--- title: "Temperature dependence of carrier concentration and mobility" subject: "Electrical Engineering Material" chapter: 6 key: "semester_3/electrical_engineering_material:temperature-carrier-mobility" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/temperature-carrier-mobility site: BCT Engineering --- # Temperature dependence of carrier concentration and mobility Electrical Engineering Material · Chapter 6: Semiconductors - Asked in: 2078 Kartik, 6 marks; 2071 Chaitra, 6 marks (asked 2 times) - Video: [Lecture 6: Effect of temp. on Carrier Conc., conductivity and mobility in semiconductor material](https://www.youtube.com/watch?v=Bs1YikVX7aQ) — Deep Shekhar - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/temperature-carrier-mobility - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6 ## Question In doped semiconductors, show that the carrier concentration and drift mobility both are highly dependent on temperature with necessary diagrams. ## Short answer Carrier concentration follows the same three-region behaviour. Mobility is limited by lattice scattering (μ ∝ T^(-3/2)) at high T and impurity scattering (μ ∝ T^(3/2)) at low T; combining via Matthiessen’s rule yields a peak at intermediate temperatures, lowered by heavier doping. ## Answer In a doped semiconductor, conductivity $\sigma = e(n\mu_e + p\mu_h)$. Both factors, the carrier concentration and the drift mobility, change with temperature. #### Carrier concentration vs temperature For n-type doping $N_d$: 1. **Ionisation region (low T):** $n = \left(\tfrac{1}{2}N_cN_d\right)^{1/2}\exp(-\Delta E/2kT)$; $n$ rises as more donors ionise. 2. **Extrinsic region:** all donors ionised, $n \approx N_d$ (constant). 3. **Intrinsic region (high T):** $n \approx n_i \propto T^{3/2}\exp(-E_g/2kT)$; rises steeply. ```text ln n |\ | \ intrinsic | \ slope -Eg/2k | \ | \__________________ n = Nd | extrinsic \ | \ ionisation | \ slope -ΔE/2k +----|------------------|------------ 1/T 1/Ti 1/Ts (high T at left, low T at right) ``` #### Drift mobility vs temperature Mobility $\mu = e\tau/m^*$ depends on the mean time between scattering events. Two scattering processes dominate: 1. **Lattice (phonon) scattering:** atomic vibrations grow with $T$, so $\tau$ falls: $$ \mu_L \propto T^{-3/2} $$ 2. **Ionised impurity scattering:** a slow carrier is deflected more by charged dopant ions; faster (hotter) carriers are deflected less: $$ \mu_I \propto \frac{T^{3/2}}{N_I} $$ where $N_I$ is the ionised impurity concentration. 3. **Matthiessen's rule** combines them: $$ \frac{1}{\mu} = \frac{1}{\mu_L} + \frac{1}{\mu_I} $$ At low $T$, impurity scattering controls $\mu$ (rises with $T$); at high $T$, lattice scattering controls it (falls with $T$). So $\mu$ has a peak, and heavier doping lowers the curve and shifts the peak to higher $T$. ```text log μ | .-''-. lightly doped | / `-. | / .--. `-. | / / `-. `-. ~T^-3/2 | / / heavily `-. | / / doped `-. |/ / ~T^3/2 +--------------------------- log T ``` #### Result Conductivity follows mainly $n$: rises at low $T$, roughly flat (falling slightly with $\mu$) in the extrinsic range, then rises sharply in the intrinsic range. So both carrier concentration and mobility are strongly temperature dependent, and device behaviour must account for both. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How does lattice (phonon) scattering limit mobility as T rises? A. mu increases as T^(3/2) B. mu is independent of T C. mu decreases as T^(-3/2) because tau falls with larger atomic vibrations D. mu decreases because the band gap increases Answer: C. Greater lattice vibrations shorten the mean free time tau, so mu_L is proportional to T^(-3/2). (Key point: Lattice scattering) 2. Why does ionised-impurity-limited mobility rise with temperature? A. Impurities are removed at high T B. Faster, hotter carriers are deflected less by charged dopant ions C. More dopants are added D. Phonons screen the ions Answer: B. Hot carriers spend less time near an ion, so mu_I is proportional to T^(3/2)/N_I. (Key point: Impurity scattering) 3. How are the scattering mobilities combined? A. mu = mu_L + mu_I B. mu = mu_L x mu_I C. 1/mu = 1/mu_L + 1/mu_I D. mu = larger of the two Answer: C. Scattering rates add, so reciprocals of mobility add; the smaller mobility dominates, giving a peak in mu versus T. 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