Homework # 8
Chapter 9 Kittel
Phys 175A
Dr. Ray Kwok
SJSU
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/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
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/a
2π/3a
2nd BZ
First BZ
Prob. 2 – Brillouin zone,rectangular lattice
Gregory Kaminsky
This is a Wigner-Seitz cell.
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 * π * k F2


4 *π

0 2
 4*2*(A ) 
2
π
kF =
2 * A0
=1
π
=
2
* 1012 cm −1
π * k F2
2*
2 = N
 2π 
 
 L
Brilloin zone
Radius of free electron fermi sphere =
π
2
π *1012 cm −1
π
2
* 1012 cm −1
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 scheme
Second Zone periodic scheme
Prob. 5 – Open Orbits
John Anzaldo
An open orbit in a monovalent tetragonal metal connects
opposite faces of the boundary of a Brillouin zone. The
faces are separated by
G = 2 ×108 cm −1.
A magnetic field B = 10 −1T
is normal to the place of the open orbit. (a) What is the
order of magnitude of the period of thek motion in
8
Take v = 10 cm / s
space?
(b) Describe in real space the motion
of an electron on this orbit in the presence of the magnetic
field.
9.5
v
v
dk
dr v
h
= q ×B
dt
dt
From Eq. 25a we have
, where I have
decided to use SI units.
v
v
v
G
d
r
q
=
−
e
dt
=
τ
h
=
−
ev
B
Letting
we get
= v , setting dk = G
τ
dt
because v ⊥ B since B is normal to the Fermi surface.
Solving for τ gives Gh = τ . Plugging in the givens we
evB
get
2
2 ⋅108 100cm
−34 kg ⋅ m
⋅
⋅ 6.62 ⋅10
Gh
cm
m
s
=
= 1.315 ⋅10 −10 s
evB 2 ⋅ π ⋅1.602 ⋅10 −19 C ⋅108 cm ⋅ 1m ⋅10 −1 kg
s 100cm
C⋅s
Part b)
The electron will travel along the Fermi surface
as shown. The velocity will change as the
electron moves along the Fermi surface.
Mike Tuffley
5/12/09
U(x)
-a/2
a/2
x
-U0
Chapter 9 Problem 7
Adam Gray
1
(a) Calculate the period ∆( B ) expected for
potassium on the free electron model.
(b) What is the area in real space of the extremal
orbit, for B = 10kG = 1T ?
Starting with equation 34:
1
2πe
∆( ) =
B
hcS
Where
S = πK
2
f
Using Table 6.1 on pg. 139, for potassium we find
kf=0.75x108cm-1 .
Plugging in:
1
2πe
∆( ) =
B hc(πK 2f )
1
2e
∆( ) =
B hcK2f
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
1
∆( ) = 5.55×10−9 G −1
B
This results in
(b) To solve this part of the problem, go back to
the equations we used for the cyclotron.
Be
ωc =
mc
r =
v
f
ωc
P = mv = hk
Solve for r
vf
v f mc
hk f c
r =
=
=
=
ω c  Be 
Be
Be


 mc 
vf
Plugging in values from before and B=10kG
r = 4.94x10-4 cm
The orbit is circular, so the area is
2
−7
πr = 7.67 ×10 cm
2