Effective mass

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 holes or anti-holes in a solid, the effective mass is usually stated in units of the rest mass of an electron, m_e (9.11×10⁻³¹ kg). In these units it is usually in the range 0.01 to 10,.

• But it can also 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.

In the most general case the effective mass is a tensor known as the inertial effective mass tensor. Its properties can be counter-intuitive:

• The effective mass tensor generally varies depending on wavevector k, meaning that the mass of the particle actually changes after it is subject to an impulse.
  • The only cases in which it remains constant are those of parabolic bands, described below.
• The effective mass tensor diverges (becomes infinite) for linear dispersion relations, such as with photons or electrons in graphene. (These particles are sometimes said to be massless, however this refers to their having zero rest mass; rest mass is a distinct concept from effective mass.)

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

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

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.

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

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

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.)

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:

• For calculation of the total density of states and the total carrier density, via the geometric mean combined with a degeneracy factor g which counts the number of valleys (in silicon g = 6):

$m_{\text{density}}^{*}={\sqrt[{3}]{g^{2}m_{x}m_{y}m_{z}}}$

(This effective mass corresponds to the density of states effective mass, described later.)

For the per-valley density of states and per-valley carrier density, the degeneracy factor is left out.

• For the purposes of calculating conductivity as in the Drude model, via the harmonic mean

$m_{\text{conductivity}}^{*}=3\left[{\frac {1}{m_{x}^{*}}}+{\frac {1}{m_{y}^{*}}}+{\frac {1}{m_{z}^{*}}}\right]^{-1}$

Since the Drude law also depends on scattering time, which varies greatly, this effective mass is rarely used; conductivity is instead usually expressed in terms of carrier density and an empirically measured parameter, carrier mobility.

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 itself. 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. This intuitive principle appears identically in semiclassical approximations derived from band structure. However, each of the symbols has a slightly modified meaning; acceleration becomes the rate of change in group velocity:

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

where ∇_k is the del operator in reciprocal space, and force gives a rate of change in crystal momentum p_crystal:

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

where ħ = h/2π is the reduced Planck constant. Combining these two equations yields

$\mathbf {a} =\nabla _{k}\left({\frac {\mathbf {F} }{\hbar }}\cdot \nabla _{k}\,\omega (\mathbf {k} )\right).$

Extracting the ith element from both sides gives

$a_{i}=\left({\frac {1}{\hbar }}\,{\frac {\partial ^{2}\omega \left(\mathbf {k} \right)}{\partial k_{i}\partial k_{j}}}\right)\!F_{j}=\left({\frac {1}{\hbar ^{2}}}\,{\frac {\partial ^{2}E\left(\mathbf {k} \right)}{\partial k_{i}\partial k_{j}}}\right)\!F_{j},$

where a_i is the ith element of a, F_j is the jth element of F, k_i and k_j are the ith and jth elements of k, respectively, and E is the total energy of the particle according to the Planck–Einstein relation. The index j is contracted by the use of Einstein notation (there is an implicit summation over j). Since Newton's second law uses the inertial mass (not the gravitational mass), we can identify the inverse of this mass in the equation above as the tensor

$\left[M_{\text{inert}}^{-1}\right]_{ij}={\frac {1}{\hbar ^{2}}}{\frac {\partial ^{2}E}{\partial k_{i}\partial k_{j}}}\,.$

This tensor expresses the change in group velocity due to a change in crystal momentum. Its inverse, M_inert, is known as the effective mass tensor.

Cyclotron effective mass

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

$\scriptstyle f_{c}\;=\;{\frac {eB}{2\pi m^{*}}}$.

Classically, a charged particle in a magnetic field moves in a helix along the magnetic field axis. The period T of its motion depends on its mass m and charge e,

$T=\left\vert {\frac {2\pi m}{eB}}\right\vert$

where B is the magnetic flux density.

For particles in asymmetrical band structures, the particle no longer moves exactly in a helix, however its motion transverse to the magnetic field still moves in a closed loop (not necessarily a circle). 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.

The semiclassical motion of the particle can be described by a closed loop in k-space. Throughout this loop, the particle maintains a constant energy, as well as a constant momentum along the magnetic field axis. By defining A to be the k-space area enclosed by this loop (this area depends on the energy E, the direction of the magnetic field, and the on-axis wavevector k_B), then it can be shown that the cyclotron effective mass depends on the band structure via the derivative of this area in energy:

$m^{*}\left(E,{\hat {B}},k_{\hat {B}}\right)={\frac {\hbar ^{2}}{2\pi }}\cdot {\frac {\partial }{\partial E}}A\left(E,{\hat {B}},k_{\hat {B}}\right)$

Typically, experiments that measure cyclotron motion (cyclotron resonance, de Haas–van Alphen effect, etc.) are restricted to only probe motion for energies near the Fermi level.

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. The cyclotron effective mass therefore is only a function of energy, and it turns out to be exactly related to the density of states at that energy via the relation

$\scriptstyle g(E)\;=\;{\frac {g_{v}m^{*}}{\pi \hbar ^{2}}}$, where g_v is the valley degeneracy.

Such a simple relationship does not apply in three-dimensional materials.

Significance

The effective mass is used in transport calculations, such as transport of electrons under the influence of fields or carrier gradients, but it also is used to calculate the carrier density and density of states in semiconductors. These masses are related but are not the same because the weightings of various directions and wavevectors are different. These differences are important, for example in thermoelectric materials, where high conductivity, generally associated with light mass, is desired at the same time as high Seebeck coefficient, generally associated with heavy mass. Methods for assessing the electronic structures of different materials in this context have been developed.

Certain group III-V compounds such as gallium arsenide (GaAs) and indium antimonide (InSb) have far smaller effective masses than tetrahedral group IV materials like silicon and germanium. In the simplest Drude picture of electronic transport, the maximum obtainable charge carrier velocity is inversely proportional to the effective mass:

${\vec {v}}\;=\;\Vert \mu \Vert \cdot {\vec {E}}\quad$ where $\left\Vert \mu \right\Vert \;=\;{\frac {e\tau }{\left\Vert m^{*}\right\Vert }}\quad$ 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.

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.