Homework # 8 Chapter 9 Kittel Phys 175A Dr. Ray Kwok SJSU
Prob. 1 – Brillouin zones of rectangular lattice
9.1 Brillouin zones of rectangular lattice. Make a plot of the first two Brillouin zones of a primitive primitive rectangular rectangular two-dimensional two-dimensional lattice with 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
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 F
4 * π 0 2 4 * 2 * ( A ) 2
2 * A
* 10 cm
* k F
Brilloin zone Radius of free electron fermi sphere =
* 10 cm
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
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
(b) Describe in real space the motion
of an electron on this orbit in the presence of the magnetic field.
v × B
From Eq. 25a we have dt dt , where I have decided to use SI units. v v v G d r = − q e = h = − ev B dt τ Letting we get = v , setting d k = G dt because v ⊥ B since B is normal to the Fermi surface. Solving for gives Gh = . Plugging in the givens we evB get
2 ⋅108 100cm
2 ⋅ π ⋅1.602 ⋅10
⋅ 6.62 ⋅10
kg ⋅ m 2 s
= 1.315 ⋅10
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
Chapter 9 Problem 7
1 ( ) expected ∆ B
(a) Calculate the period 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: ∆(
Using Table 6.1 on pg. 139, for potassium we find kf=0.75x108cm-1 .
2π e 2
hc(π K f )
)= 2 B hcK f
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
This results in
) = 5.55 ×10−9 G −1
(b) To solve this part of the problem, go back to the equations we used for the cyclotron. ω c
P = mv = hk
Solve for r r =
v f ω c
v f mc Be
h k f c
Plugging in values from before and B=10kG r = 4.94x10-4 cm The orbit is circular, so the area is