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Spin dependent disorder in a junction device with spin orbit couplings

Using the multi-probe Landauer-BUttiker formula and Green's function approach, we calculate the longitudinal conductance (LC) and spin Hall conductance (SHC) numerically in a two-dimensional junction system with the Rashba and Dresselhaus spin orbit coupling (SOC) and spin dependent disorder (SDD) in presence of both random onsite and hopping disorder strengths. It has been found that when the strengths of the RSOC and DSOC are same, the SHC vanishes. Further in presence of random onsite or hopping disorder, the SHC is still zero when the strengths of the two types of SOC, that is Rashba and Dressselhaus are the same. This indicates that the cancellation of SHC is robust even in the presence of random disorder. Only with the inclusion of SDD (onsite or hopping), a non-zero SHC is found and it increases as the strength of SDD increases. The physical implication of the existence of a non-zero SHC has been explored in this work. Finally, we have compared the effect of onsite SDD and hopping SDD on both longitudinal and spin Hall conductances.


Introduction
The notion of dissipationless spin current [1] has attracted considerable interest recently. The spin-orbit coupling produces a transverse force on a moving electron and this force tends to form a transverse spin current. In the simplest form, a spin current is about the flow of spin-up electrons in one direction, say +x, accompanied by the flow of an equal number of spin-down electrons in the opposite direction, −x. The total charge current in the x direction is therefore zero, I e = e (I ↑ + I ↓ ) = 0. The total spin current is finite: I s = e (I ↑ − I ↓ ) = 0.
In a two-dimensional electron gas (2DEG) lacking structure inversion symmetry of the confining potential and bulk inversion symmetry, the effective Hamiltonian is given by, where the second term is the Rashba spin-orbit coupling (RSOC) and the third term is the Dresselhaus spin-orbit coupling (DSOC). σ x/y is the x/y component of the Pauli matrices, and α and β are the coupling parameters which denote the strengths of RSOC and DSOC respectively. In this paper, we investigate the effect of a spin dependent random disorder on a four terminal square lattice in presence of both Rashba and Dresselhaus spin orbit interactions. Recently it has become possible to create the spin dependent disorder within a new experimental context, namely that of ultracold fermionic and bosonic atomic gases confined to optical lattices [5,6]. Few theoretical works have been done on this context [2,3,4]. However, till now it is not possible to create experimentally which facilitate studying of such phenomena in mesoscopic physics. Nevertheless, the issue deserves to be studied theoretically.
We organize our paper as follows. The theoretical formalism leading to the expressions for the spin Hall and longitudinal conductances using Landauer Bütiker formula are presented in section II. Section III includes an elaborate discussion on the results obtained for the spin Hall and longitudinal conductances that are helpful in studying the effect of spin dependent disorder on four terminal junction device in presence of spin orbit interactions. Figure 1. Cross-shaped device with four semi-infinite metallic leads. The spin-orbit coupling exists in the central region only, and the effect of the semi-infinite leads is treated exactly through the self-energy terms.

Theoretical formulation
We choose a four-probe measuring set-up as shown in Fig.1 to observe the spin Hall effect.
Here the four ideal semi-infinite leads are attached to the central conducting region which in our case is a square lattice with spin orbit interaction. An unpolarized charge current is allowed to pass through the longitudinal leads along lead-1 and lead-2 (see Fig.1) inducing spin Hall current in the transverse directions that is along lead-3 and lead-4. With the tight binding approximation for nearest neighbour interaction, Eq.(1) reads as, Here ǫ iσ is the random on-site spin dependent potential and t ij,σ is the spin dependent hopping strength chosen from a uniform rectangular distribution (−W to W ), V R =h α a 0 and V D =h β a 0 are the Rashba and Dresselhaus coupling strengths, respectively. δ x/y is the unit vector along x/y direction. For the four-probe case, where pure spin current is expected to flow through the transverse leads, due to the flow of charge current through the longitudinal leads, the longitudinal and spin Hall conductances are defined as [7], where I q 2 and I s 3 are the charge and spin current flowing through the lead-2 and lead-3 respectively. V m is the potential at the m-th lead. I ↑ 3 and I ↓ 3 are the up and down spin currents flowing in lead-3.
The calculation of electric and spin currents is based on the Landauer-Büttiker multi-probe formalism [8]. The charge and spin currents flowing through the lead m with potential V m , can where, we have defined two useful quantities as follows, Physically, the term e 2 h n =m T out nm V m is the total spin current flowing from the m-th lead with potential V m to all other n leads, while the term e 2 h n =m T in mn V n is the total spin current flowing into the m-th lead from all other n leads having potential V n . The zero temperature conductance, G σσ ′ pq that describes the spin resolved transport measurements, is related to the spin resolved transmission coefficient as [10,11], The transmission coefficient can be calculated as [12,13], p 's are the coupling matrices representing the coupling between the central region and the leads, and they are defined by the relation [14], Here Σ σ p is the retarded self-energy associated with the lead p. The self-energy contribution is computed by modeling each terminal as a semi-infinite perfect wire [15].
The retarded Green's function, G R is computed as , where E is the electron Fermi energy and H is the model Hamiltonian for the central conducting region. The advanced Green's function is, G A = G † R . Now, following the spin Hall phenomenology, in our set-up since the transverse leads are voltage probes, I q 3 = I q 4 = 0. Also, as the currents in various leads depend only on voltage differences among them, we can set one of the voltages to zero without any loss of generality. Here we set V 2 = 0. Finally, if we assume that the leads are connected to a geometrically symmetric ordered bridge, so,

Results and Discussion
We have investigated the effect of spin dependent disorder (onsite and hopping) (W ) in presence of spin orbit coupling (both Rashba and Dresselhaus terms) on the experimentally measurable quantities such as the longitudinal conductance (G L ) and the spin Hall conductance (G SH ). Before we start computing the physical quantities, we briefly describe the values of different parameters used in our calculation. Throughout our work, we have considered lattice constant, a = 1, system size, L = 20, onsite term, ǫ = ǫ L = 0, hopping term, t = t L = t C = 1. All the energies are measured in unit of t. Further we choose a unit where c = h = e = 1. The longitudinal conductance, G L is measured in unit of e 2 h . The spin Hall conductance, G SH is measured in unit of e 8π . Disorder (onsite and hopping) in the system is modeled by a random rectangular distribution between [−W, +W ]. We consider two different types of disorder: (i) Spin dependent disorder for the up spin (USDD) with W ↑ = W and W ↓ = 0, (ii) Spin dependent disorder for the down spin (DSDD) with W ↑ = 0 and W ↓ = W .
All the results obtained below are averaged over 1000 disorder configurations. For most of our numerical calculations we have used KWANT [16].
There exists a special case at the symmetric point of V R = V D . The spin Hall conductance (SHC) vanishes at that point [17,18]. In presence of onsite or hopping spin independent disorder (SID), SHC is still zero in the given symmetric case. However, the main finding of the paper is that when we introduce onsite or hopping spin dependent disorder, we found a non-zero SHC. Fig.2 shows the variation of spin Hall conductance as a function of energy E, in presence of onsite and hopping SDD when Rashba and Dresselhaus spin orbit interaction have the same strengths (V R = V D ). Here, W Oσ stands for the onsite disorder with spin σ and W Hσ is the hopping disorder with spin σ. In Fig.2(a), G SH increases in the negative region with increasing the strength of USDD.
The same thing happens with DSDD, except the value of G SH which acquires non zero values in the positive region as shown in Fig.2(b). Another interesting point is that the hopping SDD is much more efficient to increase SHC in magnitude than onsite SDD.
We can explain this non-zero SHC from the definition of the spin current (Eq. (3)). As we introduce SDD in the up spin electron, I ↑ 3 (up spin current flowing in terminal 3) will decrease and most of I ↓ 3 (down spin current flowing in terminal 3) will reach lead 3, resulting negative G SH . The same argument will render positive G SH in case of DSDD. In Fig.3, we have studied the variation of longitudinal conductance, G L as a function of energy (E) for different spin dependent disorder strengths (W ). Fig.3(a) shows G L vs E plot for onsite and hopping disorder with different USDD strengths. For W O↑ = 2 and W H↑ = 1, the value of G L remains almost same, implying that the hopping disorder is more efficient to decrease the longitudinal conductance. Fig.3(b) shows G L vs E plot for the DSDD case. Hopping disorder is again found to be more efficient to suppress longitudinal conductance than onsite disorder. Hence we can say that the variation of G L is independent of SDD, i.e., G L remains same for both USDD and DSDD irrespective of the onsite and hopping disorder.
We can explain why the longitudinal conductance is independent of spin dependent disorder from (Eq.(3)). G L is just the ratio between I s 3 (the total charge current flowing in terminal 3) and V 3 (the voltage at terminal 3). Since the expressions of G L contains the total charge will always be the same. As a result, G L is independent of SDD.   The variation of G SH as a function of disorder, W at E = 0 is shown in Fig.4. Here we set the energy at E = 0. From Fig.4(a) we see that for USDD, G SH starts to increase with W ↑ and tends to saturate in the negative region. For DSDD, G SH shows a behaviour which looks like a mirror reflection of the corresponding feature for USDD . From Fig.4(a) it is clear that G SH increases more in case of hopping disorder than that of onsite disorder. Fig.4(b) shows the variation of G L as a function of disorder strength. For both USDD and DSDD, the variation of G L is exactly the same as we saw in Fig.3. Here G L decreases with increasing disorder strength and finally tends to saturate in the higher region of W . In case of hopping disorder, G L decreases more abruptly than the onsite case. G SH becomes negative for USDD and positive for DSDD. In presence of USDD, I ↑ 3 will be suppressed and for larger value of W ↑ , I ↑ 3 ≃ 0. As a result G SH tends to saturate in the higher region of W ↑ .
In order to study the effect of system size on G SH and G L in presence of SDD, we fixed width of the scattering region to lattice of size L = 20 and set energy, E = 0. Fig.5(a) shows the variation of G SH as a function of the system size, L. Here G SH is essentially constant except when the width is lower than the length of the scattering region. G SH is almost same for W O↑ = 2 and W H↑ = 1. Fig.5(b) shows the variation of G L as a function of the system size, L. G L decreases with length and then becomes almost constant.  Figure 5. (a) G SH is plotted as a function of system size, L. (b) G L is plotted as a function of system size.
After a certain value of L, both G SH and G L remain constant with L. G SH takes appropriate sign depending upon the nature of the SDD as we discussed earlier. Whereas, the variation of G L with L is found to be independent of the nature of SDD.

Summary and Conclusions
In summary, in the present work we have studied the effect of spin dependent disorder effect on a four terminal junction device in presence of Rashba and Dresselhaus spin orbit coupling, specifically when the strengths of the two types of the spin orbit coupling are the same.
Spin dependent disorder can produce a non zero G SH even if the strengths of RSOC and DSOC are the same. Higher the SDD, larger is the G SH (in magnitude). G SH is symmetric about E = 0 and takes opposite signs for different types of SDD. Hopping disorder is more efficient to increase G SH than the onsite disorder. G L is independent of the nature of SDD. With the same disorder strength, G L decreases more in case of hopping disorder than the onsite case. With increasing the strength of SDD, both G L and G SH tend to saturate in the higher region of SDD. Further, we have found that essentially G L and G SH are independent of length (L) of the sample, at least for larger values of L.