Download Chapter 6b: Lecture Notes

Problem #4 BZ of rectangular Lattice
Phys. 555/342: Ch.6B(2007)
Problem #4 BZ of rectangular
Lattice
Phys. 555/342: Ch.6B(2007)
Shift in Chemical Potential 2-D
Density of States in 2-D
A 2m
g(")d" =
d"
2
2# h
1
f FD (") =
exp[(" # µ) /kT] + 1
!
#
N=
$ g(") f
FD
(")d"
0
h2
2"n =
2m
!
%
&
0
d#
exp[(# $ µ) /kT] + 1
h2
2"n = #F (T = 0)
2m
Phys. 555/342: Ch.6B(2007)
!
!
!
h2
2"n =
2m
%
&
0
d#
exp[(# $ µ) /kT] + 1
Change variables
x=(ε−µ)/kT
$
0
h2
kTdx
kTdx
2"n = % x
= % x
+
2m
# µ / kT e + 1
# µ / kT e + 1
0
$ 1
'
h 2 2"n
= * & x
#1)dx +
2m kT # µ / kT % e + 1 (
0
h 2 2"n
#dx
= $ #x
+
2m kT # µ / kT e + 1
%
$
0
+
*
0
$
%
0
kTdx
ex + 1
dx
+ µ /kT
x
e +1
dx
+ µ /kT
x
e +1
µ / kT
%
h 2 2"n # µ
dx
dx
= # $ #x
+ $ x
2m kT
+1 0 e +1
0 e
h 2 2"n
If kT<<µ
= µ = #F
2m kT
!
$
h 2 2"n # µ
dx
=+ % x
2m kT
µ / kT e + 1
Phys. 555/342: Ch.6B(2007)
Effective Mass m*
me is “true” mass of an electron
In a crystal we have an effective mass m*
Apply a field E for time δt
work = "eE#x
"# = $eE • vG"t
But"# $ % k#(k) • "k = hvG • "k
!
# k%(k)
with " "vG = # k$ (k) =
!
!
!
So " "h#k = $eE#t
h
•
h k = $eE
Phys. 555/342: Ch.6B(2007)
Effective Mass m*-Acceleration
2
dv i 1 d
1
$ # dk j
ai =
=
(" k#(k)) i = %
dt h dt
h j $k i$k j dt
2
dv i 1
#$
Giving " "
= 2%
(&eE)
dt h j #ki#k j
!
!
dv
#eE
Classically " " = a =
dt
me
2
h
"
(me ) ij = 2
# $ /#ki#k j
Phys. 555/342: Ch.6B(2007)
Effective Mass m*
2
h
(m ) = 2
# $ /#ki#k j
"
e ij
• Acceleration is not in direction of E
• flat bands have high effective mass
• Effective mass can and is negative!!
!
One Dimensional
Spherical bands
2
h
m = 2
2
# $ / #k
"
e
Phys. 555/342: Ch.6B(2007)
!
Warning! Different Effective Masses
Specific Heat
2
2
B
" Nk # $ V '
CV =
mCV & 2 )
2
% 3" N (
h
2/3
T
2
!
!
# 2
or " "CV = k B g($F )T
3
g(ε) is always positive the heat capacity is positive
m*CV is different from m*--in fact mCV=|m*|1/3
Phys. 555/342: Ch.6B(2007)
Metals, semiconductor, Insulators
Phys. 555/342: Ch.6B(2007)
Effective Mass m*
1-D Band Structure
" k$(k)
vG = " k# (k) =
h
Effective Mass m*
What does a negative
Effective mass mean?
!
m* diverges
m*<0
Phys. 555/342: Ch.6B(2007)
Effective Mass m*
Direct Gap Semiconductor
We are going to talk about
Holes!
Phys. 555/342: Ch.6B(2007)
Effective Mass m*
Indirect Gap Semiconductor
Direct excitation
Indirect Excitation
Needs momentum
Phys. 555/342: Ch.6B(2007)
Holes: Photoabsortpion
Absorption of a photon of energy h" and negligible wavevector takes
and electron from state ke with energy E from a valence band state to a
conduction band state Q. Total wave vector of valence band after
The valence band is treated
excitaiton is -ke.
as a hole with kh=-ke. This is
!
the wavevector ascribed to
the hole. The valence band
is describe as occupied by
one hole. Since ke=-kh this is
the same as having an
electron at G.
For the entire system the
total wavevector is zero,
ke+kh=0.
Phys. 555/342: Ch.6B(2007)
Construction of a one-Dimensional Hole band
The upper half of the figure shows thehole band that simulates
the dynamics of a hole, constructed by inversion of the vlence
band at the origin.
The wavevector and
energy of the hole are
equal, butr opposite in
sign, to the k and E of
the empty electron
orbital in the valence
band.
Phys. 555/342: Ch.6B(2007)
Time evolution of a hole
in the presence of a field E
At t=0 (a) all states are
filled except F at the top
of the band. The
velocity vx=0 .
(b) A field is applied in x
direction. The force is
in the -k direction. All
the electrons make a
transition in the -k
direction.
(c) After another
interval of time moves
to D.
Phys. 555/342: Ch.6B(2007)
Time evolution of a hole
in the presence of a field E
Same picture except in the 1-D hole band picture
Phys. 555/342: Ch.6B(2007)
Hall Effect
The Hall field EH is
established due to the
charge build-up on
opposing faces in the
dirfection normal to the
applied magnetic and
electric fields, resulting
from the Lorentz force
acting on the moving
electrons or Holes.
In steady state, the
force -e EH balances
the Lorentz force.
Phys. 555/342: Ch.6B(2007)
Hall Effect
In Steady state
Since vy=0
!
!
jx B
E y = vx B = "
ne
E = "• j
Ey B
" xy = # " xy =
=
j x ne
Hall Coefficient RH
!
!
0 = "e(E y " v x B)
E H = R H Bxj
Ey
" xy
1
RH =
=
=#
j x Bz Bz
ne
Phys. 555/342: Ch.6B(2007)