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A substitutional group-V atom such as phosphorus has one more valence electron than germanium and acts as a donor impurity, giving an n-type semiconductor. Approximate its outer electron by a hydrogenic donor state with screened Coulomb attraction and effective mass $m_e^{\ast }$. The ionization energy is the positive binding-energy magnitudeIts effective Bohr radius of a hydrogenic donor is about $3.80\mathrm{nm}$, much larger than an atomic core, consistent with a shallow extended bound state.

At $300\mathrm{K}$, $k_{B}T=25.9\mathrm{meV}$, while the mean three-dimensional classical translational kinetic energy is $3k_{B}T/2=38.8\mathrm{meV}$. Both exceed the donor binding energy, so substantial thermal ionization is expected, unlike excitation across the much larger $0.661\mathrm{eV}$ host band gap. A precise ionized fraction also depends on the chemical potential, dopant density and donor charge-state degeneracy; comparing two energy scales alone does not prove complete ionization at arbitrary concentration.

A substitutional phosphorus atom supplies one more valence electron than the silicon atom it replaces. In the shallow hydrogenic donor state model, the remaining positive impurity attracts that extra electron through the screened Coulomb potentialRescaling the hydrogen atom binding energy by the effective mass and dielectric screening gives the ionization energyThe donor level lies at $E_c-E_D^{\mathrm{bind}}$; its binding magnitude is positive. The associated effective Bohr radius of a hydrogenic donor is $a_B^{\ast }={ϵ}_r(m_e/m_e^{\ast })a_0\simeq 2.38\mathrm{nm}$, consistent with a shallow state extended over several lattice spacings.

At $1000\mathrm{K}$, $k_{B}T=0.08617\mathrm{eV}$, approximately $3.3$ times this binding energy. Dilute donors are consequently expected to be largely ionized, rather than frozen out. The actual ionized fraction also depends on the chemical potential, available conduction states and donor charge-state degeneracy; comparing $k_{B}T$ to the binding energy alone is not a normalized ionization probability. At sufficiently high temperature, intrinsic electron-hole production can dominate the total carriers even though donors are ionized. Band anisotropy and short-range impurity corrections limit the numerical hydrogenic estimate.

In the hydrogenic donor state approximation, the screened Coulomb potential energy is $-e^2/(4\pi {ϵ}_0{ϵ}_{r}r)$ and the kinetic term uses $m^{\ast }=0.067m_e$. Rescaling the hydrogen atom gives the effective Bohr radius of a hydrogenic donor and the positive binding energyThereforeThe radius here is the exponential length of the ground-state wavefunction, not its root-mean-square radius. With $\psi (r)=(\pi a_b^3{)}^{-1/2}{e}^{-r/a_b}$, radial integration gives $⟨r^2⟩=3a_b^2$ and $⟨x^2+y^2⟩=2a_b^2$.

For dilute independent donors of number density $n$, a bound spin one-half electron has magnetic moment magnitude ${\mu }_s=|g|{\mu }_B/2$, with the Bohr magneton defined using the bare electron mass. Its weak-field Curie law response isFor the orbital response, expand the effective mass Hamiltonian $(\mathbf{p}+e\mathbf{A}{)}^2/(2m^{\ast })$ in a uniform magnetic field, using $\mathbf{A}=\mathbf{B}\times \mathbf{r}/2$. The spherical ground state has zero orbital angular momentum; the quadratic term raises its energy byDifferentiating this energy twice and using $B\simeq {\mu }_{0}H$ givesThe donor-only magnetic susceptibility cancels whenThe negative sign of $g=-0.45$ drops out of the linear magnetic susceptibility. This crossover is far below $E_b/k_B\simeq 63.6\mathrm{K}$, consistent with donor binding. It assumes sufficiently weak magnetic field, negligible exchange between donors and a thermal scale above any inter-donor splitting; sufficiently dilute donors can satisfy those assumptions. The background diamagnetism from part (b) is excluded from this cancellation.

A hydrogenic donor has the effective Bohr radius of a hydrogenic donorusing the external $a_0=0.52917721067\times 10^{-10}\mathrm{m}$.^[ref] Increasing donor density causes donor states to overlap and screens their binding, allowing a metallic impurity system rather than isolated frozen-out donors.

Using the standard approximate Mott criterion, $n_D^{1/3}{a}_B^{\ast }\simeq 0.25$, givesThe factor 0.25 is an externally documented empirical criterion, not supplied exam data or an exact universal threshold.^[ref] An order-of-magnitude overlap argument without that prefactor would give $(a_B^{\ast }{)}^{-3}\simeq 1.3\times 10^{25}{\mathrm{m}}^{-3}$. Compensation, disorder and multivalley anisotropy affect a real material's transition; this estimate uses the scalar mass and permittivity actually supplied, without substituting other germanium parameters.