28 May 2025

Semiconductor Materials, Carriers, and Band Gaps

Semiconductor classification, electron and hole carriers, direct and indirect gaps, and heavy doping.

bsc semester-iv mj-7 semiconductor-devices semiconductor-materials

A crystalline semiconductor is described by allowed electron-energy bands. The highest band occupied at absolute zero is the valence band; the next allowed band is the conduction band. Their edge energies $E_v$ and $E_c$ define

\[E_g=E_c-E_v.\]

An insulator has a gap too large for appreciable thermal excitation, a metal has a partly filled band or overlapping bands, and a semiconductor has an intermediate gap whose carrier population can be controlled by temperature and impurities. Silicon, germanium, and gallium arsenide are standard semiconductor materials.

Electrons and holes

An electron promoted from the valence band to the conduction band leaves one missing valence electron. If a neighbouring valence electron fills that vacancy, the vacancy moves oppositely. The vacancy is therefore represented as a hole of charge $+q$. For an electric field $\mathbf E$,

\[\mathbf F_e=-q\mathbf E,\qquad \mathbf F_h=+q\mathbf E.\]

Both carriers contribute to conventional current in the field direction. If $n$ and $p$ are their number densities and $\mu_n,\mu_p$ their mobilities, the drift velocities are $\mathbf v_n=-\mu_n\mathbf E$ and $\mathbf v_p=\mu_p\mathbf E$. Hence

\[\mathbf J_n=(-qn)\mathbf v_n=qn\mu_n\mathbf E, \qquad \mathbf J_p=(qp)\mathbf v_p=qp\mu_p\mathbf E,\]

and

\[\boxed{\sigma=q(n\mu_n+p\mu_p)},\qquad \rho=\sigma^{-1}.\]

Here $n,p$ have units $\mathrm{m^{-3}}$, mobility has units $\mathrm{m^2\,V^{-1}s^{-1}}$, and conductivity has units $\mathrm{S\,m^{-1}}$.

Intrinsic and extrinsic material

In a pure, thermally equilibrated semiconductor every excitation produces one electron-hole pair, so

\[n=p=n_i.\]

Intentional impurity atoms create an extrinsic semiconductor. A shallow donor supplies an electron and produces n-type material; a shallow acceptor captures a valence electron and leaves a hole, producing p-type material. Charge neutrality is not lost: the mobile carrier is balanced by a fixed ionized impurity.

Direct and indirect gaps

Crystal momentum is $\hbar\mathbf k$. In a direct-gap semiconductor, the conduction-band minimum and valence-band maximum occur at the same $\mathbf k$. A vertical transition can conserve momentum while a photon supplies energy $h\nu\ge E_g$, so radiative recombination is efficient.

In an indirect-gap semiconductor, those extrema occur at different wavevectors. A photon carries negligible crystal momentum on the Brillouin-zone scale, so a phonon must supply the momentum difference. For an absorption transition, the two possibilities are

\[\begin{aligned} \text{phonon absorbed:}\quad &h\nu=E_g-\hbar\Omega_{\rm ph}, &\mathbf k_c-\mathbf k_v&=+\mathbf q_{\rm ph},\\ \text{phonon emitted:}\quad &h\nu=E_g+\hbar\Omega_{\rm ph}, &\mathbf k_c-\mathbf k_v&=-\mathbf q_{\rm ph}. \end{aligned}\]

The extra phonon requirement makes light emission less probable. GaAs is direct-gap; Si is indirect-gap.

Heavy doping

At moderate doping, donor or acceptor states are discrete and the carriers remain non-degenerate. With heavy doping the impurity wavefunctions overlap, impurity levels broaden into an impurity band, and the Fermi level may enter a principal band. Maxwell-Boltzmann statistics then fail and the full Fermi-Dirac distribution is required. Heavy doping also narrows a junction depletion region, making field-assisted tunnelling possible; it does not change the sign of the electron or hole charge.