Effective mass (solid-state physics)
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.
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.
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.
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.
The offsets k0,x, k0,y, and k0,z reflect that the conduction band minimum is no longer centered at zero wavevector.
This is because there are multiple valleys (conduction-band minima), each with effective masses rearranged along different axes.
It is possible to average the different axes' effective masses together in some way, to regain the free electron picture.
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.
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 pcrystal: where ħ = h/2π is the reduced Planck constant.
Acceleration for a wave-like particle becomes the rate of change in group velocity: where ∇k is the del operator in reciprocal space.
We see that the equivalent of the Newtonian reciprocal inertial mass for a free particle defined by a = m−1F has become a tensor quantity whose elements are This tensor allows the acceleration and force to be in different directions, and for the magnitude of the acceleration to depend on the direction of the force.
Moreover, the time to complete one of these loops still varies inversely with magnetic field, and so it is possible to define a cyclotron effective mass from the measured period, using the above equation.
In two-dimensional electron gases, the cyclotron effective mass is defined only for one magnetic field direction (perpendicular) and the out-of-plane wavevector drops out.
In semiconductors with low levels of doping, the electron concentration in the conduction band is in general given by where EF is the Fermi level, EC is the minimum energy of the conduction band, and NC is a concentration coefficient that depends on temperature.
In silicon, for example, this effective mass varies by a few percent between absolute zero and room temperature because the band structure itself slightly changes in shape.
These band structure distortions are a result of changes in electron–phonon interaction energies, with the lattice's thermal expansion playing a minor role.
Practically, this effective mass tends to vary greatly between absolute zero and room temperature in many materials (e.g., a factor of two in silicon), as there are multiple valence bands with distinct and significantly non-parabolic character, all peaking near the same energy.
Effective masses can also be estimated using the coefficient γ of the linear term in the low-temperature electronic specific heat at constant volume
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.
Some of these theoretical methods can also be used for ab initio predictions of effective mass in the absence of any experimental data, for example to study materials that have not yet been created in the laboratory.
These masses are related but, as explained in the previous sections, are not the same because the weightings of various directions and wavevectors are different.
In the simplest Drude picture of electronic transport, the maximum obtainable charge carrier velocity is inversely proportional to the effective mass:
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.
[15] In April 2017, researchers at Washington State University claimed to have created a fluid with negative effective mass inside a Bose–Einstein condensate, by engineering the dispersion relation.
