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PHYS4050 Solid State Physics

创建日期：2024-06-05 15:13:38

所属分类：物理Physical

标签： PHYS4050

Solid State Physics

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PHYS4050 Solid State Physics Problem Set 2

Due: 13 October 2023 (Friday); “T+2” = 15 October 2023 (Sunday) (15% discount)
You should submit your work in ONE CLEAR PDF file to Blackboard no later than 23:59 on the
due date.
Please work out the steps of the calculations in detail. Discussions among students are encouraged, yet
it is expected that you do your homework independently.
2.0 Reading Assignment: Don’t need to hand in anything. We are discussing lattice vibrations (see Ch.
V of class notes). The 1D monatomic and diatomic chains served as the typical cases for a crystal
with one atom in the basis and two atoms in the basis, respectively. When there are p atoms in the
basis and each atom can move in three directions, there are 3p branches in the dispersion relation.
Among them, only 3 are acoustic branches with ω → 0 as q → 0, and the remaining (3p − 3)
are optical branches with ω ̸= 0 at q = 0. The values of q can be restricted to be within the 1st
Brillouin Zone (1st B.Z.). For a finite sample consisting of N unit cells (N ≫ 1), there are exactly
N allowed values of q in the 1st B.Z. after imposing the periodic boundary conditions. These
statements are also valid in 2D and 3D systems. Quantization leads to the notions of acoustic
and optical phonons. Each phonon carries an energy h¯ωs(q⃗) and a crystal momentum h¯q⃗. These
key ideas go into the topics of: how neutron inelastic scattering and other similar problems can
be used to probe ω(q⃗), the heat capacity of insulators, the thermal conduction in insulators, and
the high reflectivity and high absorption in the IR (infrared) range of ionic solids.
References are given at the end of the chapter. We based mostly on Kittel, Introduction to
Solid State Physics Ch.4 and 5, Christman, Fundamentals of Solid State Physics, Ch. 6, Omar,
Elementary Solid State Physics, Ch. 3, and Blakemore, Solid State Physics, Sec. 2.1 - 2.5. To see
more data of the physical properties of real materials and down-to-earth explanations (minimal
mathematics), see Understanding the Properties of Matter (2nd edition) by de Podesta.
2.1 (14 points) Phase velocity and Group velocity of a 1D monatomic chain
In class notes, we worked out the dispersion relation ω(q) of the longitudinal acoustic branch of
a 1D monatomic chain. Take the result (no need to derive the result again), obtain the phase
velocity and group velocity for q ∈ 1st B.Z. Sketch the two results for q ∈ 1st B.Z.
2.2 (34 points) 1D diatomic chain with nearest-neighbor and next-nearest-neighbor interactions
[Read the set up carefully.] Consider a 1D diatomic chain. The two atoms in a basis have
masses MA and MB, respectively. When the atoms are in their equilibrium positions, the nearest
neighbors are at a distance a apart, so that the period (lattice constant) is 2a. These atoms are
connected by springs that mimic the chemical bonds. Let KAB be the spring constant of the
springs connecting nearest-neighboring A and B atoms. In addition, there are springs of spring
constant KAA connecting an A atom to the next A atom (they are next-nearest-neighbors) and
there are springs of spring constant KBB connecting an B atom to the next B atom.
(a) [24] Work out the dispersion relations of the acoustic and optical phonon branches.
[Hint: For simplicity, you may consider longitudinal motions only. One way is to follow the
steps in the class notes for a 1D diatomic chain with nearest-neighboring interactions.]
(b) [10] Plot the dispersion relations ω(q) in a well-labelled diagram for the case of MA =
2MB, KAB = 2KAA = 4KBB. Physically, it means that the nearest-neighboring interaction
KAB is the strongest.
1
2.3 (22 points) Plotting dispersion relation of a 2D system
Let’s consider a hypothetical 2D crystal. It takes on the 2D square lattice of lattice constant a and
has a basis of one atom. After some work, the dispersion relation of an acoustic phonon branch
is found to be
ω(q⃗) = ω(qx, qy) = ωm
√
sin2(
qxa
2
) + sin2(
qya
2
) (1)
where ωm sets the maximum value of the angular frequency.
[5] Sketch the 1st B.Z. and label the zone center Γ, the point at the boundary in the x-direction
(or y-direction) X, and the corner M .
[5] Give the value of ω(q⃗) at Γ, at X and at M .
[12] Plot the phonon dispersion relation given by Eq. (1) for (i) q⃗ that goes from Γ → X
along a straight line; (ii) q⃗ that goes from Γ→M along a straight line, and (iii) q⃗ that goes from
X →M along a straight line.
[Remark: The length of the horizontal axis should be made proportional to the |q⃗| from the
starting point to the end point in the three plots. The same remark also goes with the vertical
axis.]
(Optional: No bonus points) To become a pro, see if you can make a combined plot so that the
horizontal axis goes continuously from Γ → X → M → Γ, i.e. paste the three separate plots
together.
2.4 (30 points) Debye model of a 2D system
Let’s consider a hypothetical 2D system. It takes on the 2D square lattice of lattice constant a
and has a basis of one atom. For simplicity, we only allow motions in the 2D plane. Therefore, we
get two acoustic phonon branches. They are of the form of Eq. (1) in Problem 2.3. One branch
has a maximum angular frequency ωm = ω1 and the other has a maximum ωm = ω2.
Following the idea of the Debye model, we will work on the long-wavelength limit of the dispersion
relations.
(a) [5] Find the sound velocity for each of the two branches.
(b) [10] Obtain an expression of the (total) density of normal modes (sometimes also called
density of states) D(ω), where D(ω)dω is the number of modes in the angular frequency
range between ω to ω + dω within the Debye model.
[Hint: The two branches are considered in getting the total density of normal modes.]
(c) [5] Determine a cutoff in q and the corresponding cutoff in ω so that the total number of
modes is preserved.
(d) [10] Hence, extract the temperature dependence of the energy and heat capacity at
low temperatures.

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