5.5 Effective mass

Effective mass concept

Effective mass describes how electrons respond to forces inside a crystal. Instead of tracking every interaction between an electron and the periodic potential of the lattice, you assign the electron a modified mass that captures all those effects. Then you can apply familiar equations (Newton's second law, the Schrödinger equation for a free particle) using this new mass in place of the bare electron mass.

This simplification is what makes semiconductor physics tractable. Nearly every device property you care about, from carrier mobility to optical absorption, depends on effective mass.

Electrons in periodic potentials

Inside a crystal, atoms are arranged in a repeating pattern, and their combined electric fields create a periodic potential. An electron moving through this potential doesn't behave like a free electron in vacuum. It accelerates differently under an applied force because it's constantly interacting with the lattice.

Effective mass absorbs all of that complexity into a single parameter. You replace the true electron mass $m_e$ with an effective mass $m^*$, and from that point on you can treat the electron as though it were free. The value of $m^*$ depends on the specific material and the electron's position in the band structure. In silicon, for example, the electron effective mass is roughly $0.26\,m_e$ for the transverse direction and $0.98\,m_e$ for the longitudinal direction.

Analogy to free electrons

For a free electron, Newton's second law gives $F = m_e \, a$. Inside a crystal, the same equation works if you swap in the effective mass:

$F = m^* \, a$

This is powerful because it means you can reuse the entire framework of classical and quantum mechanics for free particles. The Schrödinger equation for a free particle, kinetic energy expressions, and transport equations all carry over with $m^*$ replacing $m_e$.

The trade-off is that $m^*$ is not a fixed constant of nature. It's a material-dependent quantity derived from the band structure.

Effective mass tensor

Definition and the curvature formula

The effective mass is defined through the curvature of the energy band $E(\mathbf{k})$:

$\frac{1}{m^*_{ij}} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \, \partial k_j}$

This is a second-rank tensor (a 3×3 matrix), not necessarily a single number. Each component $m^*_{ij}$ relates the force applied along direction $j$ to the acceleration produced along direction $i$.

The key relationship to remember: high curvature in $E(k)$ means small effective mass (the electron accelerates easily), while a flat band means large effective mass (the electron is sluggish).

Anisotropic effective mass

In many crystals, the curvature of $E(\mathbf{k})$ differs along different crystallographic directions, so $m^*$ depends on which way the electron is moving. Silicon is a classic example: the conduction band minima sit along the $\langle 100 \rangle$ directions and have an ellipsoidal shape, giving a longitudinal mass $m^*_l \approx 0.98\,m_e$ and a transverse mass $m^*_t \approx 0.19\,m_e$.

Materials like graphite and bismuth show even more dramatic anisotropy. You can't describe their transport properties with a single scalar mass; the full tensor is essential.

Relation to band structure

The band structure $E(\mathbf{k})$ is the starting point for any effective mass calculation. You focus on the regions near band extrema (the conduction band minimum or valence band maximum), because that's where carriers sit at typical energies.

• Near a band minimum, $\frac{\partial^2 E}{\partial k^2} > 0$, so $m^* > 0$ (electrons).
• Near a band maximum, $\frac{\partial^2 E}{\partial k^2} < 0$, so $m^* < 0$. This negative effective mass is reinterpreted as a positive-mass hole.

Band structures can be computed with methods like density functional theory (DFT) or tight-binding models, and the effective mass is then extracted by fitting a parabola to the band near the extremum.

Effective mass calculations

Tight-binding model

The tight-binding approach builds the band structure from atomic orbitals. You assume each electron's wavefunction is a linear combination of atomic orbitals (LCAO) centered on lattice sites, and you solve for the allowed energies as a function of $\mathbf{k}$.

For a simple 1D chain with nearest-neighbor hopping parameter $t$ and lattice constant $a$, the dispersion is:

$E(k) = E_0 - 2t\cos(ka)$

Taking the second derivative at the band minimum ($k = 0$):

$m^* = \frac{\hbar^2}{2ta^2}$

This shows directly that stronger hopping (larger $t$) gives smaller effective mass, which makes physical sense: electrons that hop easily between atoms move more freely.

Tight-binding is especially useful for materials with localized electrons, such as transition metal oxides and organic semiconductors.

k·p perturbation theory

The k·p method is the standard approach for semiconductors. It starts from the exact solutions at a high-symmetry point in the Brillouin zone (usually $\Gamma$, where $\mathbf{k} = 0$) and treats small deviations in $\mathbf{k}$ as a perturbation.

The steps are:

1. Solve the Schrödinger equation exactly at $\mathbf{k} = 0$ to get the zone-center energies and wavefunctions.
2. Write the Hamiltonian for nearby $\mathbf{k}$ values. The extra terms involve the momentum operator $\mathbf{p}$, which is where the name "k·p" comes from.
3. Apply second-order perturbation theory. The coupling between bands through the momentum matrix elements modifies the curvature of each band.
4. Extract the effective mass tensor from the resulting dispersion.

A key result is that bands that are close in energy and strongly coupled through $\mathbf{p}$ produce small effective masses. This is why narrow-gap semiconductors like InSb ($E_g \approx 0.17$ eV) have very small electron effective masses ($m^* \approx 0.014\,m_e$).

The k·p method is widely used for III-V semiconductors (GaAs, InP, GaN) and gives accurate effective masses and optical properties near band edges.

Conductivity effective mass

The conductivity effective mass determines how well a material conducts electricity. It combines the effective mass tensor with information about which electronic states actually carry current.

Density of states effective mass vs. conductivity effective mass

These two quantities are related but distinct. The density of states (DOS) effective mass $m^*_{\text{DOS}}$ determines how many states are available at a given energy. For a semiconductor with multiple equivalent conduction band valleys (like silicon, which has 6), the DOS effective mass includes a factor accounting for the number of valleys:

$m^*_{\text{DOS}} = N_v^{2/3}(m^*_l \, m^{*2}_t)^{1/3}$

where $N_v$ is the number of equivalent valleys.

The conductivity effective mass $m^*_c$ is what enters the Drude formula for conductivity $\sigma = ne^2\tau / m^*_c$. For silicon's ellipsoidal valleys:

$\frac{1}{m^*_c} = \frac{1}{3}\left(\frac{1}{m^*_l} + \frac{2}{m^*_t}\right)$

This harmonic-mean form reflects the fact that conductivity averages over all directions.

Fermi surface effects

In metals, the conductivity depends on electrons right at the Fermi surface. Different parts of the Fermi surface can have very different curvatures, meaning the local effective mass varies across the surface.

Copper, for instance, has a Fermi surface that is roughly spherical but with "necks" extending toward the Brillouin zone boundary along $\langle 111 \rangle$. Electrons near these necks have a different effective mass than those on the belly of the surface. The measured conductivity reflects an average over all these contributions.

Optical effective mass

The optical effective mass governs how a material interacts with light. It determines quantities like the plasma frequency and the strength of interband optical transitions.

Relation to dielectric function

For free carriers (as in a doped semiconductor or a metal), the Drude contribution to the dielectric function is:

$\epsilon(\omega) = \epsilon_\infty - \frac{ne^2}{m^*_{\text{opt}}\,\epsilon_0\,\omega^2}$

where $n$ is the carrier density and $m^*_{\text{opt}}$ is the optical effective mass. A smaller optical effective mass means a higher plasma frequency and stronger free-carrier absorption. The dielectric function can be measured experimentally using techniques like ellipsometry or infrared reflectivity.

Exciton effective mass

When a photon is absorbed in a semiconductor, it can create a bound electron-hole pair called an exciton. The exciton behaves like a hydrogen atom, and its reduced mass is:

$\frac{1}{\mu} = \frac{1}{m^*_e} + \frac{1}{m^*_h}$

The exciton binding energy scales as $E_b \propto \mu$, so materials with small effective masses (both electron and hole) have weakly bound excitons that are easy to ionize at room temperature. GaAs has an exciton binding energy of only about 4 meV.

In contrast, transition metal dichalcogenides (TMDs) like $\text{MoS}_2$ have much larger effective masses and reduced dielectric screening, giving exciton binding energies of hundreds of meV. This makes excitonic effects dominant in their optical spectra even at room temperature.

Polaron effective mass

A polaron forms when a charge carrier distorts the surrounding lattice and becomes "dressed" by a cloud of phonons. The combined quasiparticle is heavier than the bare carrier.

Electron-phonon interactions

As an electron moves through a polar crystal, it attracts nearby positive ions and repels negative ones, creating a local lattice distortion that follows the electron. This self-trapping effect increases the carrier's inertia.

The strength of the coupling is characterized by the Fröhlich coupling constant $\alpha$:

$\alpha = \frac{e^2}{4\pi\epsilon_0\hbar}\left(\frac{1}{\epsilon_\infty} - \frac{1}{\epsilon_0}\right)\sqrt{\frac{m^*}{2\hbar\omega_{\text{LO}}}}$

where $\omega_{\text{LO}}$ is the longitudinal optical phonon frequency. Larger $\alpha$ means stronger coupling and a heavier polaron.

Large vs. small polarons

• Large polarons ($\alpha \lesssim 6$): The lattice distortion extends over many unit cells. The mass enhancement is modest, roughly $m^*_{\text{pol}} \approx m^*(1 + \alpha/6)$ for weak coupling. The carrier remains mobile and moves in a band-like fashion. GaAs ($\alpha \approx 0.07$) is a typical example where polaron effects are small.
• Small polarons ($\alpha \gg 6$): The distortion is confined to roughly one unit cell. The carrier is essentially trapped and moves by thermally activated hopping rather than band transport. The effective mass can be orders of magnitude larger than the bare band mass. Materials like $\text{TiO}_2$ and $\text{Fe}_2\text{O}_3$ exhibit small polaron behavior, which limits their carrier mobility.

Experimental determination

Cyclotron resonance

Cyclotron resonance is the most direct way to measure effective mass. The idea:

1. Place the sample in a strong, uniform magnetic field $B$.
2. Electrons orbit in the plane perpendicular to $B$ at the cyclotron frequency $\omega_c = eB/m^*$.
3. Irradiate the sample with microwave or far-infrared radiation and sweep either the frequency or the magnetic field.
4. When the radiation frequency matches $\omega_c$, resonant absorption occurs.
5. From the resonance condition, extract $m^* = eB/\omega_c$.

By rotating the crystal relative to $B$, you can map out the full effective mass tensor. This technique requires high-purity samples and low temperatures so that carriers complete many orbits before scattering ($\omega_c \tau \gg 1$). It works well in high-mobility semiconductors like GaAs ($m^* \approx 0.067\,m_e$) and InSb.

Shubnikov-de Haas oscillations

In a strong magnetic field, the continuous density of states splits into discrete Landau levels separated by $\hbar\omega_c$. As you sweep the field, these levels pass through the Fermi energy one by one, causing oscillations in the resistivity.

From these oscillations you can extract:

• The oscillation period in $1/B$, which gives the extremal cross-sectional area of the Fermi surface.
• The temperature dependence of the oscillation amplitude, which yields $m^*$ through the Lifshitz-Kosevich formula.

This technique is especially valuable for 2D electron gases (found in semiconductor heterostructures) and topological insulators, where it reveals both the effective mass and the Berry phase.

Angle-resolved photoemission spectroscopy (ARPES)

ARPES directly maps the band structure $E(\mathbf{k})$, from which you can read off the effective mass by fitting a parabola near the band extremum.

How it works:

1. High-energy photons (UV or soft X-ray) eject electrons from the sample surface.
2. An analyzer measures each emitted electron's kinetic energy and emission angle.
3. Conservation of energy and parallel momentum lets you reconstruct $E(k_\parallel)$.
4. Fitting the measured dispersion near a band extremum with $E = \hbar^2 k^2 / 2m^*$ gives the effective mass.

ARPES has been used extensively on high-temperature superconductors, topological surface states, and TMDs. Its main limitation is that it's surface-sensitive and requires ultra-high vacuum and clean, flat surfaces.

Applications of effective mass

Semiconductor devices

Effective mass directly controls carrier mobility $\mu = e\tau/m^*$. A smaller $m^*$ means higher mobility, which translates to faster transistors and more efficient charge collection in solar cells.

This is why III-V semiconductors like GaAs ($m^*_e = 0.067\,m_e$) and InAs ($m^*_e = 0.023\,m_e$) are used in high-frequency electronics, while silicon ($m^*_c \approx 0.26\,m_e$) dominates where cost matters more than raw speed.

In quantum wells and superlattices, confinement modifies the effective mass, giving engineers another knob to tune device performance. LED emission wavelengths and laser threshold currents both depend on the effective masses of electrons and holes in the active region.

Thermoelectric materials

A good thermoelectric material needs high electrical conductivity, a large Seebeck coefficient, and low thermal conductivity. These requirements create a tension: high conductivity favors small $m^*$, but a large Seebeck coefficient benefits from a large density of states effective mass.

The best thermoelectric materials, like $\text{Bi}_2\text{Te}_3$ and PbTe, resolve this by having multiple band extrema (high valley degeneracy). This gives a large DOS effective mass for thermopower while keeping the conductivity effective mass reasonable. The thermoelectric figure of merit $ZT$ depends on balancing these competing demands.

Superconductors

In BCS theory, the superconducting gap and critical temperature depend on the density of states at the Fermi level, which is proportional to $m^*$. Conventional superconductors like aluminum and lead have effective masses close to $m_e$, and their superconductivity is well described by electron-phonon coupling.

Unconventional superconductors (cuprates, iron-based compounds, heavy-fermion systems) can have dramatically enhanced effective masses. In heavy-fermion materials like $\text{CeCoIn}_5$, $m^*$ can exceed $100\,m_e$ due to strong electron-electron correlations. Understanding how this mass enhancement relates to the pairing mechanism remains one of the central open questions in condensed matter physics.