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1
Fermi surfaces and Electron dynamics
Band structure calculations give E(k)
E(k) determines the dynamics of the electrons
It is E(k) at the Fermi Surface that is important
Form of Fermi surface is important
Fermi surface can be complicated due to overlapping bands.
2
Constructing Brillouin Zones
1st B. Z.
2ndB. Z.
2D Square lattice. BZ constructed from the perpendicular bisectors of the vectors joining a reciprocal lattice point to neighbouring lattice points 2/a
3
Brillouin Zones and Fermi SurfacesEmpty Lattice model (limit of weak lattice potential):
States are Bloch states.Independent states have k-vectors in first BZ.
No energy gaps at the BZ boundaries.
kx aa
E
0
E1
E2
[100]
k21/2a 0 21/2a
kx = ky
E1
E2
[110]2st B. Z. 1st B. Z.
E1
E2
ky
4
Reduced Zone scheme
PLUS
Fermi Contours in reduced Zone
Parts of Fermi circle moved into 1st BZ
from 2nd BZ
E2
Extended Zone scheme
2st B. Z. 1st B. Z.
6http://dept.physics.upenn.edu/~mele/phys518/anims/Kronig/FermiSurf1.gif
E = - – Cos[kx x] - Cos[ky y]),
2D simple square
Lattice tight binding
model.
Changing Fermi
Contour with
Increasing Fermi
Energy.
7
BZs and Fermi Surfaces with gaps
Energy gaps make the Fermi contours appear discontinuous at the BZ boundaries.
dE/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary.
0a a
E1
E2
kx
E2
E1
2st B. Z. 1st B. Z.
8
E2
E1
Energy gaps: Fermi contours appear discontinuous at the BZ boundaries.
dE/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary.
BZs and Fermi Surfaces with gaps
2st B. Z. 1st B. Z.
E1
E2
ky
No gaps With gaps
9
Fermi Surfaces with gaps “Hole like” orbits
Periodic zone picture of part of the Fermi contour at energy E1.
On this part of the Fermi contour electrons behave like positively charged “holes”. See later
10
Fermi Surfaces with gaps: “Electron like” orbits
Periodic zone picture of part of the Fermi contour at energy E2.
On this part of the Fermi contour electrons behave like negatively charged “electrons”. See later
11
Tight binding simple cubic model:Fermi Surfaces
- – Cos[kx x] - Cos[ky y] - Cos[kz z]
Increasing Fermi Energy
http://home.cc.umanitoba.ca/~loly/fermiarticle.html
19
Semi-classical model of electron dynamics
E(k), which is obtained from quantum mechanical band structure calculations, determines the electron dynamics
It is possible to move between bands but this requires a discontinuous change in the electron’s energy that can be supplied, for example, by the absorption of a photon.
In the following we will not consider such processes and will only consider the behaviour of an electron within a particular band.
The wavefunctions are eigenfunctions of the lattice potential. The lattice potential does not lead to scattering but does determine the dynamics. Scattering due to defects in and distortions of the lattice
20
Dynamics of free quantum electrons Classical free electrons F = -e (E + v B) = dp/dt and p =mev .
Quantum free electrons the eigenfunctions are ψ(r) = V-1/2 exp[i(k.r-t) ]
The wavefunction extends throughout the conductor.
Can construct localise wavefunction i.e. a wave packets
The velocity of the wave packet is
the group velocity of the waves
The expectation value of the momentum of the wave packet responds to a force according to F = d<p>/dt (Ehrenfest’s Theorem)
for E = 2k2/2me
)]t -exp[i(k.rA (r)k k
kkv
d
dE
d
d
1
ee mm
pkv
Free quantum electrons have free electron dynamics
21
Dynamics of free Bloch electrons Allowed wavefunctions are
The wavefunctions extend throughout the conductor.
Can construct localise wavefunctions
The electron velocity is the group velocity
in 3D
This can be proved from the general form of the Bloch functions (Kittel p205 and appendix E).
In the presence of the lattice potential the electrons have well defined velocities.
)(A (r)k k rk
kkv
d
dE
d
d
1
)()( . rr rk ueik
)(1
kv kE
22
Response to external forcesConsider an electron moving in 1D with velocity vx acted on by a force by a force Fx for a time interval t. The work, E, done on the electron is
and
so
In 3D the presence of electric and magnetic fields
since
Note: Momentum of an electron in a Bloch state is not k
and so the !
Because the electron is subject to forces from the crystal lattice as well as external forces
tvFE xx
dt
d xx
kF
))(1
()( BkEBvEFk
k Eee
dt
d
xx dk
dE
1
v
)(1
kv kE
tdk
dE
dk
dE
x
xx
x
F
k
dt
dpF
23
Electron effective massIn considering the response of electrons in a band to external forces it is useful to introduce an effective electron mass, m*.
Consider an electron in a band subject to an external force Fx
differentiating
Gives and
So where
An electron in a band behaves as if it has an effective mass m*.
Note magnitude of m* can depend on direction of force
dt
dk
dk
Ed1
dtdk
Ed1
dt
dv x2x
2
x
2x
xx
dt
d Fk
1
2x
22*
dk
Edm
xx dk
dE
1
v
dt
dvx*mx F
24
Dynamics of band electronsConsider, for example, a 1D tight-binding model: E(kx) = 2cos(kxa)
0 0
0
x
xx e
dt
d FE
k
aka
dk
dEx
x
sin21
vg
In a filled band the sum over all the vg
values equals zero.
A filled band can carry no current
For electrons in states near the bottom of the band a force in the positive x-direction increases k and increases vx .
For electrons in states near the bottom of the band a force in the positive x-direction increases k but decreases vx .
25
Effective MassConsider, for example, a 1D tight-binding model: E(k) = 2cos(ka)
0
0kacos
1
a2dk
Edm
2
21
2x
22*
Near the bottom of the band i.e. |k|<</a cos(ka) ~ 1 So m* ~ 2/2a2
As before.
For a = 2 x 10-10 and = 4 eV
m* =0.24 x me
States near the top of the band have negative effective masses.
Equivalently we can consider the mass to be positive and the electron charge to be positive
dt
dv)( x*mex BvEF
26
Bloch Oscillations
Consider an electron at k = 0 at t = 0
When the electron reaches k = /a
it is Bragg reflected to k = -/a. It
them moves from -/a to /a again.
Period of motion
x
xx F
Ee
dt
kd
te
t xx Ek
)(
Consider a conductor subject to an electric –Ex
Expect “Bloch oscillations” in the current current of period T
Note observed due to scattering since T >> p
aeT
x
2
E
0 1 2
0
v(t
)
t/T
F1
k(t)
/a
/a
/a
0
0
27
Conductivity
(i)p momentum relaxation time at the Fermi surface as before
(ii) m is replaced by m* at the Fermi surface
(iii) Each part filled band contributes independently to conductivity,
(iv) Filled band have zero conductivity
Conductivity is now given by
ne2p/m*
28
Motion in a magnetic fieldFree electrons
The electrons move in circles in real space and in k-space.
Bloch electrons
In both cases the Lorentz force does not change the energy of the electrons. The electrons move on contours of constant E.
BkBvF )/( mee
x kx
y ky
BkBvk
k )(2
Eee
dt
d
29
Electron and Hole orbits
dk
dE
dt
dkdk
dEdt
dk
(a) (b)
Bz
kxkx
kyky dk
dE
dt
dkdk
dEdt
dk
dk
dE
dt
dk
dk
dE
dt
dkdk
dEdt
dk
dk
dEdt
dk
(a) (b)
BzBz
kxkx
kyky
(a) Electron like orbit centred on k = 0. Electrons move anti-clockwise.
(b) Hole like orbit. Electrons move clockwise as if they have positive charge
B)k(Ee
dt
kdk2
Filled states are indicated in grey.
30
Periodic zone picture of Fermi contour ( E1 ) near bottom of a band.
Electron like orbits
Grad E
kx aa
E
0
E1
E1
31
Periodic zone picture of the Fermi contour at the top of a band
Hole like orbits
Grad E
kx aa
E
0
E2
E2
32
HolesCan consider the dynamical properties of a band in terms of the filled electron states or in terms of the empty hole states
Consider an empty state (vacancy) in a band moving due to a force.
The electrons and vacancy move in the same direction.
k
Energy
Force on Electronsdt
d xx
kF
33
Energy & k-vector of a hole
Vacancy
Energy
k
Hole
ke
kh
Ee
Eh
E = 0
Choose E = 0 to be at the top of the band.
If we remove one electron from a state of energy –Ee the total energy of the band is
increased by
Eh = -Ee
This is the energy of the hole and it is positive.
A full band has
If one electron, of k-vector ke, is missing
the total wavevector of the band is –ke.
A hole has k-vector kh = -ke
0 k
34
Charge of a holeIn an electric field the electron wavevector would respond as
since kh = -ke
So the hole behaves as a positively charged particle.
F
Ek
e
dt
d h
F
Ek
e
dt
d e
The group velocity of the missing electron is .
The sign of both the energy and the wave vector of the hole is the opposite of that of the missing electron.
Therefore the hole has the same velocity as the missing electron.
vh = ve
)(1
kv kE
35
Effective mass of a hole
The effective mass is given by
Since the sign of both the energy and the wave vector of the hole is the opposite of that of the missing electron the sign of the effective mass is also opposite.
The electron mass near the top of the band is usually negative so the hole mass is usually positive.
Holes positive charge and usually positive mass.
Can measure effective masses by cyclotron resonance.
1
2x
22*
dk
Edm
*e
*h mm