Doping in semiconductors

Last lecture:

• Semiconductors have conduction and valence band where the particles are electrons and holes.
• Electrons and holes have Hall conducance $\sigma_H = B e(n_e - n_h)$

This lecture:

• Doping (=adding atoms of different type)
• Temperature dependence of carrier density

Density of states

Electrons $$ E = E_G + {p^2}/{2m_e}$$ $$ g(E) = V(2m_e)^{3/2}\sqrt{E-E_G}/2\pi^2$$ Holes $$ E = - {p^2}/{2m_h}$$ $$ g(E) = V(2m_h)^{3/2}\sqrt{E}/2\pi^2$$

Adding an impurity to semiconductor

• Typical semiconductors are group IV (Si, Ge, GaAs).
• Unfilled shell of group V atom (donor) has 1 extra electron and its nucleus 1 extra proton
• Group III atom (acceptor) lacks 1 electron and 1 nucleus charge

Extra electron (or extra hole) is attracted to the extra charge of the nucleus.

In H the energy levels are: $$ E_n = - \frac{me^4}{8\pi^2\hbar^3\varepsilon_0n^2} = -R_E /n^2= -\frac{13.6\text{eV}}{n^2}$$

Bohr radius (size of the ground state wave function): $4 \pi \varepsilon_0 \hbar^2/m_{\mathrm{e}} e^2$

In a semiconductor $m\to m_{\text{eff}}$, $\epsilon_0 \to \epsilon\epsilon_0$.

Adding an impurity to semiconductor II

An impurity creates a very weakly bound state: $$E = -\frac{m_e}{m\varepsilon^2} R_E = -0.01 \text{eV (in Ge)}$$ $r = 4$ nm (vs $r = 0.5$ Å in H).

Binding energy smaller than room temperature (0.026 eV).

Density of states

All donor/acceptor states at the same energy: $$g_A = N_A \delta(E_A),\quad g_D = N_D \delta(E_G - E_D)$$

Finite temperature

Intrinsic semiconductor: $$E_F, E_G - E_F \gg k_BT$$

For electrons: $$f(E) \approx e^{-(E-E_F)/k_BT}$$ holes: $$f(E) \approx 1- e^{(E-E_F)/k_BT}$$

Number of carriers

Electron concentration: $$ n_e = V^{-1} \int_0^\infty f(\varepsilon)g_e(\varepsilon) d \varepsilon$$

Result: $$ n_e = N_{\rm C}(T) \exp((E_{\rm F} - E_{\rm G})/kT), \quad n_h = N_{\rm V}(T) \exp(- E_{\rm F}/kT)$$ $$ N_{\rm C}(T) = 2(2\pi m_e kT/h^2)^{3/2} \quad N_{\rm V}(T) = 2(2\pi m_h kT/h^2)^{3/2} $$

Charge conservation: $$n_e - n_h + n_D - n_A = 0$$

Summary

Doping shifts the Fermi level and creates charge carriers of a dominant type.