Density of States

Introduction

Knowing the Fermi energy tells us where a metal's occupied states end at zero temperature, but it does not tell how many states are packed into each energy interval. The density of states , or DOS, answers that counting question. Multiplying DOS by the probability of occupation tells how many electrons actually occupy a small interval. Separating availability from occupancy is essential when interpreting heat capacity, optical absorption, tunnelling measurements and semiconductor carriers.

Core explanation

Write g(E)dE for the number of allowed one-electron states between E and E + dE in a specified specimen. Alternatively define a DOS per unit length, area or volume, which requires explicitly stating that normalisation. The occupied electron count in that interval is g(E)f(E)dE, where f(E) is the Fermi–Dirac probability. A large DOS does not imply many electrons if those states are empty, and a filled region cannot be understood from occupancy alone without knowing how many states exist.

For a parabolic free-electron dispersion measured upward from a band minimum E₀, write ε = E − E₀ = ħ²k²/(2m ), with a positive effective mass m . In one spatial dimension, allowed k states lie along a line. Counting points from −k to +k gives N(ε) proportional to k, so N(ε) ∝ ε^(1/2) and g 1D(ε) = dN/dε ∝ ε^(−1/2). The ideal 1D DOS therefore rises sharply as the band begins. In two dimensions, allowed states fill a disk of area πk²; N(ε) ∝ k² ∝ ε and g 2D is constant within a parabolic subband. In three dimensions, states fill a sphere of volume 4πk³/3; N(ε) ∝ k³ ∝ ε^(3/2), so g 3D ∝ √ε. The MIT OpenCourseWare free-electron DOS lecture derives the three-dimensional counting from the sphere in k-space.

The different curves are geometric consequences of how the volume of a line segment, disk or sphere grows with radius and of how energy grows with k². They do not say every nanowire must obey an exact ε^(−1/2) curve. Real quasi-one-dimensional systems have finite width, multiple subbands, disorder and scattering, all of which broaden the sharp ideal onset. A quantum well may show a staircase DOS as successive 2D subbands open; a bulk three-dimensional semiconductor has a square-root-like onset near a simple parabolic band edge. MIT's transport lecture uses these dimensional comparisons to connect quantum states with transport.

For a three-dimensional free electron gas with spin factor two, DOS per volume is g V(E) = 1/(2π²)^(3/2)√(E−E₀), above the edge. Integrating it from E₀ to E F returns electron density. But electronic bands in crystals are not necessarily free-electron parabolas across their full range. A band may flatten along some k directions, placing many states into a narrow energy range and raising the DOS. Critical points where the gradient of E(k) vanishes can produce van Hove features. Such features can influence optical spectra and instabilities, though one must also consider occupation and transition selection rules.

Units are a frequent source of mistakes. A total DOS can be states per joule or states per electronvolt for the entire sample; dividing by volume makes it states per energy per cubic metre. Numerical values differ by a factor of 1.602 × 10⁻¹⁹ when converting between per joule and per electronvolt, and changing the sample size scales total DOS while leaving DOS per volume unchanged. An experimental DOS is also often broadened by finite temperature or instrument resolution.

Step-by-step reasoning

1. State whether DOS is total or normalised by length, area or volume. 2. Identify dispersion E(k) near the relevant band edge. 3. Count k states inside a constant-energy region in the relevant dimension. 4. Differentiate cumulative count N(E) to obtain g(E). 5. Multiply by occupation f(E) when the question asks for electrons rather than available states.

Visual explanation

Place three small diagrams side by side: a 1D line from −k to +k, a 2D disk and a 3D sphere. Beneath them draw DOS versus energy: a high sharp 1D onset decreasing as 1/√ε, a flat 2D step and a 3D curve growing as √ε. A separate shaded occupation curve shows that DOS and filled-state number are different plots.

Real-world analogy

The DOS is like the number of theatre seats at each price level; f(E) is the probability that a seat at that level is occupied. A large balcony with many empty seats gives a high seat count but not a high audience count. The geometry of the building determines how quickly seat numbers grow as more rows become available.

Real-world example

In a semiconductor quantum well, electrons are confined in one direction but move approximately freely in two. Each allowed confinement mode creates a 2D subband with a step-like DOS onset. Changing well thickness shifts those onsets, affecting optical absorption and carrier distribution. A bulk version of the same material has a smoother square-root onset near a simple three-dimensional band edge.

Why?

Why is 3D DOS zero at an ideal parabolic band minimum but 1D DOS formally divergent there? Near k = 0, a tiny energy interval corresponds to a small shell. In three dimensions its surface area shrinks rapidly; in one dimension the derivative dk/dE grows as k approaches zero. Finite lifetime and disorder smooth the ideal mathematical divergence.

Common misconception

"The density of states is the density of electrons." DOS counts available one-electron states. Electron density requires integrating DOS times an occupation probability over energy. The distinction is especially important for an empty conduction band, which can have a substantial DOS while containing almost no electrons.

Worked example

Question: For parabolic bands beginning at ε = 0, compare g at ε = 4ε₁ with g at ε₁ in one, two and three dimensions, assuming both energies lie within the same subband and no singularity other than the ideal edge.

Reasoning: In one dimension g ∝ ε^(−1/2), so the ratio is 4^(−1/2) = 1/2. In two dimensions g is constant, so the ratio is 1. In three dimensions g ∝ ε^(1/2), so the ratio is √4 = 2. These ratios compare states per energy , not occupancy, which may be entirely different at the two energies.

Answer: The ratios are 1/2 in 1D, 1 in 2D and 2 in 3D.

Quick check

1. Can a region of energy have nonzero DOS but nearly zero electron population? Answer: Yes. An almost empty conduction band has allowed states there, but its Fermi–Dirac occupation probability is small.

Exam focus

Derive each dimensional trend from k-space counting instead of memorising a curve without conditions. Specify a parabolic band and energy measured from its edge. State the DOS units and multiply by f(E) when asked for occupied electrons.

Advanced insight

Scanning tunnelling spectroscopy can probe a local density of states rather than the total bulk DOS. A peak in a tunnelling spectrum is influenced by tunnelling matrix elements and broadening, so a direct assignment to a bulk van Hove singularity needs care. In topological or strongly correlated materials, spectral weight and quasiparticle lifetime add information beyond elementary state counting.

Summary

DOS counts available states per unit energy; occupation is a separate factor. For an ideal parabolic band, the onset scales as ε^(−1/2) in 1D, a constant in 2D and √ε in 3D. Real band structure, subbands and broadening modify these forms, but k-space geometry explains their origin.

Practice questions

1. What must multiply g(E)dE to estimate electrons in that interval? Answer: The occupation probability f(E). 2. Why is the 2D parabolic DOS constant within one subband? Answer: The number of k states in a disk scales as k², which scales linearly with energy, so its derivative is constant. 3. Which ideal parabolic dimensionality has g(E) proportional to √(E−E₀)? Answer: Three dimensions. 4. Why does a 1D divergence not produce an infinite measured peak? Answer: Finite lifetime, temperature, disorder, finite resolution and departures from the ideal parabola broaden it.