Semiconductors

At $T = 0$The valence band is completely filled and the conduction band is completely empty. There is a band gap $E_g$.

At finite $T$Electrons are thermally excited across the gap. The intrinsic carrier concentration:

$n_i = p_i = \sqrt{N_c N_v}\, e^{-E_g/(2k_B T)}$

Where $N_c$ and $N_v$ are the effective density of states in the conduction and valence bands:

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

Here $m_e^*$ and $m_h^*$ are the effective masses of electrons and holes.

Fermi level in an intrinsic semiconductor: $E_F = (E_c + E_v)/2 + \frac{3}{4}k_B T\ln(m_h^*/m_e^*)$. For $m_e^* = m_h^*$: $E_F = E_g/2$.

n-type: Doping with donor atoms (e.g., P in Si) that donate electrons to the conduction band. Majority carriers: electrons.

p-type: Doping with acceptor atoms (e.g., B in Si) that accept electrons from the valence band, Creating holes. Majority carriers: holes.

For $n$-type with donor concentration $N_D$ (non-degenerate, $T$ not too high):

$n \approx N_D, \quad p = \frac{n_i^2}{N_D}$

$E_F \approx E_c - k_B T \ln\left(\frac{N_c}{N_D}\right)$

For $p$-type with acceptor concentration $N_A$:

$p \approx N_A, \quad n = \frac{n_i^2}{N_A}$

$E_F \approx E_v + k_B T \ln\left(\frac{N_v}{N_A}\right)$

Mass action law: $np = n_i^2$ holds at thermal equilibrium regardless of doping.

At the interface between p-type and n-type material:

• Depletion region: Mobile carriers diffuse, leaving behind fixed ionised donors (n-side) and acceptors (p-side), creating a built-in electric field.
• Built-in potential: $V_0 = \frac{k_B T}{e}\ln\left(\frac{N_A N_D}{n_i^2}\right)$.
• Depletion width: $W = \sqrt{\frac{2\varepsilon_s V_0}{e}\left(\frac{1}{N_A} + \frac{1}{N_D}\right)}$ where $\varepsilon_s$ is the permittivity of the semiconductor.

Current-voltage characteristic (Shockley equation):

$I = I_0\left(e^{eV/(k_B T)} - 1\right)$

Where $I_0$ is the reverse saturation current. Forward bias ($V \gt 0$) exponentially increases the Current. Reverse bias ($V \lt 0$) gives approximately $I \approx -I_0$.

Derivation of the built-in potential. In equilibrium, the Fermi level is constant. The potential Difference between the n-side (where $E_F$ is near $E_c$) and the p-side (where $E_F$ is near $E_v$) Is:

$eV_0 = E_{c,n} - E_{c,p} = E_g - (E_c - E_F)_n - (E_F - E_v)_p$

Using $n = N_c e^{-(E_c - E_F)/(k_B T)}$ and $p = N_v e^{-(E_F - E_v)/(k_B T)}$ with $np = n_i^2$:

$V_0 = \frac{k_B T}{e}\ln\left(\frac{N_A N_D}{n_i^2}\right)$

$\blacksquare$

Capacitance. The depletion region acts as a parallel-plate capacitor:

$C = \frac{\varepsilon_s A}{W} = A\sqrt{\frac{e\varepsilon_s}{2}\frac{N_A N_D}{N_A + N_D}\frac{1}{V_0 - V}}$

This $C \propto 1/\sqrt{V_0 - V}$ dependence is used experimentally to determine $N_A$ and $N_D$ (C—V profiling).

In equilibrium, the Fermi level is constant across the junction. Under forward bias, the bands on The n-side are raised relative to the p-side, reducing the barrier. Under reverse bias, the barrier Is increased.

The electronic and optical properties of semiconductors can be tailored by forming heterostructures --- junctions between different semiconductor materials.

Band offsets. When two semiconductors with different band gaps are joined, the conduction band Minimum and valence band maximum are offset. The type-I (straddling) alignment has the band gap Of one material contained within the gap of the other (e.g., GaAs/AlGaAs). The type-II (staggered) alignment has the conduction and valence band edges of different materials at different Energies (e.g., InAs/GaSb).

Quantum wells. A thin layer of a narrow-gap semiconductor (e.g., 10 nm of GaAs) sandwiched Between wide-gap barriers (e.g., AlGaAs) confines electrons and holes in one dimension. The Confinement energy for an infinite well of width $L$:

$E_n = \frac{n^2 \pi^2 \hbar^2}{2m^* L^2}$

This quantisation raises the effective band gap, allowing the optical transition energy to be tuned By varying $L$.

Quantum wires and dots. Further confinement in two dimensions (quantum wire) or three dimensions (quantum dot) leads to additional quantisation. Quantum dots have discrete, atom-like energy levels And are often called “artificial atoms.”

Strain engineering. Lattice mismatch between a thin film and its substrate induces strain, Modifying the band structure. Tensile strain reduces the band gap, while compressive Strain can lift degeneracies (e.g., splitting the heavy-hole and light-hole bands).

Absorption. A photon of energy $\hbar\omega$ can be absorbed if $\hbar\omega \geq E_g$Promoting An electron from the valence band to the conduction band.

• Direct band gap (e.g., GaAs, InP): The conduction band minimum and valence band maximum occur at the same $\mathbf{k}$. Photon absorption requires only energy conservation (the photon momentum $\hbar\omega/c \approx 0$ is negligible). The absorption coefficient rises sharply above $E_g$:

  $\alpha(\omega) \propto \sqrt{\hbar\omega - E_g}$

• Indirect band gap (e.g., Si, Ge): The band edges occur at different $\mathbf{k}$. A phonon is required to conserve momentum, making the absorption weaker and temperature-dependent:

  $\alpha(\omega) \propto \frac{(\hbar\omega - E_g - \hbar\Omega)^2}{e^{\hbar\Omega/k_BT} - 1} + \frac{(\hbar\omega - E_g + \hbar\Omega)^2}{1 - e^{-\hbar\Omega/k_BT}}$

where $\hbar\Omega$ is the phonon energy.

Excitons. The electron and hole created by photon absorption are attracted by the Coulomb Interaction, forming a bound state called an exciton with binding energy:

$E_{\mathrm{ex} = \frac{\mu e^4}{2(4\pi\varepsilon_s)^2\hbar^2} = \frac{\mu}{m_e\varepsilon_r^2} \times 13.6\ \mathrm{eV}}$

Where $\mu = m_e^* m_h^*/(m_e^* + m_h^*)$ is the reduced mass and $\varepsilon_r$ is the relative Permittivity. Excitons produce sharp absorption lines slightly below $E_g$.

Photoluminescence. When electron—hole pairs recombine radiatively, photons are emitted at Energies near $E_g$. Direct-gap materials are efficient light emitters (used in LEDs and laser Diodes). Indirect-gap materials like Si have very low radiative efficiency.

Light-emitting diodes (LEDs). Under forward bias, electrons and holes are injected into the Depletion region where they recombine radiatively. The emission wavelength is determined by the Band gap: $\lambda = hc/E_g$. GaAs ($E_g = 1.42$ eV) emits in the infrared; GaN ($E_g = 3.4$ eV) Emits in the ultraviolet; InGaN alloys span the visible spectrum.

Solar cells. A p-n junction under illumination generates electron—hole pairs. The built-in Field separates them, producing a photocurrent. The open-circuit voltage satisfies $V_{\mathrm{OC} \lt E_g/e}$ ( $V_{\mathrm{OC} \approx 0.7\,E_g/e}$). The power conversion Efficiency is limited by the Shockley—Queisser limit ($\sim 33\%$ for a single junction) Due to spectral mismatch, thermalisation, and radiative recombination losses.

Field-effect transistor (FET). A voltage applied to a gate electrode modulates the conductivity Of a semiconductor channel. In a MOSFET (metal—oxide—semiconductor FET), the gate voltage creates An inversion layer at the oxide—semiconductor interface, forming a conductive channel. The Threshold voltage $V_T$ depends on the oxide thickness, doping, and work function difference.

HEMTs and HBTs. High-electron-mobility transistors (HEMTs) use heterojunctions (e.g., AlGaAs/GaAs) to create a two-dimensional electron gas (2DEG) with very high mobility. Heterojunction Bipolar transistors (HBTs) use a wide-gap emitter to improve injection efficiency.