--- title: "Fermi level with 10¹⁶ B then 10¹⁷ Sb" subject: "Electrical Engineering Material" chapter: 6 key: "semester_3/electrical_engineering_material:boron-1e16-antimony-1e17" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/boron-1e16-antimony-1e17 site: BCT Engineering --- # Fermi level with 10¹⁶ B then 10¹⁷ Sb Electrical Engineering Material · Chapter 6: Semiconductors - Asked in: 2075 Asoj, 6 marks - Video: [Fermi Energy Level in Intrinsic and Extrinsic Semiconductors](https://www.youtube.com/watch?v=SxLX65vVMtg) — Ekeeda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/boron-1e16-antimony-1e17 - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6 ## Question A silicon wafer is uniformly doped with 10¹⁶ Boron atoms per cm³. Where will be the Fermi level compared to its intrinsic Fermi level? Where will be the Fermi level is shifted if the sample is further doped with 10¹⁷ antimony atom per cm³? ## Short answer With 10^16 cm^-3 boron (p-type), Fermi level is 0.348 eV below intrinsic. Adding 10^17 antimony compensates, yielding n-type with Fermi level 0.404 eV above intrinsic, a total upward shift of 0.75 eV. ## Answer Boron is an acceptor and antimony a donor. Use $$ E_{Fi}-E_{Fp} = kT\ln\frac{p}{n_i},\qquad E_{Fn}-E_{Fi} = kT\ln\frac{n}{n_i} $$ **Data used (Si, 300 K):** $n_i = 1.45\times10^{10}\ \text{cm}^{-3}$, $kT = 8.617\times10^{-5}\times300 = 0.02585\ \text{eV}$, $\mu_e = 1350$, $\mu_h = 450\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$, $e = 1.6\times10^{-19}\ \text{C}$. #### Step 1: Only boron, $N_a = 10^{16}\ \text{cm}^{-3}$ The wafer is p-type, $p \approx N_a = 10^{16}\ \text{cm}^{-3}$, so $E_F$ lies **below** $E_{Fi}$: $$ \begin{aligned} E_{Fi}-E_{Fp} &= 0.02585\ln\frac{10^{16}}{1.45\times10^{10}}\\ &= 0.02585\times13.44 = 0.348\ \text{eV} \end{aligned} $$ #### Step 2: Add antimony, $N_d = 10^{17}\ \text{cm}^{-3}$ Now $N_d > N_a$, so the wafer is converted to **n-type** (compensated): $$ n = N_d - N_a = 10^{17} - 10^{16} = 9\times10^{16}\ \text{cm}^{-3} $$ $$ \begin{aligned} E_{Fn}-E_{Fi} &= 0.02585\ln\frac{9\times10^{16}}{1.45\times10^{10}}\\ &= 0.02585\times15.64 = 0.404\ \text{eV} \end{aligned} $$ #### Shift $$ \Delta E_F = 0.348 + 0.404 = 0.752\ \text{eV (upwards)} $$ ```text Ec ------------------------------- EFn ........ 0.404 eV above EFi (after Sb) EFi - - - - - - - - - - - - - - - - EFp ........ 0.348 eV below EFi (B only) Ev ------------------------------- ``` **Answer: with boron alone the Fermi level is 0.348 eV below $E_{Fi}$; after adding antimony it moves to 0.404 eV above $E_{Fi}$, a total upward shift of about 0.75 eV.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With only 10^16 cm^-3 boron in silicon, where is the Fermi level relative to E_Fi? A. 0.348 eV above E_Fi B. Exactly at E_Fi C. 0.348 eV below E_Fi D. 0.404 eV below E_Fi Answer: C. Boron is an acceptor, so the wafer is p-type and E_F lies below E_Fi by kT ln(p/n_i) = 0.348 eV. (Key point: Boron makes p-type) 2. Which expression gives the distance of the Fermi level below E_Fi for the p-type wafer? A. kT ln(n_i/p) B. kT ln(p/n_i) C. kT ln(N_d/N_a) D. kT ln(p n_i) Answer: B. E_Fi - E_Fp = kT ln(p/n_i), with p about N_a. (Key point: Formula for p-type shift) 3. After adding 10^17 antimony to the 10^16 boron wafer, what is the net electron concentration? A. 10^17 + 10^16 = 1.1 x 10^17 B. 10^16 C. 10^17 D. 9 x 10^16 Answer: D. Donors exceed acceptors, so the material is compensated n-type with n = N_d - N_a = 9 x 10^16 cm^-3. (Key point: Compensation) 4. What is the total shift of the Fermi level when antimony is added to the boron wafer? A. About 0.75 eV upward B. About 0.056 eV upward C. About 0.75 eV downward D. About 0.404 eV upward Answer: A. It moves from 0.348 eV below E_Fi to 0.404 eV above, a total upward shift of 0.752 eV. (Key point: Total Fermi level shift) ## Close variants set in other exams - [Fermi level with 10¹⁶ Sb then 2×10¹⁷ B](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aantimony-1e16-boron-2e17) — Electrical Engineering Material, chapter 6, asked 6× - [Fermi level with 10¹⁰ Sb then 2×10¹⁵ B](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aantimony-1e10-boron-2e15) — Electrical Engineering Material, chapter 6, asked 6× - [Fermi level with 10¹⁷ Sb then further doping](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aantimony-1e17-fermi-shift) — Electrical Engineering Material, chapter 6, asked 6× ## Related in Electrical Engineering Material - [Si with 10¹⁵ Sb: carriers, resistance, Fermi level](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aantimony-1e15-resistance-fermi) — Electrical Engineering Material, chapter 6 - [Fermi level of Si with 10¹⁶ Sb](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aantimony-1e16-fermi) — Electrical Engineering Material, chapter 6, asked 5× - [Fermi level shift with As then B](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aarsenic-1e15-fermi-shift) — Electrical Engineering Material, chapter 6, asked 2× - [Si with 10¹⁵ As: resistance and Fermi level](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aarsenic-1e15-resistance-fermi) — Electrical Engineering Material, chapter 6, asked 2× - [Si cube with As then 10¹⁴ boron](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asi-cube-arsenic-boron-1e14) — Electrical Engineering Material, chapter 6 - [Resistance of Si cube doped with Sb](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asi-cube-antimony) — Electrical Engineering Material, chapter 6