Current-Carrier Transport with Space Charge in Semiconductors

Differential equations are given for a general formulation of current-carrier transport that includes space charge. Arbitrary dependences of diffusivities and magnitudes of drift velocities on electrostatic field are considered, and extension is made for applied magnetic field. Though excess electron and hole concentrations are not equal, the small-signal recombination rate depends on a single lifetime, the "diffusion-length lifetime," ${\tau }_0$. The formulation is applied to one-dimensional drift with recombination for an injected pulse of electron-hole pairs. The exact electron and hole distributions are obtained in closed form for the linear small-signal case. The condition for linearity is given; it is usually the same as that for substantially unperturbed applied field, $E_0$. There are two principal types of solution, essentially according to whether ${\tau }_0$ is larger or smaller than the dielectric relaxation time, ${\tau }_d$. For ${\tau }_0>{\tau }_d$, the electron and hole distributions in not too strongly extrinsic material are ultimately similar Gaussian distributions displaced by the "polarization distance," $x_P$, the distance electrons and holes drift apart in time ${({{\tau }_d}^{-1\mathrm{}}-{{\tau }_0}^{-1\mathrm{}})}^{-1\mathrm{}}$. These distributions drift at a velocity that differs from the ambipolar velocity by an amount which, besides being small for small $\frac{{\tau }_d}{{\tau }_0}$, vanishes for equal mobilities. They spread, exhibiting an apparent diffusion. A "pseudodiffusivity," $D_v$, is defined. For ${\tau }_0\gg {\tau }_d$ and constant mobilities, $D_v$ is proportional to $\frac{{\tau }_d{E_0}^2}{{{\sigma }_0}^2}$, with ${\sigma }_0$ the conductivity. The ambipolar diffusivity and $D_v$ are additive. They are equal in intrinsic material for $E_0$ equal to $\frac{\mathrm{kT}}{e}$ divided by the Debye length ${\left(\frac{\mathrm{kT}\epsilon }{8\pi n_i{e}^2}\right)}^{\frac{1}{2}}$, or 10 v/cm for silicon at 300°K. An extension to a nonlinear case involving high-level injection is given; concentration-dependent $D_v$ and velocity function are defined. For sufficiently strongly extrinsic material and ${\tau }_0>{\tau }_d$, the minority carriers drift in a delta pulse that leads the majority carriers distributed in an exponential tail of characteristic length $x_P$, which may be quite large. For nonconstant mobilities and ${\tau }_0>{\tau }_d$, ambipolar velocity in the majority-carrier or "reverse" direction may occur. For ${\tau }_d>{\tau }_0$, the other principal type of solution gives distributions that in general (and for constant mobilities) drift in the reverse direction. Involving also regions of local carrier depletion, and thus generation as well as recombination, these distributions may persist for times long compared with ${\tau }_0$, being attenuated then with time constant ${\tau }_d$.