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# Effective mass

In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. This mass is usually stated in units of the ordinary mass of an electron me (9.11×10-31 kg).

## Definition

When electron is moving inside the solid material, the force between other atoms will affect its movement and it will not be described by Newton's law. So we introduce the concept of effective mass to describe the movement of electron in Newton's law. The effective mass can be negative or different due to circumstances.

Effective mass is defined by analogy with Newton's second law $\vec{F}=m\vec{a}$. Using quantum mechanics it can be shown that for an electron in an external electric field E:

$a = {{1} \over {\hbar^2}} \cdot {{d^2 \varepsilon} \over {d k^2}} qE$

where a is acceleration, $\hbar = h/2\pi$ is reduced Planck's constant, k is the wave number (often loosely called momentum since k = $p/\hbar$ for free electrons), ε(k) is the energy as a function of k, or the dispersion relation as it is often called. From the external electric field alone, the electron would experience a force of $\vec{F}=q\vec{E}$, where q is the charge. Hence under the model that only the external electric field acts, effective mass m * becomes:

$m^{*} = \hbar^2 \cdot \left[ {{d^2 \varepsilon} \over {d k^2}} \right]^{-1}$

For a free particle, the dispersion relation is a quadratic, and so the effective mass would be constant (and equal to the real mass). In a crystal, the situation is far more complex. The dispersion relation is not even approximately quadratic, in the large scale. However, wherever a minimum occurs in the dispersion relation, the minimum can be approximated by a quadratic curve in the small region around that minimum. Hence, for electrons which have energy close to a minimum, effective mass is a useful concept.

In energy regions far away from a minimum, effective mass can be negative or even approach infinity. Effective mass, being generally dependent on direction (with respect to the crystal axes), is a tensor. However, for most calculations the various directions can be averaged out.

Effective mass should not be confused with reduced mass, which is a concept from Newtonian mechanics. Effective mass can only be understood with quantum mechanics.

## Quantum Calculation

In the free electron model, the electronic wave function can be in the form of $e^{i k \cdot z}$. For a wave packet the group velocity is given by:

$v = {{d \omega} \over {d k}}$ = ${{1} \over {\hbar}} \cdot {{d \varepsilon} \over {d k}}$

In presence of an electric field E, the energy change is:

$d \varepsilon = {{d \varepsilon} \over {d k}} {d k} = -eE{d x} = -eEv{d t} = {-eE \over {\hbar}} {{d \varepsilon} \over {d k}}{dt}$

Now we can say:

$\hbar \cdot {{d k} \over {d t}} = {{d p} \over {d t}} = m \cdot {{d v} \over {d t}}$

where p is the electron's momentum. Just put previous results in this last equation and we get:

${{\hbar} \over {m}} \cdot {{d k} \over {d t}} = {{1} \over {\hbar}} \cdot {{d} \over {d t}}{{d \varepsilon} \over {d k}} ={{1} \over {\hbar}} \cdot {{d^{2} \varepsilon} \over {d^{2} k}}{{d k}\over {d t}}$

From this follows the definition of effective mass:

${{1} \over {m}} = {{1} \over {\hbar^{2}}} \cdot {{d^{2} \varepsilon} \over {d k^2}}$

## Effective mass for some common semiconductors (for density of states calculations)

Material Electron effective mass Hole effective mass
Group IV
Si (4.2K) 1.08 me 0.56 me
Ge 0.55 me 0.37 me
III-V
GaAs 0.067 me 0.45 me
InSb 0.013 me 0.6 me
II-VI
ZnO 0.19 me 1.21 me
ZnSe 0.17me 1.44 me

Sources:
S.Z. Sze, Physics of Semiconductor Devices, ISBN 0-471-05661-8.
W.A. Harrison, Electronic Structure and the Properties of Solids, ISBN 0-486-66021-4.
This site gives the effective masses of Silicon at different temperatures.

## Experimental determination

Traditionally effective masses were measured using cyclotron resonance, a method in which microwave absorption of a semiconductor immersed in a magnetic field goes through a sharp peak when the microwave frequency equals the cyclotron frequency $\omega_c = \frac{eB}{m^* c}$. In recent years effective masses have more commonly been determined through measurement of band structures using techniques such as angle-resolved photoemission (ARPES) or, most directly, the de Haas-van Alphen effect. Effective masses can also be estimated using the coefficient γ of the linear term in the low-temperature electronic specific heat at constant volume cv. The specific heat depends on the effective mass through the density of states at the Fermi level and as such is a measure of degeneracy as well as band curvature. Very large estimates of carrier mass from specific heat measurements have given rise to the concept of Heavy Fermion materials. Since carrier mobility depends on the ratio of carrier collision lifetime τ to effective mass, masses can in principle be determined from transport measurements, but this method is not practical since carrier collision probabilities are typically not known a priori.

## Significance

As the table shows, III-V compounds based on GaAs and InSb have far smaller effective masses than tetrahedral group IV materials like Si and Ge. In the simplest Drude picture of electronic transport, the maximum obtainable charge carrier velocity is inversely proportional to the effective mass: $\vec{v} = \begin{Vmatrix}\mu\end{Vmatrix} \cdot \vec{E}$ where $\begin{Vmatrix}\mu\end{Vmatrix} = \frac{e \tau}{\begin{Vmatrix}m^*\end{Vmatrix}}$ with e being the electronic charge. The ultimate speed of integrated circuits depends on the carrier velocity, so the low effective mass is the fundamental reason that GaAs and its derivatives are used instead of Si in high-bandwidth applications like cellular telephony.