--- title: "Concept of effective mass in a crystal" subject: "Electrical Engineering Material" chapter: 1 key: "semester_3/electrical_engineering_material:effective-mass-concept-crystal" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/effective-mass-concept-crystal site: BCT Engineering --- # Concept of effective mass in a crystal Electrical Engineering Material · Chapter 1: Theory of Metals - Asked in: 2082 Kartik (new course), 5 marks; 2075 Chaitra, 6 marks; 2070 Asar, 6 marks; 2068 Baisakh, 6 marks (asked 4 times) - 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/effective-mass-concept-crystal - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1 ## Question Explain the concept of effective mass in crystal with necessary mathematical expression. ## Short answer The effective mass m* = ħ²/(d²E/dk²) describes an electron's response to external forces, incorporating the lattice's periodic potential. It depends on the curvature of the E-k band: positive at band bottom, negative at top, and infinite at inflection points, enabling a free-particle-like description. ## Answer The **effective mass** $m^*$ of an electron in a crystal is the mass it appears to have when it responds to an **external** force, after the effect of all the internal forces from the lattice ions and other electrons has been included in it. It lets us treat a crystal electron as if it were a free particle with mass $m^*$. #### Why it is needed Inside a crystal an electron feels the external force $F_{ext}$ (for example $-eE$ from an applied field) and also the internal periodic force $F_{int}$ from the lattice. Newton's law gives $$ F_{ext} + F_{int} = m_e a $$ $F_{int}$ is very complicated and changes from point to point. We therefore write $$ F_{ext} = m^* a $$ and put all the effect of the lattice into $m^*$. #### Expression for $m^*$ The electron is a wave packet that moves with the group velocity $$ v_g = \frac{d\omega}{dk} = \frac{1}{\hbar}\frac{dE}{dk} $$ Its acceleration is $$ a = \frac{dv_g}{dt} = \frac{1}{\hbar}\frac{d^2E}{dk^2}\frac{dk}{dt} $$ In time $dt$ the external force does work $dE = F_{ext}v_g\,dt$, so $\frac{dE}{dk}dk = F_{ext}\frac{1}{\hbar}\frac{dE}{dk}dt$, giving $$ F_{ext} = \hbar\frac{dk}{dt} $$ Substituting $dk/dt = F_{ext}/\hbar$: $$ a = \frac{1}{\hbar^2}\frac{d^2E}{dk^2}F_{ext} \quad\Rightarrow\quad m^* = \frac{\hbar^2}{d^2E/dk^2} $$ #### Meaning - $m^*$ depends on the **curvature** of the E–k curve. A sharply curved band gives a small $m^*$ (light, fast electron); a flat band gives a large $m^*$. - For a free electron $E = \hbar^2k^2/2m_e$, so $d^2E/dk^2 = \hbar^2/m_e$ and $m^* = m_e$. - Near the bottom of a band $m^*$ is positive; near the top it is negative (this is why holes are used); at the inflection point it is infinite. #### Typical values | Material | $m_e^*/m_e$ | $m_h^*/m_e$ | |---|---|---| | Si | 1.08 | 0.56 | | Ge | 0.55 | 0.37 | | GaAs | 0.067 | 0.48 | The small $m^*$ of electrons in GaAs is why GaAs devices are faster than Si devices. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the effective mass m* of a crystal electron represent? A. The actual rest mass of the electron changes in a crystal B. The mass of the lattice ions C. The mass it appears to have under an external force, with lattice effects included D. The mass of an electron-hole pair Answer: C. m* absorbs all internal lattice forces so that F_ext = m* a. (Key point: Meaning of m*) 2. Which formula gives m*? A. m* = hbar^2 / (d^2E/dk^2) B. m* = hbar^2 x (d^2E/dk^2) C. m* = hbar / (dE/dk) D. m* = (d^2E/dk^2) / hbar^2 Answer: A. From a = (1/hbar^2)(d^2E/dk^2)F, m* = hbar^2 / (d^2E/dk^2). (Key point: Formula) 3. How does m* depend on the E-k curve? A. Sharper curvature gives larger m* B. Sharper curvature gives smaller m* (lighter electron) C. m* is independent of curvature D. A flat band gives zero m* Answer: B. m* is inversely proportional to curvature, so a sharply curved band gives a light electron and a flat band a heavy one. (Key point: Curvature) 4. What is the sign of m* near the top of a band? A. Positive B. Zero C. Infinite D. Negative Answer: D. At the band top the curve is concave down, d^2E/dk^2 < 0, so m* is negative, which leads to the hole concept. (Key point: Sign of m*) ## 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 - [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× - [Infinite effective mass and its E-k explanation](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ainfinite-effective-mass-ek) — Electrical Engineering Material, chapter 1, asked 4× - [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 - [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 ## Related in other subjects - [Single-phase cycloconverter with mathematical expression](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asingle-phase-cyclo-math) — Power Electronics, chapter 6 - [Two-mass system X2/F by free body diagram, F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afbd-fv-m1-m2-k1-b1) — Control Systems, chapter 2 - [Optimum filter; Remez exchange expression](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aoptimum-filter-remez-expression) — Digital Signal Analysis and Processing, chapter 5, asked 5× - [Flange width concept in T and L beams](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aflange-width-concept-in-t-and) — Design of RCC Structure, chapter 4