Characteristic lengths for three-carrier transport with spin-flip and electron-hole recombination

The exact solution of the linearized, steady-state transport equation for three-carrier systems, such as can occur for semiconductors and ionic conductors, is constructed starting from the near-equilibrium entropy-production requirements of irreversible thermodynamics. Three characteristic modes are found, one associated with electrostatic screening (which is often neglected), and two modes associated with diffusion and “reactions.” For a spintronics model with up and down electrons and unpolarized holes, the “reactions” are spin-flip and electron-hole recombination. We discuss how the variations in carrier density, diffusivity, recombination rate, and spin relaxation time affect the characteristic lengths. We apply these modes to study spin-polarized surface photoabsorption.

Figure 1

Processes considered in the model. The holes are considered to be unpolarized.

Figure 2

Electric potential $\delta \varphi \left(x\right)$ as a function of distance from the illuminated surface. The (Dember) voltage at the surface is about 4.2 mV.

Figure 3

Charge density $\delta \rho \left(x\right)$ as a function of distance from the illuminated surface. Charge density profile ensures overall electroneutrality of the system.

Figure 4

Variation of carrier number densities $\delta p\left(x\right)$ and $\delta n_{\uparrow }\left(x\right)$ as a function of distance from the illuminated surface. The distinction between spin-up and spin-down densities cannot be seen on this scale, so only spin-up density is indicated.

Figure 5

Dimensionless spin polarization $p_s$, from (68), as a function of distance from the illuminated surface. The polarization simply decays exponentially with the distance from the surface.

Figure 6

Carrier number current densities as functions of distance from the illuminated surface.

Figure 7

Spin current density $j_s=j_{\uparrow }-j_{\downarrow }$ as a function of distance from the illuminated surface.