Download Review Lecture: March 15, 2013

Ch t 4:
Chapter
4 S
Summary
Solve lattice vibration equation of one atom/unit cell case.
case
Consider a set of ions M separated by a distance a ,

R  na for integral n. Let u (na ) be the displacement.
Assuming only neighboring ions interact, we have
2
1
U
 C   u( na )  u([n  1]a ) ,
2 n
Newton's second law F  Ma or
harm
U harm
du( na )
M

 C  2u( na )  u([n  1]a )  u([n  1]a )
2
dt
u( na )
Dai/PHYS 342/555 Spring 2013
Review 1-1
For each of the N values of k there are thus two solutions,
leadingg to a total of 2 N normal modes. The two  vs k curves
are two branches of the dispersion relation. Acoustic and optical
branches.
Dai/PHYS 342/555 Spring 2013
Review 1-2
Prob. 3, Consider a longitudinal wave us  u cos(t  sKa )
which propagates in a monatomic linear lattice of atoms of mass M ,
spacing a, and nearest neighbor interaction C.
a) show that the total energy of the wave is
1
1
2
2
E  M   dus / dt   C   us  us 1  ,
2
2 s
s
where s runs over all atoms.
b)By substitution of us in this expression, show that the time average
total energy per atom is
1
1
1
2 2
2
M  u  C 1  cos K
Ka  u  M  2u 2 ,
4
2
2
where the last step we have used the dispersion relation for this problem.
Dai/PHYS 342/555 Spring 2013
Review 1-3
Chapter 5: Summary
Planck distribution function:
Th average excitation
The
it ti quantum
t
# off an oscillator
ill t is:
i
s exp(  n /  )
1

 n 

 exp( s /  ) exp( /  )  1
At low temperatures,


0

1 4
 x    3  nx
dx  x     x e dx  6 4 
,
0
15
 e  1  n 1
n 1 n
12 4
T 
the heat capacity CV 
Nk B   .
5
 
3
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Review 1-4
Einstein models of the density of states
In the case of N oscillators of the same frequency 0 in 1D,
the Einstein density of states is D( )  N  (  0 )
 N  
U  N n     /
.
1 
e
e 
 U 
  
The heat capacity CV  N 
  Nk B 
  /

T

 1) 2

V

 (e
2
Dai/PHYS 342/555 Spring 2013
Review 1-5
Thermal conductivity
The thermal conductivity coefficient K of a solid is defined as,
as
dT
jU   K
, where jU is the flux of thermal energy, and
dx
x is distance. From the kinetic theory of gases we find
1
1 2
K  Cvl  Cv  , where C is the heat capacity per volume,
3
3
v is the average
a erage particle velocity,
elocit and l is the mean free
path of a particle between collisions,  1 is the phonon
collision rate.
Dai/PHYS 342/555 Spring 2013
Review 1-6
Dai/PHYS 342/555 Spring 2013
Review 1-7
Chapter 6: Free electron Fermi gas
Under quantum theory and the Pauli exclusion principle, we consider
N noninteracting electrons confined to a volume V ( L3 ). If the wave function

of single electron is  (r ), then
2 2 
2   2
2
2  


(
r
)




(
r
)


(
r
).



 2
2
2 
2m
2m  x y z 
Applying boundary condition
 ( x, y, z  L)   ( x, y, z ); ( x, y  L, z )   ( x, y, z );
 ( x  L, y, z  L)   ( x, y, z ). The solutions are
2k 2
1 ikr

 K (r ) 
e ,  (k ) 
. Note the probability of
2m
V
  2
findingg the electron somewhere in the volume is 1   dr  (r ) .
Dai/PHYS 342/555 Spring 2013
Review 1-8

Note that  K (r ) is an eigenstate of the momentum operator
operator,
 ikr
      ikr
p
  ,
 e  ke
i r i
i r



an electron in the level  K (r ) has a momentum p  k and a

 
velocity v  p / m  k / m, where   2 / k .
Periodic boundary condition requires
Periodic
2 n y
2 nx
2 nz
e  e  e  1 or k x 
, ky 
, kz 
L
L
L
Thus in a 3-D k -space, the allowed wavevectors are those along
ik x L
ik y L
ik z L
2
the three axes given integer mutiples of
.
L
Dai/PHYS 342/555 Spring 2013
Review 1-9
To calculate the allowed states in a region of k -space volume ,

V

or the number of allowed k -values per unit
3
3
(2 / L)
(2 )
volume of k -space (known as the k -space density of levels) is
V
. Because the electrons are noninteracting we can built up
3
(2 )
the N -electron ground state by placing electrons into the allowed
one-electron levels. Pauli exclusion principle allows each wavevector
to have 2 electronic levels with spins up and down.
Dai/PHYS 342/555 Spring 2013
Review 1-10
Since the energy of a one
one-electron
electron level is directly proportional
to k 2 , when N is enormous the occupied region will be indistinguishable
from a sphere
sphere. The radius of this sphere is called k F (F for Fermi),
ermi)

3
and its volume  is 4 k F / 3. The # of allowed k within the sphere is:
 4 k F3

 3
  V  k F3
  3   2 V . Since each allowed k -value leads to two
  8  6
k F3
k F3
one-electron levels, we must have N  2 2 V  2 V .
6
3
If electron density is n  N / V , then we have n  k F3 / 3 2 .
Dai/PHYS 342/555 Spring 2013
Review 1-11
The sphere of radius k F containing the occupied one eletron levels
is called the Fermi sphere.
The Surface of the Fermi sphere, which separate the occupied form
the unoccupied levels is called the Fermi surface.
1/3
 3 N 
The momentum pF  k F = 
 of the occupied
 V 
one-electron levels of highest
g
energy
gy is the Fermi momentum.
2
2


3

N
2 2
 F   k F / 2m 


2m  V 
2
vF  p F / m  (
is the Fermi energy;
1/3
  3 N 
)

m  V 
2
2/3
is the Fermi velocity.
Dai/PHYS 342/555 Spring 2013
Review 1-12
Experimental heat capacity of metals
At sufficient low temperatures, CV   T  AT 3 . Where  is the
Sommerfeld parameter. The ratio of the observed to the free
electron values of the electronic heat capacity is related to
thermal effective mass as:
mth  (observed)

m
 (free)
Dai/PHYS 342/555 Spring 2013
Review 1-13
El t i l conductivity
Electrical
d ti it and
d Ohm’s
Oh ’ law
l
Considering Newton's second law, we have



 1 
dv
dk
F m

 e( E  v  B )
dt
dt
c


The displacement of the Fermi sphere,  k  eEt / .


If collision time is  , the incremental velocity is v  eE / m.

In a constant electric field E and n electrons per volume, the




2
electric current density is j  nqv  ne E / m   E.
The electrical conductivity   ne 2 / m.
Dai/PHYS 342/555 Spring 2013
Review 1-14
Thermal conductivity of metals
Wiedemann-Franz law
Thermal conductivity for a Fermi gas
 2 nk B2T
 2 nk B2T
1
K el  Cvl 

 vF  l 
2
3
3 mvF
3m
The Wiedemann-Franz law states that for metals at not
too low temperatures the ratio of the thermal conductivity
to the electrical conductivity is directly proportional to the
temperature independent of the particular metal.
temperature,
metal
K el


 nk T / 3m   k B 

  T  LT .
3  e 
ne  / m
2
2
B
2
2
2
Lorenz number L  2.45  108 watt-ohm/deg 2
Dai/PHYS 342/555 Spring 2013
Review 1-15
Ch t 7:
Chapter
7 S
Summary
Bloch’s
Bloch
s theorem
The eigenstates  of the one-electron Hamiltonian

 2 2
 
 

H  
  U (r )  , where U (r  R)  U (r ) for all R in
 2m

a Bravias lattice, can be chosen to have the form of a plane wave
times a function with the periodicityy of the Bravias lattice:






ik r




 nk (r )  e unk (r ), where unk (r  R)  unk (r )

f all
for
ll R in
i the
th Bravias
B i lattice.
l tti or
 
 

ik  R
 (r  R)  e  (r )
Dai/PHYS 342/555 Spring 2013
Review 1-16
The effect of a periodic potential
The periodic potential has form:
2 x
, where a is lattice parameter
a
U 0 . If U1  0 then we have the free
U  U 0  U1 cos
and U1
2 k 2
electron gas case where  
2m
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Review 1-17
Wave equation of electron in periodic potential
U ( x)   U G eiGx   U G (eiGx  e  iGx )  2 U G cos Gx
G
H    (
G 0
G 0
1 2
p   U G eiGx ) ( x)  ( x)
2m
G
 ( x)   C (k )eikx , k  2 n / L.
k
1 2
2
2
ikx
p  ( x) 
k
C
(
k
)
e
,

2m
2m k
( U G eiGx ) ( x)  U G eiGx C (k )eikx ,
G
G
k
2 2
ikx
ikx
iGx
ikx
k
C
(
k
)
e

U
e
C
(
k
)
e


C
(
k
)
e
k 2m


G
k
G k
2 k 2
(
  )C (k )   U G C (k  G ) 0.
0 k   2 k 2 / 2m.
2m
G
Dai/PHYS 342/555 Spring 2013
Review 1-18
Crystal momentum of an electron
Under a crystal lattice translation we have
  
 
 
 

ik T ik r
ik T
 k (r  T )  e e uk (r  T )  e  k (r ).
If the lattice potential vanishes, the result recovers
to that of free electron gas.
gas

k is
i called
ll d the
h crystall momentum off an electron.
l

If an electron k absorbs in a collision a phonon of
   

wavevector q , the selection rule is k  q  k ' G.
Dai/PHYS 342/555 Spring 2013
Review 1-19
Approximate
pp
solution near a zone boundary
y
At the zone boundary
y the kinetic energy
gy of the waves
1
k   G are equal. (   )C (k )   U G C (k  G ) 0.
2
G
(   )C (G / 2)  UC (G / 2)  0
(   )C (G / 2)  UC (G / 2)  0
2
1 2

2
2
(   )  U ;     U 
( G)  U .
2m 2
Thus the potential energy has created an energy gap 2U at
the zone boundary
boundary.
Dai/PHYS 342/555 Spring 2013
Review 1-20
Chapter 8: Summary
A solid with an energy gap will be nonconducting at T  0
unless electric breakdown occurs or unless the AC field is
of such high frequency that  exceeds the energy gap.
However, when T  0 some electrons will be thermally
excited to unoccupied bands (conduction bands).
If the enegy gap Eg  0.25
0 25 eV
eV, the fraction of electrons across
 E / 2k T
the gap is of order e g B  102 , and observable conductivity
will occur. These materials are semiconductors.
Dai/PHYS 342/555 Spring 2013
Review 1-21
Effective Mass
The group velocity vg  d / dk ,    / , so vg   1d  / dk .
1 d 2 1  d 2 dk  1  d 2   dk  1  d 2   F 

  2
   2     2  
dt
 dkdt   dk dt    dk   dt    dk    
dvg
1
1  d 2 
F  ma, then we have *  2  2  .
m
  dk 
Effective Mass in Semiconductors
The angular rotation frequency  c of the current carriers
is:
eB
 c  * , where m* is the effective mass.
mc
Dai/PHYS 342/555 Spring 2013
Review 1-22
In an intrinsic semiconductor, n  p, Eg  Ec  Ev ,
 k BT 
n  2
2 
2




3/ 2
 me mh 
3/ 4
p[( Eg / 2k BT ]].
exp[(
Dai/PHYS 342/555 Spring 2013
Review 1-23
Chapter 9: Reduced zone scheme
 



ik 'r
For a Bloch function written as  k ' (r )  e uk ' (r ), with k '
  
outside the first Brillouin zone, we have k  k ' G.
 


 





ik 'r
ik r
 iG r
ik r
 k ' ( r )  e uk ' ( r )  e e uk ' ( r )  e uk ( r )   k ( r )


Dai/PHYS 342/555 Spring 2013
Review 1-24
Dai/PHYS 342/555 Spring 2013
Review 1-25
Electron orbits, hole orbits, and open orbits
An electron on the Fermi surface will move in a curve on
the Fermi surface,
surface because it is a surface of constant energy
energy.
Dai/PHYS 342/555 Spring 2013
Review 1-26
Tight banding method for energy bands
Tight banding approximation deals with the case in which
the ovelap of atomic wave functions is enough to require
corrections to the picture of isolated atoms, but not so
muchh as to
t render
d the
th atomic
t i description
d
i ti irrelevant.
i l
t
Dai/PHYS 342/555 Spring 2013
Review 1-27
Free electrons in a uniform magnetic
g
field
The orbital energy levels of an electron in a cubic box with sides
of length L parallel to the x-, y -, and z -axes
axes are determined in
the presence of a uniform magnetic field H along the z -direction
by two quantum numbers
numbers,  and k z :
1
eH
2 2
k z  (  )c , c 
.
  k z  
2m
2
mc
 runs through all nonmagnetic integers, and k z takes on the
same values as in the absence of a magnetic
g
field:
k z  2 nz / L for any integral nz .
The energy
gy of motion pperpendicular
p
to the field,, which would
be  2 (k x2  k y2 ) / 2m if no field were present, is quantized in
steps of c (c  eH / mc).
) This is orbit quantization.
quantization
Dai/PHYS 342/555 Spring 2013
Review 1-28
Origin of the oscillatory phenomena
Most electronic properties of metals depend on the density of levels
at the Fermi energy, g   F  . It follows that g   F  will be singular
whenever the value of H causes an extramal orbit on the Fermi surface
to satisfy the quantization condition ( + )A  Ae ( F ).
1

H
e
 2 e 1
4

.

1.34

10
K / G.

 c Ae   F  mck B
Dai/PHYS 342/555 Spring 2013
Review 1-29
For a system of N electrons at absolute zero the Landau levels
g
level s  1 will be ppartially
y
are filled to s. Orbitals at the next higher
filled. The Fermi level will lie between s and s  1. As the magnetic
field is increased the electrons move to lower levels because the area
between successive circles are increased
(k 2 )  ((2 k )(k )  2 eB / c.
Dai/PHYS 342/555 Spring 2013
Review 1-30