29 May 2025

Doping, Carrier Concentration, and Mobility

Charge neutrality, donor and acceptor doping, Fermi-level displacement, mobility, and conductivity.

bsc semester-iv mj-7 semiconductor-devices doping mobility

Let $N_D^+$ and $N_A^-$ be ionized donor and acceptor densities. Local electrical neutrality requires total positive and negative charge densities to agree:

\[\boxed{p+N_D^+=n+N_A^-}.\]

When donors and acceptors are fully ionized, $N_D^+=N_D$ and $N_A^-=N_A$. Together with $np=n_i^2$, this determines the equilibrium carriers.

n-type material

For $N_D>N_A$, neutrality becomes $n-p=N_D-N_A\equiv N$. Since $p=n_i^2/n$,

\[n-\frac{n_i^2}{n}=N,\]

so

\[n^2-Nn-n_i^2=0, \qquad \boxed{n=\frac{N+\sqrt{N^2+4n_i^2}}2}, \qquad p=\frac{n_i^2}{n}.\]

If $N\gg n_i$, then $n\simeq N_D-N_A$ and $p\simeq n_i^2/n$. Using $n=n_i\exp[(E_F-E_i)/(k_BT)]$,

\[\boxed{E_F-E_i=k_BT\ln(n/n_i)}.\]

Thus donor doping raises the Fermi level.

p-type material

For $N_A>N_D$, put $P=N_A-N_D$. Then $p-n=P$ and

\[\boxed{p=\frac{P+\sqrt{P^2+4n_i^2}}2}, \qquad n=\frac{n_i^2}{p},\]

with

\[\boxed{E_i-E_F=k_BT\ln(p/n_i)}.\]

Acceptor doping therefore lowers the Fermi level. In heavily doped material $E_F$ approaches or enters a band, and these non-degenerate exponential formulas must be replaced by Fermi-Dirac integrals.

Temperature dependence of carrier concentration

For shallow dopants there are three useful temperature ranges. At low temperature, dopants are only partly ionized (freeze-out), so the majority-carrier density rises rapidly as dopant atoms ionize. In the intermediate extrinsic range, ionization is nearly complete and the majority density is approximately the net dopant density; for example, $n\simeq N_D-N_A$ in compensated n-type material. At sufficiently high temperature, thermal pair generation dominates the dopants and

\[n\simeq p\simeq n_i =\sqrt{N_cN_v}\exp\!\left[-\frac{E_g}{2k_BT}\right].\]

Because $N_c,N_v\propto T^{3/2}$ for temperature-independent effective masses, $n_i\propto T^{3/2}e^{-E_g/(2k_BT)}$. The minority density in the extrinsic range follows $p=n_i^2/n$ or $n=n_i^2/p$, so it remains strongly temperature dependent even when the majority density is nearly fixed.

Mobility and its dependence

In the relaxation-time model, a carrier with effective mass $m^*$ obeys

\[m^*\frac{d\mathbf v}{dt}=q_c\mathbf E-\frac{m^*\mathbf v}{\tau}.\]

In steady state,

\[\mathbf v_d=\frac{q_c\tau}{m^*}\mathbf E,\]

so the positive mobility magnitude is

\[\boxed{\mu=\frac{\lvert q_c\rvert\tau}{m^*}}.\]

Lattice vibrations shorten $\tau$ as temperature rises, while ionized impurities shorten it as doping rises. If the two scattering mechanisms are independent, their rates add:

\[\boxed{\frac1\mu=\frac1{\mu_{\rm lattice}}+\frac1{\mu_{\rm impurity}}}.\]

Carrier concentration and mobility therefore affect conductivity separately:

\[\sigma=q(n\mu_n+p\mu_p).\]

Doping normally increases $n$ or $p$ by many orders of magnitude even though impurity scattering reduces mobility; the net conductivity consequently rises.

The mobility laws are material- and temperature-range dependent. In simple three-dimensional models, lattice-limited mobility often varies approximately as $T^{-3/2}$, whereas ionized-impurity-limited mobility varies approximately as $T^{3/2}/N_{\rm ion}$. These powers are approximations, not universal constants.