R?ssler U. Solid State Theory: An Introduction

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4.5 The Dielectric Function 105

The inverse dielectric function has poles at ω = ±ω

− iΓ in the

lowe r complex ω-plane (see Fig. 3.5), thus measurements of 1/ε(q,ω)pro-

vide information about the exact excitation energies ¯hω

= E

− E

. Such

experiments are p erformed as inelastic scattering of charged particles (elec-

trons), in which energy and momentum is transferred to the electron system.

This experimental technique is known as energy–loss spectroscopy.

Let us evaluate the inverse dielectric function at T = 0 K in HF approx-

imation. This is done by calculating the matrix elements of the number

ﬂuctuations between the HF ground state (which is the ﬁll ed Fermi sphere)

and excited states, which are obtained by removing a n electron from the

Fermi sphere and placing it into a state outside (see Fig. 4.14). In other words,

the excited states are the Fermi sp here plus an electron–hole excitation. The

matrix elements can be evaluated and yield



kσ

†

k+qσ

kσ

Ψ

kqσ

′

|Ψ

 =



′

|k|≤k

, |k + q| >k

0otherwise

(4.118)

and one ﬁnds (with ¯hΓ=δ)

(q,ω)

= 1 + lim

δ→0



|k|≤k

,σ

|k+q|>k



¯hω + ǫ

k+q

− ǫ

+iδ

−

¯hω − ǫ

k+q

+ ǫ

+iδ



. (4.119)

Here the energies ǫ

ought to be the HF single particle energies, but can be

replaced by ¯h

/2m which is a good approximation for r

≪ 1. The restricted

sum can be considered by using the Fermi distribution function f

and if in

addition we replace in the ﬁ rst term k + q →−k the rhs of (

4.119)reads

= 1 + lim

δ→0



k,σ



−k−q

(1 − f

−k

)

¯hω + ǫ

−k

− ǫ

−k−q

+iδ

−

(1 − f

k+q

)

¯hω − ǫ

k+q

+ ǫ

+iδ



. (4.120)

k,σ

k+q,σ

Fig. 4.14. Schematic view of an excited state: Fermi sphere plus electron–hole pair

106 4 The Free Electron Gas

Finally with f

−k

= f

, ǫ

= ǫ

−k

,andf

k+q

= 0 (one of the states is not

occupied) one obtains

(q,ω)

=1+v

(q,ω) (4.121)

where

(q,ω) = lim

δ→0



k,σ

− f

k+q

¯hω − ǫ

k+q

+ ǫ

+iδ

. (4.122)

is the polarization function of the free noninteracting electron system.

A standard approximation for the inverse longitudinal dielectric function

beyond HF is the so-called random phase appr oximation or RPA. Although its

derivation in the original literature (where also the name is justiﬁed) is quite

involved (see [

5]) it can be obtained simply by replacing the bare Coulomb

interaction v

by the screened on e v

/ε(q,ω). This gives rise to the following

formulation:

ε(q,ω)

=1+

ε(q,ω)

(q,ω)

=1+v

(q,ω)



1+v

(q,ω)(1 + v

(q,ω)(...))



=1+v

(q,ω)+



(q,ω)



+ ...

1 − v

(q,ω)

. (4.123)

The structure of this expression, reminding of the Born series known from scat-

tering theory, is typical for results obtained for interacting particles and can

be cast into a graphic representation in terms of Feynman diagrams [

4, 5, 64].

Here we give only the formula for the dielectric function in RPA, also known

as the Lindhard function,

RPA

(q,ω)=1− v

(q,ω)

=1− v

lim

δ→0



k,σ

− f

k+q

¯hω − ǫ

k+q

+ ǫ

+iδ

. (4.124)

It should be mentioned, that this result which goes beyond the HF appr oxima-

tion, is not obtained by a more accurate description o f the eigenstates rather

than by the physical argument o f screening applied to the electron–electron

interaction in the system. The same result can be obtained in an alterna-

tive way known as the self-consistent ﬁeld approximation. The idea here is to

consider the induced number density ﬂuctuations together with the external

charge as the perturbation of the system but evaluate the response with the

HF ground state. This calculation is the subject of Problem 4 .9.

4.6 Discussion of the Dielectric Function 107

4.6 Di scuss i on of the Dielectric Function

Starting from (

4.117), the exact expression for the inverse longitudinal dielec-

tric function at T =0Kcanbeseparatedwith

lim

δ→0

x ± iδ

= P

∓ iπδ(x) (4.125)

into its real and imaginary part:

ε(q,ω)

=1+v



|Ψ

|Ψ

|

2¯hω

(¯hω)

− (¯hω

)

(4.126)

ε(q,ω)

= −πv



|Ψ

|Ψ

|

{δ(¯hω −¯hω

) − δ(¯hω +¯hω

)}.

(4.127)

In Sect.

1.3 we have introduced the dynamic structure factor

S(q,ω)=

2π



∞

−∞

−iωt



(t)

−q

(0)dt (4.128)

which for T = 0 K (after resolving the Heisenberg picture for

(t), introduc-

ing a complete set of exact eigenstates, and performing the integratio n over t)

takes the form

S(q,ω)=



|Ψ

|Ψ

|

δ(ω − ω

). (4.129)

Thus it is possible to express the imaginary part of the dielectric functio n as

ε(q,ω)

= −

¯h



S(q,ω) − S(q, −ω)



. (4.130)

Moreover, the diﬀerential cross section for inelastic scattering is related with

the dynamic structure factor:

dΩdω

′



2π¯h



S(q,ω). (4.131)

This relation tells us how to extract information about the excitations in the

interacting electron system from inelastic scattering exp eriments: The cross-

section will be enhanced if the experimental settings, q and ω, corresp ond

to the energy and momentum of an excitation in the electron system. The

structure facto r or likewise the dielectric function can be expressed in terms

of the characteristic excitation energies of the system. Due to this relation

Im(1/ε(q,ω) is known also as the energy–loss function.

In order to discuss this in more d etail, we look at the dielectric function

in RPA ε

RPA

(q,ω)=ε

(q,ω)+iε

(q,ω)with

108 4 The Free Electron Gas

(q,ω)=1+v



kσ

k+q

− f

¯hω − ǫ

k+q

+ ǫ

(4.132)

(q,ω)=πv



kσ

− f

k+q

) δ(¯hω − ǫ

k+q

+ ǫ

) . (4.133)

The energy–loss function can be written in terms of ε

(q,ω)andε

(q,ω)as

ε(q,ω)

= −

(q,ω)

(ε

(q,ω))

+(ε

(q,ω))

. (4.134)

The numerator ε

(q,ω) gives nonvanishing contributions only for

¯hω = ǫ

k+q

− ǫ

¯h

(2k + q) · q (4.135)

which in view of the particle–hole excitations out of the Fermi sphere is possi-

ble at ¯hω =0forallq with 0 ≤|q|≤2k

and for ¯hω > 0 for the same range

of |q| but shifted to higher values. This excitation spectrum is known as the

particle–hole continuum (see Fig. 4.15).

Signiﬁcant contributions to the en ergy-loss function are also expected if

the denominator in (

4.134) vanishes, i.e., for ε

(q,ω)=ε

(q,ω)=0.Letus

consider the case ¯hω ≫ ǫ

k+q

− ǫ

for which ε

(q,ω) = 0 and (after replacing

k + q in the ﬁrst term of ε

(q,ω)byk)

(q,ω)=1+v



kσ



¯hω − ǫ

+ ǫ

k−q

−

¯hω − ǫ

k+q

+ ǫ



=1+v



kσ

2ǫ

− ǫ

k+q

− ǫ

k−q

(¯hω − ǫ

+ ǫ

k−q

)(¯hω − ǫ

k+q

+ ǫ

)

. (4.136)

Using ǫ

=¯h

/2m, the numerator can be rewritten as −¯h

/m,andsim-

plifying the denominator for ¯hω ≫ ǫ

k+q

− ǫ

, one obtains for suﬃciently

small |q|

plasmon

particle - hole

continuum

ω(q)

q/2k

Fig. 4.15. The spectrum of excitations derived from the energy–loss function: RPA

spectrum

4.6 Discussion of the Dielectric Function 109

(q,ω) ≃ 1 − v

¯h

(¯hω)



=1−

Vmω

(4.137)

or for q → 0

(q,ω) ≃ 1 −

,ω

. (4.138)

Apparently, the energy–loss function has a singularity at the plasma frequency

. As the typical result of RPA it represents a collective excitation, the plas-

mon, in which all particles of the electron system participate. Moreover, it

obeys a sum rule, which can be expressed in terms of ω

(Problem 4.10).

We note that (

4.138) does not contain ¯h. This indica tes the possibility of

an interpretation in terms of classical physics. In fact, the classical equation of

motion of a single electron in an electric ﬁeld E, determined by the dielectric

polarization E = −P /ε

= ner/ε

of the electron system with density n =

N/V ,reads

r = −

r or

r = −ω

r . (4.139)

In the long wavelength limit each electron of the system experiences a restoring

force characterized by the plasma frequency when (in the collision with a fast

charged particle) the system is displaced again st the jellium background. For

ﬁnite q,oneobtainsfromε

(q,ω)=0

(q)=ω

¯h

mω

+ ...

, (4.140)

where the second and higher order terms indicate quantum mechanical

corrections to the classical result.

The typical RPA sp ectrum, consisting of the particle–hole continuum and

the plasmon mod e, as depicted schematically in Fig. 4.15, can be translated

for small momentum transfer into the real and imaginary part of the dielec-

tric function as shown in Fig. 4.16: ε

(q,ω) is determined by the particle–hole

continuum giving contributions only at low frequencies, while ε

(q,ω), show-

ing a more complex frequency dependence, starts at ω = 0 with the value

deﬁned by the Thomas–Fermi screening parameter k

(see Problem 4.11),

changes sign around the upper cut-oﬀ frequency of the particle–hole excita-

tions, passes through zero at the plasma frequency, and approaches 1 for high

frequencies. These characteristic features are also found in the experimental

data, e.g., [

5, 89].

In metals with electron densities of the order of 10

−3

(see Table 4.1),

the plasmon energy ¯hω

is about 1 Ry and determines the response in an

energy-loss experiment in this energy range. In contrast, doped semiconduc-

tors represent diluted metallic systems with much lower plasmon energies,

which can be tuned by the concentration of the dopands in the range of

110 4 The Free Electron Gas

q<<

Fig. 4.16. The frequency dep endent real and imaginary part of the dielectric

function as obtained for the RPA spectrum

optical phonons at q = 0. The response of the system (measured by inelas-

tic light scattering [

118]) is then characterized by couple d plasmon–phonon

modes with frequencies determined by the zeros of the dielectric function in

the long-wavelength limit

ε(ω)=ε

∞



1 −

− ω



, (4.141)

which is the sum of the plasmon and phonon contributions (see Sect. 3.5).

Note that the plasmon contribution is modiﬁed by the dielectric constant ε

∞

of the semiconductor accomodating the diluted plasma, while the electron

mass becomes the eﬀective mass m

∗

. The biquadratic equation ε(ω)=0

has two solutions, which in dependence on the carrier concentration show a n

anticrossing b e havior typical for the coupling between th e two modes, which

is also experimentally veriﬁed.

4.7 Electronic Correlation

The mean energy per electron in the ground state at T =0K

= ǫ

+ ǫ

(4.142)

can be separated into the HF energy and the correlation energy. The former

is determined by the ﬁlli n g of the Fermi sphere and considering the exchange

interaction (as a consequence of the Pauli principle). The latter takes into

account the electron–electron interaction beyond the HF approximation and

will lead to a modiﬁed picture of the pair–distribution function of Fig. 4.12,

in which the exchange h ole is replaced by the correlation hole.

4.7 Electronic Correlation 111

It is helpful to start with the expectation value (per electron) of the

electron–electron interaction in (

4.107) for the exact ground state |Ψ





q=0

Ψ

|(N

†

− N)|Ψ

, (4.143)

where the index xc refers to exchange and correlation. It can be reformulated

by making use of the expression for the dynamic structure factor

S(q,ω)=

2π



+∞

−∞

−iωt

Ψ

(t)

−q

(0)|Ψ

dt (4.144)

which can be written also as

S(q,ω)=



|Ψ

|Ψ

|

2π



+∞

−∞

−i(ω−ω

dt. (4.145)

The integral is the δ function and thus



∞

S(q,ω)dω =



|Ψ

|Ψ

|

= Ψ

−q

|Ψ

 = NS(q) . (4.146)

Here S(q) is the static structure factor which now enters the (still exact)

expression for ǫ



q=0

(S(q) − 1). (4.147)

Due to the isotropy of the free electron system we have S(q)=S(q)andwe

may write after integration over the angles

= −

2π

−



∞

(S(q) − 1)dq

. (4.148)

By introducing the dimensionless function

γ(k

)=−



∞

(S(q) − 1)dq (4.149)

and expressing k

by the density parameter r

we ﬁnd (in Ry)

)=−

πr

9π

1/3

γ(r

) . (4.150)

The static structure factor S(q) is shown in Fig. 4.17 for diﬀerent values of r

It should be kept in mind here that the expectation value of the kinetic

energy (to be taken for the exact ground state) depends on the interaction.

112 4 The Free Electron Gas

r = 2

r = 6

r = 10

r = 0.001

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5

S(q)

q/k

Fig. 4.17. The static structure factor S(q) for diﬀerent values of the density

parameter r

(after [

119])

This is accounted for within the following consideration known as the ground

state theorem: L et the system Hamiltonian be given by

H(g)=H

kin

+ gH

int

(4.151)

where g = e

is the strength of the (electron–electron) interaction. The

eigenstates of H(g), including the ground state |Ψ

(g), depend on g and

int

(g)=Ψ

(g)|gH

int

|Ψ

(g) (4.152)

is the exact contribution of the interaction to the ground state energy E

(g).

Then, according to the theorem

(g)=E

(0) +



′

int

′

)dg

′

. (4.153)

To prove this theorem one has simply to write

(g)

= Ψ

(g)|

dH(g)

|Ψ

(g) +

Ψ

(g)|

H(g)|Ψ

(g)

+Ψ

(g)|H(g)

|Ψ

(g)

int

(g)+E

(g)

Ψ

(g)|Ψ

(g). (4.154)

The last term vanishes, because |Ψ

(g) is normalized, and the resulting dif-

ferential equation can be integrated directly to yield the statement of the

theorem.

4.7 Electronic Correlation 113

For the homogeneous electron gas we can immediately write

−

2π



γ(g

′

)dg

′

. (4.155)

Thus, in terms of the density parameter r

, the exact ground state ener gy per

electron reads (in Ry)

2.2099

−

πr

9π

1/3



γ(λr

)dλ. (4.156)

If γ or S(q)orIm1/ε(q,ω) is given, the ground state energy per electron ǫ

)

is also known. The ﬁrst term is the contribution of the kinetic energy in the

Sommerfeld (or noninteracting electron) model, the second term describes the

exchange and correlation energy (including also modiﬁcations of the kinetic

energy due to the electron–electron interaction), and the exact correlation

energy per electron can be written as

0.916

−

πr

9π

1/3



γ(λr

)dλ. (4.157)

By deﬁnition ǫ

) vanishes in the HF approximation.

Calculations of the correlation energy have been carried out in diﬀerent

approximations for ε(q,ω)(orS(q)orγ), whose accuracy depends on the

density (or r

)[

64, 119]. Typically one uses

ε(q,ω)=1−

(q,ω)

1+v

G(q)π

(q,ω)

(4.158)

with diﬀerent forms of G(q). One particularly simple form is G(q)=0,for

which the dielectric function takes the RPA form of (

4.125). But also in

general, considering correlation means some kind of screening the Coulomb

interaction. Results are visualized by plotting the pair–distribution function

(see Problem 4.12) which for the homogeneous electron system reads

g(r)=1+

2rk



∞

q sin(qr)(S(q) − 1) dq. (4.159)

As can be seen in Fig. 4.18, the HF result, with g(r → 0) = 1/2, is recovered

for dense electron systems (r

≪ 1) while in the low-density case (large r

)

g(r → 0) → 0 and the exchange or Fermi hole evolves into a correlation hole

because the electrons try to avoid each other (irrespective of their spin). We

will see in the next chapter how these concepts of exchange and correlation

can be considered in describing the electronic states of the inhomogen eous

electron systems found in solids.

Let us also brieﬂy discuss the low density limit r

≫ 1, for which the

electron–electron interaction is expected to dominate the kinetic energy. The

114 4 The Free Electron Gas

1.0

0.8

0.6

0.4

0.2

0.50 1.0 1.5 2.0 2.5

r/r

g(r)

r = 1

r = 5

r = 15

r = 0.001

r = 1

r = 5

Fig. 4.18. The pair–distribution function g(r) for diﬀerent values of the density

parameter r

(after [

119]). The symbols refer to Monte–Carlo simulations of [120]

ultimate limit, when the kinetic energy can be completely neglected, leads to

a problem of classical electrostatics: the grou nd state is deﬁned by the conﬁg-

uration of point charges with the lowest potential energy. This conﬁguration

is a crystalline one: the Wigner crystal. Its electrostatic energy shall be evalu-

ated here in the jellium model, in which each electron is assigned the volume

of the Seitz sphere with radius r

= r

.Thepotentialenergyofanelectron

in the center and t h e sphere ﬁlled with the compensating positive charge of

the jellium background is

−

= −



4πε

r = −

Ry. (4.160)

The potential energy of the posit i ve jellium background in the Seitz sphere,

, is to be calculated from the electrostatic potential of the homogeneously

charged sphere

V (r)=

−

+ C (4.161)

where C =3e

/2r

has to be chosen to establish charge neutrality, thus

V (r)=

3 −





Ry (4.162)

and



V (r)nd

r =

Ry. (4.163)