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Semiconductor Device Theory

Field: Circuit Design

Sequence of Expressions

Define the current density JJ across a PN junction as a function of applied voltage VV: J(V)=Js(eqV/nkT−1)J(V) = J_s \big(e^{qV/nkT} - 1\big), where JsJ_s is the reverse saturation current density, qq is the elementary charge, nn is the ideality factor, kk is the Boltzmann constant, and TT is the absolute temperature.
Let ICI_C be the collector current, VBEV_{BE} the base-emitter voltage, and VCBV_{CB} the collector-base voltage. The operational regions are defined by the biasing conditions:\n1. **Cutoff:** VBE<VbiV_{BE} < V_{bi} and VCB<0V_{CB} < 0, resulting in IC→0I_C \to 0.\n2. **Active:** VBE>VbiV_{BE} > V_{bi} and VCB>0V_{CB} > 0, where IC≠0I_C \neq 0 and ICI_C is governed by the transconductance model.\n3. **Saturation:** VCB<VCE(sat)V_{CB} < V_{CE(sat)}, resulting in ICI_C being limited by the external load resistance RLR_L.
The capacitance CC of a Metal-Oxide-Semiconductor (MOS) structure is defined by the geometric and material parameters: C = \frac{A \times \text{\epsilon}}{\text{Thickness}} where AA is the physical area, Thickness\text{Thickness} is the oxide thickness, and the permittivity \text{\epsilon} is given by \text{\epsilon} = \text{\epsilon}_0 \times \text{\epsilon}_r, with \text{\epsilon}_0 being the permittivity of free space and \text{\epsilon}_r being the relative permittivity.
The transconductance gmg_m quantifies the efficiency of voltage-to-current conversion in a FET. It is formally defined as the partial derivative of the drain current IDI_D with respect to the gate-source voltage VGSV_{GS}, while holding the drain-source voltage VDSV_{DS} constant: gm=∂ID∂VGS∣VDS=constg_m = \frac{\partial I_D}{\partial V_{GS}}\bigg|_{V_{DS} = \text{const}}.
At thermal equilibrium in a semiconductor material, the product of the electron concentration nn and the hole concentration pp is constant, defined by the mass action law: n×p=ni2n \times p = n_i^2 where nin_i is the intrinsic carrier concentration, which depends solely on the material's band gap energy EgE_g and temperature TT.
The electrostatic potential V(r)V(\mathbf{r}) within a semiconductor material is governed by Poisson's equation, relating the Laplacian of the potential to the net charge density ρ(r)\rho(\mathbf{r}): \nabla^2 V = -\frac{\rho}{\text{\epsilon}_r \text{\epsilon}_0} where ρ=q(p−n+ND−NA)\rho = q(p - n + N_D - N_A) is the net charge density, qq is the elementary charge, and \text{\epsilon}_r is the relative permittivity.
The total recombination rate RR (pairs per unit volume per unit time) is modeled as the sum of various mechanisms: R=Rdirect+RSRH+RAugerR = R_{direct} + R_{SRH} + R_{Auger} where RdirectR_{direct} is the radiative recombination rate, RSRHR_{SRH} is the Shockley-Read-Hall rate mediated by trap states, and RAugerR_{Auger} accounts for multi-particle recombination effects.
The band gap energy EgE_g is defined as the minimum energy difference between the conduction band minimum EcE_c and the valence band maximum EvE_v: Eg=Ec−EvE_g = E_c - E_v This value determines the material's electronic classification, where Eg→0E_g \to 0 implies a conductor, and Eg>3 eVE_g > 3 \text{ eV} implies an insulator.
The majority carrier concentration (nn or pp) in a doped semiconductor is approximated by the concentration of the intentionally introduced dopant impurities (NDN_D or NAN_A), assuming full ionization and charge neutrality: n×approx→ND (for n-type) or p×approx→NA (for p-type)n \times \text{approx} \to N_D \text{ (for n-type)} \text{ or } p \times \text{approx} \to N_A \text{ (for p-type)}.