I. Minoroty carrier concentration profile.

For a PNP bipolar transistor, the minority hole density in the n-type base is given by

dp(x) = Dp_n sinh((W_b-x)/L_p)/sinh(W_b/L_p) - p_n0 sinh(x/L_p) / sinh(W_b/L_p).

Short Base:

Long Base:

II. Current density for a PNP transistor

Emitter currents: J_e = J_ep + J_en
J_ep = minority holes, from Emitter into Base
      = - q D_p dp/dx
      = (q D_p Dp_n/L_p) cosh((W_b-x)/L_p)/sinh(W_b/L_p) + (q D_p p_n0/L_p) cosh(x/L_p)/sinh(W_b/L_p)|_{x = 0}
      = (q D_p / L_p ) [ Dp_ncosh(W_b/L_p) + p_n0] / sinh(W_b/L_p)
where D_p is chosen from the Choice box.
J_en = minority electrons, from Base into Emitter
     = q D_n dn/dx
     = q (D_n/L_n) Dn_p
where,
Dn_p = n_p0 exp(V_eb/kT) = excess electron in the Emitter.

Values used in the applet:
D_n = 35 cm²/s = electron diffusion coefficient in the Emitter.
L_n = sqrt(D_n t_n),
t_n = 10^-8 s = electron lifetime in p-type Emitter

For the NPN transistor, D_n for Base is chosen from choice box, and
D_p=   12.5 cm²/s, t_p =   10^-7 s for minority holes in the Emitter.
Base Currents:  J_b = J_e - J_c

Collector Currents:

PNP:
J_c = J_cp + J_c0, where
J_c0 = J_c0 (hole leaking from B to C) + J_c0 (electron leaking from C to B).

• a = alpha = J_c/J_e
• b = beta = J_c/J_b = J_c / (J_e - J_c) = a / (1-a)
• For PNP transistor
• B = Base transport factor = J_cp / J_ep= fraction of collected minority carrier number to the injected number.
• g = gamma = Emitter injection efficiency = J_ep / J_e = J_ep / (J_ep + J_en) » N_e / (N_e + N_b), because J_ep is proportional to D_p/(W_b N_b) and J_en to D_n/(L_n N_e), and all other factors are common to both. The injection efficiency is nearly one if N_e >> N_b, Emitter doping level is much heavier than the Base doping level.