Question bank — Compare band gaps - conductor - semiconductor - insulator

False — with every state occupied, for each electron pushed one way there is one going the opposite way, so the net current stays zero no matter how strong the field; you need empty states to accelerate into.

Effectively true — zero gap (bands overlapping or a partly-filled band) means free carriers exist at every temperature, which is the defining feature of a conductor.

False — as $T\to 0$, $e^{-E_g/2kT}\to 0$, so intrinsic carriers vanish and a pure semiconductor becomes an insulator; it is heat that supplies the "jump energy."

False — because the intrinsic Fermi level sits near mid-gap, each carrier costs only about $E_g/2$ on average, so the correct exponent is $-E_g/2kT$; the missing 2 changes results by many orders of magnitude.

False — both follow the same $n_i\propto e^{-E_g/2kT}$; the only difference is the numeric value of $E_g$, which pushes $n_i$ from "useful" to "negligible." The classification is a threshold, not a new law.

False — $E_g$ lives in an exponent, so doubling it squares the suppression factor and can change carrier count by dozens of orders of magnitude, not a factor of two.

False — metals have no gap to cross; their carriers exist regardless of temperature. Thermal excitation is the semiconductor mechanism, not the metal one.

False — $kT$ is about 40 times smaller than $E_g$, so $E_g/2kT\approx 21$ and only $\sim e^{-21}$ of the attempts succeed; Si is a weak intrinsic conductor.

$e^{-0}=1$, not $0$ — the exponential gives full (unsuppressed) carriers. Also the intrinsic formula was never meant for metals, whose free-electron count is set by band filling, not thermal promotion.

Phonon scattering hurts mobility in every material. In semiconductors the exponential rise in carrier number simply overwhelms that scattering loss; the two effects fight and carriers win.

The gap (≈ 5.5 eV) is large but finite, so $n_i=\sqrt{N_c{N}_v}{e}^{-E_g/2kT}$ is tiny but nonzero. "Insulator" means technologically negligible, not literally zero.

The order is reversed. The CB bottom $E_c$ sits above the VB top $E_v$, so $E_g=E_c-E_v>0$; the given expression would be negative.

A partly-filled band has empty states right next to occupied ones, so electrons can shift into them and produce net current. Only a completely full band cancels out.

In heavily doped (extrinsic) semiconductors the carrier count is nearly fixed over a wide range, so mobility loss can dominate and resistance can rise like a metal there. The exponential rise is the intrinsic regime story. See Intrinsic vs Extrinsic Semiconductors.

It comes from the Fermi–Dirac occupation $f(E)=1/(1+e^{(E-E_F)/kT})$; probabilities of thermal excitation are inherently exponential in energy-over-$kT$. See Fermi–Dirac Distribution.

Because $E_g$ enters as $e^{-E_g/2kT}$ and $kT$ is only ~0.026 eV, even a 0.1 eV change shifts the exponent by ~2, multiplying $n_i$ by nearly $10\times$; several tenths of an eV give many orders of magnitude.

The Fermi level sits roughly mid-gap, so a conduction electron is only $\sim E_g/2$ above $E_F$ (and the hole it leaves is $\sim E_g/2$ below), splitting the total gap energy between the two carriers. See Doping and the Fermi Level.

Metals have a fixed carrier count, so hotter lattice vibrations (phonons) just scatter electrons more → resistance up. Semiconductors gain exponentially more carriers, which dominates over extra scattering → resistance down. See Temperature Dependence of Resistance in Metals and Conductivity and Mobility.

An electric field must be able to nudge electrons into slightly higher-momentum states to build a net drift; if every nearby state is occupied (Pauli exclusion), there is nowhere to nudge them and no current flows.

The exponential varies by dozens of orders of magnitude between materials while $\sqrt{N_c{N}_v}$ changes by only a small factor, so the exponent dominates any ratio. The prefactor comes from the Density of States.

Real insulators have large but finite gaps (diamond ≈ 5.5 eV); calling them infinite would wrongly predict exactly zero carriers and hide the fact that they obey the very same equation as semiconductors.

$e^{-E_g/2kT}\to e^0=1$, so there is no thermal suppression and carriers are abundant at all temperatures — the metal (conductor) limit.

The exponent $-E_g/2kT\to -\infty$, so $n_i\to 0$; even a semiconductor freezes into an insulator at absolute zero because there is no thermal energy to lift electrons.

With $E_g/2kT$ no longer huge, the suppression weakens and even a wide-gap material starts producing appreciable carriers — it behaves semiconductor-like. The labels are always relative to the operating temperature.

They are equal — every electron promoted across the gap leaves exactly one hole behind, so $n=p=n_i$ in a pure (undoped) material. See Intrinsic vs Extrinsic Semiconductors.

No — doping shifts $E_F$ toward a band edge, so the simple $E_g/2$ "cost" no longer applies and the extrinsic carrier count is set by the dopants instead. See Doping and the Fermi Level.

A "negative gap" means the bands overlap so the CB bottom sits below the VB top; there is no forbidden region at all, and the material is metallic (a semimetal in the borderline overlap case).