--- title: "Resistance of 8 cm³ Si cube" subject: "Electrical Engineering Material" chapter: 6 key: "semester_3/electrical_engineering_material:si-cube-8cm3" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/si-cube-8cm3 site: BCT Engineering --- # Resistance of 8 cm³ Si cube Electrical Engineering Material · Chapter 6: Semiconductors - Asked in: 2079 Bhadra, 6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/si-cube-8cm3 - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6 ## Question Calculate the resistance of pure silicon cubic crystal of 8 cm³ at room temperature. What will be the resistance of the cube when it is doped with 1 arsenic in 5 × 10⁹ Si atom? Take atomic concentration of Si is 5 × 10²² cm⁻³. ## Short answer 8 cm³ cube has L = 2 cm, A = 4 cm². Pure Si: σi = 4.176×10⁻⁶ S/cm, ρ = 2.395×10⁵ Ω·cm, R = ρ/L ≈ 1.20×10⁵ Ω. Doping with 1 As per 5×10⁹ Si gives Nd = 10¹³ cm⁻³, σ ≈ e Nd μe = 2.16×10⁻³ S/cm, ρ = 463 Ω·cm, R ≈ 231.5 Ω. ## Answer The cube volume is 8 cm³, so each side is $L = 2$ cm and the face area is $A = 4\ \text{cm}^2$. For a cube $$ R = \frac{\rho L}{A} = \frac{\rho}{L} = \frac{\rho}{2\ \text{cm}} $$ **Data used (Si, 300 K):** $n_i = 1.45\times10^{10}\ \text{cm}^{-3}$, $kT = 8.617\times10^{-5}\times300 = 0.02585\ \text{eV}$, $\mu_e = 1350$, $\mu_h = 450\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$, $e = 1.6\times10^{-19}\ \text{C}$. #### Pure (intrinsic) silicon $$ \begin{aligned} \sigma_i &= n_i e(\mu_e+\mu_h) \\ &= 1.45\times10^{10}\times1.6\times10^{-19}\times1800 \\ &= 4.176\times10^{-6}\ \text{S cm}^{-1}\\ \rho_i &= 1/\sigma_i = 2.395\times10^{5}\ \Omega\,\text{cm}\\ R_i &= \frac{2.395\times10^{5}}{2} = 1.197\times10^{5}\ \Omega \end{aligned} $$ #### Doped with 1 As per $5\times10^9$ Si atoms Arsenic is a donor: $$ N_d = \frac{5\times10^{22}}{5\times10^{9}} = 10^{13}\ \text{cm}^{-3} $$ Since $N_d \gg n_i$, $n \approx N_d = 10^{13}\ \text{cm}^{-3}$ and $p = n_i^2/N_d = 2.1\times10^{7}\ \text{cm}^{-3}$ (negligible). $$ \begin{aligned} \sigma &= e(n\mu_e + p\mu_h) \approx eN_d\mu_e \\ &= 1.6\times10^{-19}\times10^{13}\times1350 \\ &= 2.16\times10^{-3}\ \text{S cm}^{-1}\\ \rho &= 463\ \Omega\,\text{cm}\\ R &= \frac{463}{2} = 231.5\ \Omega \end{aligned} $$ **Answer: pure cube $R \approx 1.20\times10^{5}\ \Omega$; doped cube $R \approx 231.5\ \Omega$** — about 517 times smaller, even though only one atom in five billion was replaced. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For an 8 cm^3 cube, what are L and A, and how does R relate to rho? A. L = 2 cm, A = 4 cm^2, R = rho/2 B. L = 8 cm, A = 1 cm^2, R = 8 rho C. L = 2 cm, A = 2 cm^2, R = rho D. L = 4 cm, A = 4 cm^2, R = rho Answer: A. R = rho L / A = rho / L for a cube, so rho/2 cm. (Key point: Cube geometry) 2. What is R of the pure cube? A. About 2.4x10^5 ohm B. About 1.20x10^5 ohm C. About 4.8x10^5 ohm D. About 231 ohm Answer: B. rho = 2.395x10^5 ohm cm, halved to 1.197x10^5 ohm. (Key point: Intrinsic R) 3. For 1 As in 5x10^9 Si atoms, what is Nd? A. 2.5x10^13 cm^-3 B. 5x10^13 cm^-3 C. 10^13 cm^-3 D. 10^10 cm^-3 Answer: C. 5x10^22 / 5x10^9 = 10^13 cm^-3. (Key point: Donor density) 4. What is R of the doped cube? A. About 231.5 ohm, from sigma = e Nd mu_e = 2.16x10^-3 S/cm giving rho 463 ohm cm and R = rho/2 B. 463 ohm C. 1.2x10^5 ohm D. About 92 ohm Answer: A. The hole term is negligible; R = 463/2 = 231.5 ohm. (Key point: Doped R) ## Close variants set in other exams - [Resistance of Si cube doped with As and B](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asi-cube-arsenic-boron-billion) — Electrical Engineering Material, chapter 6, asked 6× ## Related in Electrical Engineering Material - [Si cube doped 1 As in 10⁶](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asi-cube-arsenic-1e6) — Electrical Engineering Material, chapter 6, asked 5× - [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 - [Si cube with As and B per million](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asi-cube-boron-million) — Electrical Engineering Material, chapter 6 - [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 - [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 - [Resistance of arsenic-doped germanium wire](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Agermanium-arsenic-wire) — Electrical Engineering Material, chapter 6