R?ssler U. Solid State Theory: An Introduction

Подождите немного. Документ загружается.

4.7 Electronic Correlation 115

The electrostatic energy per electron in the Wigner crystal amounts to

= ǫ

−

+ ǫ

= −

1.8

Ry. (4.164)

This expression corresponds to the sum of exchange and correlation energy.

With the latter being given by ǫ

= −0.916/r

Ry, the correlation energy per

electron in the Wigner crystal turns out to be

≫ 1) = −

0.88

. (4.165)

We may ﬁnally compare with the electrostatic energy of the smeared (instead

of the point-like) electron in the Seitz sphere: it is given by

jell

−

= −2ǫ

= −

2.4

Ry (4.166)

which is larger than ǫ

−

. Thus, the Wigner crystal is the predicted ground

state of the strongly diluted homogeneous electron system. The experimental

veriﬁcation of the Wigner crystal represents a tremendous challenge b ecause

the realization of a diluted homogeneous electron system turns out to be very

diﬃcult.

Problems

4.1 Calculate the interaction energy of a homogeneous electron gas (density

n = N/V ) with a homogeneous jellium background of positive charges

with the same density and the electrostatic po tential energy of this

background. Hint: Make use of the Fourier transform of the Coulomb

interaction.

4.2 According to (

4.25) the chemical potential μ(T ) is determined by the

particle density n. For the three-dimensional electron gas, calculate the

temperature dependence of μ for low temperatures (k

T ≪ E

)upto

the order (k

T/E

)

4.3 Calculate the density of states for a system of electrons, which can move

freely in a plane (two-dimensional electron gas) or along a straight line

(1-dimensional electron gas). Give an expression for the density of states

of bound electrons (zero-dimensional electron gas). Plot and discuss these

results.

4.4 Typical electron densities are n ≃ 10

−3

for metals and n ≃

−3

for doped semicondu ctors (see Table 4.1) Consider the mag-

netic qua ntum limit, w h en all electrons are in the lowest Landau level,

and derive for this case a relation between the magnetic ﬁeld and n.For

which of the two systems is it po ssible to reach the magnetic quantum

limit in a laboratory?

116 4 The Free Electron Gas

4.5 The Pauli spin-paramagnetism is determined by the spin-splitting of

the electronic energy spectrum due to the Zeeman term (see Fig.

4.7).

Express this contribution to the magnetization by the number of non-

compensated electron spins and compare the result with (

4.53).

4.6 Discuss the quality of the HF approximation starting from ǫ

(

4.93)

for metals and semiconductors (see parameters given in Table 4.1). In

terms of the density parameter r

, which system is of higher density?

4.7 In order to go beyond the HF approximation (which is a ﬁrst order per-

turbation calculation) one may consider the second order perturbation

correction E

due to the Coulomb interaction. (a) Make use of the Fermi

sphere to ﬁnd excited states which yield a contribution to E

.(b)Dis-

tinguish between direct and exchange contributions to E

and show that

the direct term can be formulated as

dir

= −4m



kqp

(1 − n

p+k

(1 − n

q+k

)

¯h

k · (k + q + p)

(4.167)

where n

is the occupation number. (c) Show by expanding n

p+k

for

small k that



(1 − n

p+k

) ∼ k and use this result to ﬁnd that (for

small k) E

dir

is logarithmically divergent. Interpret this result!

4.8 Field operators are deﬁned by

Ψ(r)=



(r), Ψ

†

(r)=



†

∗

(r) (4.168)

with fermion operators c

†

and a complete orthonormal set of single

particle wave functions ψ

(r) (for free electrons ψ

(r)=exp(ik · r)/

√

V ).

(a) Discuss the meaning of Ψ(r)andΨ

†

(r). (b) Derive the commutation

relations for Ψ(r)andΨ

†

(r). (c) The density operator is given by ˆn(r)=

†

(r)Ψ(r). Give for free electrons the Fourier components ˆn

deﬁned by

ˆn(r)=



iq·r

ˆn

. What is the meaning of ˆn

for q =0andq =0?

d) Express the Coulomb i nteraction in terms of the number ﬂuctuation

operators

= V ˆn

4.9 Calculate the thermal expectation value of the numb er ﬂuctuation

operator 

 by starting from the exact expression



 = lim

Γ →0

i¯h



∞

iωτ−Γτ

[

(τ),

−q

(0)]

ext

dτ (4.169)

by replacing the bare external charge eN

ext

by e(N

ext

+ 



RPA

)with

the induced number ﬂu ctuation 



RPA

in RPA and evaluating the

expectation value of the commutator not in the exact ground state of

the interacting system but in the ground state of the system without

interaction. This is the self-c onsistent ﬁeld approximation.

Problems 117

4.10 Prove the sum rule



∞

ǫ(q,ω)

ωdω = −

(4.170)

where ω

is the plasma frequency. Hint: Calculate the ground state

expectation value of [[H

jell

, ˆn

], ˆn

−q

] by making use of the commutation

relations of c

and c

†

and by intro ducing a complete set of eigenstates

of H

jell

4.11 Evaluate the real part of the dielectric constant (

4.132) for the static

case (ω = 0) in the long-wavelength limit |q|→0 (Thomas–Fermi

approximation). It will depend on the parameter k

deﬁned by

3Ne

2ε

. (4.171)

Consider the screened Coulomb potential v

/ε

(q, 0): How does the

Fourier transform of this potential look like and what is the obvious

meaning of k

−1

4.12 The ground state density–density correlation function is deﬁned as

p(r)=



Ψ

|ˆn(r + r

′

)ˆn(r

′

)|Ψ

dr

′

. (4.172)

The Fourier transform of p(r)isS(k)=

ik·r

p(r)d

r.Itcanbe

obtained from the dynamic form factor S(k,ω)with

S(k)=



S(k,ω)dω. (4.173)

Use the relation between S(k,ω)andIm(1/ǫ(q,ω)) to calculate p(r)in

HF approximation.

Electrons in a Periodic Potential

In Chap.

4, the crystal structure of the solid has been suppressed by smear-

ing out the p eriodic conﬁguration of ions in the jellium model, because the

many-body aspects of the electron–electron interaction stood in the focus of

interest. Now, our attention will be, in addition, at the eﬀect of the periodic

potential formed by the ions in the conﬁguration {R

n,τ

} of a crystal lattice.

Therefore, we reverse the introduction of the jellium term in Sect.

4.3 with

the replacement

⇒



n,τ ,l

v(r

− R

n,τ



n,n

′

τ ,τ

′

V (R

nτ

− R

′



V (r

). (5.1)

Note the simpliﬁed notation of the single-particle potential V (r

), which does

not explicitly refer to the ion positions. It is invariant under lattice transla-

tions and is the same for all electrons. The Hamiltonian for a system of N

electrons in the crystal volume



l=1

+ V (r

)



k,l=1

k=l

v(r

− r

), (5.2)

with the Coulomb potential

v(r

− r

κ|r

− r

,κ=4πε

(5.3)

deﬁnes the starting point for this chapter.

The complexity of the eigenvalue problem of H

comes from the simul-

taneous presence of the periodic lattice potential and the electron–electron

interaction. As we have already seen in Chap.

4, the latter prevents a rigorous

solution of the eigenvalue problem and some approximate treatment has to

be applied. In some textbooks, e.g., [

6, 7, 13, 121], the potential energy terms

of H

,namely



V (r

) a n d the electron–electron interaction, are replaced

U. R¨ossler, Solid State Theory,

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

 Springer-Verlag Berlin Heidelberg 2009

120 5 Electrons in a Periodic Potential

by an eﬀective single-particle potential:

⇒



+ V

eﬀ

)

(5.4)

which incorporates the many-body aspect in an approximate way. Other text-

books, e.g., [

8, 9, 122], provide arguments, on how this replacement can be jus-

tiﬁed and we shall follow this line in Sect.

5.1. Given such a potential, electrons

can be understood as independent particles and in a wave picture, are expected

to undergo Bragg reﬂections due to the periodic potential (as in the case of

elastic or electromagnetic waves). The consequence is that the energy spec-

trum becomes a band structure with energy intervals for which propagation

of electrons is possible. They are separated by gaps, where this is not the case.

In this approximation, the Hamiltonian is a sum of identical sing l e-particle

terms and can be separated with a product Ansatz into one and the same

single-particle Schroedinger equation for all electrons:

−

¯h

∆ + V

eﬀ

(r)

(r)=E

(r). (5.5)

Here, α denotes a complete set of single-particle quantum numbers, which, as

will be outlined in Sect.

5.2, are the band index n and the crystal momentum

or wave vector k in the ﬁrst Brillouin zone. Spin–orbit coupling and a spin

index can be added if required. The energy eigenvalues E

= E

(k)formthe

energy band structure of electrons. The foremost task is the justiﬁcation of this

approach and the deﬁnition of the eﬀective single-particle potential V

eﬀ

(r).

5.1 Density Functional Theory

The many-particle problem is deﬁned by the Hamiltonian H

in (

5.2). The

eigenvalue equation H

= EΨ

,isthesameas(

2.17)or(2.25)withthe

electron–ion interaction considered for the crystalline equilibrium conﬁgura-

tion of the ions. But it corresponds also to that of the jellium model treated in

Chap.

4 with the exception, now that the crystal structure of the ions is consid-

ered. Thus, we may use the concepts develop ed for the homogeneous electron

gas. The simplest approach would be to apply the Hartree approximation with

the product Ansatz

...r

α=1

). (5.6)

The single-particle wave functions ψ

have to be determined from the condi-

tion that the expectation value of H

takes a minimum under the constraint

of their normalization. This leads to Ritz’ variational problem (see Sect.

4.1)

Ψ

|Ψ

−





ψ

|ψ

−1



=0 (5.7)

5.1 Density Functional Theory 121

where the normalization condition enters with the Lagrangian parameters E

The expression in the curly brackets is a functional of the single-particle wave

functions ψ

, thus the variation is to be performed as the functional derivative

with respect to ψ

or ψ

∗

.Ifthisisdonewith

Ψ

|Ψ

 =



α=1



rψ

∗

(r)

+ V (r)

(r)



α,β =1

α=β



′

∗

(r)ψ

∗

′

)v(r − r

′

)ψ

(r)ψ

′

)

(5.8)

one obtains the Euler–Lagrange equations of the var iational problem (note

that the variational derivative gives two contributions from the double sum

of the interaction term)



−

¯h

∆ + V (r)+



′



β=α

|ψ

′

κ|r − r

′



(r)=E

(r). (5.9)

They have th e form of (

5.5) with an eﬀective potential composed of the poten-

tial V (r) due to the periodic ion conﬁguration and a contribution from the

electron–electron interaction, which can be identiﬁed as the Hartree poten-

tial: it describes the direct Coulomb interaction of the electron in the state α

with the charge distribution of all the other electrons. Remember, that in the

jellium model the Hartree contribution exactly compensates the jelli u m term,

to which V (r) simp liﬁ es in this case.

Three aspects are to be mentioned here: the Hartree potential for ψ

H,α

(r)=



′

κ|r − r

′



β=α

occ

|ψ

′

(5.10)

depends on all the other occupied single-particle states with β = α with the

consequence that (

5.9) can be solved only if the solutions (the eigenfunc-

tions) are known. How such a self-consistent solution can be achieved, will be

explained at the end of this section. Moreover, the Hartree potential depends

also on the state α, and one has to deal with equations containing diﬀerent

potentials. Finally, as we know from Sect.

4.3, the product Ansatz does not

prevent double occupation of single-particle states, thus being in contrast with

the Pauli principle. Nevertheless, the Hartree approximation indicates in some

way, how to arrive at the eﬀective single-particle potential we are looking for,

without reaching it.

In the next step, we assume Ψ

in the form o f a Slater determinant

as in (

4.61) constructed from single-particle wave functions ψ

(x), where x

comprises space and spin variables and α is a complete set of quantum num-

bers including spin, but proceed as before. By carrying out the variation

122 5 Electrons in a Periodic Potential

(Problem 5.1) we arrive at the Hartree equations (

5.9) augmented by an

additional term

x,α

(r)ψ

(r)=−



′

v(r − r

′

)



β=α,

occ

∗

′

)ψ

′

)ψ

(r). (5.11)

This exchange term results from the fact that the expansion of the Slater deter-

minant contains products of single-particle wave functions with interchanged

particle coordinates, which do not exist in the simple product Ansatz of the

Hartree approximation. Here, V

x,α

(r) is not a simple p otential acting as a fac-

tor on ψ

(r) but a nonlocal integral op erator, because ψ

′

) appears under

the integral. The -sign in the sum under the integral indicates the restric-

tion to contributions from states β = α with parallel spins and results from

the summation over the spin variables. We may include the contributions for

α = β in the Hartree and the exchange terms, which cancel each other, to

obtain the Hartree–Fock equations



−

¯h

∆ + V (r)+



′

n(r

′

)v(r − r

′

)

−



′

(r, r

′

)v(r − r

′

)



(r)=E

(r). (5.12)

Here we have introduced the density

n(r)=



β,occ

|ψ

(r)|

(5.13)

and the Hartree–Fock (HF) or exchange density

(r, r

′



β,

occ

∗

′

)ψ

′

)ψ

∗

(r)ψ

(r)

∗

(r)ψ

(r)

. (5.14)

For free electrons with ψ

(r)=e

ik·r

√

(here we denote the crystal volume

by V

)(

5.14) leads back to the results in Sect. 4.4 (Problem 5.2).

As the Hartree potential (

5.10), also the exchange potential is state-

depen d ent and the HF equations have to be solved self-consistently. However,

by replacing the state-dependent HF density by its average over all occupied

states

(r, r

′

) ⇒ ¯n

(r, r

′



α=1

(r, r

′

), (5.15)

we obtain an eﬀective single-particle potential

eﬀ

(r)=V (r)+



′



n(r

′

) − ¯n

(r, r

′

)



κ|r − r

′

. (5.16)

5.1 Density Functional Theory 123

which is the same for all electrons. Besides the electrostatic interaction with

the periodic ion conﬁguration, it contains the electron–electron interaction in

approximate form. It should be noted, however, that although deriving from

the Ansatz with the Slater determinant, it is not exactly the HF approximation

because the averaging of the e xchange density in (

5.15) represents a non-

systematic step.

The question arises if one could go systematically beyond HF and con-

sider in V

eﬀ

(r) cor relation contributions besides exchange also as discussed in

Chap.

4. The answer to this question comes from the density functional theory

(DFT) [

9, 122–126], which since its formulation in the sixties has developed

into the most frequently used concept in calculating the electronic structure of

atoms, molecules, and solids and earned in 1998 the Nobel prize in Chemistry

shared by Kohn and Pople. Here we outline the three essential steps of DFT:

1. The Hohenberg–Kohn

theorem [127]

2. The Kohn–Sham equations [

128]

3. The loc a l density approximation (LDA)

The basic idea of the Hohenberg–Kohn theorem is that the ground state

energy of the N electron system for a given external pot ential (here that of the

periodic conﬁguration of the ions V (r)) is a unique functional of the single-

particle density n(r ) . This idea is conceivable: when adding the electrons to

the external potential, they will arrange in a unique way to establish the state

with lowest energy. This state will be characterized by a many-particle wave

function Ψ

({r

}) and a single-particle density (the spin degree of freedom is

suppressed here)

n(r)=



...



...d

|Ψ

(r, r

...r

,N=



r n(r). (5.17)

The statement of the theorem is:

Let n(r) be the (inhomogeneous) single particle density for the ground state

of a system of interacting electrons in an external potential V (r)andletthe

density n

′

(r) have the same relation to the external potential V

′

(r). Then it

follows from n(r)=n

′

(r)thatV (r)=V

′

(r)uptoaconstant.

The pr oof of the theorem is indirect:

Assume two systems with external potentials V (r) = V

′

(r), which diﬀer by

more than just a constant, but have identical densities n(r)=n

′

(r)inthe

ground state. Then one has the ground state energies

′

= Ψ

′

|T + V

′

+ U|Ψ

′

,E

= Ψ

|T + V + U |Ψ

 (5.18)

where T and U denote the kinetic energy and the electron–electron interaction,

respectively, and |Ψ

 and |Ψ

′

 the ground states of the system with external

potential V (r)andV

′

(r), respectively. As |Ψ

 is not the ground state of

the system with the external potential V

′

(and |Ψ

′

 not that of the system

Walther Kohn *1923, shared the Nobel prize in chemistry 1998 with J. Pople.

124 5 Electrons in a Periodic Potential

with V ), we may formulate the following relations

′

< Ψ

|T + V

′

+ U|Ψ

 = E

+ Ψ

′

− V |Ψ

 (5.19)

and

< Ψ

′

|T + V + U |Ψ

′

 = E

′

+ Ψ

′

|V − V

′

|Ψ

′

. (5.20)

Due to the assumption of identical densities, the last terms in both relations

are identical (except for the sign) and by taking the sum of these expressions

we ﬁnd the contradictory relation E

+ E

′

+ E

′

. Thus, the assumption

must be incorrect, while the statement of the theorem is correct. (We note

that this proof applies, if the ground state is not degenerate. A more general

proof was given by Levy [

124].)

The Hohenb erg–Kohn theorem can be formulated also by saying that for

the given external potential V (r) the exact ground state energy E

is a unique

functional of the exact ground state density n(r):

= Ψ

|T + V + U |Ψ

 = E

[n(r)] . (5.21)

It tells us, that in order to ﬁnd the ground state energy of the N -particle

problem with H

, it is not required to ﬁnd the exact many-particle wave

function Ψ

(which is a function of the coordinates of N electrons), it suﬃces

to ﬁnd the exact single-particle density n(r) (which depends only on the

coordinates of one particle). Note, that the same statement as for the ground

state energy E

can b e made for the expectation value of any observable in

this state (and this includes also the response functions). One has to keep in

mind, however, that the theorem is restricted to the system ground state.

Now, the problem to be solved is to ﬁnd E

as min i mum of the energy

functional

[n(r)] = T [n(r)] +



V (r)n(r)d



n(r)n(r

′

)

κ|r − r

′

+ E

[n(r)], (5.22)

where the ﬁrst and last terms describe the exact functionals of the kinetic and

exchange–correlation energy, respectively, of the interacting electron system,

while the second and third term are the electrostatic energy of the electron

density en(r) in the external potential V (r) and the Hartree energy for this

charge density. In mathematical language (

5.23) represents, for a given exter-

nal potential, a mapping of the densities onto the energies of which we have

to ﬁnd the minimum. T [n]andE

[n] ar e not exactly known and will be

considered in an approximate way, as outlined in the following.

The variational problem can be treated by assuming the representation o f

the density by a complete set of single-particle wave functions

5.1 Density Functional Theory 125

n(r)=



α=1

|ψ

(r)|

, ψ

|ψ

 =1,N=



n(r)d

r. (5.23)

Taking the functional derivatives with respect to the ψ

∗

(which corresponds

to the variation with respect to the density n(r)) an d assuming T [n]asfor

non-interacting electrons

T [n (r)] ≃ T

[n(r)] =





∗

(r)

−

¯h

∆

(r)d

r (5.24)

one arrives at a set of Schroedinger equations for the single-particle

functions ψ



−

¯h

∆ + V (r)+



n(r

′

)

κ|r − r

′

+ V

(r)



(r)=E

(r), (5.25)

the Kohn–Sham equations. They have the form of (

5.5), but are integro–

diﬀerential equations due to the fact that, because of

(r)ψ

(r)=

δn

[n]

δn

δψ

∗

, (5.26)

(r) is a nonlocal integral operator and we may write similar to (

5.11)

(r)ψ

(r)=



′

(r, r

′

; E

)ψ

′

). (5.27)

Here, Σ

(r, r

′

; E

) is the exchange–correlation self-energy or mass operator.

In the last step, the Kohn–Sham equations can be reduced to the form of

(

5.5) by applying the local density approximation (LDA) with the replacement

[n(r)] ⇒ E

LDA

[n(r)] =



n(r)ǫ



n(r)



r, (5.28)

where ǫ

(n) is the xc energy per electron of a homogeneous system with

density n.Foreachr of the inhomogeneous system (with external potential

V (r) = const) the xc energy of the jellium model (Chap.

4)istakenwith

the local density n = n(r). This allows one to write the xc term as a local

single-particle potential

LDA

(r)=



nǫ

(n)



n=n(r)

(5.29)

with ǫ

(n)=ǫ

)from(

4.150).Thus,wehavefoundtheeﬀectivesingle-

particle potential to be used in (

5.5)

LDA

eﬀ

(r)=V (r)+



n(r

′

)v(r − r

′

+ V

LDA

(r). (5.30)