Download Solid State Physics II

Lecture IX
dr hab. Ewa popko
Electron dynamics
Band structure calculations give E(k)
E(k) determines the dynamics of the electrons
1
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
2
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-wt) ]
The wavefunction extends throughout the conductor.
Can construct localise wavefunction i.e. a wave packets
(r)  k A k exp[i(k.r - wt )]
The velocity of the wave packet is
the group velocity of the waves
v
dw 1 dE

dk  dk
for E = 2k2/2me
v
k p

me me
The expectation value of the momentum of the wave packet responds
to a force according to F = d<p>/dt
(Ehrenfest’s Theorem)
Free quantum electrons have free electron dynamics
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Dynamics of free Bloch electrons
Allowed wavefunctions are
 k (r)  eik .r u (r)
The wavefunctions extend throughout the conductor.
Can construct localise wavefunctions
(r)  k A k k (r )
The electron velocity is the group velocity
dw 1 dE
v

dk  dk
in 3D
1
v   k E (k )

This can be proved from the general form of the Bloch functions
(Kittel p205 ).
In the presence of the lattice potential the electrons have well defined
velocities.
4
Response to external forces
Consider an electron moving in 1D with velocity vx acted on by a
force Fx for a time interval dt. The work, dE, done on the electron is
and
1 dE
dE  Fx vx dt
so
F dE
dE
dE  x
dt 
dk x
 dk x
dk x
vx 
 dk x
dk x
Fx  
dt
In 3D the presence of electric and magnetic fields
1
dk F
e
e
1
   (E  v  B)   (E   k E (k )  B) since v   k E (k )
dt 




Note: Momentum of an electron in a Bloch state is not k
and so the
dp
F
dt
!
Because the electron is subject to forces from the crystal lattice as
well as external forces
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Electron effective mass
In 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
So
1 dE
vx 
 dk x
dv x 1 d 2 E
1 d 2 E dk x


dt
 dtdk x  dk 2x dt
* dv x
Fx  m
dt
and
dk x Fx

dt

d E 
m   2
 dk x 
2
where
*
1
2
An electron in a band behaves as if it has an effective mass m*.
Note magnitude of m* can depend on direction of force
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Dynamics of band electrons
Consider, for example, a 1D tight-binding model: E(kx) = a 2gcos(kxa)
1 dE 2ag
vg 

sin k x a
E
 dk x

vg
In a filled band the
sum over all the vg
values equals zero.
0
A filled band can
carry no current
0
dk x
Fx
e
  Ex 
dt


k
/a
0
k
/a
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 .
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Effective Mass
Consider, for example, a 1D tight-binding model: E(k) = a 2gcos(ka)
vg
E
m*
1
d E 
2
1
m   2  2
2a g cos ka
 dk x 
2
*
k
2
/a
k
/a
0
0
k
/a
Near the bottom of the band i.e. |k|<</a cos(ka) ~ 1 So m* ~ 2/2a2g
States near the top of the band have negative
effective masses.
dv x
Fx  e(E  v  B)  m
dt
*
Equivalently we can consider the mass to be
positive and the electron charge to be positive
As before.
For a = 2 x 10-10
and g = 4 eV
m* =0.24 x me
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Bloch Oscillations
E
Consider a conductor subject to an electric –Ex
dk x e
F
 Ex  x
dt


Consider an electron at k = 0 at t = 0
/a
0
/a
k
F1
e
k x (t )  E x t

/a
it is Bragg reflected to k = -/a. It
them moves from -/a to /a again.
v(t)
When the electron reaches k = /a
0
Period of motion T   2
eE x a
0
t/T
1
2
/a
Expect “Bloch oscillations” in the current current of period T
Not observed due to scattering since T >> tp
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Conductivity
Conductivity is
now given by
s ne2 tp/m*
(i) tp 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, s
(iv) Filled band have zero conductivity
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Motion in a magnetic field
Free electrons
F  ev  B  (e / m)k  B
The electrons move in circles in real space and in k-space.
Bloch electrons
dk
e
e
  v  B   2  k E (k )  B
dt


In both cases the Lorentz force does not change the energy of
the electrons. The electrons move on contours of constant E.
y
ky
x
kx
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Electron and Hole orbits
dk
e
  2  k E (k )  B
dt

ky
Filled states are indicated in grey.
dk
dt
dE
dk
dk
dt
dE
dk
ky
Bz
(b)
(a)
kx
kx
(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
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Electron like orbits
Periodic zone
picture of Fermi
contour ( E1 ) near
bottom of a band.
E1
E
Grad E
E1
/a
0 kx
/a
13
Hole like
orbits
Periodic zone
picture of the
Fermi contour at
the top of a band
Grad E
E2
E
E2
/a
0 kx
/a
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Holes
Can consider the dynamical properties of a band in terms of the
filled electron states or in terms of the empty hole states
dk x
Fx  
dt
Force on
Electrons
Energy
k
Consider an empty state (vacancy) in a band moving due to a force.
The electrons and vacancy move in the same direction.
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Energy & k-vector of a hole
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
Energy
Hole
kh
Eh = -Ee
Eh
This is the energy of the hole and it is
E=0
positive.
A full band has
 k 0
k
Ee
ke
Vacancy
If one electron, of k-vector ke, is missing
the total wavevector of the band is –ke.
A hole has k-vector kh = -ke
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Charge of a hole
In an electric field the electron wavevector would respond as
dk e
e
F
 E
dt


since kh = -ke
dk h
e
F
 E
dt


So the hole behaves as a positively charged particle.
The group velocity of the missing electron is .
1
v   k E (k )

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
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Effective mass of a hole
d E 
m   2
 dk x 
2
The effective mass is given by
*
1
2
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.
m *h  m *e
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.
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