In solid state physics, a particle's effective mass (often denoted ${\textstyle m^{*}}$) is the mass that it seems to have when responding to forces, or the mass that it seems to have when interacting with other identical particles in a thermal distribution. One of the results from the band theory of solids is that the movement of particles in a periodic potential, over long distances larger than the lattice spacing, can be very different from their motion in a vacuum. The effective mass is a quantity that is used to simplify band structures by modeling the behavior of a free particle with that mass. For some purposes and some materials, the effective mass can be considered to be a simple constant of a material. In general, however, the value of effective mass depends on the purpose for which it is used, and can vary depending on a number of factors.

For electrons or electron holes in a solid, the effective mass is usually stated as a factor multiplying the rest mass of an electron, m_e (9.11 × 10⁻³¹ kg). This factor is usually in the range 0.01 to 10, but can be lower or higher—for example, reaching 1,000 in exotic heavy fermion materials, or anywhere from zero to infinity (depending on definition) in graphene. As it simplifies the more general band theory, the electronic effective mass can be seen as an important basic parameter that influences measurable properties of a solid, including everything from the efficiency of a solar cell to the speed of an integrated circuit.

Simple case: parabolic, isotropic dispersion relation

At the highest energies of the valence band in many semiconductors (Ge, Si, GaAs, ...), and the lowest energies of the conduction band in some semiconductors (GaAs, ...), the band structure E(k) can be locally approximated as

${\displaystyle E(\mathbf {k} )=E_{0}+{\frac {\hbar ^{2}\mathbf {k} ^{2}}{2m^{*}}}}$

where E(k) is the energy of an electron at wavevector k in that band, E₀ is a constant giving the edge of energy of that band, and m^* is a constant (the effective mass).

It can be shown that the electrons placed in these bands behave as free electrons except with a different mass, as long as their energy stays within the range of validity of the approximation above. As a result, the electron mass in models such as the Drude model must be replaced with the effective mass.

One remarkable property is that the effective mass can become negative, when the band curves downwards away from a maximum. As a result of the negative mass, the electrons respond to electric and magnetic forces by gaining velocity in the opposite direction compared to normal; even though these electrons have negative charge, they move in trajectories as if they had positive charge (and positive mass). This explains the existence of valence-band holes, the positive-charge, positive-mass quasiparticles that can be found in semiconductors.^[1]

In any case, if the band structure has the simple parabolic form described above, then the value of effective mass is unambiguous. Unfortunately, this parabolic form is not valid for describing most materials. In such complex materials there is no single definition of "effective mass" but instead multiple definitions, each suited to a particular purpose. The rest of the article describes these effective masses in detail.

Intermediate case: parabolic, anisotropic dispersion relation

In some important semiconductors (notably, silicon) the lowest energies of the conduction band are not symmetrical, as the constant-energy surfaces are now ellipsoids, rather than the spheres in the isotropic case. Each conduction band minimum can be approximated only by

${\displaystyle E\left(\mathbf {k} \right)=E_{0}+{\frac {\hbar ^{2}}{2m_{x}^{*}}}\left(k_{x}-k_{0,x}\right)^{2}+{\frac {\hbar ^{2}}{2m_{y}^{*}}}\left(k_{y}-k_{0,y}\right)^{2}+{\frac {\hbar ^{2}}{2m_{z}^{*}}}\left(k_{z}-k_{0,z}\right)^{2}}$

where x, y, and z axes are aligned to the principal axes of the ellipsoids, and m^*
_x, m^*
_y and m^*
_z are the inertial effective masses along these different axes. The offsets k_0,x, k_0,y, and k_0,z reflect that the conduction band minimum is no longer centered at zero wavevector. (These effective masses correspond to the principal components of the inertial effective mass tensor, described later.^[3])

In this case, the electron motion is no longer directly comparable to a free electron; the speed of an electron will depend on its direction, and it will accelerate to a different degree depending on the direction of the force. Still, in crystals such as silicon the overall properties such as conductivity appear to be isotropic. This is because there are multiple valleys (conduction-band minima), each with effective masses rearranged along different axes. The valleys collectively act together to give an isotropic conductivity. It is possible to average the different axes' effective masses together in some way, to regain the free electron picture. However, the averaging method turns out to depend on the purpose:^[4]

General case

In general the dispersion relation cannot be approximated as parabolic, and in such cases the effective mass should be precisely defined if it is to be used at all. Here a commonly stated definition of effective mass is the inertial effective mass tensor defined below; however, in general it is a matrix-valued function of the wavevector, and even more complex than the band structure. Other effective masses are more relevant to directly measurable phenomena.

Inertial effective mass tensor

A classical particle under the influence of a force accelerates according to Newton's second law, a = m⁻¹F, or alternatively, the momentum changes according to d/dtp = F. This intuitive principle appears identically in semiclassical approximations derived from band structure when interband transitions can be ignored for sufficiently weak external fields.^[5][6] The force gives a rate of change in crystal momentum p_crystal:

${\displaystyle \mathbf {F} ={\frac {\operatorname {d} \mathbf {p} _{\text{crystal}}}{\operatorname {d} t}}=\hbar {\frac {\operatorname {d} \mathbf {k} }{\operatorname {d} t}},}$

where ħ = h/2π is the reduced Planck constant.

Acceleration for a wave-like particle becomes the rate of change in group velocity:

${\displaystyle \mathbf {a} ={\frac {\operatorname {d} }{\operatorname {d} t}}\,\mathbf {v} _{\text{g}}={\frac {\operatorname {d} }{\operatorname {d} t}}\left(\nabla _{k}\,\omega \left(\mathbf {k} \right)\right)=\nabla _{k}{\frac {\operatorname {d} \omega \left(\mathbf {k} \right)}{\operatorname {d} t}}=\nabla _{k}\left({\frac {\operatorname {d} \mathbf {k} }{\operatorname {d} t}}\cdot \nabla _{k}\,\omega (\mathbf {k} )\right),}$

where ∇_k is the del operator in reciprocal space. The last step follows from using the chain rule for a total derivative for a quantity with indirect dependencies, because the direct result of the force is the change in k(t) given above, which indirectly results in a change in E(k)=ħω(k). Combining these two equations yields

${\displaystyle \mathbf {a} =\nabla _{k}\left({\frac {\mathbf {F} }{\hbar }}\cdot \nabla _{k}\,{\frac {E(\mathbf {k} )}{\hbar }}\right)={\frac {1}{\hbar ^{2}}}\left(\nabla _{k}\left(\nabla _{k}\,E(\mathbf {k} )\right)\right)\cdot \mathbf {F} =M_{\text{inert}}^{-1}\cdot \mathbf {F} }$

using the dot product rule with a uniform force (∇_kF=0). ${\displaystyle {\mathrm{\nabla <!--\; \nabla \; -->}}_k({\mathrm{\nabla <!--\; \nabla \; -->}}_k}$Zdroj:https://en.wikipedia.org?pojem=Effective_mass_(solid-state_physics)
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