Semiconductor

Mobility

The carrier mobility is defined by

μvD/E \mu \equiv v_D / E

From Drude model for conductivity, we have

σ=J/E=nevDE=neμ \sigma = J / E = \frac{nev_D}{E} = ne\mu

And μ=σ/ne=eτ/m\mu = \sigma / ne = e\tau/m^*

Here in semiconductors, the carrier density nn is tuned.

Temperature effect

In metals, conductivity decreases with increasing temperature. This is generally caused by the reduced relaxation time τ\tau because of the increasing lattice vibrations.

However, in semiconductors, the conductivity will be overwhelmed by the huge increase in carrier density (generally, n(T)Nexp(E/KT)n(T) \approx N\exp(-E/KT)).

Density

Density of the occupying electrons ρ\rho is the density of states D(E)D(E) times state occupancy f(E)f(E).

ρ(E)=dn(E)/dE=D(E)f(E) \rho(E) = \mathrm{d}n(E) / \mathrm{d}E = D(E)f(E)

Direct and indirect bandgap

To direct bandgap semiconductor can emit photon directly, indirect semiconductors must have phonon involve.

Direct and indirect band gap. (via Ashcroft / Mermin)

Number of holes/vacancies

n=Egf(E)D(E)dEandp=0Eg[1f(E)]D(E)dE n = \int^\infty_{E_\text{g}} f(E) D(E) \mathrm{d}E \quad \text{and} \quad p = \int^{E_\text{g}}_0 [1-f(E)] D(E) \mathrm{d}E

with Fermi-Dirac distribution function f(E)=1/(1+expEμkT)f(E) = 1 / (1 + \exp \frac{E - \mu}{kT}) and D(E)=DeEEgD(E) = D_e\sqrt{E - E_\text{g}} from electron gas model.

n=Egf(E)D(E)dEandp=0Eg[1f(E)]D(E)dE n = \int^\infty_{E_\text{g}} f(E) D(E) \mathrm{d}E \quad \text{and} \quad p = \int^{E_\text{g}}_0 [1-f(E)] D(E) \mathrm{d}E

Where Nc,Nv(mc,v)3/2T3/2N_\text{c}, N_\text{v} \propto (m_{\text{c,v}})^{3/2} T^{3/2}

Then conveniently we have

np=NcNveEg/kT np = N_\text{c}N_\text{v}e^{-E_\text{g}/kT}

Intrinsic case

n=p=ni n = p = n_\text{i}

We have n=p=NcNveEg/2kTn = p = \sqrt{N_\text{c}N_\text{v}}e^{-E_\text{g}/2kT}

Intrinsic and extrinsic semiconductor