Exercises — Compound semiconductors (GaN, GaAs, SiC) overview

We read the group of each element from the periodic table: Ga = group III, As = group V, N = group V, Si = C = group IV.

• (a) GaAs → Ga(III) + As(V) → III–V
• (b) SiC → Si(IV) + C(IV) → IV–IV
• (c) Si → one element, group IV → elemental (IV)
• (d) GaN → Ga(III) + N(V) → III–V

Why this is the whole skill at L1: recognising the "III+V" and "IV+IV" patterns is exactly what tells you the tetrahedral covalent scheme can survive — the valence average lands on 4.

Direct-gap: GaAs and GaN. Indirect: Si and SiC. In a direct gap the lowest-energy electron and the empty hole sit at the same momentum, so an electron can drop across the gap and hand all its energy to a single photon — clean, efficient light. In an indirect gap they sit at different momenta, so a lattice vibration (phonon) must also join to conserve momentum; that three-body meeting is rare, so light emission is feeble. See LEDs and laser diodes.

Use tool 2, because the question links an energy to a colour, and $\lambda =1240/E_g$ is exactly the bridge (photon energy = the gap it fell across): $\lambda =\frac{1240}{3.4}\approx 365\text{nm}.$ 365 nm is just below the violet edge of visible light — near-ultraviolet. Real blue LEDs sit at ~450 nm because commercial devices emit slightly below the full gap (from doped/indium-alloyed layers), but the bare-gap answer is 365 nm.

$\lambda =\frac{1240}{1.12}\approx 1107\text{nm}.$ Visible light runs ~400–700 nm. 1107 nm is infrared — invisible to the eye. So even ignoring silicon's indirect gap, its gap is too narrow: a smaller gap means a lower-energy (longer-wavelength) photon. To reach visible you need a bigger gap, which is precisely why we jump to wide-gap GaN.

Invert tool 2 (same formula, solved for $E_g$), because now we know the colour and want the energy: $E_g=\frac{1240}{\lambda }=\frac{1240}{850}\approx 1.46\text{eV}.$ This is essentially GaAs's $E_g=1.42$ eV — consistent. The tiny difference is rounding; a real device is pinned by the material gap.

Treat the formula as (group-III site) + (group-V site). The III-site group number is the average of Al and Ga, both = 3, so it is 3 for any $x$. The V-site is As = 5. The per-atom average over the 2-atom unit is: $\bar{g}=\frac{[x(3)+(1-x)(3)]+5}{2}=\frac{3+5}{2}=4.$ The $x$ cancels because we only swapped one group-III atom for another group-III atom. Physical meaning: the tetrahedral bonding is untouched, so the crystal stays stable for all $x$ — which is exactly why AlGaAs is a workhorse tunable alloy. We are moving the bandgap (via Al fraction) without breaking the lattice.

We use tool 3 because the question asks us to derive one material's field from another's gap — the ratio form removes the unknown proportionality constant: $\frac{E_{crit}^{\text{GaN}}}{E_{crit}^{\text{Si}}}={\left(\frac{E_g^{\text{GaN}}}{E_g^{\text{Si}}}\right)}^2={\left(\frac{3.4}{1.12}\right)}^2\approx (3.036{)}^2\approx \mathrm{9.2.}$ $E_{crit}^{\text{GaN}}\approx 9.2\times 0.3\approx 2.8\text{MV/cm}.$ The table lists 3.3 MV/cm — same order, ~15% low. The simple $n=2$ rule is only a sketch (the real exponent drifts toward ~2.5 for very wide gaps), but it correctly predicts the ~10× jump over silicon that makes GaN a power-switching material.

Tool 4, because "how badly do two crystals disagree" is exactly what $f$ measures: $f=\frac{a_{\text{layer}}-a_{\text{substrate}}}{a_{\text{substrate}}}=\frac{3.19-2.75}{2.75}\approx +0.16=+16\%.$ A +16% mismatch is enormous — the atoms cannot line up one-to-one, strain builds, and past a critical thickness the film relieves it by spawning threading dislocations (defect lines that pierce the device). The fix, from the parent note and Epitaxy and crystal growth, is a buffer layer (a thin AlN/graded nucleation layer) that absorbs the misfit before the active GaN grows.

Winner: GaN. Reasoning by property:

• Breakdown field (voltage survival): 48 V needs a decent field tolerance. Si ($0.3$) is workable but thick and lossy; GaN ($3.3$) and SiC ($3.0$) block the same 48 V in ~1/10 the thickness → far lower on-resistance.
• Mobility (speed): the "few-GHz" requirement rewards fast electrons. GaN ($2000$, and much higher in a 2-D channel, see High Electron Mobility Transistor (HEMT)) beats SiC ($900$). GaAs ($8500$) is fastest but its $E_{crit}=0.4$ is far too low for 48 V blocking.
• So GaAs is out (can't hold the voltage), SiC is fine but slower and overkill for only 48 V, Si is bulky/lossy. GaN matches both the modest voltage and the high frequency — exactly the lateral-HEMT sweet spot. Rank: GaN > SiC > Si > GaAs (for this job).

Winner: SiC. The property that flips it is thermal + high-voltage robustness, not raw speed:

• At 900 V and high temperature, you need a material that (i) blocks huge fields — SiC ($3.0$) and GaN ($3.3$) both qualify, but (ii) conducts heat away and has a robust native oxide and bulk substrate. SiC conducts heat ~3× better than GaN and grows on native SiC wafers, so it survives the hottest, highest-voltage duty.
• GaN's lateral devices and weaker heat spreading make 900V+ harder; GaAs and Si are eliminated (breakdown far too low for 900 V in any sane thickness). Rank: SiC > GaN > Si > GaAs. Lesson: frequency chose GaN at 48 V; voltage + heat chose SiC at 900 V — you match the material's superpower to the operating point, echoing the parent's "GaAs Runs, GaN Lights, SiC Fights."

(a) Colour → energy uses tool 2 inverted (we know the wavelength, want the gap): $E_g=\frac{1240}{530}\approx 2.34\text{eV}.$ (b) Solve the interpolation for $x$. Expand: $E_g(x)=3.4-2.7x$. Set equal to 2.34: $3.4-2.7x=2.34⟹2.7x=1.06⟹x\approx \mathrm{0.39.}$ So about 39% indium (${\text{In}}_{0.39}{\text{Ga}}_{0.61}\text{N}$). Check the valence average is still 4: In and Ga are both group III, N is group V, so exactly as in L3.1 the average stays $((3)+5)/2=4$ for any $x$ — the lattice remains a valid semiconductor. (c) High indium fraction means a large lattice-constant change vs the GaN template (indium atoms are bigger), so strain and mismatch (tool 4) grow, causing composition clustering and defects. This is the famous "green gap" — green InGaN LEDs are notoriously less efficient than blue ones precisely because that indium-rich alloy is hard to grow cleanly. See Doping and carrier concentration and Epitaxy and crystal growth.

Chain it:

1. Blue means ~450–470 nm. By tool 2, $E_g=1240/460\approx 2.7$ eV — you need a wide gap.
2. Si ($1.12$ eV) and GaAs ($1.42$ eV) are too narrow — they emit infrared, so they are eliminated on gap value alone.
3. That leaves the wide-gap compounds SiC ($3.26$) and GaN ($3.4$). SiC is indirect → it emits light too feebly to be a bright LED (the phonon requirement from L1.2). Eliminated on gap type.
4. GaN is both wide and direct → the only survivor that can emit bright, high-energy blue photons.
5. The reason it took until the 1990s (and a Nobel Prize): GaN had no native substrate and a +16% mismatch on sapphire (L3.3), so growing defect-free GaN and p-type doping it were the real barriers — not the physics of the gap, but the crystal growth. One-line synthesis: blue light demands wide + direct + growable, and GaN is the unique material that clears all three bars — which is exactly the "GaN Lights" role in the parent's mnemonic.

Colour needs a direct gap of the right width ($\lambda =1240/E_g$); voltage needs a wide gap ($E_{crit}\propto E_g^2$); speed needs mobility; and all of it needs a growable crystal (low $|f|$). Match the material's superpower to the job.