Generation of carriers and conductivity

Generation of electrons and holes

In a pure semiconductor such as Si, every atom forms four covalent bonds with its neighbours. At 0 K all valence electrons are held in bonds; the valence band is full, the conduction band is empty, and the crystal is an insulator.

At a higher temperature the lattice vibrates. When a bond electron gains energy $\ge E_g$ (1.1 eV for Si) from thermal vibration (or from a photon, $h\nu \ge E_g$), it breaks free and jumps to the conduction band. This leaves an empty bond, a hole, which acts as a positive charge $+e$ in the valence band. So carriers are always produced in pairs: electron–hole pair generation.

 Ec ----- e-  (free electron) ----
          ^
          |  thermal energy >= Eg
          |
 Ev ----- h+  (hole) -------------

At the same time, free electrons meet holes and recombine. In equilibrium the generation rate equals the recombination rate, giving a steady concentration $n = p = n_i$, where $n_i \propto T^{3/2}e^{-E_g/2kT}$ grows rapidly with temperature.

Conduction

When a field $\mathcal{E}$ is applied:

• Free electrons in the CB drift opposite to $\mathcal{E}$.
• A neighbouring bond electron jumps into a hole, so the hole moves in the direction of $\mathcal{E}$. Holes behave like positive particles.

Both movements give current in the same direction (along $\mathcal{E}$), so the currents add.

Derivation of conductivity

Let $n$, $p$ be the concentrations and $v_{de}$, $v_{dh}$ the drift velocities. The current density due to electrons is $J_e = env_{de}$, and due to holes $J_h = epv_{dh}$. With $v_{de} = \mu_e\mathcal{E}$ and $v_{dh} = \mu_h\mathcal{E}$:

$$\begin{aligned} J &= J_e + J_h = en\mu_e\mathcal{E} + ep\mu_h\mathcal{E}\\ &= e(n\mu_e + p\mu_h)\mathcal{E} \end{aligned}$$

Comparing with Ohm's law $J = \sigma\mathcal{E}$:

$$\sigma = e(n\mu_e + p\mu_h)$$

Special cases:

• Intrinsic: $\sigma_i = en_i(\mu_e+\mu_h)$
• n-type: $\sigma \approx eN_d\mu_e$
• p-type: $\sigma \approx eN_a\mu_h$

Example: intrinsic Si, $\sigma_i = 1.45\times10^{10}\times1.6\times10^{-19}\times1800 \approx 4.2\times10^{-6}$ S cm⁻¹.