Advanced Semiconductor Physics (Continued)

16.1 Quantum Hall Effect

When a 2D electron gas (2DEG) is placed in a strong perpendicular magnetic field at low temperature, the Hall resistance shows quantised plateaux:

$R_{xy} = \frac{h}{\nu e^2} = \frac{R_K}{\nu}$

Where $\nu = 1, 2, 3, \ldots$ is an integer and $R_K = h/e^2 \approx 25812.8\,\Omega$ is the von Klitzing constant.

Integer Quantum Hall Effect (IQHE) (von Klitzing, 1980):

• Occurs when the filling factor $\nu = n_{2D}h/(eB)$ is an integer
• At these plateaux, the longitudinal resistance $R_{xx} = 0$ (dissipationless transport)
• The quantisation is exact to better than 1 part in $10^{10}$Providing the resistance standard

Fractional Quantum Hall Effect (FQHE) (Tsui, Stormer, Gossard, 1982):

• Plateaux at $\nu = 1/3, 2/3, 2/5, 3/7, \ldots$
• Arises from electron—electron correlations (Laughlin wavefunction)
• Described by Chern—Simons topological field theory

Composite fermions: At $\nu = 1/2$The FQHE electrons bind two flux quanta to become “composite fermions” that see zero effective field. The FQHE of electrons maps to the IQHE of composite fermions, elegantly explaining the observed sequence of fractions.

16.2 Mesoscopic Physics

Mesoscopic systems are intermediate in size between microscopic (atomic) and macroscopic (bulk). Key length scales:

• Phase coherence length $L_\phi$: distance over which the electron maintains phase coherence ( $1$—$10\,\mu$M at low $T$)
• Mean free path $\ell$: distance between elastic scattering events
• Thermal length $L_T = \hbar v_F/(k_BT)$

When the sample size $L < L_\phi$Quantum interference effects become observable:

• Aharonov—Bohm oscillations: Periodic oscillations in magnetoresistance as $B$ varies, with period $\Delta B = \Phi_0/A$ where $A$ is the area enclosed by the paths.
• Weak localisation: Quantum interference of backscattered paths enhances the probability of returning to the origin, increasing the resistance. This is destroyed by a magnetic field (negative magnetoresistance).
• Universal conductance fluctuations: Sample-specific, reproducible fluctuations in conductance of order $e^2/h$.

16.3 Thermoelectric Effects

Seebeck effect: A temperature gradient $\nabla T$ produces an electric field $\mathbf{E} = S\nabla T$ where $S$ is the Seebeck coefficient.

Peltier effect: A current $I$ through a junction produces heat flow $\dot{Q} = \Pi I$ where $\Pi = ST$ (Kelvin relation).

Figure of merit: $ZT = S^2\sigma T/\kappa$ where $\sigma$ is electrical conductivity and $\kappa$ is thermal conductivity.

The Mott formula relates the Seebeck coefficient to the energy derivative of the conductivity:

$S = -\frac{\pi^2 k_B^2 T}{3e}\frac{d\ln\sigma(\epsilon)}{d\epsilon}\bigg|_{\epsilon_F}$

Best thermoelectric materials: Bi$_2$Te$_3$ ($ZT \approx 1$ at 300 K), PbTe ($ZT \approx 1.5$ at 700 K), SnSe ($ZT \approx 2.6$ at 923 K).

Worked Examples

Example 1: Bragg diffraction

Problem. X-rays of wavelength $0.154 \mathrm{ nm}$ are diffracted by a crystal with interplanar spacing $d = 0.2 \mathrm{ nm}$. Find the first-order diffraction angle.

Solution. $2d\sin\theta = n\lambda \implies \sin\theta = \frac{0.154}{2 \times 0.2} = 0.385 \implies \theta = 22.7°$.

$\blacksquare$

Example 2: Band gap

Problem. A semiconductor has a band gap of $1.1 \mathrm{ eV}$. Find the minimum wavelength of light that can excite an electron across the gap.

Solution. {\lambda = \frac{hc}{E_g} = \frac{1240 \mathrm{ eV\cdot} nm}}{1.1 \mathrm{ eV}} = 1127 \mathrm{ nm} (infrared).

$\blacksquare$

Common Pitfalls

• Confusing reciprocal and real space. The reciprocal lattice is the Fourier transform of the real-space lattice; its vectors have dimensions of inverse length. Fix: $\vec{b}_1 = 2\pi \frac{\vec{a}_2 \times \vec{a}_3}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}$.
• Wrong effective mass interpretation. The effective mass $m^*$ can be negative near the top of a band; it reflects the curvature of $E(k)$. Fix: $1/m^* = \frac{1}{\hbar^2}\frac{d^2E}{dk^2}$; negative curvature gives negative effective mass.
• Confusing metals, semiconductors, and insulators. Metals: partially filled band. Semiconductors: small band gap ($\sim 1 \mathrm{ eV}$). Insulators: large band gap ($> 4 \mathrm{ eV}$). Fix: Band gap determines electrical properties; temperature can excite carriers across semiconductor gaps.

Summary

• Crystal structure: Bravais lattices, reciprocal lattice, Miller indices.
• Bragg”s law: $2d\sin\theta = n\lambda$; determines crystal structure from diffraction patterns.
• Band theory: metals (partially filled bands), semiconductors (small gap), insulators (large gap).
• Effective mass: $m^* = \hbar^2/(d^2E/dk^2)$; describes carrier response to external fields.

Cross-References