--- title: "Temperature effect on carrier concentration" subject: "Electrical Engineering Material" chapter: 6 key: "semester_3/electrical_engineering_material:temperature-carrier-concentration" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/temperature-carrier-concentration site: BCT Engineering --- # Temperature effect on carrier concentration Electrical Engineering Material · Chapter 6: Semiconductors - Asked in: 2079 Bhadra, 6 marks; 2073 Chaitra, 6 marks (asked 2 times) - Video: [Solid State Electronics | Temperature Dependence of Carrier Concentration (Extrinsic)](https://www.youtube.com/watch?v=H2L6N-HKAA4) — India Engineered - 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-concentration - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6 ## Question Explain how does temperature affects the formation of carrier concentration in semiconductor. ## Short answer In intrinsic semiconductors, carrier concentration rises exponentially with temperature: ni ∝ T^(3/2) exp(-Eg/2kT). In doped semiconductors, three regions emerge: at low T ionisation gives n ∝ exp(-ΔE/2kT); at medium T saturation n ≈ Nd constant; at high T intrinsic generation dominates, n ≈ ni again. ## Answer The carrier concentration in a semiconductor depends strongly on temperature because carriers are created thermally, either by ionising impurities or by breaking covalent bonds across the band gap. #### Intrinsic semiconductor Electron-hole pairs are created when thermal energy lifts electrons across $E_g$: $$ n_i = \sqrt{N_cN_v}\,\exp\left(-\frac{E_g}{2kT}\right), \quad N_c, N_v \propto T^{3/2} $$ so $n_i \propto T^{3/2}e^{-E_g/2kT}$. The exponential dominates: for Si, $n_i$ roughly doubles for every 8 to 10 K rise near room temperature. #### Extrinsic (n-type) semiconductor A plot of $\ln n$ against $1/T$ shows three regions: 1. **Low temperature (ionisation or freeze-out region):** only some donors are ionised. $$ n = \left(\frac{1}{2}N_cN_d\right)^{1/2}\exp\left(-\frac{\Delta E}{2kT}\right), \quad \Delta E = E_c - E_d $$ 2. **Medium temperature (extrinsic or saturation region):** all donors are ionised, $n \approx N_d$, nearly constant. This range (about 150 K to 500 K for Si with $N_d = 10^{15}$ cm$^{-3}$) is where devices work. It begins at the saturation temperature $T_s$, where the donors become fully ionised. 3. **High temperature (intrinsic region):** thermal generation across $E_g$ exceeds $N_d$; $n \approx n_i$ and the material behaves as intrinsic. The changeover temperature $T_i$ is where $n_i \approx N_d$. ```text ln n |\ | \ intrinsic region | \ slope -Eg/2k | \ | \____________________ n = Nd | extrinsic region \ | \ ionisation region | \ slope -ΔE/2k +-----|--------------------|-------- 1/T 1/Ti 1/Ts (high T at left, low T at right) ``` #### Fermi level As $T$ rises, $E_F$ moves from near $E_d$ (low $T$) towards mid-gap (high $T$), because the extrinsic carriers become a smaller fraction of the total. The upper limit of device operation is set by $T_i$; wide band gap materials (GaAs, SiC) have higher $T_i$ and can work at higher temperatures. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What mainly controls the rise of n_i with temperature? A. The T^(3/2) factor in Nc and Nv B. The exponential factor exp(-Eg/2kT) C. A change in the mobility D. A change in donor concentration Answer: B. n_i is proportional to T^(3/2) exp(-Eg/2kT), and the exponential dominates, so n_i doubles every 8 to 10 K near room temperature. (Key point: Intrinsic temperature dependence) 2. What is the electron concentration in the medium-temperature (extrinsic) region of an n-type sample? A. n is about ni and rises steeply B. n is about Nd, nearly constant C. n is zero D. n falls exponentially Answer: B. All donors are ionised and intrinsic generation is still small, so n is about Nd; devices operate here. (Key point: Saturation region) 3. What happens at high temperature above Ti? A. Donors freeze out B. n stays constant at Nd C. Holes disappear D. Thermal generation across Eg exceeds Nd, so n is about ni and the material becomes intrinsic Answer: D. Band-to-band generation dominates and the slope of ln n versus 1/T becomes -Eg/2k. (Key point: High temperature region) ## Related in Electrical Engineering Material - [n-type carrier concentration versus temperature](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Antype-concentration-temperature) — Electrical Engineering Material, chapter 6, asked 2× - [Temperature dependence of carrier concentration and mobility](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Atemperature-carrier-mobility) — Electrical Engineering Material, chapter 6, asked 2× - [Electron concentration in an extrinsic semiconductor](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aextrinsic-electron-concentration) — Electrical Engineering Material, chapter 6 - [Intrinsic concentration and its temperature dependence](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aintrinsic-concentration-derivation) — Electrical Engineering Material, chapter 6 - [Intrinsic Fermi level of pure semiconductor](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aintrinsic-fermi-level) — Electrical Engineering Material, chapter 6 - [Compensated semiconductor concentrations and conductivity](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acompensated-conductivity) — Electrical Engineering Material, chapter 6 ## Related in other subjects - [Stress concentration and its effect in brittle material](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astress-concentration-brittle-effect) — Strength of Materials, chapter 3 - [Tensile properties and effect of temperature](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Aproperties-from-tensile-test-and-temperature) — Material Science, chapter 3 - [Material response, isotropy and temperature effect](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Aelastic-response-isotropy-temperature) — Mechanics of Solids, chapter 1 - [Temperature coefficient and effect on resistance](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Atemperature-coefficient-effect-on-resistance) — Basic Electrical Engineering, chapter 1