--- title: "n-type carrier concentration versus temperature" subject: "Electrical Engineering Material" chapter: 6 key: "semester_3/electrical_engineering_material:ntype-concentration-temperature" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/ntype-concentration-temperature site: BCT Engineering --- # n-type carrier concentration versus temperature Electrical Engineering Material · Chapter 6: Semiconductors - Asked in: 2074 Asoj, 6 marks; 2068 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/ntype-concentration-temperature - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6 ## Question Explain how carrier concentration of an n-type extrinsic semiconductor depends on temperature with necessary diagram and graphs. ## Short answer At low temperature, carrier concentration rises exponentially as donors ionise with slope -ΔE/(2k). In the saturation (extrinsic) region, n ≈ Nd and is nearly constant. At high temperature, intrinsic generation dominates, and n rises steeply with slope -Eg/(2k). ## Answer In an n-type semiconductor (donor concentration $N_d$, donor level $\Delta E = E_c - E_d$ below the conduction band), the electron concentration changes with temperature through three distinct regions. A plot of $\ln n$ against $1/T$ shows them clearly. ```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) ``` ```text n | / | / intrinsic | ________________/ Nd | / extrinsic (n = Nd) | / | / ionisation |___/________________________ T ~100K Ts Ti (~500K for Si) ``` #### 1. Ionisation (freeze-out) region: low T At very low $T$, electrons are bound to donors. As $T$ rises, donors ionise: $$ n = \left(\frac{1}{2}N_cN_d\right)^{1/2}\exp\left(-\frac{\Delta E}{2kT}\right) $$ Slope of $\ln n$ vs $1/T$ is $-\Delta E/2k$. $E_F$ lies between $E_d$ and $E_c$. #### 2. Extrinsic (saturation) region: medium T Above $T_s$ almost all donors are ionised, and intrinsic generation is still small: $$ n \approx N_d $$ The concentration stays nearly constant. $E_F = E_c - kT\ln(N_c/N_d)$ moves slowly towards mid-gap. Devices are designed to operate here (about 150 K to 450 K for Si with $N_d = 10^{15}$ cm$^{-3}$). #### 3. Intrinsic region: high T Above $T_i$, electrons excited across the band gap outnumber those from donors: $$ n \approx n_i = \sqrt{N_cN_v}\exp\left(-\frac{E_g}{2kT}\right) $$ Slope is $-E_g/2k$, much steeper. $E_F$ approaches $E_{Fi}$ (mid-gap) and $n \approx p$; the material loses its n-type character. #### Fermi level vs temperature ```text E Ec |------------------------- | EF near Ed | \_ | `-._ Ei |---------`-.__________ (EF -> Ei) | Ev |------------------------- T ``` Summary: $n$ rises exponentially, flattens at $N_d$, then rises again exponentially at high $T$. $T_i$ increases with doping and with band gap. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the slope of ln n versus 1/T in the ionisation region? A. -Eg/2k B. -Eg/k C. -DeltaE/2k, where DeltaE = Ec - Ed D. Zero Answer: C. n is proportional to exp(-DeltaE/2kT), so the slope is -DeltaE/2k. (Key point: Freeze-out slope) 2. In the extrinsic region the electron concentration is A. roughly constant at Nd B. roughly ni C. growing exponentially D. zero Answer: A. All donors are ionised and intrinsic generation is small, so n is about Nd. (Key point: Saturation region) 3. What happens to EF and n at high temperature? A. EF moves up to Ec B. n stays at Nd and EF stays at Ed C. n falls and EF moves to Ev D. n is about ni, with slope -Eg/2k, and EF approaches mid-gap Answer: D. Band-to-band generation dominates, the material loses n-type character and EF goes toward Ei. (Key point: Intrinsic region and Fermi level) ## Related in Electrical Engineering Material - [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 - [Temperature effect on carrier concentration](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Atemperature-carrier-concentration) — 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× - [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 - [Minority charge suppression in extrinsic semiconductors](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aminority-charge-suppression) — Electrical Engineering Material, chapter 6, asked 2× - [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 ## Related in other subjects - [Direct shear on loose and dense sand graphs](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Adirect-shear-sand-graphs) — Soil Mechanics, chapter 10