2.7 Density of states

Definition and significance

The density of states (DOS) quantifies how many energy states are available to electrons at each energy level in a material. Think of it as a histogram: for a given energy, how many quantum states exist that an electron could potentially occupy? This single function drives a huge range of material properties, from electrical conductivity to optical absorption to heat capacity.

DOS is typically denoted $g(E)$ or $D(E)$, where $E$ is energy. It varies depending on the dimensionality and crystal structure of the material, and it directly governs how electrons distribute themselves across energy levels.

Why DOS matters in solid-state physics

• Electrical conductivity: The DOS at the Fermi level determines how many electrons can participate in conduction.
• Optical properties: Absorption and emission spectra in semiconductors depend on the DOS in the valence and conduction bands.
• Thermal properties: The electronic contribution to heat capacity scales with the DOS at the Fermi energy.
• Device design: Engineering the DOS is central to technologies like solar cells, LEDs, quantum well lasers, and thermoelectrics.

Mathematical formulation

General expression for DOS

The DOS is defined as the number of states per unit energy per unit volume:

$g(E) = \frac{dN}{dE}$

where $N$ is the total number of states up to energy $E$. To derive $g(E)$ for a given system, you typically:

1. Start with the dispersion relation $E(\mathbf{k})$, which relates energy to wavevector.
2. Count the number of allowed $\mathbf{k}$-states enclosed by a constant-energy surface in $k$-space.
3. Convert that count from $k$-space to energy space using the dispersion relation.
4. Differentiate with respect to energy to get $g(E)$.

This procedure often involves integrating over constant-energy surfaces in the Brillouin zone.

Units and dimensions

The units of DOS depend on the dimensionality of the system:

DOS in different dimensions

Dimensionality has a dramatic effect on the shape of $g(E)$. Reducing the number of spatial dimensions confines electrons, quantizes energy levels, and reshapes the DOS in characteristic ways.

Three-dimensional systems

For free electrons in 3D, the DOS grows as:

$g(E) \propto \sqrt{E}$

This smooth, monotonically increasing function describes the behavior of simple metals reasonably well. In real materials, band structure effects create peaks, dips, and other features on top of this baseline shape.

Two-dimensional systems

In 2D (e.g., quantum wells, graphene), the DOS becomes a step function. Within each quantized subband, $g(E)$ is constant and independent of energy. Each new subband that becomes accessible adds another step. This flat DOS within subbands is one reason 2D electron gases show such clean quantum phenomena, including the quantum Hall effect.

One-dimensional systems

In 1D systems like nanowires and carbon nanotubes, the DOS diverges at each subband edge:

$g(E) \propto \frac{1}{\sqrt{E - E_n}}$

where $E_n$ is the subband edge energy. These divergences are called Van Hove singularities. They produce sharp peaks that strongly enhance optical absorption and electron-hole interactions at specific energies.

Zero-dimensional systems (quantum dots)

With confinement in all three directions, energy levels become fully discrete. The DOS reduces to a series of delta functions, resembling the energy spectrum of an atom. This is why quantum dots are sometimes called "artificial atoms."

Free electron model

Deriving the 3D free electron DOS

The free electron model treats conduction electrons as non-interacting particles in a constant potential (a "box"). Despite its simplicity, it captures many metallic properties. Here's the derivation outline:

1. Apply periodic boundary conditions to a cubic box of side $L$. Allowed wavevectors are $\mathbf{k} = \frac{2\pi}{L}(n_x, n_y, n_z)$.
2. The dispersion relation is parabolic: $E = \frac{\hbar^2 k^2}{2m}$.
3. Count the number of states with energy less than $E$. In $k$-space, these states fill a sphere of radius $k = \sqrt{2mE/\hbar^2}$.
4. The volume of that sphere is $\frac{4}{3}\pi k^3$. Each state occupies a volume $(2\pi/L)^3$ in $k$-space, and the factor of 2 for spin doubles the count.
5. Differentiate $N(E)$ with respect to $E$ to get:

$g(E) = \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2} \sqrt{E}$

This $\sqrt{E}$ dependence is the hallmark of the 3D free electron DOS.

Fermi energy and DOS

The Fermi energy $E_F$ is the energy of the highest occupied state at absolute zero. Its relationship to the DOS determines key material properties:

• In metals, $E_F$ sits inside a band where $g(E_F) > 0$. A finite DOS at the Fermi level means plenty of electrons can respond to applied fields, giving metals their high conductivity.
• In semiconductors and insulators, $E_F$ falls within the band gap where $g(E) = 0$. No states are available at the Fermi energy, so conduction requires thermal excitation of carriers across the gap.

The value of $g(E_F)$ shows up directly in expressions for electronic heat capacity, Pauli paramagnetism, and superconducting transition temperatures.

Band structure and DOS

Relationship to energy bands

The DOS is essentially a projection of the full band structure $E(\mathbf{k})$ onto the energy axis. The connection is intuitive:

• Flat bands (where $E$ changes slowly with $\mathbf{k}$) pack many states into a narrow energy range, producing peaks in the DOS.
• Steep bands (where $E$ changes rapidly with $\mathbf{k}$) spread states over a wide energy range, giving a low DOS.
• Van Hove singularities occur at critical points in the Brillouin zone where $\nabla_\mathbf{k} E = 0$, such as band edges, saddle points, and zone boundaries. These produce kinks, jumps, or divergences in $g(E)$ depending on the dimensionality.

Effect of band gaps

Band gaps create energy windows where $g(E) = 0$. No electronic states exist in these regions. The size of the gap between the valence band and conduction band determines whether a material is a semiconductor (gap ~ 0.1–3 eV) or an insulator (gap > 3 eV). The DOS shape near the band edges controls:

• Carrier concentrations at finite temperature
• Optical absorption onset and emission wavelengths
• The effectiveness of doping

DOS in semiconductors

Effective mass approximation

Near band extrema, the dispersion relation is approximately parabolic, so you can write:

$E(\mathbf{k}) \approx E_0 + \frac{\hbar^2 k^2}{2m^*}$

where $m^*$ is the effective mass, which encodes how the electron responds to forces within the crystal lattice. With this approximation, the DOS near a band edge takes the same $\sqrt{E}$ form as the free electron model, but with $m^*$ replacing the bare electron mass $m$:

$g(E) \propto (m^*)^{3/2}\sqrt{E - E_{\text{edge}}}$

Electrons and holes generally have different effective masses, so their DOS curves differ. A heavier effective mass means a higher DOS near the band edge.

Temperature dependence

At finite temperature, the Fermi-Dirac distribution $f(E)$ determines which states are actually occupied:

$f(E) = \frac{1}{e^{(E - \mu)/k_BT} + 1}$

As temperature increases, the sharp cutoff at $E_F$ broadens over a range of roughly $\pm k_BT$. This thermal smearing populates conduction band states and depopulates valence band states, creating free carriers. The product $g(E) \cdot f(E)$ gives the actual distribution of occupied states, which is what determines measurable quantities like carrier concentration and conductivity.

DOS in low-dimensional systems

Quantum wells (2D confinement)

Confining electrons in one direction (say $z$) quantizes the motion along that axis into discrete subbands labeled by quantum number $n$. Within each subband, electrons move freely in the $xy$-plane, giving a constant DOS per subband. The total DOS is a staircase function, with each step appearing at the energy of a new subband.

By adjusting the well width and barrier composition, you can tune the subband energies and tailor the optical and electronic properties. Quantum well lasers exploit this to achieve low threshold currents and narrow emission linewidths.

Quantum wires (1D confinement)

Confining electrons in two directions leaves free motion along only one axis. The DOS for each subband diverges as $1/\sqrt{E - E_n}$ at the subband edge. These Van Hove singularities concentrate the DOS at specific energies, enhancing optical transitions and electron-phonon coupling. Carbon nanotubes are a well-studied example, where the sharp DOS peaks produce distinctive optical absorption features.

Quantum dots (0D confinement)

Full 3D confinement yields discrete, atom-like energy levels. The DOS is a set of delta functions (broadened in practice by temperature and disorder). Because the energy levels are discrete and tunable via dot size, quantum dots find applications in:

• Quantum dot lasers with ultra-low threshold currents
• Single-photon sources for quantum communication
• Biological imaging using size-tunable fluorescence
• Qubits for quantum computing

Experimental techniques

Photoemission spectroscopy

Photoemission spectroscopy (PES) directly measures the occupied DOS below the Fermi level. The basic process:

1. Shine photons of known energy $h\nu$ onto the sample surface.

2. Electrons absorb photons and escape the material with kinetic energy $E_k = h\nu - E_B - \phi$, where $E_B$ is the binding energy and $\phi$ is the work function.

3. Measure the kinetic energy distribution of emitted electrons.

4. Map this distribution back to binding energy to reconstruct the occupied DOS.

Angle-resolved photoemission spectroscopy (ARPES) adds momentum resolution by tracking the emission angle, allowing you to map out the full band structure $E(\mathbf{k})$ rather than just the energy-projected DOS.

Tunneling spectroscopy

Scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) probe the local DOS with atomic spatial resolution. The tunneling current between the STM tip and sample is proportional to the convolution of the tip and sample DOS. By sweeping the bias voltage and measuring $dI/dV$, you get a signal proportional to the local DOS at that energy.

This technique has been especially powerful for studying superconductors (revealing the energy gap), surface states, and defect-induced states in nanostructures.

Applications of DOS

Electronic properties

• Carrier concentrations: Integrating $g(E) \cdot f(E)$ over the conduction or valence band gives the electron or hole density.
• Conductivity: The Drude and Boltzmann transport formalisms both depend on $g(E_F)$ in metals, or on the DOS near band edges in semiconductors.
• Thermoelectrics: The Seebeck coefficient depends on the energy derivative of the DOS near $E_F$. Materials with sharp DOS features near the Fermi level tend to have larger thermoelectric responses.

Optical properties

The rate of optical transitions between bands depends on the joint density of states (see Advanced Concepts below), which combines the DOS of initial and final states. This determines:

• The shape of absorption edges in semiconductors
• Emission spectra and efficiency of LEDs
• The spectral response of photodetectors and solar cells

Computational methods

Numerical calculation of DOS

For real materials with complex band structures, the DOS is computed numerically from the calculated $E(\mathbf{k})$:

1. Sample the Brillouin zone on a fine grid of $\mathbf{k}$-points.
2. Calculate $E(\mathbf{k})$ at each point (from a tight-binding model, pseudopotential method, or DFT).
3. Bin the energies into a histogram, or use the tetrahedron method (which interpolates between $\mathbf{k}$-points) for smoother, more accurate results.
4. Apply Gaussian or Lorentzian broadening to smooth out numerical noise if needed.

Convergence testing is critical: you need enough $\mathbf{k}$-points that the DOS doesn't change when you add more.

DOS from first principles

Density Functional Theory (DFT) calculates the electronic structure from the atomic positions alone, with no empirical fitting parameters. DFT-derived DOS predictions are highly accurate for many materials and allow you to explore how the DOS changes under pressure, doping, strain, or alloying. The main limitation is computational cost, especially for large unit cells or strongly correlated systems where standard DFT approximations break down.

DOS and thermodynamic properties

Heat capacity

The electronic contribution to heat capacity in metals is:

$C_{\text{el}} = \frac{\pi^2}{3} k_B^2 T \, g(E_F)$

This linear-in-$T$ dependence is a direct consequence of having a finite DOS at the Fermi level. Only electrons within about $k_BT$ of $E_F$ can be thermally excited, and $g(E_F)$ tells you how many such electrons there are.

For lattice (phonon) contributions, the analogous quantity is the phonon DOS, which enters the Debye model of heat capacity.

Magnetic susceptibility

The Pauli paramagnetic susceptibility of a metal is:

$\chi_{\text{Pauli}} = \mu_0 \mu_B^2 \, g(E_F)$

A larger DOS at the Fermi level means more electrons can flip their spins in response to an applied magnetic field, producing a stronger paramagnetic response. This is why transition metals with their high $g(E_F)$ from narrow $d$-bands tend to be more strongly paramagnetic than simple metals like aluminum.

Advanced concepts

Joint density of states

The joint density of states (JDOS) counts the number of pairs of states (one in the valence band, one in the conduction band) separated by a given energy $\hbar\omega$:

$J(\hbar\omega) = \int \delta(E_c(\mathbf{k}) - E_v(\mathbf{k}) - \hbar\omega) \, d\mathbf{k}$

The JDOS determines the spectral shape of interband optical absorption. Peaks in the JDOS correspond to energies where many transitions occur simultaneously, producing strong absorption features. The optical absorption coefficient is proportional to the JDOS weighted by the transition matrix element.

Local density of states

The local density of states (LDOS) gives the DOS resolved in real space:

$\rho(\mathbf{r}, E) = \sum_n |\psi_n(\mathbf{r})|^2 \, \delta(E - E_n)$

Unlike the total DOS, the LDOS tells you where in the material the states at a given energy are concentrated. This is directly what STM measures, and it's essential for understanding surface states, impurity states, and the electronic structure of interfaces and nanostructures.