B.tech - ECE/VLSI - Physics of Semiconductor Devices

Cheatsheet Content

```html Semiconductor Materials Elemental Semiconductors: Group IV elements (e.g., Silicon (Si), Germanium (Ge)). Compound Semiconductors: Combinations of Group III and Group V elements (e.g., Gallium Arsenide (GaAs), Gallium Phosphide (GaP)). Also Group II-VI. Ternary Compounds: e.g., $Al_xGa_{1-x}As$. Crystal Structures Amorphous: Order only within a few atomic dimensions. Polycrystalline: High degree of order over many atomic dimensions (grains separated by grain boundaries). Single Crystal: High degree of order throughout the entire volume. Lattice: Periodic arrangement of atoms in a crystal. Unit Cell: Smallest volume that can reproduce the entire crystal. Primitive Cell: Smallest unit cell. Basic Crystal Structures Simple Cubic (SC): Atom at each corner. Body-Centered Cubic (BCC): Atom at each corner + one in the center. Face-Centered Cubic (FCC): Atom at each corner + one on each face. Volume Density of Atoms $$ \text{Volume Density} = \frac{\text{# atoms per unit cell}}{\text{volume of unit cell}} $$ Crystal Planes and Miller Indices (hkl) Determine intercepts along axes ($\vec{a}, \vec{b}, \vec{c}$). Take reciprocals of intercepts. Multiply by the lowest common denominator to get integers (hkl). Parallel planes are equivalent (e.g., {100} set of planes). Surface Density of Atoms $$ \text{Surface Density} = \frac{\text{# of atoms per lattice plane}}{\text{area of lattice plane}} $$ Directions in Crystals Expressed as a set of three integers [uvw] representing vector components. In cubic lattices, [hkl] direction is perpendicular to (hkl) plane. Diamond and Zincblende Structures Diamond Structure: For Si, Ge. Each atom has four nearest neighbors in a tetrahedral configuration. Zincblende Structure: For compound semiconductors like GaAs. Similar to diamond but with two different atom types. Atomic Bonding Ionic Bond: Electrostatic attraction between oppositely charged ions (e.g., NaCl). Atoms gain/lose valence electrons. Covalent Bond: Sharing of valence electrons (e.g., H₂ molecule, Si crystal). Achieve closed valence shells. Metallic Bond: Positive metallic ions surrounded by a "sea" of negative electrons (e.g., solid Sodium). Van der Waals Bond: Weakest bond, due to interaction of electric dipoles. Imperfections and Impurities in Solids Lattice Vibrations (Phonons): Thermal motion of atoms disrupting perfect periodicity. Point Defects: Vacancy: Missing atom from lattice site. Interstitial: Atom located between lattice sites. Frenkel Defect: Vacancy-interstitial pair. Line Defects (Dislocations): Entire row of atoms missing. Impurities: Foreign atoms. Substitutional Impurities: Impurity atom at normal lattice site. Interstitial Impurities: Impurity atom between normal sites. Doping: Adding controlled amounts of specific impurity atoms to alter conductivity. Impurity Diffusion: High-temperature process for introducing dopants. Ion Implantation: High-energy beam of dopant ions; lower temperature, precise control. Growth of Semiconductor Materials Czochralski Method: Growing single crystals from a melt (boule). Zone Refining: Purification technique for boules. Epitaxial Growth: Growing a thin, single-crystal layer on a single-crystal substrate. Homoepitaxy: Layer and substrate are the same material (e.g., Si on Si). Heteroepitaxy: Layer and substrate are different materials (e.g., AlGaAs on GaAs). Chemical Vapor-Phase Deposition (CVD): Deposition from chemical vapor. Liquid-Phase Epitaxy: Growth from a liquid compound. Molecular Beam Epitaxy (MBE): Deposition in vacuum; precise doping control. Principles of Quantum Mechanics Energy Quanta: Energy is in discrete packets (photons). $E = h\nu$. Photoelectric Effect: $T = \frac{1}{2}mv^2 = h\nu - \phi = h\nu - h\nu_0$. Wave-Particle Duality: Particles exhibit wave-like properties. Momentum of photon: $p = \frac{h}{\lambda}$. De Broglie Wavelength: $\lambda = \frac{h}{p}$. Uncertainty Principle: Impossible to simultaneously know position and momentum, or energy and time, with absolute accuracy. $\Delta p \Delta x \ge \hbar$. $\Delta E \Delta t \ge \hbar$. $\hbar = h / 2\pi$. Schrodinger's Wave Equation Time-Dependent: $-\frac{\hbar^2}{2m} \frac{\partial^2 \Psi(x,t)}{\partial x^2} + V(x) \Psi(x,t) = j\hbar \frac{\partial \Psi(x,t)}{\partial t}$. Time-Independent (1D): $-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x)$. Wave Function ($\Psi(x,t) = \psi(x)\phi(t)$): $\phi(t) = e^{-j(E/\hbar)t}$. Physical Meaning: $|\Psi(x,t)|^2 dx$ is the probability of finding the particle between $x$ and $x+dx$. Boundary Conditions: $\psi(x)$ must be finite, single-valued, and continuous. $d\psi(x)/dx$ must be finite, single-valued, and continuous (unless $V(x)$ is infinite). Normalization: $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$. Enter content here... Applications of Schrodinger's Wave Equation Electron in Free Space ($V(x)=0$): $\psi(x) = A e^{jkx} + B e^{-jkx}$, where $k = \frac{\sqrt{2mE}}{\hbar}$. Represents a traveling wave. Infinite Potential Well ($V(x)=0$ for $0 \le x \le a$, $\infty$ elsewhere): $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right)$, where $n=1,2,3,...$. Quantized Energy: $E_n = \frac{n^2\pi^2\hbar^2}{2ma^2}$. Represents a standing wave. Step Potential Function ($V(x)=0$ for $x For $E For $E > V_0$: Partial reflection and transmission. Potential Barrier and Tunneling ($V(x)=0$ for $x a$): For $E Transmission Coefficient: $T \approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right) \exp(-2k_2a)$, where $k_2 = \frac{\sqrt{2m(V_0-E)}}{\hbar}$. One-Electron Atom Potential: $V(r) = -\frac{e^2}{4\pi\epsilon_0 r}$. Quantized Energy: $E_n = -\frac{m_0e^4}{(4\pi\epsilon_0)^2 2\hbar^2 n^2}$. Bohr Radius: $a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m_0e^2} \approx 0.529 \, \text{Å}$. Quantum Numbers: Principal (n): $n=1, 2, 3, ...$. Determines energy level. Azimuthal (l): $l=0, 1, ..., n-1$. Determines orbital shape. Magnetic (m): $m=-l, ..., 0, ..., l$. Determines orbital orientation. Spin (s): $s = \pm \frac{1}{2}$. Electron spin. Pauli Exclusion Principle: No two electrons can occupy the same quantum state (same set of n, l, m, s). Allowed and Forbidden Energy Bands Formation: As atoms come together to form a crystal, discrete atomic energy levels split into bands of allowed energies (due to Pauli Exclusion Principle and interatomic interaction). Forbidden Energy Bands (Bandgaps): Regions of energy where no electron states exist. Kronig-Penney Model: 1D periodic potential used to model crystal lattice. Predicts allowed/forbidden energy bands. Conduction Band: Upper allowed energy band, typically empty at $T=0K$. Valence Band: Lower allowed energy band, typically full at $T=0K$. Bandgap Energy ($E_g$): Energy difference between conduction band minimum and valence band maximum. Electrical Conduction in Solids Drift Current: Net flow of charge due to an applied electric field. E-k Diagrams: Energy (E) vs. wavevector (k) diagram. Determines effective mass and band characteristics. Effective Mass ($m^*$): $m^* = \frac{\hbar^2}{d^2E/dk^2}$. Accounts for internal forces in the crystal. Near conduction band minimum: $m^* > 0$. Near valence band maximum: $m^* Hole: Concept of a positive charge carrier in the valence band, representing an empty electron state. Has positive effective mass. Materials Classification based on Band Structure Insulator: Large $E_g$ (typically $> 3.5\text{eV}$). Valence band full, conduction band empty. Very low conductivity. Semiconductor: Moderate $E_g$ (typically $0.1\text{eV}$ to $3.5\text{eV}$). Valence band almost full, conduction band almost empty at $T>0K$. Conductivity can be controlled. Metal: No $E_g$ (partially filled band or overlapping valence/conduction bands). High conductivity. Three-Dimensional Extension Direct Bandgap Semiconductor: Valence band maximum and conduction band minimum occur at the same k-value (e.g., GaAs). Efficient for optical devices. Indirect Bandgap Semiconductor: Valence band maximum and conduction band minimum occur at different k-values (e.g., Si, Ge). Transitions require phonon interaction. Density of States Function $g(E)$ Number of available quantum states per unit volume per unit energy. For free electron model: $g(E) = \frac{4\pi(2m)^{3/2}}{h^3} \sqrt{E}$. Conduction Band: $g_C(E) = \frac{4\pi(2m_n^*)^{3/2}}{h^3} \sqrt{E - E_C}$ for $E \ge E_C$. Valence Band: $g_V(E) = \frac{4\pi(2m_p^*)^{3/2}}{h^3} \sqrt{E_V - E}$ for $E \le E_V$. Statistical Mechanics Fermi-Dirac Probability Function ($f_F(E)$): Probability that a quantum state at energy E is occupied by an electron. $f_F(E) = \frac{1}{1 + \exp\left(\frac{E - E_F}{kT}\right)}$. Fermi Energy ($E_F$): Energy level where $f_F(E_F) = 0.5$. At $T=0K$, all states below $E_F$ are filled, all above are empty. Maxwell-Boltzmann Approximation: $f_F(E) \approx \exp\left(-\frac{E - E_F}{kT}\right)$ when $E - E_F \gg kT$. Charge Carriers in Semiconductors Thermal-Equilibrium Electron Concentration ($n_0$): $n_0 = N_C \exp\left(-\frac{E_C - E_F}{kT}\right)$. Thermal-Equilibrium Hole Concentration ($p_0$): $p_0 = N_V \exp\left(-\frac{E_F - E_V}{kT}\right)$. $N_C = 2\left(\frac{2\pi m_n^* kT}{h^2}\right)^{3/2}$ (Effective density of states in conduction band). $N_V = 2\left(\frac{2\pi m_p^* kT}{h^2}\right)^{3/2}$ (Effective density of states in valence band). Intrinsic Carrier Concentration ($n_i$): In intrinsic (pure) semiconductors, $n_0 = p_0 = n_i$. $n_i^2 = N_C N_V \exp\left(-\frac{E_g}{kT}\right)$. $n_0 p_0 = n_i^2$ (Mass Action Law, valid in thermal equilibrium under Boltzmann approx.). Intrinsic Fermi-Level Position ($E_{Fi}$): $E_{Fi} = \frac{E_C + E_V}{2} + \frac{3}{4}kT \ln\left(\frac{m_p^*}{m_n^*}\right)$. Dopant Atoms and Energy Levels Donor Atoms (Group V): Add electrons to conduction band (e.g., P in Si). Create donor energy level ($E_d$) just below $E_C$. Acceptor Atoms (Group III): Create holes in valence band (e.g., B in Si). Create acceptor energy level ($E_a$) just above $E_V$. Ionization Energy: Energy required to excite a donor electron to $E_C$ or an acceptor electron from $E_V$. n-type Semiconductor: Doped with donors, $n_0 > p_0$. $E_F$ moves closer to $E_C$. p-type Semiconductor: Doped with acceptors, $p_0 > n_0$. $E_F$ moves closer to $E_V$. Extrinsic Semiconductor $n_0 = n_i \exp\left(\frac{E_F - E_{Fi}}{kT}\right)$. $p_0 = n_i \exp\left(-\frac{E_F - E_{Fi}}{kT}\right)$. Fermi-Dirac Integral: Used when Boltzmann approximation is not valid (e.g., degenerate semiconductors). Degenerate Semiconductor: Doping concentration is very high, $E_F$ lies within $E_C$ (n-type) or $E_V$ (p-type). Nondegenerate Semiconductor: $E_F$ lies within the bandgap. Statistics of Donors and Acceptors Probability of Donor State Occupancy: $n_d = \frac{N_d}{1 + \frac{1}{2}\exp\left(\frac{E_d - E_F}{kT}\right)}$. Probability of Acceptor State Occupancy (by electron): $p_a = \frac{N_a}{1 + \frac{1}{g}\exp\left(\frac{E_F - E_a}{kT}\right)}$. Complete Ionization: At room temperature, most dopants donate/accept carriers ($N_d^+ \approx N_d$, $N_a^- \approx N_a$). Freeze-Out: At very low temperatures, dopants are not ionized. Charge Neutrality In thermal equilibrium, the total positive charge equals the total negative charge. $p_0 + N_d^+ = n_0 + N_a^-$. For complete ionization ($N_d^+ \approx N_d$, $N_a^- \approx N_a$): $n_0 = \frac{(N_d - N_a)}{2} + \sqrt{\left(\frac{N_d - N_a}{2}\right)^2 + n_i^2}$. $p_0 = \frac{(N_a - N_d)}{2} + \sqrt{\left(\frac{N_a - N_d}{2}\right)^2 + n_i^2}$. Compensated Semiconductor: Contains both donor and acceptor atoms. Fermi Energy Level Position $E_C - E_F = kT \ln\left(\frac{N_C}{n_0}\right)$. $E_F - E_V = kT \ln\left(\frac{N_V}{p_0}\right)$. $E_F - E_{Fi} = kT \ln\left(\frac{n_0}{n_i}\right)$. $E_{Fi} - E_F = kT \ln\left(\frac{p_0}{n_i}\right)$. Relevance: In thermal equilibrium, $E_F$ is constant throughout a system. Carrier Drift Drift Current Density: $J_{drift} = e(\mu_n n + \mu_p p)E$. Drift Velocity: $v_d = \mu E$. For electrons: $v_n = -\mu_n E$. For holes: $v_p = \mu_p E$. Mobility ($\mu$): $\mu = \frac{e\tau_c}{m^*}$. Describes how well a particle moves in an electric field. Lattice (Phonon) Scattering: $\mu_L \propto T^{-3/2}$. Dominates at low doping, high T. Ionized Impurity Scattering: $\mu_I \propto \frac{T^{3/2}}{N_I}$. Dominates at high doping, low T. Net Mobility: $\frac{1}{\mu} = \frac{1}{\mu_I} + \frac{1}{\mu_L}$. Conductivity ($\sigma$): $\sigma = e(\mu_n n + \mu_p p)$. Resistivity ($\rho$): $\rho = \frac{1}{\sigma}$. Ohm's Law: $V = I R = \rho \frac{L}{A} I$. Velocity Saturation: At high electric fields, drift velocity reaches a constant maximum ($v_{sat} \approx 10^7 \text{ cm/s}$). Carrier Diffusion Flow of charge due to a concentration gradient. Electron Diffusion Current Density: $J_n = e D_n \frac{dn}{dx}$. Hole Diffusion Current Density: $J_p = -e D_p \frac{dp}{dx}$. Diffusion Coefficient (D): $D = \frac{1}{2} v_{th} l$. Total Current Density $J_{total} = J_{n,drift} + J_{p,drift} + J_{n,diff} + J_{p,diff}$. $J_{total} = e\mu_n n E + e\mu_p p E + e D_n \frac{dn}{dx} - e D_p \frac{dp}{dx}$. Graded Impurity Distribution Induced Electric Field: A non-uniform doping profile in thermal equilibrium leads to an internal electric field to prevent net current flow. $E_x = -\frac{kT}{e} \frac{1}{N_d(x)} \frac{dN_d(x)}{dx}$. Einstein Relation: Relates diffusion coefficient and mobility. $\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{e} = V_t$. Hall Effect Measures majority carrier concentration and mobility. Force on moving charge in magnetic field: $F = q v \times B$. Hall Voltage ($V_H$): Induced voltage perpendicular to current and magnetic field. Polarity indicates carrier type. For p-type: $p = \frac{I_x B_z}{e d V_H}$. For n-type: $n = -\frac{I_x B_z}{e d V_H}$. Carrier Generation and Recombination Generation: Process of creating electron-hole pairs (EHPs). Recombination: Process of annihilating EHPs. Excess Carriers ($\Delta n, \Delta p$): Carriers above thermal equilibrium concentrations ($n = n_0 + \Delta n, p = p_0 + \Delta p$). In thermal equilibrium: Generation Rate ($G_0$) = Recombination Rate ($R_0$). In non-equilibrium: $\frac{dn}{dt} = G - R$. Low-Level Injection: $\Delta n \ll n_0$ (majority carrier concentration). Excess Minority Carrier Lifetime ($\tau$): Time constant for excess carriers to decay. $\Delta n(t) = \Delta n(0) e^{-t/\tau}$. Recombination Rate: $R_{\Delta n} = \frac{\Delta n}{\tau}$. Ambipolar Transport Excess electrons and holes drift and diffuse together due to an internal electric field that prevents separation. Ambipolar Transport Equation (1D): $D_a \frac{\partial^2 (\Delta n)}{\partial x^2} + \mu_a E \frac{\partial (\Delta n)}{\partial x} + G_L - \frac{\Delta n}{\tau} = \frac{\partial (\Delta n)}{\partial t}$. Ambipolar Diffusion Coefficient ($D_a$): $D_a = \frac{D_n D_p (n+p)}{D_n n + D_p p}$. Ambipolar Mobility ($\mu_a$): $\mu_a = \frac{\mu_n \mu_p (p-n)}{\mu_n n + \mu_p p}$. Low-Level Injection Simplification: Ambipolar parameters reduce to minority carrier parameters. For p-type: $D_a \approx D_n, \mu_a \approx \mu_n$. For n-type: $D_a \approx D_p, \mu_a \approx -\mu_p$. Dielectric Relaxation Time Constant ($\tau_d$): $\tau_d = \frac{\epsilon}{\sigma}$. Time for a net charge density to neutralize. Very fast (ps). Haynes-Shockley Experiment: Measures minority carrier mobility, lifetime, and diffusion coefficient. Quasi-Fermi Energy Levels In non-equilibrium, $E_F$ is not defined. Instead, separate quasi-Fermi levels are defined for electrons ($E_{Fn}$) and holes ($E_{Fp}$). $n = n_i \exp\left(\frac{E_{Fn} - E_{Fi}}{kT}\right)$. $p = n_i \exp\left(-\frac{E_{Fp} - E_{Fi}}{kT}\right)$. The splitting of $E_{Fn}$ and $E_{Fp}$ indicates deviation from thermal equilibrium. Excess Carrier Lifetime Shockley-Read-Hall (SRH) Theory: Recombination via trap levels ($E_t$) within the bandgap. Recombination Rate: $R = \frac{np - n_i^2}{\tau_{p0}(n+n') + \tau_{n0}(p+p')}$. $n' = N_C \exp\left(-\frac{E_C - E_t}{kT}\right)$, $p' = N_V \exp\left(-\frac{E_t - E_V}{kT}\right)$. Minority Carrier Lifetime: For low injection in extrinsic material, recombination is dominated by minority carrier lifetime (e.g., $\tau_{p0}$ in n-type). Surface Effects Surface States: Disruption of periodicity at surface creates allowed energy states in bandgap, acting as recombination centers. Surface Recombination Velocity (s): Measures recombination rate at the surface. $-D_p \frac{d(\Delta p)}{dx} \Big|_{\text{surf}} = s \Delta p_{\text{surf}}$. pn Junction in Thermal Equilibrium Metallurgical Junction: Interface between p and n regions. Depletion Region (Space Charge Region): Region near junction depleted of mobile carriers, revealing ionized dopants. Built-in Electric Field: Created by exposed ionized dopants, pointing from n to p. Built-in Potential Barrier ($V_{bi}$): Potential difference across depletion region. $V_{bi} = V_t \ln\left(\frac{N_a N_d}{n_i^2}\right)$. Electric Field: Linear in uniformly doped abrupt junction. $E_{max}$ at metallurgical junction. Space Charge Width ($W$): Total width of depletion region. $W = \left[\frac{2\epsilon_s V_{bi}}{e} \left(\frac{N_a + N_d}{N_a N_d}\right)\right]^{1/2}$. $x_n = \frac{N_a}{N_a+N_d} W$, $x_p = \frac{N_d}{N_a+N_d} W$. ($x_n N_d = x_p N_a$). pn Junction Under Reverse Bias Positive voltage applied to n-region relative to p-region. Potential Barrier: Increases to $V_{bi} + V_R$. Depletion Width: Increases as $W \propto (V_{bi} + V_R)^{1/2}$. Electric Field: Increases, $E_{max} = \frac{2(V_{bi}+V_R)}{W}$. Junction Capacitance ($C_j$): $C_j = \frac{\epsilon_s}{W} = \left[\frac{e\epsilon_s N_a N_d}{2(V_{bi}+V_R)(N_a+N_d)}\right]^{1/2}$. One-Sided Junction: If $N_a \gg N_d$ (p$^+$n junction), $W \approx x_n$, $C_j \propto (N_d)^{1/2}$. Plotting $1/C_j^2$ vs $V_R$ yields $V_{bi}$ and $N_d$. Junction Breakdown Zener Breakdown: Tunneling of electrons from valence band in p-side to conduction band in n-side due to very high electric fields in highly doped junctions. Avalanche Breakdown: Impact ionization of lattice atoms by energetic carriers, creating new EHPs. Breakdown Voltage ($V_B$): Voltage at which breakdown occurs. For one-sided junction: $V_B = \frac{\epsilon_s E_{crit}^2}{2e N_B}$. Nonuniformly Doped Junctions Linearly Graded Junction: Doping changes linearly with distance across the junction. Space charge density: $\rho(x) = eax$. Electric field: Quadratic function of distance. $C_j \propto (V_{bi} + V_R)^{-1/3}$. Hyperabrupt Junction: Doping decreases away from metallurgical junction. $N = Bx^m$ with $m $C_j \propto (V_{bi} + V_R)^{-1/(m+2)}$. Used for varactor diodes. pn Junction Under Forward Bias Positive voltage applied to p-region relative to n-region. Potential Barrier: Decreases to $V_{bi} - V_a$. Carrier Injection: Majority carriers cross the junction, becoming excess minority carriers. $n_p(-x_p) = n_{p0} \exp\left(\frac{eV_a}{kT}\right)$. $p_n(x_n) = p_{n0} \exp\left(\frac{eV_a}{kT}\right)$. Minority Carrier Distribution: Excess minority carriers decay exponentially into the neutral regions (long diode assumption). Ideal pn Junction Current Dominated by diffusion of minority carriers across the depletion region. Ideal Diode Equation ($J = J_s (\exp(\frac{eV_a}{kT}) - 1)$): $J_s = e\left(\frac{D_n n_{p0}}{L_n} + \frac{D_p p_{n0}}{L_p}\right)$. $L_n = \sqrt{D_n \tau_{n0}}$, $L_p = \sqrt{D_p \tau_{p0}}$. $J_s$ is the reverse saturation current density. Temperature Effects: $J_s$ is highly temperature-dependent due to $n_i^2$. Short Diode: If neutral region length ($W_n$ or $W_p$) is much smaller than diffusion length, carrier profile is linear, and current is proportional to $1/W_n$. Nonideal Effects in pn Junctions Reverse-Biased Generation Current ($J_{gen}$): Thermal generation of EHPs in the depletion region (via traps) contributes to reverse current. $J_{gen} = \frac{e n_i W}{2\tau_0}$. Often dominates $J_s$ in Si. Forward-Bias Recombination Current ($J_{rec}$): Recombination of injected carriers within the depletion region. $J_{rec} \propto n_i W \exp\left(\frac{eV_a}{2kT}\right)$. Dominates at low forward bias. High-Level Injection: $\Delta n \approx n_0$. $I \propto \exp\left(\frac{eV_a}{2kT}\right)$. Ideality Factor ($n$): $I = I_s (\exp(\frac{eV_a}{nkT}) - 1)$. $n=1$ for ideal diffusion, $n=2$ for recombination dominance. Small-Signal Model of pn Junction Diffusion Resistance ($r_d$): $r_d = \frac{V_t}{I_{DQ}}$. Inverse of slope of I-V curve at DC bias point. Diffusion Capacitance ($C_d$): Due to storage of excess minority carriers. $C_d = \frac{1}{2V_t}(I_{p0}\tau_{p0} + I_{n0}\tau_{n0})$. Dominates in forward bias. Junction Capacitance ($C_j$): Depletion capacitance, dominates in reverse bias. Charge Storage and Diode Transients Turn-Off Transient: Switching from forward to reverse bias. Storage Time ($t_s$): Time to remove stored excess minority charge. $t_s \propto \tau$. Fall Time ($t_f$): Time for current to fall to 10% of initial value. Turn-On Transient: Switching from off to on state. Tunnel Diode Degenerately doped pn junction. Exhibits negative differential resistance due to tunneling of electrons. Majority carrier device, very fast switching. Metal-Semiconductor Junctions Schottky Barrier Diode: Rectifying contact between metal and semiconductor. Schottky Barrier Height ($\phi_B$): Potential barrier for electrons from metal to semiconductor. $\phi_B = \phi_m - \chi$ (ideal for n-type). Built-in Potential ($V_{bi}$): $V_{bi} = \phi_B - V_n$. Current Transport: Majority carrier thermionic emission over the barrier. I-V Relationship: $J = A^* T^2 \exp(-\frac{e\phi_B}{kT}) (\exp(\frac{eV_a}{kT}) - 1)$. Schottky Barrier Lowering (Image Force Effect): Electric field lowers the effective barrier height. Interface States: Can pin the Fermi level and modify barrier height. Comparison with pn Diode: Higher $J_s$, faster switching (no minority carrier storage). Ohmic Contact: Low-resistance, non-rectifying contact. Formed when $\phi_m \chi + E_g/e$ (for p-type). Can also be formed by heavy doping to allow tunneling through a thin barrier. Specific Contact Resistance ($R_c$): Measures quality of ohmic contact. $R_c = \left(\frac{\partial J}{\partial V}\right)^{-1} \Big|_{V=0}$. Heterojunctions Junction between two different semiconductor materials. Band Alignment: Discontinuities in conduction ($\Delta E_C$) and valence ($\Delta E_V$) bands. $\Delta E_C + \Delta E_V = \Delta E_g$. Electron Affinity Rule: $\Delta E_C = \chi_1 - \chi_2$. Two-Dimensional Electron Gas (2-DEG): In an n-AlGaAs/i-GaAs heterojunction, electrons accumulate in a potential well at the interface, forming a 2-DEG. This separates carriers from dopants, reducing scattering and increasing mobility. MOSFET Basic Structure MOS Capacitor: Metal-Oxide-Semiconductor structure. Terminals: Gate (G), Source (S), Drain (D), Body/Substrate (B). Oxide Capacitance ($C_{ox}$): $C_{ox} = \frac{\epsilon_{ox}}{t_{ox}}$. MOS Capacitor Operation Accumulation: Gate voltage attracts majority carriers to semiconductor-oxide interface. Depletion: Gate voltage repels majority carriers, exposing ionized dopants and forming a depletion region. Depletion width ($x_d$) increases with voltage. Inversion: Gate voltage attracts minority carriers to form an inversion layer at the interface. Threshold Inversion Point: Surface potential ($\phi_s$) is $2\phi_f$. Depletion width reaches maximum ($x_{dT}$). Flat-Band Voltage ($V_{FB}$): Gate voltage required for zero band bending in the semiconductor. $V_{FB} = \phi_{ms} - \frac{Q'_{ss}}{C_{ox}}$. ($\phi_{ms}$ is metal-semiconductor work function difference, $Q'_{ss}$ is fixed oxide charge). Threshold Voltage ($V_T$): Gate voltage to achieve threshold inversion. For n-channel (p-substrate): $V_T = V_{FB} + 2\phi_f + \frac{eN_a x_{dT}}{C_{ox}}$. C-V Characteristics: Capacitance vs. Gate Voltage. Accumulation: $C = C_{ox}$. Depletion: $C = \frac{C_{ox} C_{dep}}{C_{ox} + C_{dep}}$. $C_{dep} = \frac{\epsilon_s}{x_d}$. Inversion (low frequency): $C = C_{ox}$. Inversion (high frequency): $C = C_{min} = \frac{C_{ox} C_{maxdep}}{C_{ox} + C_{maxdep}}$. MOSFET Operation and I-V Characteristics Enhancement Mode: Channel is off at $V_{GS}=0$. Requires $V_{GS} > V_T$ (n-channel) or $V_{GS} Depletion Mode: Channel is on at $V_{GS}=0$. Requires $V_{GS}$ to turn off. Nonsaturation (Linear) Region ($V_{DS} $I_D = \mu_n C_{ox} \frac{W}{L} \left((V_{GS}-V_T)V_{DS} - \frac{1}{2}V_{DS}^2\right)$. Saturation Region ($V_{DS} \ge V_{GS} - V_T$): $I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS}-V_T)^2$. $V_{DS(sat)} = V_{GS} - V_T$. Transconductance ($g_m$): $g_m = \frac{\partial I_D}{\partial V_{GS}}$. Measures gain. Body Effect: Change in $V_T$ due to source-to-body voltage ($V_{SB}$). $V_T = V_{T0} + \gamma (\sqrt{2\phi_f + V_{SB}} - \sqrt{2\phi_f})$. MOSFET Nonideal Effects Subthreshold Conduction: Current flow below $V_T$ due to weak inversion. $I_D \propto \exp(V_{GS})$. Channel Length Modulation: Effective channel length $L$ decreases in saturation as $V_{DS}$ increases, leading to finite output resistance. Mobility Variation: $\mu$ decreases with increasing transverse electric field (surface scattering) and high lateral electric fields (velocity saturation). Velocity Saturation: At high fields, carriers reach $v_{sat}$, making $I_D$ linear with $V_{GS}$ in saturation. Ballistic Transport: Carriers travel without scattering in very short channels ($L \approx$ mean free path). MOSFET Scaling and Threshold Voltage Modifications Constant-Field Scaling: Scale dimensions (L, W, $t_{ox}$) and voltages ($V_{GS}, V_{DS}$) by a factor $k Short-Channel Effects: $V_T$ decreases as $L$ decreases due to charge sharing by source/drain depletion regions. Narrow-Channel Effects: $V_T$ increases as $W$ decreases due to additional depletion charge at channel edges. Threshold Adjustment: Ion implantation modifies channel doping to set desired $V_T$. MOSFET Breakdown and Reliability Oxide Breakdown: Catastrophic failure due to high electric field in $SiO_2$. Avalanche Breakdown: Impact ionization in drain depletion region. Punch-Through: Drain depletion region extends to source region, causing rapid current increase. Snapback Breakdown: Regenerative process involving parasitic BJT, causing negative resistance. Lightly Doped Drain (LDD): Design to reduce peak electric fields at drain, mitigating breakdown and hot-electron effects. Radiation Effects: Ionizing radiation creates electron-hole pairs in oxide, leading to fixed oxide charge ($V_T$ shift) and interface states (mobility degradation). Hot-Electron Effects: Energetic carriers inject into oxide, causing trapped charge and interface state generation. MOSFET Frequency Limitations Small-Signal Equivalent Circuit (Hybrid-$\pi$): Includes transconductance ($g_m$), output resistance ($r_{ds}$), and parasitic capacitances ($C_{gs}, C_{gd}, C_{ds}$). Cutoff Frequency ($f_T$): Frequency where current gain is unity. $f_T = \frac{g_m}{2\pi (C_{gs} + C_{gd})}$. Bipolar Junction Transistor (BJT) Basic Structure Three doped regions: Emitter (E), Base (B), Collector (C). Two pn junctions (E-B, B-C). Base width ($x_B$) is narrow. Emitter doping ($N_E$) is highest, Collector doping ($N_C$) is lowest. NPN and PNP are complementary. BJT Operation Modes Forward-Active: E-B forward biased, B-C reverse biased. (Amplification). Cutoff: Both junctions reverse biased. ($I_C \approx 0$). Saturation: Both junctions forward biased. ($I_C$ not controlled by $I_B$). Inverse-Active: E-B reverse biased, B-C forward biased. (Roles of E and C reversed). BJT Currents and Gain Minority Carrier Distribution: Governed by diffusion in neutral regions. Collector Current ($I_C$): $I_C \propto \frac{e D_n n_{B0}}{x_B} \exp\left(\frac{eV_{BE}}{kT}\right)$. Common-Base Current Gain ($\alpha$): $\alpha = \frac{\Delta I_C}{\Delta I_E}$. $\alpha $\alpha = \gamma \alpha_T \delta$. Emitter Injection Efficiency ($\gamma$): $\gamma = \frac{J_{nE}}{J_{nE} + J_{pE}}$. Maximize $N_E \gg N_B$. Base Transport Factor ($\alpha_T$): $\alpha_T = \frac{J_{nC}}{J_{nE}}$. Maximize by $x_B \ll L_B$. Recombination Factor ($\delta$): Accounts for recombination in E-B space charge region. Maximize by minimizing $J_R$. Common-Emitter Current Gain ($\beta$): $\beta = \frac{\Delta I_C}{\Delta I_B} = \frac{\alpha}{1-\alpha}$. $\beta \gg 1$. BJT Nonideal Effects Base Width Modulation (Early Effect): Neutral base width ($x_B$) changes with $V_{CB}$, leading to finite output resistance and Early Voltage ($V_A$). High Injection: At high $I_C$, $\Delta n_B \approx N_B$. $\beta$ decreases. Emitter Bandgap Narrowing: In highly doped emitters, $E_g$ effectively decreases, increasing $n_i^2$ and reducing $\gamma$. Current Crowding: Non-uniform current in emitter due to base resistance. Nonuniform Base Doping: Can create an accelerating electric field in the base. Breakdown Voltage: Punch-Through: B-C depletion region extends through base to E-B junction. Avalanche Breakdown ($BV_{CEO}$): $BV_{CEO} = \frac{BV_{CBO}}{\sqrt[n]{\beta}}$. (Lower than $BV_{CBO}$). BJT Equivalent Circuit Models Ebers-Moll Model: Applicable in all operating modes, based on interacting diodes. $I_C = \alpha_F I_{ES}(\exp(\frac{eV_{BE}}{kT})-1) - I_{CS}(\exp(\frac{eV_{BC}}{kT})-1)$. Gummel-Poon Model: More physics-based, handles non-uniform doping and non-ideal effects. Uses Gummel numbers (integrated charges). Hybrid-Pi Model: Small-signal model for amplification. Includes resistances ($r_{\pi}, r_0$) and capacitances ($C_\pi, C_\mu$). BJT Frequency Limitations Total Emitter-to-Collector Delay Time ($\tau_{ec}$): $\tau_{ec} = \tau_e + \tau_b + \tau_d + \tau_c$. $\tau_e$: Emitter-base capacitance charging time. $\tau_b$: Base transit time. $\tau_b = \frac{x_B^2}{2D_n}$. $\tau_d$: Collector depletion region transit time. $\tau_c$: Collector capacitance charging time. Cutoff Frequency ($f_T$): $f_T = \frac{1}{2\pi\tau_{ec}}$. Frequency where $\beta$ drops to unity. BJT Large-Signal Switching Delay Time ($t_d$): Time for E-B junction to forward bias. Rise Time ($t_r$): Time for $I_C$ to rise (0.1 to 0.9 $I_{C(sat)}$). Storage Time ($t_s$): Time to remove excess base charge when switching from saturation. Fall Time ($t_f$): Time for $I_C$ to fall (0.9 to 0.1 $I_{C(sat)}$). Schottky-Clamped Transistor: Schottky diode prevents deep saturation, reducing $t_s$. JFET Basic Structure and Operation pn Junction FET (JFET): Uses a pn junction to modulate channel conductance. MESFET: Uses a Schottky barrier junction to modulate channel conductance. Terminals: Gate (G), Source (S), Drain (D). Channel: Region where majority carriers flow between source and drain. Pinchoff: Gate voltage causes depletion region to completely fill the channel, turning off the device. Depletion Mode: Channel is on at $V_{GS}=0$. Requires voltage to turn off. Enhancement Mode: Channel is off at $V_{GS}=0$. Requires voltage to turn on. JFET Device Characteristics Internal Pinchoff Voltage ($V_{p0}$): Total potential across the gate junction to achieve pinchoff. $V_{p0} = \frac{e a^2 N_d}{2\epsilon_s}$. Pinchoff Voltage ($V_p$ or $V_T$): Gate-to-source voltage to achieve pinchoff. $V_p = V_{bi} - V_{p0}$. (For n-channel depletion mode, $V_p$ is negative). Drain-to-Source Saturation Voltage ($V_{DS(sat)}$): $V_{DS(sat)} = V_{p0} - (V_{bi} + V_{GS})$. I-V Relationship (Nonsaturation): $I_D \propto V_{DS}$. I-V Relationship (Saturation): $I_D = I_{DSS} \left(1 - \frac{V_{GS}}{V_p}\right)^2$. ($I_{DSS}$ is saturation current at $V_{GS}=0$). Transconductance ($g_m$): $g_m = \frac{\partial I_D}{\partial V_{GS}}$. ($g_m \propto (1 - V_{GS}/V_p)$ in saturation). JFET Nonideal Effects Channel Length Modulation: Similar to MOSFET, finite output resistance in saturation. Velocity Saturation: Carriers reach saturation velocity in short channels, modifying I-V curves. Subthreshold Conduction: Current below pinchoff voltage. Gate Current: Reverse leakage current through the gate junction. JFET Frequency Limitations Cutoff Frequency ($f_T$): $f_T = \frac{g_m}{2\pi C_G}$, where $C_G$ is total gate capacitance. In short channels, $f_T \approx \frac{\mu_n V_{p0}}{2\pi L^2}$. High Electron Mobility Transistor (HEMT) Utilizes heterojunction (e.g., AlGaAs/GaAs) to create a 2-DEG channel. Separates carriers from dopants, reducing scattering and increasing mobility. High frequency, low noise, low power dissipation. Optical Absorption Photon Energy: $E = h\nu = \frac{hc}{\lambda} = \frac{1.24}{\lambda (\mu m)}$ eV. Absorption Coefficient ($\alpha$): $I_\nu(x) = I_{\nu0} e^{-\alpha x}$. $\alpha$ increases rapidly for $h\nu > E_g$. Electron-Hole Pair Generation Rate ($G_L$): $G_L = \frac{\alpha I_\nu(x)}{h\nu}$. Solar Cells pn junction converts optical power to electrical power. I-V Characteristic: $I = I_L - I_s (\exp(\frac{eV}{kT}) - 1)$. Short-Circuit Current ($I_{SC}$): $I_{SC} = I_L$ (at $V=0$). Open-Circuit Voltage ($V_{OC}$): $V_{OC} = V_t \ln\left(1 + \frac{I_L}{I_s}\right)$ (at $I=0$). Conversion Efficiency ($\eta$): $\eta = \frac{P_m}{P_{in}} = \frac{I_m V_m}{P_{in}}$. Fill Factor (FF): $FF = \frac{I_m V_m}{I_{SC} V_{OC}}$. Heterojunction Solar Cells: Use wide bandgap material as an optical window to reduce surface recombination and improve efficiency. Amorphous Silicon Solar Cells: Low cost, large area. High absorption coefficient. PIN structure. Photodetectors Convert optical signals to electrical signals. Photoconductor: Change in conductivity due to excess carriers. Photocurrent: $I_L = e G_L \tau_p (\mu_n + \mu_p) A E$. Photoconductor Gain: $G_{ph} = \frac{\tau_p}{t_n} (1 + \frac{\mu_p}{\mu_n})$. Photodiode: Reverse-biased pn junction. Photocurrent due to EHPs generated in depletion region and diffusion from neutral regions. Prompt Photocurrent: From depletion region. $I_{L1} = e G_L W$. Delayed Photocurrent: From neutral regions (diffusion). PIN Photodiode: Wide intrinsic region for larger depletion width, higher speed. Avalanche Photodiode: Operated at high reverse bias to use impact ionization for current gain. Phototransistor: BJT with open base. Photocurrent amplified by transistor gain ($I_C = (1+\beta)I_L$). Photoluminescence and Electroluminescence Luminescence: Light emission. Photoluminescence: Light emission from recombination of EHPs generated by photon absorption. Electroluminescence: Light emission from recombination of EHPs generated by electric current (e.g., injection across pn junction). Quantum Efficiency ($\eta_q$): $\eta_q = \frac{\text{Radiative Recombination Rate}}{\text{Total Recombination Rate}}$. Light Emitting Diodes (LEDs) Forward-biased pn junction, typically direct bandgap material (e.g., GaAs, GaAsP, AlGaAs). Light Generation: Spontaneous radiative recombination of injected minority carriers. Wavelength $\lambda = \frac{1.24}{E_g}$. Internal Quantum Efficiency: $\eta_i = \eta_{inj} \eta_{rad}$. ($\eta_{inj}$ is injection efficiency, $\eta_{rad}$ is radiative efficiency). External Quantum Efficiency: $\eta_{ext} = \eta_i \eta_{extract}$. Accounts for reabsorption, Fresnel loss, critical angle loss. Laser Diodes Light Amplification by Stimulated Emission of Radiation (LASER). Stimulated Emission: Incident photon causes electron in higher energy state to emit an identical photon. (Requires population inversion). Population Inversion: More electrons in higher energy state than lower. Achieved in degenerately doped pn junction under forward bias. Optical Cavity (Fabry-Perot Resonator): Two parallel mirrors provide optical feedback, leading to coherent output. Threshold Current ($J_{th}$): Minimum current for lasing. Gain equals losses. Double Heterojunction Laser: Uses heterojunctions to confine carriers and photons, improving efficiency and reducing $J_{th}$. Microwave Devices Tunnel Diode: Degenerately doped pn junction. Exhibits negative differential resistance (NDR) due to tunneling. Used in oscillators. Gunn Diode (Transferred Electron Device - TED): N-type GaAs or InP. Exhibits NDR due to "transferred electron effect" (electrons scatter from high-mobility lower valley to low-mobility upper valley in conduction band at high electric fields). Forms high-field domains that drift, producing current oscillations. IMPATT Diode (Impact Ionization Avalanche Transit-Time): Reverse-biased p$^+$n-i-n$^+$ structure. Uses avalanche multiplication and transit time delay to produce dynamic NDR at microwave frequencies. Power Bipolar Transistors Vertical Structure: Collector terminal at bottom, maximizing current path. Wide Base: Prevents punch-through at high $V_{BC}$. Low Collector Doping: Supports high blocking voltage. Interdigitated Structure: Many narrow emitters in parallel to handle high current and prevent current crowding. Characteristics: Lower $\beta$, lower $f_T$ than small-signal BJTs. Limited by $I_{C,max}$, $V_{CE,sus}$, $P_D$. Safe Operating Area (SOA): Defines safe regions of $I_C$ vs $V_{CE}$ operation. Second Breakdown: Thermal runaway due to localized heating. Darlington Pair: Two BJTs cascaded to achieve high effective current gain. Power MOSFETs Structures: DMOS (Double-Diffused MOS): Vertical structure, channel formed by double diffusion. VMOS (V-Groove MOS): Vertical channel in a V-shaped groove. HEXFET: Repetitive pattern of small DMOS cells in parallel. On-Resistance ($R_{on}$): $R_{on} = R_S + R_{CH} + R_D$. $R_{on}$ has positive temperature coefficient, providing thermal stability and current sharing in parallel devices. Advantages over Power BJTs: Faster switching, no second breakdown, stable gain over temperature. Parasitic BJT: Power MOSFETs have an inherent parasitic BJT. Must be kept off to prevent snapback breakdown. Thyristor General class of pnpn switching devices with bistable regenerative characteristics. SCR (Semiconductor Controlled Rectifier): Three-terminal thyristor (Anode, Cathode, Gate). Operation: Forward Blocking State: High voltage, low current. Center junction (J2) reverse biased. Forward Conducting State: Low voltage, high current. All junctions forward biased (device latches ON). Triggering: Breakdown Triggering: J2 avalanche breakdown. Gate Current: Small current at gate injects carriers, turning ON the device. dV/dt Triggering: Rapid change in voltage causes transient current, turning ON the device. Turn-Off: Reduce anode current below "holding current" (current required to sustain ON state). Triac: Bilateral thyristor, switches symmetrically for AC. MOS Gated Thyristor: Uses MOS gate to control turn-on. ```