Metals, Semiconductors, and Insulators

Learning Objectives

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

1. Explain why the metal/semiconductor/insulator classification is not a property of the chemical element but of the combination of band filling and band gap — and why that single sentence is more powerful than every periodic-table mnemonic you have ever met.
2. State the Pauli-principle accounting that fixes the position of the Fermi level: each Bloch state holds two electrons (spin up + spin down), and electrons fill from the bottom of the band catalogue upward.
3. Decide, from a band structure alone, whether a material is a metal, a semiconductor, or an insulator — and recognise the three giveaway edge cases (semimetal, half-metal, and the "intrinsic gap is deceptively small" one).
4. Quantify the thermal window $k_B T \approx 25\,\text{meV}$ at room temperature, and use it to estimate intrinsic carrier density and how the conductivity of each class varies with temperature.
5. Describe how doping — substituting a donor or acceptor on a host site — slides the Fermi level, why this is the single most consequential trick in semiconductor engineering, and how a VASP calculation reports the resulting class through $E_F$, the band gap, and the DOS at $E_F$.

One-line preview: a metal is a crystal whose Fermi level cuts through a band; an insulator is one whose Fermi level sits in a wide gap; a semiconductor is the same with a narrow gap. Every electrical property of every solid you have ever held — copper wire, silicon chip, glass tumbler — is one of those three sentences with the word gap filled in.

An Old Puzzle: Why Do Some Things Conduct?

Before quantum mechanics, the difference between copper and quartz was a mystery. Drude's 1900 model — electrons as tiny billiard balls bouncing through a fixed positive lattice — gave the right order-of-magnitude conductivity for metals, but it had no good answer for why the same electrons, in a different host, refuse to move at all. Worse, classical statistics predicted a heat capacity $C_v = \tfrac{3}{2} N k_B$ from the electron gas alone, ten times larger than what experiments actually measured. The model worked for the wrong reasons.

The fix arrived in two steps. Sommerfeld (1928) noted that electrons obey Fermi–Dirac statistics, so only those within $k_B T$ of the Fermi level can absorb heat — that solved the heat-capacity puzzle. Bloch (1928) and Wilson (1931) went further: in a periodic potential, the eigenstates are not free particles but Bloch waves, their energies organised into bands separated by gaps. Combining the two observations gave the modern classification: it is not the number of electrons that decides whether a solid conducts but the geometry of the band structure they fill.

The Pauli Principle Sets the Filling Rules

Recall from Chapter 5.1 that for each band index $n$ the energy $E_n(\mathbf{k})$ is defined on the first Brillouin zone. In a finite crystal of $N_c$ primitive cells, the allowed values of $\mathbf{k}$ form a discrete grid with exactly $N_c$ points. With spin-1/2 electrons, that means each band holds

Pour $Z$ valence electrons per primitive cell into the catalogue. They fill states from the bottom up, two at a time. At $T = 0$ the highest occupied energy is the Fermi level $E_F$. The classification falls out of one simple test:

The Three Cases — Set By Filling, Not By Element

It is worth seeing the three cases drawn together, because the visual is what most students remember long after the algebra is forgotten. Imagine the same band-structure diagram three times, with the Fermi level pencilled in three different places.

Interactive — Filling, Gap, and Class

Drag the two sliders. The left slider sets electrons per primitive cell. The right slider sets the band gap. Cyan = filled (electrons living here), gray = empty. The amber dashed line is the Fermi level. The badge at the top tells you what kind of material you have built.

Things to try, in order:

1. Park the filling at 1.0 e⁻/cell, gap = 1.5 eV. The lower band is half-filled — the Fermi level sits inside a band, and the badge reads METAL. The gap above does not matter; nothing the electrons see at $E_F$ cares that there is a gap two volts up.
2. Now slide the filling to exactly 2.0 e⁻/cell. The lower band fills; $E_F$ jumps into the gap. With gap = 1.5 eV the badge reads SEMICONDUCTOR. Drag the gap up to 4 eV — same band, same filling, but now INSULATOR. Same crystal structure, different gap, completely different physics.
3. Set filling = 2 and shrink the gap to 0. The bands touch. The badge becomes SEMIMETAL. There are no carriers to speak of, but no gap either: a vanishing DOS at $E_F$. Graphite and bismuth live here.
4. Push filling past 2. Now the upper band starts to fill — METAL again, but this time the carriers are at the bottom of the conduction band rather than half-way up the valence band.

Why a Full Band Cannot Carry Current

Here is one of the most beautiful arguments in solid-state physics. Why, exactly, does a completely filled band carry zero current — even though it has zillions of electrons in it, all moving?

The current contributed by a single Bloch electron in band $n$ at wavevector $\mathbf{k}$ is $-e\,\mathbf{v}_n(\mathbf{k})$. Summing over every occupied state gives the total current density,

Time-reversal symmetry guarantees $E_n(-\mathbf{k}) = E_n(\mathbf{k})$, so $\mathbf{v}_n(-\mathbf{k}) = -\mathbf{v}_n(\mathbf{k})$: every state at $\mathbf{k}$ has a partner at $-\mathbf{k}$ moving in the opposite direction at exactly the same speed. If both partners are filled — which is exactly what happens in a full band — their currents cancel state by state. The integral over the whole BZ is then zero. A completely filled band carries zero current.

An electric field tries to break that pairing by accelerating electrons in the direction of $-\mathbf{F}$ (because $\hbar\dot{\mathbf{k}} = \mathbf{F}$), but in a full band there is nowhere for the electrons to go: every target state is already occupied, and Pauli forbids double-occupancy. The crystal momentum gets handed back to the lattice through Bragg reflection at the zone boundary, and the band returns to its symmetric equilibrium. That is why a glass tumbler is an insulator even though it contains an Avogadro's number of electrons.

The Edge Cases: Semimetals and Half-Metals

Real materials don't always cooperate with the clean three-way sort. Two classes of edge cases are worth naming.

Semimetals

A semimetal has a tiny indirect overlap: the conduction band minimum at one $\mathbf{k}$ sits a few meV below the valence band maximum at a different $\mathbf{k}$. So a small pocket of electrons and a small pocket of holes coexist, with the Fermi level inside both. Total carrier density is small ($\sim 10^{18}\,\text{cm}^{-3}$ in bismuth versus $\sim 10^{22}$ in copper), and the DOS at $E_F$ vanishes linearly. Famous examples: bismuth, antimony, graphite (along the c-axis), and graphene at the Dirac point. The last is special because the dispersion is linear, $E \propto |\mathbf{k}|$, giving electrons the formal properties of massless Dirac fermions — which earned a Nobel Prize.

Half-metals

A half-metal is metallic for one spin channel and insulating for the other: spin-up bands cross the Fermi level, spin-down bands have a gap. The conduction electrons are then 100% spin-polarised. The prototypical examples are CrO₂, Heusler alloys like NiMnSb, and the magnetite Fe₃O₄. Half-metals are the physical foundation of spintronics — every magnetic-tunnel-junction read head in your laptop hard drive uses one. We will see this concretely when we add Mn to CdSe in Chapter 7 and ask whether the doped material is half-metallic.

The Thermal Window — Why kT Decides Everything

At finite temperature the Fermi function smears out occupation around $E_F$ with characteristic width $k_B T$. At room temperature $T = 300\,\text{K}$,

That number — 25 meV — is the single most useful constant in semiconductor physics. Memorise it. It is the energy scale on which thermal excitations operate. The intrinsic carrier density of a non-degenerate semiconductor scales as

Plug in numbers and you discover why semiconductors and insulators live in different worlds:

The exponential is brutal. Doubling the gap from Si to GaN drops the Boltzmann factor by 19 orders of magnitude, and intrinsic conductivity with it. This is why the metal–semiconductor–insulator boundary sits near $E_g \sim 3\,\text{eV}$: below it, room-T thermal energy can produce measurable carriers; above it, you would need temperatures comparable to the Sun's surface before intrinsic conduction wakes up.

Doping: Turning the Fermi Level Into a Design Knob

An intrinsic Si crystal at room temperature has $n_i \sim 10^{10}\,\text{cm}^{-3}$. Replace one Si atom in $10^7$ with a phosphorus atom and the carrier density jumps to $\sim 10^{15}\,\text{cm}^{-3}$ — five orders of magnitude with a vanishing structural perturbation. That is the miracle of doping.

The mechanism is best read off the band diagram. P (Group V) on a Si (Group IV) site brings one extra electron and one extra proton on the same site. The proton attracts the extra electron in a hydrogenic orbital with binding energy $\sim 45\,\text{meV}$ — far less than the gap, so the donor level $E_D$ sits a hair below the conduction band minimum. At room temperature $k_B T \approx 25\,\text{meV}$ is enough to ionise nearly every donor, and the freed electron lives in the conduction band. The Fermi level slides up toward $E_D$: the crystal becomes n-type.

The mirror story uses Group III. Boron on a Si site is missing one electron — it has an empty acceptor level $E_A$ just above the valence band. Thermal electrons hop up from the VBM into $E_A$, leaving a hole behind. The Fermi level slides down toward $E_A$: the crystal becomes p-type.

Interactive — Donors, Acceptors, and the E_F Slide

Drag the doping slider. Negative values add acceptors (pink), positive values add donors (cyan). Watch the Fermi level physically slidewithin the gap, and the carrier population in the conduction or valence band change with it.

Two specific things to convince yourself of:

1. The gap does not change. Only the Fermi level moves. The CBM and VBM stay where they are; the donor/acceptor level appears as a thin horizontal in the gap. The carriers actually conducting are not on the donor level — they have fallen off into the conduction band (n-type) or jumped up into the valence band leaving a hole (p-type).
2. Doping is asymmetric in concentration scale. Even one part-per-billion of phosphorus can dominate over the intrinsic $n_i \sim 10^{10}\,\text{cm}^{-3}$ in Si. The slider exaggerates the visual density to make the population obvious — in a real crystal, dopants are extraordinarily dilute.

Reading the Class From a VASP Calculation

Given a converged VASP calculation, three numbers — the Fermi level, the gap, and the DOS at the Fermi level — tell you the class of the material. The Python below uses pymatgen to extract them from a single vasprun.xml.

vr.final_structure → pymatgen Structure

Vasprun("vasprun.xml")

EF = -2.34  # eV

{'energy': 1.12, 'direct': False, 'transition': 'Γ → X'}

energies = [-3.0, -2.95, ..., 0.00, 0.05, ...]

window = array([False, False, ..., True, True, ..., False])

dos_at_EF = 1.4e-08  # → insulator

E_F = 3.21 eV / Gap = 1.12 eV (indirect) / VBM @ Γ, CBM @ X / DOS(E_F) = 1.4e-08 / => SEMICONDUCTOR

1from pymatgen.io.vasp.outputs import Vasprun
2import numpy as np
3
4vr = Vasprun("vasprun.xml", parse_dos=True, parse_eigen=True)
5bs = vr.get_band_structure()
6dos = vr.complete_dos
7EF = vr.efermi
8
9gap_info = bs.get_band_gap()
10gap = gap_info["energy"]      # eV; 0.0 if metallic
11direct = gap_info["direct"]   # bool
12vbm_k = bs.get_vbm()["kpoint"].label if bs.get_vbm() else None
13cbm_k = bs.get_cbm()["kpoint"].label if bs.get_cbm() else None
14
15# DOS at EF, smeared over a small window
16energies = np.array(dos.energies) - EF
17window = np.abs(energies) < 0.05      # ±50 meV
18dos_at_EF = float(np.mean(dos.densities[1][window])) if window.any() else 0.0
19
20# Classify
21if gap < 1e-3 or dos_at_EF > 1e-2:
22    cls = "METAL"
23elif gap < 0.05:
24    cls = "SEMIMETAL"
25elif gap < 3.0:
26    cls = "SEMICONDUCTOR"
27else:
28    cls = "INSULATOR"
29
30print(f"E_F      = {EF:.3f} eV")
31print(f"Gap      = {gap:.3f} eV  ({'direct' if direct else 'indirect'})")
32print(f"VBM @ {vbm_k},  CBM @ {cbm_k}")
33print(f"DOS(E_F) = {dos_at_EF:.3e} states/eV")
34print(f"=> {cls}")

What the INCAR has to be for this to work

The classification is only as trustworthy as the calculation behind it. Three INCAR settings dominate:

1# INCAR — minimum to get a defensible class
2PREC   = Accurate
3ENCUT  = 1.3 * max(ENMAX of POTCARs)   # converge to <1 meV/atom
4EDIFF  = 1E-6
5ISMEAR = -5     # tetrahedron with Blöchl corrections; best for accurate DOS
6                # (use 0 + small SIGMA only for metals or relaxations)
7LORBIT = 11     # writes orbital-projected DOS — needed for atom-resolved class
8NEDOS  = 3001   # dense DOS grid; a coarse one will smear away small features

Summary

• The metal/semiconductor/insulator distinction is not a chemical property but a geometric property of the band structure: where the Fermi level lands and how big a gap surrounds it.
• Each Bloch band holds $2$ electrons per primitive cell. Pour $Z$ valence electrons into the catalogue from the bottom; if $Z$ is odd or two bands overlap, the material is a metal. If $Z$ is even and the topmost occupied band is below a gap, the material is a non-metal — a semiconductor or an insulator depending on whether the gap is $\lesssim 3\,\text{eV}$ or wider.
• A completely filled band cannot carry current: states at $+\mathbf{k}$ and $-\mathbf{k}$ have opposite group velocities and exact pairwise cancellation. This is the most fundamental reason insulators exist.
• The thermal scale $k_B T \approx 25\,\text{meV}$ at 300 K, combined with the exponential factor $\exp(-E_g/2 k_B T)$, explains why a 24-orders-of-magnitude conductivity range fits onto a single E vs k diagram.
• Doping introduces shallow donor or acceptor levels close to a band edge. With $k_B T$ able to ionise them at room temperature, doping slides the Fermi level up (n-type) or down (p-type) inside the gap — without changing the bands themselves.
• Edge cases — semimetals (Bi, graphene), half-metals (CrO₂), Mott insulators (NiO), topological insulators (Bi₂Se₃) — sit outside the simple three-way sort but are handled by enriched versions of the same band picture.
• From a VASP calculation, the class is read off three numbers in vasprun.xml: the Fermi level, the gap (and its directness), and the DOS at $E_F$. The 30-line script in this section returns one of {METAL, SEMIMETAL, SEMICONDUCTOR, INSULATOR} for any pre-computed material.

Coming next: Section 5.4 — Effective Mass and Carrier Transport — where the curvature of the bands we have just classified turns into the mobility, conductivity, and Hall coefficient that experimentalists actually measure.