5.2 Spin Transport Equations

If the spin degree of freedom of the electrons and the holes is considered, one can write the same set of transport equations (as was mentioned in the section 5.1) for the corresponding up(down)-spin in silicon [173174134]. In this work the analysis for only the electron spin is discussed, as the analysis for the holes can be perfromed in an analogous way. Thus, the acceptor doping is neglected and the Poisson equation Equation 5.1 can be rewritten as

∇  ⋅ ˜ϵ ⋅ ∇V = q(n + n  - N  ),
                ↑    ↓     D
(5.15)

where the up(down)-spin concentration is expressed as n(n). The electron concentration n and the spin density s can be expressed as

n = n ↑ + n ↓,
(5.16a)

s = n  - n .
     ↑    ↓
(5.16b)

5.2.1 Spin Continuity Equation

The continuity equations for the up(down)-spin electrons in the channel including the spin-flip term [173] is revealed below. An additional electron generation-recombination process can be neglected because the considered system is n-doped only (unipolar).

1         δn↑   δn ↓   ∂δn ↑
-∇ ⋅ J↑ - ----+ ---- = -----,
q         τ↑↓    τ↓↑     ∂ ˜t
(5.17a)

1         δn↓   δn ↑   ∂δn ↓
-∇ ⋅ J↓ - ----+ ---- = ---˜-.
q         τ↓↑    τ↑↓     ∂ t
(5.17b)

Here, J(J) represents the up(down)-spin current density, τ↑↓-1 is the rate at which up-spin flips to down-spin, and τ↓↑-1 is the rate at which down-spin flips to up-spin. δn (δn) represents the up(down)-spin density deviation from its thermal equilibrium value, denoted by neql(n eql).

The structure under scrutiny consists of a ferromagnetic semiconductor (FMS), which inherently has an effective spin polarization, sharing a junction with silicon which does not. Therefore, one must introduce a non-zero bulk spin polarization P in the FMS. The total electron concentration at the thermal equilibrium is n = ND by considering all the dopants to be ionized. The equilibrium up(down)-spin concentration can thus be expressed as n()eql = 0.5(1 + (-)P)N D in the FMS, and n()eql = 0.5N D in Si. Then the up(down)-spin and the total spin density deviation (δs) can be expressed as

             eql
δn ↑ = n↑ - n↑ ,
(5.18a)

             eql
δn ↓ = n↓ - n↓ ,
(5.18b)

δs =  δn↑ - δn↓.
(5.18c)

The expressions for the electron (carrier) current density (Jn) and the spin current density (Js) are given by [173]

Jn =  J↑ + J↓,
(5.19a)

Js =  J↑ - J↓,
(5.19b)

One can relate the spin-flip rates τ↑↓-1 and τ ↓↑-1 with the total spin relaxation time by using the Matthiessen rule

 -1    - 1    -1
τs  = τ↑↓ + τ↓↑ .
(5.20)

The continuity equation for the spin current density is obtained by subtracting Equation 5.17b from Equation 5.17a and can be expressed as

1         δs    ∂δs
-∇  ⋅ Js ----=  ---.
q         τs    ∂ ˜t
(5.21)

5.2.2 Spin Drift-Diffusion

In accordance with Equation 5.14a, the up(down)-spin current density can be expressed as

             ( E        )            (δn  )
J ↑ = qn↑μ↑∇   -C,↑-- V  +  qD ↑NC ∇  ---↑ ,
                q                     NC
(5.22a)

             ( E        )            (δn  )
J ↓ = qn↓μ↓∇   -C,↓-- V  +  qD ↓NC ∇  ---↓ .
                q                     NC
(5.22b)

Here, the up(down)-spin diffusion coefficient is D (D), and the corresponding mobility is μ (μ). These are related by D = VT μ (D = VT μ). EC, (EC,) represents the conduction band edge for the up(down)-spin.

5.2.3 Discretized Form

In order to discretize the in general multi-dimensional transport equations the following steps are used.

Poisson Equation

To solve the partial differential equations numerically, they must be discretized. For that reason, the domain V where the equations are posed has to be partitioned into a finite number of sub-domains Vi, which are usually obtained by a Voronoi tessellation. In order to obtain the solution with a desired accuracy, the equation system is approximated in each of these sub-domains by algebraic equations. The unknowns of this system are approximations of the continuous solutions at the discrete grid points in the domain [182]. It has been found to be advantageous to apply the finite boxes discretization scheme for semiconductor device simulation [182] [185]. This method considers the integral form of the equation for each sub-domain, which is the so-called control volume Vi associated with the node the point i.

One can rewrite the Poisson equation (c.f. Equation 5.15) by using the Gauss’ integral theorem.

∫                  ∫
   ∇  ⋅ (˜ϵ∇V )dV˜ =   q(n - ND )dV˜,
∫ V                ∫V
               ˜                  ˜
    (˜ϵ∇V ) ⋅ ˆndA =    q(n - ND )dV .
  ∂V                V
(5.23)

where ˆn is the outward pointing local normal vector of the enclosing surface Ã.

Finally, the discretized equation for point i with neighbor points j can be written implicitly as

∑        - ( V  - V )
       - ϵij  -j----i Ai,j = q(ni - ND,i)˜Vi,
   j∈Ni        di,j
(5.24)

where V i(V j) is the potential at the ith(jth) node, N D,i is the doping at the ith node, d i,j is the distance between ith and jth node points, and A i,j is the interface area between the domains Vi and Vj. ˜Vi is the volume of the domain Vi. For a position dependent permittivity, one can use an average, e.g. ϵij = ˜ϵi+˜ϵj
 2, where ˜ϵ
i (˜ϵ
 j) is the permittivity at the node point i (j).


PIC

Figure 5.1: The two-dimensional representation of the Voronoi box is shown. The domain ˜
Vi contains the mesh point i and is surrounded by six adjacent mesh points. Ai,j represents the boundary between the box around the ith node and its neighbor j. di,j implies the distance between the ith and the jth nodes.


A two-dimensional realization of the above mentioned box integration process can be found in Figure 5.1. In this case, V˜i representes the area surrounding the ith node, di,j represents the distance between ith and an adjacent jth node point, and Ai,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, ∂n↑(↓)
--∂˜t- = 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.

∫                ∫  (          )
  (∇  ⋅ J )d˜V = q     δn↑---δn↓- d˜V ,
 V      ↑          V     2τs
∫                ∫  ( δn  - δn )
   (J↑ ⋅ ˆn)dA˜ = q    --↑-----↓- d˜V .
 ∂V                V     2τs
(5.25a)

∫                ∫  ( δn  - δn )
  (∇  ⋅ J↓)d˜V = q     --↓-----↑- d˜V ,
∫V               ∫ V     2τs
            ˜       ( δn↓---δn↑)  ˜
   (J↓ ⋅ ˆn)dA = q        2τ      dV .
 ∂V                V       s
(5.25b)

Then, considering the same Voronoi box around the ith node,

∑                  ( δn↑,i---δn↓,i) ˜
   j∈Ni J↑,i,jAi,j = q     2τ       Vi,
                           s
(5.26a)

∑                    ( δn  -  δn  )
       J ↓,i,jAi,j = - q  --↑,i-----↓,i- ˜Vi.
   j∈Ni                    2τs
(5.26b)

Here, J,i,j (J,i,j) represents the up(down)-spin current density flowing from ith to jth node. δn ,i (δn,i) represents the up(down)-spin density deviation at the ith 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 [187183]. 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 EC, = EC,. 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 neql and n eql.

       qμ-   VT (                          )
J↑,i,j = ---n,i,j--- n↑,jB (△ (1i,)j ) - n↑,iB(- △ (i1,j) )
          di,j
       ( qμn,i,jVT-)    ---
     -     d      (0.5N D,i,j)(Pj - Pi),
            i,j
(5.27a)

        --
       qμn,i,jVT-(        (1)             (1))
J↓,i,j =    di,j    n↓,jB (△ i,j ) - n↓,iB(- △ i,j )
       (  --     )    ---
     -   qμn,i,jVT--(0.5N D,i,j)(Pj - Pi),
           di,j
(5.27b)

with

  (1)           ( Nc,i)
△ i,j = △i,j + ln  ---- ,
                  Nc,j
(5.28a)

       Vj - Vi
△i,j = -------.
         VT
(5.28b)

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

             x
B (x) = -----------.
        exp (x) - 1
(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


SVG-Viewer needed.


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 = -W-
 2) is grounded, and the right boundary (x = W-
 2) is under the voltage bias (Uc). 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, -W-
 2<x<0) and silicon (Si, 0<x<W-
2), 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.




Parameters

Values



Electron mobility in Si

μn=1400cm-2V-1s-1 [50]

Donor doping in Si

ND=1016cm-3

Intrinsic spin diffusion length

Li=1μ[181]

Temperature

T=300K

Doping ratio between the ferromagnetic semiconductor and Si

K1

Ratio between the electronic density of states among the ferromagnetic semiconductor and Si

K2




Table 5.1: The simulation parameters for the spin drift-diffusion equations are listed.

For a one-dimensional channel ˜Vi = d   +d
-i,i-12-i,i+1, where di,j is the distance between ith and jth 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 1
q [189]. λD in an n-doped semiconductor can be expressed as

      ∘ ------
        ϵSiVT-
λD =     qND  ,
(5.30)

where ϵSi 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. di,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 ND,i,j = ND,i+ND,j
    2 can be considered. The mobility is assumed to be homogeneous in the channel.