Semiconductors and Band Gaps Introduction

⚡ Key Takeaways

Semiconductors are crystalline solids whose electronic conductivity lies between that of metals and insulators, controlled by a small band gap $E_g$ (typically 0.1–4 eV) separating filled valence states from empty conduction states. Key result: the number of mobile charge carriers depends exponentially on temperature and gap, $\propto e^{-E_g/2k_BT}$ , making conductivity tunable by heat, light, and doping. Where: transistors, diodes, solar cells, LEDs, photodetectors. Who/when: band theory developed by Bloch, Wilson, and others (1928–1931); silicon devices from the 1940s onward.

💡 The Core Insight

When atoms assemble into a periodic crystal, the discrete atomic energy levels broaden into continuous bands of allowed states separated by gaps of forbidden energies. Whether a solid is a metal, insulator, or semiconductor is decided by where the electrons stop filling: if the highest occupied band is only partly full, the material is a metal; if a band is exactly filled and the next empty band lies a small energy $E_g$ above it, the material is a semiconductor. The exponential sensitivity of carrier number to $E_g/k_BT$ is what makes semiconductors uniquely controllable.

DefinitionBand Gap and Carrier Statistics

The band gap $E_g$ is the energy difference between the top of the valence band $E_v$ and the bottom of the conduction band $E_c$ :

1

\quad E_g = E_c - E_v

Derivation hint: read directly off the band-structure diagram $E(\mathbf{k})$ as the minimum vertical separation between the lowest unoccupied and highest occupied bands. The density of allowed states per unit volume near a parabolic band edge (effective mass $m^*$ ) is

2

\quad g(E) = \frac{1}{2\pi^2}\left(\frac{2m^*}{\hbar^2}\right)^{3/2}\sqrt{E-E_c}, \qquad E \ge E_c

Derivation hint: count $\mathbf{k}$ -states in a sphere of radius $k$ in reciprocal space, $N = 2\cdot \frac{(4/3)\pi k^3}{(2\pi/L)^3}$ , use $E - E_c = \hbar^2 k^2/2m^*$ , and differentiate. The occupation probability of a state at energy $E$ is the Fermi–Dirac distribution:

3

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

Derivation hint: maximize entropy of a fermion gas under fixed particle number and energy; $E_F$ is the Lagrange multiplier (chemical potential). Symbols: $E_g$ band gap (J or eV), $E_c$ / $E_v$ conduction/valence band edges, $m^*$ carrier effective mass (kg), $\hbar$ reduced Planck constant, $E_F$ Fermi level, $k_B$ Boltzmann constant, $T$ absolute temperature.

TheoremIntrinsic Carrier Concentration

For an intrinsic (undoped) semiconductor with $E_g \gg k_BT$ , the equilibrium concentration of conduction electrons $n$ equals the hole concentration $p$ , and

4

\quad n_i = \sqrt{N_c N_v}\; e^{-E_g/2k_BT}

where the effective densities of states are

5

\quad N_c = 2\left(\frac{m_e^* k_B T}{2\pi\hbar^2}\right)^{3/2}, \qquad N_v = 2\left(\frac{m_h^* k_B T}{2\pi\hbar^2}\right)^{3/2}

Derivation hint: integrate $g(E)f(E)$ over the conduction band using the Boltzmann tail $f \approx e^{-(E-E_F)/k_BT}$ ; the Gaussian-type integral $\int_0^\infty \sqrt{x}\,e^{-x/k_BT}dx$ yields $N_c$ . Multiply $n\cdot p$ and impose $n=p$ .

ProofBand Formation and the Exponential Carrier Law

Step 1 — Periodic potential gives Bloch states. In a crystal the potential satisfies $V(\mathbf{r}+\mathbf{R}) = V(\mathbf{r})$ for lattice vectors $\mathbf{R}$ . Bloch's theorem (see Bloch's Theorem) forces solutions of the form $\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k}}(\mathbf{r})$ , with $u_{\mathbf{k}}$ lattice-periodic.

Step 2 — Gaps open at zone boundaries. Near a Brillouin-zone boundary, degenerate plane waves $e^{ikx}$ and $e^{i(k-G)x}$ are coupled by the Fourier component $V_G$ of the potential. Diagonalizing the $2\times 2$ degenerate-perturbation matrix splits the energy by $2|V_G|$ :

6

\quad E_\pm = \frac{\hbar^2}{2m}\left(k^2 + (k-G)^2\right)/2 \pm |V_G|

This split is the band gap: $E_g \approx 2|V_G|$ in the nearly-free-electron limit.

Step 3 — Filling determines material class. Each band holds $2N$ electrons ( $N$ unit cells, factor 2 for spin). With an even number of valence electrons per cell, bands fill exactly: the valence band is full, the conduction band empty. The Fermi level sits in the gap.

Step 4 — Thermal excitation. Electrons in the Boltzmann tail of $f(E)$ are promoted across $E_g$ . Integrating $g(E)f(E)$ over conduction states and $g(E)[1-f(E)]$ over valence states (Step 1 of equation [4]) gives $n = N_c e^{-(E_c-E_F)/k_BT}$ and $p = N_v e^{-(E_F-E_v)/k_BT}$ .

Step 5 — Mass-action and the gap. Multiplying:

7

\quad np = N_c N_v\, e^{-E_g/k_BT}

independent of $E_F$ . For intrinsic material $n=p=n_i$ , taking the square root yields equation [4]. The factor of 2 in the exponent ( $E_g/2$ ) reflects that exciting one electron creates two carriers (an electron and a hole). $\blacksquare$

∎

AH

Alan Herries Wilson

1906–1995

In two landmark 1931 papers, The Theory of Electronic Semi-Conductors, Wilson applied the new band theory to explain the distinction between metals, insulators, and semiconductors. He recognized that conductivity arises from electrons thermally excited across a gap and from impurity levels lying within it — the conceptual birth of the band-gap picture of semiconductors. His work built on Bloch's 1928 theory of electrons in periodic lattices and resolved the puzzle of why some solids with filled bands could still conduct weakly.

🧠 Intuition

Think of energy levels as parking floors in a garage. In a metal, the top occupied floor is half-empty — cars (electrons) can shuffle around freely, so current flows. In an insulator, every floor is either completely full or completely empty, with a huge wall ( $E_g \sim 5$ eV) to the next empty floor; no shuffling is possible. A semiconductor is the same as an insulator but the wall is short ( $E_g \sim 1$ eV) — at room temperature a few thermally energetic electrons can hop over the wall, leaving empty spaces ("holes") behind on the lower floor. Now both floors can have movement: electrons on top, holes below. The exponential $e^{-E_g/2k_BT}$ is just the Boltzmann odds of finding an electron with enough energy to make the jump.

Figure

Band structure of a metal, semiconductor, and insulator. The conduction band (top), forbidden gap (middle), and valence band (bottom) are shown; the gold curve traces the Fermi–Dirac occupation.

ExampleIntrinsic carrier concentration of 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, $m_0 = 9.11\times10^{-31}$ kg.

Step 1 — Compute $N_c$ . Using equation [5] with the prefactor $2(m_0 k_B T/2\pi\hbar^2)^{3/2} = 2.51\times10^{19}\,\text{cm}^{-3}$ at 300 K:

N_c = 2.51\times10^{19}\,(1.08)^{3/2} = 2.82\times10^{19}\,\text{cm}^{-3}.

Step 2 — Compute $N_v$ .

N_v = 2.51\times10^{19}\,(0.81)^{3/2} = 1.83\times10^{19}\,\text{cm}^{-3}.

Step 3 — Exponential factor.

\frac{E_g}{2k_BT} = \frac{1.12}{2\times0.02585} = 21.66, \qquad e^{-21.66} = 3.9\times10^{-10}.

Step 4 — Combine (equation [4]).

n_i = \sqrt{(2.82\times10^{19})(1.83\times10^{19})}\times 3.9\times10^{-10}.

\sqrt{N_cN_v} = 2.27\times10^{19}\,\text{cm}^{-3}.

\boxed{n_i \approx 9.0\times10^{9}\,\text{cm}^{-3}}

(The accepted value is $\sim 1.0\times10^{10}\,\text{cm}^{-3}$ ; the small difference comes from temperature-dependence of $E_g$ and rounding.)

ExampleSensitivity to band gap

Given: Compare intrinsic carrier concentrations of germanium ( $E_g = 0.66$ eV) and diamond ( $E_g = 5.47$ eV) at 300 K, taking equal prefactors $\sqrt{N_cN_v} \approx 2\times10^{19}\,\text{cm}^{-3}$ .

Step 1 — Germanium exponent.

\frac{0.66}{2(0.02585)} = 12.77, \quad e^{-12.77} = 2.85\times10^{-6}.

n_i(\text{Ge}) \approx 2\times10^{19}\times2.85\times10^{-6} = 5.7\times10^{13}\,\text{cm}^{-3}.

Step 2 — Diamond exponent.

\frac{5.47}{2(0.02585)} = 105.8, \quad e^{-105.8} \approx 10^{-46}.

n_i(\text{diamond}) \approx 2\times10^{19}\times10^{-46} = 2\times10^{-27}\,\text{cm}^{-3}.

Step 3 — Interpret. Diamond's predicted intrinsic carrier density is below one carrier in the entire observable universe of crystal — it behaves as a perfect insulator.

\boxed{n_i(\text{Ge})/n_i(\text{diamond}) \sim 10^{40}}

A factor of 8 in $E_g$ produces 40 orders of magnitude in conductivity — the exponential law at work.

⚠️ Common mistake

"Semiconductors conduct better than insulators because they have more electrons." They do not have more electrons — a pure semiconductor at $T=0$ has exactly the same filled-band structure as an insulator and conducts nothing. The difference is purely the size of the gap, which sets the exponential probability of thermal excitation. Likewise, "holes" are not real particles but the absence of an electron in the valence band, behaving as positive charge carriers.

Counterexample

A small gap does not guarantee semiconducting behaviour. Consider a hypothetical material with $E_g = 0$ where the conduction-band minimum touches the valence-band maximum — a semimetal (e.g. bismuth, graphene). Here carriers exist at all temperatures and the $e^{-E_g/2k_BT}$ law degenerates; conductivity does not vanish at $T\to 0$ . The clean exponential carrier law strictly requires $E_g \gg k_BT > 0$ . Conversely, an "insulator" like diamond is structurally a wide-gap semiconductor — the metal/semiconductor/insulator boundary is one of degree, not of kind.

connection

The band-gap picture rests on Bloch's Theorem, which guarantees that electrons in a periodic potential occupy extended states labelled by crystal momentum $\mathbf{k}$ . The detailed shape $E(\mathbf{k})$ comes from the Nearly Free Electron Model and Tight Binding Model, two complementary approximations. Carrier statistics use Fermi-Dirac Statistics and the concept of the Fermi Level. The exponential law is a special case of the Boltzmann Distribution applied to a two-level (valence/conduction) system, and doping introduces ideas from Donor and Acceptor States. Optical absorption across the gap connects to Photon Absorption and Emission.

🔬 Where this appears

⏳ History

- 1821 — Thomas Seebeck observes thermoelectric effects in semiconducting compounds (unexplained at the time).

- 1833 — Michael Faraday notes 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 point-contact diode).

- 1928 — Felix Bloch proves Bloch's theorem, establishing electron bands in periodic crystals.

- 1930–1931 — Rudolf Peierls and Léon Brillouin develop the zone scheme and explain gap formation at zone boundaries.

- 1931 — Alan Wilson publishes the band theory of semiconductors, defining intrinsic vs. extrinsic conduction.

- 1939 — Boris Davydov and Walter Schottky develop the theory of the p–n junction barrier.

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

- 1954 — First practical silicon solar cell (Chapin, Fuller, Pearson); the integrated circuit follows in 1958–59 (Kilby, Noyce).

open-problem

Real band gaps are more subtle than the simple two-band picture suggests. Direct vs. indirect gaps: in silicon the conduction-band minimum lies at a different $\mathbf{k}$ than the valence maximum, so optical transitions require a phonon — making silicon a poor light emitter despite decades of effort to build a silicon laser. Temperature dependence: $E_g$ itself shrinks with $T$ (Varshni's empirical law), correcting the naive exponential. Many-body effects: the single-particle band gap measured by photoemission (the "quasiparticle gap") differs from the optical gap by the exciton binding energy, and standard density-functional theory systematically underestimates gaps — the "band-gap problem" still drives research in computational materials science (GW approximation, hybrid functionals). Topological materials and Mott insulators further show that band-filling alone does not determine conductivity once strong correlations or topology enter.

Semiconductors and Band Gaps Introduction

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also