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PH611 2014S Lecture 19
Measuring the Fermi Surface
(A&M Ch. 14)
1. Quantization of orbits in H field leads to experimentally observable effects
related to only fundamental constants and geometrical properties of the orbits.
Most direct experimental demonstration that Fermi surface exists.
Most direct measurement of the shape of the Fermi surface in k-space.
Purely quantum effect (Bohr van Leeuwen theorem that no equilibrium property
of a classical system can depend upon magnetic field H)
2. The same principles as the famous Aharonov-Bohm effect (Phys. Rev. 115, 485
(1959); See review in Rev. Mod. Phys. 57, 339 (1985)); an electron path
encircling a magnetic flux interfere constructively if the flux is an integral
multiple of = flux quantum = .
H field threading ring
wave functions interfere
constructive if phase
around circle is
3. In two dimensions electron energies in a free gas
become
discrete Landau levels in H field with energy , where
.
The number of states per unit area in a Landau level is
where density of states in absence of H field
Result: There is one orbit per flux quantum , i.e. .
4. In three dimensions there is still motion parallel to the field (the z direction) sothat the electron energies are
. The energies form a
continuum with a one-dimensional density of states for each Landau level, i.e.
, which diverges at each Landau energy .
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5. All properties of the system change as the peaks in move through the
Fermi energy: specific heat, resistance, magnetic susceptibility, ... all vary as a
function of H.
Leads to oscillations with period in 1/H.
where
= flux quantum = ;
is the
area of an orbit in real space; and is the area of an orbit in k space.
In 2d is the area of the orbit in k space.
In 3d oscillations as function of 1/H occur where is area of extremal orbit.
(Non-extremal areas add together to give smooth background.)
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Measuring the Fermi Surface
A&M Ch. 14, Kittel Ch. 9
What is the experimental evidence that the Fermi surface actually exists in a
metal?
1. Electrical conductivity (and related thermal conductivity, thermopower, ...) shows
there must be states for electrons which are extended (i.e. the wavefunctions
extend across large, macroscopic bodies) and can be excited by arbitrarily small
energies.
Filled Empty
Our description of independent electrons in the semiclassical approximation are
consistent with the conductivity.
2. The detailed shape of the surface in k-space that separates filled and empty
states the Fermi surface is experimentally determined by measurements of
quantized orbits in H field.
In a magnetic field electrons do not have well-defined k states, but rather theymove in orbits about the magnetic field. Here we discuss the proper
quantum-mechanical description of electrons in a magnetic field and the
experimental consequences.
We have seen that in the classical free electron approximation the electrons move
in circles with angular frequency (Ch. 1)
This still holds in the semiclassical theory.
If one quantizes the orbits in a magnetic field, the energies become those of a
harmonic oscillator
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Note on harmonic oscillator solution for electron in B field (Kittel, Prob. 9-11).
Let .
In the Landau gauge, .
The Hamiltonian is
Clearly the z dependence is decoupled so the wave function is
Also in this gauge
The solution is the set of Harmonic oscillator functions with energies
with
.
Two dimensions
Suppose the electrons move in the plane and the magnetic field is in the
direction. Then the energies are quantized unlike the continuous spectrum for
H=0.
functions
Landau
Levels
Since the number of states is conserved the Landau levels must be degenerate.Each level must contain the number of states
(# of energyarea) (energy)
density of states
for H=0
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Now for a two-dimensional electron gas is
(We have not included spin. Each spin is considered separately in the magnetic
field.)
If
, this gives
per unit energy per unit area.
Thus the degeneracy of each state is
= flux quantum =
Note: the degeneracy increases with B exactly as the spacing increases, so
that the total number of states remains constant.(See discussion in Kittel, p.242, 252)
Three dimensions
In three dimensions the motion along the direction parallel to
is not affected
by the magnetic field, so the energies are
Thus each Landau level gives a continuum of states (exactly like a one
dimensional spectrum)
can beshifted
change of
susceptibility,
conductivity
for each n
add 1-D feature
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As the magnetic field is changed the peaks in the density of states move through
the Fermi energy. The properties of the metal change. (oscillation as a function of
(1/H)), as each peak moves through .
De Haas van Alphen effect
moment vs. B (magnetic susceptibility)
De Haas Shubnikov effect
resistivity vs. B
Observation at low T
The characteristic T must be in order for the structure to be seen.
This gives
So must be .
(1-10 Tesla)
in order for experiments to be done at .
Low , high field.
The oscillations occur when
or
So
Thus effects should be plotted as a function of 1/H and should be periodic with
period
fundamental
constant
extremum area of
orbit on Fermi Surface
extremum area in k
space for cross
section of Fermi
surface to
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Why is this formula correct for extremum orbits in a general crystal?
1. The description in Ch. 12 of electron dynamics is sufficient to show that the
electron moves in orbits on the Fermi surface to
at fixed .
2. On p. 271 and 272, it is shown that for high Landau orbits (large n) the area A
enclosed by the orbit enters exactly as the form
3. The subtle point is that only extremal orbits give a definite observable signal.
This is because the continuous range of A for non-extremal orbits cancels the
oscillations and one is left only with the extremal areas that make a finite
contribution.
two extremal orbit, for
But for H at on angle there
may be only one
As we rotate relative to the crystal axes, the Fermi surface may be mapped
out in great detail.
This is the way the Fermi surfaces shown in A&M are actually measured
51 oscillations
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We have seen (dynamics of electrons, #16) that the orbit in k space and in r
space are the same except rotated and the orbit in r space scaled by a factor
compared to k space.
First, we show that the area of the orbit in real space is quantized:
From the Bohr-Sommerfeld condition
Now
flux through area
So
or
(r space orbit encloses quantized flux values.
: flux quantum.)
Now the orbit in k space has area (cf. scaling rule)
Now we have periodicity in the field for values with given , i.e.,
or
Thus the inverse of the area of the orbit in k space is given by
subtle to get
the value 1/2
(cf. Berrys phase discussion)
2 area
area in r space flux quantum
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Aharonov Bohm Effect
Phys. Rev. 115, 485 (1959)
The interference of electron wave functions around metal loops can be observed:
H field inside ring
Interference of the wave function around the two paths gives resistance which
oscillates with magnetic field. The wavefunctions interfere constructively if the total
phase around the loop is integer
H changes by a flux quantum .
Resistivity fluctuates as the phase difference of
the electron wavefunctions passing through
two paths depends on the magnetic flux.
AB effect of electrons passing through and away of the toroidal magnetic field
completely confined within the superconducting Nb cladding.