Figure 5.1: The two-dimensional representation of the Voronoi box is shown. The domain contains the mesh point i and is surrounded by six adjacent mesh points. A_i,j represents the boundary between the box around the i^th node and its neighbor j. d_i,j implies the distance between the i^th and the j^th nodes.

A two-dimensional realization of the above mentioned box integration process can be found in Figure 5.1. In this case, representes the area surrounding the i^th node, d_i,j represents the distance between i^th and an adjacent j^th node point, and A_i,j is the interface length as shown.

Spin Continuity Equation

One can also apply the Gauss’ integral theorem to rewrite the continuity equations Equation 5.17a and Equation 5.17b. If steady-state analysis is under investigation, = 0. Again, in order to derive a rather simplified analytical solution for the spin density later in this chapter to make comparisons with the simulated results, τ_↑↓ = τ_↓↑ = 2τs is considered (c.f. Equation 5.20). Because each spin flip contributes to relaxation of both up- and down-spin, the spin relaxation becomes twice as fast.

(5.25a)

(5.25b)

Then, considering the same Voronoi box around the i^th node,

(5.26a)

(5.26b)

Here, J_↑,i,j (J_↓,i,j) represents the up(down)-spin current density flowing from i^th to j^th node. δn _↑,i (δn_↓,i) represents the up(down)-spin density deviation at the i^th node.

Spin Drift-Diffusion

The Scharfetter-Gummel (SG) discretization scheme for the carrier current density provides an optimum way to discretize the drift-diffusion equation for particle transport [186]. This discretization scheme can be applied as well to rewrite the expression for the up(down)-spin current densities (c.f. Equation 5.22a and Equation 5.22b) into their discretized form. One can discretize the spin drift-diffusion equations by following the steps as explained in [187, 183]. In the non-degenerate transport regime, the diffusion constant, the mobility, and the conduction band edge are spin-independent [174]. Therefore, D_↑ = D_↓, μ_↑ = μ_↓, and E_C,↑ = E_C,↓. For simplicity one can assume a homogeneous position-independent electronic conduction band edge in the channel. Again, the bulk spin polarization P term can be inserted in the SG discretization scheme via the expressions of n_↑^eql and n _↓^eql.

(5.27a)

(5.27b)

with

(5.28a)

(5.28b)

Here, P_i is the bulk spin polarization at the i^th node (=P for the FMS, =0 for silicon), μ_n,i,j represents the average mobility between i^th and j^th node, and a good approximation is μ_n,i,j = . The effective density of states of electrons (NC) in i^th (j^th) node is represented as N _c,i (N_c,j). The average equilibrium electron concentration (i.e. the donor concentration) between two neighboring points i and j is represented by N_D,i,j, and thus the average equilibrium up(down)-spin concentration can be considered as 0.5N_D,i,j. The expression for the term N_D,i,j is described in the next section. B denotes the Bernoulli function

(5.29)

When spin transport only in a silicon bar is considered, the electron density of states NC becomes position-independent, and thus △_i,j⁽¹⁾ = △ _i,j.

5.2.4 Transport Channel

Figure 5.2: The schematic shows the simulation set up for the spin injection in Si from a ferromagnetic semiconductor (FMS). The left boundary (x = -) is grounded, and the right boundary (x = ) is under the voltage bias (U_c). The interface is shown as a dotted line. The doping (effective density of states) in the Si side is ND (NC), and in the FMS is K₁ND (K₂NC).

In order to predict the impact of a space-charge layer on the spin transport in silicon, a one-dimensional transport channel is assumed. However, the same predictions will be valid for a multi-dimensional structure as well. The semiconducting channel of length W contains a semiconductor ferromagnet (FMS, -<x<0) and silicon (Si, 0<x<), sharing a junction at x=0 (c.f. Figure 5.2). Now, to enable a violated or restored charge neutral source of spin injection in silicon, the following two assumptions are under consideration in this work.

• The ferromagnetic semiconductor FMS can be considered to be doped to a concentration, which is a factor K₁ of the value in Si. Thus, when K₁>1 (K₁<1), one can introduce a charge accumulated (depleted) source of spin injection into Si (c.f. text in Figure 5.2). Indeed when K₁=1, the charge neutrality condition is restored.
• One can consider a homogeneous electronic density of states in the FMS, which is a factor K₂ of the homogeneous density of states in Si. Thus, when K₂>1 (K₂<1), one can introduce a charge depleted (accumulated) source of spin injection into Si.

For a one-dimensional channel = , where d_i,j is the distance between i^th and j^th nodes, and A _i,j=1. As the space-charge effect on the spin transport is under scrutiny the charge screening length, also known as the Debye length (λD), puts a strict limitation on the choice of the mesh size near the interface in both the ferromagnetic and silicon sides [188]. The Debye length relates to the measure of a charge carrier’s net electrostatic effect, and it is the length over which the carrier density in a semiconductor changes by a factor  [189]. λD in an n-doped semiconductor can be expressed as

(5.30)

where ϵ_Si is the silicon permittivity. The value of λD is 40nm for the parameters listed in Table 5.1. Thus, a good approximation is to use a step value (i.e. d_i,j) of 10nm near the interface in this particular simulation set up. In such a case, the node points i and j are close enough and N_D,i,j = can be considered. The mobility is assumed to be homogeneous in the channel.