--- title: "Fermi level of Si with 10¹⁶ Sb" subject: "Electrical Engineering Material" chapter: 6 key: "semester_3/electrical_engineering_material:antimony-1e16-fermi" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/antimony-1e16-fermi site: BCT Engineering --- # Fermi level of Si with 10¹⁶ Sb Electrical Engineering Material · Chapter 6: Semiconductors - Asked in: 2076 Asoj, 6 marks - Video: [Fermi level and effect of doping on Fermi level IN JUST 5 MINUTES](https://www.youtube.com/watch?v=gRHzR4X_RWc) — Learning on the way! - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/antimony-1e16-fermi - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6 ## Question An n-type silicon wafer is uniformly doped with 10¹⁶ antimony atoms per cm³. Where will be the Fermi level compared to its intrinsic Fermi level? ## Short answer All donors ionized, n ≈ 10^16 cm⁻³. E_F − E_i = kT ln(n/n_i) = 0.02585 × ln(10^16/1.45×10^10) ≈ 0.348 eV. The Fermi level lies 0.348 eV above the intrinsic Fermi level. ## Answer Antimony is a group V **donor**, so the wafer is n-type and the Fermi level moves **above** the intrinsic Fermi level $E_{Fi}$. #### Relation used For a non-degenerate semiconductor $$ n = n_i\exp\left(\frac{E_{Fn}-E_{Fi}}{kT}\right) \;\Rightarrow\; E_{Fn}-E_{Fi} = kT\ln\frac{n}{n_i} $$ **Data used (Si, 300 K):** $n_i = 1.45\times10^{10}\ \text{cm}^{-3}$, $kT = 8.617\times10^{-5}\times300 = 0.02585\ \text{eV}$, $\mu_e = 1350$, $\mu_h = 450\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$, $e = 1.6\times10^{-19}\ \text{C}$. At room temperature all donors are ionised and $N_d = 10^{16} \gg n_i$, so $$ n \approx N_d = 10^{16}\ \text{cm}^{-3},\qquad p = \frac{n_i^2}{N_d} = 2.1\times10^{4}\ \text{cm}^{-3} $$ #### Calculation $$ \begin{aligned} E_{Fn}-E_{Fi} &= 0.02585\ln\frac{10^{16}}{1.45\times10^{10}}\\ &= 0.02585\ln(6.90\times10^{5})\\ &= 0.02585\times13.44\\ &= 0.348\ \text{eV} \end{aligned} $$ **Answer: the Fermi level lies about 0.348 eV above the intrinsic Fermi level** (about 0.2 eV below $E_c$, since $E_c - E_{Fi} \approx 0.55$ eV). ```text Ec ----------------------------- ~0.20 eV EFn ............................ 0.348 eV EFi - - - - - - - - - - - - - - - ~0.55 eV Ev ----------------------------- ``` The level is still well below $E_c$ (more than $3kT$), so the non-degenerate formula used is valid. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is E_F above E_Fi for Sb-doped Si? A. Sb is a donor, so n > ni B. Sb is an acceptor C. Sb lowers ni D. E_F is always above E_Fi Answer: A. Majority electrons raise E_F. (Key point: Direction) 2. Which relation is used? A. E_Fn - E_Fi = kT ln(ni/n) B. E_Fn - E_Fi = kT ln(n/ni) with n about Nd C. E_Fn - E_Fi = Eg D. E_Fn - E_Fi = kT n/ni Answer: B. Derived from n = ni exp((E_Fn - E_Fi)/kT). (Key point: Formula) 3. What is E_F - E_Fi for 10^16 cm^-3? A. 0.348 eV above B. 0.407 eV above C. 0.348 eV below D. 0.2 eV above Answer: A. 0.02585 x ln(6.9x10^5) = 0.02585 x 13.44. (Key point: Value) 4. Why is the non-degenerate formula valid here? A. E_F is inside the CB B. N_d exceeds Nc C. E_F is more than 3kT below Ec (about 0.2 eV) D. Because T is 0 K Answer: C. Boltzmann statistics hold when E_F is well within the gap. (Key point: Validity) ## Close variants set in other exams - [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× - [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× ## 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 with 10¹⁶ B then 10¹⁷ Sb](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aboron-1e16-antimony-1e17) — Electrical Engineering Material, chapter 6, asked 5× - [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× - [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 - [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 10¹⁶ As: carriers, conductivity, Fermi shift](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aarsenic-1e16-carriers-fermi) — Electrical Engineering Material, chapter 6