Chapter 26 Quiz: QM in Condensed Matter

Instructions: This quiz covers the core concepts from Chapter 26. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. Bloch's theorem states that the wavefunctions of an electron in a periodic potential can be written as:

(a) $\psi_k(x) = e^{ikx}$ — a pure plane wave (b) $\psi_k(x) = u_k(x)$ — a purely periodic function (c) $\psi_k(x) = e^{ikx} u_k(x)$ where $u_k(x)$ has the periodicity of the lattice (d) $\psi_k(x) = e^{-\kappa x}$ — an exponentially decaying function

Q2. In the nearly free electron model, band gaps open at:

(a) The center of the Brillouin zone ($k = 0$) (b) The Brillouin zone boundaries ($k = \pm n\pi/a$) (c) Random $k$-values determined by the potential strength (d) Every $k$-value simultaneously

Q3. The magnitude of the first band gap in the nearly free electron model is:

(a) $|V_1|$ — the first Fourier component of the potential (b) $2|V_1|$ — twice the first Fourier component (c) $|V_1|^2$ — the square of the first Fourier component (d) $\pi|V_1|/a$ — depends on the lattice constant

Q4. A material with an even number of electrons per unit cell is:

(a) Always a metal (b) Always an insulator (c) Necessarily a semiconductor (d) Possibly a metal (if bands overlap) or possibly an insulator (if they do not)

Q5. In the tight-binding model for a 1D chain with nearest-neighbor hopping $t$ and on-site energy $\epsilon_0$, the bandwidth is:

(a) $t$ (b) $2t$ (c) $4t$ (d) $\pi t$

Q6. The "Dirac cones" in graphene refer to:

(a) The conical shape of the carbon atoms' $sp^2$ orbitals (b) The linear energy-momentum dispersion near the K points of the Brillouin zone (c) The shape of the Brillouin zone itself (d) The conical cross-section of the $\sigma$ bonds

Q7. In a BCS superconductor, electrical resistance is exactly zero because:

(a) Electrons do not scatter off impurities (b) The crystal lattice is perfectly ordered at low temperatures (c) Cooper pairs form a coherent quantum state with an energy gap that prevents scattering into lower-energy states (d) Magnetic fields are expelled, removing all sources of resistance

Q8. The integer quantum Hall effect demonstrates that the Hall resistance is quantized as:

(a) $R_{xy} = h/(ie^2)$ where $i$ is an integer (b) $R_{xy} = ie^2/h$ where $i$ is an integer (c) $R_{xy} = h/(ie)$ where $i$ is an integer (d) $R_{xy} = hie^2$ where $i$ is an integer

Q9. Crystal momentum $\hbar k$ differs from true momentum $\hbar k_{\text{true}}$ in that:

(a) Crystal momentum is always smaller than true momentum (b) Crystal momentum is conserved only modulo a reciprocal lattice vector $\hbar G$ (c) Crystal momentum is a continuous variable while true momentum is quantized (d) Crystal momentum has units of energy while true momentum has units of momentum

Q10. The effective mass of an electron near the top of a band is:

(a) Positive and equal to the free electron mass (b) Positive and smaller than the free electron mass (c) Negative, which is why these states are described as "holes" with positive mass (d) Zero, because the electron is at a band edge

True/False (4 questions)

For each statement, indicate True or False and provide a brief justification (1-2 sentences).

Q11. True or False: In a crystal with one atom per unit cell and one valence electron per atom, the material must be a metal.

Q12. True or False: The distinction between a "semiconductor" and an "insulator" is a sharp, qualitative one based on fundamentally different physics.

Q13. True or False: The Kronig-Penney model can only be applied to 1D systems and has no relevance to real 3D crystals.

Q14. True or False: In BCS theory, Cooper pairs are bosons (total spin 0), so they can all occupy the same quantum state without violating the Pauli exclusion principle.

Short Answer (4 questions)

Q15. Explain why doping silicon with a pentavalent impurity (like phosphorus) dramatically increases its electrical conductivity, even at doping levels of one impurity atom per million silicon atoms. Your answer should reference band structure and the Fermi energy.

Q16. The density of states of a 1D tight-binding band has Van Hove singularities (divergences) at the band edges. Explain physically why the density of states diverges where the band is flat (i.e., where $dE/dk = 0$).

Q17. Explain how the isotope effect in superconductors ($T_c \propto M^{-1/2}$, where $M$ is the isotopic mass) provides evidence that phonons are involved in the pairing mechanism. Why would a purely electronic mechanism not show this isotope dependence?

Q18. The quantum Hall effect is often described as "topologically protected." In simple terms, what does this mean? Why is the quantization of $R_{xy}$ robust against disorder, impurities, and variations in sample geometry?

Applied Scenarios (2 questions)

Q19. A 1D crystal has lattice constant $a = 4.0\,\text{\AA}$ and a periodic potential with first Fourier component $V_1 = 1.5$ eV.

(a) Calculate the first band gap.

(b) Calculate the free-electron kinetic energy at the zone boundary $k = \pi/a$ using $\hbar^2/(2m_e) = 7.62\,\text{eV}\cdot\text{\AA}^2$.

(c) What are the approximate energies of the top of the first band and the bottom of the second band?

(d) If this material has 2 electrons per unit cell (filling the first band completely), and the Fermi energy lies in the middle of the gap, estimate the fraction of electrons thermally excited across the gap at room temperature ($k_BT = 0.026$ eV).

Q20. A 2D electron system in GaAs ($m^* = 0.067\,m_e$) has an electron density $n = 2.0 \times 10^{11}\,\text{cm}^{-2}$ and is placed in a perpendicular magnetic field.

(a) At what magnetic field $B$ are exactly 3 Landau levels filled? (Use $\nu = nh/(eB)$ and set $\nu = 3$.)

(b) What is the quantized Hall resistance at this filling?

(c) What is the Landau level spacing $\hbar\omega_c$ at this field?

(d) Is the quantum Hall effect observable at room temperature for this system? Justify your answer by comparing $\hbar\omega_c$ to $k_BT$.

Answer Key

Q1: (c) — This is the statement of Bloch's theorem.

Q2: (b) — Gaps open at BZ boundaries where free-electron states are degenerate and mix via the periodic potential (Bragg reflection).

Q3: (b) — Degenerate perturbation theory gives a splitting of $2|V_G|$ where $G = 2\pi/a$.

Q4: (d) — An even electron count is necessary but not sufficient for insulating behavior; band overlap can make it metallic.

Q5: (c) — $E(k) = \epsilon_0 - 2t\cos(ka)$ ranges from $\epsilon_0 - 2t$ to $\epsilon_0 + 2t$, bandwidth $= 4t$.

Q6: (b) — The linear $E \propto |\mathbf{q}|$ dispersion near the K points resembles the dispersion of massless Dirac fermions.

Q7: (c) — The energy gap in the BCS state prevents quasiparticle excitations, so there are no states for Cooper pairs to scatter into.

Q8: (a) — $R_{xy} = h/(ie^2)$ with $i = 1, 2, 3, \ldots$ is the von Klitzing quantization.

Q9: (b) — Crystal momentum is conserved modulo $\hbar G$; true momentum is not conserved at all in a periodic potential.

Q10: (c) — The negative curvature $d^2E/dk^2 < 0$ gives $m^* < 0$; the "hole" picture restores a positive effective mass.

Q11: True. With 1 electron per cell and each band holding 2 electrons (spin), the first band is half-filled, guaranteeing a partially filled band and metallic behavior.

Q12: False. The distinction is one of degree, not kind. Both have a band gap; "semiconductor" conventionally means $E_g \lesssim 3$-4 eV. The physics is identical; only the magnitude of $e^{-E_g/2k_BT}$ differs.

Q13: False. While the Kronig-Penney model is a 1D model, it captures the essential physics (Bloch's theorem, band gaps, band widths) that applies in all dimensions. It is a pedagogical tool whose qualitative predictions are valid for real 3D crystals.

Q14: True. Cooper pairs have total spin $S = 0$ (singlet) and total orbital angular momentum that makes the pair effectively bosonic. They undergo Bose-Einstein-like condensation into a single macroscopic quantum state. (Technically, Cooper pairs are not elementary bosons and their condensation is more subtle than BEC, but the bosonic character of the pair is correct.)

Q15: Phosphorus has 5 valence electrons vs. silicon's 4. The extra electron occupies a donor level just below the conduction band edge (~0.045 eV below). At room temperature ($k_BT = 0.026$ eV), the donor is easily ionized, providing a free electron in the conduction band. Even at 1 ppm doping ($\sim 5 \times 10^{16}\,\text{cm}^{-3}$), this vastly exceeds the intrinsic carrier concentration ($\sim 10^{10}\,\text{cm}^{-3}$), increasing conductivity by ~7 orders of magnitude. The Fermi energy shifts from mid-gap toward the conduction band.

Q16: The density of states counts how many $k$-states have energy in a given range: $g(E) \propto 1/|dE/dk|$. Where the band is flat ($dE/dk = 0$), many $k$-states are crammed into a tiny energy interval, so $g(E)$ diverges. Physically, electrons near a band edge have very low group velocity and "pile up" in energy.

Q17: The isotope effect shows $T_c \propto M^{-1/2}$. Phonon frequencies scale as $\omega \propto \sqrt{k/M} \propto M^{-1/2}$ (heavier atoms vibrate more slowly). If the attractive interaction is mediated by phonons, the coupling strength and hence $T_c$ depend on phonon frequencies. A purely electronic mechanism would involve only electron masses and charge, which do not change with isotope substitution, so $T_c$ would be independent of $M$.

Q18: "Topologically protected" means the quantization of $R_{xy}$ arises from a topological invariant (the Chern number), which is an integer by mathematical necessity. Continuous deformations of the system — adding disorder, impurities, or changing the sample geometry — cannot change an integer without closing the energy gap. As long as the Fermi energy remains in the gap between Landau levels, the Chern number is fixed and the Hall conductance remains exactly quantized.

Q19: (a) First band gap = $2|V_1| = 2 \times 1.5 = 3.0$ eV. (b) $E_{\text{free}} = \hbar^2 k^2/(2m) = 7.62 \times (\pi/4.0)^2 = 7.62 \times 0.617 = 4.70$ eV. (c) Top of band 1: $E_{\text{free}} - |V_1| = 4.70 - 1.5 = 3.20$ eV. Bottom of band 2: $E_{\text{free}} + |V_1| = 4.70 + 1.5 = 6.20$ eV. (d) Fraction $\sim \exp(-E_g/2k_BT) = \exp(-3.0/(2 \times 0.026)) = \exp(-57.7) \approx 10^{-25}$. Essentially zero — this is a good insulator.

Q20: (a) $\nu = nh/(eB) = 3 \Rightarrow B = nh/(3e) = (2.0 \times 10^{15}\,\text{m}^{-2})(6.626 \times 10^{-34})/(3 \times 1.602 \times 10^{-19}) = 2.76$ T. (b) $R_{xy} = h/(3e^2) = 25812.8/3 = 8604.3\,\Omega$. (c) $\omega_c = eB/m^* = (1.602 \times 10^{-19})(2.76)/(0.067 \times 9.109 \times 10^{-31}) = 7.24 \times 10^{12}\,\text{rad/s}$. $\hbar\omega_c = 4.77$ meV. (d) $k_BT$ at 300 K $= 25.9$ meV $\gg \hbar\omega_c = 4.77$ meV. Thermal broadening smears out the Landau levels. QHE is NOT observable at room temperature for this system; it requires $T \lesssim \hbar\omega_c/k_B \approx 55$ K.