2.3 Effective mass and density of states

Effective mass concept

Effective mass and density of states describe how electrons behave inside a crystal and how many energy states are available for them to occupy. These two ideas are central to understanding carrier transport, optical properties, and device performance in semiconductors.

In free space, an electron has a well-defined rest mass. Inside a crystal lattice, the periodic potential from the atoms modifies how the electron accelerates in response to external forces. Rather than solving the full quantum problem every time, we assign the electron an effective mass that captures the lattice's influence in a single parameter.

Electrons in periodic potential

Electrons in a semiconductor don't move through empty space. They interact continuously with the periodic potential of the crystal lattice. This interaction is what creates energy bands in the first place, and it changes how electrons respond to applied forces compared to free electrons in vacuum.

The key insight is that the electron's response depends on the curvature of the energy band $E(\mathbf{k})$ in k-space. The effective mass packages that curvature into a quantity you can plug into familiar equations like $F = m^*a$.

Defining effective mass from band curvature

The effective mass is defined through the second derivative of the energy dispersion relation:

$\frac{1}{m^*} = \frac{1}{\hbar^2} \frac{d^2 E}{dk^2}$

This formula tells you:

• High curvature (steep parabola in $E$ vs. $k$) → small effective mass → electrons accelerate easily
• Low curvature (flat parabola) → large effective mass → electrons are sluggish

Near the bottom of the conduction band and the top of the valence band, the bands are often well-approximated as parabolic, so this definition works cleanly. Farther from the band edges, the parabolic approximation breaks down and higher-order corrections may be needed.

Analogy to free electrons

The whole point of effective mass is simplification. Once you know $m^*$, you can treat the electron as if it were a free particle with mass $m^*$ moving through the crystal. Semiclassical equations of motion, drift-diffusion models, and scattering rate calculations all use $m^*$ in place of the free electron mass $m_0$.

This works because the lattice interactions are already baked into $m^*$. You don't need to re-solve the Schrödinger equation for the full periodic potential every time you want to know how an electron responds to an electric field.

Density of states (DOS)

The density of states, $g(E)$, counts the number of available electronic states per unit volume per unit energy. It answers the question: at a given energy $E$, how many states can electrons actually occupy?

This matters because carrier concentration, conductivity, and optical absorption all depend not just on whether states exist, but on how many states exist at each energy.

DOS formula for 3D semiconductors

For a bulk (3D) semiconductor with parabolic bands, the DOS near the conduction band edge is:

$g_{3D}(E) = \frac{1}{2\pi^2} \left(\frac{2m^*}{\hbar^2}\right)^{3/2} \sqrt{E - E_c}$

where $E_c$ is the conduction band edge. Notice two things:

• The DOS goes as $\sqrt{E - E_c}$, so it starts at zero right at the band edge and increases with energy.
• A larger effective mass means a higher DOS. More states are packed into each energy interval when carriers are heavier.

A similar expression applies near the valence band edge for holes, using the hole effective mass.

Derivation outline

The derivation follows a clear logic:

1. Start with the parabolic dispersion relation: $E = E_c + \frac{\hbar^2 k^2}{2m^*}$
2. Count the number of allowed k-states in a spherical shell of thickness $dk$ in k-space. The volume of that shell is $4\pi k^2 \, dk$.
3. Account for spin degeneracy (factor of 2) and the volume per allowed k-state, $(2\pi)^3 / V$.
4. Convert from $dk$ to $dE$ using the dispersion relation, which introduces the $\sqrt{E - E_c}$ dependence.

DOS in different dimensions

The dimensionality of the system changes the DOS shape dramatically:

DOS vs. energy plots

Plotting $g(E)$ against $E$ is one of the most useful visualizations in semiconductor physics. These plots let you:

• Identify band edges where the DOS turns on
• Spot van Hove singularities (sharp features in the DOS from flat regions of the band structure)
• Compare how different dimensionalities redistribute the available states

For bulk semiconductors, the plot looks like a pair of $\sqrt{E}$ curves opening away from the band gap in both directions (one for electrons, one for holes).

Effective mass in semiconductors

Real semiconductors rarely have perfectly isotropic bands. The effective mass is generally a tensor quantity that varies with crystallographic direction.

Effective mass tensor

Because the band curvature can differ along different crystal axes, the effective mass is described by a 3×3 tensor:

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

For crystals with sufficient symmetry, the off-diagonal elements vanish and you're left with up to three distinct effective masses along the principal axes.

Longitudinal vs. transverse mass

In materials like silicon, the conduction band minima sit along specific crystallographic directions (the $\langle 100 \rangle$ directions for Si). The effective mass splits into:

• Longitudinal mass ($m_l^*$): along the axis pointing toward the band minimum. In Si, $m_l^* \approx 0.98 \, m_0$.
• Transverse mass ($m_t^*$): perpendicular to that axis. In Si, $m_t^* \approx 0.19 \, m_0$.

The large difference between these two values reflects the elongated (ellipsoidal) shape of the constant-energy surfaces near the conduction band minimum in silicon.

Light vs. heavy holes

The valence band in most semiconductors has two overlapping bands at the $\Gamma$ point:

• Heavy hole band: large effective mass, lower curvature. Dominates the DOS because more states are available.
• Light hole band: small effective mass, higher curvature. Carriers here have higher mobility.

There's also a split-off band pushed down in energy by spin-orbit coupling. The interplay of heavy and light holes affects optical absorption spectra, hole mobility, and the design of p-type devices.

Conductivity effective mass

For transport calculations, you often need a single scalar effective mass rather than the full tensor. The conductivity effective mass is defined as:

$\frac{1}{m_{\text{cond}}^*} = \frac{1}{3}\left(\frac{1}{m_1^*} + \frac{1}{m_2^*} + \frac{1}{m_3^*}\right)$

This is a harmonic average over the three principal directions. For silicon's six equivalent conduction band valleys, the conductivity effective mass works out to:

$\frac{1}{m_{\text{cond}}^*} = \frac{1}{3}\left(\frac{1}{m_l^*} + \frac{2}{m_t^*}\right)$

A separate quantity, the density-of-states effective mass, is used when calculating carrier concentrations. Don't confuse the two; they weight the directional masses differently.

Temperature effects

Temperature changes the occupation of states and shifts the Fermi level, directly affecting carrier concentrations and device behavior.

Fermi-Dirac distribution

The probability that a state at energy $E$ is occupied is given by:

$f(E) = \frac{1}{1 + \exp\left(\frac{E - E_F}{k_B T}\right)}$

At $T = 0$, this is a sharp step: all states below $E_F$ are filled, all above are empty. As temperature rises, the step smears out over an energy range of roughly $\pm \text{few} \, k_B T$ around $E_F$.

Carrier concentration calculations

The electron concentration in the conduction band is found by integrating the product of the DOS and the Fermi-Dirac distribution:

$n = \int_{E_c}^{\infty} g(E) \, f(E) \, dE$

When $E_F$ is more than a few $k_BT$ below $E_c$ (the non-degenerate case), the Fermi-Dirac function can be approximated by a Boltzmann exponential, giving:

$n = N_c \exp\left(-\frac{E_c - E_F}{k_B T}\right)$

where $N_c$ is the effective density of states in the conduction band:

$N_c = 2\left(\frac{m_e^* k_B T}{2\pi \hbar^2}\right)^{3/2}$

A similar expression holds for holes in the valence band using $N_v$ and the hole effective mass. Notice that $N_c$ depends on both $m^*$ and $T$, tying effective mass and DOS directly to carrier concentration.

Intrinsic vs. extrinsic semiconductors

• Intrinsic: no intentional doping. Electron and hole concentrations are equal ($n = p = n_i$), determined by the band gap and temperature. The intrinsic carrier concentration rises exponentially with temperature.
• Extrinsic: doped with donors (n-type) or acceptors (p-type). At moderate temperatures, the majority carrier concentration is set by the dopant density. At very high temperatures, intrinsic carriers overwhelm the dopants and the material behaves intrinsically again.

Fermi level temperature dependence

In an intrinsic semiconductor, the Fermi level sits near mid-gap. It shifts slightly toward the band with the smaller effective mass:

$E_F = \frac{E_c + E_v}{2} + \frac{3}{4} k_B T \ln\left(\frac{m_h^*}{m_e^*}\right)$

In extrinsic semiconductors, $E_F$ starts near the dopant level at low temperatures, then migrates toward mid-gap as temperature increases and intrinsic carriers take over. Tracking this shift is essential for predicting device behavior across operating temperatures.

Applications of effective mass and DOS

Carrier transport properties

Carrier mobility is inversely related to effective mass: $\mu = \frac{q\tau}{m^*}$, where $\tau$ is the average scattering time. Materials with small effective masses (like GaAs, where $m_e^* \approx 0.067 \, m_0$) have high electron mobilities, making them attractive for high-frequency transistors.

The anisotropy of the effective mass tensor also means mobility can differ along crystal directions, which matters for device orientation on a wafer.

Optical absorption and emission

The joint density of states (combining conduction and valence band DOS) determines the shape of the optical absorption spectrum. The effective mass enters through the reduced mass of the electron-hole pair:

$\frac{1}{\mu_r} = \frac{1}{m_e^*} + \frac{1}{m_h^*}$

This reduced mass also sets the exciton binding energy and the threshold for interband absorption. Materials with smaller reduced masses tend to have sharper absorption edges.

Quantum confinement effects

When a semiconductor structure shrinks to dimensions comparable to the de Broglie wavelength of carriers, quantum confinement modifies both the effective mass and the DOS. Quantum wells, wires, and dots each produce distinct DOS profiles (see the table above), enabling engineers to:

• Tune emission wavelengths by adjusting well width
• Increase optical gain in quantum well lasers through the step-like DOS
• Create discrete, atom-like energy levels in quantum dots for single-photon emitters

Semiconductor device modeling

Accurate device simulation depends on getting $m^*$ and $g(E)$ right. In practice:

• Drift-diffusion models use the conductivity effective mass to compute mobility and diffusion coefficients.
• Band structure solvers (k·p method, tight-binding) calculate the full $E(\mathbf{k})$ dispersion from which effective masses and DOS are extracted.
• Monte Carlo transport simulations track individual carriers through k-space, using the band curvature (and thus effective mass) at each point to determine scattering rates and velocities.

Getting these parameters wrong can lead to significant errors in predicted current-voltage characteristics, switching speeds, and power dissipation.