Solution

5.3 Solution

5.3.1 Solution with Charge Neutrality Constraint

Analytical Approach

When the charge current flows through the interface, the spin accumulation in the semiconductor appears. When K₁=K₂=1 (c.f. Table 5.1), the charge neutrality is restored. To solve the spin transport equations analytically for the structure shown in Figure 5.2, the general solution for the spin density deviation δs in both the FMS and the Si sides must be assumed according to [173, 190]

(- x - W- ) ( x + W-) δsFMS = a1ND exp -------2- + a2ND exp ----2-- , Ld Lu

(5.31a)

( W-) ( W-) δsSi = b1ND exp --x----2- + b2ND exp x-+--2- , Ld Lu

(5.31b)

where a₁(a₂) and b₁(b₂) are the constants. The spin density can be expressed as s = δs + (n_↑^eql - n_↓^eql) (c.f. Equation 5.18a to Equation 5.18c). The electron concentration is n = ND. Based on these two, one can derive the up- and down-spin concentrations and the spin current density (Js) as well from Equation 5.22a and Equation 5.22b.

The external electric field (Ẽ) modifies the intrinsic spin diffusion length Li in any semiconductor [173]. Two distinct spin diffusion lengths (L_u is the up-stream, and L_d is the down-stream) characterize the spin motion, and those strongly depend on Ẽ

Lu = ----------┌---1--------------, ││ ( )2 -q|˜E-|-+ ∘ -q|˜E|-- + -1- 2KBT 2KBT L2i

(5.32a)

---------------1--------------- Ld = ┌│ (-------)2-------, q|˜E | │∘ q|˜E| 1 - -------+ ------- + -2- 2KBT 2KBT Li

(5.32b)

which is related to the intrinsic spin diffusion length Li by

L2 = LuLd. i

(5.33)

Li is related to the spin relaxation time via [173]

∘ ----- Li = Dn τs.

(5.34)

In order to solve the transport equations, one can formulate the boundary conditions as elucidated below.

In a doped semiconductor with homogeneous carrier concentration the up(down)-spin chemical potential [173, 174, 181], represented by M_↑ (M_↓), is related to its concentration. The spin chemical potential remains continuous at the interface, hence [173]

$( n0- ) ( n0+ ) ln -------↑------ = ln ---↑--- - G--, ---0.5(1 +-P-)ND-- ◟---0.5ND◝◜-----VT◞ ◟ Chem. p◝o◜t. (FMS) ◞ Chem. pot. (Si)$ (5.35a)

$(-----n0-↓------) (--n0↓+--) G-- ln 0.5(1 - P )N = ln 0.5N - V . ◟--------◝◜-----D-◞ ◟-------◝D◜------T◞ Chem. pot. (FMS) Chem. pot. (Si)$ (5.35b)

Here, G is a constant describing as the spin chemical potential drop at the interface, and is responsible for spin injection from the FMS to the Si side. Therefore, G must depend on both the bulk spin polarization P and the electric field Ẽ.
In the language of chemical potentials, the expression for the up(down)-spin current density can be reformulated as [173, 190]

$J↑ = qμnn↑ dM-↑, dx$ (5.36a)

$J↓ = qμnn↓ dM-↓. dx$ (5.36b)

As the up(down)-spin current density J_↑ (J_↓) is continuous at the interface, the above equations yield to

$0- dM-0↑-- 0+ dM-↑0+- qμnn↑ dx = qμnn ↑ dx ,$ (5.37a)

$0- 0+ 0- dM-↓-- 0+ dM-↓-- qμnn↓ dx = qμnn ↓ dx .$ (5.37b)
The spin density deviation δs is zero at the left and the right boundary, maintaining the thermal equilibrium. Hence,
$( ) ( ) δs x = - W-- = δs x = W-- = 0. 2 2$ (5.38)

In order to derive the solution for the spin deviation and the spin current density, one must solve for 5 unknown parameters viz. a₁, a₂, b₁, b₂, and G. The spin injection efficiency can be formulated from two different approaches [173], one by the polarization of the spin current density (α) and the other by the polarization of the spin density (β)

Js α = J-, n

(5.39)

β = s-. n

(5.40)

Therefore, at the silicon junction, α₀ = J (0+)
-sJn- and β₀ = s(0+)
n(0+) .

The analytical expressions for α₀ and β₀ are cumbersome. The simplified expressions, valid for lower values of the bulk spin polarization P are [190]

Ld α0 = P ------------2----, Ld + (1 - P )Lu

(5.41)

( Lu ) β0 = 1 - --- α0. Ld

(5.42)

The expression for constant G is

G = VT ln (1 + β0P ).

(5.43)

Equation 5.42 indicates that α₀ is always larger than β₀ irrespective of the value of the applied field. If the electric field is very small, both the up- and down-spin diffusion lengths L_u and L_d tend to the intrinsic value and hence from Equation 5.41, α₀ → -P---
2-P2 . This also means β₀=0 and from Equation 5.43 G=0. At the strong field limit, the electrons move with the drift velocity and so does the spin polarization [173]. L_d is simply the distance over which the carriers move within the spin lifetime. Thus, when the electric field increases, L_d → ˜
qK|ET|
B Li², L_u → KBT˜-
q|E| [173]. This makes the ratio LLu-
d →0 as Ẽ →∞, and both α₀ and β₀ tend to reach a saturation value fixed by the bulk spin polarization P at the ferromagnetic semiconductor. The maximum and minimum values of the mentioned parameters are listed in Table 5.2. Therefore, for small values of P, the spin injection efficiency in Si can be predicted analytically.

Figure 5.3: The analytically calculated spin densities in the channel are shown, when the applied voltage (U_c) is used as a parameter. The bulk spin polarization in the ferromagnetic semiconductor (FMS) is P=20%.

Figure 5.4: The spin current density (α₀) and spin density (β₀) injection efficiencies at the Si interface are shown as a function of the applied electric field Ẽ. Lines→ theory and dots→ simulation (P=10%).

Figure 5.5: The spin current density (α₀) and spin density (β₀) injection efficiencies at the Si interface as a function of the applied electric field Ẽ (c.f. Figure 5.4) is shown (P=50%).

Figure 5.6: A plot of M_up = VT ln

and M_down = VT ln

through the bar (P=10%, |qE˜|
KBT-

=2μm^-1 where Ẽ is the applied electric field) is depicted, showing a discontinuity at the junction, which gives the term G (c.f. Equation 5.35a and Equation 5.35b). n_↑^eql (n_↓^eql) is the up(down)-spin concentration at the thermal equilibrium.


Parameters	Minimum value	Maximum value

α₀		P
β₀	0	P
G	0	VT (1 + P²)

Table 5.2: The spin injection parameters with their optimum values are shown.

Comparison between Analytical and Simulated Results

Figure 5.3 shows how spin piles up at the junction with the applied voltage U_c, when K₁ = K₂=1 (c.f. Table 5.1), and the bulk polarization P value is low (P = 20%). The figure depicts how the applied bias improves the spin injection efficiency.

Even though for low values of bulk polarization P the analytical model for the spin current density (spin density) injection efficiency α₀ (β₀) is quite precise (c.f. Figure 5.4), as soon as the value of P increases the analytical solution is no longer accurate enough and the error increases significantly (c.f. Figure 5.5). Therefore, it is also important to solve the drift-diffusion equation set numerically to lift the restriction to high values of P. It is noted in Figure 5.5 that the α₀ and β₀ values remain upper limited by P. Nevertheless, once up(down)-spin concentration is calculated from the simulations, the spin chemical potential drop G (c.f. Equation 5.35a and Equation 5.35b) can be obtained as described in Figure ??.

5.3.2 Solutions without Charge Neutrality Constraint

Figure 5.7: The spin density accumulation near the junction over a channel of 4μm is shown, when the current density (J_n) is fixed to 23.4MA/m² with P=20%. K₁ is used as a parameter (c.f. Table 5.1). λD represents the Debye length (c.f. Equation 5.30).

Figure 5.8: The variation of the electric potential through the channel is described, related to Figure 5.7.

Figure 5.9: The spin current density through the channel is shown, with the same conditions as in Figure 5.7. With the notations as described in Equation 5.16b and Equation 5.19b, the direction of the spin current is from the ferromagnetic semiconductor FMS towards Si.

Figure 5.10: The spin density accumulation through the channel is depicted, with the same conditions as in Figure 5.7. Both K₁ and K₂ are used as parameters (c.f. Table 5.1).

Figure 5.11: The spin density and the spin current density injection efficiencies (β_D and α_D respectively) are shown, taken at a Debye length away from the interface towards Si, with the same conditions as in Figure 5.7.

By varying K₁, K₂ (c.f. Table 5.1), one can remove the charge neutrality constraint. Charge injection to silicon or charge release always cause a non-zero current (hence the flow of charge) through the junction, even at the absence of an external electric field. This flow of charge causes spin accumulation/depletion at the junction. Therefore, using current as the external control parameter, rather than the applied voltage, is more convenient, because the current is always constant throughout the channel. On the other hand, if one uses voltage as the control parameter, one has to consider the potential profile in different parts of the conducting channel.

A constant current can be maintained in the channel by tuning the applied voltage U_c at certain values for K₁, K₂. At a constant current the distribution for the spin density deviation, the spin current density, and also the potential profile are now analyzed. Strong nonlinear effects due to the conductivity variation in the space-charge layer close to the interface cause deviations of the compensating voltage, which has to be properly considered.

It is noticed that the spin density deviation δs behaves differently at the interface and in the bulk of silicon, c.f. Figure 5.7. When K₁>1, δs gradually piles up in the bulk FMS and drops down in the bulk silicon, compared to the charge neutrality condition. On the other hand when K₁<1, δs gradually drops down in the FMS bulk and piles up in the Si bulk. This phenomenon happens due to the differences in the material conductivities (proportional to the doping concentration) and the electric field Ẽ at the bulk, which eventually alters the spin diffusion lengths both in the FMS and Si part of the channel. However, the behavior at the interface is noted to be completely different. When K₁>1 (K₁<1), δs develops a dip (peak) at the FMS interface followed by a sharp peak (dip) at Si interface. These features can be correlated with the presence of charge depletion (accumulation) at the ferromagnetic/nonmagnetic interface, which results in the formation of a potential profile with a barrier for electrons (c.f. Figure 5.8). Indeed, these interface effects give rise to a very small alteration in the spin current density Js (particularly for K₁=5) as observed in Figure 5.9. Nonetheless, the interface effects only persist up to the classical charge screening length λD (c.f. Equation 5.30), and completely vanish beyond that limit.

In order to investigate if the alteration of K₂ has an additional effect on the observed results, one can look into Figure 5.10. It is observed that under the constraint K₁=K₂ the charge neutrality condition is restored. This is the reason why the additional interface effects on the spin density deviation δs is noted to be absent compared to Figure 5.7. When K₁≠K₂, the charge neutrality condition is violated at the interface. It is noted that when K₂<1 (K₂>1), δs develops a sharp dip (peak) in the FMS side and sharp peak (dip) in the silicon side for K₁=1. However, Figure 5.10 also depicts how the spin signal remains unaltered in the silicon bulk if the value of K₁ is kept fixed and the value of K₂ is varied. This confirms that the tuning of the parameter K₂ does not impact the bulk spin signal in silicon.

The estimated values of α_D and β_D, which are corresponding α and β at a distance λD away from the interface in silicon, are shown in Figure 5.11. It is revealted that α_D remains greater than β_D, and both are enhanced when spin is injected from a charge depleted ferromagnetic semiconductor compared to the charge neutral one. However, both α_D and β_D are always upper limited by the bulk spin polarization P.

[next] [prev] [prev-tail] [front] [up]