Forward biased pn junction band diagram

Forward bias means connecting the p side to the positive terminal and the n side to the negative terminal. The applied voltage $V$ opposes the built-in potential $V_0$, lowers the barrier to $e(V_0 - V)$, and allows a large diffusion current of majority carriers.

Energy band diagram

In equilibrium $E_F$ is flat and the bands bend by $eV_0$. Under forward bias the n-side bands are raised by $eV$ relative to the p side (the n side is at lower potential), so the Fermi levels separate by $eV$ and the bending is reduced to $e(V_0 - V)$.

 Equilibrium            Forward bias V
 p          n           p           n
 Ec_                    Ec__
    \___ Ec                 \__ Ec
 EF ------- EF          EFp ---   -.- EFn
 Ev_            barrier:      eV gap
    \___ Ev   eV0       Ev__
                            \__ Ev
                         barrier e(V0-V)

Mathematical expressions

1. Barrier and width

$$W = \sqrt{\frac{2\varepsilon(N_a+N_d)(V_0-V)}{eN_aN_d}}$$

2. Minority carrier injection (law of the junction). Using the Boltzmann relation across the lowered barrier:

$$p_n(0) = p_{n0}\exp\left(\frac{eV}{kT}\right),\qquad n_p(0) = n_{p0}\exp\left(\frac{eV}{kT}\right)$$

with $p_{n0} = n_i^2/N_d$ and $n_{p0} = n_i^2/N_a$.

3. Excess carriers decay by recombination in the neutral n region:

$$\Delta p_n(x) = \Delta p_n(0)\,e^{-x/L_h},\qquad L_h = \sqrt{D_h\tau_h}$$

4. Diffusion current of holes at the edge $x = 0$:

$$J_h = -eD_h\frac{d\Delta p_n}{dx}\Big|_{0} = \frac{eD_h}{L_h}\,\frac{n_i^2}{N_d}\left(e^{eV/kT}-1\right)$$

Similarly for electrons in the p region:

$$J_e = \frac{eD_e}{L_e}\,\frac{n_i^2}{N_a}\left(e^{eV/kT}-1\right)$$

5. Total current — Shockley diode equation

$$I = I_0\left(e^{eV/\eta kT}-1\right),\qquad I_0 = Aen_i^2\left(\frac{D_h}{L_hN_d}+\frac{D_e}{L_eN_a}\right)$$

where $\eta = 1$ for ideal diffusion current (Ge) and $\eta \approx 2$ when recombination in the depletion layer dominates (Si at low current).

Key results

• The current rises exponentially with V; for $V \gg kT/e$, $I \approx I_0e^{eV/kT}$.
• Each 60 mV increase in V raises the current about ten times at 300 K.
• Noticeable conduction starts at the cut-in voltage (~0.7 V Si, ~0.3 V Ge).