Carrier transport formulas

Electrons drift opposite to $E$ (hence the minus sign) and holes drift along $E$. ${\mu }_n,{\mu }_p$ are the electron and hole mobilities in ${\text{cm}}^2/\text{V-s}$. ${\tau }_m$ is the mean time between scattering events and $m^{\ast }$ is the conductivity effective mass. The linear relation $v_d=\mu E$ holds only at low fields (see velocity saturation below).

$q=1.6\times 10^{-19}$ C. With $n,p$ in ${\text{cm}}^{-3}$ and $E$ in V/cm, $J$ is in ${\text{A/cm}}^2$. Electron and hole drift currents both point along $E$ (conventional current), so they add.

For a fully ionized, uncompensated extrinsic sample with doping much larger than $n_i$, one term dominates: $\sigma \approx q{N}_D{\mu }_n$ (n-type) or $\sigma \approx q{N}_A{\mu }_p$ (p-type). In a compensated sample, use the net doping, e.g. $n\approx N_D-N_A$ when $N_D-N_A\gg n_i$. The mobility is still reduced by all $N_D+N_A$ ionized impurities. For an intrinsic sample, ${\sigma }_i=q{n}_i({\mu }_n+{\mu }_p)$. $L$ is the length along the current and $A$ the cross-section.

This uses $np=n_i^2$ (thermal equilibrium, non-degenerate). Since ${\mu }_n>{\mu }_p$ in Si, the minimum lies at $n<n_i$, i.e. in slightly p-type material, not at the intrinsic point. ${\sigma }_{min}<{\sigma }_i$ because $2\sqrt{{\mu }_n{\mu }_p}\le {\mu }_n+{\mu }_p$.

$D_n,D_p$ are diffusion coefficients in ${\text{cm}}^2/\text{s}$. Both carriers diffuse from high to low concentration. Electrons are negative, so the electron diffusion current points up the electron gradient (it has the sign of $dn/dx$). Holes are positive, so the hole diffusion current points down the hole gradient (it has the sign of $-dp/dx$).

The drift-diffusion equations for each carrier type. They are the inputs to the continuity equation.

Holds for a non-degenerate semiconductor. It is derived in thermal equilibrium and used at low fields in the drift-diffusion model. $V_T\approx 0.02585$ V at $T=300$ K, often rounded to $0.0259$ V or $0.026$ V. State the value you use, because the rounding shifts the third significant figure.

Conditions: thermal equilibrium (zero net current), full ionization, and quasi-neutrality ($n\approx N_D(x)$ or $p\approx N_A(x)$). Drift must cancel diffusion, so a doping gradient produces a field. Example: $N_D=N_0{e}^{-x/\lambda }$ gives a uniform field $E=V_T/\lambda$ along $+x$.

$I$ flows along the sample length and $B$ is perpendicular to $I$. $t$ is the sample thickness measured along $B$, $W$ is the width across which $V_H$ is measured, and $J=I/(Wt)$. $V_H$ does not depend on $W$. For a fixed geometry, $V_H$ has opposite polarity for holes and for electrons, so the sign of $V_H$ (or $R_H$) identifies the majority carrier. These simple forms assume one dominant carrier type and take the Hall scattering factor as 1. From a measured $V_H$: $p=IB/(q{V}_{H}t)$ or $n=IB/(q|V_H|t)$.

Valid when one carrier type dominates. It combines the measured Hall coefficient and the measured conductivity to give the majority-carrier mobility.

Valid for low-level injection, where the excess carrier density is much smaller than the majority-carrier density. $\delta p=p-p_0$ and $\delta n=n-n_0$ are the excess concentrations. In a neutral region with pair generation, $\delta n=\delta p$. ${\tau }_p,{\tau }_n$ are the minority-carrier lifetimes; they depend on the material and its defects and are given as constants in problems.

Conditions: a uniformly illuminated n-type sample, low-level injection, and no concentration gradients (so no diffusion). The continuity equation then reduces to $d(\delta p)/dt=G_L-\delta p/{\tau }_p$. $G_L$ is the optical generation rate in ${\text{cm}}^{-3}{\text{s}}^{-1}$. Since $\delta n=\delta p$, both carriers add to the conductivity.

The characteristic distance a minority carrier diffuses before it recombines. In a long sample it is the $1/e$ decay length of the steady-state excess-carrier profile (see the continuity-equation section).

$G$ and $R$ are the generation and recombination rates. The opposite signs come from the charge of each carrier: $J_n=-q\times$(electron flux) and $J_p=+q\times$(hole flux). Substituting the drift-diffusion expressions for $J_n$ and $J_p$ gives the minority-carrier equation below.

Assumes uniform doping, a uniform field $E$ and low-level injection. $G_{ext}$ is any externally driven generation, e.g. optical. With $E=0$ the drift term drops out. Setting the time derivative to zero gives the steady-state form.

Conditions: no generation for $x>0$, the excess concentration held at $\delta p(0)$ (e.g. by an injecting junction), and $\delta p\to 0$ as $x\to \infty$ (sample length much greater than $L_p$). The diffusion current decays with the same length $L_p$ as the profile.

The same equation with $\delta p(W)=0$, as at an ohmic contact or a reverse-biased junction at $x=W$. When $W\ll L_p$, almost no carriers recombine in transit. The profile is then linear, and the diffusion current $q{D}_p\delta p(0)/W$ is nearly uniform. This is the short-base picture used for BJTs and short diodes.

$E=-dV/dx$. ${\rho }_{net}$ here is the space-charge density, not resistivity. ${ϵ}_s={ϵ}_r{ϵ}_0$ is the semiconductor permittivity, with ${ϵ}_r\approx 11.7\text{–}11.9$ for Si and ${ϵ}_0=8.85\times 10^{-14}$ F/cm. With the depletion approximation ($n,p\approx 0$ inside the depletion region), it gives the field and potential profiles and the depletion width of a junction. For example, on the n-side of a step junction $dE/dx=q{N}_D/{ϵ}_s$, so $E$ varies linearly with $x$.

Valid for a non-degenerate semiconductor. With no excess generation or injection, both reduce to the single equilibrium Fermi level ($E_{Fn}=E_{Fp}=E_F$, $np=n_i^2$). Quoted values of $n_i$ for Si at 300 K range from about $9.65\times 10^9$ to $1.5\times 10^{10}{\text{cm}}^{-3}$. Use the value the question gives, and state it whenever a numerical answer depends on it.