Electronic Structure of Semiconductors

Manual: General · Subject: Semiconductor Science

Build a rigorous band-theory foundation for understanding how semiconductor electronic structure governs device behavior.

Why Band Structure Matters

From atoms to bands

Semiconductor science begins with the transition from discrete atomic orbitals to delocalized Bloch states in a crystal. Periodicity leads to energy bands $E_n(\mathbf{k})$ , band gaps $E_g$ , and effective masses $m^*$ . These quantities determine transport, optical absorption, recombination, and device switching.

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Central idea

Most macroscopic semiconductor properties can be traced back to just a few microscopic descriptors: band gap, effective mass, density of states, and scattering mechanisms.

Crystal Symmetry and Bloch States

Bloch theorem

For a periodic potential $V(\mathbf{r}) = V(\mathbf{r}+\mathbf{R})$ , electronic eigenstates can be written as $\psi_{n\mathbf{k}}(\mathbf{r}) = u_{n\mathbf{k}}(\mathbf{r}) e^{i\mathbf{k}\cdot\mathbf{r}}$ , where $u_{n\mathbf{k}}$ has the lattice periodicity. This representation naturally introduces the Brillouin zone, reciprocal lattice vectors, and band extrema at symmetry points.

Which statement best captures the consequence of lattice periodicity for electron states?

Electron energies become continuous and independent of momentum Electron eigenstates can be chosen as Bloch waves labeled by crystal momentum Only localized atomic orbitals remain physically relevant The density of states becomes zero everywhere except at band edges

State Bloch's theorem in one sentence.

Band Gap Types and Effective Mass

Direct vs indirect band gaps

Property	Direct gap semiconductor	Indirect gap semiconductor
Band-edge momentum	Conduction-band minimum and valence-band maximum at same $\mathbf{k}$	Band edges at different $\mathbf{k}$
Optical transition	Strong radiative recombination	Phonon-assisted recombination dominates
Examples	GaAs, InP	Si, Ge
Device relevance	Lasers, LEDs, photodetectors	CMOS electronics, many sensors

Effective mass approximation

Near a band extremum, the dispersion can often be approximated by $E(\mathbf{k}) \approx E_0 + \frac{\hbar^2 k^2}{2m^*}$ . More generally, $m^*$ is a tensor derived from the curvature of $E_n(\mathbf{k})$ . A small effective mass implies high mobility when scattering is not dominant.

A small electron effective mass generally implies:

Lower carrier mobility in all cases A flatter band dispersion near the minimum Higher acceleration response to an applied electric field A larger band gap by necessity

Why are direct band gaps important for light emission?

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