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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.