Chapter 26 Exercises: QM in Condensed Matter
Part A: Conceptual Questions (7 problems)
These questions test your understanding of the core ideas. No calculations required unless stated.
A.1 Explain in your own words why Bloch's theorem holds. What is the essential symmetry that leads to it? Why is it the discrete translation symmetry of the lattice, rather than continuous translation symmetry, that matters?
A.2 A student claims: "Crystal momentum $\hbar k$ is just the ordinary momentum of the electron in the crystal." Correct this misconception. What is crystal momentum? Why is ordinary momentum not conserved in a periodic potential? Give an example of a physical process where the distinction matters.
A.3 Explain the physical origin of band gaps in the nearly free electron model. Why do gaps open specifically at the Brillouin zone boundaries? Relate your answer to Bragg reflection.
A.4 Why does the effective mass $m^*$ become negative near the top of a band? What physical behavior does a negative effective mass correspond to, and why is it conventional to describe these states in terms of "holes" with positive mass and positive charge?
A.5 A crystal has 2 atoms per unit cell, each contributing 3 valence electrons. Is this material a metal, semiconductor, or insulator? Explain your reasoning using band-filling arguments. (Assume no band overlap.)
A.6 Explain why the electrical conductivity of a metal decreases with increasing temperature, while the conductivity of a semiconductor increases with temperature. Both effects are quantum mechanical in origin — what is the key difference?
A.7 In BCS theory, Cooper pairs form between electrons with opposite momenta and opposite spins: $(k\uparrow, -k\downarrow)$. Why is it energetically favorable for the pairs to have zero total momentum? What role does the Pauli exclusion principle play in the formation of Cooper pairs?
Part B: Applied Problems (10 problems)
These problems require direct application of the chapter's key equations.
B.1: Free Electron Bands in Reduced Zone Scheme
For free electrons ($V = 0$) in a 1D lattice with period $a$:
(a) Write down the energy $E(k)$ for the first three bands in the reduced zone scheme ($-\pi/a < k \leq \pi/a$). Express your answers in terms of $E_0 = \hbar^2\pi^2/(2ma^2)$.
(b) At what $k$-values do the first and second bands become degenerate? What is the energy at these degeneracies?
(c) Sketch the band structure in the reduced zone scheme for $-\pi/a < k \leq \pi/a$, labeling the first four bands.
B.2: Nearly Free Electrons — Band Gap
A 1D crystal with lattice constant $a = 3.0\,\text{\AA}$ has a weak periodic potential with first Fourier component $V_1 = 0.5$ eV.
(a) What is the band gap at the first Brillouin zone boundary?
(b) What is the free-electron energy at $k = \pi/a$? (Use $\hbar^2/(2m_e) = 7.62\,\text{eV}\cdot\text{\AA}^2$.)
(c) Write the wavefunctions of the two states at the zone boundary. Which state has higher energy and why?
(d) If the lattice constant is increased to $a = 5.0\,\text{\AA}$ while keeping $V_1$ constant, how does the gap change? How does the free-electron energy at the zone boundary change?
B.3: Kronig-Penney: Delta Function Limit
For the Kronig-Penney model in the delta-function limit (the Dirac comb), the dispersion relation is:
$$\cos(ka) = \cos(\alpha a) + \frac{P}{\alpha a}\sin(\alpha a)$$
where $\alpha = \sqrt{2mE}/\hbar$ and $P$ is a dimensionless barrier strength parameter.
(a) Show that for $P = 0$ (no barriers), the dispersion reduces to the free-electron result $E = \hbar^2 k^2/(2m)$.
(b) Find the energies at the bottom and top of the first band for $P = 3\pi/2$. Express your answers in units of $\hbar^2\pi^2/(2ma^2)$.
(c) Show that in the limit $P \to \infty$, the allowed bands shrink to discrete points and the energies approach $E_n = \hbar^2 n^2\pi^2/(2ma^2)$ — the infinite square well energies for a well of width $a$. Why is this physically expected?
B.4: 1D Tight-Binding Band Structure
A 1D chain of atoms has lattice constant $a = 2.5\,\text{\AA}$, on-site energy $\epsilon_0 = -5.0$ eV, and nearest-neighbor hopping $t = 1.2$ eV.
(a) Write down the band dispersion $E(k)$ and compute the bandwidth.
(b) Compute the effective mass $m^*$ at $k = 0$ (the band bottom). Express it as a multiple of the free electron mass $m_e$.
(c) Compute the group velocity at $k = \pi/(2a)$.
(d) If there is one electron per atom (half-filled band), what is the Fermi energy?
(e) Is this system a metal or an insulator? Explain.
B.5: Tight-Binding with Two Orbitals per Site
Consider a 1D chain where each atom has two relevant orbitals, $|s_n\rangle$ and $|p_n\rangle$, with on-site energies $\epsilon_s = -8$ eV and $\epsilon_p = -3$ eV. The only non-zero hopping parameters are $t_{ss} = 1.0$ eV (s-s hopping) and $t_{pp} = 1.5$ eV (p-p hopping) between nearest neighbors. (Assume $t_{sp} = 0$ for simplicity.)
(a) Write down the two band dispersions $E_s(k)$ and $E_p(k)$.
(b) What is the band gap between the $s$-band and the $p$-band? Does the gap depend on $k$?
(c) If there are 2 electrons per atom, which band(s) are filled? Is this system a metal or insulator?
(d) What happens to the gap if we allow $t_{sp} \neq 0$? (Answer qualitatively — you need not solve the full $2\times 2$ problem.)
B.6: Graphene — Dirac Point Dispersion
Starting from the graphene tight-binding dispersion $E_\pm(\mathbf{k}) = \pm t|f(\mathbf{k})|$:
(a) Show that near the K point, $f(\mathbf{K} + \mathbf{q}) \approx -\frac{3a_{\text{cc}}}{2}(q_x - iq_y)$ for small $|\mathbf{q}|$, where $a_{\text{cc}} = 1.42\,\text{\AA}$ is the carbon-carbon distance. (You will need to expand the exponentials $e^{i(\mathbf{K}+\mathbf{q})\cdot\boldsymbol{\delta}_j}$ to first order in $\mathbf{q}$.)
(b) Hence show that $E_\pm(\mathbf{q}) \approx \pm \hbar v_F |\mathbf{q}|$ with $v_F = 3t a_{\text{cc}}/(2\hbar)$.
(c) Compute $v_F$ numerically using $t = 2.7$ eV. Express it in m/s and as a fraction of $c$.
(d) What is the density of states at the Dirac point? Why is this unusual?
B.7: Semiconductor Physics
Silicon has a band gap $E_g = 1.12$ eV at 300 K. The intrinsic carrier concentration is given by:
$$n_i = \sqrt{N_c N_v}\,\exp\left(-\frac{E_g}{2k_BT}\right)$$
where $N_c = 2.8 \times 10^{19}\,\text{cm}^{-3}$ and $N_v = 1.04 \times 10^{19}\,\text{cm}^{-3}$ are the effective densities of states for the conduction and valence bands.
(a) Compute $n_i$ at 300 K.
(b) Compute $n_i$ at 400 K (assume $E_g$, $N_c$, $N_v$ are approximately temperature-independent for this estimate).
(c) By what factor does $n_i$ change between 300 K and 400 K?
(d) If the silicon is doped with phosphorus at a concentration $N_d = 10^{16}\,\text{cm}^{-3}$, what is the electron concentration at 300 K? (Assume complete ionization.) Compare to $n_i$.
B.8: Effective Mass in a Real Semiconductor
The conduction band minimum of GaAs is at the $\Gamma$ point with effective mass $m^* = 0.067\,m_e$.
(a) For an electron at the conduction band minimum with crystal momentum $\hbar k = 0.01\,\text{\AA}^{-1} \times \hbar$, what is its kinetic energy above the band edge?
(b) Compare this to the kinetic energy a free electron would have with the same momentum.
(c) In a uniform electric field $\mathcal{E} = 1000$ V/m, what is the acceleration of this electron? Compare to the acceleration of a free electron.
(d) Why is the effective mass so much smaller than $m_e$? What does this tell you about the band curvature in GaAs?
B.9: Landau Levels
An electron in a 2D system (effective mass $m^* = 0.067\,m_e$, as in GaAs) is placed in a perpendicular magnetic field $B = 10$ T.
(a) Calculate the cyclotron frequency $\omega_c = eB/m^*$.
(b) Calculate the Landau level spacing $\hbar\omega_c$ in meV.
(c) At what temperature is $k_BT = \hbar\omega_c$? Below this temperature, Landau level quantization dominates.
(d) How many Landau levels are filled if the 2D electron density is $n = 3 \times 10^{11}\,\text{cm}^{-2}$? (Recall: each Landau level holds $eB/h$ electrons per unit area.)
(e) What is the integer quantum Hall resistance for this filling?
B.10: Superconductor Energy Gap
The BCS theory predicts $\Delta_0 = 1.76\,k_BT_c$ for the zero-temperature energy gap.
(a) Compute $\Delta_0$ (in meV) for aluminum ($T_c = 1.18$ K), lead ($T_c = 7.19$ K), and niobium ($T_c = 9.26$ K).
(b) The minimum energy to break a Cooper pair is $2\Delta_0$. What photon wavelength corresponds to this energy for each material? In what part of the electromagnetic spectrum do these wavelengths fall?
(c) Below $T_c$, the electronic specific heat of a superconductor is proportional to $\exp(-\Delta_0/k_BT)$. Explain why this exponential behavior is evidence for a gap.
Part C: Advanced Problems (7 problems)
These problems require synthesis across sections or connections to previous chapters.
C.1: Bloch's Theorem from Fourier Analysis
Starting from the Schrodinger equation for a 1D periodic potential:
$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi(x) = E\psi(x)$$
with $V(x+a) = V(x)$:
(a) Expand $V(x) = \sum_G V_G e^{iGx}$ and $\psi(x) = \sum_q c_q e^{iqx}$. Substitute into the Schrodinger equation and derive the coupled equations for $c_q$.
(b) Show that the equations couple $c_q$ only to $c_{q-G}$ for reciprocal lattice vectors $G$.
(c) Explain why this means the wavefunction $\psi_k(x) = e^{ikx} u_k(x)$ with $u_k$ periodic. (This is the Fourier-analysis proof of Bloch's theorem.)
(d) For a potential with only the $G = \pm 2\pi/a$ Fourier components, write down the coupled equations as a matrix eigenvalue problem. Explain the connection to the nearly free electron model.
C.2: Kronig-Penney — Full Solution
Derive the Kronig-Penney dispersion relation from scratch:
(a) Write the general solution $\psi(x)$ in the well ($0 < x < b$) and barrier ($b < x < a$) regions.
(b) Apply continuity of $\psi$ and $\psi'$ at $x = b$.
(c) Apply Bloch's theorem at $x = 0$ and $x = a$: $\psi(a) = e^{ika}\psi(0)$ and $\psi'(a) = e^{ika}\psi'(0)$.
(d) Derive the transcendental equation: $$\cos(ka) = \cos(\alpha b)\cosh(\beta w) + \frac{\alpha^2 - \beta^2}{2\alpha\beta}\sin(\alpha b)\sinh(\beta w)$$
(e) Take the delta-function limit ($w \to 0$, $V_0 \to \infty$, $V_0 w = \text{const}$) and derive the Dirac comb result.
C.3: Graphene — Sublattice Symmetry and Berry Phase
(a) The graphene Hamiltonian near the K point can be written as $H = \hbar v_F(\sigma_x q_x + \sigma_y q_y) = \hbar v_F \boldsymbol{\sigma} \cdot \mathbf{q}$, where $\sigma_x, \sigma_y$ are Pauli matrices acting on the sublattice degree of freedom. Show that this has the same form as the Dirac Hamiltonian for a massless relativistic particle.
(b) The eigenstates of this Hamiltonian can be written as: $$|\mathbf{q}, \pm\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ \pm e^{i\theta_\mathbf{q}} \end{pmatrix}$$ where $\theta_\mathbf{q}} = \arctan(q_y/q_x)$. Verify that these are eigenstates with eigenvalues $\pm\hbar v_F|\mathbf{q}|$.
(c) Compute the Berry phase acquired by transporting $|\mathbf{q}, +\rangle$ around a closed loop encircling the K point (i.e., $\theta_\mathbf{q}$ goes from 0 to $2\pi$). You should find $\gamma = \pi$.
(d) Explain why this Berry phase of $\pi$ implies that backscattering ($\mathbf{q} \to -\mathbf{q}$) is forbidden for massless Dirac fermions in graphene. (This is called "Klein tunneling" by analogy with relativistic quantum mechanics.)
C.4: Metal-Insulator Transition
Consider a 1D Kronig-Penney model with one electron per unit cell (half-filled band).
(a) For any non-zero barrier strength $V_0 > 0$, show that the Fermi energy lies inside the first band (not in a gap). This system is always a metal in 1D with one electron per cell.
(b) Now consider 2 electrons per unit cell. Show that the Fermi energy lies at the top of the first band — precisely at the gap. Whether this is a metal or insulator depends on whether there is a gap. Explain.
(c) As $V_0$ is increased from zero, the gap opens and grows. At what value of $V_0$ does the system transition from metal to insulator? (Trick question: think carefully about what "metal" means.)
(d) Connect your answer to the general rule: an even number of electrons per unit cell is necessary (but not sufficient) for insulating behavior.
C.5: Perturbation Theory and Band Gaps (Connection to Chapter 17)
The nearly free electron model is an application of degenerate perturbation theory.
(a) Identify: what are the unperturbed states? What is the perturbation? Why is degenerate perturbation theory needed at the zone boundary?
(b) At $k = \pi/a$, the two degenerate free-electron states are $|k\rangle$ and $|k - G\rangle$ with $G = 2\pi/a$. Compute the matrix elements $\langle k|\hat{V}|k\rangle$, $\langle k - G|\hat{V}|k - G\rangle$, and $\langle k|\hat{V}|k - G\rangle$ in terms of the Fourier coefficients $V_G$.
(c) Write down and solve the $2 \times 2$ secular equation. Show that the energy splitting is $2|V_G|$.
(d) Compute the second-order correction to the energy at a general $k$ (away from the zone boundary) using non-degenerate perturbation theory. Show that it is proportional to $|V_G|^2/(E_k^{(0)} - E_{k-G}^{(0)})$ and is small when $k$ is far from the zone boundary.
C.6: Superconducting Energy Gap and Specific Heat
(a) In BCS theory, the density of quasiparticle states above the gap is: $$g_s(E) = g_n(E_F) \frac{E}{\sqrt{E^2 - \Delta^2}} \quad \text{for } E > \Delta$$ and $g_s(E) = 0$ for $E < \Delta$, where $g_n(E_F)$ is the normal-state density of states. Sketch $g_s(E)$.
(b) Show that the number of thermally excited quasiparticles at temperature $T \ll T_c$ is proportional to $\exp(-\Delta/k_BT)$. (Hint: use the Fermi-Dirac distribution and the fact that $\Delta \gg k_BT$.)
(c) Explain why the electronic specific heat in a superconductor is exponentially small at low $T$, in contrast to the linear-$T$ specific heat of a normal metal.
(d) What happens at $T = T_c$? The gap goes to zero — describe the change in the density of states and the specific heat.
C.7: Quantum Hall Effect — Landau Level Counting
(a) In a 2D system of area $A$ in a magnetic field $B$, show that the degeneracy of each Landau level is $N_\phi = BA/\Phi_0$ where $\Phi_0 = h/e$ is the flux quantum.
(b) The filling factor is $\nu = n/(eB/h)$ where $n$ is the 2D electron density. Show that $\nu$ is the number of filled Landau levels (for integer $\nu$).
(c) Show that the Hall resistance $R_{xy} = B/(ne) = h/(\nu e^2)$ when exactly $\nu$ Landau levels are filled.
(d) Explain qualitatively why the Hall resistance remains quantized even when the filling factor is not exactly an integer (i.e., why the plateaus have finite width). Hint: consider the role of disorder and localized states.
(e) The von Klitzing constant is $R_K = h/e^2 = 25{,}812.807\ldots\,\Omega$. If the Hall resistance is measured as $R_{xy} = 12{,}906.40\,\Omega$, what is the filling factor?
Part D: Computational Problems (4 problems)
These problems require numerical computation. Use the code files code/example-01-bands.py and code/project-checkpoint.py as starting points.
D.1: Kronig-Penney Band Structure Explorer
Modify example-01-bands.py to:
(a) Sweep the barrier height $V_0$ from 0 to 100 (natural units) and plot the width of the first three band gaps as a function of $V_0$. Verify that gap width increases with $V_0$.
(b) Sweep the barrier width $w$ from 0 to $a/2$ (keeping $b + w = a$) at fixed $V_0 = 20$ and plot band gap vs $w$. What happens as $w \to 0$? As $w \to a/2$?
(c) For $V_0 = 15$, $b = 0.8$, $w = 0.2$, compute the effective mass at the bottom of the first three bands and compare to the free electron mass.
D.2: 1D Tight-Binding with Second-Neighbor Hopping
Extend the TightBinding1D class to include second-nearest-neighbor hopping $t_2$:
$$E(k) = \epsilon_0 - 2t_1\cos(ka) - 2t_2\cos(2ka)$$
(a) Plot the band structure for $t_1 = 1.0$ eV and $t_2 = 0, 0.2, 0.5$ eV. How does $t_2$ modify the band shape?
(b) For $t_2/t_1 = 0.25$, find the $k$-value at which $E(k)$ is minimized. Is it still at $k = 0$?
(c) Compute and plot the density of states for each case. How does $t_2$ change the Van Hove singularities?
D.3: Graphene — Full Brillouin Zone Visualization
Using the GrapheneBands class:
(a) Create a contour plot of $E_+(\mathbf{k})$ over the hexagonal Brillouin zone. Mark the K, K', M, and $\Gamma$ points.
(b) Compute the density of states $g(E)$ numerically by histogramming the band energies over a fine $k$-grid. Verify that $g(E) \to 0$ linearly at the Dirac point ($E = 0$).
(c) Add a staggered sublattice potential $\Delta$ (so on-site energies are $+\Delta$ on sublattice A, $-\Delta$ on sublattice B). This opens a gap of $2\Delta$ at the Dirac points. Plot the band structure for $\Delta = 0, 0.5, 1.0$ eV and verify the gap opening. This models hexagonal boron nitride (different atoms on the two sublattices).
D.4: Landau Level Visualization
Write a Python script that:
(a) Computes and plots the density of states for a 2D electron gas as a function of energy for $B = 0$ (smooth parabolic DOS: $g(E) \propto \text{const}$) and for $B = 5, 10, 20$ T (discrete Landau levels broadened by a Gaussian of width $\Gamma$).
(b) For $B = 10$ T and $m^* = 0.067\,m_e$, mark the first 5 Landau levels and shade them to indicate degeneracy.
(c) Plot the filling factor $\nu$ as a function of $B$ for a fixed electron density $n = 2.5 \times 10^{11}\,\text{cm}^{-2}$. Mark the fields at which $\nu$ is an integer — these are where the QHE plateaus occur.