GATE Physics Solid State Physics Syllabus

Distinguish intraband and interband electronic transitions.

Intraband transitions occur within a single (partly filled) band, require a momentum-changing process (e.g. phonon assistance, as in free-carrier/Drude absorption at low energies). Interband transitions move an electron from one band to another across an energy gap, dominating optical absorption above the threshold $\hbar\omega \geq E_g$.

What distinguishes a direct from an indirect interband transition?

In a direct transition the conduction-band minimum and valence-band maximum occur at the same $\vec{k}$, so a photon (negligible momentum) alone causes the transition. In an indirect transition they are at different $\vec{k}$, requiring a phonon to conserve crystal momentum, making the process weaker.

Write the complex dielectric function and identify the physical meaning of its parts.

$$\varepsilon(\omega) = \varepsilon_1(\omega) + i\,\varepsilon_2(\omega)$$ The real part $\varepsilon_1$ governs dispersion (polarization, refraction); the imaginary part $\varepsilon_2$ governs absorption (energy dissipation). The complex refractive index satisfies $\varepsilon = (n+i\kappa)^{2}$.

Give the dielectric function of a free-electron (Drude) metal and define the plasma frequency.

$$\varepsilon(\omega) = 1 - \frac{\omega_p^{2}}{\omega^{2}}$$ (collisionless limit) with plasma frequency $$\omega_p = \sqrt{\frac{n e^{2}}{\varepsilon_0 m^{*}}}$$ For $\omega < \omega_p$, $\varepsilon < 0$ and the metal reflects; for $\omega > \omega_p$ it becomes transparent.

Define electronic polarizability $\alpha$ and relate it to the induced dipole moment.

Polarizability relates induced dipole moment to local field: $$\vec{p} = \alpha \vec{E}_{\text{local}}$$ The macroscopic polarization is $\vec{P} = N\alpha\vec{E}_{\text{local}}$, where $N$ is the number density of polarizable units.

State the Clausius–Mossotti relation linking dielectric constant to polarizability.

$$\frac{\varepsilon_r - 1}{\varepsilon_r + 2} = \frac{N\alpha}{3\varepsilon_0}$$ where $\varepsilon_r$ is the relative permittivity, $N$ the number density of polarizable units, and $\alpha$ the total polarizability per unit.

List the main contributions to total polarizability and their characteristic frequency response.

Electronic (electron cloud displacement, up to optical/UV frequencies), ionic/atomic (relative ion displacement, up to IR), and orientational/dipolar (alignment of permanent dipoles, up to microwave). Total $\alpha = \alpha_e + \alpha_i + \alpha_d$; higher-frequency mechanisms drop out as frequency increases.

Define a ferroelectric material and its key distinguishing property.

A ferroelectric exhibits a spontaneous electric polarization (below the Curie temperature $T_C$) that can be reversed by an applied electric field, producing a polarization–field hysteresis loop. Example: $\ce{BaTiO3}$.

What characterizes a ferroelectric hysteresis loop? Name the key quantities.

The $P$–$E$ loop shows remanent polarization $P_r$ (polarization at $E=0$), coercive field $E_c$ (field needed to bring $P$ to zero), and saturation polarization $P_s$. It reflects irreversible domain switching.

What happens to a ferroelectric at its Curie temperature $T_C$?

Above $T_C$ the spontaneous polarization vanishes and the material becomes paraelectric. The susceptibility follows a Curie–Weiss law $$\chi = \frac{C}{T - T_C}$$ and the crystal typically transforms to a higher-symmetry (e.g. cubic) phase.

Classify the five main magnetic behaviors by their susceptibility $\chi$ sign and magnitude.

Diamagnetic: $\chi < 0$, small (weak repulsion). Paramagnetic: $\chi > 0$, small. Ferromagnetic: $\chi \gg 0$, large, spontaneous magnetization. Antiferromagnetic: $\chi > 0$, small, peaks at Néel temperature. Ferrimagnetic: $\chi \gg 0$, large net moment from unequal opposing sublattices.

What is the microscopic origin of diamagnetism, and what is the sign of its susceptibility?

Diamagnetism arises from the Lenz-law response of orbital electron motion to an applied field, inducing a moment opposing the field. It gives a small negative susceptibility $\chi < 0$, is present in all materials, and is temperature-independent.

State the Curie law for paramagnetic susceptibility.

$$\chi = \frac{C}{T}, \qquad C = \frac{N \mu_0 \mu_{\text{eff}}^{2}}{3 k_B}$$ where $C$ is the Curie constant, $N$ the moment density, and $\mu_{\text{eff}}$ the effective magnetic moment per atom. Susceptibility decreases as temperature rises.

State the Curie–Weiss law for a ferromagnet above its Curie temperature.

$$\chi = \frac{C}{T - \theta}$$ where $\theta$ (positive, $\approx T_C$) is the Weiss/paramagnetic temperature. The divergence at $T = \theta$ signals the onset of spontaneous ferromagnetic ordering.

Compare the spin arrangements in ferromagnetic, antiferromagnetic, and ferrimagnetic ordering.

Ferromagnetic: neighboring moments parallel, large net magnetization. Antiferromagnetic: adjacent moments antiparallel and equal, zero net magnetization. Ferrimagnetic: adjacent moments antiparallel but unequal (two sublattices), giving a nonzero net magnetization.

What is the Néel temperature and how does antiferromagnetic susceptibility behave around it?

The Néel temperature $T_N$ is where an antiferromagnet orders. As $T$ falls toward $T_N$, $\chi$ rises to a maximum at $T_N$, then decreases below it. Above $T_N$ it follows $\chi = C/(T+\theta)$ with negative intercept.

Why do ferromagnetic materials form magnetic domains?

Domains form to minimize the total energy, chiefly the magnetostatic (demagnetizing) energy of stray external fields. By splitting into regions of differing magnetization direction, the external field energy is reduced, at the cost of domain-wall energy.

What is magnetic anisotropy and what are 'easy' and 'hard' axes?

Magnetocrystalline anisotropy is the dependence of magnetic energy on the orientation of magnetization relative to crystal axes. The 'easy axis' is the direction along which magnetization spontaneously lies (lowest energy); the 'hard axis' requires the most field energy to magnetize.

What is a Bloch domain wall and what determines its width?

A Bloch wall is the transition region between two domains where spins rotate gradually out of the plane. Its width results from a balance between exchange energy (favors wide walls, gradual rotation) and anisotropy energy (favors narrow walls). Wall thickness $\delta \sim \sqrt{J/K}$ where $J$ is exchange and $K$ anisotropy constant.

Distinguish Type-I and Type-II superconductors by their magnetic response.

Type-I: single critical field $H_c$; complete Meissner effect (full flux expulsion) up to $H_c$, then abrupt transition to normal state. Type-II: two critical fields $H_{c1} < H_{c2}$; complete Meissner below $H_{c1}$, then a mixed (vortex) state with partial flux penetration up to $H_{c2}$.

What determines whether a superconductor is Type-I or Type-II in terms of penetration depth and coherence length?

The Ginzburg–Landau parameter $\kappa = \lambda/\xi$ (penetration depth over coherence length). Type-I if $\kappa < 1/\sqrt{2}$ (positive surface energy); Type-II if $\kappa > 1/\sqrt{2}$ (negative surface energy, favoring vortices).

Describe the mixed (vortex) state of a Type-II superconductor.

Between $H_{c1}$ and $H_{c2}$, magnetic flux penetrates as quantized flux tubes (vortices/fluxoids), each carrying one flux quantum $\Phi_0$, surrounded by circulating supercurrents with a normal core. The vortices form a triangular (Abrikosov) lattice; superconductivity coexists with partial flux.

State the Meissner effect.

When a material is cooled below its critical temperature in a magnetic field, it actively expels the magnetic flux from its interior ($\vec{B} = 0$ inside), behaving as a perfect diamagnet. This is a distinct equilibrium property, not merely a consequence of perfect conductivity.

Why is the Meissner effect not simply explained by perfect conductivity ($\rho=0$)?

A perfect conductor would only keep $d\vec{B}/dt = 0$, trapping whatever flux was present when resistance vanished. The Meissner effect requires $\vec{B} = 0$ inside regardless of the field-cooling history, showing superconductivity is a true thermodynamic phase, not just zero resistance.

State the two London equations.

First (acceleration): $$\frac{\partial \vec{J}_s}{\partial t} = \frac{n_s e^{2}}{m}\vec{E}$$ Second (Meissner): $$\nabla \times \vec{J}_s = -\frac{n_s e^{2}}{m}\vec{B}$$ where $n_s$ is the superfluid density. Together they describe zero resistance and flux expulsion.

Define the London penetration depth and give its expression.

The London penetration depth $\lambda_L$ is the characteristic length over which an external magnetic field decays inside a superconductor, $B(x) = B(0)e^{-x/\lambda_L}$. $$\lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^{2}}}$$ where $n_s$ is the superconducting electron density.

What is the central physical idea of BCS theory?

At low temperature, an attractive electron–electron interaction mediated by phonons (lattice vibrations) causes electrons of opposite momentum and spin to bind into Cooper pairs. These bosonic pairs condense into a single coherent quantum ground state, producing superconductivity with an energy gap.

What is a Cooper pair and what is the energy gap in BCS theory?

A Cooper pair is a bound pair of electrons with opposite momenta and spins $(\vec{k}\uparrow, -\vec{k}\downarrow)$ held together by phonon-mediated attraction. BCS predicts an energy gap $2\Delta$ to break a pair, with $$2\Delta(0) \approx 3.52\, k_B T_c$$

What isotope effect does BCS theory predict, and what does it confirm?

BCS predicts $T_c \propto M^{-\alpha}$ with $\alpha \approx 1/2$, i.e. $T_c\sqrt{M} = \text{const}$, where $M$ is the isotopic mass. The dependence of $T_c$ on lattice ion mass confirms that phonons (lattice vibrations) mediate the pairing interaction.

State the result of flux quantization in a superconducting ring and give the flux quantum.

The magnetic flux threading a superconducting ring is quantized in integer multiples of the flux quantum: $$\Phi = n\Phi_0, \qquad \Phi_0 = \frac{h}{2e} \approx 2.07\times10^{-15}\ \text{Wb}$$ The factor $2e$ (not $e$) is direct evidence of electron pairing (Cooper pairs).

What does the factor of $2e$ in the flux quantum $\Phi_0 = h/2e$ reveal about superconductivity?

It shows the charge carriers responsible for supercurrent are Cooper pairs of charge $2e$, not single electrons. Flux quantization arises from requiring the macroscopic pair wavefunction to be single-valued around the ring.