Donor impurity

An impurity that can supply an electron to the conduction band. A shallow donor has a localized occupied impurity level in a crystal near and below the conduction-band edge; thermal or other ionization releases its electron. The binding energy depends on the host dielectric response and effective mass.

Let an isolated donor impurity admit an empty ionized state with statistical weight $g_0$ and a singly occupied neutral state with statistical weight $g_1$, with double occupation excluded. Its grand canonical ensemble weights are $g_0$ and $g_1{e}^{-\beta ({ϵ}_D-\mu )}$, so $r=g_1/g_0$ multiplies the occupied-to-empty ratio. A nondegenerate empty state and two allowed electron spin orientations give $r=2$; a simplified single-state occupation model uses $r=1$. With negligible electron holes and uncompensated donor impurities, charge neutrality and nondegenerate conduction occupation imply $n^2/(N_D-n)=(N_c/r)e^{-E_i/(k_{B}T)}$, where $E_i$ is the donor binding energy and $N_c$ the effective density of states.

A nondegenerate donor-bound state of energy zero and an excited representative conduction level of energy $\Delta >0$ and degeneracy $g$ have the displayed bound probability by canonical state counting. The inferred degeneracy from one measurement is $g=(P_b^{-1}-1)e^{\Delta /(k_{B}T)}$. At finite fixed $g$, the probability decreases from1 toward $1/(1+g)$ as temperature grows. A temperature-independent effective degeneracy is an assumed two-level approximation; a full conduction-band density of states can behave differently.

In a scalar effective-mass dielectric model, the donor radius is $a_B^{\ast }={ϵ}_r(m_e/m^{\ast })a_0$. The effective mass reduces the kinetic-energy scale and dielectric screening weakens attraction. The corresponding binding energy scales as $(m^{\ast }/m_e){ϵ}_r^{-2}$ times the vacuum hydrogen binding energy.

For shallow hydrogenic donor states of radius $a_B^{\ast }$ and random donor number density $N_D$, the mean spacing is of order $N_D^{-1/3}$. Taking this spacing comparable to $a_B^{\ast }$ gives $N_D\sim (a_B^{\ast }{)}^{-3}$; a geometric diameter criterion instead gives $N_D\sim (2a_B^{\ast }{)}^{-3}$. The orbitals have exponentially decaying tails rather than hard boundaries, so there is no exact geometric onset. Appreciable overlap also limits the independent-donor ionisation model; a genuine transport transition needs additional electronic physics.

A shallow donor impurity in an isotropic semiconductor can be approximated by an electron of effective mass $m^{\ast }$ in a Coulomb potential screened by relative permittivity ${ϵ}_r$. Its ground-state binding energy and radius areThe donor level lies $E_B$ below the conduction band. These are magnitudes: the bound electron's energy relative to the band edge is negative. Band anisotropy, short-distance impurity structure and screening by mobile carriers can invalidate the simple Bohr model rescaling.

An electron confined to a plane in the screened Coulomb potential $-\kappa /r$, $\kappa =e^2/(4\pi {ϵ}_0{ϵ}_r)$, has the radial ground state $\psi =A{e}^{-br}$. The planar radial Laplacian is $(b^2-b/r)\psi$. Cancelling the $1/r$ term in the Schrödinger equation gives $b=2m^{\ast }\kappa /{\hslash }^2$ and $E=-{\hslash }^2{b}^2/(2m^{\ast })$. The binding magnitude is four times the three-dimensional effective Rydberg. Normalization in the plane gives $A=b\sqrt{2/\pi }$. The ordinary three-dimensional electric field is retained; a logarithmic two-dimensional interaction is a different problem.