Chapter 2

The Fermi Method

2.1 Statistical Mechanics

At finite temperatures there are thermal excitations of the electronic system, i.e., inthermodynamic equilibrium not only the ground state (Ee

0, Φ0({rkσ})) of He is present,but also excited states. Thus, due to thermal fluctuations all states (Ee

ν , Φν) are realizedwith a certain probability. Assuming that the number of particles and the temperatureare determined by external conditions, we have a canonical ensemble and the probabilityP (Ee

ν , T ) for the occupation of state (Eeν , Φν) is proportional to exp(−Ee

ν/kBT ). Here kB

is the Boltzmann constant. The ensemble is described by the density operator

ρ =∑

ν

P (Eeν , T )|Φν >< Φν | . (2.1)

Of course the ensemble of all states has to be normalized to 1 and therefore we have

∑

ν

P (Eeν , T ) = 1 =

1

Ze

∑

ν

exp(−Eeν/kBT ) . (2.2)

One obtainsZe =

∑

ν

exp(−Eeν/kBT ) = Tr(exp(−He/kBT )) . (2.3)

Ze is the partition function of the electrons and is related to the Helmholtz free energy:

−kBT ln Ze = F e = U e − TSe , (2.4)

where U e and Se are the internal energy and the entropy of the electronic systems, i.eof the electron-hole excitations. Below at Eq. (2.6) we will come back to this point.Consequently, the probability of a thermal occupation of a certain state (Eν , Φν) is

P (Eeν , T ) =

1

Zeexp(−Ee

ν/kBT ) = exp[−(Eeν − F e)/kBT ] . (2.5)

At finite temperature we therefore need the full energy spectrum of the many-body Hamil-ton operator. Then we can calculate the partition function (Eq. (2.3)) and the free energy(Eq. (2.4)). Let us now discuss briefly, how the internal energy and the entropy can bedetermined separately1: The internal energy is what up to now we have called total energyat finite temperature:

U e(T ) =∑

ν

Eeν(T )P (Ee

ν , T ) . (2.6)

In the general case, i.e., when also atomic vibrations are excited, we have U = U e + Uvib,not just U e.

1cf. e.g. N.D. Mermin, Phys. Rev. 137, A 1441 (1969); M. Weinert and J.W. Davenport, Phys. Rev. B45, 13709 (1992); M.G. Gillan, J. Phys. Condens. Matter 1 689 (1989); J. Neugebauer and M. Scheffler,Phys. Rev. B 46, 16067 (1992); F. Wagner, T. Laloyaux, and M. Scheffler, Phys. Rev. B 57, 2102 (1998).

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From the laws of thermodynamics [(∂u/∂T )V = T (∂s/∂T )V ] and from the third law ofthermodynamics (s → 0 if T → 0) we obtain

se =Se

V= −kB

∑

i

[

f(ǫi, T ) ln f(ǫi, T ) + (1 − f(ǫi, T )) ln (1 − f(ǫi, T ))]

. (2.7)

Here we used the energy and entropy per unit volume (u = U/V, s = S/V ), and f(ǫi, T )is the Fermi function (see below). The derivation is particularly simple, if one assumesthat we are dealing with independent particles (Eq. (2.11), (2.12), below).

From Eq. (2.4) or (2.7) we obtain the specific heat

cv =1

V

(

∂U

∂T

)

V

(2.8)

=T

V

(

∂S

∂T

)

V

. (2.9)

Here we removed the superscript e and in fact mean U = U e + Uvib and S = Se + Svib.The calculation of cv of metals is an important example of the importance of Fermi-Diracstatistics of the electrons (cf. Ashcroft-Mermin p. 43, 47, 54).

2.2 Fermi Statistics of the Electrons

Let us assume that the N electrons of our many-body problem occupy single particlelevels. Then we also know that due to the Pauli principle each single particle level canbe occupied with two electrons at most (one electron with spin up and one electron withspin down). With this assumption it follows (for T = 0 K) that the N lowest energy levelsǫi are occupied:

Ee(T = 0K) = Ee0 =

N∑

i=1

ǫi + ∆ , (2.10)

where ∆ is a correction describing the electron-electron interaction. For independent par-ticles ∆ is zero, but for the many-body problem it is very important (see Chapter 3).

The ǫi then are eigenvalues of an effective single-particle Hamiltonian

h =−h2

2m▽2 +V eff(r) .

Employing the above description in terms of the density matrix (cf. Marder, Chapter6.4 and Landau-Lifshitz, Vol. IV) to a situation of independent particles gives for finitetemperature the lowest energy that is compatible with the Pauli principle as

Ee(T ) =∞∑

i=1

ǫif(ǫi, T ) + ∆ . (2.11)

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f(ǫ)

µ

T = 0

T 6= 0

ǫ

1.0

0.5

0.0

Figure 2.1: The Fermi distribution function [Equation (2.12)].

The index i is running over all single particle states.

The occupation probability (cf. e.g. Ashcroft-Mermin, Eq. (2.41) - (2.49) or Marder, Chap-ter 6.4) of the ith single particle level ǫi is given by the Fermi function:

f(ǫ, T ) =1

exp[(ǫ − µ)/kBT ] + 1. (2.12)

Here kB is the Boltzmann constant and µ is the chemical potential of the electrons, i.e.,the lowest energy, which is required to remove a particle from the system:

−µ = Ee(N − 1) − Ee(N) . (2.13)

How are µ and its temperature dependence determined? The number of electrons is N ,and it is independent of the temperature. Therefore, we have

N =∞∑

i=1

f(ǫi, T ; µ) . (2.14)

For a given temperature this equation contains only one unknown quantity, the chemicalpotential µ. If all ǫi are known, µ(T ) can be calculated.

2.3 Some Definitions

We will now introduce some definitions and constrain ourselves to a so-called “jellium”system. The most simple way (i.e., the crudest approximation to the atomic structure) toinvestigate the Schrodinger equation of the Hamilton operator

He = T e + V e−Ion + V e−e (2.15)

29

is obtained when setting V e−Ion +V e−e as a constant function of the electron coordinates.We note that this crude approximation provides reasonable and helpful results for someproblems. A system with V e−Ion = constant is called “jellium”, and if V e−Ion as a functionof the electronic coordinates is constant, then it can easily be shown that also V e−e isconstant2 We like to consider here a system without spin-orbit interaction. Thus spin andposition coordinates can be separated.

Φ({rkσk}) = Φ({rk})χ({σk}) . (2.16)

Without introducing a new approximation the zero point of the energy is chosen in a waythat the constant potential V e−Ion + V e−e vanishes. Then the Hamilton operator of theelectrons has the simple form

He = T e =N∑

k=1

− h2

2m∇2

rk, (2.17)

and the many-body Schrodinger equation decomposes into a number of N single particleequations

− h2

2m∇2ϕj(r) = ǫjϕj(r) . (2.18)

The solutions of Eq. (2.17) are plane waves

ϕk(r) = eikr , (2.19)

and the energy eigenvalues are

ǫ(k) =h2k2

2m, (2.20)

where the vectors k and the components kx, ky, kz have to be interpreted as quantumnumbers, up to now noted as index j in (ǫj, ϕj): the state of an electron of the Hamiltonoperator (2.17) is labeled by the quantum number k and the spin s. The wave length

λ = 2π/k (2.21)

is called de Broglie wave length.

The wave functions in Eq. (2.19) are not normalized (or they are normalized with respectto δ functions). In order to obtain a simpler mathematical discussion often it is useful, orhelpful, to constrain the electrons to a finite volume. This volume is called the base region,Vg, and it shall be large enough to obtain results independent of its size.3 The base regionVg shall contain N electrons and M atoms. The shape of the base region in principle ismeaningless. For simplicity here we chose a box of the dimensions Lx, Ly, Lz (cf. Ashcroft,

2For systems with very low densities, however, electrons will localize themselves at T = 0 K due tothe Coulomb repulsion. This is called Wigner crystallization and was predicted in 1930.

3For external magnetic fields the introduction of a base region can give rise to difficulties, becausethen physical effects often depend significantly on the border.

30

Mermin: Exercise for more complex shapes). For the wave function we could chose analmost arbitrary constraint (because Vg shall be large enough). It is advantageous to useperiodic boundary conditions

ϕ(r) = ϕ(r + Lxex) = ϕ(r + Lyey) = ϕ(r + Lzez) . (2.22)

Here ex, ey, ez are the unit vectors in the three Cartesian directions. This is also calledthe Born-von Karman boundary condition.

As long as Vg, or Lx ×Ly ×Lz is large enough, all physical results do not depend on thistreatment. Sometimes also anti-cyclic boundary conditions are chosen in order to checkthe independence of the results of the choice of the base region.

Using Eq. (2.22) and the normalization condition∫

Vg

ϕ∗

k(r)ϕk′(r)d3r = δk,k′ (2.23)

we obtain

ϕk(r) =1√V g

eikr . (2.24)

Because of Eq. (2.22), i.e., because of the periodicity, only discrete values are allowed forthe quantum numbers k, i.e., k · Liei = 2πni and therefore

k =

(

2πnx

Lx

,2πny

Ly

,2πnz

Lz

)

, (2.25)

with ni being arbitrary integer numbers. Thus, the number of vectors k is countable andfinite. Each k point therefore has the volume

(2π)3

Vg

(2.26)

in k-space.

Each state ϕk(r) can be occupied by two electrons. In the ground state at T = 0 K theN/2 k points of lowest energy are occupied by two electrons each. Because ǫ dependsonly on the absolute value of k, these points fill (for non-interacting electrons) a spherein k-space of radius kF (the “Fermi sphere”). We have

N = 24

3πk3

F

Vg

(2π)3=

1

3π2k3

FVg . (2.27)

Here the spin of the electron (factor 2) has been taken into account, and Vg/(2π)3 is thedensity of the k-points (cf. Eq. (2.26). The particle density of the electrons in jellium isconstant:

n(r) = n =N

Vg

=1

3π2k3

F , (2.28)

31

and the charge density of the electrons is −en, and kF =3√

3π2n.

For the single particle of the highest energy (in the ground state at T = 0 K) we get

ǫF =h2

2mk2

F =h2

2m(3π2n)2/3 . (2.29)

Often for jellium-like systems the electron density is given by the density parameter rs.This is defined by a sphere 4π

3r3s , which contains exactly one electron. One obtains

4π

3r3s = Vg/N = 1/n . (2.30)

The density parameter rs is typically given in bohr units.

For metals rs is typically around 2 bohr (remember: this only refers to the valence elec-trons), and therefore kF is approximately 1 bohr−1, or 2 A−1, respectively.

Later, we will often apply Eq. (2.29) and (2.30) because some formulas can be presentedand interpreted more easily, if ǫF, kF and n(r) are expressed in this way.

Now we introduce the (electronic) density of states:

N(ǫ)dǫ = number of states in the energy interval [ǫ, ǫ + dǫ] .

For the total number of electrons in the base region we have:

N =∫ +∞

−∞

N(ǫ)f(ǫ, T )dǫ . (2.31)

For free electrons (jellium) we have for the density of states:

N(ǫ) = 2Vg

(2π)3

∫

d3k δ(ǫ − ǫk)

=2Vg4π

(2π)3

∫

k2dk δ(ǫ − ǫk)

=Vg

π2

∫ dǫk|∇kǫk|

2mǫk

h2δ(ǫ − ǫk)

=Vg

π2

∫

dǫk

√m

h√

2ǫk

2mǫk

h2δ(ǫ − ǫk)

=mVg

π2h3

√2mǫ . (2.32)

For ǫ < 0 we have N(ǫ) = 0. The density of states for two- and one dimensional systemsis discussed in the exercises (cf. also Marder).

Figure 2.2 shows the density of states and the occupation at T = 0 K and at finitetemperature. The density of states at the Fermi level is

N(ǫF)

Vg

=3

2

N

Vg

1

ǫF

=m

h2π2kF (2.33)

The figure shows that at finite temperature holes below µ and electrons above µ are gen-erated.

32

occupied

states

occupied

stateselectrons

holes

Figure 2.2: Density of states of free electrons√

(ǫ)f(ǫ, T ) and the separation in occupiedand unoccupied states for two temperatures.

33