Solid State Physics I First... Nearly Free Electrons in a Weak Periodic Potential 1.

Solid State Physics I
First Period (Moad A)
9/2/06
D. Rich
BGU
Please answer the following four questions. The number of points for each question is
shown. There are three pages to this exam.
Nearly Free Electrons in a Weak Periodic Potential (25 points):
1. Consider electrons in two-dimensions subject to a weak periodic potential coming from a
square lattice having a lattice constant a. The potential is given by
 2π x   2π y 
 . For k vectors far from the Brillouin zone
 cos
U ( x, y ) = −U o cos
 a   a 
boundaries, the wavefunction is well described by plane waves.
(a) Determine the form of the Fourier expansion of the crystal potential,
r r
r
U ( x, y ) = U (r ) = ∑U Gr eiG ⋅r ,
r
G
by finding the appropriate Fourier vectors G
corresponding to the non-zero Fourier coefficients UG. What are the UG.?
L
(b) The two-dimensional first Brillouin-zone (BZ) is
schematically shown on the left. Note that Γ represents the
X
usual BZ center position where k = 0. Calculate the energy
gap at the boundary points X and points L by (i) writing down
the full Hamiltonian, (ii) identifying the basis states and
wavefunctions involved in the degeneracies, and (iii) solving
the appropriate secular equation(s) for the eigenstates (energies and wavefunctions)
that diagonalize the Hamiltonian. Hint: The largest determinant that you will need
to solve is 2×2.
5 2π
5 2π
(c) Plot the energy band dispersion, E(k), for the range −
≤k≤+
along
2a
2a
the ΓL direction. Show the plots in both (i) the extended zone scheme and (ii) the
reduced zone scheme (i.e., representation in the first Brillouin Zone) .
Γ
Semiconductor structure and transport (25 points):
2. Consider a crystal of GaAs (gallium arsenide), which crystallizes in the zinc blende
crystal structure.
(a) Describe the (i) lattice and (ii) basis for this crystal. Draw the lattice. (iii) How many
nearest neighbor As atoms are there for a single Ga atom and what is the approximate
angle between each Ga-As chemical bond?
(b) Write down (i) the set of primitive translation vectors (Ti) in real space and the set of
primitive reciprocal translation vectors (Gi).
(c) Consider a single electron in the conduction band that obeys the effective mass
approximation (i.e., E (k ) = h 2 k 2 / 2meff ; meff=0.067mo) for all k. Initially, the
electron has a wavevector k=0 at t=0. Suppose that the electron experiences now an
electric field (E) of 10 kV/cm along the [100] direction at time t = 0. Determine the
energy of the electron (relative to the conduction band minimum) and its velocity at
time intervals t of 1, 2, and 10 picoseconds (ps). Using the reduced zone scheme,
what are the corresponding wavevectors k at these time intervals? Use the tables that
are provided to find the appropriate constants.
(d) The direction of the electric field (E) is now changed. Again the field is turned on at t=0
when k=0, but now along a new direction. (i) Along which direction must the field be
pointed in order for the electron to leave the first Brillouin Zone in the least amount of
time (t)? What is this time t? (ii) Along which direction must the field be pointed in
order for the electron to leave the first Brillouin Zone in the greatest amount of time (t)?
Again, what is this time t?
Vibrational Properties of a Square Lattice (36 points):
3. Part A (24 points). A simple, monoatomic, two-dimensional square lattice is
shown below. The lattice constant is a, the mass of each atom is m, the force
constant for the nearest neighbors is C1 (along the edges of the cell) and the force
constant for the next nearest neighbors is C2 (along the diagonal of the cell). All
other force interactions can be neglected. Assume that there are N-rows of atoms and
M-columns of atoms. Consider atomic motion only in the x-y plane of the 2D lattice.
The displacement vector of an atom from its
i,j+1
C1
equilibrium
position
is
denoted
as
r
uij = xˆu xij + yˆ u yij , where i is the atomic position
C2
along a column and j is the position along a row.
C1
i-1,j
i,j
i,j -1
i+1,j
(a) Plot rschematically the phonon dispersion
r
relations, w(k ) vs kx for k = (k x , 0) in the first Brillouin
Zone, showing and labeling all braches properly. Do not
solve the equations of motions.
(b) Suppose that the system is in motion such that
r
k = π , 0 . If the mode is longitudinal, construct the
a
following four plots involving the components of the
displacement vector during a typical snapshot in time: (i)
uxij vs i; (ii) uyij vs i; (iii) uxij vs j and (iv) uyij vs j. You
may start your plots at any arbitrary i and j positions.
(
y
x
)
(c) Repeat the question in part (b) for the transverse mode.
(d) Derive an expression for the (i) total kinetic energy (EK) and (ii) total
potential energy (EP) of this 2D crystal for the longitudinal mode.
(e) Repeat the question in part (d) for the transverse mode.
(f) Determine the frequency ω L of the longitudinal mode from part (d).
(g) Determine the frequency ωT of the transverse mode from part (e).
Part B (12 points). Consider the Thermal Properties of this 2D Crystal
r
(a) How many independent k vectors exist for this crystal? What is the distance
in k-space (∆k) between the kx and ky components?
(b) Suppose that the propagation velocity in this lattice is a costant, vo, in the
long-wavelength limit (i.e. 2π/k >> a) and assume that it is constant for all kdirections. Determine the 2D phonon density of states, D(ω ) and calculate
the Debye Frequency, ωD .
(c) Determine an expression for the Heat Capacity, cV, for this crystal. You may
leave the expression in an integral form. Demonstrate that cV ∝ T 2 at very
low temperatures without the use of an integral table.
X-ray Diffraction and the Structure Factor (14 points):
4. (a) A cubic crystal is formed containing atoms (all of the same type and labeled α) so
that the conventional cell has a volume of a3, contains two atoms, and has a
primitive cell whose volume is a3/2. Describe the lattice and basis for this crystal.
Draw the crystal structure.
(b) Determine the structure factor of this crystral for x-ray scattering. Given a set of
planes defined by (v1 v2 v3), for which planes involved in the x-ray scattering will
there be (i) maxima and (ii) minima in the detected x-ray intensity? Explain.
(c) Suppose now that we need to modify this crystal by forming an alloy that contains a
second type of atoms, labeled β. The new cubic crystal will contain an equal number
of α and β atoms and possess the same lattice type as before. Where must the β
atoms be located in the conventional cubic cell if we require the (200), (110), and
(222) x-ray reflections to vanish. Base your arguments on the Structure Factor.