The spin density distribution s in the channel is shown in Figure 5.13, with the parameters as listed in Table 5.1. When the applied bias is positive (negative), the electric field increases (decreases) the effective spin diffusion length [173]. This is why the signal s is high in the channel when U_c>0. For the same conditions, the spin current density Js through the channel is shown in Figure 5.14. With the notations as described in Equation 5.16b and Equation 5.19b, the direction of the spin current is towards the positive x-axis (c.f. Figure 5.12). The figure also depicts the direction of the drift and diffusion components of Js, and it is observed that Js is higher when the spin flow is in the direction of the electron (charge) flow (i.e. U_c>0). In contrast to Figure 5.3 one observes the non-vanishing value of s (and hence Js) in the channel even when U_c=0. This signifies that the spin flow due to its diffusive component is still possible in this structure, even when Ẽ is absent.

Figure 5.15 examines the variation of the spin current density Js for different values of the spin density polarization at the injecting point (β₀), when no voltage is applied (i.e. U_c=0, and thus the carrier current is absent). Js through the channel increases with β₀ proportionally, reaching its maximum when β₀ is maximum (i.e. β₀=1). Thus from the above discussions it is confirmed that the spin behavior is completely decoupled from charge, and the spin diffusion lengths (L_u and L_d) are solely determined by the intrinsic spin diffusion length Li and the electric field Ẽ.

Spin Extraction and Critical Current Density

A functional spintronic device also involves the reverse process of spin injection, i.e. the extraction of spin-polarized electrons from the semiconductor to a ferromagnetic structure. One can investigate spin extraction from a non-magnetic semiconductor like silicon into the ferromagnet in the regime where the degree of spin polarization is very high. For the analysis of the spin extraction phenomenon, the detailed structure of the interface is not very important, and one can solve the spin transport equations for the semiconductor region instead [191]. The electrons, flowing from the silicon side and entering in the magnet, are supposed to be unpolarized. Now, if the structure is a ferromagnetic half-metal [103], then it accepts only one spin orientation (e.g. up-spin electrons). In such a case, a cloud of down-spin electrons is formed in silicon near the relevant boundary, as those can not enter in the ferromagnet unless undergoing a spin flip. The current in the bar increases the cloud, and eventually reaches its maximum, when the silicon region near the junction becomes completely depleted of the electrons with the same direction of spins (i.e. up-spins) as in the ferromagnet. This phenomenon is known as spin-blockade [191]. This signifies that a further increase of the steady-state current through the junction will not be possible.

In order to estimate this critical current density represented as J_c,cr, one can use the general solution of the spin density from Equation 5.44. The spin extraction scheme is portrayed in Figure 5.16. For the simplicity of analytical calculations one can assume the bar to be long enough to negate the impact of the right boundary. In such a case and for the structure as depicted in Figure 5.16, the up-stream diffusion length given by L^′ = solely determines the transport [191]. Then, one can assume the spin density for the spin extraction model in the form of [191] s = -A₁ exp , where A₁ is a positive quantity. Then from Equation 5.46,

(5.48)

Henceforth at the spin extraction boundary,

(5.49)

The spin current density and the charge current density are related by

(5.50)

Because the electric field Ẽ is homogeneous and linearly coupled to the total current density, it can be assumed that, the polarization for the spin current density α₀ is fixed by the ferromagnet and thus does not depend on the channel current. Then,

(5.51)

which eventually yields to

(5.52)

and therefore,

(5.53)

If the expression for the up-stream diffusion length is put in Equation 5.53, A₁ can be estimated as A₁ = . Since the maximum possible spin polarization can only be 100%, the maximum possible value for A₁ is the doping concentration ND, and one can write

(5.54)

Figure 5.17: The normalized critical current density is shown for spin-blockade as a function of the boundary current density spin polarization.

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Finally, one can write the expression for the critical current density

(5.55)

A fully polarized (unpolarized) spin current pertains to α₀=1 (0). In this range, J_c,cr decreases with increasing α₀. Equation 5.55 also reveals that spin-blockade of the current is more important in materials with long spin relaxation times (particularly Si), hence this phenomenon must be considered in order to design efficient silicon-based spintronic devices.

Figure 5.17 depicts the critical current density at the spin-blockade. J_c,cr increases slowly by decreasing α₀ from 1 to around 0.3. Therefore, the spin-blockade phenomenon is also important in junctions with ordinary ferromagnets.

5.4.2 Solution without Charge Neutrality Constraint

In order to lift the charge neutrality constraint, one can adjust the up(down)-spin concentrations n_↑0 and n_↓0 in such a manner that the net electron concentration at the left boundary n₀≠ND. It is hereby mentioned that by tuning up- and down-spin current densities J_↑0 and J_↓0 one can lift the charge neutrality constraint as well. For simplicity of analysis the former method is adopted. One can fix the charge density with a single parameter M_Ch (this term can be attributed to the charge chemical potential) as given below.

(5.56)

Now, if the maximum value for the spin density polarization β₀=1 (i.e. n₀=s₀) is considered

(5.57)

and thus making

(5.58)

Therefore, one can inject (release) up-spin and hence charge at the same time. This way it is possible to describe

• spin injection from a charge accumulated source (M_Ch>0),
• spin injection from a charge depleted source (M_Ch<0),
• restore charge neutrality (M_Ch=0).

Via Equation 5.57 a considerable spin and charge accumulation (depletion) at the interface can be introduced and spin will diffuse out of this region. Non-zero values of M_Ch always cause a flow of charge even though the applied voltage is zero. The carrier current density J_n, the spin density s, and the spin current density Js in the channel can be tuned by varying both M_Ch and the applied voltage U_c. One can set any non-zero value for M_Ch and adjust U_c to keep a fixed charge flow (J_n=constant) in the channel as described in Figure 5.18.

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Figure 5.18: The variation of the electron current density with M_Ch (c.f. Equation 5.56) and the bias voltage U_c is depicted, when the channel length is 4μm.

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Figure 5.19: The electron concentration is shown near the charge injection boundary (c.f. Equation 5.56). The current density is 11.9MAm^-2, and the channel length 4μm. λD represents the Debye length.

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Figure 5.20: The variation of the spin current density is delineated with the same conditions as in Figure 5.19.

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Figure 5.21: The spin current density is shown for up to 5 times the Debye length λD from the left boundary, when spin is injected from a charge neutral and charge accumulated source. The carrier current is absent. The left boundary for (1) is set with n_↑0=900ND and n_↓0=100ND, and for (2) c.f. Equation 5.57.

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Figure 5.19 depicts how the charge accumulation (depletion) causes the pile up (reduction) of the carriers near the left boundary and through the channel. As found previosuly, this pile up persists only up to the screening region characterized by the Debye length λD (c.f. Equation 5.30). The spin carriers follow a similar profile, which is shown in Figure 5.20 and illustrates that an abundance of spin carriers during the accumulation enhances the spin current density Js only close to the interface, while in contrast a lack of spin carriers in depletion causes a very strong diminution of the same, both at the interface and the bulk. In order to complete the comparison, the charge neutrality condition is also shown. Figure 5.20 reveals that under charge accumulation Js shows an upper threshold [181]. The amount of Js which leaks from the accumulation region almost does not change in high accumulation, regardless of the high values of the spin density s and Js near the interface. This means that an effort to boost Js by increasing the boundary spin density s₀ to inject more spin polarized electrons does not result in a substantial increment of Js in the bulk [192].

To scrutinize any further enhancement in the spin current density Js at the injection boundary or in the bulk due to a charge accumulation, the variation of Js for different values of spin density polarization β₀ at the injection boundary is shown in Figure 5.21. Indeed for β₀=1, Js close to the spin injection interface is significantly higher at charge accumulation compared to charge neutrality. In contrast, at a distance of about one Debye length from the interface, Js becomes the same to that at the interface under the charge neutrality conditions. Now, one must check the behavior of Js near the boundary and the bulk, when β₀<1. Figure 5.21 also shows if β₀=0.8, it is not possible to obtain a value for Js in the bulk as high as for β₀=1, even under higher charge accumulation at the left interface and the current is close to that under the charge neutrality condition [192].

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Figure 5.22: The charge and the spin distribution over the channel are shown under the charge accumulation and with the boundary condition Equation 5.57 (M_Ch=100mV, U_c=-300mV). The inset figure shows the spin density polarization β near the spin injecting interface. The current density 11.9MAm^-2 is maintained over a 4μm channel.

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Figure 5.23: The spin and spin current densities in depletion (M_Ch=-100mV, U_c=-140mV) and for charge neutrality (U_c=-204mV) with the boundary as in Equation 5.57 are shown. The inset figure shows the spin density polarization β. The current density 11.9MAm^-2 is maintained over a 4μm channel.

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Figure 5.24: The spin current densities with their up(down)-spin components are depicted, when the channel is in depletion (M_Ch=-100mV) with the boundary as in Equation 5.57. The current density 7.9MAm^-2 is maintained.

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Figure 5.25: The spin current density for accumulation, charge neutrality, and depletion with the boundary condition Equation 5.57 are shown. The current density 7.9MAm^-2 is maintained. The channel length is 3μm.

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Behavior of the Spin Density Polarization

Up to now it has been predicted that at a fixed boundary spin density polarization β₀ and at a fixed charge current, the spin density s and the spin current density Js show an increment at the spin injection interface by injecting more charge, but the bulk signals are determined by the charge neutrality condition. On the contrary, the spin signals are diminished dramatically both at the interface and the bulk when spin is injected from a charge depleted source. One must examine the peculiarities of the spin density polarization β, the electron concentration n, and the spin density s close to the left boundary in order to explain this behavior.

• Figure 5.22 shows both n and s in the same scale to allow a comparison. One can notice that β remains approximately constant through the accumulation layer, while the charge decrease from its high value at the interface to the equilibrium value determined by the donor concentration ND. Therefore, s also decreases substantially within the accumulation region. Thus, Js in the bulk is determined by the value of s at the end of the accumulation layer, where the charge neutrality condition is restored, and is thus determined by Js at the charge neutrality condition with the same spin density polarization at the injection boundary β₀.
• Figure 5.23 highlights s and Js in the same plot in order to make a comparison, when spin is injected from a charge depleted source. When the spin diffusion is along the current like in Figure 5.23, a substantial decrease of s and hence Js is noticed, both at the interface and the bulk as compared to their values at the charge neutrality condition. This behavior can be correlated with a significant increase of the minority (down-spin) spin current density (J_↓) in the depletion layer (c.f. Figure 5.24). This current is due to two contributions, drift and diffusion, which add constructively in this case and cause β to decrease substantially over a very short distance close to the interface. At the same time, Js is noticed to be nearly constant through the depletion layer. Indeed, in this case the spin diffusion length is increased due to the high value of the electric field Ẽ at the depletion region. At the end of the space-charge layer β is thus significantly smaller than at the interface [193], explaining the degradation of Js in the bulk as compared to the charge neutrality condition. The sharp decrease of β is due to the high spin minority current close to the interface. Should this current be reduced, for instance by applying a voltage of opposite polarity and inverting the current (c.f. Figure 5.24), Js in the bulk is enhanced, but still remains below the level determined by the charge neutrality condition (c.f. Figure 5.25).

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