Density of States

Learning Objectives

By the end of this section you should be able to:

1. Define the density of states $g(E)$ in plain language — "how many electronic states are available at energy E" — and write it as a delta-function sum over the band structure of Section 5.1.
2. Explain the inverse-velocity rule: $g(E) \propto 1/|\nabla_{\mathbf{k}} E_n|$. See instantly why flat bands give big DOS and steep bands give small DOS.
3. Recognise the three universal free-electron shapes and remember which one belongs to which dimension: $1/\sqrt{E}$ (1-D), constant (2-D), and $\sqrt{E}$ (3-D).
4. Identify van Hove singularities on a DOS plot and connect them to critical points of $E_n(\mathbf{k})$ — band edges, saddle points, ridges.
5. Read a projected DOS and answer in one sentence: "the VBM is mostly Se 4p, the CBM is mostly Cd 5s."
6. Run a VASP DOS calculation: pick the right ISMEAR (tetrahedron $-5$ for static DOS, Gaussian $0$ for semiconductors during SCF, Methfessel–Paxton $1$ for metals), set NEDOS, write LORBIT=11 for projections, and plot the result with five lines of pymatgen.

One-line preview: the band structure tells you where in reciprocal space the energies live; the density of states tells you how many of them sit at each energy. Almost every experimental probe — heat capacity, photoemission, optical absorption, Pauli susceptibility — measures an integral over $g(E)$, not $E_n(\mathbf{k})$ directly. So this is the form of electronic structure that the laboratory actually sees.

The Question DOS Answers

Last section we built the band structure $E_n(\mathbf{k})$ — a beautiful object, but a difficult one. It is a stack of 3-D surfaces in a 3-D Brillouin zone, and no experiment ever sees it directly. Photoemission gets you a smeared cross-section. Heat capacity gets you one number. Optical absorption gets you a 1-D spectrum. What every one of these probes secretly cares about is the same simpler question:

That number — divided by the volume of the crystal and by $dE$ — is the density of states $g(E)$. It is one number per energy. Plot it on a sheet of paper and you have what every DFT post-processing tool spits out as DOSCAR: a 1-D function that contains an enormous amount of information about the crystal's electrons.

Why one number per energy is enough for so many properties is the magic. Two examples to anchor your intuition:

• Specific heat at low temperature in a metal is $C_V = \tfrac{\pi^2}{3}\,k_B^2\,T\,g(E_F)$. The whole answer is set by one number: $g(E_F)$, the DOS evaluated right at the Fermi level.
• Optical absorption at photon energy $\hbar\omega$ is, in the simplest dipole approximation, an integral $\int g_v(E)\,g_c(E + \hbar\omega)\,|M|^2\,dE$ — the joint density of states, weighted by a matrix element. The shape of an absorption spectrum is, to leading order, a convolution of two DOS curves.

Definition — Counting States Per Energy

Formally, the density of states (per unit volume, per spin) is

Read it slowly. The Dirac delta $\delta(E - E_n(\mathbf{k}))$ is an infinitely-sharp filter that fires when the band energy $E_n(\mathbf{k})$ hits the energy $E$ we are asking about. Integrating over the Brillouin zone counts every wavevector at which a state with that energy exists. Summing over band index $n$ bundles together all bands. Dividing by the crystal volume $V$ turns the count into a density.

Interactive — From E(k) to g(E)

The cleanest way to internalise the formula is to see it executed on a one-dimensional band. Below, the left panel is the familiar tight-binding cosine $E(k) = -2t\cos k$. The right panel is the same data, binned into 90 small energy windows: a histogram of $E(k_i)$ over 2400 evenly-spaced k-points. Drag $t$ to see the band stretch and the histogram scale with it. Toggle "add a second band" to watch new peaks appear when a gap opens.

Three observations that are not obvious until you have played for a minute:

1. The peaks of the histogram sit exactly at the band edges. At $k = 0$ and $k = \pm \pi$ the slope of the band vanishes. The energy spends "extra time" near those values, so many k-points end up in those bins.
2. The middle of the band is shallow. Near $k = \pm \pi/2$ the band is at its steepest (group velocity $v = 2t \sin k$ is maximum). The histogram dips because the band rushes through those energies and relatively few k-points stop there.
3. The band width $4t$ scales the whole picture. Doubling $t$ doubles the band width and halves the peak DOS — the area under $g(E)$ is conserved (one state per atom always).

The Inverse-Velocity Rule

The histogram you just watched is a numerical approximation of the formula

In words: the density of states at energy $E$ is the inverse of how fast the band moves through $E$. Mathematically the same thing as the constant-energy-surface formula above, but in 1-D it collapses to something you can reason about with one finger:

Free Electrons in 1, 2, and 3 Dimensions

Before crystals, before bands, the simplest model is the free-electron gas: $E = \hbar^2 k^2 / 2m$, a parabola. Even for this baby model the DOS depends dramatically on dimensionality because the constant-energy surface $E(\mathbf{k}) = E$ has different geometry in different dimensions.

1-D wire

In 1-D the constant-energy "surface" is just two points, $k = \pm \sqrt{2mE}/\hbar$. Plug into the general formula:

States pile up at the band bottom — a true $1/\sqrt{E}$ divergence. This is the signature of single-walled carbon nanotubes and quantum wires.

2-D sheet

In 2-D the constant-energy surface is a circle of radius $k = \sqrt{2mE}/\hbar$ with circumference $2\pi k$. The gradient $|\nabla E| = \hbar^2 k/m$. The two cancel:

A perfect step function at $E = 0$. Each free-electron subband in a 2-D quantum well adds another step — the famous staircase DOS seen in semiconductor heterostructures.

3-D bulk

In 3-D the constant-energy surface is a sphere of radius $k$ with surface area $4\pi k^2$. After dividing by $(2\pi)^3 |\nabla E|$:

Starts at zero and grows as $\sqrt{E}$. This is the curve every solid-state textbook prints when introducing free electrons in a metal — and the leading edge of every parabolic 3-D band, which is why it dominates the bottom of the conduction band in most semiconductors.

Interactive — DOS by Dimensionality

Click between the three dimensions to see the shapes side by side. Drop a Fermi level on the plot to read off how many states sit below it — the electron filling is the area under $g(E)$ up to $E_F$.

Van Hove Singularities

Real bands are not parabolas. They have multiple maxima, minima, and saddle points — places where $\nabla_{\mathbf{k}} E_n = 0$. Léon van Hove showed in 1953 that every such critical point produces a non-analytic feature in $g(E)$. The shape of the feature depends on dimension and on the nature of the critical point:

The 2-D logarithmic singularity is the famous van Hove peak in graphene: the saddle point at the M-point of the hexagonal Brillouin zone gives a logarithmic divergence in the DOS at $\pm t$. Every twisted-bilayer-graphene paper that talks about "flat-band physics" near a magic angle is secretly talking about the same physics: pushing van Hove peaks down to the Fermi level, where the divergence triggers superconductivity or magnetism through diverging response functions.

Reading a DOS Plot

A standard DOS plot is a 1-D graph: energy on the x-axis (almost always with $E_F$ shifted to zero), DOS on the y-axis. Sometimes the plot is rotated 90° so the energy axis is vertical and aligned with a band-structure plot beside it — that combined plot is the single most informative figure a DFT calculation produces.

Every one of those readings is a habit worth building. A practiced eye can spot the band gap, judge the metal-vs-semiconductor question, and guess the orbital character of the band edges in less than ten seconds from a DOS plot alone — without ever touching the band-structure plot.

Projected DOS — Asking Which Atom and Which Orbital

The total DOS counts all states. But the most useful question is usually finer: which atoms contribute at the Fermi level? which orbitals make up the valence band maximum? The answer is the projected DOS:

where $\phi_{\alpha,\ell}$ is an atomic-like orbital localised on atom $\alpha$ with angular momentum $\ell$. In words: project every Bloch state onto a chosen atomic orbital, square the amplitude, and bin the result into the DOS sum.

For our flagship system CdSe, the standard story readable in a single PDOS plot is:

• Valence band maximum ≈ Se 4p — the anion gives the holes.
• Conduction band minimum ≈ Cd 5s — the cation accepts the electrons.
• Cd 4d sits as a narrow band ~7 eV below VBM (a sharp PDOS peak — Cd's closed semicore d-shell).
• Mn 3d in doped Mn:CdSe (Chapter 7) appears as mid-gap impurity peaks split by the crystal field — the entire reason Mn:CdSe has tunable luminescence.

Computing DOS in VASP — Tetrahedron and Smearing

A DOS calculation in VASP is the same two-step pattern as a band structure: a converged SCF run, followed by a non-SCF run with a denser k-grid. The differences are only in how you tell VASP to integrate the delta function — that is the role of ISMEAR.

Step 1 — converged SCF

1# INCAR — SCF for CdSe
2SYSTEM = CdSe SCF for DOS
3PREC   = Accurate
4ENCUT  = 400          # converged cutoff (eV)
5EDIFF  = 1E-6
6ISMEAR = 0            # Gaussian during SCF — safe for semiconductors
7SIGMA  = 0.05
8LCHARG = .TRUE.       # write CHGCAR for step 2
9LWAVE  = .FALSE.

1# KPOINTS — uniform grid for SCF
2Monkhorst-Pack
30
4Gamma
58 8 8
60 0 0

Step 2 — non-SCF DOS run (denser k-grid + tetrahedron)

1# INCAR — DOS run
2SYSTEM   = CdSe DOS
3PREC     = Accurate
4ENCUT    = 400
5ICHARG   = 11         # read CHGCAR, do not update density
6ISMEAR   = -5         # tetrahedron with Blöchl corrections
7NEDOS    = 3001       # number of energy points in DOSCAR
8EMIN     = -15        # lowest energy in DOS window (eV)
9EMAX     = 15         # highest energy in DOS window (eV)
10LORBIT   = 11         # write per-atom, per-orbital projections
11NBANDS   = 32

1# KPOINTS — denser uniform grid
2Monkhorst-Pack
30
4Gamma
516 16 16
60 0 0

Output files

Plotting DOS with pymatgen

With those files in hand, generating a publication-quality DOS plot — total + element-projected + orbital-projected — takes about thirty lines of pymatgen. Click any line below to see what every variable, function argument, and library call is doing in plain language.

1from pymatgen.io.vasp.outputs import Vasprun
2from pymatgen.electronic_structure.core import OrbitalType
3from pymatgen.core.periodic_table import Element
4from pymatgen.electronic_structure.plotter import DosPlotter
5import matplotlib.pyplot as plt
6
7vr = Vasprun("vasprun.xml", parse_dos=True)
8
9dos = vr.complete_dos
10e_fermi = dos.efermi
11print(f"Fermi level: {e_fermi:.3f} eV")
12
13plotter = DosPlotter(zero_at_efermi=True, stack=False, sigma=0.05)
14
15plotter.add_dos("Total DOS", dos)
16
17element_dos = dos.get_element_dos()
18for element, edos in element_dos.items():
19    plotter.add_dos(f"{element.symbol} (all orbitals)", edos)
20
21cd_spd = dos.get_element_spd_dos(Element("Cd"))
22plotter.add_dos("Cd 5s", cd_spd[OrbitalType.s])
23
24se_spd = dos.get_element_spd_dos(Element("Se"))
25plotter.add_dos("Se 4p", se_spd[OrbitalType.p])
26
27ax = plotter.get_plot(xlim=(-6, 6), ylim=(0, 18))
28plt.axvline(0, color="black", linestyle="--", linewidth=0.6)
29plt.savefig("dos_cdse.png", dpi=200, bbox_inches="tight")

Run it, open dos_cdse.png, and you should see the canonical CdSe story drawn for you: a clean gap at $E = 0$, a dominant Se 4p shoulder just below zero, a Cd 5s peak just above zero, and a sharp narrow Cd 4d band a few eV below the VBM. Every feature in that plot is a sentence in the electronic-structure story of the material.

Summary

• The density of states $g(E) = \frac{1}{V}\sum_n \int_\text{BZ} \delta(E-E_n(\mathbf{k}))\,\frac{d^3k}{(2\pi)^3}$ counts how many electronic states sit at each energy. It is what experimental probes (heat capacity, photoemission, optics) actually see — the lab's view of the band structure.
• Equivalently, $g(E) = \sum_n \int_{S_n(E)} dS / [(2\pi)^3 |\nabla_{\mathbf{k}} E_n|]$: a constant-energy-surface integral weighted by the inverse of the band gradient. Flat bands → big DOS, steep bands → small DOS.
• For free electrons, the DOS is $1/\sqrt{E}$ (1-D), $\text{const}$ (2-D), and $\sqrt{E}$ (3-D) — the dimensionality fingerprint that quantum-confined nanostructures inherit.
• Van Hove singularities are the kinks, peaks, and divergences in $g(E)$ at critical points of $E_n(\mathbf{k})$. The 2-D logarithmic peak (saddle point) is responsible for graphene's famous van Hove singularity and the magic-angle bilayer flat-band physics.
• Projected DOS decomposes $g(E)$ by atom and angular momentum. For CdSe: VBM is Se 4p, CBM is Cd 5s, semicore Cd 4d sits ~7 eV below VBM. Mn dopants add mid-gap d-states.
• In VASP: do an SCF with $\texttt{ISMEAR}=0$, then a non-SCF run with $\texttt{ICHARG}=11$, $\texttt{ISMEAR}=-5$ (tetrahedron), $\texttt{LORBIT}=11$, and a denser k-grid. Plot with five lines of pymatgen.

Coming next: Section 5.3 — Metals, Semiconductors, and Insulators — where the integral of $g(E)$ up to $E_F$ tells us whether the material conducts, insulates, or sits on the boundary between the two.