Effective Mass

Effective mass is a local response coefficient extracted from the dispersion of a band or quasiparticle. It tells how band velocity changes when crystal momentum changes. Near a smooth band extremum, this response makes a Bloch electron behave approximately like a free particle with a modified, generally anisotropic mass:

$$\varepsilon_n(\mathbf k_0+\mathbf q) \approx \varepsilon_n(\mathbf k_0) + \frac{\hbar^2}{2} \mathbf q^{\mathsf T} \mathsf M_n^{-1} \mathbf q.$$

The tensor $\mathsf M_n$ is not the electron’s bare mass $m_e$. It summarizes the combined effect of kinetic energy, the periodic potential, orbital hybridization, and, when a quasiparticle dispersion is used, interaction renormalization. Its value depends on the band, expansion point, direction, energy window, and observable.

Required background. Band Theory Overview supplies smooth band dispersions, band indices, occupations, and the band-versus-quasiparticle distinction used here.

Helpful background. Density of States supplies spectral state counting, while Fermi Surface supplies constant-energy geometry and closed-orbit language.

Why a Crystal Can Change Inertia

For a free electron,

$$\varepsilon(\mathbf k) = \frac{\hbar^2 k^2}{2m_e},$$

so the curvature is fixed by $m_e$. In a crystal, Bragg scattering and orbital mixing reshape the dispersion. A large local curvature magnitude gives a small mass magnitude, while a locally flat band has a small inverse mass. The total bandwidth alone does not determine the curvature at a particular momentum. Curvature can even change sign within one band.

The crystal does not literally alter the rest mass or charge of the electron. Instead, an external force changes the electron’s crystal momentum, while the periodic lattice continually redistributes momentum between the wave packet and the crystal. Effective mass packages the resulting acceleration into a free-particle-like law over a limited region of momentum space.

Several quantities are called an effective mass:

Name	Defined by	Most directly relevant to
curvature tensor	Hessian of $\varepsilon_n(\mathbf k)$	local acceleration in a smooth band
density-of-states mass	state-counting equivalence	carrier density and thermodynamics
conductivity mass	current response with a scattering model	dc transport and mobility
cyclotron mass	energy derivative of orbit area	quantum oscillations and cyclotron resonance
optical mass	intraband spectral weight	Drude response and plasma frequency
quasiparticle mass	curvature or velocity of an interacting pole	low-energy interacting excitations

In an independent-carrier model with one isotropic parabolic band and a constant relaxation time, these mass definitions reduce to the same band parameter. Interacting response functions can differ even when the quasiparticle dispersion is approximately parabolic. Outside that special case, a quoted value without a definition is incomplete.

Local Expansion of a Band

Let $\mathbf k_0$ be a point in a smooth, nondegenerate band and write

$$\mathbf q = \mathbf k-\mathbf k_0.$$

The Taylor expansion is

$$\varepsilon_n(\mathbf k_0+\mathbf q) ={} \varepsilon_{n0} + \hbar\, \mathbf v_{n0}\cdot\mathbf q + \frac{\hbar^2}{2} \sum_{ij} q_i \left( \mathsf M_n^{-1} \right)_{ij} q_j + O(q^3),$$

where

$$\mathbf v_{n0} = \frac{1}{\hbar} \nabla_{\mathbf k} \varepsilon_n(\mathbf k) \biggr\rvert_{\mathbf k_0}$$

and

$$\left( \mathsf M_n^{-1} \right)_{ij} = \frac{1}{\hbar^2} \frac{\partial^2\varepsilon_n} {\partial k_i\,\partial k_j} \biggr\rvert_{\mathbf k_0}.$$

It is the inverse mass tensor that is directly equal to the Hessian divided by $\hbar^2$. The Hessian is real and symmetric for a smooth real-valued band energy, so it can be diagonalized by an orthogonal change of axes.

At a nondegenerate quadratic band extremum, $\mathbf v_{n0}=0$, and the quadratic term is the leading momentum dependence. Away from an extremum, the linear term must be retained. Silently dropping it changes both the reference velocity and the physical meaning of the approximation.

One dimension

In one dimension,

$$\frac{1}{m^*(k_0)} = \frac{1}{\hbar^2} \frac{d^2\varepsilon}{dk^2} \biggr\rvert_{k_0},$$

or

$$m^*(k_0) = \frac{\hbar^2} {d^2\varepsilon/dk^2\rvert_{k_0}}.$$

Positive curvature gives positive electron effective mass. Negative curvature gives negative electron effective mass. At an inflection point the inverse mass vanishes; writing $m^*=\infty$ is a shorthand for zero local acceleration under a force, not evidence that the wave packet has acquired infinite rest energy.

Principal masses

In principal-axis coordinates,

$$\mathsf M^{-1} = \operatorname{diag} \left( \frac{1}{m_1}, \frac{1}{m_2}, \frac{1}{m_3} \right).$$

A nondegenerate quadratic minimum has finite $m_i>0$ in every direction. A nondegenerate quadratic maximum has $m_i<0$ in the electron description. A quadratic saddle has mixed signs. At a higher-order critical point, a Hessian eigenvalue can vanish even for a smooth isolated band: the corresponding inverse mass is zero and higher-order dispersion is needed. This does not by itself require a multiband Hamiltonian. A degeneracy between bands or a band crossing is a separate issue; an individual-band Hessian can then fail to be smooth, and a coupled-subspace effective Hamiltonian is generally the correct object.

Effective mass is local and tensorial. In (a), one band contains positive, zero, and negative curvature. In (b), $\mathsf M=\operatorname{diag}(m,4m)$ gives a constant-energy ellipse twice as tall as it is wide. A force at $45^\circ$ produces acceleration at $\arctan(1/4)\simeq14^\circ$. The arrows compare directions, not lengths in common units.

Semiclassical Acceleration

The group velocity of a wave packet in band $n$ is

$$v_i = \frac{1}{\hbar} \frac{\partial\varepsilon_n} {\partial k_i}.$$

If a slowly varying perturbation produces a semiclassical force

$$\hbar\dot{\mathbf k} = \mathbf F,$$

then the chain rule gives

$$\begin{aligned} \dot v_i &= \frac{1}{\hbar} \sum_j \frac{\partial^2\varepsilon_n} {\partial k_i\,\partial k_j} \dot k_j \\ &= \sum_j \left( \mathsf M_n^{-1} \right)_{ij} F_j. \end{aligned}$$

Thus

$$\dot{\mathbf v} = \mathsf M_n^{-1}\mathbf F.$$

This is the dynamical origin of the curvature mass. If $\mathsf M$ is anisotropic, force and acceleration need not be parallel. For an electron of charge $-e$ in electric and magnetic fields, the ordinary band-force term is

$$\mathbf F = -e \left( \mathbf E+\mathbf v\times\mathbf B \right).$$

The derivation assumes a wave packet confined to one smooth band and fields varying slowly on microscopic scales. Semiclassical Dynamics of Bloch Electrons derives the wave-packet equations and explains when interband motion or magnetic quantization matters. Berry curvature can add an anomalous velocity, magnetic moments can shift the energy, and strong fields can drive interband tunneling. Those effects are corrections to the simple acceleration law, not alternative definitions of the Hessian retained here.

What the Tensor Means

Consider an ellipsoidal conduction-band minimum:

$$\varepsilon(\mathbf k_0+\mathbf q) = E_c + \frac{\hbar^2}{2} \left( \frac{q_1^2}{m_1} + \frac{q_2^2}{m_2} + \frac{q_3^2}{m_3} \right).$$

Constant-energy surfaces are ellipsoids. Along principal axis $i$,

$$v_i = \frac{\hbar q_i}{m_i}, \qquad \dot v_i = \frac{F_i}{m_i}.$$

If the force is not along a principal axis, each component is rescaled differently. A scalar mass is therefore meaningful only after specifying a direction, symmetry average, or measurement.

Symmetry constraints

The little group that preserves the selected extremum $\mathbf k_0$ modulo a reciprocal vector constrains its mass tensor:

• a cubic little group forces a nondegenerate quadratic extremum to have $\mathsf M=m^*\mathsf I$;
• uniaxial symmetry permits longitudinal and transverse masses;
• lower symmetry permits three distinct principal masses and rotated principal axes;
• symmetry-protected degeneracy generally requires a matrix-valued multiband Hamiltonian rather than one scalar dispersion.

A globally cubic crystal can still have anisotropic valley masses when a valley’s little group is smaller, as at a silicon conduction valley. A symmetry operation that moves one valley to another rotates its tensor rather than requiring either individual tensor to be isotropic.

The tensor transforms under a spatial rotation $\mathsf R$ as

$$\mathsf M^{-1} \longmapsto \mathsf R \mathsf M^{-1} \mathsf R^{\mathsf T}.$$

Its eigenvalues are coordinate independent, but its displayed components are not.

Band Mixing and the k·p View

The curvature of a band is not determined only by the bare kinetic term. For a local periodic potential and an isolated nondegenerate band, perturbation theory for the cell-periodic Hamiltonian gives a representative $k\cdot p$ expression,

$$\left( \mathsf M_n^{-1} \right)_{ij} ={} \frac{\delta_{ij}}{m_e} + \frac{2}{m_e^2} \sum_{m\ne n} \frac{ \operatorname{Re} \left[ \langle n|p_i|m\rangle \langle m|p_j|n\rangle \right] }{ \varepsilon_n-\varepsilon_m }.$$

The states are the cell-periodic eigenvectors at $\mathbf k_0$. For $H_{\mathbf k}=(\mathbf p+\hbar\mathbf k)^2/(2m_e)+V(\mathbf r)$, differentiation produces $p_i+\hbar k_{0i}$; the constant term has zero matrix element between distinct bands, leaving the displayed $p_i$ in the interband sum. Thus the formula also applies at nonzero $\mathbf k_0$ under these local-potential assumptions. Remote bands alter the curvature through virtual mixing. Nearby bands usually produce the strongest renormalization because their energy denominators are small.

This formula explains why effective masses can be much lighter or heavier than $m_e$, why signs can differ between conduction and valence bands, and why narrow gaps often accompany strong nonparabolicity. It also exposes a danger: when two bands become degenerate, nondegenerate perturbation theory is singular. One must retain the coupled subspace and construct a multiband effective Hamiltonian.

Spin–orbit coupling, nonlocal pseudopotentials, and many-body self-energies modify the velocity matrix elements or the effective Hamiltonian. The displayed equation is a structural guide, not a universal computational formula for every electronic-structure method.

Electrons and Holes

Near a valence-band maximum $\mathbf k_v$, write

$$\varepsilon_v(\mathbf k_v+\mathbf q) \approx E_v - \frac{\hbar^2}{2} \mathbf q^{\mathsf T} \mathsf M_h^{-1} \mathbf q,$$

where $\mathsf M_h$ is positive definite. The electron curvature tensor at the maximum is negative:

$$\mathsf M_{e,v}^{-1} = -\mathsf M_h^{-1}.$$

A nearly full valence band is more simply described by its missing electrons. A hole carries charge $+e$, positive excitation energy measured downward from the valence maximum, and positive hole mass $\mathsf M_h$. This change of variables converts the collective response of many occupied negative-curvature electron states into the motion of a small number of positive carriers.

Negative electron effective mass does not mean negative rest mass, negative kinetic energy, or motion opposite to velocity. It means that the electron-band acceleration produced by a force has the opposite sign locally. Holes derives the operator transformation and the consistent charge, momentum and current conventions.

Density-of-States Mass

For the three-dimensional ellipsoidal minimum above, rescale momenta by

$$Q_i = \frac{q_i}{\sqrt{m_i}}.$$

The energy becomes spherical in $\mathbf Q$ space, while the volume element acquires the Jacobian

$$d^3q = \sqrt{m_1m_2m_3}\, d^3Q.$$

Define the three-dimensional density-of-states mass by

$$m_{\mathrm{DOS}} = \left( m_1m_2m_3 \right)^{1/3}.$$

For total degeneracy $g$, including only degeneracies not already counted as separate bands, the density of states per physical volume is

$$\rho(E) = \frac{g}{4\pi^2} \left( \frac{2m_{\mathrm{DOS}}}{\hbar^2} \right)^{3/2} \sqrt{E-E_c}\, \Theta(E-E_c).$$

In two dimensions,

$$m_{\mathrm{DOS}}^{(2D)} = \sqrt{m_1m_2},$$

and the parabolic-band density of states is constant:

$$\rho_{2D}(E) = \frac{g\,m_{\mathrm{DOS}}^{(2D)}} {2\pi\hbar^2} \Theta(E-E_c).$$

The geometric mean appears because density of states counts momentum-space volume. Valley multiplicity is a separate factor. Folding a degeneracy into both $g$ and $m_{\mathrm{DOS}}$ double counts states.

The Density of States page develops normalization, dimensional threshold laws, van Hove singularities, projected densities, and numerical methods.

Conductivity Mass

In a one-valley parabolic model with a scalar, momentum-independent relaxation time $\tau$, the Drude conductivity tensor is

$$\boldsymbol{\sigma} = nq^2\tau\, \mathsf M^{-1}.$$

Drude Theory derives this response and its relaxation-time assumptions.

For an electric field along a unit vector $\hat{\mathbf n}$, the projection of current onto that direction defines

$$\frac{1} {m_{\mathrm{cond}}(\hat{\mathbf n})} = \hat{\mathbf n}^{\mathsf T} \mathsf M^{-1} \hat{\mathbf n}.$$

The current can also have a transverse component. If the experimental boundaries instead constrain the current to lie along $\hat{\mathbf n}$, the field adjusts and the measured longitudinal resistivity is

$$\rho_{nn}=\hat{\mathbf n}^{\mathsf T}\boldsymbol\sigma^{-1}\hat{\mathbf n} =\frac{\hat{\mathbf n}^{\mathsf T}\mathsf M\hat{\mathbf n}}{nq^2\tau}.$$

This generally differs from $1/\sigma_{nn}$. Boundary conditions matter even at zero magnetic field in an anisotropic sample.

In principal axes,

$$\sigma_{ii} = \frac{nq^2\tau}{m_i}.$$

For equally populated equivalent valleys in a cubic material, each with one longitudinal mass $m_l$ and two transverse masses $m_t$, the orientationally averaged conductivity mass is

$$m_{\mathrm{cond}} = \frac{3} {m_l^{-1}+2m_t^{-1}}.$$

This harmonic-type average differs from the density-of-states mass. In a real multiband material, the conductivity also weights velocities, lifetimes, carrier densities, and scattering anisotropy over the Fermi surface. A mobility measurement gives $\mu=|q|\tau/m_{\mathrm{cond}}$ only within a specified scattering model; it cannot determine mass without information about $\tau$.

Cyclotron Mass

In a magnetic field, a semiclassical orbit follows the intersection of a constant-energy surface with a plane perpendicular to $\mathbf B$. Let $A(E,k_\parallel)$ be the enclosed $k$-space area at fixed momentum component along the field. The positive cyclotron-mass magnitude extracted from temperature damping is

$$m_c = \frac{\hbar^2}{2\pi} \left\lvert \frac{\partial A}{\partial E} \right\rvert_{k_\parallel}.$$

The magnitude of this derivative controls the temperature damping measured in Quantum Oscillations through the Lifshitz–Kosevich factor. A signed orbital mass can be retained separately to encode electron-like or hole-like circulation, but the Lifshitz–Kosevich mass quoted from damping is positive. It is therefore an orbit property, not simply one Cartesian component of the curvature tensor.

For a parabolic ellipsoid with $\mathbf B$ along principal axis $3$,

$$m_c = \sqrt{m_1m_2}.$$

More generally, for a positive mass tensor and field direction $\hat{\mathbf n}$,

$$m_c(\hat{\mathbf n}) = \sqrt{ \frac{\det\mathsf M} {\hat{\mathbf n}^{\mathsf T} \mathsf M \hat{\mathbf n}} }.$$

All parallel closed sections of a parabolic ellipsoid have this same area derivative; the central section is the extremal orbit relevant to quantum oscillations. For a general Fermi surface, one must calculate the actual orbit area and its energy derivative. Multiple extremal orbits can produce several measured masses even within one band. Landau Levels in Solids uses the local tensor to derive the quantized levels and their range of validity.

Optical and Plasma Masses

For an ideal parabolic carrier gas,

$$\left( \boldsymbol{\omega}_{p}^{\,2} \right)_{ij} = \frac{nq^2}{\epsilon_0} \left( \mathsf M^{-1} \right)_{ij}.$$

This is the unscreened Drude plasma-frequency parameter. For a scalar, nearly constant background dielectric response $\epsilon_\infty$, the weakly damped plasma resonance instead lies near $\omega_p/\sqrt{\epsilon_\infty}$. The intraband spectral weight defines an optical or Drude mass. In a lattice, that weight is an occupation-weighted average of band curvature or Fermi-surface velocities, and interband transitions redistribute spectral weight. The total continuum $f$-sum rule involves the bare electron mass, while a low-frequency partial sum can involve a band or quasiparticle mass. Confusing the partial Drude weight with the full sum rule is a common source of apparently contradictory masses.

Interacting Quasiparticle Mass

For one coherent band, write the retarded inverse Green function as

$$G^{-1}(\mathbf k,E)=E+\mu+i0^+-\varepsilon_b(\mathbf k)-\Sigma^R(\mathbf k,E).$$

Here $E$ is energy relative to the chemical potential, and $\Sigma^R$ is a scalar self-energy for this band. Differentiating the real pole equation at $E=0$ gives

$$Z=\left[1-\partial_E\operatorname{Re}\Sigma^R\right]_{\mathrm F}^{-1},$$ $$\mathbf v_{\mathrm F}^{*}=Z\left[ \mathbf v_b+\frac{1}{\hbar}\nabla_{\mathbf k}\operatorname{Re}\Sigma^R \right]_{\mathrm F}.$$

A frequency variable in angular-frequency units would require the corresponding factors of $\hbar$. A multiorbital problem also requires projection onto the appropriate pole eigenvector; taking the real part of matrix entries is not a substitute for that calculation.

An isotropic velocity mass can then be defined by

$$m_v^* = \frac{\hbar k_{\mathrm F}} {v_{\mathrm F}^{*}}.$$

This need not equal a simple Hessian mass away from a parabolic extremum. The specific-heat mass weights the interacting density of states; quantum oscillations measure an orbit mass; optical and dc responses can include vertex corrections and backflow. In a Galilean-invariant fluid, interactions can strongly renormalize the quasiparticle mass while conservation laws constrain the uniform current response.

Fermi Liquid Theory explains residues, Landau parameters, backflow and thermodynamic mass. Polarons Preview treats mass enhancement from a particle dragging a deformable environmental cloud.

Worked Band Examples

One-dimensional cosine band

For nearest-neighbor hopping,

$$\varepsilon(k) = -2t\cos(ka),$$

the velocity and inverse mass are

$$v(k) = \frac{2ta}{\hbar} \sin(ka),$$ $$\frac{1}{m^*(k)} = \frac{2ta^2}{\hbar^2} \cos(ka).$$

Near $k=0$ the mass is positive:

$$m^*(0) = \frac{\hbar^2}{2ta^2}.$$

Near $k=\pi/a$ it is negative in the electron description. At $ka=\pi/2$, the inverse mass vanishes while the speed is maximal. A single constant $m^*$ cannot represent the whole band.

Nonparabolic semiconductor band

A common isotropic approximation motivated by two-band mixing is

$$E(1+\alpha E) = \frac{\hbar^2k^2} {2m_{\mathrm{edge}}},$$

where $E$ is measured from the band edge and the branch has $1+2\alpha E>0$. The isotropic cyclotron or velocity mass becomes

$$m_c(E) = m_{\mathrm{edge}} \left( 1+2\alpha E \right).$$

The radial curvature gives a different result,

$$m_{\mathrm{rad}}(E)=\frac{\hbar^2}{d^2E/dk^2} =m_{\mathrm{edge}}(1+2\alpha E)^3.$$

The tangential Hessian eigenvalues involve $(1/k)dE/dk$ and give the same mass as $m_c$. Thus rotational symmetry of a nonparabolic dispersion does not make its Hessian proportional to the identity at every nonzero momentum. Both masses approach $m_{\mathrm{edge}}$ at the edge; the exercise derives their difference at finite energy.

Dirac cone

For an isotropic cone,

$$\varepsilon_{\pm}(k) = \pm\hbar v_D k,$$

there is no constant band-edge curvature mass. Nevertheless, a circular orbit at energy $E$ has

$$A(E) = \pi \left( \frac{E}{\hbar v_D} \right)^2,$$

and hence

$$m_c(E) = \frac{|E|}{v_D^2}.$$

The radial curvature vanishes, but the tangential curvature is $(1/k)dE/dk=\hbar v_D/k$ on the positive branch. The corresponding tangential mass equals $E/v_D^2$. This explains how a cone can have a finite cyclotron mass despite zero radial curvature.

Extracting a Mass from Data

From a numerical band

If the curvature in one direction is reported as

$$C = \frac{d^2E}{dk^2}$$

in units of $\mathrm{eV\,\mathring A^2}$, then

$$\frac{m^*}{m_e} \approx \frac{7.62}{C}.$$

The sign of $C$ must be retained. If a fit is written as

$$E(k) = E_0 + A(k-k_0)^2,$$

then $C=2A$ and

$$\frac{m^*}{m_e} \approx \frac{3.81}{A},$$

with $A$ in $\mathrm{eV\,\mathring A^2}$.

For a tensor fit:

1. choose a physically justified expansion point and energy window;
2. express all components of $\mathbf k$ in the same reciprocal-length units;
3. fit the full symmetric quadratic form, including cross terms;
4. symmetrize the numerical Hessian;
5. diagonalize it and report principal axes as well as eigenvalues;
6. repeat the fit over several windows to expose nonparabolicity and numerical noise.

A polynomial can always return a number. Stability under reasonable changes of window is part of the evidence that the number represents an effective mass.

From experiment

• ARPES measures a spectral dispersion and can infer local velocities or curvature. Resolution, matrix elements, surface sensitivity, and self-energy structure affect the fit.
• Quantum oscillations determine extremal orbit areas from frequencies and cyclotron masses from temperature damping. The result is orbit specific.
• Cyclotron resonance probes the dynamical orbit mass and can separate carrier types under favorable conditions.
• Specific heat weights the quasiparticle density of states near the chemical potential and often yields a thermodynamic mass enhancement.
• Optical conductivity constrains intraband spectral weight and scattering rate, permitting a Drude mass only with a carrier-density and multiband model.
• Transport measures combinations of carrier density, velocity, lifetime, and mass. Hall data do not generally return an effective mass directly.

Different well-performed experiments can therefore report different masses without disagreement. The comparison is meaningful only after matching the band, orbit, direction, energy, temperature, and response definition.

Where the Approximation Breaks Down

The constant effective-mass approximation is reliable when:

• the relevant band is smooth and isolated over the occupied wave-packet region;
• the momentum or energy window is small enough that cubic and higher terms are negligible;
• the wave packet remains in one band under the applied fields;
• the fields vary slowly compared with lattice scales;
• the quasiparticle peak is sufficiently coherent if interactions are present;
• the chosen tensor or scalar matches the response being modeled.

It becomes unreliable or ambiguous in several important regimes.

Crossings and near-degeneracies

At an exact crossing, an individual eigenvalue can be nonanalytic and an individual eigenvector is basis dependent inside the degenerate subspace. Near an avoided crossing, curvature can vary rapidly. A multiband $k\cdot p$ model is usually more faithful than fitting one branch to a parabola.

Saddles and van Hove points

A saddle has positive curvature in some directions and negative curvature in others. Its mass tensor is indefinite, and a scalar average can conceal the geometry responsible for a van Hove singularity.

Inflection points

At an inflection point one principal inverse mass vanishes. Higher derivatives control the dynamics in that direction. Treating the divergent scalar mass as a globally heavy particle misses the finite velocity and rapid variation nearby.

Strong fields and interband motion

If a field moves the packet through a significant fraction of the Brillouin zone, the local tensor changes along the trajectory. Near small gaps, Landau–Zener transitions invalidate one-band dynamics. Magnetic fields strong enough to quantize motion require Landau-level or magnetic-breakdown descriptions.

Geometric corrections

Berry curvature produces an anomalous velocity transverse to the applied force, even without an off-diagonal mass tensor. Orbital magnetic moments shift the band energy in a field. These geometric effects should not be absorbed into an invented scalar mass.

Incoherent or strongly reconstructed spectra

If no sharp quasiparticle pole exists, assigning a band curvature to a broad spectral feature can be misleading. Symmetry breaking, strong correlations, disorder, or coupling to collective modes can also reconstruct the low-energy dispersion and make the mass temperature, field, or frequency dependent.

Exercises

1. Anisotropic acceleration and two transport geometries

Let $\mathsf M=\operatorname{diag}(m,4m)$ and apply $\mathbf F=F(1,1)/\sqrt{2}$. Find $\mathbf a$ and determine whether it is parallel to $\mathbf F$. For the same mass tensor and a constant relaxation time, compare the effective masses inferred from an electric field constrained along $45^\circ$ and a current constrained along $45^\circ$.

Solution

The inverse tensor is

$$\mathsf M^{-1} = \frac{1}{m} \operatorname{diag} \left( 1,\frac{1}{4} \right).$$

Hence

$$\mathbf a = \mathsf M^{-1}\mathbf F = \frac{F}{\sqrt{2}m} \left( 1,\frac{1}{4} \right).$$

The force points at $45^\circ$, while the acceleration obeys

$$\tan\theta_a = \frac{a_y}{a_x} = \frac{1}{4}.$$

Thus $\theta_a\approx14.0^\circ$, and the vectors are not parallel.

For $\hat{\mathbf n}=(1,1)/\sqrt2$, the field-directed current projection gives

$$m_{\mathrm{cond}}=\left(\frac{1}{2m}+\frac{1}{8m}\right)^{-1}=\frac85m.$$

When the current is constrained to that direction, the longitudinal resistivity instead contains $\hat{\mathbf n}^{\mathsf T}\mathsf M\hat{\mathbf n}=5m/2$. A transverse electric field enforces the zero transverse current. These numbers describe different boundary conditions, not conflicting band masses.

2. Density-of-states mass of an ellipsoid

For principal masses $m_1=m$, $m_2=2m$, and $m_3=8m$, find the three-dimensional density-of-states mass. If there are six equivalent valleys and spin degeneracy remains unresolved, what total degeneracy multiplies the one-valley, one-spin density of states?

Solution

The density-of-states mass is

$$\begin{aligned} m_{\mathrm{DOS}} &= \left( m_1m_2m_3 \right)^{1/3} \\ &= \left( 16m^3 \right)^{1/3} \\ &= 16^{1/3}m \approx 2.52m. \end{aligned}$$

Six valleys and two spin states give

$$g = 6\times2 = 12.$$

The factor $12$ is kept separate from $m_{\mathrm{DOS}}$.

3. Nonparabolic radial and cyclotron masses

Starting from

$$E(1+\alpha E) = \frac{\hbar^2k^2}{2m_{\mathrm{edge}}},$$

derive both $m_c(E)$ and the radial curvature mass. Compare them at $\alpha E=1/4$.

Solution

The orbit area is

$$A(E) = \pi k^2 = \frac{2\pi m_{\mathrm{edge}}}{\hbar^2} E(1+\alpha E).$$

Using the orbit definition,

$$\begin{aligned} m_c(E) &= \frac{\hbar^2}{2\pi} \frac{\partial A}{\partial E} \\ &= m_{\mathrm{edge}} \left( 1+2\alpha E \right). \end{aligned}$$

For the radial curvature, first differentiate the dispersion:

$$(1+2\alpha E)\frac{dE}{dk}=\frac{\hbar^2 k}{m_{\mathrm{edge}}}.$$

A second derivative, followed by elimination of $k^2$ using the original relation, gives

$$\frac{d^2E}{dk^2} =\frac{\hbar^2}{m_{\mathrm{edge}}(1+2\alpha E)^3}.$$

Hence $m_{\mathrm{rad}}=m_{\mathrm{edge}}(1+2\alpha E)^3$. At $\alpha E=1/4$, $m_c=1.5m_{\mathrm{edge}}$ but $m_{\mathrm{rad}}=3.375m_{\mathrm{edge}}$. The tangential curvature, rather than the radial one, determines the circular orbit mass.

4. Converting a fitted curvature

A calculated conduction band is fitted near its minimum by

$$E(k) = E_c + 12.0\, \mathrm{eV\,\mathring A^2} \left( k-k_0 \right)^2.$$

Estimate $m^*/m_e$.

Solution

The coefficient is $A=12.0\,\mathrm{eV\,\mathring A^2}$, so

$$\frac{m^*}{m_e} \approx \frac{3.81}{A} = \frac{3.81}{12.0} \approx 0.318.$$

Equivalently, the curvature is $C=2A=24.0\,\mathrm{eV\,\mathring A^2}$ and $7.62/C$ gives the same result.

References

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