Semiconductors & Band Gaps

⚡ Key Takeaways

Semiconductors are crystalline materials whose electrical conductivity lies between metals and insulators, governed by a small energy band gap ( $E_g \sim 0.1$ – $3$ eV) separating filled valence states from empty conduction states. At $T=0$ K they are insulators; thermal or optical excitation across the gap creates mobile charge carriers. They apply to all of solid-state electronics — diodes, transistors, LEDs, solar cells — and the modern theory dates from the 1930s (Bloch, Wilson, Mott).

💡 The Core Insight

In a periodic crystal the allowed electron energies form continuous bands separated by gaps. Whether a material conducts depends not on how many electrons it has but on whether bands are partially or completely filled. A completely filled band carries no current (every forward state cancels a backward one). Conduction therefore requires either a partly filled band (metal) or thermal/optical promotion of electrons across a gap into an empty band (semiconductor). The size of the gap, not the number of electrons, decides everything.

DefinitionBand Gap

For a crystalline solid with single-electron energy spectrum $E_n(\mathbf{k})$ (band index $n$ , crystal momentum $\hbar\mathbf{k}$ ), define the valence band maximum $E_v = \max_{\mathbf{k}} E_{\text{val}}(\mathbf{k})$ and conduction band minimum $E_c = \min_{\mathbf{k}} E_{\text{cond}}(\mathbf{k})$ . The band gap is

1

\quad E_g = E_c - E_v

Derivation hint: $E_v$ and $E_c$ are extrema of adjacent bands; $E_g>0$ means no allowed states in $(E_v, E_c)$ . If the extrema occur at the same $\mathbf{k}$ , the gap is direct; otherwise indirect. A material is a semiconductor when $0 < E_g \lesssim 3$ eV.

DefinitionEffective Mass

Near a band extremum the dispersion is approximately parabolic:

2

\quad E(\mathbf{k}) \approx E_c + \frac{\hbar^2 (\mathbf{k}-\mathbf{k}_0)^2}{2m^*}, \qquad \frac{1}{m^*} = \frac{1}{\hbar^2}\frac{\partial^2 E}{\partial k^2}

Derivation hint: Taylor-expand $E(\mathbf{k})$ about the extremum $\mathbf{k}_0$ ; the linear term vanishes, the curvature defines $m^*$ . Here $m^*$ is the effective mass, encoding how the periodic potential modifies the carrier's inertial response. $\mathbf{k}_0$ is the location of the extremum, $E_c$ the band edge energy.

TheoremIntrinsic Carrier Concentration

For an intrinsic (undoped) semiconductor in thermal equilibrium, the product of electron ( $n$ ) and hole ( $p$ ) concentrations is independent of Fermi level:

3

\quad n\,p = n_i^2 = N_c N_v \, e^{-E_g/k_BT}

where $N_c = 2\left(\frac{m_e^* k_B T}{2\pi\hbar^2}\right)^{3/2}$ and $N_v = 2\left(\frac{m_h^* k_B T}{2\pi\hbar^2}\right)^{3/2}$ are the effective densities of states. For intrinsic material $n = p = n_i$ .

ProofOrigin of Bands and Gaps — Nearly Free Electron Model

Step 1. Start with free electrons, $E = \hbar^2 k^2/2m$ , a single parabola. Introduce a weak periodic potential $V(x) = V(x+a)$ with Fourier components $V_G$ on reciprocal lattice vectors $G = 2\pi n/a$ .

Step 2. Bloch's theorem (see Bloch's Theorem) requires eigenstates $\psi_k(x) = e^{ikx}u_k(x)$ with $u_k(x+a)=u_k(x)$ . Energies are periodic and labelled by $k$ in the first Brillouin zone $[-\pi/a, \pi/a]$ .

Step 3. Degenerate perturbation theory: plane waves $|k\rangle$ and $|k-G\rangle$ become degenerate when $k = G/2 = \pi/a$ (zone boundary). The potential couples them, lifting the degeneracy.

Step 4. Diagonalize the $2\times 2$ block:

4

\quad E_\pm = \frac{\hbar^2}{2m}\left(\frac{\pi}{a}\right)^2 \pm |V_G|

Derivation hint: solve $\det\begin{pmatrix}E_0-E & V_G\\ V_G^* & E_0-E\end{pmatrix}=0$ at the zone boundary.

Step 5. The two solutions differ by $\Delta E = 2|V_G|$ : a forbidden gap opens at the zone boundary. The standing waves $\cos(\pi x/a)$ (pile charge on ions, lower energy) and $\sin(\pi x/a)$ (charge between ions, higher energy) account for the splitting. This is the band gap. $\blacksquare$

∎

ProofWhy a Filled Band Carries No Current

Step 1. Current density $j = -\frac{e}{V}\sum_{\text{occ}} v(k)$ , with group velocity $v(k) = \frac{1}{\hbar}\frac{dE}{dk}$ .

Step 2. For a filled band the sum runs over all $k$ in the Brillouin zone. Since $E(k) = E(-k)$ (time-reversal symmetry), $v(-k) = -v(k)$ .

Step 3. Each occupied state at $+k$ is paired with one at $-k$ carrying opposite velocity; the sum cancels exactly: $j = 0$ , even in an applied field (the field shifts $k$ uniformly, but a full band maps onto itself). Hence only partly filled bands conduct. $\blacksquare$

∎

AH

Alan Herries Wilson

1906–1995

British mathematical physicist who in 1931 provided the first quantum-mechanical theory distinguishing metals, insulators, and semiconductors purely by band filling. Building on Bloch's 1928 band theory, Wilson showed that a semiconductor is an insulator with a small gap, and that its conductivity arises from thermally excited carriers and impurities — explaining the puzzling exponential temperature dependence. His two 1931 papers in Proc. Roy. Soc. essentially founded the band theory of semiconductors and introduced the concept of intrinsic versus extrinsic (doped) conduction.

🧠 Intuition

Picture electrons in a crowded theatre. In a metal, the floor (band) is half-empty — people shuffle sideways freely when nudged: current flows. In an insulator, the floor is packed solid and the balcony is far above and empty — nobody can move; no current. A semiconductor is a packed floor with a balcony just barely out of reach: a little thermal jostling lets a few people leap up to the empty balcony. Now two things can move — the person on the balcony (electron) and the empty seat they left behind, which neighbours keep shuffling into (the hole, an effective positive carrier moving the opposite way). Doping is like building a small staircase (donor/acceptor level) inside the gap so people reach the balcony — or leave seats empty — at almost no energy cost.

Figure

Band structure of metal, semiconductor, and insulator. Shaded = filled (valence), key curve marks the gap.

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ExampleIntrinsic Carrier Concentration in Silicon

Given: Silicon at $T = 300$ K, $E_g = 1.12$ eV, $m_e^* = 1.08\,m_0$ , $m_h^* = 0.81\,m_0$ , $k_BT = 0.02585$ eV.

Step 1 — Effective densities of states. Using $N_{c,v} = 2(m^* k_BT/2\pi\hbar^2)^{3/2}$ . A useful reference value: $2(m_0 k_BT/2\pi\hbar^2)^{3/2} = 2.51\times10^{19}\,\text{cm}^{-3}$ at 300 K. $N_c = 2.51\times10^{19}\,(1.08)^{3/2} = 2.82\times10^{19}\,\text{cm}^{-3}$ $N_v = 2.51\times10^{19}\,(0.81)^{3/2} = 1.83\times10^{19}\,\text{cm}^{-3}$

Step 2 — Boltzmann factor. $E_g/k_BT = 1.12/0.02585 = 43.3$ , so $e^{-43.3} = 1.5\times10^{-19}$ .

Step 3 — Apply equation [3]. $n_i = \sqrt{N_c N_v}\,e^{-E_g/2k_BT} = \sqrt{2.82\times10^{19}\cdot1.83\times10^{19}}\cdot e^{-21.65}$ $n_i = 2.27\times10^{19}\cdot 3.9\times10^{-10}$

\boxed{n_i \approx 9 \times 10^{9}\ \text{cm}^{-3}}

(Accepted value $\approx 1.0\times10^{10}\,\text{cm}^{-3}$ ; small differences arise from $T$ -dependence of $E_g$ .)

ExampleDirect-Gap Absorption Edge Wavelength

Given: Gallium arsenide, direct gap $E_g = 1.42$ eV. Find the longest wavelength of light it can absorb.

Step 1 — Photon energy must exceed the gap. Absorption requires $h\nu \ge E_g$ , so the threshold is $h\nu = E_g$ .

Step 2 — Convert via $\lambda = hc/E$ . Use the convenient constant $hc = 1240\ \text{eV}\,\cdot\,\text{nm}$ :

\lambda_{\max} = \frac{1240\ \text{eV}\,\cdot\,\text{nm}}{1.42\ \text{eV}}

\boxed{\lambda_{\max} \approx 873\ \text{nm (near-infrared)}}

GaAs is transparent to longer wavelengths and absorbs visible/near-IR light — the basis of its use in LEDs and solar cells.

⚠️ Common mistake

"Conductivity depends on the number of electrons." False. A filled band carries zero current regardless of how many electrons it holds (see proof above). What matters is band filling and the gap. Likewise, semiconductors are not metals with fewer electrons — they are insulators with a small enough gap that thermal excitation matters. Also note: the band gap is temperature-dependent (it generally shrinks as $T$ rises, via the Varshni relation), so quoting a single $E_g$ without a temperature is imprecise.

CounterexampleIndirect Gap Defeats Naïve Optical Absorption

One might expect any semiconductor with $h\nu > E_g$ to absorb light strongly. Silicon ( $E_g = 1.12$ eV) does not. Its conduction-band minimum lies at a different $\mathbf{k}$ than its valence maximum (indirect gap). A photon carries negligible momentum, so the transition requires absorbing/emitting a phonon to conserve crystal momentum — a second-order, weak process. This is why silicon is a poor light emitter (no efficient LEDs/lasers) despite dominating electronics, while direct-gap GaAs glows brightly. The gap value alone does not predict optical behaviour; the gap's $\mathbf{k}$ -space geometry is decisive.

connection

The band gap concept sits at the crossroads of several frameworks. Bloch's Theorem supplies the periodic eigenstates whose energies form bands. The Kronig-Penney Model is the exactly solvable 1-D potential that demonstrates gap formation explicitly. Fermi-Dirac Statistics determines how carriers populate the bands and fixes the Fermi level. The Density of States in three dimensions controls $N_c$ , $N_v$ , and hence $n_i$ . The reciprocal-space scaffolding comes from Brillouin Zones and the Reciprocal Lattice, where zone boundaries are precisely where gaps open. Doping connects to Hydrogenic Impurity Model for shallow donor/acceptor levels.

🔬 Where this appears

⏳ History

- 1833 — Michael Faraday observes that silver sulphide's resistance decreases with temperature, opposite to metals — the first hint of semiconducting behaviour.

- 1874 — Ferdinand Braun discovers rectification at metal–semiconductor contacts (the cat's-whisker detector), foundational to early radio.

- 1928 — Felix Bloch proves Bloch's theorem; electrons in periodic potentials occupy energy bands.

- 1930–31 — Rudolf Peierls and Léon Brillouin establish that gaps open at zone boundaries.

- 1931 — Alan Wilson formulates the band theory of metals, semiconductors, and insulators by band filling; introduces intrinsic/extrinsic distinction.

- 1939 — Nevill Mott and Walter Schottky develop the theory of the rectifying barrier (depletion layer).

- 1947 — Bardeen, Brattain, and Shockley invent the point-contact transistor at Bell Labs (Nobel Prize 1956).

- 1954 — First practical silicon solar cell (Bell Labs); silicon begins its dominance.

- 1962–2014 — Era of compound semiconductors: GaAs LEDs/lasers, then GaN blue LEDs (Akasaki, Amano, Nakamura — Nobel 2014).

open-problem

The single-particle band picture is an approximation. Many-body corrections (the GW approximation) are needed because density-functional theory (DFT) with standard functionals systematically underestimates $E_g$ by 30–100% — the notorious "band-gap problem." Excitonic effects (bound electron–hole pairs) shift optical absorption below the single-particle gap, especially in low-dimensional and organic semiconductors. Topological insulators force a rethink: bulk-gapped materials with protected conducting surface states blur the metal/insulator dichotomy. Predicting gaps in disordered, amorphous, or strongly correlated materials (Mott insulators, where electron–electron repulsion, not band structure, opens the gap) remains open. Finally, the gap's precise temperature and pressure dependence, and gap engineering in 2-D materials (graphene's zero gap, tunable in bilayers), are active research frontiers.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also