GATE Physics Solid State Physics Syllabus

Solid State Physics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Solid State Physics in GATE Physics, not a summary of it.

1. 1

   Elements of Crystallography

   1 topic

   • Diffraction Methods for Structure Determination
2. 2

   Bonding in Solids

   overview

   Examined as a single unit within Solid State Physics — no further topic split in the official outline.

3. 3

   Lattice Vibrations and Thermal Properties of Solids

   overview

   Examined as a single unit within Solid State Physics — no further topic split in the official outline.

4. 4

   Free Electron Theory

   overview

   Examined as a single unit within Solid State Physics — no further topic split in the official outline.

5. 5

   Band Theory of Solids

   2 topics

   • Nearly Free Electron Model
   • Tight Binding Model
6. 6

   Metals, Semiconductors and Insulators

   1 topic

   • Conductivity, Mobility and Effective Mass
7. 7

   Optical Properties of Solids

   2 topics

   • Kramer's-Kronig Relation
   • Intra- and Inter-band Transitions
8. 8

   Dielectric Properties of Solids

   3 topics

   • Dielectric Function
   • Polarizability
   • Ferroelectricity
9. 9

   Magnetic Properties of Solids

   1 topic

   • Dia, Para, Ferro, Antiferro and Ferri-Magnetism
     • Domains and Magnetic Anisotropy
10. 10

    Superconductivity

    1 topic

    • Type-I and Type-II Superconductors
      • Meissner Effect
      • London Equation
      • BCS Theory
      • Flux Quantization

Solid State Physics flashcards for GATE Physics

18 of 51 cards from the Solid State Physics deck — real questions with worked answers.

1. State Bragg's law for X-ray diffraction and define each symbol.

   $$n\lambda = 2d\sin\theta$$ where $n$ is the diffraction order, $\lambda$ the X-ray wavelength, $d$ the interplanar spacing, and $\theta$ the glancing (Bragg) angle measured from the crystal plane.

2. What is the Laue condition for diffraction in terms of the scattering vector and reciprocal lattice?

   Diffraction occurs when the scattering vector equals a reciprocal lattice vector: $\Delta\vec{k} = \vec{k}' - \vec{k} = \vec{G}$, where $\vec{G}$ is a reciprocal lattice vector. Equivalently $2\vec{k}\cdot\vec{G} = G^{2}$.

3. Name the three principal X-ray diffraction methods and the variable each scans to satisfy Bragg's law.

   Laue method (fixed crystal, continuous $\lambda$ spectrum — varies $\lambda$); rotating-crystal method (fixed $\lambda$, rotates crystal — varies $\theta$); powder (Debye–Scherrer) method (fixed $\lambda$, many random orientations — effectively varies $\theta$).

4. What is the structure factor $F_{hkl}$ and why is it important in diffraction?

   $$F_{hkl} = \sum_{j} f_{j}\, e^{\,2\pi i (h x_{j} + k y_{j} + l z_{j})}$$ where $f_{j}$ is the atomic form factor of atom $j$ at fractional position $(x_j,y_j,z_j)$. The diffracted intensity $\propto |F_{hkl}|^{2}$; it determines which reflections are allowed and gives systematic absences that reveal the lattice type.

5. State the systematic absence (selection) rules for BCC and FCC lattices in diffraction.

   BCC: reflections present only when $h+k+l$ is even. FCC: reflections present only when $h,k,l$ are all even or all odd (unmixed parity).

6. What does the Ewald sphere construction represent in diffraction?

   A sphere of radius $|\vec{k}| = 2\pi/\lambda$ drawn in reciprocal space; a reflection occurs whenever a reciprocal lattice point lies on the surface of the sphere, satisfying $\Delta\vec{k} = \vec{G}$.

7. In the nearly free electron model, what happens to the electron energy at the Brillouin zone boundary?

   A periodic potential mixes the degenerate plane-wave states $\pm k$ at the zone boundary, lifting the degeneracy and opening an energy gap. The bands flatten ($dE/dk = 0$) at the boundary.

8. In the nearly free electron model, what is the magnitude of the energy gap at a Brillouin zone boundary?

   The gap equals twice the relevant Fourier component of the periodic potential: $$E_{g} = 2|U_{G}|$$ where $U_{G}$ is the Fourier coefficient of the lattice potential at reciprocal vector $\vec{G}$.

9. State Bloch's theorem for an electron in a periodic potential.

   The eigenstates take 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: $u_{\vec{k}}(\vec{r}+\vec{R}) = u_{\vec{k}}(\vec{r})$.

10. What is the central physical assumption of the tight-binding model?

    Electrons are tightly bound to their atoms; the crystal wavefunction is built as a linear combination of atomic orbitals (LCAO), and only overlap between neighboring atomic orbitals is significant, allowing electrons to hop between adjacent sites.

11. Give the tight-binding dispersion relation for a simple cubic lattice (s-band) with lattice constant $a$.

    $$E(\vec{k}) = E_{0} - \alpha - 2\gamma\left(\cos k_x a + \cos k_y a + \cos k_z a\right)$$ where $\alpha$ is the on-site shift and $\gamma$ the nearest-neighbor overlap (hopping) integral.

12. In the tight-binding model, what is the total bandwidth of an s-band in a simple cubic lattice and what controls it?

    The bandwidth is $12\gamma$ (from $-2\gamma$ to $+2\gamma$ along each of three directions). It is controlled by the nearest-neighbor overlap/hopping integral $\gamma$: larger overlap gives a wider band.

13. How does the energy band width relate to electron localization in the tight-binding picture?

    Strong overlap between neighbors (large hopping integral) gives wide bands and delocalized, mobile electrons; weak overlap gives narrow bands and more localized electrons. Atomic limit ($\gamma \to 0$) gives discrete atomic levels.

14. Define the effective mass of an electron in a band in terms of the dispersion $E(k)$.

    $$\frac{1}{m^{*}} = \frac{1}{\hbar^{2}}\frac{d^{2}E}{dk^{2}}$$ (in 1D). It is inversely proportional to the band curvature; flat bands give large $m^{*}$, sharply curved bands give small $m^{*}$.

15. Why is the effective mass negative near the top of an energy band?

    Near the top of a band the dispersion $E(k)$ curves downward, so $d^{2}E/dk^{2} < 0$, giving $m^{*} < 0$. These states behave as positively charged holes.

16. Define carrier mobility $\mu$ and give its relation to conductivity.

    Mobility is drift velocity per unit field: $\mu = v_d/E = e\tau/m^{*}$. Conductivity relates to it via $$\sigma = n e \mu$$ where $n$ is the carrier density and $\tau$ the relaxation time.

17. Write the Drude expression for electrical conductivity $\sigma$ of a metal.

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

18. How do conductivity and carrier mobility typically vary with temperature in a metal versus an intrinsic semiconductor?

    In a metal, $\mu$ and $\sigma$ decrease with rising $T$ (more phonon scattering, $n$ roughly constant). In an intrinsic semiconductor, $\sigma$ increases strongly with $T$ because the exponential rise in carrier density $n \propto e^{-E_g/2k_BT}$ dominates over the mild mobility decrease.

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