In solid-state physics, the concept of effective mass is used to describe the behavior of electrons in a crystal lattice. The effective mass of an electron in a solid is the mass that the electron appears to have when it moves through the crystal lattice under the influence of an external force.

The effective mass of an electron in a solid is generally different from its free-space mass because of the presence of the crystal lattice. The crystal lattice can affect the motion of electrons in several ways, including:

The lattice can cause the electron to experience an effective force, which can alter its motion.
The electron can interact with the lattice atoms, causing it to scatter and lose energy.
The electron can be confined to certain regions of the crystal lattice, leading to a modification of its wavefunction.

The effective mass of an electron is defined as the curvature of the energy-momentum relation near the band edge. In other words, it is the second derivative of the energy of the electron with respect to its momentum.

The effective mass can be calculated using experimental techniques, such as cyclotron resonance, or theoretical methods, such as band structure calculations. The effective mass is an important quantity in solid-state physics because it affects many physical properties of the solid, including its electrical conductivity, optical properties, and thermal conductivity.

We know,

Vg = dw/dk

Comparing F = ma

It gives effective mass of electron on the crystal. The ratio of rest mass of free electron m to its effective mass m* in the crystal in k state is represented by

The variation of effective mass with wave vector k is shown in figure.

Effective mass depends upon d²E/ dk² and is a function of K since v∝k and v∝ dE/dk. This also gives variation of dE/dk with k. d²E/dk² is positive which represents lower half of energy band. For k = - π/a to -ko and k = ko to + π/a, d²E/ dk² is negative which represents upper half of energy band. In other words, effective mass m* is positive at bottom and negative at the top.

This note is a part of the Physics Repository.

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