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\noindent
\begin{center}
{\bf Problems and exams of the Course \\ {\large Electrons in Crystals} \\ Universit\`a di Trieste}
\end{center}

\vspace{1truecm}
The problems here belong to three distinct classes

\begin{itemize}
\item[(i)] problems given to the students for the final exam or for an
  intermediate exam are  denoted with a * in the list; 
\item[(ii)] Problem solved during tutorials;
\item[(iii)] Longer problems for homework.
\end{itemize}


A typical exam  requires solving two problems in three hours

The student should give all the details of calculation, as well as
motivating the route chosen to solve the problem. 

Problems (questions in a problem) for which only the final results is given
are normally not counted in establishing the score.  

Numerical evaluations should be given with 3 significant figures if not
otherwise indicated. 

Other exercises may be found on the N. Ashcroft e N.~D. Mermin, Solid State
Physics, Saunders-College (1976) [AM] and on the C. Kittel, Introduction to
Solid State Physics, Wiley (1996). 

Here we give  a list of problems from AM: : Chap. {\bf 1}, n. 5; {\bf 2}, 1;
{\bf 4}, 6; {\bf 8}, 1 and 2; {\bf 10}, 1, 2 and  3; {\bf 12}, 1, 2, 6 and  7; {\bf 13}, 1; {\bf 16}, 2 and  3.

Another book with problems is L. Mih\'aly e M.~C. Martin, Solid State Physics:
Problems and Solutions (Wiley, 1996).

\newpage

\tableofcontents

\newpage

\section{\ * Crystalline organic  conductor}

\bigskip

A crystalline organic  conductor has a plasma frequency   $\omega_p=2.50\times
10^{15}s^{-1}$ and a relaxation time  $\tau=5.03\times 10^{-15}$, at room  temperature.

\bigskip

{\bf 1.} Calculate the static electric conductivity $\sigma$ of this
conductor using  the data given above.  Use  c.g.s. units.

{\bf 2.} Express  $\sigma$ in MKSA, that is in (ohm
cm)$^{-1}$.\\ {\it Note that :} One may use: 1 Volt = 1/300 statvolt (cgs
potential unit ) and  1 Ampere = 3$\times$10$^{9}$ statampere (cgs  current unit).

{\bf 3.} Assuming that transport is due to one kind of carrier (electrons),with density $n=7.5\times 10^{20}
cm^{-3}$, obtain the effective mass $m^*$ of the carriers (i) in
grams and  (ii) in units of  the electron mass.

{\bf 4.} Calculate the  $r_s$  parameter for the carriers, using the definition $1/n= 4 \pi r_s^3 a_0^3/3$.

{\bf 5.} Calculate the  Fermi  energy of the carriers in eV.

{\bf 6.} Calculate  the Fermi temperature.

{\bf 7.} Calculate the specific heat  of these carriers treated as free
electrons with  mass $m^*$, assuming that $T\ll T_F$). Give $C_v$ in
$calorie/(mole~^oK)$. Is the assumption  $T\ll T_F$ corrects?

\newpage

\section{\ * Surface plasmon}

\bigskip

Consider an homogeneous metal occupying the region $z>0$, with the remaining
half space $z<0$ being vacuum. Use Drude formula for the dielectric function
of the metal $\varepsilon(\omega)$,  any spacial dependence being neglected.

Consider density waves in the metal with no volume charge density, $\rho(z)=0$,
$z>0$. They will satisfy  Poisson equation 
\begin{equation}\nabla^2 V({\bf r})=0 \label{poisson}
\end{equation} 
both for  $z>0$ and for  $z<0$. Use the notation  ${\bf s}\equiv (x,y)$,
so that  ${\bf r}=({\bf s},z)$.

\bigskip

{\bf 1.} Consider, for $z>0$,   $V({\bf r})= V_0
e^{-\kappa z} cos({\bf Q}\cdot {\bf s})$.  Find which is the relation that there
must be between ${\bf Q}$ and $\kappa$ in order to satisfy Eq. \ref{poisson}.

{\bf 2.} Calculate the  electric field for the potential above. 

{\bf 3.} Consider, for $z<0$, $V'({\bf r})= V_0' e^{\kappa' z} cos({\bf Q}'\cdot
{\bf s})$.  Again, what is t he relation between ${\bf Q}$ and $\kappa$ in order
to satisfy Eq. \ref{poisson}.


{\bf 4.}  Calculate the  electric field for the potential above. 

{\bf 5.} Using the continuity of  the parallel electric field  ${\bf
E}_p\equiv (E_x,E_y,0)$ at $z=0$,  obtain the  relation between  $V_0$ and 
$V_0'$ and  between $\kappa$ e $\kappa'$.

{\bf 6.} Using the continuity of  the transverse displacement  ${\bf
D}_t\equiv (0,0,D_z)$ a $z=0$, obtain a condition on the frequency  $\omega$ of
the waves.

{\bf 7.} For rare collisions, that is $\omega \tau\gg 1$, obtain an explicit
expression of the propagation frequency of the waves in terms of the electron 
average density $n$,  their  mass $m$ and charge $e$.

{\bf 8.} Compute the frequency for aluminum at room conditions. 

\newpage

\section{\ * Plasmon in a  {\sl jellium } sphere at the  density of Sr}

\bigskip

Let us consider a {\sl jellium} sphere, that is (i) a sphere of radius $R$
centered at the origin with a uniform positive charge density [the {\sl
  background}] and (ii) $N$ electrons moving in the electric potential of the
background. The system is globally uncharged (electrically neutral) and the
electrons are distributed with a uniform density on a sphere of radius  $R$
which at rest is also centered at the origin.

Describe the oscillatory motion   [{\sl plasmon}] corresponding to small
rigid displacements of the electronic sphere with respect to the background
sphere, that is of its center with respect to the origin.

\begin{enumerate}

\item Give the charge density of the background, using the parameters and
  informations given above, and denoting the absolute value of the electron
  charge with  $e$.

\item Using  Gauss  theorem, calculate the electric field  due to the
  background both inside and outside the background sphere. 

\item Using the result of the previous point give the electric field
due to the electronic sphere inside it, when centered in  ${\bf d}$.

\item Calculate the force on a test charge internal to both spheres,
  when the  electronic sphere is centered in  ${\bf d}$ and $d< 2R$.

\item Consider now $d \ll R$ and neglect the role of the electrons which fall
  outside of the background sphere.  Using the answer to the previous point
  give the force on an electron internal to both sphere when the displacement
  of the electronic sphere is ${\bf d}$ e write down its equation of motion.

\item Solve the equation above and evaluate (numerically) the frequency of 
oscillation (plasmon frequency) for $ R = 5 \AA $ and  $N=125$.

\item Can the above mode be excited by light any of the following ranges:
infrared, visible, ultraviolet.[Optional question].

\end{enumerate}



\newpage

\section{One-dimensional electron gas}

\bigskip

Consider a system of  N noninteracting point fermions  moving in 1D in [0,L]. Let's
study the problem with each  of the following boundary conditions:
\begin{itemize}
\item[(a)] Born --- von Karman (also know as PBC):
$\psi(x+L)=\psi(x)$
\item[(b)] Hard wall
$\psi(L)=\psi(0)=0$.
\end{itemize}

\noindent

In each of cases  (a) and  (b):

\begin{enumerate}
\item Give eigenfunctions and eigenvalues.

\item Calculate the one body density $n(x)$ giving also a qualitative plot and
  discussing what happens in the thermodynamic limit
  ($N,~L\longrightarrow\infty$; $N/L=n$).

\item Calculate the Fermi energy $E_F(N,L)$ and its behavior in the
  thermodynamic limit. 

\item Calculate the total energy $E_{tot}(N,L)$. Discuss whether it is possible
  to break it  in a  {\it volume} term 
\[E_V(N,L) = N {\cal E}_V\left(\displaystyle{N\over L}\right)\]
and a  {\it surface} term
\[E_S(N,L) = {\cal E}_S\left(\displaystyle{N\over L}\right),\] plus terms that
vanish in the thermodynamic limit. Check if there is a relation between the
total energy and the Fermi energy.
\end{enumerate}
\bigskip

{\bf Suggestion }
\begin{enumerate}
\item It might be useful to remember:
\[\sum_{n=1}^m n^2 = \displaystyle{1\over 6} m(m+1)(2m+1)\]
\[\sum_{n=1}^m a^n = \displaystyle{a-a^{m+1}\over 1-a} \]
\item and  the definition of the  Bessel function:
\[j_0(x) = \displaystyle{ \sin{x}\over x}.\]

\end{enumerate}

\newpage

\section{Boundary conditions and thermodynamic limit}

\bigskip
Let's consider $N$ free  noninteracting electrons in one dimension and moving on
 $0\le x\le L$. The average particle density is  $n=N/L$. In the following for
 simplicity we assume  $N=4M+2$, with  $M$ a positive integer. \\[3mm]

\begin{enumerate}

\item First, obtain the one particle orbitals  $\phi_k(x)$ with periodic
  boundary conditions  $\phi_k(x+L)=\phi_k(x)$. Calculate the allowed $k$'s, then 
  considering that for each $k$ there are two possible spin projections
  calculate:

\begin{itemize}

\item the occupied  $k$'s; 

\item the energy per particle [use the identities below].

\item Discuss what is the  relation between the result of the previous point 
[for $N$ electrons on the segment  $L$] and that for a system in the
thermodynamic limit [i.e., $N\rightarrow \infty$ ed $L\rightarrow\infty$ with $N/L=n$=constant].
\end{itemize}

\item Solve now the problem at (1) with the boundary conditions
  $\phi_k(x+L)=e^{i 2\pi \alpha}\phi_k(x)$, just for the allowed $k$'s and the
  occupied $k$'s.

\begin{itemize}

\item Consider [without attempting a direct calculation -- f is generic] the
  average value of $f(k)$ over the occupied  $k$'s, 
$$
\langle f \rangle =\frac{1}{N}\sum_{k,occup.} f(k).
$$
This quantity depends on $\alpha$ since the allowed $k$ 's do. Thus, $\langle f
\rangle = g(\alpha)$.

\item Show that $\overline{g}$, the average of $g(\alpha)$ over $\alpha$,
$$
\overline{g} = \int_{-1/2}^{1/2} d\alpha \; g(\alpha),
$$
with $N$ finite,   coincides with  $\langle f \rangle$ in the thermodynamic
limit and $ n=N/L$. This is true for any  $f$, not only for the energy.
 [It is suggested to exploit the fact that the the integral over $\alpha$ of
 the $k$-sum is a sum of integrals that covers the interval .....]

\item To be able to obtain the result above it is necessary to be able to write
  $\langle f \rangle$ in the thermodynamic  limit as an integral.

\end{itemize} 

\end{enumerate}

In this simple case, the average on the boundary conditions for  $N$ and  $L$
finite provides a result equivalent to the one for a system in the thermodynamic
 limit and at  density $n=N/L$!

\newpage

\section{He$^3$ film.}

\bigskip

An He$^3$ film is a system in which helium atoms  (1/2 spin Fermions) 
move in a two dimensional word (on a plane). Assume to be close to zero
temperature. The areal  density is $\rho=2.24\times 10^{-9} g/cm^2$ and set the
atomic mass of helium to 3. 

\bigskip 

{\bf 1.} Evaluate the areal density  $n$ in $atoms/cm^2$ for the above film.

{\bf 2.} Calculate $r_s$, the radius  (in units of Bohr radii ) of the disk
that contains one  atom in average, as function of  $n$. Estimate  $r_s$.

{\bf 3.} Calculate the radius $k_F$ of Fermi disk, in terms of $n$ and then
express it as function of  $r_s$.

{\bf 4.} Calculate the Fermi energy  $\epsilon_F$ in terms of $r_s$, in Ry.

{\bf 5.} Evaluate Fermi  energy for the present film in  eV.

{\bf 6.} Evaluate Fermi temperature in  $^o$K.

\newpage 

\section{\ * Simple Metal}

\bigskip

A metal has a specific heat of  1879 erg cm$^{-3}$K$^{-1}$ at
T=3 K. Describe the metal as an electron gas.

\bigskip 

{\bf 1.} Calculate the electronic density $n$ in cm$^{-3}$.

{\bf 2.} Calculate the  Fermi energy in eV.

{\bf 3.} Calculate $\omega_P$ in s$^{-1}$.

{\bf 4.} Is it larger the Fermi energy or the plasmon energy $\hbar\omega_p$?

{\bf 5.} A T=77 K, $\omega_p\tau=1644$. What is the value of the real part of
the dielectric function (order of magnitude).

{\bf 6.} At the same temperature, what is the order of magnitude of the
imaginary part of the  dielectric function. 

\newpage



\section{\ * Hexagonal lattice}

\bigskip

Consider the honeycomb lattice (a collection of hexagonal tiles) with the
lattice points at the vertexes of the hexagons.  One of the hexagon sides is
parallel to the y axis. The lower and  upper vertexes of  such a side are
occupied respectively by atoms of type A and B. 
 
\begin{enumerate}

\item The lattice is a Bravais with a basis: what is the Bravais? Give a pair
 of  primitive vectors Bravais (${\bf a}_1$ ed ${\bf a}_2$) for such a Bravais.

\item Give the basis vectors  (${\bf b}_1$ ed ${\bf b}_2$).

\item Say what is the reciprocal lattice of this Bravais and its primitive
  vectors (${\bf g}_1$ ed ${\bf g}_2$).

\item Calculate the geometrical form factor at the generic vector of the
  reciprocal lattice ${\bf G}=n_1 {\bf g}_1+n_2 {\bf g}_2$ , when  $f_A$ and 
$f_B$ (real numbers)  are the atomic form factors, which here are independent
from the wave vector [we are thinking of neutron diffraction].

\item For which  ${\bf G}$ the geometrical form factor $F({\bf G})$ is real,
  regardless of the value of the ratio  $\alpha=f_A/f_B$?

\item If  $f_B=-f_A$,  are there ${\bf G}$'s for which  $F({\bf G})=0$?
If so say which ones and show them in a sketch if the reciprocal lattice. 

\end{enumerate}

\newpage

\section{\ * Square lattice with a basis}

\bigskip

Consider a square lattice with side (of the square) $a$, and basis $\vec{0}$,
$\vec{ d}=a(1/2,1/2)$. The basis locations are occupied  by atoms with potential
 $\phi_1(r)$ and  $\phi_2(r)$.

\bigskip

{\bf 1.} Write the crystalline potential $U({\bf r})$  as a lattice sum.

{\bf 2.} Write $U({\bf r})$ in terms of its Fourier components.

{\bf 3.} Set $ \phi_1(q) =[\alpha(q)+\beta(q)]/2 $ and  $ \phi_2(q)
=[\alpha(q)-\beta(q)]/2$ and express the  Fourier  components of 
$U({\bf r})$ in terms of  $\alpha$ and $\beta$.

{\bf 4.} What are the simplifications that arise from the assumption above for
the Fourier components $U_{\bf G}$ of the potential. How many subsets of \{${\bf
  G}$\}  is possible to distinguish? 

{\bf 5.} If  $\alpha=0$, in reciprocal space the potential has the symmetry of
which lattice? With which lattice parameter and which axes.

{\bf 6.} As above but for   $\beta=0$.

{\bf 7.} Write the scattering amplitude of such a  lattice setting for the
atomic form factors of the two kinds of atom $ f_1(q) =[\gamma(q)+\delta(q)]/2$
and $ f_2(q) =[\gamma(q)-\delta(q)]/2$. If  $\gamma=0$, which operations
(rotations and translations and in which order) bring the  reciprocal lattice to
a known  Bravais?

\newpage

\section{\ * Electrons in 2 dimensions}

\bigskip

Consider electrons in 2D in a potential $$ U({\bf r}) = -2U\left\{\cos{[g x]}
+2 \cos{\left[\frac{gx}{2}\right]}\cos{\left[\frac{\sqrt{3}}{2} g
y\right]}\right\},$$ con $U>0$.

\bigskip

{\bf 1.} Write $U({\bf r})$ in terms of its Fourier components. How many are
the non vanishing components and at which wavevectors ${\bf G}$? Give the
value of such $U_{\bf G}$?

{\bf 2.} Give a basis for this Bravais. Sketch both the Bravais and its
reciprocal lattice.

{\bf 3.} What's the value of the potential at (i) the origin, (ii) midway
between the origin and a first neighbor, (iii) at any of the farthest points
from the origin within the Wigner-Seitz cell.

{\bf 4.} Assume that there are two electrons in the primitive cell and that
the electrons are independent.  For $U=0$ (free electrons) express $k_F$ and
the energy per electron $\epsilon_{el}$ in terms of $g$.
 
{\bf 5.} Again for $U=0$, build up the ground sate {\it occupying} the
wavevectors in the first Brillouin zone, rather than inside the Fermi
disk. Express the energy per electrons $\epsilon_{rv}$ in terms of $g$.

[Use symmetry to express the energy as a sum over 1/12 of the FBZ (for
instance in the irreducible corner $0<\phi< \pi/6)$.]

{\bf 6.} What are the ratios (i) $k_F/(g/2)$ e (ii)
$\epsilon_{el}/\epsilon_{rv}$?

{\bf 7.} If $g=7.25\AA^{-1}$, what is (in $\AA$) the lattice parameter of the
Bravais?  {\bf 8.} And the value of $\epsilon_{el}$ in Ry?


\newpage


\section{\ * Band filling in a 2D square lattice.}

\bigskip

Consider Schr\"odinger's single particle equation  for an electron moving on a
plane under in a potential  $$U(x,y)=
-2U\left[cos\left(\frac{2\pi}{a}x\right)+
cos\left(\frac{2\pi}{a}y\right)\right].$$

\bigskip

{\bf 1.} First say where are minima and maxima of  $U(x,y)$. Give a sketch of
Bravais formed by all the minima (maxima).

{\bf 2.} Which are the non vanishing  $U_{\bf G}$ and their value? ({\it Hint:}:
write trigonometric functions in terms of exponentials).

{\bf 3.} Sketch the First Brillouin Zone(FBZ).

{\bf 4.} Assume U ``small'' and calculate with first order perturbation theory
the change in the plane waves (energies) at the corners of the FBZ,
$k=(\pi/a)(\pm 1,\pm 1)$. Define as belonging to   first band the solution with
the lowest energy.

{\bf 5.} Repeat the calculation for the points at the center of the sides of the
FBZ. The two solution found for the energy give give the first two bands at
these points. 

{\bf 6.} What is the condition under which the second band at the side center
becomes lower in energy than the first band at the corner of the FBZ?

{\bf 7.} In a system with 2 electrons  in the primitive cell and when the
condition found  immediately above is fulfilled, how many bands are occupied?
Would you be able to sketch the Fermi surface in  the FBZ when  $U\rightarrow 0$?
 
\bigskip

{\bf Equivalent exercise:} change the potential into $$U(x,y)=
-2U\left[cos\left(\frac{2\pi}{a}x\right)
cos\left(\frac{2\pi}{a}y\right)\right].$$

N.B. For the alternative choice of the potential  the coordinates of the corners
will need to be changed from those of the first choice.

\newpage

\section{\ * 1D Tight binding}

\bigskip

Consider electrons moving in 1D. First tackle the problem of an electron in the
potential  $V(x)=-e^2 U \delta(x/a_B)$, ($U>0$), with 
$e$ e $a_B$ the electron charge and the Bohr radius.

\bigskip

{\bf 1.} What are the units of  $U$? 

{\bf 2.} Write Schr\"odinger's equation for the electron in the potential above
in atomic unit ($e=\hbar=m=1$), which you will use throughout the exercise.

{\bf 3.} Find the normalized wavefunction  of the bound state exploiting (i)
the continuity of the wavefunction and  (ii) the know discontinuity of its first
derivative at $x=0$. Are there any other bound states, in addition  to the one
that you found? 

{\bf 4.} Consider now a 1D Bravais with a periodic potential
$v(x)=-U\sum_n\delta(x-n)$ and treat it by tight binding. Using the textbook
definitions for a system with a 1S orbital calculate the  shift $\beta$ as
functions of $U$.

{\bf 5.} calculate the $\alpha(n)$ as function of $U$.

{\bf 6.} Calculate the so-called transfer integral $\gamma(n)$, as function of
$U$.

{\bf 7.} Calculate the effective mass at the band minimum.

\newpage

\section{* Tight binding in a triangular lattice}

\bigskip

Consider a triangular lattice with an atom per site with only an $s$ orbital
and treat the problem within the {\it tight binding} approximation. Neglect
overlap and assume that only first neighbors are important so that only $\beta$
e the first transfer $\gamma$ need be considered, $\beta,\gamma>0$
following the convention in AM. The orbital unperturbed energy is  $-\epsilon_0$,
with  $\epsilon_0>0$.

Use as real space basis  ${\bf a}_1=a(1,0)$, ${\bf
a}_2=a(1/2,\sqrt{3}/2)$ and as reciprocal lattice basis  ${\bf
g}_1=(4\pi/\sqrt{3}a)(\sqrt{3}/2,-1/2)$ and  ${\bf
g}_2=(4\pi/\sqrt{3}a)(0,1)$

\bigskip 

{\bf 1.} Sketch the First Brillouin Zone (FBZ). Use the notation 
$\Gamma=(0,0)$, $A=(4\pi/3a,0)$, $B=(0,2\pi/\sqrt{3}a)$

{\bf 2.} Calculate the energy dispersion $\epsilon({\bf k})$ of the tight
binding energy band and express functions in such a way that only real functions
appear.

{\bf 3.} Sketch the band energy along the segment in k-space $\overline{\Gamma
A}$, specifying the value of the energy at  $\Gamma$ and at $A$ and respecting
the qualitative features of the dispersion at these points.

{\bf 4.} As above but along $\overline{\Gamma B}$.

{\bf 5.} Calculate the elements of the inverse  mass tensor at any of the
points above where it makes sense. 

{\bf 6.} What are the values of the off-diagonal elements of the mass tensor at
the points found in 5)?

{\bf 7.} Say at which of the points found at 5) is found the heaviest
effective mass. 

\newpage

\section{\ * Orbits of  Bloch electrons in a magnetic field}

\bigskip

Consider a solid in the simple cubic structure, with  a single 1s orbital per
site, that has a tight binding band $\epsilon({\bf q}) = -2 \gamma [cos(q_x
a)+cos(q_y a)+cos(q_z a)]$, with $a$  the lattice parameter and suitably chosen
zero of energy. A uniform static magnetic filed  ${\bf H}=H \hat z$ is present.

\bigskip

{\bf 1.} Write the Bloch electron velocity in  the plane (x,y).
 
{\bf 2.} Write the condition satisfied by an orbit (in ${\bf q}$ space) for
$q_z=0$ and energy $E$,  $\epsilon({\bf q}) = E$.

{\bf 3.} What's the orbit when  $E= -2\gamma [ 3-\delta^2/2]$ and
$\delta\ll 1$?

{\bf 4.} Give an expression for the period of the orbit when $\gamma a^2
=\hbar^2/2 m^\ast$?

{\bf 5.} Consider now the case $E=-2\gamma$ and due to the symmetry of the
problem restrict your attention to the region satisfying ($q_x\ge 0$, $q_y\ge
0$). Obtain in this region the relation between  $q_x$ and  $q_y$ on the orbit
and sketch it in the First Brillouin Zone (FBZ). 

{\bf 6.} Indicate the direction of motion along the orbit.

{\bf 7.} Rewrite the velocity in the  (x,y) plane  calculated at 1) in terms of
$q_x$, only exploiting the previous results.

{\bf 8.} Use the previous answer to integrate $q_x(t)$ on one quarter of the
orbit to get $T/4$, with  $T$ the period of the orbit. [{\sl Note:
$\int dx/sin(x)=ln(tan(x/2))$}]. Whats the value of  $T$?

\bigskip
 
{\bf Alternative choice :} Choose the zero of energy in such a way that the
minimum of the band is  -12$\gamma$.

\bigskip

{\bf 1.}  Write the Bloch electron velocity in  the plane (y,z).
 
{\bf 2.}  Write the condition satisfied by an orbit (in ${\bf q}$ space) for
$q_z=0$ and energy $E$,  $\epsilon({\bf q}) = E$.

{\bf 3.} What's the orbit when $E= -4\gamma [ 3-\delta^2/4]$ and $\delta\ll 1$?

{\bf 4.} What's the orbit period in terms of the quantities given so far?

{\bf 5.} What should be $\gamma$, in eV, in order to have the same period as for
a free electron?

{\bf 6.} What the equation of the orbit when $q_z=2\pi/a$ and  $E=-4\gamma[-1+
\delta^2]$, with $\delta\ll 1$ .

{\bf 7.} What's the point of the orbit closest to the origin of the last orbit considered?

{\bf 8.} Are you able to sketch qualitatively the orbit on the face of the FBZ
containing  the orbit?

\newpage

\section{\ * Proprieties of an anisotropic metal.}

\bigskip

Consider a metal, with one valence electron per primitive cell, with a band:
$$
\varepsilon({\bf k}) = \varepsilon_0 + \frac{\hbar^2}{2}
 \left( \frac{k^2_x+k^2_y}{m_t}+\frac{k^2_z}{m_l} \right).
$$

\bigskip

{\bf 1.} Calculate the effective mass tensor${\bf M}$.

{\bf 2.} Calculate  the energy density of states $g(\varepsilon)$.

{\bf 3.} Denoting the electronic number density, in  cm$^{-3}$, with {\it n}, 
 calculate the  Fermi energy density of states.

{\bf 4.} If $n=10^{22}cm^{-3}$ and  $m_l=2m_e=4m_t$, evaluate
(i) the Fermi energy in $^o$K and  (ii) the  density of states at the Fermi
energy in $cm^{-3} eV^{-1}$.

\bigskip

{\bf ~} Apply a magnetic field ${\bf H} = H(\sin\theta,0,\cos\theta)$ to such a
solid.

\bigskip

{\bf 5.} Write and solve the semi classical equation s of motion for Bloch
electrons.

{\bf 6.} Give the expression of the cyclotron effective mass $m^*_c$ that one
can obtain from {\bf 5)} above, imposing that the period has the same form as for free
electrons with $m^*_c$ replacing $m_e$.

\bigskip

{\bf ~} Consider now  $\theta=0$.

\bigskip

{\bf 7.} For such choice of $\theta$ give the  ${\bf k}$-space orbits,exploiting
the energy conservation and  verifying that the results that you obtain is
equivalent to the one at {\bf 5}.

{\bf 8.} Verify that the value of $m^*_c$ found at {\bf 6} equals 
$\frac{\hbar^2}{2 \pi} \frac{\partial A}{\partial \varepsilon}$, where $A$ is 
the area enclosed in the orbit. 

\newpage

\section{\ * Hall effect for an e-h system}

\bigskip

Consider a conductor with electron density $n$ and home density $p$ in an
electric field ${\bf E}=(E_x,0,0)$ and a magnetic filed ${\bf H}= (0,0,H)$, in
a Hall geometry. Holes and electrons have masses $m_e$ e $m_b$ and collision
times $\tau_e$ e $\tau_b$. Let us set $b=(\tau_e/m_e)/(\tau_b/m_b)$.

\bigskip 

{\bf 1.} Write the evolution equation of the average electron and hole momenta at stationarity, for each Cartesian component. 

{\bf 2.} As we have an open circuit along  $y$,
$j_y=j_y^{(e)}+j_y^{(b)}=0$. Use this condition to obtain two equations that contain only  $j_x^{(e)}$ and $j_x^{(b)}$.

{\bf 3.} Derive a relation between  $j_x^{(e)}$ and $j_x^{(b)}$
when $n=p$.

{\bf 4.} Calculate the  Hall coefficient $E_y/(j_x H)$ when $n=p$.

{\bf 5.} What's the value of  $E_y$ when the mobilities of electrons and hole are equal, ($\mu =e \tau/m$), and  e $n=p$?

\newpage


\newpage

\section{* Conductivity tensor in a metal}

\bigskip
Consider the expression for the conductivity tensor in the relaxation time approximation
$$
\sigma_{ij} = e^2 \int \frac{d {\bf k}}{4 \pi^3} \tau(\epsilon({\bf
k}))\left[\frac{-\partial g^o}{\partial \epsilon({\bf k})}\right]
v_i({\bf k}) v_j({\bf k})
$$
for a metal. Assume that there is only one band crossed by the Fermi surface.

\bigskip

{\bf 1.} Assume  $T_F\gg T$ (and that the  Fermi energy is far from the band extrema ). Take the  $\tau$ out of the integration on the basis that it is dependent on the energy. Say why you can do it.

{\bf 2.} Exploiting the fact that the velocity is the gradient of the energy
rewrite 'Eq. 1 so as to display the {\bf k } gradient of $g^o(\epsilon({\bf
  k}))$.

{\bf 3.}  Integrate by part the previous expression, taking into account the fact that all the function appearing in the integral are periodic and that the  integral is over the whole Brillouin zone.

{\bf 4.} Rearrange the previous expression in such a way that the inverse mass tensor $\left[\frac{1}{m({\bf
k})}\right]_{ij}$ appears.

{\bf 5.} Assuming that you are close to an extremum and that  $m_{ij}({\bf
k})=m_{ij}$ is  independent from   ${\bf k}$ over the integration region, calculate the integral to obtain  $\sigma_{ij}$ in terms of 
$n$ (the carrier density in the band ), $\tau$, and the mass  tensor.

{\bf 6.} If the mass tensor is  diagonal, $m_{ij}=\delta_{ij}m^*$, what is
the form of the conductivity  tensor?

{\bf 7.} Utilizing the expression obtained in  4) and the fact that the mass tensor is the gradient of a function periodic in {\bf k} rewrite the conductivity in terms of the holes rather that electrons.

\newpage

\section{* Linear chain of H  atoms: {\it s}  and  {\it p} bands  within the
  ``tight-binding'' approximation}

\bigskip

Consider a linear chain  of equispaced H atoms along the $z$ axis: 
${\bf R}_n=(0,0,z_n)=(0,0,na)$, $n$=0, $\pm 1$,
$\pm 2$, \dots.  Our aim is to study the bands deriving from the first
degenerate excited states, that is $2s$ and $2p$. It is convenient
to consider $p_x$, $p_y$, and $p_z$ (rather than $p_{0,\pm 1}$),
so that they can be written as $\psi_{2p_x}=x\phi_p(r)$,
$\psi_{2p_y}=y\phi_p(r)$, and $\psi_{2p_z}=z\phi_p(r)$ respectively.
In the ``tight-binding'' method the energies of the 4 bands coming from
these states are given by (neglecting all overlap integrals):
$$
\left |({\cal E}(k) - E_0)\delta_{ij} + \beta_{ij} +\widetilde
\gamma_{ij}(k) \right |=0
$$
where $E_0$ is the energy of the atomic $2s$ and $2p$ levels, and:
\begin{eqnarray*}
\widetilde \gamma_{ij}(k)&=&\sum_n e^{\imath kz_n}
\gamma_{ij}(z_n)\approx \gamma_{ij}(a)e^{\imath ka}+
\gamma_{ij}(-a)e^{-\imath ka} \hfill
~~~~(first-neighbour approx.)\nonumber\\ \gamma_{ij}(z_n)&=&-\int
d{\bf r} \; \psi_i({\bf r}) \psi_j({\bf r-R}_n) \Delta U({\bf r})
\hfill ~~~~~~~~~~~~~~ (``hopping'' integral) \nonumber\\
\beta_{ij}&=&\gamma_{ij}(0)
\end{eqnarray*}

and the indexes $i$, $j$ run over the 4 atomic basis orbitals.

\bigskip

{\bf 1.} Write the expression of the function $\Delta U({\bf r})$
(apply the independent electron approximation, and consider the
crystal potential as due only to the pointlike nuclei).  Discuss the
symmetry of $\Delta U({\bf r})$, and show that the expansion of
$\Delta U({\bf r})$ around the origin and up to third-order terms
included is given by
$$
\Delta U({\bf r}) \approx c_0 + c_1(x^2+y^2) + c_2 z^2.
$$

{\bf 2.} Show that the matrix $\beta_{ij}$ is diagonal, determine how
many and which are its independent entries. ({\it N.B.:
the explicit calculation of the integral is not required)}.

{\bf 3.} Show that also the hopping integrals $\gamma_{ij}({\bf R})$
and $\gamma_{ij}(-{\bf R})$ can be expressed in terms of a few
independent entries, that in the tight-binding jargon are called:
$\langle ss\sigma\rangle=\gamma_{ss}(a)$, $\langle
sp\sigma\rangle=\gamma_{sp_z}(a)$, $\langle
pp\sigma\rangle=\gamma_{p_zp_z}(a)$, $\langle
pp\pi\rangle=\gamma_{p_xp_x}(a)$. ({\it N.B.: again, explicit
calculation of integrals is not required})

{\bf 4.} Show that the matrix $\widetilde \gamma_{ij}$ is block-diagonal,
write how many and which are the independent entries, and write them
down explicitly as a function of the above mentioned
$\langle ss\sigma\rangle$, $\langle sp\sigma\rangle$, $\langle
pp\sigma\rangle$, and $\langle pp\pi\rangle$.

{\bf 5.} Write down explicitly the equation that determines the band
dispersion, and discuss its solutions, pointing out their
degeneration, provided there is. Draw a qualitative plot in the first
Brillouin zone. Which parameters determine the band {\it amplitude?}

{\bf 6.} What would change, within the same approximations, if we had
helium atoms instead of hydrogen?

\newpage

\section{Density-density response function for a gas of
  non-interacting electrons}

\bigskip

Consider a set of $N$ non-interacting electrons in a volume
$V=L^d$ in $d$-dimensions, with periodic boundary conditions
(PBC). Applying an external potential $U({\bf r})=\int \frac{d{\bf
q}}{(2\pi)^d}U({\bf q}) e^{i{\bf q}\cdot{\bf r}}$ the electron density
changes from $n({\bf r})=n_0\equiv\frac{N}{V}$ to $n({\bf
r})=n_0+n_1({\bf r})+\cdots$, $n_1$ being linear in $U$.

\begin{enumerate}

\item Compute the density-density linear response function
$\chi(q)$, which links the Fourier transform of the first-order
density change $n_1({\bf r})$ to the Fourier components of
$U$, $$n_1({\bf q})=\chi(q) U({\bf q}).$$ To this end, proceed as
suggested below:

\begin{enumerate}

\item Use (static) perturbation theory to compute the first-order change
$\psi_{\bf k}^{1}({\bf r})$ of the non-interacting electrons 
$$ \psi_{\bf k}({\bf r}) = \psi_{\bf k}^{0}({\bf r}) + \psi_{\bf
k}^{1}({\bf r}) +\cdots.$$ By definition $\psi_{\bf k}^{1}({\bf r})$
is linear in $U$.

\item Compute the density change $n_1({\bf r})$ induced by
the potential $U$ using $$ n({\bf r}) = \sum_{\bf k} f_{\bf k}
\psi_{\bf k}({\bf r})^2 = n_0+n_1({\bf r})+\cdots,$$  $f_{\bf k}$ being
the occupation number of the orbital ${\bf k}$.

\item Compute the Fourier transform of $n_1({\bf r})$, obtained
at the previous point and obtain from it
\begin{eqnarray}
 \chi(q) &=& 
 V^{-1} \sum_{\bf k} f_{\bf k} 
 ( \frac{1}{\epsilon_{\bf k} - \epsilon_{{\bf k}+ {\bf q}} } 
 + \frac{1}{\epsilon_{\bf k} - \epsilon_{{\bf k}- {\bf q}} } ) 
 \nonumber \\ &=&
 V^{-1}\sum_{\bf k} 
 \frac{ f_{ {\bf k}-{\bf q}/2 }- f_{ {\bf k}+{\bf q}/2 } } 
 { \epsilon_{ {\bf k}- {\bf q}/2 } - \epsilon_{ {\bf k}+ {\bf q}/2} }.
 \label{chi_k}
\end{eqnarray} 

\item In doing all the algebraic manipulations in the previous points
do bear in mind that $\psi_{\bf k}^0= \frac{1}{\sqrt{V}}e^{i{\bf
q}\cdot{\bf r}}$. $\epsilon_{\bf k}$ is the energy of the orbital
$\psi_{\bf k}^0$.

\end{enumerate}

\item Specialize the treatment at the case $d=2$ and compute explicitly
$\chi(q)$ for unpolarized electrons ($f_{\bf k} =2$ for $k\leq k_F$)
[Stern function] proceeding as follows:

\begin{enumerate}

\item Use the relation \ref{chi_k} and rewrite the sum over the
discrete ${\bf k}$ compatible with the PBC as an integral in polar coordinates.

\item First compute the angular integral.

\item Then compute the radial integral [i.e., the integral in $k$].

\item You should obtain:
\begin{eqnarray*} 
 \chi(q) &=& - \frac{m}{\pi \hbar^2}, ~~~~~q<2q_F \\
 \chi(q) &=& - \frac{m}{\pi \hbar^2}[1-\sqrt{1-4q_F^2/q^2}] ,
 ~~q>2q_F
\end{eqnarray*}

\end{enumerate}

\end{enumerate}

\bigskip

{\bf Note:} you need to be able to integrate the functions
$\int dt/(1+t^2)$ and $\int dt/(1-t^2)$.

\newpage

\section{\ * Peierls instability.}

\bigskip

Consider a 1D non-interacting electron gas.
The definition of the (density-density) static response function 
in Fourier transform space is $$
\chi_0(q)=\frac{2}{L}\sum_{k}\left[\frac{f(k-q/2)-f(k+q/2)}{\varepsilon(k-q/2)-
\varepsilon(k+q/2)}\right],$$ $\varepsilon(k)$ being the energy level
of the non-interacting electrons, and $f(k)=\theta(k_F-|k|)$ the Fermi
functions at $T=0$.
 
\bigskip  
\begin{enumerate}

\item Compute $\chi_0(q)$ starting from the above formula, writing down
  all relevant passages.

\item Draw a semi-quantitative plot of $\chi_0(q)$ as a function
of $Q=q/k_F$, $Q\ge 0$. Comment on the shape of the plot for
$Q\le 0$, without drawing it.

\item The density response to an external energy field $V$ is
given by $\rho(q)=\chi_0(q)V(q)$. What do you expect it happens if one
applies an arbitrarily small field with $V(2k_F)\ne 0$? What will be
the periodicity of the perturbed system? (Remember that in 1D
$k_F=\pi n_0/2$, $n_0=N/L$.)

\item The kinetic energy per electron of an homogeneous electron gas
in 1D is $t(n_0)=C n_0^2$, $C=(\hbar^2\pi^2)/(24
m)$. Write down $E^{LDA}[n]$, the total energy functional in the Local
Density Approximation (LDA) for an inhomogeneous system.

\item Using the following alternative definition for the response
function of a homogeneous 1D fluid:
$$\chi^{-1}_0(x,y) = -\left[\frac{\delta^2E[n]}{\delta n(x)\delta
n(y)}\right]_{n(x)=n_0},$$ where $\chi^{-1}_0$ is such that
$$\int dz\; \chi_0(x,z)\chi^{-1}_0(z,y) = \delta(x,y),$$ compute
the LDA response function in k-space.

\item Compare the result of the previous point with that of the first,
on the plot done at point 2, and identify in which limit the two
results coincide and why.
 
\end{enumerate}

\bigskip

{\bf Note:} Remember that in a translationally invariant system 
$\chi_0(x,y) = \chi_0(x-y)$.

\newpage

\section{1D chain of rare-gas atoms}

\bigskip

Consider a linear chain of Krypton atoms and assume that they interact
with a pair force of the Lennard-Jones form $\phi(|x|) =
4\epsilon [(\sigma/x)^{12} - (\sigma/x)^{6}]$.

\begin{enumerate}

\item Write the potential energy per particle $u(a)$ when the particles
lie regularly at a spacing of $a$, i.e. when
$x_n=na$. Compute: (i) the equilibrium distance $a_0$ as a function of
$\sigma$ and (ii) the cohesive energy as a function of $\epsilon$.

\item Compute the phonon dispersion relation [i.e. $\omega(q)$ as a
function of $q$], under the hypothesis that interactions beyond
first-neighbours can be neglected. Express explicitly the dispersion
law as a function of $\epsilon, \sigma, M$, where $M$ is the mas of
the rare gas atom.

All the atoms have zero speed at $t=0$. In symbols, if $u(n,t)$ is the
displacement at time $t$ of the n-th atom from the equilibrium
position $na_0$, we are using the initial conditions
\begin{eqnarray*}
 u(n,0) = l
\delta_{n,0},\cr \dot{u}(n,0) = 0, \forall n. 
\end{eqnarray*}

\begin{itemize}

\item[A.] Compute the time evolution of the chain at $t>0$.

\item[B.] Compute at which time $t_1>0$ [in seconds] the 0-th
atoms is to be found again in its equilibrium position [$u(0,t_1)=0$].

\item[C.] Compute the speed [in cm/s] which the 0-th atom has
when it goes through the equilibrium position at $t=t_1$.

\end{itemize}

\end{enumerate}

\bigskip

{\bf Hints}: 

\begin{itemize}

\item The parameters for the Lennard-Jones potential for the rare
gases can be found on Ashcroft-Mermin [AS].

\item Numerical values of the Riemann zeta functions
[$\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$] can be found, for instance, in
the book [HMF]:{\it Handbook of Mathematical Functions}, Edited by
M. Abramowitz and I. A. Stegun (Dover, New York).

\item Remember that the phonons are the normal modes of oscillation of
the chain, in the harmonic approximation. Hence any displacement of
the atoms from the equilibrium position [provided it is sufficiently
small that the harmonic approximation still holds] can be analyzed in
terms of phonons. One has then to find the linear combination of
normal oscillations that satisfies the conditions listed in (1).

\item Remember the identity $\sum_q exp(iqna_0)= N \delta_{n,0}$
where the sum over $q$ is on the 1st BZ and one considers a chain of
$N$ atoms with periodic boundary conditions.

\item Remember the definition of Bessel Function
$$
 J_n(z) = \frac{1}{\pi} \int _0^{\pi}d x \; cos(nx - zsinx).
$$
Zeros and other properties of the Bessel Functions can be found in
the HMF book above mentioned, or in other specific books.

\end{itemize}

\newpage

\section{* Stability of monatomic crystals at $T>0$}

\bigskip

\noindent
Consider a harmonic crystal with one atom per cell, in $D$
dimensions (we are clearly interested especially in $D=1,2,3$.)

\bigskip

{\bf 1.} Show that the phonon density of states at low frequencies
($\omega_s({\bf k})=c_s(\hat{\bf k}) k$ per $\omega<\omega_c$) has
an $g(\omega)= A\omega^{D-1}$ behaviour.
 
[To be more precise one should find $A=[D\Omega_D]/[(2\pi)^D c^D]$,
with $\Omega_D$ the {\it solid angle} in $D$ dimensions and $c$ a
suitable average over angles and polarizations of a function of
$c_s(\hat{\bf k})$: which one?]

{\bf 2.} Consider now the square mean displacement of the atoms from
the equilibrium position, defined as $$ \overline{{\bf u}^2} =
\frac{1}{N}\sum_{\bf R} \langle{\bf u}^2({\bf R})\rangle, $$ with
$\langle \dots \rangle$ the average at a given temperature. Express $
\overline{{\bf u}^2}$ as a sum over the normal-modes frequencies.

[We suggest to use the following:\hfill\break (i) ${\bf u}({\bf
R})=(1/\sqrt{N})\sum_{{\bf k},s} {\bf u}_s({\bf k})e^{i{\bf
k}\cdot{\bf R}}$, with ${\bf u}_s({\bf k})$ {\it normal} coordinates
and ${\bf k} \in {\rm FBZ}$; (ii) ${\bf u}_s({\bf k})\propto
\mbox{\boldmath{$\epsilon$}}_s({\bf k})$, and
$\mbox{\boldmath{$\epsilon$}}_s({\bf
k})=\mbox{\boldmath{$\epsilon$}}_s(-{\bf k})$; (iii) the mean value at
a given temperature of the potential energy of a normal oscillator is
clearly equal to half of the average value of the total energy
$\hbar \omega_s({\bf k}) (n_s({\bf k})+1/2)$.]

{\bf 3.} Rewrite $\overline{{\bf u}^2}$ as an integral over the
frequencies, introducing the density of states.

{\bf 4.} Specialize the previous formula to a finite system, simply by
limiting the integral over the frequencies to $\omega> \omega_t=
2 \pi c /L$, where the cutoff frequency $\omega_t$ is $L=
N^{1/D}/\rho^{1/D}$, with $ \rho$ the atomic density and within
adimensional constants of order 1.

{\bf 5.} Estimate, for $T>0$, the dominating contribution to $\overline{{\bf
u}^2}$ due to the low-frequency modes ($\omega_t<\omega<\omega_c$),
in the regime $\beta\hbar\omega\ll 1$ .

{\bf 6.} What happens to $\overline{{\bf u}^2}$ for $D=2$ when
$N\rightarrow \infty$?

{\bf 7.} As above but for $D=1$.

\newpage

\section{1D chains with long-range interactions: vibration modes and stability}

\bigskip

Consider a system of particles forced to move in 1D [for instance the
$x$ axis] interacting via a pair potential $\phi(|x_n -x_m|; |n-m|)$.

\begin{enumerate}

\item Write the potential energy per particle $u(a)$ when the
particles are arranged in a regular fashion spaced by $a$, that in
when $x_n=na$.

\item Consider small oscillations of the particles around the regular
arrangement described at point 1, that is $x_n=na+\delta_n$ with
$|\delta_n|\ll a$. Write then the total potential energy in powers of
the $\delta_n$, up to the second order included. The zero-order term
of course coincides with $Nu(a)$, if you consider a piece of the
chain with N particles [and Born - von Karman boundary conditions].
 
\begin{itemize}

\item[2.a)] Show that the first-order term is always missing, so that
the configuration $x_n=na$ is always an extreme of the energy for each
$a$.

\item[2.b)] Writhe the dynamical matrix $D(|n-m|)$, determining the
second order term, as a function of the $\phi^{\prime\prime}(|n|a;
|n|)$, where the double prime denotes the second derivative of $\phi$
with respect to its first argument.

\item[2.c)] Using the result of point 2.b) write the dispersion relation
for the small oscillations [harmonic vibrations] as a function of
the $\phi^{\prime\prime}(|n|a; |n|)$, assuming that the particles mass
is $M$; that is, write explicitly $\omega^2(k)$.

\end{itemize}

\item Consider a pair potential of the type
$$ \phi(|x|; |n|) = \frac{C}{|x|^{\alpha}} +\frac{e^2}{|x|} (-1)^n,$$
which includes both repulsive [$C>0$ and let us consider for the
time being $\alpha>0$ so that we have a repulsion that declines with
distance] and Coulombic interactions.

\begin{itemize}

\item[3.a)] Write the dispersion relation in this specific case, using
the notation: $G=\alpha(\alpha+1)C/a^{\alpha+2}$, with $G$ being a
force constant, $\sigma=e^2/(Ga^3)$ an adimensional parameter
measuring the relative importance of the Coulombic interaction with
the repulsive interaction, and $\omega_0^2 =4G/M$. One should obtain
$$\frac{\omega^2(k)}{\omega^2_0} = \sum_{n=1}^{\infty}
\frac{1}{n^{\alpha+2}} sin^2(\frac{kna}{2}) +\sigma
\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3} [1-cos(kna)]$$.

\item[3.b)] Consider the relation obtained in 3.a) at the zone boundary,
i.e. for $k=\pi/a$. To this end introduce the Riemann Zeta function
$\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$. Bear also in mind the identity
$\sum_{n=1}^{\infty} (-1)^n n^{-s}= -\zeta(s)
(1-2^{1-s})$, easily provable separating the sum in even and odd terms
and by summing and subtracting once the even terms.

\item[3.c)] Using the result of 3.b) show that the square of the frequency
becomes negative for $\sigma \geq \sigma_c^{\pi}= 4
\zeta(\alpha+2) /(7 \zeta(3))$. This indicates an instability of the
1D crystal. That is, the configuration assumed as the equilibrium one
is just an extremum and is not a minimum.

\item[3.d)] Consider the dispersion for small $k$, i.e. for
$k\rightarrow 0$. Assuming that $\alpha>1$, compute explicitly the
speed of sound. Show that it becomes imaginary for $\sigma
\geq \sigma_c^0= \zeta(\alpha) /(2 ln 2)$, drawing conclusions
similar to the previous point.

\end{itemize}
 
\item Clearly for $\sigma \geq \sigma_c^{\pi}$ one finds imaginary
frequencies close to the zone boundary and for $\sigma \geq
\sigma_c^0$ also at the zone center. This type of instability, anyway,
concerns crystal deformations at fixed {\it density}, that is at fixed
$a$.

\begin{itemize}
 
\item[4.a)] Try to study the stability with respect to the changes in
$a$, considering the equilibrium energy $u(a)$ of point 1. Show that
the requirement that $u(a)$ is a minumum yields an equilibrium
parameter $a_0 = [\alpha C \zeta(\alpha)/(e^2 ln 2)]^{1/(\alpha-1)}$,
with the additional condition $\alpha>1$. [Compare with the assumption
at 3.d)].

\item[4.b)] Consider small oscillations around the equilibrium points
obtained at (4.a). Then, at $a_0$ there will be one $G_0$ and thus
$\sigma_0 =\zeta(\alpha)/((\alpha+1)\ln 2)$. Show that in this case
the speed of sound is always a real quantity. With reference to point
3.c, try to discuss what happens at the zone boundary for integer
values of $\alpha$, with $\alpha>1$. [If one wishes to attack this
last point it will be necessary to compute numerically $\zeta(n)$,
which can be found tabulated in the book {\it
Handbook of Mathematical Functions}, edit. M. Abramowitz and
I. Stegun, (Dover, NY, ~ 1972).]

\end{itemize} 

\end{enumerate}

\newpage

\section{\ * Kohn anomaly}

\bigskip

Consider a linear chain of identical atoms with a equilibrium spacing
$a$. Let $u(n)$ be the displacement from the equilibrium position
($na$) of the n--th atom. There are springs acting over all atoms, so
that the force on the n--th atom is given by $F(n) =\sum_m
C_m[u(n+m)-u(na)]$.

\bigskip 

{\bf 1.} Write th equation of motion of a generic atom.

{\bf 2.} Show that solutions of the form $e^{i(qna -\omega_q t)}$
satisfy the equations of motion, obtaining an expression for
$\omega_q^2$.

{\bf 3.} Assume that $$C_m=C \frac{sin(mk_0a)}{ma}$$ and compute
$d\omega_q^2/dq$.

{\bf 4.} How much is $d\omega_q^2/dq$ at $q=k_0$?

{\bf 5.} Draw a qualitative plot of $\omega_q$ as a function of $q$
around $q=k_0$.

\newpage

\section{\ * 1D chain with further-neighbours interactions}

\bigskip

Consider the oscillation modes of a linear atomic chain having mass
$M$, in the case in which there exist interactions beyond first neighbours.

\bigskip

{\bf 1.} Write the energy of the chain denoting with $u(na)$ the
displacement from the equilibrium position $na$ of the $n-$th atom and
with $C_m$ the force constant of order $m$, $m\geq 1$.\\ Remember that
the interaction energy between the $n-$th atom and its $m-$th neighbour
is $(1/2)C_m\left[u(na)-u[(n+m)a]\right]^2$.

{\bf 2.} Write the dispersion relation $\omega(k)$.

{\bf 3.} Consider $C_1=C$, with $C_m=0$ for $m>1$, and plot $\tilde
\omega(q)\equiv \omega \sqrt{M/C}/2$, where $q\equiv ka/2$, for $0\leq
q\leq \pi/2$.

{\bf 4.} Consider now $C_1=C$, $C_2=C/4$ with $C_m=0$ for $m>2$ and
plot again $\tilde \omega(q)$. [It is enough to concentrate on the
differences from the case at the point 3 for small $q$ and for
$q=\pi/2$.]

{\bf 5.} Consider now $C_1=C$, $C_2=C/4$, $C_3=-C/5$ with $C_m=0$ for
$m>3$, and plot again $\tilde \omega(q)$. [It is suggested to consider
explicitly the behaviour at small $q$ and the points $q=\pi/6,\pi/3,
\pi/2$.]

\newpage

\section{\ * Square harmonic crystal}

\bigskip

Consider a square crystal wit lattice parameter
$a$. Each atom has mass $M$ and is connected to its first neighbors
via springs having elastic constant $G_x$ in the $x$ direction and
$G_y$ in the $y$ direction.
 
\bigskip

{\bf 1.} Write the potential energy $U$ of the crystal when the atoms
perform small oscillations around the lattice sites.
[The springs have rest length $a$ and the motion is strictly 2D.]

{\bf 2.} Write the equation of motion for a generic atom.

{\bf 3.} Solve the equation obtained in (2) using a plane wave ansatz
and characterize the two modes so obtained.

{\bf 4.} Here and in what follows take $G_y=4G_x$. Plot the two modes:
(i) along the path (in units of $\pi/a$) (0,0) $\rightarrow$ (1,0)
$\rightarrow$ (1,1) $\rightarrow$ (0,1) $\rightarrow$ (0,0) and (ii)
along the path (again in units of $\pi/a$) (0,0) $\rightarrow$
(1,1). [The paths are meant to be straight lines connecting the given
points.]

{\bf 5.} Write the dispersion relations for the two modes when
$\sqrt{q_x^2+q_y^2} \ a\ll 1$.

{\bf 6.} Compute the phonon specific heat at low $T$,
in a similar way to the way used in 3D and considering that
$\int_0^{\infty} dx \; x/(e^x-1)=\pi^2/6$.

{\bf 7.} Write the dynamical matrix as a matrix respect to the
Cartesian indexes (x,y) and lattice indexes (m,n). [{\sl Note:}
the Bravais translations are ${\bf R}=(m,n)a$].

\newpage

\section{\ * Effective masses in a semiconductor}

\bigskip

The experimental analysis of a certain semiconductor shows that the
density of states in energy of a specific band has, close to the
band edge ${\bf k}^*$ of that band, the expected square root behaviour
$g(\epsilon) = C \sqrt{\epsilon}$, where $\epsilon=|E -E^*|$ and $C=6.97\times
10^{20}\,cm^{-3}/eV^{3/2}$. In what follows we shall assume that the
laboratory coordinate system coincides with the principal axes system 
$(\hat{x}_1,\, \hat{x}_2,\, \hat{x}_3)$ of the mass tensor at ${\bf k}^*$.

\bigskip

{\bf 1.} How much is the determinant of the mass tensor in units of
$m_e^3$?

{\bf 2.} Performing experiments in a magnetic field one finds that
rotating ${\bf H}$ around the $\hat{x}_3$ axis the cyclotron effective mass
does not change. How many different eigenvalues has, at most, the mass
tensor? Argue your answer in full details.

{\bf 3.} Knowing that for a magnetic field orthogonal to the
$\hat{x}_3$ axis the cyclotron mass is $m^*({\bf H})= 0.253m_e$
compute the eigenvalues of the mass tensor (in units of $m_e$).

{\bf 4.} Do you have an idea on which semiconductor could it be?

{\bf 5.} What is the shape of the orbits for a magnetic field lying
along the $\hat{x}_3$ axis?

{\bf 6.} And for a magnetic field lying along the $\hat{x}_1$ axis?

{\bf 7.} What will be the measured cyclotron frequency (in
cm$^{-1}$ ) for a magnetic field with
$\hat{H}=(1/\sqrt{2},0,1/\sqrt{2})$?

\newpage

\section{\ * Divalent impurities}

\bigskip

Consider a semiconductor with a dielectric constant
$\epsilon=12.5$ and conduction effective mass $m=0.067m_e$, doped with
few divalent donors, that can thus be regarded as
independent. Remember that the total binding energy of the
$He$ atom is $E=5.81Ry$.

\bigskip

\begin{enumerate}

\item Write the expression and the numerical value (in eV) of the
binding energy $\varepsilon_1$, respect to the conduction band for a
single electron on the donor.

\item Write the expression and the numerical value (in eV) of the
binding energy $\varepsilon_2$, after the addition of a second
electron to the donor.

\item Estimate the effective Bohr radius (in $\AA$) of the impurity
with one electron and judge if the ``hydrogenic'' approximation is reasonable.
Do you expect that with two electrons the approximation gets better or worse?

\item Compute the average occupation of these (independent) donor
centers computing a suitable average. How much is it at $T=0$?

\item Determine the position of the chemical potential at
$T\rightarrow 0$, in the hypothesis that only the mentioned donors are
there.

\item If $N_d$ is the density of the donor dopants and one adds
(monovalent) acceptor dopants as well with density $N_a=N_d$,
where will the chemical potential will go as $T\rightarrow 0$?

\end{enumerate}

\newpage

\section{\ * Impurity-induced conductivity}

\bigskip

Consider the crystalline semiconductor GaAs (gallium arsenide).
This is a direct-gap semiconductor ($E_g\simeq 1.52\ eV$ at room
$T$: neglect the dependence of $E_g$ on $T$). The conduction band has
a measured effective mass of $m_c=0.068m_e$ and the valence band of
$m_v=0.41m_e$. The measured dielectric constant is $\epsilon=14.6$.

Assume that the semiconductor is doped with donor and assume that one
can describe the state of the excess electrons thus introduced using
the {\it hydrogenic} model.

\bigskip

{\bf 1.} Compute the binding energy $E_d$ (in eV) of the excess
electrons with respect to the conduction band.

{\bf 2.} Compute the effective Bohr radius $a_B^*$ (in $\AA$) of such
electrons, assuming that each one is in the {\it hydrogenic} ground
state centered on a donor.

{\bf 3.} Is the value computed for $a_B^*$ compatible with the
hydrogenic model used?

{\bf 4.} Assume that when the average distance between the donor
centres becomes comparable with $a_B^*$ the electrons {\it screen}
each other, ceasing to be bound and ending up in the conduction
band. For which critical value $n_{cr}$ of the donor (electron)
density will this happen? Compute $n_{cr}$ in $cm^{-3}$.\\~ [Hint:
write the average donor/electron density as $n_d=[(4\pi/3)a^3]^{-1}$
and take $a$ as a measure of the average distance between
donors. Other possible methods to estimate $n_{cr}$ are not, however,
discouraged.]

{\bf 5.} Estimate the carrier density ($n_c+p_v$, in $cm^{-3}$)
at $T=10K^o$ in the intrinsic (undoped) semiconductor using the data
provided in the introduction.

{\bf 6.} Compare the carrier densities obtained at point 4 and 5 and
write whether at $T= 100K^o$ and for $n_d>n_{cr}$ the doped semiconductor
is in the extrinsic (dopant-dominated) or intrinsic regime.

{\bf 7.} At $T=0$ and for a donor density $n_d>n_{cr}$, is doped $GaAs$
an insulator, a semiconductor or a metal? Why?

\newpage

\section{* Electrons in GaAs/AlGaAs-based quantum wells}

\bigskip

It is possible to make in the laboratory GaAs/AlGaAs-based
heterostructures in which an (almost) 2D electron gas can be created.

It is known that the dielectric constant of these systems is
$\epsilon=12.5$. Moreover, cyclotron resonance experiments using a
field $B=10^4gauss$ reveal a cyclotron frequency $\omega_c=1.76\times
10^{11}s^{-1}$ for the electrons.

One can assume that the said electrons move in a dielectric {\it
continuum}.

\bigskip

{\bf 1.} Compute the electron effective mass $m^*$ in units of
the electron mass $m_e$.

{\bf 2.} How much is the energy per electron $\varepsilon$ and the Fermi
energy $E_F$ (in eV) for such a 2D electron gas if the density is
$n=3.18\times 10^9 cm^{-2}$?

{\bf 3.} How much is $T_F$ in K?

{\bf 4.} Draw a qualitative plot of the Fermi function as a function
of the energy for $0\leq E\leq 0.1 eV$, plotting $f(E)$ as a function
of $E/E_F$.

{\bf 5.} The importance of electron interaction can be measured via
the ratio $\gamma$ between (i) the potential energy of two electrons
at the typical average distance $a= \sqrt{1/(\pi n)}$ and (ii)
the Fermi energy $E_F$. How much is $\gamma$ in this electron gas?

\newpage

\section{\ * Charge transfer in coupled quantum wells}

\bigskip

Consider two strictly 2D symmetrical quantum wells at a distance $d$.

[Each quantum well traps electrons forcing them to move exclusively on
a plane. The first well (A) can be modeled as the plane perpendicular
to the $z$-axis an passing through $z=0$, the second (B) is parallel
to the first and passes through $z=d$. Initially each plane is
electrically neutral with an electron (number) areal density
$N/A=\rho$ neutralized by a uniform positive background with charge
areal density $e\rho$. Note that the substrates are fixed, while the
electrons are free to transfer from a plane to the other.]

Without any loss of generality we shall consider the possibility that
electron transfer from B to A. The electron charge distributions
remain always uniform in each plane.

\bigskip 

{\bf 1.} Write the variation of the electrostatic energy per unit area
$E_H(n,d)$ caused by the transfer of $n \leq \rho$ electron (per unit
area) from B to A. [Hint: the charge transfer leads to a net charge
density $-en$ on plane A and $+ en$ on plane B. Consider the energy
per unit area accumulated in the electric field, or equivalently the
work per unit area needed to separate the two charged planes from
distance 0 to a distance $d$.]

{\bf 2.} In the Hartree-Fock scheme the electrons in each plane
interact with each other with an energy per unit area of the form
$E_I(\rho_I)=C\rho_I^2 -D\rho_I^{3/2}$, I=A, B, with $C= e^2\pi a_B/2$
and $D=e^2 (4/3)\sqrt{2/\pi}$; evidently $\rho_B=\rho-n$ and
$\rho_A=\rho+n$. Write the energies per unit area due to the
interactions in the two planes $E_A(\rho+n)$ and $E_B(\rho-n)$.

{\bf 3.} Write the total energy per unit area
$E_T(\rho,n,d)=E_A+E_B+E_H$

{\bf 4.} Write (i) the change in total energy corresponding to a
charge transfer from B to A $\Delta(\rho,n,d)=E_T(\rho,n,d)
-E_T(\rho,0,d)$ and (ii) consider specifically the case of total
transfer $n=\rho$, namely compute $\Delta_t=\Delta(\rho,\rho,d)$.

{\bf 5.} Are there, at a fixed distance $d$, situations where the
total transfer is energetically favourable (i.e., $\Delta_t <0$)? If
so, write the condition that must be satisfied by the density $\rho$.

{\bf 6.} In case of an affirmative answer at the previous point,
consider the special case in which $d=200\AA$ and the effective Bohr
radius $a_B=100\AA$, and estimate the maximum density $\rho^*$ (in
$cm^{-2}$) at which the transfer can be observed.

{\bf 7.} How much is the radius $r_s^*$ of the Wigner disk
corresponding to the density of the previous point? [$\pi (r_s^*
a_B)^2=\rho^{*-1}$]

\newpage

\section{\ * Properties of molecules with a triplet excited state.}

\bigskip

Consider molecules with a triplet excited state at an energy superior of
$\Delta$ with respect to the singlet ground state. Both states have
zero orbital angular momentum. All other excited states can be
neglected at the temperatures of interest. Imagine a non-interacting
gas of such molecules at temperature $T$.

\bigskip  

\begin{enumerate}
 
\item Write the Helmholtz free energy per molecule of such a gas
in the presence of a magnetic field ${\bf B}=B\hat{z}$. 

\item Compute the average magnetization per molecule.

\item Calculate the molecular magnetic susceptibility.

\item Write the limiting expression for the magnetic susceptibility
when (i) $\beta\Delta\ll 1$ and (ii) $\beta\Delta\gg 1$.

\item Compute the entropy per molecule at $B=0$ and try to draw a
qualitative plot of it as a function of $K_BT/\Delta$.

\item Compute the molecular specific heat and draw a qualitative plot
of it as a function of $K_BT/\Delta$.

\end{enumerate}

\newpage

\section{\ * Magnetism of the electron gas and Stoner condition.}

\bigskip

A 3D homogeneous electron gas is subject to a magnetic field $B$
pointing in the $z$ direction. Consider a stat in which $N_+$
electrons have positive spin component (along $z$) and $N_-$ negative
and define the spin polarization $\zeta=(N_+-N_-)/N$ with
$N=N_++N_-$. The energy of each spin component ($\sigma=\pm$) is made
up by three contributions: (i) kinetic energy $N_{\sigma}(3/5)(\hbar^2
K_{F,\sigma}^2)/(2m)$, (ii) exchange energy $N_{\sigma}(3 e^2
K_{F,\sigma})/(4 \pi)$, and the coupling energy between the spin
magnetic moments and $B$; clearly $K_{F,\sigma}$ is the Fermi
wavevector of the spin component with projection $\sigma$.

\bigskip  

\begin{enumerate}

\item Write the energy of the spin-up electrons.

\item Write the energy of the spin-down electrons.

\item Write the total energy and write down the condition that determines
the equilibrium polarization at constant $N$ and $V$.

\item Specialize the condition found at the previous point to the case
$\zeta \ll 1$ and give an explicit formula the equilibrium $\zeta$.

\item Find for which values of $r_s$, considering $B$ of about $10^4$
gauss, the value of $\zeta$ found at the previous point is indeed
small.
 
\item What is the condition for which the state with $\zeta=0$ becomes
unstable (a maximum for the energy!).

\end{enumerate}

\newpage

\section{Specific heat in a Heisenberg ferromagnet.}

\bigskip

Consider a 3D Heisenberg ferromagnet, in the absence of external
magnetic fields. Assume that the spins have {\it magnitude $S$} and
lie on a BCC lattice with lattice parameter $a_L$. Also assume
{\it only nearest-neighbours} exchange interactions, with
exchange coupling constant $J$.

\begin{enumerate}

\item Consider excitations from the ground state in the form of spin
waves. Find the dispersion relations of the excitation energies for
small values of the wave vector ${\bf k}$, writing explicitly the
excitation energy $\epsilon({\bf k})$ of the spin wave as a function
of $S, a_L, J, k$.

\item { Consider the specific heat per unit volume, at constant
volume, $c_v^M$, due to the magnetic excitations, {\it in the limit of
low temperatures}, assuming that the low-energy excitations
are due to spin waves and have the form found in the previous point.\\
Express $c_v^M$ in terms of $S, a_L, J,
K_B, T$. [{\sl Note: the problem is very similar to that of the
low-temperature specific heat due to lattice vibrations, where one
considers spin waves (or magnons) in place of phonons.}] }

\item Taken as known the lattice specific heat $c_v^L$ at low $T$
[in terms of $K_B, T, \hbar, c$, with $c$ a suitable average
of the speed of sound].\\ Assuming that the crystal considered here is
an insulator and thus that the total specific heat $c_v
=c_v^M+c_v^L$ comes only from the lattice and magnetic contributions,
estimate at which temperature $T^*$ [written in terms of $K_B, \hbar,
c, S, a_L, J$] one has $c_v^M=c_v^L$.
 
\item With reference to point 3, write down the condition that must be
obeyed by the temperature so that the magnetic specific heat dominates
on the lattice specific heat and thus is easily observable.

\end{enumerate}

\bigskip

{\bf Note:} The Riemann zeta function, $\zeta(s) =
\sum_{1,\infty} n^{-s}$, $n$ an integer, can be found tabulated in the
book: Jahnke-Emde-L\"osche, {\it Tables of higher functions},
McGraw-Hill, NY 1960.

\newpage

\section{* Peierls instability in a 1D conductor.}

\bigskip

Consider a system of ions and electrons forced to move along a line.
The ions have unit charge and there is one electron per ion, so that
the system is globally neutral. Assume that the ions lie in the
positions $X_n^0=na$ ant that the electron energy
levels$\epsilon^0(q)$ are known. As a simplification we shall assume
$\epsilon^0(q)=\hbar^2q^2/2m$. In the ground state all and only the
levels with $|q|\leq q_F$ are occupied, with $q_F=\pi/2a$ and the
energy per electron turns out to be $E_e^0=\epsilon_F/3$.

Consider a lattice deformation
\begin{eqnarray*}
X_n&=&X_n^0+u(n),\cr
u(n)&=&\Delta\, cos(2q_Fna)=\Delta\, cos(\pi n)
\end{eqnarray*}
where $\Delta$ is a parameter.

\bigskip

{\bf 1.} What is the periodicity of the deformed lattice?

{\bf 2.} Compute the variation of the ion-ion energy (per ion)
$E_a$, describing the interionic interaction as due to springs
having an elastic constant $C$ (and rest length $a$), connecting
each ion to its first neighbors.

{\bf 3.} The lattice deformation induces an additional electron-ion
potential of the form
\begin{eqnarray*}
U_{ei}&=&2A\Delta cos(2q_F x)=A\Delta(e^{i 2q_Fx} +e^{-i 2q_Fx}).
\end{eqnarray*}
$U_{ei}$ couples states with wavevectors differing by 
$2q_F\equiv G_1$, with an amplitude $U_{G_1}=A\Delta$.

In the two-level scheme, each eigenvalue $\epsilon^0(q)$ originates
two perturbed levels. Clearly at each $q$ the electrons occupy the
lowest level $\epsilon(q)\equiv \epsilon_-(q)$.
Compute the energy per electron in the deformed lattice
\begin{eqnarray*}
E_e=\frac{1}{q_F}\int_0^{q_F} dq \; \epsilon(q).
\end{eqnarray*}
(In order to compute the integral easily it is suggested to change the
integration variable to $y=q_F-q$).

{\bf 4.} Compute the equilibrium $\Delta^*$ by minimizing the total
energy $E(\Delta)=E_a+E_e$ with respect to $\Delta$, namely by
imposing that
\begin{eqnarray*}
\frac{dE(\Delta)}{d\Delta}=0.\cr
\end{eqnarray*}

{\bf 5.} Exploit the expression for $\Delta^*$ to show that
$E(\Delta^*)-E(0)<0$. This shows that the deformed system has lower
energy and hence is the equilibrium state. Is this deformed system a
metal or an insulator?

{\bf 6.} Simplify the expression of $\Delta^*$ assuming that
$C\epsilon_F/A^2$ is much greater than one, so that $\sinh(x) \approx
e^x/2$. Then rewrite $C\epsilon_F/A^2$ as a function of the density of
states in $ q$ at the Fermi level, $N_0= g(\epsilon_F)/2$, and of the
effective electron-electron potential $V_0= A^2a/2C$.  Does the
expression found remind you of something?

{\bf Notes:}

{\bf 1.} Remember that: $\int dt \; \sqrt{1+t^2}= (1/2)[t\sqrt{1+t^2}
+ \mbox{arcsinh}(t)]$, and $(d/dx)\mbox{arcsinh}(x) = 1/\sqrt{1+x^2}.$

{\bf 2.} Remember that in $D$ dimensions the free-electron density of
states is $$g(\epsilon)=\frac{Dn}{2
\epsilon_F}\left[\frac{\epsilon}{\epsilon_F}\right]^{D/2-1},$$ $n$
being the density.

\end {document}