Chapter 15, Electrons in a Periodic Potential Video Solutions, The Oxford Solid State Basics

Problem 1

†Nearly Free Electron Model
Consider an electron in a weak periodic potential in one dimension $V(x)=V(x+a)$. Write the periodic potential as
$$
V(x)=\sum_{G} e^{i G x} V_{G}
$$
where the sum is over the reciprocal lattice $G=$ $2 \pi n / a$, and $V_{G}^{*}=V_{-G}$ assures that the potential $V(x)$ is real.
(a) Explain why for $k$ near to a Brillouin zone boundary (such as $k$ near $\pi / a$ ) the electron wavefunction should be taken to be
$$
\psi=A e^{i k x}+B e^{i(k+G) x}
$$
where $G$ is a reciprocal lattice vector such that $|k|$ is close to $|k+G|$.
(b) For an electron of mass $m$ with $k$ exactly at a zone boundary, use the above form of the wavefunction to show that the eigenenergies at this wavevector are
$$
E=\frac{\hbar^{2} k^{2}}{2 m}+V_{0} \pm\left|V_{G}\right|
$$
where $G$ is chosen so $|k|=|k+G|$.
$D$ Give a qualitative explanation of why these two states are separated in energy by $2\left|V_{G}\right|$.
D. Give a sketch (don't do a full calculation) of the energy as a function of $k$ in both the extended and the reduced zone schemes.
(c) ${ }^{*}$ Now consider $k$ close to, but not exactly at, the zone boundary. Give an expression for the energy $E(k)$ correct to order $(\delta k)^{2}$ where $\delta k$ is the wavevector difference from $k$ to the zone boundary wavevector.
$\triangleright$ Calculate the effective mass of an electron at this wavevector.

Ameer Said

Numerade Educator

02:42

Problem 2

Periodic Functions
Consider a lattice of points $\{\mathbf{R}\}$ and a function $\rho(\mathbf{x})$ which has the periodicity of the lattice $\rho(\mathbf{x})=$ $\rho(\mathbf{x}+\mathbf{R})$. Show that $\rho$ can be written as
$$
\rho(\mathbf{x})=\sum_{G} \rho_{G} e^{i G \cdot \mathbf{x}}
$$
where the sum is over points $\mathbf{G}$ in the reciprocal lattice.

Nick Johnson

Numerade Educator

05:19

Problem 3

Tight Binding Bloch Wavefunctions
Analogous to the wavefunction introduced in Chapter 11 , consider a tight-binding wave ansatz of the form
$$
|\psi\rangle=\sum_{\mathbf{R}} e^{i \mathbf{k} \cdot \mathbf{R}}|\mathbf{R}\rangle
$$
where the sum is over the points $\mathbf{R}$ of a lattice, and |R) is the ground-state wavefunction of an electron bound to a nucleus on site $\mathbf{R}$. In real space this ansatz can be expressed as
$$
\psi(\mathbf{r})=\sum_{\mathbf{R}} e^{i \mathbf{k} \cdot \mathbf{R}} \varphi(\mathbf{r}-\mathbf{R})
$$
Show that this wavefunction is of the form required by Bloch's theorem (i.e., show it is a modified plane wave).

Eduard Sanchez

Numerade Educator

05:43

Problem 4

*Nearly Free Electrons in Two Dimensions Consider the nearly free electron model for a square lattice with lattice constant $a$. Suppose the periodic potential is given by
$$
\begin{aligned}
V(x, y) &=2 V_{10}[\cos (2 \pi x / a)+\cos (2 \pi y / a)] \\
&+4 V_{11}[\cos (2 \pi x / a) \cos (2 \pi y / a)]
\end{aligned}
$$
(a) Use the nearly free electron model to find the energies of states at wavevector $\mathbf{G}=(\pi / a, 0)$.
(b) Calculate the energies of the states at wavevector $\mathbf{G}=(\pi / a, \pi / a)$. (Hint: You should write down a 4 by 4 secular determinant, which looks difficult, but actually factors nicely. Make use of adding together rows or columns of the determinant before trying to evaluate it!)

Ameer Said

Numerade Educator

03:11

Problem 5

Decaying Waves
As we saw in this chapter, in one dimension, a periodic potential opens a band gap such that there are no plane-wave eigenstates between energies $\epsilon_{0}(G / 2)-\left|V_{G}\right|$ and $\epsilon_{0}(G / 2)+\left|V_{G}\right|$ with $G$ a reciprocal lattice vector. However, at these forbidden energies, decaying (evanescent) waves still exist. Assume the form
$$
\psi(x)=e^{i k x-\kappa x}
$$
with $0<\kappa \ll k$ and $\kappa$ real. Find $\kappa$ as a function of energy for $k=G / 2$. For what range of $V_{C}$ and $E$ is your result valid?

Ameer Said

Numerade Educator

11:38

Problem 6

Kronig-Penney Model*
Consider electrons of mass $m$ in a so-called "deltafunction comb" potential in one dimension
$$
V(x)=a U \sum_{n} \delta(x-n a)
$$
(a) Argue using the Schroedinger equation that inbetween delta functions, an eigenstate of energy $E$
is always of a plane wave form $e^{i q E x}$ with
$$
q_{E}=\sqrt{2 m E} / \hbar .
$$
Using Bloch's theorem conclude that we can write an eigenstate with energy $E$ as
$$
\psi(x)=e^{i k x} u_{E}(x)
$$
where $u_{E}(x)$ is a periodic function defined as
$$
u_{E}(x)=A \sin \left(q_{E} x\right)+B \cos \left(q_{E} x\right) \quad 0<x<a
$$
and $u_{E}(x)=u_{E}(x+a)$ defines $u$ outside of this interval.
(b) Using continuity of the wavefunction at $x=0$ derive
$$
B=e^{-i k a}\left[A \sin \left(q_{E} a\right)+B \cos \left(q_{E} a\right)\right],
$$
and using the Schroedinger equation to fix the discontinuity in slope at $x=0$ derive
$$
q_{E} A-e^{i k a} k\left[A \cos \left(q_{E} a\right)-B \sin \left(q_{E} a\right)\right]=2 \operatorname{maU} B / \hbar^{2}
$$
Solve these two equations to obtain
$$
\cos (k a)=\cos \left(q_{E} a\right)+\frac{m U a}{\hbar^{2} q_{E}} \sin \left(q_{E} a\right)
$$
The left-hand side of this equation is always between $-1$ and 1 , but the right-hand side is not. Conclude that there must be values of $E$ for which there are no solutions of the Schroedinger equation-hence concluding there are gaps in the spectrum.
(c) For small values of the potential $U$ show that this result agrees with the predictions of the nearly free electron model (i.e., determine the size of the gap at the zone boundary).

Ameer Said

Numerade Educator