--- title: "Fermi level with 10¹⁷ Sb then further doping" subject: "Electrical Engineering Material" chapter: 6 key: "semester_3/electrical_engineering_material:antimony-1e17-fermi-shift" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/antimony-1e17-fermi-shift site: BCT Engineering --- # Fermi level with 10¹⁷ Sb then further doping Electrical Engineering Material · Chapter 6: Semiconductors - Asked in: 2080 Baisakh, 8 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-1e17-fermi-shift - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6 ## Question An n-type silicon wafer is uniformly doped with 10¹⁷ antimony per cm³. Where will the Fermi level compared to its intrinsic Fermi level? Where will the Fermi level be shifted if the sample is further doped with 2×10¹⁶ atoms per cm³? ## Short answer With 10¹⁷ Sb/cm³, n ≈ Nd, EF – EFi = 0.407 eV above intrinsic level. Further doping with 2×10¹⁶ acceptors compensates, leaving n = 8×10¹⁶ cm⁻³; EF – EFi = 0.401 eV, shifting down by 0.006 eV but still above EFi. ## Answer Antimony (group V) is a donor, so the wafer is n-type and the Fermi level $E_{Fn}$ lies **above** the intrinsic level $E_{Fi}$. For a non-degenerate semiconductor $n = n_i\,e^{(E_{Fn}-E_{Fi})/kT}$, so $$ E_{Fn}-E_{Fi} = kT\ln\frac{n}{n_i} $$ **Data used (Si, 300 K):** $n_i = 1.45\times10^{10}\ \text{cm}^{-3}$, $kT = 8.617\times10^{-5}\times300 = 0.02585\ \text{eV}$, $\mu_e = 1350$, $\mu_h = 450\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$, $e = 1.6\times10^{-19}\ \text{C}$. #### Case 1: $N_d = 10^{17}\ \text{cm}^{-3}$ antimony At room temperature all donors are ionised and $N_d \gg n_i$, so $n \approx N_d = 10^{17}\ \text{cm}^{-3}$. $$ \begin{aligned} E_{Fn}-E_{Fi} &= 0.02585\ln\frac{10^{17}}{1.45\times10^{10}} \\ &= 0.02585\times15.75 = 0.407\ \text{eV} \end{aligned} $$ **Answer: the Fermi level is 0.407 eV above $E_{Fi}$.** #### Case 2: further doped with $2\times10^{16}\ \text{cm}^{-3}$ atoms The type of the second dopant is not printed; the usual reading is that it is an **acceptor (e.g. boron)**, $N_a = 2\times10^{16}\ \text{cm}^{-3}$. The acceptors capture an equal number of the donated electrons (compensation): $$ n = N_d - N_a = 10^{17} - 2\times10^{16} = 8\times10^{16}\ \text{cm}^{-3} $$ $$ \begin{aligned} E_{Fn}-E_{Fi} &= 0.02585\ln\frac{8\times10^{16}}{1.45\times10^{10}} \\ &= 0.401\ \text{eV} \end{aligned} $$ **Answer: the Fermi level is still above $E_{Fi}$, now 0.401 eV above it, i.e. it moves down by about 0.006 eV towards $E_{Fi}$.** The sample stays n-type because $N_d > N_a$. (If the extra $2\times10^{16}$ atoms were more antimony, $n = 1.2\times10^{17}$ and $E_F - E_{Fi} = 0.412$ eV, a shift of 0.005 eV upward.) ```text Ec --------------------------- EFn ......... (0.407 eV) 1st EFn ......... (0.401 eV) 2nd EFi - - - - - - - - - - - - - Ev --------------------------- ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why does antimony put E_F above E_Fi? A. It is a group V donor, so electrons are the majority carriers B. It is a group III acceptor C. It reduces the band gap D. It adds holes Answer: A. Donors raise n above ni and so raise E_F. (Key point: Dopant type) 2. What is E_F - E_Fi for Nd = 10^17 cm^-3? A. About 0.407 eV below E_Fi B. About 0.407 eV above E_Fi C. About 0.348 eV above E_Fi D. About 0.55 eV above E_Fi Answer: B. 0.02585 x ln(10^17/1.45x10^10) = 0.02585 x 15.75 = 0.407 eV. (Key point: Case 1 value) 3. If the added 2x10^16 atoms are acceptors, what is n? A. n = 1.2x10^17 cm^-3 B. n = 2x10^16 cm^-3 C. n = Nd - Na = 8x10^16 cm^-3 D. n = 10^17 cm^-3 unchanged Answer: C. Acceptors capture an equal number of donor electrons. (Key point: Compensation) 4. What happens to the Fermi level after the acceptor doping? A. It crosses below E_Fi, making the sample p-type B. It moves up by 0.1 eV C. It moves to mid-gap exactly D. It stays above E_Fi at 0.401 eV, moving down about 0.006 eV, still n-type Answer: D. Nd > Na so the sample is still n-type, only slightly less so. (Key point: 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 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× - [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× ## 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 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 crystal resistance with As then B doping](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asi-arsenic-boron-resistance) — Electrical Engineering Material, chapter 6 - [Doping to raise Fermi level by 0.7 eV](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Araise-fermi-0-7ev) — 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