--- title: "Infinite effective mass and its E-k explanation" subject: "Electrical Engineering Material" chapter: 1 key: "semester_3/electrical_engineering_material:infinite-effective-mass-ek" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/infinite-effective-mass-ek site: BCT Engineering --- # Infinite effective mass and its E-k explanation Electrical Engineering Material · Chapter 1: Theory of Metals - Asked in: 2080 Baisakh, 8 marks - Video: [Effective mass of Electron (Solid state Physics) PHYSICS (in हिन्दी)](https://www.youtube.com/watch?v=__nOVPc7FIU) — Gautam Varde - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/infinite-effective-mass-ek - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1 ## Question What do you understand by infinite effective mass? Derive the expression for effective mass of electron and show that it can be positive as well as negative, explain with the help of E-k diagram. ## Short answer Infinite effective mass occurs where d²E/dk²=0; external force gives zero acceleration. Effective mass m* = ħ²/(d²E/dk²). In an E-k diagram, lower band has m*>0, inflection point m*=∞, upper band m*<0. ## Answer **Effective mass** $m^* = \hbar^2/(d^2E/dk^2)$ is the mass an electron in a crystal appears to have under an external force. **Infinite effective mass** means $d^2E/dk^2 = 0$ (the inflection point of the E–k curve): there the external force produces no acceleration at all, because the force from the lattice exactly balances it. #### Derivation of $m^*$ The electron is a wave packet moving with group velocity ($E = \hbar\omega$): $$ v_g = \frac{d\omega}{dk} = \frac{1}{\hbar}\frac{dE}{dk} $$ An external force $F$ acting for time $dt$ does work $dE = F v_g\,dt$: $$ \frac{dE}{dk}dk = F\cdot\frac{1}{\hbar}\frac{dE}{dk}dt \quad\Rightarrow\quad F = \hbar\frac{dk}{dt} $$ Acceleration: $$ \begin{aligned} a &= \frac{dv_g}{dt} = \frac{1}{\hbar}\frac{d^2E}{dk^2}\frac{dk}{dt} = \frac{1}{\hbar^2}\frac{d^2E}{dk^2}F \end{aligned} $$ Comparing with $F = m^*a$: $$ m^* = \frac{\hbar^2}{d^2E/dk^2} $$ #### Positive, negative and infinite $m^*$ from the E–k diagram In a crystal the band has a cosine-like shape. Taking $E = E_0 - 2\gamma\cos ka$, $$ \frac{dE}{dk} = 2\gamma a\sin ka, \qquad \frac{d^2E}{dk^2} = 2\gamma a^2\cos ka $$ ```text E (a) E vs k ^ __ __ | \ m*<0 m*<0 / | \ / | * <- inflection, -> * | \ m* = infinite / | \ / | `--.__ m*>0 __.--' +----|------------|------------|----> k -pi/a 0 +pi/a v_g ~ dE/dk (b) velocity ^ max at +pi/2a | .--. | / \ +----------*--------*---------> k | \ / 0 +pi/a | '--' ``` - **Lower half of band** ($|k| < \pi/2a$): curve concave up, $d^2E/dk^2 > 0$, **$m^* > 0$**. Velocity rises with $k$; the electron accelerates in the direction of the force, like a normal particle. - **Inflection point** ($|k| = \pi/2a$): $d^2E/dk^2 = 0$, **$m^* = \infty$**. Velocity is maximum; increasing $k$ does not increase $v_g$. The applied force gives zero acceleration. - **Upper half of band** ($\pi/2a < |k| < \pi/a$): curve concave down, $d^2E/dk^2 < 0$, **$m^* < 0$**. Velocity decreases as $k$ increases, so the electron accelerates opposite to the applied force. Physically, the electron is being Bragg-reflected by the lattice and the lattice pushes back harder than the external force. - At $k = \pm\pi/a$ (band edge), $v_g = 0$: a standing wave. #### Significance - Near the bottom of the conduction band electrons have positive $m^*$ and behave like free electrons. - Near the top of the valence band electrons have negative $m^*$. An electron missing from there behaves as a **hole** with positive charge and positive mass, which is the basis of hole conduction in semiconductors. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Effective mass of an electron in a crystal is defined as: A. m* = h^2 / (dE/dk) B. m* = hbar^2 / (d^2E/dk^2) C. m* = hbar * dE/dk D. m* = (d^2E/dk^2) / hbar^2 Answer: B. m* is set by the curvature of the E-k curve: m* = hbar^2/(d^2E/dk^2). (Key point: Definition of m*) 2. When is the effective mass infinite? A. When dE/dk = 0 B. When k = 0 C. When d^2E/dk^2 = 0 (inflection point) D. When d^2E/dk^2 is very large Answer: C. Zero curvature (inflection point) makes m* infinite, so the force gives no acceleration. (Key point: Infinite effective mass) 3. For E = E0 - 2*gamma*cos(ka), where is m* negative? A. For |k| < pi/2a, where the curve is concave up B. Only at k = 0 C. Exactly at |k| = pi/2a D. For pi/2a < |k| < pi/a, where the curve is concave down Answer: D. There d^2E/dk^2 < 0, so m* < 0 and the electron accelerates opposite to the force. (Key point: Negative m*) 4. Which relation, combined with v_g = (1/hbar) dE/dk, gives a = F/m*? A. F = hbar dk/dt B. F = hbar k C. F = dE/dt D. F = h/lambda Answer: A. Work done dE = F v_g dt gives F = hbar dk/dt; then a = (1/hbar^2)(d^2E/dk^2)F. (Key point: Derivation step) ## Close variants set in other exams - [Effective mass can be positive or negative](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aeffective-mass-positive-negative) — Electrical Engineering Material, chapter 1, asked 4× ## Related in Electrical Engineering Material - [Negative and infinite effective mass](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Anegative-infinite-effective-mass) — Electrical Engineering Material, chapter 1 - [Negative effective mass at top of valence band](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Anegative-mass-valence-band-top) — Electrical Engineering Material, chapter 1 - [Effective mass of a free electron](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afree-electron-effective-mass) — Electrical Engineering Material, chapter 1, asked 3× - [Effective mass of electron in gold](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Agold-effective-mass-justify) — Electrical Engineering Material, chapter 1 - [Electron in 0.1 nm well: ground energy](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Awell-numerical-ground-energy) — Electrical Engineering Material, chapter 1, asked 4× - [Electron in 8.5 nm well: fourth level](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Awell-numerical-8nm-fourth-level) — Electrical Engineering Material, chapter 1, asked 4× ## Related in other subjects - [Energy of particle in infinite well](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aparticle-infinite-well-energy-eigenvalue) — Engineering Physics, chapter 10, asked 7× - [Overhang beam: ILD, max positive/negative moment at D](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aoverhang-ild-moment-d-three-loads) — Theory of Structures I, chapter 5 - [Positive XOR equals negative XNOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Apositive-xor-negative-xnor-define) — Digital Logic, chapter 2, asked 3× - [Prove positive XOR equals negative XNOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Apositive-xor-negative-xnor-prove) — Digital Logic, chapter 2, asked 3×