Post on 12-Nov-2014
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Homework # 8
Chapter 9 KittelPhys 175A
Dr. Ray Kwok
SJSU
2π/a
2π/3a
9.1 Brillouin zones of rectangular lattice. Make a plot of the first two
Brillouin zones of a primitive rectangular two-dimensional lattice with axes
a, b=3a
Prob. 1 – Brillouin zones of
rectangular lattice Daniel Wolpert
9.1 Brillouin zones of rectangular lattice. Make a plot of the first two
Brillouin zones of a primitive rectangular two-dimensional lattice with axes
a, b=3a
2π/3a
2π/a
First BZ
2nd BZ
9.1 Brillouin zones of rectangular lattice. Make a plot of the first two
Brillouin zones of a primitive rectangular two-dimensional lattice with axes
a, b=3a
This is a Wigner-Seitz cell.
Prob. 2 – Brillouin zone,rectangular latticeGregory Kaminsky
A two-dimensional metal has one atom of valency one in a simple
rectangular primitive cell a = 2 A0 ; b = 4 A0.
� a) Draw the first Brillouin zone. Give it’s dimensions in cm-1.
� b) Calculate the radius of the free electron fermi sphere.
� c) Draw this sphere to scale ona drawing of the first
Brillouin zone.
Calculation of the radius of
the Fermi sphere
2
4
4 2
1
2
2
0 2
* *
*
* *( )
π
π
k
A
F
=
kA
cmF = =−
π π
2 2100
12 1
**
22
2
2**π
π
k
L
NF
=
π
21012 1* cm−
π *1012 1
cm−
Radius of free electron fermi sphere = π
2
Brilloin zone
Make another sketch to show the first few
periods of the free electron band in the periodic
zone scheme, for both the first and second energy
bands. Assume there is a small energy gap at the
zone boundary.
This is the first energy band
Second energy band
Prob. 4 – Brillouin Zones of Two-Dimensional Divalent Metal
Victor Chikhani
A two dimensional metal in the form of a square lattice has two conduction electrons per atom. In
the almost free electron approximation, sketch carefully the electron and hole energy surfaces. For
the electrons choose a zone scheme such that the Fermi surface is shown as closed.
Hole Energy surface
Electron Energy Surface
BZ periodic schemeSecond Zone periodic scheme
Prob. 5 – Open Orbits
An open orbit in a monovalent tetragonal metal connects
opposite faces of the boundary of a Brillouin zone. The
faces are separated by . A magnetic field
is normal to the place of the open orbit. (a) What is the
order of magnitude of the period of the motion in space?
Take (b) Describe in real space the motion
of an electron on this orbit in the presence of the magnetic
field.
18102 −×= cmG TB
110−=
k
scmv /108=
John Anzaldo
9.5
� From Eq. 25a we have , where I have decided to use SI units.
� Letting , setting we get
because since is normal to the Fermi surface.
� Solving for gives . Plugging in the givens we get
� Part b)
Bdt
rdq
dt
kd vvv
h ×=
vdt
rd=
v
Gkd =v
τ=dteq −= BevG v
h −=τ
Bv ⊥ B
τ τ=evB
Gh
The electron will travel along the Fermi surface
as shown. The velocity will change as the
electron moves along the Fermi surface.
s
sC
kg
cm
m
s
cmC
s
mkg
m
cm
cm
evB
G 10
1819
234
8
10315.1
10100
11010602.12
1062.6100102
−
−−
−
⋅=
⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅
⋅
=
π
h
Mike Tuffley
5/12/09
-a/2 a/2
-U0
x
U(x)
Chapter 9 Problem 7 Adam Gray
(a) Calculate the period expected for
potassium on the free electron model.
(b) What is the area in real space of the extremal
orbit, for B = 10kG = 1T ?
)1
(B
∆
Starting with equation 34:
Where
Using Table 6.1 on pg. 139, for potassium we find
kf=0.75x108cm-1 .
cS
e
B h
π2)
1( =∆
2
fKS π=
Plugging in:
Note: The equation 34 was for cgs units, so all
values used with this equation must be in this
form.
c=3x1010 cm/s
h=1.05459x10-27 erg s
e=4.803x10-10 erg1/2 cm1/2
)(
2)
1(
2
fKc
e
B π
π
h=∆
2
2)
1(
fcK
e
B h=∆
This results in
(b) To solve this part of the problem, go back to
the equations we used for the cyclotron.
Solve for r
191055.5)1
( −−×=∆ G
B
kmvP h==c
fvr
ω=
mc
Bec =ω
Be
ck
Be
mcv
mc
Be
vvr
fff
c
f h==
==
ω
Plugging in values from before and B=10kG
r = 4.94x10-4 cm
The orbit is circular, so the area is
272 cm1067.7 −×=rπ