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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