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). ${\displaystyle \nabla _{k}\left(\nabla _{k}\,E(\mathbf {k} )\right)}$ is the Hessian matrix of E(k) 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

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

whose elements are

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

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.

• For parabolic bands, the off-diagonal elements of M_inert⁻¹ are zero, and the diagonal elements are constants
• For isotropic bands the diagonal elements must all be equal and the off-diagonal elements must all be equal.
• For parabolic isotropic bands, M_inert⁻¹ = 1/m^*I, where m^* is a scalar effective mass and I is the identity.
• In general, the elements of M_inert⁻¹ are functions of k.
• The inverse, M_inert = (M_inert⁻¹)⁻¹, is known as the effective mass tensor. Note that it is not always possible to invert M_inert⁻¹

For bands with linear dispersion ${\displaystyle E\propto k}$ such as with photons or electrons in graphene, the group velocity is fixed, i.e. electrons travelling with parallel with k to the force direction F cannot be accelerated and the diagonal elements of M_inert⁻¹ are obviously zero. However, electrons travelling with a component perpendicular to the force can be accelerated in the direction of the force, and the off-diagonal elements of M_inert⁻¹ are non-zero. In fact the off-diagonal elements scale inversely with k, i.e. they diverge (become infinite) for small k. This is why the electrons in graphene are sometimes said to have infinite mass (due to the zeros on the diagonal of M_inert⁻¹) and sometimes said to be massless (due to the divergence on the off-diagonals). ^[7]

Cyclotron effective mass

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,

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

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:

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

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 ${\displaystyle \scriptstyle g(E)\;=\;{\frac {g_{v}m^{*}}{\pi \hbar ^{2}}}}$, where g_v is the valley degeneracy. Such a simple relationship does not apply in three-dimensional materials.

Density of states effective masses (lightly doped semiconductors)

In semiconductors with low levels of doping, the electron concentration in the conduction band is in general given by

${\displaystyle n_{\text{e}}=N_{\text{C}}\exp \left(-{\frac {E_{\text{C}}-E_{\text{F}}}{kT}}\right)}$

where E_F is the Fermi level, E_C is the minimum energy of the conduction band, and N_C is a concentration coefficient that depends on temperature. The above relationship for n_e can be shown to apply for any conduction band shape (including non-parabolic, asymmetric bands), provided the doping is weak (E_C − E_F ≫ kT); this is a consequence of Fermi–Dirac statistics limiting towards Maxwell–Boltzmann statistics.

The concept of effective mass is useful to model the temperature dependence of N_C, thereby allowing the above relationship to be used over a range of temperatures. In an idealized three-dimensional material with a parabolic band, the concentration coefficient is given by

${\displaystyle \quad N_{\text{C}}=2\left({\frac {2\pi m_{\text{e}}^{*}kT}{h^{2}}}\right)^{\frac {3}{2}}}$

In semiconductors with non-simple band structures, this relationship is used to define an effective mass, known as the density of states effective mass of electrons. The name "density of states effective mass" is used since the above expression for N_C is derived via the density of states for a parabolic band.

In practice, the effective mass extracted in this way is not quite constant in temperature (N_C does not exactly vary as T^3/2). 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.^[8]

Similarly, the number of holes in the valence band, and the density of states effective mass of holes are defined by:

${\displaystyle n_{\text{h}}=N_{\text{V}}\exp \left(-{\frac {E_{\text{F}}-E_{\text{V}}}{kT}}\right),\quad N_{\text{V}}=2\left({\frac {2\pi m_{\text{h}}^{*}kT}{h^{2}}}\right)^{\frac {3}{2}}}$

where E_V is the maximum energy of the valence band. 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.^[8]

Experimental

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 ${\displaystyle \scriptstyle f_{c}\;=\;{\frac {eB}{2\pi m^{*}}}}$. In recent years effective masses have more commonly been determined through measurement of band structures using techniques such as angle-resolved photoemission spectroscopy (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 ${\displaystyle \scriptstyle c_{v}}$. 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 ${\displaystyle \tau }$ 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. The optical Hall effect is an emerging technique for measuring the free charge carrier density, effective mass and mobility parameters in semiconductors. The optical Hall effect measures the analogue of the quasi-static electric-field-induced electrical Hall effect at optical frequencies in conductive and complex layered materials. The optical Hall effect also permits characterization of the anisotropy (tensor character) of the effective mass and mobility parameters.^[12][13]

Theoretical

A variety of theoretical methods including density functional theory, k·p perturbation theory, and others are used to supplement and support the various experimental measurements described in the previous section, including interpreting, fitting, and extrapolating these measurements. 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.