Advanced Semiconductor Physics (Continued)

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.

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 $e^2/h$.

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).

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

$\blacksquare$

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).

$\blacksquare$

• 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.
• Forgetting spin degeneracy in the 2DEG. Each Landau level holds $eB/h$ states per unit area per spin. With spin degeneracy, each level holds $2eB/h$. The filling factor $\nu = n_{2D}h/(eB)$ counts filled spin-resolved levels.
• Misidentifying the quantum Hall regime. The IQHE requires: (1) high magnetic field so that $\hbar\omega_c \gg k_BT$ (Landau levels are resolved), (2) high mobility so that the scattering time $\tau$ satisfies $\omega_c\tau \gg 1$ (cyclotron orbits complete before scattering), and (3) low temperature to suppress thermal broadening.
• Confusing thermopower with thermal conductivity. The Seebeck coefficient $S$ measures the voltage generated per unit temperature difference; thermal conductivity $\kappa$ measures heat flow. They appear together in the figure of merit $ZT = S^2\sigma T/\kappa$ but are independent transport properties.

• 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.
• IQHE: Hall resistance quantised as $R_{xy} = h/(ne^2)$; filling factor $\nu = n_{2D}h/(eB)$.
• Thermoelectrics: figure of merit $ZT = S^2\sigma T/\kappa$; best materials achieve $ZT > 2$.

These topics form the foundation of modern semiconductor device physics and are essential for understanding electronic, photonic, and energy-harvesting technologies.

The IQHE is used to define the ohm internationally; $R_K$ is exact by definition since 2019.

These topics are closely related: quantum mechanics provides the foundation for band theory, thermal physics governs carrier statistics and thermoelectric performance, and solid state physics provides the crystal structure context. | Solid State Physics — MIT 6.720 | MIT OCW | View |