--- title: "Degenerate semiconductors; Si vs GaAs E-k" subject: "Electrical Engineering Material" chapter: 6 key: "semester_3/electrical_engineering_material:degenerate-si-gaas" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/degenerate-si-gaas site: BCT Engineering --- # Degenerate semiconductors; Si vs GaAs E-k Electrical Engineering Material · Chapter 6: Semiconductors - Asked in: 2080 Bhadra, 8 marks - Video: [Direct Bandgap Semiconductors and Indirect Bandgap Semiconductors | Electronics Devices and Circuits](https://www.youtube.com/watch?v=hEYs-utWe7E) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/degenerate-si-gaas - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6 ## Question Differentiate between non-degenerate and degenerate semiconductor. Compare between Si and GaAs semiconductor with their respective E-K curves. ## Short answer Non-degenerate: moderate doping, EF in band gap, Boltzmann statistics. Degenerate: heavy doping (~10^19 cm⁻³), EF inside band, Fermi-Dirac statistics, metallic behavior. Si has indirect gap (1.12 eV), CB minimum at k≠0, poor light emission. GaAs has direct gap (1.42 eV), both band edges at k=0, efficient photon emission, high electron mobility. ## Answer #### Non-degenerate vs degenerate semiconductor A **non-degenerate** semiconductor is moderately doped ($n \ll N_c$), so $E_F$ lies inside the band gap; a **degenerate** one is so heavily doped (about $10^{19}$ cm$^{-3}$ or more in Si) that $E_F$ enters the conduction or valence band. | Point | Non-degenerate | Degenerate | |---|---|---| | Doping | $N_d \ll N_c$ | $N_d \gtrsim N_c$ | | Fermi level | In gap, over $3kT$ from band edges | Inside CB (n$^+$) or VB (p$^+$) | | Statistics | Boltzmann; $np = n_i^2$ | Fermi-Dirac; $np \ne n_i^2$ | | Impurity levels | Discrete | Merge into an impurity band | | Behaviour | Semiconductor | Metal-like, weak T dependence | | Uses | Normal diodes, transistors | Tunnel diodes, lasers, ohmic contacts | #### Comparison of Si and GaAs | Property | Si | GaAs | |---|---|---| | Band gap | 1.12 eV, indirect | 1.42 eV, direct | | Electron mobility | about 1350 cm$^2$/Vs | about 8500 cm$^2$/Vs | | Electron effective mass | $1.08m_e$ (dos) | $0.067m_e$ | | $n_i$ at 300 K | about $10^{10}$ cm$^{-3}$ | about $2\times10^{6}$ cm$^{-3}$ | | Structure | Diamond | Zinc blende | | Native oxide | SiO$_2$ | None useful | | Use | ICs, power devices, solar cells | LEDs, lasers, microwave ICs | #### E-k curves ```text Si (indirect) GaAs (direct) E E | \ / CB | \ / CB | \_/ | \_/ | ^ min at k0 | | min at k=0 | Eg / (phonon) | Eg | photon | /\ | /\ | / \ VB | / \ VB +---0----k0---- k +-----0------ k ``` - **Si:** CB minimum at $k_0 \ne 0$, VB maximum at $k = 0$. A transition needs a phonon for momentum, so light emission is poor. - **GaAs:** CB minimum and VB maximum both at $k = 0$. Electrons drop directly and emit photons ($\lambda \approx 1.24/1.42 = 0.87$ $\mu$m). Sharp CB curvature gives small $m_e^* = 0.067m_e$ and high mobility. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What makes a semiconductor degenerate? A. Doping so heavy (about 10^19 cm^-3 or more in Si) that E_F enters the CB or VB B. Doping below 10^10 cm^-3 C. E_F exactly at mid-gap D. A direct band gap Answer: A. Heavy doping moves E_F into a band, giving metal-like behaviour. (Key point: Degenerate definition) 2. Which pairing is correct? A. Non-degenerate: Fermi-Dirac; degenerate: Boltzmann B. Non-degenerate: Boltzmann with np = ni^2; degenerate: Fermi-Dirac with np not equal to ni^2 C. Both obey np = ni^2 D. Degenerate: discrete impurity levels Answer: B. Only non-degenerate material obeys np = ni^2; degenerate impurity levels merge into a band. (Key point: Statistics) 3. How do Si and GaAs differ in band structure? A. Si is direct; GaAs is indirect B. Both are indirect C. Si has an indirect gap (1.12 eV); GaAs has a direct gap (1.42 eV) D. Both are direct with the same gap Answer: C. In Si the CB minimum is at k0 not 0; in GaAs both extremes are at k = 0. (Key point: Si vs GaAs gap) 4. Why is electron mobility in GaAs (about 8500) much higher than in Si? A. GaAs has more donors B. GaAs has a larger effective mass C. GaAs has a native oxide D. Sharp CB curvature gives a small effective mass, 0.067 m_e Answer: D. Mobility = e tau / m*, so a small m* gives high mobility. 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