---
title: "Negative and infinite effective mass"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:negative-infinite-effective-mass"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/negative-infinite-effective-mass
site: BCT Engineering
---

# Negative and infinite effective mass

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2073 Shrawan, 6 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/negative-infinite-effective-mass
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

Define and explain the effective mass of electron within a crystal. How do you understand negative and infinite mass of electron?

## Answer

The **effective mass** $m^*$ of an electron in a crystal is the mass that makes Newton's law $F_{ext} = m^*a$ hold for the external force alone. It includes the effect of the internal forces from the lattice, which are too complex to write directly.

#### Expression
With group velocity $v_g = \frac{1}{\hbar}\frac{dE}{dk}$ and $F_{ext} = \hbar\frac{dk}{dt}$:

$$
a = \frac{dv_g}{dt} = \frac{1}{\hbar}\frac{d^2E}{dk^2}\frac{dk}{dt} = \frac{1}{\hbar^2}\frac{d^2E}{dk^2}F_{ext}
$$

$$
m^* = \frac{\hbar^2}{d^2E/dk^2}
$$

So $m^*$ is set by the **curvature** of the E–k curve. For a free electron ($E = \hbar^2k^2/2m_e$), $m^* = m_e$.

#### E–k diagram of a band

```text
   E
   ^   __                           __
   |     \   m* < 0       m* < 0   /
   |      \                       /
   |       x  <- m* = infinite -> x
   |        \   (inflection)     /
   |         \                  /
   |          `-.__ m* > 0 __.-'
   +----|--------------|--------------|---> k
      -pi/a            0            +pi/a
```

#### Negative effective mass
Near the **top of a band** (close to $k = \pm\pi/a$) the curve bends downward, $d^2E/dk^2 < 0$, so $m^* < 0$. Here the electron wave is close to the Bragg condition and is strongly reflected by the lattice. When a force pushes it, the lattice pushes back even harder, so the electron accelerates **opposite** to the applied force, as if its mass were negative. An electron with negative charge and negative mass behaves like a positive charge with positive mass: this is the basis of the **hole** concept for the top of the valence band.

#### Infinite effective mass
At the **inflection point** of the curve ($k = \pm\pi/2a$ for a cosine band) $d^2E/dk^2 = 0$, so $m^* \to \infty$. The velocity $v_g$ is maximum there; an applied force cannot increase it, so the acceleration is zero. The external force is exactly balanced by the lattice force; the electron behaves as an infinitely heavy particle.

Negative and infinite mass are not real changes in the electron's mass; they are ways of describing how the lattice modifies its response to an external force.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Effective mass is the mass that makes which equation valid?
   A. F_ext + F_int = m a
   B. F_ext = m* a
   C. F_int = m* a
   D. F_ext = m_e a
   Answer: B. The lattice effect is absorbed in m*. (Key point: Effective mass)

2. m* is negative near the top of the band because:
   A. The electron has negative charge
   B. dE/dk is zero there
   C. The curve bends downward, d^2E/dk^2 < 0
   D. The electron is free
   Answer: C. Near Bragg reflection the lattice pushes back harder than the applied force. (Key point: Negative mass)

3. At the inflection point of a cosine band:
   A. v_g = 0 and m* = 0
   B. v_g is maximum and acceleration is zero
   C. v_g is minimum and m* is negative
   D. m* equals m_e
   Answer: B. d^2E/dk^2 = 0 so m* is infinite and the force cannot increase the velocity. (Key point: Infinite mass)

4. An electron with negative charge and negative mass behaves like:
   A. A negative charge with positive mass
   B. A positive charge with positive mass (a hole)
   C. A neutral particle
   D. A photon
   Answer: B. (-e)/(-|m*|) = (+e)/(+|m*|). (Key point: Hole concept)

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