Sources and References

Primary textbook — Donald A. Neamen, Semiconductor Physics and Devices, 4th ed., McGraw-Hill, 2012. Online resources — MIT OCW 6.012 Microelectronic Devices and Circuits; Pierret Semiconductor Device Fundamentals; Streetman & Banerjee Solid State Electronic Devices; Sze & Ng Physics of Semiconductor Devices, 3rd ed.

---

Chapter 1: Crystal Structure and the Quantum Theory of Solids

1.1 Crystal Lattices and Unit Cells

A crystalline solid is one in which atoms are arranged in a periodic, repeating pattern extending throughout three dimensions. This long-range order distinguishes crystalline materials — including all commercially important semiconductors — from amorphous solids and liquids.

The mathematical scaffolding for describing this periodicity is the Bravais lattice, an infinite array of discrete points generated by three primitive translation vectors \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\). Any lattice point can be reached from any other by a vector of the form

\[ \mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3, \quad n_i \in \mathbb{Z}. \]

The unit cell is the smallest repeating volume that, when tiled by Bravais lattice translations, fills all of space. The primitive unit cell contains exactly one lattice point; larger (conventional) unit cells may contain two or more but are often chosen for geometrical convenience.

1.1.1 Cubic Lattices

Three cubic Bravais lattices are important in semiconductor physics.

The simple cubic (SC) lattice places atoms only at the corners of a cube of side \(a\). Counting shared corners, there is exactly one atom per unit cell. SC is rare in nature because the packing is inefficient; polonium is the only elemental example.

The body-centred cubic (BCC) lattice adds one atom at the centre of the cube, giving two atoms per conventional unit cell. Many metals (W, Mo, Fe at room temperature) adopt BCC structure.

The face-centred cubic (FCC) lattice places atoms at all six face centres in addition to the corners, yielding four atoms per conventional unit cell. FCC is the densest packing of equal hard spheres (along with hexagonal close-packed) and is adopted by Al, Cu, Ag, Au, and — crucially — forms the underlying Bravais lattice for both the diamond cubic and zinc-blende structures of semiconductor crystals.

1.1.2 The Diamond Cubic Structure

Silicon, germanium, and carbon (diamond) crystallise in the diamond cubic structure. It can be constructed by placing two interpenetrating FCC lattices, the second displaced from the first by \((\tfrac{a}{4}, \tfrac{a}{4}, \tfrac{a}{4})\). The conventional unit cell contains eight atoms. Each atom is tetrahedrally coordinated to four nearest neighbours, reflecting the \(sp^3\) hybrid covalent bonding of Group IV elements. The nearest-neighbour distance is \(d = a\sqrt{3}/4\); for silicon, \(a = 5.43\) Å and \(d \approx 2.35\) Å.

The diamond structure has a relatively low packing fraction of 0.34, lower than FCC (0.74) or BCC (0.68). Nonetheless, the directional covalent bonds make silicon mechanically hard and give it exceptional thermal stability.

1.1.3 The Zinc-Blende Structure

Gallium arsenide (GaAs), indium phosphide (InP), and most III-V compound semiconductors crystallise in the zinc-blende (sphalerite) structure, which is identical in geometry to diamond cubic but with the two interpenetrating FCC sub-lattices occupied by different atomic species (e.g., Ga on one, As on the other). Whereas diamond has a centre of inversion symmetry, zinc-blende does not — a distinction with important consequences for optical selection rules and the piezoelectric effect.

1.1.4 Miller Indices

Miller indices provide a compact notation for crystal planes and directions.

To find the Miller index \((hkl)\) of a plane:

1. Identify where the plane intersects the three crystallographic axes, expressed in units of the lattice parameters: intercepts at \(p\mathbf{a}_1\), \(q\mathbf{a}_2\), \(r\mathbf{a}_3\).
2. Take the reciprocals: \(1/p\), \(1/q\), \(1/r\).
3. Scale to the smallest integers with the same ratio.
4. Negative intercepts are denoted by an overbar: \(\bar{1}\) rather than \(-1\).

The direction perpendicular to the plane \((hkl)\) in a cubic lattice has direction indices \([hkl]\).

Key planes in the cubic system: \((100)\) (cleavage plane in Si); \((110)\) (preferred cleavage in GaAs); \((111)\) (the closest-packed plane in FCC). The family of equivalent planes is denoted by curly braces: \(\{100\}\) includes all six face planes of the cube.

1.2 Quantum Mechanics Review

The behaviour of electrons in a crystal cannot be understood without quantum mechanics. Two fundamental departures from classical physics are central.

Wave-particle duality: A particle of momentum \(\mathbf{p}\) has an associated de Broglie wavelength \(\lambda = h/|\mathbf{p}|\), where \(h = 6.626 \times 10^{-34}\) J·s is Planck’s constant. For an electron accelerated through 1 V, \(\lambda \approx 1.23\) nm — comparable to atomic bond lengths, so interference and diffraction effects are significant.

The Schrödinger equation: The time-independent Schrödinger equation governing the spatial part \(\psi(\mathbf{r})\) of an electron’s wave function is

\[ -\frac{\hbar^2}{2m}\nabla^2 \psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r}) = E\psi(\mathbf{r}), \]

where \(\hbar = h/(2\pi)\), \(m\) is the electron mass, \(V(\mathbf{r})\) is the potential energy, and \(E\) is the total energy. The physical interpretation is that \(|\psi(\mathbf{r})|^2\) gives the probability density of finding the electron at position \(\mathbf{r}\).

Energy quantisation: Solving the Schrödinger equation with appropriate boundary conditions yields discrete (quantised) allowed energies. The classic example is the infinite square well of width \(L\):

\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}, \quad n = 1, 2, 3, \ldots \]

This result illustrates that confinement forces energy quantisation — a principle that underlies quantum wells, quantum wires, and quantum dots used in modern semiconductor devices.

1.3 Energy Bands in Crystals

1.3.1 Bloch’s Theorem

An electron moving in the perfectly periodic potential \(V(\mathbf{r}) = V(\mathbf{r} + \mathbf{R})\) of a crystal lattice has wave functions of the form

\[ \psi_{n\mathbf{k}}(\mathbf{r}) = u_{n\mathbf{k}}(\mathbf{r})\, e^{i\mathbf{k}\cdot\mathbf{r}}, \]

where \(u_{n\mathbf{k}}(\mathbf{r})\) has the same periodicity as the lattice and \(\mathbf{k}\) is the crystal momentum wave vector. This is Bloch’s theorem. The quantum number \(n\) is the band index, and the set of energies \(E_n(\mathbf{k})\) as a function of \(\mathbf{k}\) forms the \(n\)-th energy band.

1.3.2 Brillouin Zones

The first Brillouin zone (BZ) is the Wigner-Seitz primitive cell of the reciprocal lattice — the set of wave vectors \(\mathbf{k}\) closer to the reciprocal lattice origin than to any other reciprocal lattice point. Because \(E_n(\mathbf{k})\) is periodic in the reciprocal lattice, it suffices to know the band structure within the first BZ.

For the FCC real-space lattice of silicon or GaAs, the reciprocal lattice is BCC and the first BZ is a truncated octahedron. Key high-symmetry points within this BZ carry conventional labels: \(\Gamma\) is the zone centre (\(\mathbf{k} = 0\)); \(X\) and \(L\) are zone-boundary points along \(\langle 100\rangle\) and \(\langle 111\rangle\), respectively.

1.3.3 E-k Diagrams and Effective Mass

The E-k diagram plots electron energy against wave vector along high-symmetry directions. Near a band extremum the dispersion is approximately parabolic:

\[ E(\mathbf{k}) \approx E_c + \frac{\hbar^2 k^2}{2 m_n^*}, \]

where \(E_c\) is the conduction band edge and \(m_n^*\) is the electron effective mass, defined by

\[ \frac{1}{m^*} = \frac{1}{\hbar^2}\frac{d^2 E}{dk^2}. \]

The effective mass absorbs the effect of the periodic lattice potential; an electron in a band behaves like a free particle of mass \(m^*\) responding to external forces. For silicon, the conduction band minima lie along the six equivalent \(\langle 100\rangle\) directions (\(\Delta\) valleys) and the longitudinal/transverse effective masses are \(m_l^* = 0.98\,m_0\) and \(m_t^* = 0.19\,m_0\).

Holes in the valence band are treated as positively charged quasiparticles with effective mass \(m_p^*\). Silicon has both heavy-hole and light-hole bands degenerate at \(\Gamma\).

1.3.4 Direct and Indirect Bandgap

In a direct-gap semiconductor (GaAs, InP, GaN), the conduction band minimum and valence band maximum occur at the same \(\mathbf{k}\) point (both at \(\Gamma\)). An electron-hole recombination event can therefore conserve both energy and momentum by emitting a photon — momentum conservation is easily satisfied because photon momenta \(\hbar\omega/c\) are negligible on the scale of the BZ.

In an indirect-gap semiconductor (Si, Ge), the conduction band minimum and valence band maximum occur at different \(\mathbf{k}\) values. Recombination requires simultaneous participation of a phonon to supply the momentum difference, making radiative recombination a second-order process with far lower probability.

1.3.5 Metals, Insulators, and Semiconductors

The distinction between metals, insulators, and semiconductors follows from band theory and electron filling.

At \(T = 0\) K, electrons fill the lowest-energy states up to the Fermi level. If the Fermi level lies within a band (partially filled band), the material is a metal — electrons can accelerate into nearby empty states. If the Fermi level lies within a bandgap, the material is an insulator (large gap, \(E_g > \sim 4\) eV) or a semiconductor (moderate gap, \(E_g \lesssim 3\) eV). The distinction is one of degree: at room temperature, thermal excitation across the gap produces a measurable carrier concentration in semiconductors but a negligibly small one in insulators.

---

Chapter 2: Semiconductors in Equilibrium — Carrier Statistics and Doping

2.1 Fermi-Dirac Statistics

Electrons are fermions: no two may occupy the same quantum state (Pauli exclusion principle). The probability that a state at energy \(E\) is occupied in thermal equilibrium is given by the Fermi-Dirac distribution function:

\[ f(E) = \frac{1}{1 + \exp\!\left[\dfrac{E - E_F}{kT}\right]}, \]

where \(k = 8.617 \times 10^{-5}\) eV/K is Boltzmann’s constant, \(T\) is absolute temperature, and \(E_F\) is the Fermi energy.

At \(E = E_F\), \(f(E_F) = 1/2\) regardless of temperature. For \(E - E_F \gg kT\) (several \(kT\) above the Fermi level), the exponential dominates and the Fermi-Dirac distribution reduces to the Boltzmann approximation:

\[ f(E) \approx \exp\!\left[-\frac{E - E_F}{kT}\right], \quad E - E_F \gg kT. \]

This approximation is valid for non-degenerate semiconductors — those in which \(E_F\) lies at least \(3kT\) below \(E_c\) (for electrons) or above \(E_v\) (for holes).

2.2 Density of States

The density of states (DOS) \(g(E)\) counts the number of available quantum states per unit energy per unit volume. Near a band edge with parabolic dispersion, the three-dimensional DOS is:

\[ g_c(E) = \frac{4\pi(2m_n^*)^{3/2}}{h^3}\sqrt{E - E_c}, \quad E \geq E_c, \]\[ g_v(E) = \frac{4\pi(2m_p^*)^{3/2}}{h^3}\sqrt{E_v - E}, \quad E \leq E_v. \]

The quantities \(m_n^*\) and \(m_p^*\) appearing here are the density-of-states effective masses, which account for multiple equivalent valleys (e.g., the six \(\Delta\)-valleys of silicon) and are distinct from the conductivity effective mass relevant to transport.

2.3 Intrinsic Carrier Concentration

2.4 Extrinsic Semiconductors: Doping

The practical power of semiconductors lies in the ability to control carrier concentrations over many orders of magnitude through doping — the deliberate introduction of impurity atoms.

Donors (n-type doping): Group V atoms (P, As, Sb in Si) substitute on silicon lattice sites, contributing four bonds identical to Si and leaving one extra electron weakly bound to the positively charged impurity core. The binding energy is small (\(\sim 45\) meV in Si), so donors are fully ionised at room temperature. The ionised donor contributes one electron to the conduction band and leaves behind a fixed positive charge \(N_d^+\).

Acceptors (p-type doping): Group III atoms (B, Al, Ga in Si) introduce a missing electron (a hole) weakly bound to the negatively ionised acceptor. At room temperature, acceptors capture electrons from the valence band (equivalently, donate holes to the valence band), contributing one hole each. The ionised acceptor carries fixed negative charge \(N_a^-\).

2.4.1 Charge Neutrality and Carrier Concentrations

In equilibrium, the semiconductor is electrically neutral:

\[ p_0 + N_d^+ = n_0 + N_a^-. \]

Assuming complete ionisation (\(N_d^+ = N_d\), \(N_a^- = N_a\)) and using the mass-action law \(n_0 p_0 = n_i^2\):

For a net n-type sample (\(N_d \gg N_a\)):

\[ n_0 = \frac{(N_d - N_a)}{2} + \sqrt{\left(\frac{N_d - N_a}{2}\right)^2 + n_i^2}. \]

When \(N_d - N_a \gg n_i\) (typical at room temperature):

\[ n_0 \approx N_d - N_a, \quad p_0 = \frac{n_i^2}{N_d - N_a}. \]

2.4.2 Fermi Level Position with Doping

From \(n_0 = N_c e^{-(E_c - E_F)/kT}\):

\[ E_F = E_c - kT \ln\!\left(\frac{N_c}{n_0}\right). \]

Doping moves the Fermi level toward the conduction band (n-type) or valence band (p-type). In heavily doped (degenerate) semiconductors, \(E_F\) may enter the band itself.

---

Chapter 3: Carrier Transport — Drift, Diffusion, and the Hall Effect

3.1 Drift and Mobility

When an electric field \(\mathcal{E}\) is applied to a semiconductor, free carriers are accelerated. Frequent scattering events (from lattice vibrations and ionised impurities) limit the acceleration to a steady-state drift velocity:

\[ v_{dn} = -\mu_n \mathcal{E} \quad (\text{electrons}), \qquad v_{dp} = \mu_p \mathcal{E} \quad (\text{holes}), \]

where the negative sign for electrons reflects their charge.

The drift current densities are:

\[ J_n^{(\text{drift})} = qn\mu_n \mathcal{E}, \qquad J_p^{(\text{drift})} = qp\mu_p \mathcal{E}. \]

Both carrier types contribute to conventional current in the same direction (electrons move opposite to field but carry negative charge). The total drift current is:

\[ J_{\text{drift}} = q(n\mu_n + p\mu_p)\mathcal{E} = \sigma \mathcal{E}, \]

where \(\sigma = q(n\mu_n + p\mu_p)\) is the conductivity and \(\rho = 1/\sigma\) is the resistivity.

3.1.1 Temperature Dependence of Mobility

Two principal scattering mechanisms compete:

• Lattice (phonon) scattering: At higher temperature, more phonons are present and the mean free path decreases. \(\mu_L \propto T^{-3/2}\). This mechanism dominates at elevated temperatures and in lightly doped material.
• Ionised impurity scattering: Coulomb deflection from ionised donors/acceptors. Faster electrons are deflected less, giving \(\mu_I \propto T^{3/2}/N_{\text{imp}}\). This dominates at low temperature and high doping.

The total mobility follows Matthiessen’s rule: \(1/\mu = 1/\mu_L + 1/\mu_I\).

3.2 Diffusion Current and the Einstein Relation

Even without an electric field, a non-uniform carrier distribution drives a diffusion current proportional to the concentration gradient:

\[ J_n^{(\text{diff})} = qD_n \frac{dn}{dx}, \qquad J_p^{(\text{diff})} = -qD_p \frac{dp}{dx}, \]

where \(D_n\) and \(D_p\) are the electron and hole diffusion coefficients.

For silicon at 300 K: \(D_n = \mu_n V_T = 1350 \times 0.02585 \approx 34.9\) cm\(^2\)/s and \(D_p \approx 12.4\) cm\(^2\)/s.

3.3 The Hall Effect

The Hall effect is the primary experimental technique for determining both the sign and concentration of majority carriers in a semiconductor.

When a current \(I_x\) flows in the \(x\)-direction through a rectangular semiconductor bar and a magnetic field \(B_z\) is applied in the \(z\)-direction, the Lorentz force \(\mathbf{F} = q\mathbf{v} \times \mathbf{B}\) deflects carriers in the \(y\)-direction. Carriers accumulate at one face until the resulting transverse electric field (the Hall field \(\mathcal{E}_H\)) balances the Lorentz force. In steady state:

\[ q\mathcal{E}_H = qv_d B_z \implies \mathcal{E}_H = \frac{J_x B_z}{qn} = R_H J_x B_z, \]

where the Hall coefficient is \(R_H = -1/(qn)\) for n-type (negative, because electrons accumulate on the positive-\(y\) face) and \(R_H = +1/(qp)\) for p-type material. Measurement of the sign of \(R_H\) directly identifies the majority carrier type, and its magnitude gives the carrier concentration.

---

Chapter 4: Non-equilibrium Excess Carriers

4.1 Generation and Recombination Mechanisms

Optical illumination, forward biasing, or any other perturbation drives a semiconductor away from equilibrium, creating excess carriers \(\delta n = n - n_0\) and \(\delta p = p - p_0\). The system returns to equilibrium through recombination processes.

Direct (band-to-band) recombination: An electron in the conduction band falls directly into a valence band hole, releasing energy as a photon (radiative) or multiple phonons (non-radiative). The net recombination rate is:

\[ U_{\text{direct}} = \beta(np - n_i^2), \]

where \(\beta\) is the recombination coefficient. In direct-gap GaAs, \(\beta \approx 10^{-10}\) cm\(^3\)/s, making radiative recombination fast and efficient.

Shockley-Read-Hall (SRH) recombination: Indirect-gap semiconductors recombine predominantly through trap states (deep levels) at energy \(E_t\) within the bandgap, introduced by crystal defects or certain impurities (e.g., gold in Si). The net SRH recombination rate is:

Auger recombination: An electron-hole pair recombines and transfers its energy not to a photon but to a third carrier, which then thermalises. The rate scales as \(n^2 p\) or \(np^2\), making Auger recombination important only at very high carrier densities (laser operation, concentrator solar cells).

4.2 Minority Carrier Lifetime

4.3 The Continuity Equation and Minority Carrier Diffusion

The time evolution of excess minority carriers is governed by the continuity equation, which accounts for generation, recombination, and spatial transport:

\[ \frac{\partial(\delta p)}{\partial t} = D_p \frac{\partial^2(\delta p)}{\partial x^2} - \mu_p \mathcal{E}\frac{\partial(\delta p)}{\partial x} - \frac{\delta p}{\tau_p} + G_L, \]

where \(G_L\) is an optical generation rate. For a neutral region with no electric field (the quasi-neutral approximation) and no external generation, the steady-state equation reduces to the minority carrier diffusion equation:

\[ D_p \frac{d^2(\delta p)}{dx^2} - \frac{\delta p}{\tau_p} = 0. \]

The general solution is \(\delta p(x) = A e^{x/L_p} + B e^{-x/L_p}\), where

\[ L_p = \sqrt{D_p \tau_p} \]

is the diffusion length — the average distance a minority carrier travels before recombining.

The Haynes-Shockley experiment is a classic measurement that simultaneously determines \(\mu\), \(D\), and \(\tau\) for minority carriers in a single transit-time experiment: minority carriers are injected at one point, drift under an applied field, and are detected at a downstream probe. The peak amplitude decays as \(e^{-t/\tau}\), the peak position gives \(v_d = \mu\mathcal{E}\), and the spreading of the pulse gives \(D\).

---

Chapter 5: The PN Junction

5.1 Formation and the Built-In Potential

When p-type and n-type semiconductor regions are brought into contact, majority carriers diffuse across the junction: electrons from n to p, holes from p to n. This diffusion exposes fixed ionised impurities — positive donors on the n-side, negative acceptors on the p-side — creating a space charge region (SCR) or depletion region depleted of mobile carriers. The resulting electric field opposes further diffusion until equilibrium is established.

5.2 The Depletion Approximation

The depletion approximation assumes that within the SCR, the semiconductor is completely depleted of mobile carriers (\(n \approx p \approx 0\)), while outside the SCR the material is charge-neutral. This creates a rectangular (box) charge distribution:

\[ \rho(x) = \begin{cases} -qN_a, & -x_p < x < 0 \\ +qN_d, & 0 < x < x_n \end{cases} \]

where \(x_p\) and \(x_n\) are the depletion widths on the p- and n-sides respectively.

Charge neutrality requires \(N_a x_p = N_d x_n\) — the total exposed charge on each side must balance.

Electric field: Integrating Poisson’s equation \(d\mathcal{E}/dx = \rho/\varepsilon_s\) (where \(\varepsilon_s = \varepsilon_r \varepsilon_0\) is the semiconductor permittivity; \(\varepsilon_r = 11.7\) for Si):

\[ \mathcal{E}(x) = \begin{cases} -\dfrac{qN_a}{\varepsilon_s}(x + x_p), & -x_p < x < 0 \\[6pt] -\dfrac{qN_d}{\varepsilon_s}(x_n - x), & 0 < x < x_n \end{cases} \]

The peak field occurs at the metallurgical junction (\(x = 0\)):

\[ \mathcal{E}_{\max} = -\frac{qN_d x_n}{\varepsilon_s} = -\frac{qN_a x_p}{\varepsilon_s}. \]

Potential: Integrating the electric field, subject to the boundary condition that the total potential drop equals \(V_{bi}\):

\[ V_{bi} = \frac{1}{2}|\mathcal{E}_{\max}|(x_n + x_p). \]

Total depletion width:

\[ W = x_n + x_p = \sqrt{\frac{2\varepsilon_s}{q}\left(\frac{N_a + N_d}{N_a N_d}\right)V_{bi}}. \]

5.3 Bias Dependence

Applying a forward bias \(V_F > 0\) (p-side positive) reduces the potential barrier to \(V_{bi} - V_F\), narrowing the depletion region:

\[ W(V) = \sqrt{\frac{2\varepsilon_s(V_{bi} - V)}{q}\cdot\frac{N_a + N_d}{N_a N_d}}, \]

where \(V\) is the applied voltage (positive for forward bias, negative for reverse bias). Reverse bias increases \(W\) and the barrier height.

5.4 Junction Capacitance

The depletion region, with positive charge on the n-side and negative charge on the p-side, acts as a parallel-plate capacitor. The depletion (junction) capacitance per unit area is:

\[ C_j = \frac{\varepsilon_s}{W} = \sqrt{\frac{q\varepsilon_s}{2(V_{bi} - V)}\cdot\frac{N_a N_d}{N_a + N_d}}. \]

A Mott-Schottky plot of \(1/C_j^2\) vs. \(V\) gives a straight line whose slope yields \(N_d\) (or \(N_a\)) and whose \(x\)-intercept gives \(V_{bi}\) — a standard technique for characterising junctions.

Under forward bias, injected minority carriers stored in the quasi-neutral regions give rise to a second capacitance: the diffusion capacitance \(C_d \propto e^{qV/kT}\). This term dominates at forward bias and limits the switching speed of diodes.

---

Chapter 6: PN Junction Diode — I-V Characteristics and Breakdown

6.1 Minority Carrier Injection

Under forward bias, the potential barrier is reduced by \(V\), raising minority carrier concentrations at the edges of the depletion region according to the law of the junction:

\[ p_n(x_n) = p_{n0}\, e^{qV/kT}, \qquad n_p(-x_p) = n_{p0}\, e^{qV/kT}, \]

where \(p_{n0} = n_i^2/N_d\) and \(n_{p0} = n_i^2/N_a\) are the equilibrium minority carrier concentrations. The exponential factor \(e^{qV/kT}\) arises because the junction injects carriers in proportion to the Boltzmann factor of the reduced barrier.

6.2 The Ideal Diode Equation

6.3 Non-ideal Effects

Generation-recombination current: Within the depletion region, SRH processes generate electron-hole pairs under reverse bias (generation current) or enhance recombination under forward bias. The forward-bias recombination current density is approximately:

\[ J_{\text{rec}} \approx \frac{qn_i W}{2\tau_0}\, e^{qV/2kT}, \]

which has an ideality factor \(n = 2\) rather than the ideal value of 1. At low forward bias, this term dominates because \(I_s \propto n_i^2\) while \(J_{\text{rec}} \propto n_i\). In silicon at room temperature (\(n_i\) is small), the \(n=2\) component is observable below \(\sim 0.4\) V; above \(\sim 0.5\) V the ideal \(n=1\) term dominates.

Series resistance: At high forward currents, ohmic drops across the quasi-neutral regions and contacts add a series resistance \(R_s\), causing the I-V to depart from exponential at high currents.

High-level injection: When injected carrier concentrations approach the majority carrier concentration, the simple ideal diode analysis breaks down and current grows more slowly with bias.

6.4 Reverse Breakdown

Two distinct mechanisms can cause the junction current to increase rapidly under large reverse bias:

Zener breakdown: In heavily doped, narrow junctions, the large electric field (\(\sim 10^6\) V/cm) enables quantum mechanical band-to-band tunnelling. Electrons tunnel directly from the valence band on the p-side to the conduction band on the n-side. This mechanism dominates for \(V_{\text{break}} < \sim 5\) V and has a negative temperature coefficient (breakdown voltage decreases with temperature, because the narrower bandgap at higher temperature increases tunnelling).

Avalanche breakdown: In more lightly doped junctions with larger depletion widths, carriers gain sufficient kinetic energy between collisions to ionise lattice atoms — impact ionisation — creating new electron-hole pairs, which then trigger further ionisation in a multiplicative cascade. The multiplication factor diverges at the breakdown voltage \(V_{br}\). Avalanche breakdown has a positive temperature coefficient (higher temperature reduces mobility and mean free path, requiring a larger field to trigger ionisation).

---

Chapter 7: The MOS Capacitor and MOSFET

7.1 The MOS Capacitor

The metal-oxide-semiconductor (MOS) capacitor is the fundamental building block of CMOS technology. It consists of a metallic gate electrode separated from the semiconductor body by a thin insulating oxide (SiO\(_2\) in classical CMOS, high-\(\kappa\) dielectrics in modern nodes). In a p-type substrate, the three operating regimes are determined by the gate voltage \(V_G\).

Accumulation (\(V_G < V_{FB}\)): A negative gate voltage (for p-type body) attracts majority holes to the oxide-semiconductor interface. The hole concentration increases above its equilibrium value. No inversion layer forms.

Depletion (\(V_{FB} < V_G < V_T\)): A positive gate voltage repels holes from the surface. A depletion region of width \(x_d\) forms, exposing negatively charged ionised acceptors.

Inversion (\(V_G > V_T\)): Further increase of \(V_G\) bends the bands sufficiently to attract minority carriers (electrons in p-type) to the surface. When the surface electron concentration equals the bulk hole concentration (\(\phi_s = 2\phi_F\), strong inversion), an inversion layer (the MOSFET channel) forms.

The flat-band voltage \(V_{FB}\) is the gate voltage required to eliminate all band bending:

\[ V_{FB} = \phi_{ms} - \frac{Q_{ox}}{C_{ox}}, \]

where \(\phi_{ms} = \phi_m - \phi_s\) is the metal-semiconductor work function difference and \(Q_{ox}\) is the areal density of fixed oxide charges (predominantly at the Si-SiO\(_2\) interface).

7.2 Threshold Voltage

7.3 MOSFET I-V Characteristics

The MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) controls the current between drain (D) and source (S) terminals by modulating the charge in an inversion channel via the gate (G) voltage. For an NMOS transistor with gate length \(L\) and width \(W\):

7.3.1 Subthreshold Operation

Below threshold (\(V_{GS} < V_T\)), the channel is in depletion but a weak inversion layer exists. The subthreshold drain current is exponential in gate voltage:

\[ I_D \propto \exp\!\left(\frac{qV_{GS}}{nkT}\right), \]

where \(n \geq 1\) is the subthreshold ideality factor (body factor). The subthreshold slope \(S\) is defined as the gate voltage required to change \(I_D\) by one decade:

\[ S = \frac{dV_{GS}}{d(\log_{10} I_D)} = n \cdot \frac{kT}{q}\ln 10 \approx n \times 60\ \text{mV/decade at 300 K}. \]

The theoretical minimum at room temperature is 60 mV/decade (for \(n = 1\)), achieved only when the gate capacitance dominates completely over parasitic capacitances. This fundamental limit motivates the use of high-\(\kappa\) dielectrics, thin oxides, and novel device geometries (FinFETs, GAA nanowires) to minimise \(n\).

---

Chapter 8: Advanced MOSFET Concepts

8.1 Short-Channel Effects

As transistor gate lengths are scaled below \(\sim 100\) nm, several phenomena degrade ideal long-channel behaviour.

Velocity saturation: At high fields (\(\mathcal{E} > \sim 10^4\) V/cm in Si), carrier velocity saturates at \(v_{\text{sat}} \approx 10^7\) cm/s rather than increasing linearly with field. The drain current in short-channel saturation approaches:

\[ I_{D,\text{sat}} \approx W C_{ox}(V_{GS} - V_T) v_{\text{sat}}, \]

which is linear in \(V_{GS} - V_T\) rather than quadratic. This reduces transconductance gain from the ideal value but makes the device faster.

Drain-Induced Barrier Lowering (DIBL): In short-channel devices, the drain depletion region encroaches on the source end of the channel, lowering the source-to-channel potential barrier. This effectively reduces the threshold voltage at high \(V_{DS}\):

\[ \Delta V_T = -\text{DIBL coefficient} \times V_{DS}. \]

DIBL causes \(V_T\) to depend on \(V_{DS}\), increasing off-state leakage current.

Channel length modulation: Even in long-channel devices, the effective channel length decreases slightly with increasing \(V_{DS}\) above saturation, because the pinch-off point moves toward the source. This causes \(I_D\) to increase linearly with \(V_{DS}\) in saturation:

\[ I_D = I_{D,\text{sat}}\left(1 + \frac{V_{DS} - V_{DS,\text{sat}}}{V_A}\right), \]

where \(V_A\) is the Early voltage. The output conductance \(g_o = I_{D,\text{sat}}/V_A\) limits the intrinsic voltage gain to \(A_v = g_m/g_o\).

8.2 The Body Effect

In MOSFETs with a non-zero source-to-body voltage \(V_{SB}\) (e.g., when the source is not connected to the body), the depletion charge increases and \(V_T\) shifts:

\[ V_T(V_{SB}) = V_{T0} + \gamma\!\left(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}\right), \]

where \(V_{T0}\) is the zero-body-bias threshold voltage and

\[ \gamma = \frac{\sqrt{2q\varepsilon_s N_a}}{C_{ox}} \]

is the body-effect coefficient (units: V\(^{1/2}\)). This is the back-gate effect: the body acts as a secondary gate. In circuits, the body effect degrades performance of stacked transistors (source-follower or cascode configurations) because \(V_{SB}\) varies with input signal.

8.3 Small-Signal Model

For small signals superimposed on a DC bias point, the MOSFET is characterised by its small-signal equivalent circuit. The key parameters are:

• \(g_m = \mu_n C_{ox}(W/L)(V_{GS} - V_T)\): transconductance (drain current response to gate-voltage perturbation)
• \(r_o = V_A / I_D\): output resistance (from channel-length modulation)
• \(C_{gs} \approx \frac{2}{3}WLC_{ox}\): gate-source capacitance (in saturation, includes 2/3 of channel charge)
• \(C_{gd} \approx WL_{\text{ov}}C_{ox}\): gate-drain overlap capacitance (parasitic, from gate-drain overlap region)
• \(C_{sb}\), \(C_{db}\): source-body and drain-body junction capacitances

The unity-gain frequency (transition frequency) is:

\[ f_T = \frac{g_m}{2\pi(C_{gs} + C_{gd})} \approx \frac{\mu_n(V_{GS} - V_T)}{2\pi L^2} \cdot \frac{3}{2}, \]

which increases with shorter channels and higher overdrive voltage — the primary motivation for scaling \(L\).

8.4 The CMOS Inverter

The CMOS inverter pairs an NMOS and a PMOS transistor sharing a common gate (input) and drain (output). When the input is low (\(V_{\text{in}} \approx 0\)), the PMOS is on and the NMOS is off; the output is pulled to \(V_{DD}\). When the input is high (\(V_{\text{in}} \approx V_{DD}\)), the NMOS is on and the PMOS is off; the output is pulled to 0.

The switching threshold \(V_M\) (where \(V_{\text{out}} = V_{\text{in}}\) in the DC transfer characteristic) is set by the ratio of NMOS and PMOS drive strengths \(\mu_n C_{ox}(W/L)_n\) and \(\mu_p C_{ox}(W/L)_p\). For \(V_M = V_{DD}/2\), one sets \((W/L)_p / (W/L)_n = \mu_n/\mu_p \approx 2\text{--}3\).

The key advantage of CMOS is near-zero static power dissipation: in both logic states, one of the two transistors is off, so the DC supply-to-ground path is open. Dynamic (switching) power \(P = \alpha C_L V_{DD}^2 f\) (where \(\alpha\) is the activity factor) dominates and scales as \(V_{DD}^2\) — the primary motivation for supply voltage reduction in each process generation.

---

Chapter 9: Optical Devices — Photodetectors, LEDs, and Solar Cells

9.1 Optical Absorption

When light is incident on a semiconductor, photons with energy \(h\nu > E_g\) are absorbed by exciting electrons across the bandgap. The intensity \(I(x)\) decays exponentially with depth:

\[ I(x) = I_0\, e^{-\alpha x}, \]

where \(\alpha\) (cm\(^{-1}\)) is the optical absorption coefficient. For \(h\nu\) just above \(E_g\), \(\alpha\) rises sharply for direct-gap materials (due to the large joint density of states) and more gradually for indirect-gap materials (because phonon participation broadens the onset). The absorption length \(1/\alpha\) sets the minimum device thickness for efficient photon collection; for Si at 600 nm, \(\alpha \approx 3 \times 10^4\) cm\(^{-1}\) giving \(1/\alpha \approx 0.3\ \mu\)m.

The bandgap-wavelength relation connects the minimum photon energy for absorption to wavelength:

\[ \lambda_c(\mu\text{m}) = \frac{1.24}{E_g(\text{eV})}. \]

For Si (\(E_g = 1.12\) eV): \(\lambda_c \approx 1.11\ \mu\)m (near-infrared cutoff). For GaAs (\(E_g = 1.42\) eV): \(\lambda_c \approx 0.873\ \mu\)m.

9.2 Photodetectors

9.2.1 Photoconductors

The simplest photodetector is a bulk semiconductor bar with ohmic contacts. Absorbed photons generate excess electron-hole pairs, increasing conductivity:

\[ \Delta\sigma = q(\mu_n \delta n + \mu_p \delta p) = q(\mu_n + \mu_p)\, G_L \tau, \]

where \(G_L\) is the optical generation rate and \(\tau\) is the carrier lifetime. The photoconductor gain is \(\Gamma = \tau / t_{\text{transit}}\), where \(t_{\text{transit}} = L^2/(\mu_n V)\) is the carrier transit time. Gain can exceed unity (photon-to-carrier conversion ratio \(> 1\)), but this comes at the cost of bandwidth: \(\text{BW} \propto 1/\tau\).

9.2.2 PIN Photodiode

The PIN photodiode consists of a lightly doped (intrinsic, I) region sandwiched between p\(^+\) and n\(^+\) regions, operated under reverse bias. The reverse bias fully depletes the I-region, creating a large, uniform electric field throughout. Photons absorbed in the I-region generate electron-hole pairs that are swept rapidly to the contacts by the field, without the delay associated with minority carrier diffusion.

The quantum efficiency is:

\[ \eta = (1 - R)(1 - e^{-\alpha W}), \]

where \(R\) is the surface reflectivity and \(W\) is the I-region width. The tradeoff: wider \(W\) improves quantum efficiency but increases carrier transit time, limiting bandwidth. The bandwidth-efficiency product is approximately constant for a given material.

9.2.3 Avalanche Photodiode (APD)

APDs operate at high reverse bias close to breakdown, where photogenerated carriers trigger impact ionisation. The resulting internal gain \(M\) amplifies the photocurrent, improving sensitivity. However, the ionisation process is statistical, introducing excess noise characterised by the noise factor \(F = k M + (2 - 1/M)(1-k)\), where \(k = \alpha_p/\alpha_n\) is the ratio of hole to electron ionisation coefficients. Materials with \(k \ll 1\) (GaAs: \(k \approx 0.1\); Si: \(k \approx 0.02\)) are preferred for low-noise APDs.

9.3 Solar Cells

A solar cell is a large-area pn junction designed to convert photon flux into electrical power.

Principle of operation: Sunlight absorbed throughout the p- and n-type regions generates excess electron-hole pairs. Minority carriers that diffuse to the junction depletion region are swept across by the built-in electric field, generating a photocurrent \(I_{ph}\) that flows opposite to the normal diode forward current. The terminal I-V relationship is:

\[ I = I_s(e^{qV/kT} - 1) - I_{ph}. \]

At open circuit (\(I = 0\)), the open-circuit voltage is:

\[ V_{oc} = \frac{kT}{q}\ln\!\left(\frac{I_{ph}}{I_s} + 1\right) \approx \frac{kT}{q}\ln\!\left(\frac{I_{ph}}{I_s}\right). \]

At short circuit (\(V = 0\)), \(I_{sc} = I_{ph}\) — equal to the photogenerated current.

The fill factor FF is defined as:

\[ \text{FF} = \frac{P_{\max}}{I_{sc} V_{oc}} = \frac{I_{mp} V_{mp}}{I_{sc} V_{oc}}, \]

where \(P_{\max}\) is the maximum power point. FF quantifies how “square” the I-V curve is; typical values are 0.75–0.85 for well-fabricated cells.

The power conversion efficiency is:

\[ \eta = \frac{P_{\max}}{P_{\text{in}}} = \frac{\text{FF} \cdot I_{sc} \cdot V_{oc}}{P_{\text{in}}}. \]

9.4 Light-Emitting Diodes

A light-emitting diode (LED) exploits radiative electron-hole recombination under forward bias in a direct-gap semiconductor.

Under forward bias, minority carriers are injected into the quasi-neutral regions. In a direct-gap material, a significant fraction recombines radiatively, emitting photons of energy \(h\nu \approx E_g\). The internal quantum efficiency is:

\[ \eta_{\text{int}} = \frac{R_{\text{rad}}}{R_{\text{rad}} + R_{\text{non-rad}}} = \frac{1/\tau_r}{1/\tau_r + 1/\tau_{nr}} = \frac{\tau}{\tau_r}, \]

where \(\tau_r\) and \(\tau_{nr}\) are radiative and non-radiative lifetimes and \(\tau = (\tau_r^{-1} + \tau_{nr}^{-1})^{-1}\) is the total lifetime.

The external quantum efficiency (EQE) also accounts for photon extraction from the device:

\[ \eta_{\text{ext}} = \eta_{\text{int}} \times \eta_{\text{extraction}}. \]

Photon extraction is limited by total internal reflection: only photons within the escape cone of half-angle \(\theta_c = \sin^{-1}(1/n_r)\) (where \(n_r \approx 3.5\) for GaAs) escape. This limits the extraction efficiency to \(\sim 2\%\) for a planar device, motivating the use of shaped chips, surface texturing, and resonant cavity structures. Modern high-brightness LEDs achieve EQE \(> 80\%\) using advanced packaging and chip shaping.

Heterojunction LEDs: Double-heterostructure (DH) designs confine both carriers and photons to a thin active layer of lower bandgap sandwiched between higher-gap cladding layers. The cladding layers act as mirrors for photons (via the refractive index step) and as barriers for carriers (via the band offsets), dramatically increasing recombination efficiency in the active layer. This concept underpins all modern semiconductor LEDs and lasers.

9.5 Laser Diodes

A laser diode produces stimulated, coherent emission. Three conditions must be satisfied simultaneously.

Population inversion: More electrons must reside in the conduction band (lower quasi-Fermi level \(E_{Fc}\)) than in the valence band near the photon energy of interest. In a semiconductor, this requires heavy injection until the separation of quasi-Fermi levels \(E_{Fc} - E_{Fv} > E_g\).

Optical gain: The stimulated emission rate must exceed the absorption rate. The threshold condition requires that gain \(g\) equals total loss per unit length:

\[ \Gamma g_{\text{th}} = \alpha_i + \frac{1}{2L}\ln\!\left(\frac{1}{R_1 R_2}\right), \]

where \(\Gamma\) is the optical confinement factor, \(\alpha_i\) is the internal loss, and the second term is the mirror loss of the Fabry-Pérot cavity (length \(L\), facet reflectivities \(R_1\), \(R_2\)). For cleaved GaAs facets, \(R \approx 0.32\) (from Fresnel reflection at the semiconductor-air interface).

Optical feedback: The Fabry-Pérot cavity formed by cleaved facets or distributed Bragg reflectors (DBR/DFB structures) provides the resonant condition. Lasing modes satisfy \(2nL = m\lambda\), where \(n\) is the refractive index and \(m\) is an integer.

---

Summary and Interconnections

The nine chapters of these notes trace a logical progression from atomic-scale crystal structure to functional devices:

Crystal structure and quantum mechanics (Chapter 1) establish that electrons in a periodic lattice adopt Bloch wave states organised into energy bands. The distinction between direct- and indirect-gap semiconductors — set by the relative positions of band extrema in k-space — determines which materials can be efficient light emitters.

Carrier statistics and doping (Chapter 2) quantify the equilibrium populations of electrons and holes through Fermi-Dirac statistics and the density of states. The mass-action law \(n_0 p_0 = n_i^2\) and the exponential sensitivity of \(n_i\) to temperature and bandgap are foundational. Doping shifts the Fermi level and controls conductivity over many decades.

Transport (Chapter 3) adds dynamics: drift driven by electric fields and diffusion driven by concentration gradients. The Einstein relation connects these two modes. The Hall effect provides the primary experimental probe of carrier type and concentration.

Excess carriers (Chapter 4) describe the non-equilibrium response to injection or optical excitation. Recombination lifetime and diffusion length set the spatial and temporal scales governing minority carrier behaviour — critical for diode, transistor, and solar cell design.

The pn junction (Chapter 5) emerges from placing n- and p-type regions in contact. The depletion approximation yields the built-in potential, electric field profile, depletion width, and junction capacitance — the key structural parameters.

The pn junction diode (Chapter 6) operates by minority carrier injection under forward bias. The ideal diode equation follows rigorously from solving the minority carrier diffusion equation. Non-ideal effects (generation-recombination current, high-level injection, series resistance) and breakdown mechanisms (Zener, avalanche) complete the picture.

The MOS capacitor and MOSFET (Chapters 7 and 8) introduce the field-effect principle: a gate electrode modulates the surface charge of a semiconductor through the oxide capacitance. Threshold voltage, the long-channel I-V law, and transconductance emerge from the charge-sheet model. Short-channel effects (velocity saturation, DIBL, subthreshold slope degradation), the body effect, and the small-signal model are essential for understanding modern CMOS.

Optical devices (Chapter 9) close the loop between photons and electronic states. Solar cells, photodetectors, LEDs, and laser diodes all depend on the same generation-recombination physics established in Chapter 4, combined with the junction electrostatics of Chapters 5 and 6, applied to optically active semiconductors.

Throughout, the quantitative tools — Fermi-Dirac statistics, the depletion approximation, the minority carrier diffusion equation, and the charge-sheet MOSFET model — are not disconnected techniques but aspects of a single, coherent framework for describing electrons, holes, and photons in semiconductor materials.