Materials & Atomic Structure

Level 5 (Mastery): Cross-domain — Physics · Mathematics · Coding · Proof Time limit: 90 minutes Total marks: 60

Instructions: Answer all THREE questions. Show full reasoning. Use $...$ for inline and $...$ for displayed mathematics. Physical constants where needed: $k_B = 1.381\times10^{-23}\,\text{J/K}$, $q = 1.602\times10^{-19}\,\text{C}$, $1\,\text{eV}=1.602\times10^{-19}\,\text{J}$.

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Question 1 — Bonding, lattice geometry & atomic accounting (20 marks)

Silicon crystallizes in the diamond cubic structure with lattice constant $a = 5.431\,\text{Å}$.

(a) Starting from the Bohr model, explain why the silicon atom ($Z=14$) has exactly 4 valence electrons, and connect this directly to the fact that each Si atom forms 4 covalent bonds in the crystal. (4)

(b) The diamond cubic cell contains 8 atoms per conventional cubic cell. Derive the atomic number density $N$ (atoms/m³) as a function of $a$, then evaluate it numerically. (4)

(c) In the diamond structure the nearest-neighbour distance is $d = \dfrac{\sqrt{3}}{4}a$. Derive this from the geometry (a corner atom bonding to the atom at $(\tfrac14,\tfrac14,\tfrac14)a$) and evaluate $d$ in Å. (4)

(d) Compute the atomic packing fraction (APF) of the diamond cubic lattice, treating atoms as hard spheres of radius $r = d/2$ that touch along the bond. Show your working and comment on why this value (~0.34) is low compared with FCC (~0.74). (5)

(e) Explain, in terms of valence electrons and bonding, why a perfect intrinsic silicon crystal at $0\,\text{K}$ behaves as an insulator. (3)

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Question 2 — Intrinsic carriers, thermal generation & a proof (22 marks)

The intrinsic carrier concentration of a semiconductor obeys

$n_i(T) = C\,T^{3/2}\exp\!\left(-\frac{E_g}{2k_B T}\right)$

where $C$ is a material constant and $E_g$ the bandgap.

(a) Prove that the logarithmic temperature sensitivity satisfies

$\frac{d(\ln n_i)}{dT} = \frac{3}{2T} + \frac{E_g}{2k_B T^2}.$

State clearly which term dominates near room temperature for silicon ($E_g=1.12\,\text{eV}$, $T=300\,\text{K}$) by computing the ratio of the two terms. (6)

(b) For intrinsic silicon $n_i(300\,\text{K}) \approx 1.0\times10^{10}\,\text{cm}^{-3}$. Using the model above (ignore the weak $T^{3/2}$ prefactor for this part, i.e. treat $n_i \propto \exp(-E_g/2k_BT)$), estimate $n_i$ at $T = 400\,\text{K}$. Give the result to one significant figure. (6)

(c) Germanium has $E_g = 0.66\,\text{eV}$. Using the same approximation, compute the ratio $n_i^{Ge}/n_i^{Si}$ at $300\,\text{K}$ assuming equal prefactors $C$. Hence argue quantitatively why silicon is preferred over germanium for devices operating at elevated temperature. (5)

(d) Sketch (describe) qualitatively how the conductivity $\sigma = q n_i(\mu_n+\mu_p)$ of intrinsic Si varies with $T$, noting that mobility falls roughly as $\mu \propto T^{-3/2}$ while $n_i$ rises exponentially. Which effect wins, and what does this imply for the temperature coefficient of resistance of intrinsic silicon? (5)

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Question 3 — Extrinsic material, mobility & a small simulation (18 marks)

(a) Distinguish intrinsic from extrinsic semiconductors, then define carrier mobility $\mu$ via drift velocity $v_d = \mu E$. State the units of $\mu$. (4)

(b) A silicon sample is doped with $N_D = 1\times10^{16}\,\text{cm}^{-3}$ donors (fully ionized). Using the mass-action law $np = n_i^2$ with $n_i=1.0\times10^{10}\,\text{cm}^{-3}$, compute the equilibrium electron ($n$) and hole ($p$) concentrations. (4)

(c) For this sample $\mu_n = 1350\,\text{cm}^2/\text{V·s}$ and $\mu_p = 480\,\text{cm}^2/\text{V·s}$. Compute the resistivity $\rho = 1/\sigma$ where $\sigma = q(n\mu_n + p\mu_p)$. Comment on which carrier dominates conduction and why. (5)

(d) Coding task. Write a short Python function n_i(T, Eg=1.12, C=<your choice>) that returns intrinsic carrier concentration using the full model of Q2. Then, in pseudocode or Python, describe a loop that finds (by scanning $T$) the temperature at which the thermally generated $n_i$ becomes comparable (within 10%) to the doping $N_D=10^{16}\,\text{cm}^{-3}$ — the onset of loss of extrinsic behaviour. Explain the device significance of this temperature. (5)