magnetic ﬁeld, i.e. there is a spontaneous magnetization below a

critical temperature. Energy bands as those of Fig.

6.1 result from an exten-

sion of the DFT-LDA to the SDFT-LSDA (spin–density functional theory in

the local spin–density approximation) [123, 182].

Turning back to the free electron system of Chap.

4, we recollect the expres-

sion for the ground state energy in HF approximation (without magnetic ﬁeld)

with each state with |k|≤k

occupied by a pair of ↑, ↓ spins (see (

4.91))

0↑↓

¯h



 

−

16π



 

. (6.1)

Remember the origin of the second term; it is the exchange part of the

Coulomb interaction which results from parallel spins. Instead of occupying

each state in the Fermi sphere with a pair of ↑ , ↓ electrons one could align

all s p i n s to gain exchange energy, but this can be done only at the cost of

increasing the k inetic energy because the size of the Fermi sphere has to be

doubled, i.e. by now occupying all states with |k|≤2

1/3

(see Fig.

6.3).

The HF ground state energy for this ferromagnetic order (the completely

spin-polarized free electron gas) is

0↑↑

2/3

− 2

1/3

(6.2)

164 6 Spin Waves: Magnons

2 k

1/3

Fig. 6.3. Fermi spheres for a spin-unpolarized and a completely spin-polarized

electron system

with a and b from (

6.1). In order to have E

0↑↑

0↑↓

, k

has to fulﬁll the

condition

1/3

− 1

2/3

− 1

=0.44

1.1

−1

(6.3)

with the Bohr radius a

or, with k

=(9π/4)

1/3

1/r

, this yields for the

density par ameter

9π

1/3

1.1

=5.45. (6.4)

According to this estimate the ferromagnetic ground state should be favored

in d i l uted el ectron systems with r

> 5.45, which is still in the region of

electron densities in metals (2 <r

< 6 ). However, this e stimate should not

be taken for a proof of the existence of a ferromagnetic ground state as it does

not include the eﬀects of correlation or of the periodic potential.

For a more rigorous treatment, we start from the N-electron Hamiltonian

(

5.2) that includes the periodic potential. It can be written with the help of

(

4.76)and(4.77) in terms of fermion operators as

H =



†



αβα

′

αββ

′

†

′

, (6.5)

where α, β, α

′

, and β

′

are complete sets of single-particle quantum num-

bers, which here are those of Bloch states, e.g. α = nkσ and ǫ

are the

corresponding sing l e-particle energy values. The potential matrix elements

αββ

′



′

†

(x)ψ

†

′

)

4πε

|r − r

′

(x)ψ

′

) (6.6)

canbeexpressedintermsofthesingle-electron wave functions ψ

(x)

(Problem 6.1), where x stands for space and spin coordinates. Summing up

6.1 Preliminaries 165

the spin coordinates results in getting the same spin quantum numbers for

the states α, β

′

and for the states β, α

′

. In the following, we shall separate the

Hamiltonian (

6.5) according to H = H

+ H

int

into its single-particle and

interaction parts.

Magnetism is known, from the introductory courses on solid state physics

[

29–31], to arise in materials with incomplete d or f shells. The corresponding

electrons form narrow bands (see e.g. Fig.

6.1), for which the free electron

picture hardly applies. We take this situation into account by switching to

the LCAO (or tight-binding) approximation introduced in Sect.

5.4.Inthis

picture, contrary to Fig.

6.3, the electron s pin is not attached to delocalized

electrons, and magnetism results from the magnetic moments of the total spin

carried by local atomic d or f orbitals.

Let us c onsider electrons in an energy band deriving from one such orbital.

The Bloch function can be expressed (as in Sect.

5.5)by

kσ

(r)=

√



ik·R

(r − R), (6.7)

where we have dropped the band index to simplify nota tion while the index

σ refers to the spin eigenstate. It is adva ntageous here to adopt the Wannier

representation with localized orbitals, which are orthogonal for diﬀerent lattice

sites, although they may still overlap (Problem 6.2). This allows one to switch

with

kσ

√



ik·R

Rσ

(6.8)

to fermion operators and write the single-particle term in the form



kσ

†

kσ



′



ik·(R−R

′

)



 

′

†

′

Rσ



′

†

′

Rσ

. (6.9)

The dispersio n of the energy band ǫ

(assumed to be indep endent of σ)isnow

expressed in terms of the hopping or transfer matrix elements t

′

, R = R

′

which is a two-center integral of the type given in (

5.74), while the term

with R = R

′

gives the atomic energy level from which the band derives (see

Problem 6.2) .

Likewise the electron–electron interaction takes the form (Problem 6.3)

int



′

†

′

(6.10)

Gregory Hugh Wannier 1911–1983.

166 6 Spin Waves: Magnons

with the interaction matrix element being now the four-center integral

′



′

∗

(r − R

)φ

∗

′

− R

)

4πε

|r − r

′

φ(r − R

′

)φ(r

′

− R

′

). (6.11)

Note, that the Wannier (or atomic) orbitals and the single particle energies

are assumed to be independent of the spin quantum number σ. This means

to neglect all spin-dependent eﬀects deriving from spin–orbit coupling and

from the electrons in all other occupied bands of the solid, while here all spin

related eﬀects derive from the interaction term.

6.2 The Heisenberg Hamiltonian

The extreme case of strongly localized orbitals would lead to vanishing hop-

ping matrix elements and to a single-particle part of the Hamiltonian, just

counting the occupation of the sites multiplied by the atomic level energy.

Contrary to the free-electron case (see Fig.

6.3), in a ﬂat band there is no

increase in the kinetic energy when all spins are aligned, which is the favorite

conﬁguration w ith respect to the exchange interaction. But this ferromagnetic

conﬁguration, with each site being occupied by a single electro n with g i ven

spin (Fig.

6.4), minimizes the Coulomb repulsion also, because each two elec-

trons are separated in space as much as possible in the given crystal structure.

Thus, this conﬁguration can be considered as the ground state.

We have to keep in mind here that the assumption of strongly localized

atomic orbitals does not apply to metallic ferromagnets such as the transition

metals, which form d bands with a width of the order of one eV, (see Figs.

5.14

and 6.1) due to the overlap between nearest neighb ors in a close-packed crystal

structure (bcc or fcc). This overlap is, however, essentially reduced in transi-

tion meta l compounds like MnAs, EuS, EuS with a larger spacing between the

metallic ions in a lattice with basis. They appear as ferromagnetic insulators

and are the materials, to which the mo del of Fig.

6.4 applies.

With respect to elementary excitations, this ferromagnetic ground state

plays the same role as the ﬁlled Fermi sphere for the free electrons. As in

Chap.

4, when considering the HF approximation the interaction term in

application to this ground state, we can distinguish between a direct and an

R –d R R+d

Fig. 6.4. Ferromagnetic conﬁguration of spins on a linear chain