Injected Minority Carrier Charge

       Consider a long semiconductor bar which is doped uniformly with donor atoms, as shown in the Fig..1.

Fig.1  Light falls upon end of long n-type semiconductor bar

      The concentration n ≈ N_D and is independent of position as bar is uniformly doped.

       The end on which the light is incident is demoted as x = 0.

       Due to photoexcitation, the covalent bonds are broken at the surface, which is radiated. Due to this, the electron-hole pairs are generated and are injected into the bar at x = 0.

       The bar is n-type and free electrons are large in number. Thus generated electron-hole pairs mainly affect the minority charge concentration i.e. concentration of holes. Let us analyze the behaviour of steady state minority carrier concentration p against the distance x in the bar.

       Let p' = concentration of injected minority charge carriers.

Assumption : It is assumed that the concentration p' is much smaller than the doping level i.e. p' <<n. This condition stating that minority concentration is much smaller than majority concentration and is called low level injection condiction.

       It is known that J = (n μ_n + p μ_p ) q E, thus the drift current can be neglected compared to electron drift current.

       Now p = p' + p_o << n  hence the hole drift current can be neglected compared to electron drift current.

Note : It can be assumed that the hole current I_p is entirely because of diffusion.

       According yo continuity equation, the controlling differential equation for minority carrier concentration p is,

       The diffusion length for the holes is L_p and given by,

       Hence the differential equation for the injected hole concentration p' = p - p_o is given by,

       Solution of this differential equation is given by,

       where K₁, K₂  = Constants of integration 

       Now as x → ∞, the concentration can not become infinite. Thus the second term must be zero in the equation (4), for which K₂ = 0.

       At x = 0, the injected concentration is p'(0). Hence using in the equation (4),

       p'(0) = K₁ e^o     i.e. K₁ = p'(0),

       

       The equation (6) represents that the hoe concentration decreases exponentially with the distance. This is shown in the Fig. 2.

Fig. 2   Exponential behavious of hole concentration in a n-type semiconductor bar

1.1 Diffusion Length (L_p)

       From the graph, the diffusion length L_p represents the distance into the semiconductor at which the injected concentration drops to 1/e times its value at x = 0. Thus L_p is the diffusion length which is the distance x, at which the injected carrier concentration is 36.78 % of its value at x = 0.

1.2 Diffusion Currents

       The hole diffusion current is given by,

       Where A = Area of cross-section of the bar

                   J_p = Hole diffusion current density 

       From the diffusion current denstiy,

       Using equations (8) and (9) in (7),

          p(x) = p(0)   (at x = 0) = p'(0) + p_o                          .............. From Fig.1

       This is the diffusion hole current which decreases exponentially with distance.

Note : This equation plays an important role in the derivation of diode current equation.

       Let  I_n = Electron diffusion current = + A J_n 

        According to equation of charge neutrality which is preserved under low level injection, n' = p' i.e.

       Where no and p_o are the thermal equilibrium concentrations which are independent of the position x. Differentiating the equation (14),

       Thus the ratio of majority to minority diffusion current is given by -D_n /D_P.

       From the values of D_n and D_P, the magnitude of majority to minority diffusion current is about 2.11 for germanium and is about 2.61 for silicon.

1.3 Drift Currents

       In the bar considered above, the external voltage applied is zero and the bar is open circuited. Hence the net current through the bar must be zero everywhere.

.^..        Net current = Electron currents + Hole currents

                          0 = I_nd + I_n + I_p

       Where         I_nd = Electron drift current 

                           I_n = Electron diffusion current = -(D_n /D_P) I_p

                           I_p = Hole diffusion current

.^..                        I_nd  - (D_n/D_P) Ip+ Ip = 0

       Thus as Ip decreases exponentially, the electron drift current also decreases exponentially with the distance.

        To exist a drift current, there must exist an electric field E. Hence internally an electric field E is developed in the bar, for electron drift current to exist.

                                    J_pd  = p q μ_p  E = I_pd /A

Note : All the equations derived are based on the assumption that hole drift current is zero.

       But hole drift current I_pd can be obtained as,

                                   J_pd  = p q μ_p E = I_pd/A

       But as p << n, I_{pd <<} I_p which justifies the assumption that hole drift current is negligibly small and the entire injected minority carrier current is a diffusion current.

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