Opleidingsonderdeel G0J16A Solid State Physics I

Beyond the independent electron approximation

[see chapter 17 in Solid State Physics by N.W. Ashcroft and N.D. Mermin]

06 + 13 October 2011

Born-Oppenheimer approximation I

Born - Oppenheimer or adiabatic approximation (1928): the crystal ions

(nuclei + core electrons) and the conduction electrons (valence electrons)

can be treated separately.

Hamiltonian for conduction electrons with density ~ 1023/cm3:

Born-Oppenheimer approximation II

free electrons

Fermi liquid

describe superconductivity

Bloch electrons

Hamiltonian for ions with density ~ 1023/cm3:

Born-Oppenheimer approximation III

The sum 1 + 2 violates charge neutrality

→ add uniform positive background

“jellium” Hamiltonian

represents a uniform negative charge background

→ charge neutrality

We have to deal with a liquid rather than with a gas !

Metals have a high heat capacity:

cV = π2 · kB2T · g(εF) / 3 = π2 · (kBT/εF) · nkB / 2

Metals have a high electrical conductivity:

σ = ne2τ / m = e2 ·g(εF) · vF2 · τ / 3

Free electron gas

High density of states in k-space for macroscopic sample size

→ High density of states as a function of energy:

In 1928 Hartree proposed to approximate the electron-electron

interaction by considering the interaction of one electron

(“independent electron approximation”) with the charge cloud

that is caused by all the other electrons:

Hartree approximation I

This way one obtains a Hartree (Schrödinger) equation for each

of the electrons that move into the periodic potential caused by the

positive ions:

Hartree approximation II

The Hartree approximation assumes that the many-electron wave function

is the product of the one-electron wave functions

→ The Hartree approximation is a “classical” approximation that neglects

the Pauli principle, i.e. the fact that the total wave function needs to be

anti-symmetric

In 1930 Fock and Slater extended the Hartree approximation by

using an anti-symmetric wave function. Such an anti-symmetric

wave function, which includes the spatial as well as the spin

coordinates, is the Slater determinant:

Hartree-Fock approximation I

In most of the cases the Slater determinant will not be an eigen

function of the Hamiltonian

→ a variational approach is needed

Look for a minimum of the expectation value of the Hamiltonian

(minimum of the total electron energy):

Hartree-Fock approximation II

Calculate the expectation value for the Hamiltonian:

Minimalize the expectation value of the Hamiltonian using a

variational approach (Lagrange multiplicators) to obtain a set

of equations of the form:

Hartree-Fock approximation III

In the above Hartree-Fock equations the εi are the Lagrange

multiplicators. It is, however, very tempting to treat the above

set of equations as one-electron Schrödinger equations.

Hartree-Fock approximation IV

The theorema of Koopmans implies that the energy needed to transfer an

electron from the state with label i to the state with label j is given by

the difference εj – εi

→ The Hartree-Fock equations can be considered as a set of Schrödinger

equations that need to be solved self-consistently

Introducing the Slater determinant for the many-electron wave

function causes the appearance of an extra “exchange term” in the

Schrödinger equation

The exchange term results in a lowering of the electron energy !

The exchange term involves only electrons having the same spin !

Hartree-Fock approximation for free electrons I

Look for normalized plane wave solutions ψ(r) = exp(ik·r) / V1/2

that are plugged into the Hartree-Fock equations:

Hartree-Fock approximation for free electrons II

To obtain this result we rely on the fact that for plane waves Uel

(Hartree term) is exactly canceled by the uniform positive

background that we need to introduce to conserve the charge

neutrality! We then proceed by expanding the Coulomb interaction

potential VC(r”) ∝ 1/r” into a Fourier series with Fourier components

VC(q) ∝ 1/q2:

Hartree-Fock approximation for free electrons III

Hartree-Fock approximation for free electrons IV

F(x) is a continuous function of x = k/kF and 0 < F(x) ≤ 1

The derivative F’(x) has a logarithmic singularity for x = 1, i.e. for k = kF

→ the Fermi velocity vF = ∂ε/∂k(ε=εF)/ħ → ∞

The heat capacity cV ∝ T/|lnT| will deviate from a linear temperature

dependence, which clearly is in conflict with experiment!

On the other hand, the concept of a Fermi sea, which is densely filled

with electrons up to the Fermi level, remains intact

Hartree-Fock approximation for free electrons V

Hartree-Fock approximation for free electrons VI

The energy levels are considerably lowered

The width of the band increases by a factor 2.3

BANDWIDTH

Hartree-Fock approximation for free electrons VII

Within the Hartree-Fock approximation the total energy E of the free

electron gas is given by

The summations over the allowed k-vectors can be replaced with an

integral. When taking properly into account the logarithmic divergence

of the second term, the total energy is given by

Hartree-Fock approximation for free electrons VIII

The Hartree-Fock equation for the electron with wave function ψi(r)

can be interpreted in terms of two electron charge densities. The first

density ρel is the uniform negative density that follows from the Hartree

approximation:

The second charge density is a positive density that can be associated

with the exchange term. For the electron with wave function ψi(r) it is

given by

Hartree-Fock approximation for free electrons IX

Each electron drags along an exchange hole with the same spin!

charge

density

The extra charge density resulting from the exchange can be simply

visualized if we average over all electrons i and replace the summation

with an integration. We then plot the absolute value of the difference

between ρel and the averaged density related to the exchange:

The non-locality (dependence on r and r’) of the exchange term makes

a self-consistent solution of the Hartree-Fock equations impossible for

electrons that move in a periodic potential !

While it is straightforward to write down appropriate equations, it is

unfortunately not possible to solve them for large numbers of electrons.

This was already noted in 1929 by Dirac.

Even with the most powerful computers numerical solutions become

impossible for N around 10.

While for atoms and molecules one tries to do better than Hartree-Fock,

even more crude approximations are required for treating the many

electrons in a solid.

→ We can get a first idea of how crude the approximations

are from the discussion of equations (17.25) and (17.26) in

chapter 17 in the book of Ashcroft and Mermin.

Hartree-Fock approximation for free electrons X

We introduce an external charge density ρext(r) in a gas of free

electrons → The free electron gas reacts by creating a charge

density ρind(r) and the total charge density ρ(r) is given by

Screening I

ρ(r) and ρext(r) are linked to potentials Φ(r) and Φext(r) via

Poisson’s equation:

Screening results in a reduction of the potential that can be

described in terms of a (relative) dielectric function Єr(r,r’):

For a homogeneous medium we have that Єr(r,r’) = Єr(r-r’)

→ Switch to k-space:

Screening II

This way, we have a wave vector dependent dielectric constant of the

metal that is related to the wave vector dependent potentials:

From a computational point of view it is more convenient to calculate

the induced charge density ρind(r) (use a linear approximation):

Introduce the Fourier transforms of the Poisson equations to link the

wave vector dependent dielectric constant to the wave vector

dependent induced charge density:

Screening III

We obtain the following equation for the dielectric function that

needs to be solved self-consistently:

Screening IV

The Thomas-Fermi approximation I

In principle we need to solve the Schrödinger equation:

determines the charge density ρ(r)

Provided Φ(r) varies very slowly on a scale corresponding to λF,

we can according to Ehrenfest’s theorem rely on a classical approach:

The spatial variation of the electron density n(r) is given by

The Thomas-Fermi approximation II

We need to take into account the positive charge background:

We can formally rewrite ρind(r) as

We assume that Φ(r) is a small perturbation

→ Use a series expansion for n0:

The susceptibility χ(q) is then a constant that is independent of q

and is given by

The Fourier components of the Coulomb potential need to be adapted

according to

There no longer occurs a divergence for q → 0 !

The Thomas-Fermi approximation III

Finally, the dependence on q of the dielectric constant is given by

The Thomas-Fermi approximation IV

We can then investigate how a point charge is screened:

The screening of the Coulomb potential can be compared to the

screened Yukawa potential in nuclear physics !

The Thomas-Fermi approximation V

There occurs a very effective screening of an external charge at a

distance r >> k0-1:

→

a0 represents the Bohr radius (≈ 0.5 Ǻ)

The Lindhard approximation I

For this approximation we need to solve the Schrödinger equation !

→ We assume a linear relation between ρind and Φ:

When q → 0, the Fermi-Dirac distribution function can be

approximated by a series expansion:

The linear term reproduces the Thomas-Fermi approximation. For

arbitrary q we need to solve the integral. For T → 0 the solution is

The Lindhard approximation II

In the above expression for the susceptibility χ we have defined

x = q/2kF.

For r >> kF-1 we can calculate how a point charge is screened by

the electron gas:

The long-range charge density oscillations that surround a point charge

impurity are know as Friedel oscillations

The physical origin of the Friedel oscillations can be directly linked to

the sharpness of the Fermi surface: Fourier components of the Coulomb

potential with wave vector q > kF cannot be screened since there are no

electrons with k > kF !

The extra factor of 2 in the cosine dependence results from the fact that

the charge density is determined by the square of the wave function

Fourier-transform STM I

Friedel charge density

oscillations will also be

present for the two-

dimensional (2D) electron

gas at the (111) surface of

noble metals (surface

states that are decoupled

from the “bulk” states)

STM images the electron

charge density → From

a Fourier analysis of the

charge density oscillations

the “Fermi circle” can be

reconstructed !

Fermi surface of bulk (3D) copper (measured by the de Haas – van Alphen effect)

There are no allowed bulk states within the “necks” along the [111] directions!

[See chapter 9 in the book of Kittel on Solid State Physics]

Surface states, which are decoupled from the bulk states, can be formed within

the energy gaps of the “projected bulk states”

Fourier-transform STM II

Look at surface electrons for more complicated,

highly deformed Fermi surfaces

Fourier-transform STM III

From the Fourier transform of the

charge density oscillations at a

step edge on the Be(10-10) surface

the allowed states in the surface

Brillouin zone (SBZ) can be

reconstructed

A pronounced anisotropy of the

screening is present, consistent

with the theoretical expectations