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 ≈ ND 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 μn + p μp ) q E, thus the drift current can be neglected compared to electron drift current.
       Now p = p' + po << n  hence the hole drift current can be neglected compared to electron drift current.
Note : It can be assumed that the hole current Ip 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 Lp and given by,
       Hence the differential equation for the injected hole concentration p' = p - po 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 eo     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 (Lp)
       From the graph, the diffusion length Lp represents the distance into the semiconductor at which the injected concentration drops to 1/e times its value at x = 0. Thus Lp 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
                   Jp = 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) + po                          .............. 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  In = Electron diffusion current = + A J
        According to equation of charge neutrality which is preserved under low level injection, n' = p' i.e.
       Where no and po 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 -Dn /DP.
       From the values of Dn and DP, 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 = Ind + In + Ip
       Where         Ind = Electron drift current 
                           In = Electron diffusion current = -(Dn /DP) Ip
                           Ip = Hole diffusion current
...                        Ind  - (Dn/DP) 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.
                                    Jpd  = p q μ E = Ipd /A
Note : All the equations derived are based on the assumption that hole drift current is zero.
       But hole drift current Ipd can be obtained as,
                                   Jpd  = p q μp E = Ipd/A
       But as p << n, Ipd << Ip 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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