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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 electrons and holes in a crystal respond to electric and magnetic fields almost as if they were particles with a mass dependence in their direction of travel, an effective mass tensor.[1] In a simplified picture that ignores crystal anisotropies, they behave as 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). In these units it is usually in the range 0.01 to 10, but can also be lower or higher, for example reaching 1000 in exotic heavy fermion materials, and reaching 0 at the so-called Dirac points in graphene.

The effective mass has important effects on the properties of a solid, including everything from the efficiency of a solar cell to the speed of an integrated circuit.
Constant energy ellipsoids in silicon near the six conduction band minima. The longitudinal and transverse effective masses are mℓ =0.92 m and mt = 0.19 m with m the free electron mass.[2]

Definition

When an electron is moving inside a 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. Generally, in the absence of an electric or magnetic field, the concept of effective mass does not apply.
Bulk band structure for Si,Ge,GaAs and InAs generated with tight binding model. Note that Si and Ge are indirect with minima at X and L, while GaAs and InAs are direct band gap materials.

Effective mass is defined by analogy with Newton's second law F = m a. Using quantum mechanics it can be shown that for an electron in an external electric field E, the acceleration aℓ along coordinate direction ℓ is:

$$a_{\ell} = {{1} \over {\hbar^2}} \sum_{ m} \cdot {{\partial^2 \varepsilon (\boldsymbol{k})} \over {\partial k_{\ell} \partial k_ m}} qE_m$$

where ћ = h/2π is reduced Planck's constant, k is the wave vector (often loosely called momentum since k = p / ћ for free electrons), ε (k) is the energy as a function of k, or the dispersion relation as it is often called.

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, for example:

$$\frac{1}{\hbar^2} \varepsilon (\boldsymbol{k}) \approx \frac{1}{\hbar^2} \varepsilon (\boldsymbol{k}=0)+ \frac{1}{m_x} k_x^2 + \frac{1}{m_y} k_y^2 + \frac{1}{m_z} k_z^2 \ ,$$

where the minimum is assumed to occur at k=0. In many semiconductors the minimum does not occur at k=0. For example, in silicon the conduction band has six symmetrically located minima along the Δ = [100] symmetry lines in k-space. The constant energy surfaces at these minima are ellipsoids oriented along the k-space axes (see figure).[3]

In contrast, the holes at the top of the silicon valence band are classified as light and heavy and the band energies for the two types are given by a complicated relation:[4]

$$\varepsilon (\boldsymbol{k}) = Ak^2 \pm \sqrt{ B^2 k^4 + C^2 \left( k_x^2 k_y^2+ k_y^2 k_z^2+k_z^2 k_x^2 \right)} \ ,$$

leading to what is termed warped energy surfaces. Parameters A, B and C are wavevector independent constants. This behavior is introduced here to alert the reader that it is common for carriers to have rather non-parabolic energy-wavevector relations.

A simplification can be made, however, for electrons which have energy close to a minimum, and where the effective mass is the same in all directions, the mass can be approximated as a scalar m*:

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

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 many 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 be understood only with quantum mechanics.
Derivation

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

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

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. Substitute the expression for the group velocity into this last equation and we get:

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

From this follows the definition of effective mass:

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

An alternative derivation can be given by considering the Hamiltonian of a free particle and using the de Broglie relation:

$${H={{p^{2}} \over {2m}} = {{\hbar^2 k^2} \over {2m}}}$$

The same result is obtained, identifying the Hamiltonian with the total energy:

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

Effective mass for some common semiconductors
Effective mass (me)[5][6][7] Group Material Electron me Hole mh
IV Si (300K) 1.08 0.56
Ge 0.55 0.37
III-V GaAs 0.067 0.45
InSb 0.013 0.6
II-VI ZnO 0.29 1.21
ZnSe 0.17 1.44

The effective mass is used in transport calculations, such as transport of electrons under the influence of fields or carrier gradients, but also is used to calculate the density of states. These masses are related, but are not the same because the weighting of various directions and wavevectors are different. The density-of-states effective masses are tabulated below for a few cases.
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^* }. In recent years effective masses have more commonly been determined through measurement of band structures using techniques such as angle-resolved photo emission (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 c_v. 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 \tau 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.

k·p perturbation theory
Electrical conduction
Kronig-Penney model
Tight-binding model

Footnotes

^ Charles Kittel (1996). Introduction to Solid State Physics (7th Edition ed.). Wiley. p. Eq. 29, p. 210. ISBN 0-471-11181-3.
^ Charles Kittel. op. cit.. p. 216. ISBN 0-471-11181-3.
^ Peter Y Yu and Manuel Cardona (2001). Fundamentals of Semiconductors: Physical and material properties (3rd Edition ed.). Springer. Figure 2.10, p. 53. ISBN 3-540-25470-6.
^ See Charles Kittel. op. cit.. p. 214. ISBN 0-471-11181-3.
^ 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.

References

Pastori Parravicini, G. (1975). Electronic States and Optical Transitions in Solids. Pergamon Press. ISBN 0-08-016846-9. This book contains an exhaustive but accessible discussion of the topic with extensive comparison between calculations and experiment.
S. Pekar, The method of effective electron mass in crystals, Zh. Eksp. Teor. Fiz. 16, 933 (1946).

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