Spin filter effects in an Aharonov–Bohm ring with double quantum dots under general Rashba spin–orbit interactions

Many researchers have reported on spin filters using linear Rashba spin–orbit interactions (SOI). However, spin filters using square and cubic Rashba SOIs have not yet been reported. We consider that this is because the Aharonov–Casher (AC) phases acquired under square and cubic Rashba SOIs are ambiguous. In this study, we try to derive the AC phases acquired under square and cubic Rashba SOIs from the viewpoint of non-Abelian SU(2) gauge theory. These AC phases can be derived successfully from the non-Abelian SU(2) gauge theory without the completing square methods. Using the results, we investigate the spin filtering in a double quantum dot (QD) Aharonov–Bohm (AB) ring under linear, square, and cubic Rashba SOIs. This AB ring consists of elongated QDs and quasi-one-dimensional quantum nanowires under an external magnetic field. The spin transport is investigated from the left nanowire to the right nanowire in the above structure within the tight-binding approximation. In particular, we focus on the difference of spin filtering among linear, square, and cubic Rashba SOIs. The calculation is performed for the spin polarization by changing the penetrating magnetic flux for the AB ring subject to linear, square, and cubic Rashba SOIs. It is found that perfect spin filtering is achieved for all of the Rashba SOIs. This result indicates that this AB ring under general Rashba SOIs can be a promising device for spin current generation. Moreover, the AB rings under general Rashba SOIs behave in totally different ways in response to penetrating magnetic flux, which is attributed to linear, square, and cubic behaviors in the in-plane momentum. This result enables us to make a clear distinction between linear, square, and cubic Rashba SOIs according to the peak position of the perfect spin filtering.


Introduction
Recently, many researchers have paid much attention to spintronics for its application to spin functional devices and quantum computing. One of the main purposes of spintronics is to make spin transistors and magnetoresistive devices by utilizing electron spin [1][2][3][4][5][6][7]. The generation of spin current is significant in order to realize these devices in the future. Moreover, spin current generation without ferromagnetic metals is a key issue for spintronics devices. The spin-orbit interaction (SOI) is expected to be a promising ingredient for this purpose. Since it has been confirmed that the strength of the Rashba SOI can be easily controlled by an external electric field [4], the theory of devices using SOI as a spin filter is very important in the field of spintronics. The spin-filtering devices can be used to generate and detect spin polarized currents [8,9]. There exist spin filters using ring-type conductors or two path devices [10][11][12][13][14][15][16]. This opens up the field of spin-dependent Aharonov-Bohm (AB) physics. The AB ring-type spin-filter devices frequently utilize linear Rashba and/or Dresselhaus SOIs. Generally, the spin-splitting of electrons and light hole states is proportional to wavenumber k and the splitting of heavy hole states is proportional to k 3 [17]. Until now, only linear Rashba SOI was taken into consideration for electrons. However, cubic Rashba spin-splitting has recently been found in a quasi-twodimensional electron system [18]. Also, the square Rashba effect has been found in bilayer graphene [19]. Therefore, we must take into consideration the square and cubic Rashba effects in spin-filtering devices.
However, the Aharonov-Casher (AC) phases acquired under square and cubic Rashba SOIs are ambiguous. In this study, we try to derive the AC phases acquired under square and cubic Rashba SOIs from the viewpoint of non-Abelian SU(2) gauge theory without the completing square methods. Using the results, we investigate the spin filtering in a double quantum dot (QD) AB ring under general Rashba SOIs.

Modeling and tight-binding approximation
The double QD AB ring consists of two elongated QDs and quasi-one-dimensional quantum nanowires (QNs) under an external magnetic field B ext  as shown in figure 1(a). The two QDs and the four oblique QNs are subject to general Rashba SOI and the left and right straight QNs are free of Rashba SOI. The spin transport is investigated from the left nanowire to the right nanowire in this structure within the framework of tight-binding approximation as shown in figure 1(b).
Within the framework of the tight-binding approximation, the Schrödinger equation for the wave function n |Y ñat site n is described as follows [14]: where ò is the eigenenergy, ò n is the energy of site n, V nm is defined as v n n m | | f f á ñ, and J nm is the hopping energy between site n and site m which is defined as n m Here, m e is the electron mass, n |f ñ is the atomic wave function of site n without interaction, and v n is the single-site atomic potential of site n. In (1), U nm is the 2×2 unitary matrix describing the AB [20] and AC phase [21] that an electron acquires when it travels from site m to site n. The AB phase is attributed to the magnetic flux penetrating the ring and the AC phase is attributed to the SOI like Rashba effect. In order to know the AC phase acquired by the SOI, we need to know the gauge field. The method of completing the square of the Hamiltonian is only useful for linear Rashba and linear Dresselhaus SOIs [22]. This method is useless for square and cubic Rashba SOIs.

Effective
where ÿ is the Planck constant divided by 2π, k R1 , k R2 , and k R3 are the SOI strengths of the linear, square, and cubic Rashba effects, respectively, k F is the Fermi wavenumber determined by the chemical potential, are effective magnetic fields for their SOIs, k x and k y are the wavenumbers in the xand the y-directions, respectively, k ± =k x ±ik y , and i 2 Here, x s and y s are the x-and ycomponents of Pauli matrix vector s  . These SOIs affect the motion of electrons as effective magnetic fields as described in equations (2)-(4). These effective magnetic fields are explicitly written by the following equations: The above effective magnetic fields are generated by non-Abelian gauge fields [23][24][25]. Figure 2 shows the contour maps of the above effective fields.
We find that the spin of the electron rotates once, twice, and three times around the origin in k-space under the linear, square, and cubic Rashba effects, respectively.

SU(2) non-Abelian gauge and AC phase
It is convenient to redefine the axes at each point k  such that the reference k z -axis is aligned to the direction of the local effective magnetic fields. In order to perform such a rotation, we need a unitary matrix U k g ( ) . This rotation is a kind of local gauge transformation. Under the local gauge transformation U k g ( )


, these Hamiltonians transform to the following equations, In general, the local gauge transformation U k g ( )  is given by Here, θ and f are defined as B B k sin cos , sin sin , cos is the magnitude of the each effective magnetic field. Since this is the local gauge transformation, we can obtain the next SU(2) non-Abelian gauge field A k i in reciprocal space.
where k i =k x , k y , and k z . The gauge field is explicitly written in a vector notation with respect to the spin index α by the following equation: sin sin cos cos sin sin 1 cos The corresponding field strength F mn is defined by where e˜is the coupling constant of the SU(2) gauge field, which contains the SOI strengths implicitly [22]. In fact, since the A k i is a pure gauge, the field strength equals zero identically. Therefore, we have to apply an approximation to the non-Abelian gauge in order to derive the finite effective magnetic field. Now, we impose the adiabatic approximation (AA), which corresponds to keeping only the diagonal terms of A k i mathematically [26]. As a result, we obtain an Abelian gauge field known as the spin Berry connection A k s , ad ( )   , where s=±1 corresponds to the majority and minority spins, respectively. The Abelian gauge field A k s , ad ( )   is explicitly given by In this study, we can set 2 q = p since the Rashba effect works in-plane space. Then, equation (16) reduces to Using equation (17), the Abelian gauge fields of the linear, square, and cubic Rashba SOIs for the majority spin are given by the next equations, respectively, is the magnitude of in-plane wavenumber. In reference [22], the method of completing the square was used in order to derive the AC phase using A m n n m m n n m m n n m abg m a n b g m n n m ab m a n b m n n m ab m a n b ab m a n b where ò αβ γ is the Levi-Civita symbol. This equation always holds in any space. Hereafter, we do not restrict μ and ν in k-space. Moreover, note that z s is the diagonal matrix and that both x s and y s do not have the diagonal elements. Therefore, this identity equation should result in the following equations: These equations are exact for a pure gauge. Although the AA keeps only the diagonal element (A z  ), it is found that the AA corresponds to keeping only the diagonal elements of the field strength F mn of equation (15) from the above equations. Namely, while keeping equation (22), it is only necessary to remove the left-hand side or the right-hand side of equation (23). Therefore, there exist two methods for the AA. The first method is to . As a result, the field strength F mn of equation (15) in the AA can be rewritten by the following equation: Although the first term or the second term of equation (24) is zero in the AA, equation (24) is intentionally expressed in explicit form. Equation (24) is equal to zero identically by equation (23) if an approximation is not used. In the AA, the field strength F mn can obtain a finite value by eliminating A x  and A y  (the first method). Then, the field strength F ad mn is given by the following equation: This is because the second term of equation (24) is set to zero. Conversely speaking, the same field strength F ad mn in the AA can also be obtained by eliminating A z  while keeping both A x  and A y  under the above mentioned constraints (the second method), which means that the first term of equation (24) is set to zero. Then, the field strength F ad mn is given by the following equation ) do not hold. On the other hand, this situation happens in the case of the linear Rashba SOI in real space [22]. Namely, the scenario of the AA by A x  and A y  is realized naturally in the case of the linear Rashba SOI in real space. Since the field strength F ad mn is a 2-rank tensor, the rule for the field strength tensor transformation from some b  -space into another a  -space is generally as follows [26]: When we regard a  as r  and regard b  as k,  we can map the field strength F k ad ( ) mn  in k-space to the field strength F r ij ad ( )  in real space. This means that we can obtain equation (26) in real space from the field strength F k ad ( ) mn  in k-space using the adequate transformation even though the AA of the second method happens naturally for the linear Rashba SOI in real space. As a result, we can conclude the following: the AA of the second method can be also realized for the square and cubic Rashba effects in real space using the same transformation as that for the linear Rashba effect, taking into consideration that A A where d 2 k (= dk x dk y ) is the two-dimensional k-space differential element. Here, equations (18)- (20) are used. Also, we assumed that the SOI strengths k R1 , k R2 , and k R3 are equal and that the rotation axis of the spin is in the z-direction, which do not lose generality. As a result, it is found that we can derive the unitary matrices for AC effects in real space under square and cubic Rashba effects easily once we know the unitary matrix for AC effects under the linear Rashba effect in real space. Then, it is well known that the linear Rashba SOI in real space has the chargeθ=2k R1 with respect to the non-Abelian gauge in the notation of equation (2) [22] and that the unitary matrix U AC1 for the AC effect induced by linear Rashba SOI in real space can be expressed by the following equation [14,22]: where D is the length of the path in real space, and d x and d y are the x-and y-components of the unit vector d  in the direction of the path. This equation describes the rotation about the z-axis in real space, which can be proved using the Baker-Campbell-Hausdorff formula [22]. Also, double and triple spin-orbit strengths correspond to the double and triple rotation angles. Therefore, taking into consideration equations (28)- (32) and different SOI strengths, we can obtain the following 2×2 unitary matrix U AC for AC effects induced by general Rashba SOIs in real space: is defined by the following equation for linear, square, and cubic Rashba SOIs, respectively.
k L l l l l , , 0 , , 0 , for linear Rashba, 34 l l l l 2 , 2 , 0 2 , 2 , 0 , for square Rashba, 35 l l l l 3 , 3 , 0 3 , 3 , 0 for cubic Rashba. 36 Here, α 1 =k R1 L, α 2 =k R2 L, and α 3 =k R3 L. Also, l x and l y are the x-and y-components of the unit vector l  in the direction of the path which an electron propagates from site m to site n. L is the length of the path. Therefore, the unitary matrix U nm for both the AB and AC effects in real space is written by where A ext  is the vector potential of the external magnetic field B h e , ext 0 F =  is the flux quantum, e is the elementary charge, and l d  is the line element vector. As a result, the wave function acquires an AB phase f=2πΦ/Φ 0 , where Φ is the magnetic flux penetrating the AB ring, when an electron goes around the AB ring shown in figure 1.
Finally, we discuss the validity of the AA in a realistic situation. The AA used in deriving the finite field strength of the Rashba effects means that the interband transition does not occur between the two split bands with opposite spins. The interband transition induces the rapid spin-flip, which violates the AA. However, the process can be neglected for the following reason. We consider that the interband transition can be neglected if the following criterion is satisfied: where ΔE is the interband gap for the general Rashba effect, k B is the Boltzmann constant, and T is the absolute temperature. The interband gaps ΔE for general Rashba effects are reported to be about 0.1 meV ∼30 meV by many experiments [4,18,27,28]. In a realistic situation, the present experimental technique enables us to measure the physical properties at a very low temperature of about 40 mK (3.4 μeV). Equation (38) is completely cleared. As a result, we consider that interband transition can be neglected in most experiments. Moreover, the possibility of the spin-flip in the same band has to be considered. The spin relaxation mechanism consists mainly of Elliot-Yafet and D'yakonov-Perel (DP) mechanisms [29,30]. In particular, the DP mechanism is dominant in the III-V semiconductors used in making quantum rings and QNs [29][30][31]. The spin relaxation time τ S in the DP mechanism is given by the equation where Q is the dimensionless factor ranging from 0.8 to 2.7, α is the spin-orbit coupling constant, E g is the band gap of the materials, and τ m is the momentum relaxation time. Equation (39) indicates that the spin relaxation time τ S becomes longer as both the temperature T and the momentum relaxation time τ m decrease. In realistic experiments, the spin relaxation time τ S ranges from 0.3 to 4 ns [31]. On the other hand, the momentum relaxation time τ m is at most 38 ps at a high electron mobility of 10 2 m 2 (V·s) -1 in highly doped (N s =2. 25×10 15 m −2 ) GaAs-GaAlAs heterostructures [32]. Then, the Fermi velocity v F is 2.05×10 5 m s -1 and the mean free path l F (=v F τ m ) is about 7.8 μm. Therefore, the spin of the electron seldom relaxes in the propagation of the ring in our system because the spin relaxation time is much longer than the momentum relaxation time. Therefore, the AA is guaranteed. As a result, we can expect that the following numerical results can be observed in realistic systems.

Application of the AC phase of general Rashba SOIs to an AB-ring
Now the expression of U nm has been obtained explicitly and we are going to solve equation (1) after the example of reference [14]. Only the hopping energy between nearest neighbors is considered to solve equation (1). Except for the sites '0, 1, a, b, c, d', the hopping energy between the other sites on the leads is a constant energy of V 0 , which means J 12 =J 21 =J 0−1 =J −10 =V 0 . In this study, we set the other sites energies on the leads to be zero. Using the Bloch theorem and equation (1) where R is the 2×2 reflection coefficient matrix, T is the 2×2 transmission coefficient matrix, and in |h ñ is the spin components of incoming electrons. When equations (45) and (46) are inserted into the system of equations (41), R and T can be explicitly obtained as follows: where ρ ± are eigenvalues of YY † . Therefore, the spin polarization P of the spin filter is defined by Moreover, the site energies of a, b, c, d are assumed to be the same and we set J 0a =J 0c =2.0, J ab =J cd =5.0, and J b1 =J d1 =2.0 in units of V 0 . Figure 3 shows the magnetic flux dependences of the spin polarization P of a double QD AB ring with changing angle β under the specific value of α 1 =α 2 =α 3 =α=π/12 for each Rashba SOI. Here, the angle β is shown in figure 1(b) and determines the shape of the double QD AB ring. From figure 3, it is found that the perfect spin filtering is achieved for all the Rashba SOIs. This result indicates that this AB ring under general Rashba SOIs can be a promising device for spin current generation. Moreover, they behave in totally different ways in response to penetrating magnetic flux, which is attributed to linear, square, and cubic behaviors in the in-plane momentum. Figure 4 shows the magnetic flux dependences of the spin polarization P of a double QD AB ring with changing SOI coupling constant α=α 1 =α 2 =α 3 under the specific value of the angle β=π/2. It is found that the polarization curve becomes the same form for the adequate pairs of α=π/4 and β=π/2 for linear and cubic Rashba SOIs. This is due to the following reason: the spin rotates by 3π/4 under the cubic Rashba SOI when the spin rotates by π/4 under the linear Rashba SOI. Since the the spin polarization P is a periodic function with period π, the rotation angle of π/4 is equivalent to the rotation angle of 3π/4 with respect to the phase. Generally, the different α generate the different behaviors and the different peaks of the spin polarization P even at the same angle β. These results enable us to make a clear distinction between linear, square, and cubic Rashba SOIs according to the peak position of the perfect spin filtering. This method will be very useful for determining the k-dependence of Rashba SOIs using experiments.

Conclusion
In this study, we have obtained the correct expression of AC phases for square and cubic Rashba SOIs, which were obstacles to investigating spin filters using square and cubic Rashba SOIs. Then, the spin filtering was investigated in a double QD AB ring under linear, square, and cubic Rashba SOIs by applying the AC phase to the system. As a result, it was found that the perfect spin filtering is achieved for all the Rashba SOIs at the specific penetrating magnetic flux. This means that this AB ring under general Rashba SOIs can be a promising device for spin current generation. Also, the peak positions for linear, square, and cubic Rashba SOIs behave in completely different ways. This result enables us to make a clear distinction among linear, square, and cubic Rashba SOIs according to the peak position of the perfect spin filtering. As a result, it will be easy to determine the kdependence of Rashba SOIs using experiments.