CSIR NET Physical Sciences Condensed Matter Physics Syllabus

What is the Madelung constant?

A dimensionless geometric factor $\alpha$ that gives the net electrostatic (Coulomb) energy of an ion in an ionic crystal: $U_{Coulomb} = -\dfrac{\alpha q^{2}}{4\pi\epsilon_{0} r}$. For NaCl, $\alpha \approx 1.748$.

What characterizes covalent bonding and gives examples?

Covalent bonding involves shared electron pairs forming strong, highly directional bonds with saturation. Examples: diamond (C), silicon, and germanium, which adopt the tetrahedral diamond structure.

What is the general form of the interatomic potential combining attraction and repulsion?

$$U(r) = -\frac{A}{r^{m}} + \frac{B}{r^{n}}$$ where the first (attractive) and second (repulsive) terms satisfy $n > m$; equilibrium spacing $r_{0}$ is where $dU/dr = 0$.

How is the equilibrium interatomic separation $r_{0}$ found from the potential $U(r)$, and what condition gives a stable bond?

$r_{0}$ satisfies $\left.\dfrac{dU}{dr}\right|_{r_{0}} = 0$ (zero net force). Stability requires $\left.\dfrac{d^{2}U}{dr^{2}}\right|_{r_{0}} > 0$, i.e. a potential minimum.

Relate the elastic (Young's) modulus to the interatomic potential at equilibrium.

The bond stiffness is the curvature of $U(r)$ at $r_{0}$: $k = \left.\dfrac{d^{2}U}{dr^{2}}\right|_{r_{0}}$, and Young's modulus is approximately $Y \approx \dfrac{k}{r_{0}}$.

Define Young's modulus and bulk modulus.

Young's modulus: $Y = \dfrac{\text{tensile stress}}{\text{tensile strain}} = \dfrac{F/A}{\Delta L/L}$. Bulk modulus: $B = -V\dfrac{dP}{dV}$, the resistance to uniform compression.

What is a phonon?

A quantized normal mode of lattice vibration; a quantum of vibrational energy carrying energy $\hbar\omega$ and crystal momentum $\hbar\vec{q}$. Phonons are bosons obeying Bose–Einstein statistics.

For a 1D monatomic chain (mass $m$, spring constant $C$, spacing $a$), write the phonon dispersion relation.

$$\omega(q) = 2\sqrt{\frac{C}{m}}\,\left|\sin\left(\frac{qa}{2}\right)\right|$$

What are acoustic and optical phonon branches, and when do they arise?

A lattice with $p$ atoms per primitive basis has $3$ acoustic branches ($\omega \to 0$ as $q \to 0$, atoms move in phase) and $3(p-1)$ optical branches (atoms move out of phase, finite $\omega$ at $q=0$). Optical branches require a basis of two or more atoms.

State the Dulong–Petit law for the high-temperature lattice specific heat.

At high temperature the molar lattice heat capacity approaches a constant $C_{V} = 3R \approx 24.9\ \text{J mol}^{-1}\text{K}^{-1}$, independent of temperature.

What is the temperature dependence of the lattice (phonon) specific heat at low temperature in the Debye model?

$$C_{V} \propto T^{3}$$ the Debye $T^{3}$ law, valid for $T \ll \Theta_{D}$ (the Debye temperature).

Define the Debye temperature $\Theta_{D}$.

$\Theta_{D} = \dfrac{\hbar\omega_{D}}{k_{B}}$, where $\omega_{D}$ is the Debye cutoff frequency. It marks the temperature above which all phonon modes are excited and Dulong–Petit behavior sets in.

What are the key assumptions of the free electron (Sommerfeld) theory of metals?

Valence electrons form a free, non-interacting gas in a constant background potential; they obey the Pauli exclusion principle and Fermi–Dirac statistics, filling states up to the Fermi energy.

Write the Fermi energy of a 3D free electron gas in terms of electron density $n$.

$$E_{F} = \frac{\hbar^{2}}{2m}\left(3\pi^{2} n\right)^{2/3}$$

Write the Fermi–Dirac distribution function.

$$f(E) = \frac{1}{e^{(E-\mu)/k_{B}T} + 1}$$ where $\mu$ is the chemical potential ($\approx E_{F}$ at low $T$).

What is the density of states $g(E)$ for a 3D free electron gas?

$$g(E) = \frac{1}{2\pi^{2}}\left(\frac{2m}{\hbar^{2}}\right)^{3/2}\sqrt{E} \propto \sqrt{E}$$

Why is the electronic specific heat much smaller than the classical prediction, and what is its temperature dependence?

Only electrons within $\sim k_{B}T$ of the Fermi surface can be thermally excited (Pauli blocking), so $C_{el} = \gamma T$, linear in $T$, with $\gamma = \dfrac{\pi^{2}}{2}\dfrac{N k_{B}^{2}}{E_{F}}$.

At low temperature, how do the total specific heat contributions of a metal combine?

$$C = \gamma T + \beta T^{3}$$ where $\gamma T$ is the electronic term and $\beta T^{3}$ the Debye lattice term. A plot of $C/T$ versus $T^{2}$ is linear with intercept $\gamma$ and slope $\beta$.

According to band theory, how do energy bands form in a solid?

When $N$ atoms come together, each atomic energy level splits into $N$ closely spaced levels forming a quasi-continuous energy band; allowed bands are separated by forbidden energy gaps arising from the periodic potential (Bragg reflection at zone boundaries).

State Bloch's theorem.

In a periodic potential the electron eigenstates have the form $\psi_{\vec{k}}(\vec{r}) = e^{i\vec{k}\cdot\vec{r}}\,u_{\vec{k}}(\vec{r})$, where $u_{\vec{k}}(\vec{r})$ has the periodicity of the lattice: $u_{\vec{k}}(\vec{r}+\vec{R}) = u_{\vec{k}}(\vec{r})$.

In the Kronig–Penney model, why do energy gaps open at the Brillouin zone boundaries?

At zone boundaries ($k = \pm n\pi/a$) the electron waves satisfy the Bragg condition and form standing waves with different energies (concentrated at ion cores versus between them), opening forbidden gaps in the dispersion.

How does band theory distinguish metals, insulators, and semiconductors?

Metals have a partially filled band (or overlapping bands); insulators have a filled valence band with a large gap ($E_{g} \gtrsim 3$–$4$ eV); semiconductors have a filled valence band with a small gap ($E_{g} \lesssim 1$–$2$ eV) allowing thermal excitation.

Give the approximate band gaps of Si, Ge, and an insulator like diamond.

Si: $E_{g} \approx 1.1$ eV; Ge: $E_{g} \approx 0.67$ eV; diamond (insulator): $E_{g} \approx 5.5$ eV (all at room temperature).

Distinguish intrinsic and extrinsic semiconductors.

Intrinsic: pure semiconductor where $n = p = n_{i}$ from thermal excitation across the gap. Extrinsic: doped semiconductor where impurities dominate carriers — donors (n-type, e.g. P in Si) or acceptors (p-type, e.g. B in Si).

Write the intrinsic carrier concentration dependence on temperature and gap.

$$n_{i} \propto T^{3/2}\, e^{-E_{g}/(2k_{B}T)}$$

What is the law of mass action for semiconductor carrier concentrations?

$$n\,p = n_{i}^{2}$$ independent of doping, where $n$ and $p$ are electron and hole concentrations and $n_{i}$ is the intrinsic concentration at that temperature.

How does electrical conductivity of metals and of intrinsic semiconductors vary with temperature?

Metals: conductivity decreases with increasing $T$ (more phonon scattering). Intrinsic semiconductors: conductivity increases with $T$ because the exponential growth in carriers dominates.

In the Drude model, write the DC electrical conductivity.

$$\sigma = \frac{n e^{2}\tau}{m}$$ where $n$ is electron density, $\tau$ the mean relaxation (collision) time, $e$ the electron charge, and $m$ the electron mass.

Define carrier mobility and relate conductivity to it.

Mobility $\mu = \dfrac{e\tau}{m} = \dfrac{|v_{d}|}{|E|}$ (drift velocity per unit field). Conductivity $\sigma = ne\mu$; for two carrier types $\sigma = e(n\mu_{e} + p\mu_{h})$.

State the Wiedemann–Franz law from the Drude/Sommerfeld model.

$$\frac{\kappa}{\sigma T} = L$$ the Lorenz number, a near-universal constant for metals; the Sommerfeld value is $L = \dfrac{\pi^{2}}{3}\left(\dfrac{k_{B}}{e}\right)^{2} \approx 2.44\times10^{-8}\ \text{W}\,\Omega\,\text{K}^{-2}$.

What is the Meissner effect in superconductors?

A superconductor expels magnetic flux from its interior ($B = 0$ inside) when cooled below $T_{c}$, behaving as a perfect diamagnet ($\chi = -1$). This is distinct from, and stronger than, mere perfect conductivity.

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

Type-I: single critical field $H_{c}$; complete Meissner effect then abrupt loss of superconductivity. Type-II: two critical fields $H_{c1}$ and $H_{c2}$; between them a mixed (vortex) state allows partial flux penetration while remaining superconducting.

What is the mixed (vortex) state of a Type-II superconductor, and what penetrates the material?

Between $H_{c1}$ and $H_{c2}$, magnetic flux enters as quantized flux lines (Abrikosov vortices), each carrying one flux quantum $\Phi_{0} = \dfrac{h}{2e} \approx 2.07\times10^{-15}\ \text{Wb}$, arranged in a flux lattice with normal cores.

What distinguishes Type-I from Type-II superconductors microscopically (coherence length vs penetration depth)?

The ratio $\kappa = \lambda/\xi$ (Ginzburg–Landau parameter) decides: Type-I has $\kappa < 1/\sqrt{2}$ (positive surface energy), Type-II has $\kappa > 1/\sqrt{2}$ (negative surface energy, favoring vortices).

According to BCS theory, what is responsible for superconductivity?

Electrons near the Fermi surface form Cooper pairs (bound pairs of opposite momentum and spin) via an attractive electron–phonon interaction; these pairs condense into a single coherent quantum state with an energy gap $\Delta$.

What is the BCS prediction relating the energy gap at $T=0$ to $T_{c}$?

$$2\Delta(0) = 3.52\, k_{B} T_{c}$$ i.e. $\dfrac{2\Delta(0)}{k_{B}T_{c}} \approx 3.52$.

What is a Josephson junction?

Two superconductors separated by a thin insulating (or weak-link) barrier through which Cooper pairs tunnel coherently, exhibiting the Josephson effect with a supercurrent depending on the phase difference $\phi$ between the two superconductors.

Write the DC and AC Josephson relations.

DC: $I = I_{c}\sin\phi$ (zero-voltage supercurrent). AC: $\dfrac{d\phi}{dt} = \dfrac{2eV}{\hbar}$, so an applied voltage $V$ produces an oscillating current of frequency $\nu = \dfrac{2eV}{h}$.

What is the Josephson frequency-to-voltage ratio, and what device exploits Josephson junctions?

$\dfrac{\nu}{V} = \dfrac{2e}{h} \approx 483.6\ \text{MHz}/\mu\text{V}$. SQUIDs (Superconducting Quantum Interference Devices), using two Josephson junctions in a loop, exploit this for extremely sensitive magnetic flux measurement.

What is the isotope effect in superconductors and what does it support?

$T_{c} \propto M^{-\alpha}$ with $\alpha \approx 1/2$ for many elements, where $M$ is the isotopic mass. It demonstrates that lattice (phonon) vibrations are involved in pairing, supporting the electron–phonon mechanism of BCS theory.

Compare the conductivity behavior predicted by the Drude model that succeeds versus where it fails.

Succeeds: explains Ohm's law, DC/AC conductivity $\sigma = ne^{2}\tau/m$, and order of magnitude of Wiedemann–Franz ratio. Fails: predicts a classical $\tfrac{3}{2}k_{B}$ electronic specific heat per electron (vastly too large) and wrong temperature dependence — fixed by Fermi–Dirac statistics in Sommerfeld theory.