Simple general analytical solution to the minority carrier transport in heavily doped semiconductors

Abstract

A rigorous analytical evaluation of minority carrier current in a heavily doped region (such as emitter) of a semiconductor device is presented that includes position-dependent band-gap narrowing, position-dependent mobility, and position-dependent lifetime. In addition, the analysis takes into account the possible finite surface recombination velocity.

A New Solution for Minority-Carrier Injection into the Heavily Doped Emitter of Silicon Devices

“…Here, the empirical parameter y 0 is determined such that the error of this approximated analytical solution is minimized. As a result, with the use of CN s instead of the constant C S , being N s -independent [4], Selvakumar's approach [4] can now be performed. Then, from (8), (10) and (11), we obtain for u r y [or J S y an analytical solution we denote by the zeroth-order approximated solution as that given in [12] u rY 0 y sinh Py h cosh Pyasinh P1 h cosh P1 Y 13…”

“…In addition, this constant C S is an empirical parameter, determined by a particular functional dependence of the minority-carrier lifetime tN. This approach [4] or the similar one [7,10,12] may be questionable because CN is a function of y and N s .…”

“…Second, the minority-carrier continuity equation is defined by [4,11] dJNadx ÀrN À r 0 NatN Àr 0 N uNatN Y that can also be expressed in terms of the function FN CN 2 determined in (3), as…”

“…In our recent papers [12], we have used Selvakumar's approach [4] to obtain analytical solutions for JyY N s Y WY S and for the effective minority-carrier transit time, assuming that the function CN, defined in (2) and given in (6), is equal to a constant C S , independent of xor y and N s . In addition, this constant C S is an empirical parameter, determined by a particular functional dependence of the minority-carrier lifetime tN.…”

“…First, the minority-carrier current density JN-equation is defined by [4,11] JN Àgq r 0 N DN duadx Y where q 1X602 Â 10 À19 C is the hole charge, g is identical to 1 À1) for minority-hole (electron) carriers, injected into the n(p)-type heavily doped emitter region, respectively, and uN denotes a normalized minority-carrier density defined by uN rNar 0 N À 1 with rN being the minority-carrier density. Then, JN can also be expressed in terms of the transport parameter FN determined in (1), as…”