Band Theory & Carrier Physics

Q1. (4 marks)

• Valence band: the highest range of electron energy levels that are normally occupied (filled) by electrons at 0 K. (1)
• Conduction band: the range of energy levels above the valence band that are normally empty; electrons here are free to move and conduct current. (1)
• Conduction depends on whether electrons can reach the conduction band / on the size of the band gap separating the two bands. (1)
• If the band gap is small or zero, electrons populate the conduction band easily → material conducts. (1)

Q2. (6 marks)

• Correct gap magnitudes: 1 mark each (3). Correct examples: 1 mark each (3). (6)
• Why: larger gap → fewer thermally excited carriers → lower conductivity.

Q3. (4 marks)

• Mass action law: $np = n_i^2$ (in thermal equilibrium). (2)
• $p = \dfrac{n_i^2}{n} = \dfrac{(1.5\times10^{10})^2}{5\times10^{16}}$ (1)
• $p = \dfrac{2.25\times10^{20}}{5\times10^{16}} = 4.5\times10^{3}\ \text{cm}^{-3}$ (1)

Q4. (5 marks)

• $f(E) = \dfrac{1}{1 + \exp\!\left(\dfrac{E-E_F}{k_BT}\right)}$ (2)
• At $E=E_F$: exponent $=0$, $\exp(0)=1$, so $f = \dfrac{1}{1+1} = 0.5$. (2)
• As $T\to 0$: $f(E)=1$ for $E<E_F$ and $f(E)=0$ for $E>E_F$ (step function). (1)

Q5. (5 marks)

• Direct gap: the conduction-band minimum and valence-band maximum occur at the same crystal momentum $k$; electron–hole recombination emits a photon directly (no phonon needed). (2)
• Indirect gap: the minimum and maximum are at different $k$; recombination requires a phonon to conserve momentum, so light emission is inefficient. (1)
• LEDs prefer direct gap materials. (1)
• Examples: Direct — GaAs; Indirect — Silicon (or Ge). (1)

Q6. (5 marks)

• Drift current: carrier motion caused by an applied electric field. (1.5)
• Diffusion current: carrier motion caused by a concentration gradient (from high to low concentration). (1.5)
• Expression: $J_{n,\text{drift}} = q\,n\,\mu_n\,E$ (or $\sigma E$ with $\sigma = qn\mu_n$). (2)

Q7. (5 marks)

• Einstein relation: $\dfrac{D}{\mu} = \dfrac{k_BT}{q}$, so $D = \mu \cdot \dfrac{k_BT}{q}$. (2)
• $D_n = 1350 \times 0.0259$ (1)
• $D_n = 34.97 \approx 35\ \text{cm}^2/\text{s}$ (2)

Q8. (5 marks)

• Majority carriers: electrons; Minority carriers: holes. (2)
• $n \approx N_D = 1\times10^{17}\ \text{cm}^{-3}$. (1)
• $p = \dfrac{n_i^2}{N_D} = \dfrac{(1.5\times10^{10})^2}{1\times10^{17}}$ (1)
• $p = \dfrac{2.25\times10^{20}}{1\times10^{17}} = 2.25\times10^{3}\ \text{cm}^{-3}$ (1)

Q9. (4 marks)

• Generation: creation of electron–hole pairs (e.g. thermally or by light) — electron moves from valence to conduction band. (1.5)
• Recombination: an electron in the conduction band falls back and combines with a hole, removing a carrier pair. (1.5)
• At thermal equilibrium the generation rate equals the recombination rate → carrier concentrations stay constant. (1)

Q10. (7 marks)

• As temperature rises, more thermal energy is available to excite electrons across the band gap. (2)
• More electron–hole pairs are generated → $n_i$ increases. (1)
• Relationship: $n_i \propto T^{3/2}\exp\!\left(-\dfrac{E_g}{2k_BT}\right)$. (3)
• The exponential term dominates → $n_i$ rises rapidly (roughly exponentially) with $T$. (1)

[
  {"claim":"Q3: p = ni^2/n = 4.5e3 cm^-3", "code":"ni=1.5e10; n=5e16; p=ni**2/n; result = abs(p-4.5e3) < 1"},
  {"claim":"Q7: D_n = mu * (kT/q) = 35 cm^2/s", "code":"mu=1350; vt=0.0259; D=mu*vt; result = abs(D-34.965) < 0.1"},
  {"claim":"Q8: p = ni^2/ND = 2.25e3 cm^-3", "code":"ni=1.5e10; ND=1e17; p=ni**2/ND; result = abs(p-2.25e3) < 1"},
  {"claim":"Q4: f(E_F)=0.5", "code":"from sympy import exp,Rational; f=1/(1+exp(0)); result = f == Rational(1,2)"}
]