• # A1.14

(i) Each particle in a system of $N$ identical fermions has a set of energy levels $E_{i}$ with degeneracy $g_{i}$, where $i=1,2, \ldots$. Derive the expression

$\bar{N}_{i}=\frac{g_{i}}{e^{\beta\left(E_{i}-\mu\right)}+1},$

for the mean number of particles $\bar{N}_{i}$ with energy $E_{i}$. Explain the physical significance of the parameters $\beta$ and $\mu$.

(ii) The spatial eigenfunctions of energy for an electron of mass $m$ moving in two dimensions and confined to a square box of side $L$ are

$\psi_{n_{1} n_{2}}(\mathbf{x})=\frac{2}{L} \sin \left(\frac{n_{1} \pi x}{L}\right) \sin \left(\frac{n_{2} \pi y}{L}\right)$

where $n_{i}=1,2, \ldots(i=1,2)$. Calculate the associated energies.

Hence show that when $L$ is large the number of states in energy range $E \rightarrow E+d E$ is

$\frac{m L^{2}}{2 \pi \hbar^{2}} d E$

How is this formula modified when electron spin is taken into account?

The box is filled with $N$ electrons in equilibrium at temperature $T$. Show that the chemical potential $\mu$ is given by

$\mu=\frac{1}{\beta} \log \left(e^{\beta \pi \hbar^{2} \rho / m}-1\right)$

where $\rho$ is the number of particles per unit area in the box.

What is the value of $\mu$ in the limit $T \rightarrow 0$ ?

Calculate the total energy of the lowest state of the system of particles as a function of $N$ and $L$.

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• # A2.14

(i) A simple model of a crystal consists of an infinite linear array of sites equally spaced with separation $b$. The probability amplitude for an electron to be at the $n$-th site is $c_{n}, n=0, \pm 1, \pm 2, \ldots$. The Schrödinger equation for the $\left\{c_{n}\right\}$ is

$E c_{n}=E_{0} c_{n}-A\left(c_{n-1}+c_{n+1}\right)$

where $A$ is real and positive. Show that the allowed energies $E$ of the electron must lie in a band $\left|E-E_{0}\right| \leq 2 A$, and that the dispersion relation for $E$ written in terms of a certain parameter $k$ is given by

$E=E_{0}-2 A \cos k b$

What is the physical interpretation of $E_{0}, A$ and $k$ ?

(ii) Explain briefly the idea of group velocity and show that it is given by

$v=\frac{1}{\hbar} \frac{d E(k)}{d k},$

for an electron of momentum $\hbar k$ and energy $E(k)$.

An electron of charge $q$ confined to one dimension moves in a periodic potential under the influence of an electric field $\mathcal{E}$. Show that the equation of motion for the electron is

$\dot{v}=\frac{q \mathcal{E}}{\hbar^{2}} \frac{d^{2} E}{d k^{2}},$

where $v(t)$ is the group velocity of the electron at time $t$. Explain why

$m^{*}=\hbar^{2}\left(\frac{d^{2} E}{d k^{2}}\right)^{-1}$

can be interpreted as an effective mass.

Show briefly how the absence from a band of an electron of charge $q$ and effective mass $m^{*}<0$ can be interpreted as the presence of a 'hole' carrier of charge $-q$ and effective mass $-m^{*}$.

In the model of Part (i) show that

(a) for $k^{2} \ll 12 / b^{2}$ an electron behaves like a free particle of mass $\hbar^{2} /\left(2 A b^{2}\right)$;

(b) for $(\pi / b-k)^{2} \ll 12 / b^{2}$ a hole behaves like a free particle of mass $\hbar^{2} /\left(2 A b^{2}\right)$.

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• # A4.16

Explain the operation of the $n p$ junction. Your account should include a discussion of the following topics:

(a) the rôle of doping and the fermi-energy;

(b) the rôle of majority and minority carriers;

(c) the contact potential;

(d) the relationship $I(V)$ between the current $I$ flowing through the junction and the external voltage $V$ applied across the junction;

(e) the property of rectification.

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• # A1.14

(i) An electron of mass $m$ and spin $\frac{1}{2}$ moves freely inside a cubical box of side $L$. Verify that the energy eigenstates of the system are $\phi_{l m n}(\mathbf{r}) \chi_{\pm}$where the spatial wavefunction is given by

$\phi_{l m n}(\mathbf{r})=\left(\frac{2}{L}\right)^{3 / 2} \sin \left(\frac{l \pi x}{L}\right) \sin \left(\frac{m \pi y}{L}\right) \sin \left(\frac{n \pi z}{L}\right)$

and

$\chi_{+}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \quad \chi_{-}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$

Give the corresponding energy eigenvalues.

A second electron is inserted into the box. Explain how the Pauli principle determines the structure of the wavefunctions associated with the lowest energy level and the first excited energy level. What are the values of the energy in these two levels and what are the corresponding degeneracies?

(ii) When the side of the box, $L$, is large, the number of eigenstates available to the electron with energy in the range $E \rightarrow E+d E$ is $\rho(E) d E$. Show that

$\rho(E)=\frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3} E}$

A large number, $N$, of electrons are inserted into the box. Explain how the ground state is constructed and define the Fermi energy, $E_{F}$. Show that in the ground state

$N=\frac{2}{3} \frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3}}\left(E_{F}\right)^{3 / 2}$

When a magnetic field $H$ in the $z$-direction is applied to the system, an electron with spin up acquires an additional energy $+\mu H$ and an electron with spin down an energy $-\mu H$, where $-\mu$ is the magnetic moment of the electron and $\mu>0$. Describe, for the case $E_{F}>\mu H$, the structure of the ground state of the system of $N$ electrons in the box and show that

$N=\frac{1}{3} \frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3}}\left(\left(E_{F}+\mu H\right)^{3 / 2}+\left(E_{F}-\mu H\right)^{3 / 2}\right) \text {. }$

Calculate the induced magnetic moment, $M$, of the ground state of the system and show that for a weak magnetic field the magnetic moment is given by

$M \approx \frac{3}{2} N \frac{\mu^{2} H}{E_{F}}$

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• # A2.14

(i) A system of $N$ distinguishable non-interacting particles has energy levels $E_{i}$ with degeneracy $g_{i}, 1 \leqslant i<\infty$, for each particle. Show that in thermal equilibrium the number of particles $N_{i}$ with energy $E_{i}$ is given by

$N_{i}=g_{i} e^{-\beta\left(E_{i}-\mu\right)}$

where $\beta$ and $\mu$ are parameters whose physical significance should be briefly explained.

A gas comprises a set of atoms with non-degenerate energy levels $E_{i}, 1 \leqslant i<\infty$. Assume that the gas is dilute and the motion of the atoms can be neglected. For such a gas the atoms can be treated as distinguishable. Show that when the system is at temperature $T$, the number of atoms $N_{i}$ at level $i$ and the number $N_{j}$ at level $j$ satisfy

$\frac{N_{i}}{N_{j}}=e^{-\left(E_{i}-E_{j}\right) / k T}$

where $k$ is Boltzmann's constant.

(ii) A system of bosons has a set of energy levels $W_{a}$ with degeneracy $f_{a}, 1 \leqslant a<\infty$, for each particle. In thermal equilibrium at temperature $T$ the number $n_{a}$ of particles in level $a$ is

$n_{a}=\frac{f_{a}}{e^{\left(W_{a}-\mu\right) / k T}-1} \text {. }$

What is the value of $\mu$ when the particles are photons?

Given that the density of states $\rho(\omega)$ for photons of frequency $\omega$ in a cubical box of side $L$ is

$\rho(\omega)=L^{3} \frac{\omega^{2}}{\pi^{2} c^{3}}$

where $c$ is the speed of light, show that at temperature $T$ the density of photons in the frequency range $\omega \rightarrow \omega+d \omega$ is $n(\omega) d \omega$ where

$n(\omega)=\frac{\omega^{2}}{\pi^{2} c^{3}} \frac{1}{e^{\hbar \omega / k T}-1}$

Deduce the energy density, $\epsilon(\omega)$, for photons of frequency $\omega$.

The cubical box is occupied by the gas of atoms described in Part (i) in the presence of photons at temperature $T$. Consider the two atomic levels $i$ and $j$ where $E_{i}>E_{j}$ and $E_{i}-E_{j}=\hbar \omega$. The rate of spontaneous photon emission for the transition $i \rightarrow j$ is $A_{i j}$. The rate of absorption is $B_{j i} \epsilon(\omega)$ and the rate of stimulated emission is $B_{i j} \epsilon(\omega)$. Show that the requirement that these processes maintain the atoms and photons in thermal equilibrium yields the relations

$B_{i j}=B_{j i}$

and

$A_{i j}=\left(\frac{\hbar \omega^{3}}{\pi^{2} c^{3}}\right) B_{i j}$

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• # A4.16

Describe the energy band structure available to electrons moving in crystalline materials. How can it be used to explain the properties of crystalline materials that are conductors, insulators and semiconductors?

Where does the Fermi energy lie in an intrinsic semiconductor? Describe the process of doping of semiconductors and explain the difference between $n$-type and $p$-type doping. What is the effect of the doping on the position of the Fermi energy in the two cases?

Why is there a potential difference across a junction of $n$-type and $p$-type semiconductors?

Derive the relation

$I=I_{0}\left(1-e^{-q V / k T}\right)$

between the current, $I$, and the voltage, $V$, across an $n p$ junction, where $I_{0}$ is the total minority current in the semiconductor and $q$ is the charge on the electron, $T$ is the temperature and $k$ is Boltzmann's constant. Your derivation should include an explanation of the terms majority current and minority current.

Why can the $n p$ junction act as a rectifier?

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• # A1.14

(i) A system of $N$ identical non-interacting bosons has energy levels $E_{i}$ with degeneracy $g_{i}, 1 \leq i<\infty$, for each particle. Show that in thermal equilibrium the number of particles $N_{i}$ with energy $E_{i}$ is given by

$N_{i}=\frac{g_{i}}{e^{\beta\left(E_{i}-\mu\right)}-1}$

where $\beta$ and $\mu$ are parameters whose physical significance should be briefly explained.

(ii) A photon moves in a cubical box of side $L$. Assuming periodic boundary conditions, show that, for large $L$, the number of photon states lying in the frequency range $\omega \rightarrow \omega+d \omega$ is $\rho(\omega) d \omega$ where

$\rho(\omega)=L^{3}\left(\frac{\omega^{2}}{\pi^{2} c^{3}}\right)$

If the box is filled with thermal radiation at temperature $T$, show that the number of photons per unit volume in the frequency range $\omega \rightarrow \omega+d \omega$ is $n(\omega) d \omega$ where

$n(\omega)=\left(\frac{\omega^{2}}{\pi^{2} c^{3}}\right) \frac{1}{e^{\hbar \omega / k T}-1} .$

Calculate the energy density $W$ of the thermal radiation. Show that the pressure $P$ exerted on the surface of the box satisfies

$P=\frac{1}{3} W$

[You may use the result $\int_{0}^{\infty} \frac{x^{3} d x}{e^{x}-1}=\frac{\pi^{4}}{15}$.]

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• # A2.14

(i) A simple model of a one-dimensional crystal consists of an infinite array of sites equally spaced with separation $a$. An electron occupies the $n$th site with a probability amplitude $c_{n}$. The time-dependent Schrödinger equation governing these amplitudes is

$i \hbar \frac{d c_{n}}{d t}=E_{0} c_{n}-A\left(c_{n-1}+c_{n+1}\right)$

where $E_{0}$ is the energy of an electron at an isolated site and the amplitude for transition between neighbouring sites is $A>0$. By examining a solution of the form

$c_{n}=e^{i k a n-i E t / \hbar}$

show that $E$, the energy of the electron in the crystal, lies in a band

$E_{0}-2 A \leq E \leq E_{0}+2 A$

Identify the Brillouin zone for this model and explain its significance.

(ii) In the above model the electron is now subject to an electric field $\mathcal{E}$ in the direction of increasing $n$. Given that the charge on the electron is $-e$ write down the norm of the time-dependent Schrödinger equation for the probability amplitudes. Show that it has a solution of the form

$c_{n}=\exp \left\{-\frac{i}{\hbar} \int_{0}^{t} \epsilon\left(t^{\prime}\right) d t^{\prime}+i\left(k-\frac{e \mathcal{E} t}{\hbar}\right) n a\right\}$

where

$\epsilon(t)=E_{0}-2 A \cos \left(\left(k-\frac{e \mathcal{E} t}{\hbar}\right) a\right)$

Explain briefly how to interpret this result and use it to show that the dynamical behaviour of an electron near the bottom of the energy band is the same as that for a free particle in the presence of an electric field with an effective mass $m^{*}=\hbar^{2} /\left(2 A a^{2}\right)$.

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• # A4.16

Explain how the energy band structure for electrons determines the conductivity properties of crystalline materials.

A semiconductor has a conduction band with a lower edge $E_{c}$ and a valence band with an upper edge $E_{v}$. Assuming that the density of states for electrons in the conduction band is

$\rho_{c}(E)=B_{c}\left(E-E_{c}\right)^{\frac{1}{2}}, \quad E>E_{c}$

and in the valence band is

$\rho_{v}(E)=B_{v}\left(E_{v}-E\right)^{\frac{1}{2}}, \quad E

where $B_{c}$ and $B_{v}$ are constants characteristic of the semiconductor, explain why at low temperatures the chemical potential for electrons lies close to the mid-point of the gap between the two bands.

Describe what is meant by the doping of a semiconductor and explain the distinction between $n$-type and $p$-type semiconductors, and discuss the low temperature limit of the chemical potential in both cases. Show that, whatever the degree and type of doping,

$n_{e} n_{p}=B_{c} B_{v}[\Gamma(3 / 2)]^{2}(k T)^{3} e^{-\left(E_{c}-E_{v}\right) / k T}$

where $n_{e}$ is the density of electrons in the conduction band and $n_{p}$ is the density of holes in the valence band.

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• # A1.14

(i) A spinless quantum mechanical particle of mass $m$ moving in two dimensions is confined to a square box with sides of length $L$. Write down the energy eigenfunctions for the particle and the associated energies.

Show that, for large $L$, the number of states in the energy range $E \rightarrow E+d E$ is $\rho(E) d E$, where

$\rho(E)=\frac{m L^{2}}{2 \pi \hbar^{2}}$

(ii) If, instead, the particle is an electron with magnetic moment $\mu$ moving in an external magnetic field, $H$, show that

$\begin{array}{rlr} \rho(E) & =\frac{m L^{2}}{2 \pi \hbar^{2}}, & -\mu H

Let there be $N$ electrons in the box. Explain briefly how to construct the ground state of the system. Let $E_{F}$ be the Fermi energy. Show that when $E_{F}>\mu H$,

$N=\frac{m L^{2}}{\pi \hbar^{2}} E_{F}$

Show also that the magnetic moment, $M$, of the system in the ground state is

$M=\frac{\mu^{2} m L^{2}}{\pi \hbar^{2}} H$

and that the ground state energy is

$\frac{1}{2} \frac{\pi \hbar^{2}}{m L^{2}} N^{2}-\frac{1}{2} M H$

Part II

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• # A2.14

(i) Each particle in a system of $N$ identical fermions has a set of energy levels, $E_{i}$, with degeneracy $g_{i}$, where $1 \leq i<\infty$. Explain why, in thermal equilibrium, the average number of particles with energy $E_{i}$ is

$N_{i}=\frac{g_{i}}{e^{\beta\left(E_{i}-\mu\right)}+1} .$

The physical significance of the parameters $\beta$ and $\mu$ should be made clear.

(ii) A simple model of a crystal consists of a linear array of sites with separation $a$. At the $n$th site an electron may occupy either of two states with probability amplitudes $b_{n}$ and $c_{n}$, respectively. The time-dependent Schrödinger equation governing the amplitudes gives

\begin{aligned} &i \hbar \dot{b}_{n}=E_{0} b_{n}-A\left(b_{n+1}+b_{n-1}+c_{n+1}+c_{n-1}\right), \\ &i \hbar \dot{c}_{n}=E_{1} c_{n}-A\left(b_{n+1}+b_{n-1}+c_{n+1}+c_{n-1}\right) \end{aligned}

where $A>0$.

By examining solutions of the form

$\left(\begin{array}{l} b_{n} \\ c_{n} \end{array}\right)=\left(\begin{array}{l} B \\ C \end{array}\right) e^{i(k n a-E t / \hbar)},$

show that the energies of the electron fall into two bands given by

$E=\frac{1}{2}\left(E_{0}+E_{1}-4 A \cos k a\right) \pm \frac{1}{2} \sqrt{\left(E_{0}-E_{1}\right)^{2}+16 A^{2} \cos ^{2} k a}$

Describe briefly how the energy band structure for electrons in real crystalline materials can be used to explain why they are insulators, conductors or semiconductors.

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• # A4.16

A harmonic oscillator of frequency $\omega$ is in thermal equilibrium with a heat bath at temperature $T$. Show that the mean number of quanta $n$ in the oscillator is

$n=\frac{1}{e^{\hbar \omega / k T}-1} .$

Use this result to show that the density of photons of frequency $\omega$ for cavity radiation at temperature $T$ is

$n(\omega)=\frac{\omega^{2}}{\pi^{2} c^{3}} \frac{1}{e^{\hbar \omega / k T}-1}$

By considering this system in thermal equilibrium with a set of distinguishable atoms, derive formulae for the Einstein $A$ and $B$ coefficients.

Give a brief description of the operation of a laser.

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