What is the density of states?

The density of states g(E) gives the number of allowed quantum states per unit energy per unit volume. It tells you how many electrons can occupy states near a given energy level, which controls electrical, optical, and thermal properties of solids.

Why does g(E) scale as √E in 3D?

In three dimensions, the volume of k-space enclosed within a sphere of radius k grows as k³. Since E ∝ k², differentiating with respect to E introduces a factor of k ∝ √E, giving the characteristic square-root dependence unique to 3D systems (1D gives a 1/√E divergence; 2D gives a step function).

What is the effective mass m*?

The effective mass accounts for the periodic crystal potential that electrons feel. Electrons in a real material behave as if they have a mass m* ≠ mₑ. For example, electrons in GaAs have m* ≈ 0.067 mₑ, making their DOS much smaller than for free electrons with m* = 1.0 mₑ.

What units does this calculator use?

Energy is in eV, effective mass is in units of the bare electron mass mₑ = 9.109×10⁻³¹ kg. The result is given in states per eV per m³ (SI) and per cm³ for practical use. Internally all calculations use SI base units (J, kg, m) and then convert.

Is this formula valid for all materials?

It applies to the parabolic band approximation, which is accurate near band extrema (low energies). At higher energies or in materials with strongly anisotropic bands, non-parabolic corrections or numerical band-structure methods are needed.

Density of States Calculator

Calculate the 3D free-electron density of states g(E) at any energy using the standard quantum-mechanical formula.

Inputs

Energy E (eV)

Energy at which to evaluate g(E). Must be > 0.

Effective mass m* (× m₀)

Ratio m*/m₀ — e.g. Si ≈ 1.08, GaAs ≈ 0.067, free electron = 1.0.

Common presets

Result

Enter values above and click Calculate.

g(E) vs Energy

E max (eV):

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Summary

The density of states (DOS) g(E) describes how many electron quantum states exist per unit energy per unit volume in a material. For a 3D free-electron gas, it follows g(E) = (1/2π²) × (2m*/ℏ²)^(3/2) × √E, where m* is the effective mass and ℏ is the reduced Planck constant. This calculator computes g(E) in units of states per eV per m³, which is the form most commonly used in solid-state physics and semiconductor device analysis.

How it works

Enter the energy E in electron-volts (eV) at which to evaluate the DOS.
Enter the electron effective mass as a multiple of the bare electron mass m₀.
Click Calculate to apply the 3D free-electron DOS formula: g(E) = (1/2π²)(2m*/ℏ²)^(3/2) √E.
Read the result in states/eV/m³ and the equivalent states/eV/cm³ for convenience.
Use the energy sweep to plot g(E) vs E and visualize the characteristic √E shape.

Use cases

Determine the DOS at the conduction band edge for semiconductor analysis.
Compare DOS for different effective masses across material families.
Verify hand calculations for solid-state physics coursework.
Estimate the number of available states in a given energy window.
Visualize the parabolic band approximation across an energy range.
Support Fermi–Dirac occupation and carrier concentration calculations.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu