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Logical spin-filtering in a triangular network of quantum nanorings with a Rashba spin-orbit interaction
Author’s Accepted Manuscript Logical spin-filtering in a triangular network of quantum nanorings with a Rashba spin-orbit interaction E. Dehghan, D. Sanavi Khoshnoud, A.S. Naeimi www.elsevier.com/locate/physb
PII: DOI: Reference:
S0921-4526(17)30671-3 http://dx.doi.org/10.1016/j.physb.2017.09.076 PHYSB310302
To appear in: Physica B: Physics of Condensed Matter Received date: 2 May 2017 Revised date: 16 September 2017 Accepted date: 18 September 2017 Cite this article as: E. Dehghan, D. Sanavi Khoshnoud and A.S. Naeimi, Logical spin-filtering in a triangular network of quantum nanorings with a Rashba spinorbit interaction, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.09.076 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Logical spin-filtering in a triangular network of quantum nanorings with a Rashba spin-orbit interaction E. Dehghan1, D. Sanavi Khoshnoud1,*, A.S. Naeimi2 1
2
Department of Physics, Semnan University, Semnan 35195-363, Iran
Department of Physics, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran
Abstract The spin-resolved electron transport through a triangular network of quantum nanorings is studied in the presence of Rashba spin-orbit interaction (RSOI) and a magnetic flux using quantum waveguide theory. This study illustrates that, by tuning Rashba constant, magnetic flux and incoming electron energy, the triangular network of quantum rings can act as a perfect logical spin-filtering with high efficiency. By changing in the energy of incoming electron, at a proper value of the Rashba constant and magnetic flux, a reverse in the direction of spin can take place in the triangular network of quantum nanorings. Furthermore, the triangular network of quantum nanorings can be designed as a device and shows several simultaneous spintronic properties such as spin-splitter and spin-inverter. This spin-splitting is dependent on the energy of the incoming electron. Additionally, different polarizations can be achieved in the two outgoing leads from an originally incoming spin state that simulates a Stern-Gerlach apparatus.
1
* Corresponding author. Tel.: +982331533254; fax: +982333654081 E-mail addresses: [email protected]
Keywords: Spintronics, Quantum nanoring, Rashba spin-orbit interaction, logical spin-filtering, Stern-Gerlach apparatus 1. Introduction Nowadays, improvement in technology, has led to a growing interest in a new product of lowdimensional quantum nanostructure [1-3]. Semiconductor-based devices are reliable for processing application and data storage, thereby as well as electronic, have a large contribution in the future of spintronics [4-6]. Low-dimensional quantum nanostructures such as confined zero-dimensional quantum dot (QD), quantum well (QW) and quantum ring (QR) have attracted enormous research interests in recent years and play an important role in microelectronic [7, 8]. In 1999, for the first-time quantum ring of the RSOI were indicated as a confining potential for spin interferometer device [9]. Afterward, specification of a quantum ring has been estimated in several articles [10, 11]. In recent years, some of the theoretical researchers posed quantum rings as a multipurpose and flexible spintronic device. For instance, Lee et al. introduced a new spin filter based on spinresolved Fano resonances coupled with a quantum wire (QW), in accordance with spin-split levels in a quantum ring (QR) [12]. In 2006, Citro et al investigated the effect of RSOI on the spin-dependent electron transport in a one-dimensional open semiconductor quantum ring. They explored a feasible spin-filtering model in the ring with tunable spin-orbit interaction. The proposed spin-filtering of Citro et al. is not perfect, and its efficiency dose not exceeding 70% 2
[13]. In another study, Foldi et al. analyzed the spin-orbit effect in an open quantum ring with one incoming and two outgoing leads. They also showed that such system can operate as a spinsplitter [14]. Later, a rectangular array of quantum rings exposed to the RSOI of a perpendicular magnetic field was explored by Kalman et al. (2008). They showed that, by changing various Rashba spin-orbit couplings (RSOC) in each individual ring [15], the system can control the spin-dependent electronic transmission. As a new study, Naeimi et al. explored spin-filtering properties of transmitted electrons through a quantum ring in the presence of RSOI and magnetic flux [16]. Additionally, in another work, they have studied spin-dependent electron transport in an open nonmagnetic double quantum ring, in the presence of RSOI; and showed that, the double quantum ring can act as a perfect electron spin-inverter with very high efficiency and spin-switch [17]. Newly, Faizabadi et al. have studied the radius effect of a triangular network of quantum nanorings in the presence of RSOI on the spintronic properties. They examined the effect of changing radius of the rings in the network and concluded that, such system can be considered as a multipurpose spintronic device [18]. As we know, any change in the radius of the ring, is not the most proper way for the network, because the geometry of a network will change as the radius of the rings experience a variation. An appropriate way to manipulate and control the electron spin is the Rashba spin-orbit interaction (RSOI) and magnetic flux. The strength of RSOI and magnetic flux can be easily controlled by an external electric field and a gate voltage. In this paper, we have studied the spin-dependent electron transport in a triangular network of quantum nanorings in the presence of RSOI and magnetic flux using quantum waveguide theory. It is shown that, for proper values of the RSOI and energy of incoming electrons, the designed system will indicate several spintronic properties such as logical spin-filtering, spin-inverter, and spin-splitter simultaneously. Moreover, we have introduced a new device which is able to be
3
considered as a spintronic analog of the Stern-Gerlach apparatus. The incoming electrons, dependent on their energy, are forced to split into two different spatial parts with the geometrical construction of the semiconductor device [19-21]. This paper is as follows: The theoretical framework the model and formalism with RSOI is described in Section 2, the two-step numerical calculation is presented in Section 3, and the paper is concluded by a summary of the main results in Section 4. 2. Theoretical model As shown in Fig.1, we have considered a triangular quantum network which consists of three quantum ring attached to one incoming and two outgoing leads. Typically, in comparison with the radial dimensions, the thickness of the ring is small enough. Thus, the array is considered as a one-dimensional system, without transverse modes; so, with a pretty good approximation, one can consider only the lowest level of electron sub-band [22]. Furthermore, we ignore electron–electron and electron–phonon interactions, while the temperature of the system is considered to be zero. In addition, there are tunable magnetic fluxes through the center of each ring such that there is no magnetic field on them. At the first step, we will solve the problem for a single narrow quantum ring, and then will generalize the results for the triangular network of quantum rings. The relevant dimensionless Hamiltonian can be written as [23]
̂
[(
where
)
is the azimuthal angle,
]
(1) ⁄ m a is the dimensionless kinetic energy,
reduced Planck constant, m* is the effective mass of electron, a is the radius of the ring, 4
is the is the
r-component of the Pauli spin matrix in cylindrical coordinates (i.e., is the magnetic flux passing through the ring,
),
is flux quantum, and
≡
⁄ a
is the
normalized Rashba constant. The Rashba constant is assumed to (2) be tunable with the average electric field perpendicularly applied to the ring [24]. The Eigen values of the Hamiltonian [Eq. (1)] are given by [25] ψ(K, )
e
χ( ) ( )
where
1,2 referred to the Eigen-basis
〉 and
〉 z component of the spin,
respectively. The orthogonal spinors, χ( ) ( ) in [Eq. (2)], can be expressed in terms of the 1 0 eigenvectors ( ) and ( ) of Pauli matrix 0 1
χ( ) ( )
as
θ
e 1 ( √ π e
θ
(3)
)
And
χ( ) ( )
1 √ π
(
θ
e
θ
e ar ta (
where θ
(4)
)
). By use of Eq. (2), the corresponding energy eigenvalues are
obtained as follow [26]
E
[(K
1
Φ π
π
)
]
(5)
5
where Φ
( 1) √1
π[1
and K ≡ ( m E⁄
] is the so-called Aharonov-Casher (AC) phase [26],
) is electron wavenumber. Using Eq. (5), the possible values of electron
wavenumber k can be calculated as [27] ( 1)
K
q
(6) [( ⁄ )
where q
E⁄
]
⁄
and j=1,2 are correspond to the clockwise and
counterclockwise motions of an electron through the ring, respectively. We can write the state of the electron using of quantum waveguide theory in the upper and lower arms of one ring as [2830]. ψ
( )
,
ψ ( )
(7)
χ( ) ( )
∑ ∑a e ,
,
(8)
χ( ) ( )
∑ ∑b e ,
where the signs up and low denote the upper and lower arms of the ring, and a and b are unknown coefficients, which should be determined. In this paper, we apply Eqs. (7) And (8) for upper and lower arms of the three similar rings. According to Fig. 1, the electron wave function in the incoming, middle and outgoing leads, are as follows: f ( )e f
ψ (x)
ψ (x)
(
t t
t t
(
)e
r r
r )e r
(
r r
r r
(9)
)e
(10)
6
t t
t t
)e
(
t t
t t
)e
(
ψ (x)
(
ψ (x)
(
r r
r r
)e
(11)
r r
r r
)e
(12)
And
(13) ψ
(x)
(
t t
t )e t
ψ
(x)
(
t t
t )e t
where f (
(14)
, ) is the amplitude of the injected electron with spin , t and
transmission amplitudes of the electron with incoming and outgoing spins from the B (C) ring, and t
(t
,t
,)
and outgoing spin
) are the
respectively
is the transmission amplitude from the A (A, B) ring
forward to the B (C, C) ring, and similarly r incoming spin
(t
is the reflection amplitude of electron with
from the (A) ring and r
(r
,r
)is the reflection
amplitude from the A (A, B) ring backward to the B (C, C) ring. We have applied the boundary conditions to explain that wave function must be continues and the spin current density must be conserved at the junctions of the rings and the leads. Using Griffith’s boundary conditions [31], and employing the Gaussian elimination method [32], according to Eqs. (7) And (8) we can obtain
all
a , b , a , b , a , b , t
the ,t
, t
, t
unknown , t
7
, r
,r
,r
, and
coefficients r
,
for three rings A, B
and C. Finally, the transmission coefficient for outgoing leads B and C can be determined by T
t
and T
t
.
3. Results and discussion A numerical study on spin-resolved properties of a triangular network of quantum nanorings with RSOI is presented using derived equations in Sec. 2. Here, the radius and the Rashba constant of the three rings are considered to be equal (i.e. aA=aB=aC and
). As the
Fig.1 shows, the angle between the leads in the system is considered to be constant. We choose the angles π⁄
and and
π⁄ and
of the rings as follows: for ring {A}, π⁄ , and ring {C},
π⁄
and
π⁄ , ring {B},
π⁄ . The effects of the
magnetic flux , the strength of the RSOI and energy of the incoming electron on spindependent electron transport are studied for different conditions. The advantage of the chosen triangular network allowed to derive simple analytical results and get their clear physical explanation. Comparing our results to the case of a single ring with three leads in Ref [14] it can be seen that for appropriate values of tunable parameters such as Rashba strength and magnetic flux, the system only can act as a perfect spin-splitter and it was separated incoming spin direction. While the main characteristics of triangular network are that, it acts as a multipurpose spintronic device such as logical spin-filtering, spin-inverting, spin-splitting and Stern-Gerlach apparatus at the same time. This network can be polarized spin direction in each output with high efficiency. As well as one of the most important advantages of this system is that, we can increase the system efficiency by changing the strength of Rashba spin-orbit intraction and the magnetic flux, that can be easily controlled by an external magnetic field and a gate voltage. In Fig. 2, the spin-resolved electron transmission coefficients T and T and the electron spinpolarization P are shown as a function of the energy of incoming electron, where T 8
is the
electron transmission coefficient without spin-inversion, and T is the electron transmission coefficient with spin-inversion from down to up. Here, the spin of incoming electrons are considered to be down and the normalized magnetic flux is illustrates the transmission coefficients T a
T
⁄
0. . Fig.2 (a) and (b)
as a function of the normalized electron
energy E for outgoing leads B and C, for an arbitrary value of Rashba constant ( . e. 1.
), respectively. As Fig. 2(a) shows, in a wide range of normalized electron energy (i.e.
0 < E < 1), the transmission coefficient for outgoing lead B, T (transmission without spininversion) is zero, while for all values of normalized electron energy the value (transmission with spin-inversion) is about 0.7, proving that spin-inversion can take place in presence of RSOI. According to Fig. 2(b) in a wide range of normalized electron energy (i.e. 0 < E < 1), the transmission coefficient T
is 0.7 (which is not a perfect transmission), as long as, for all values
of normalized electron energy T is zero (which shows a totally blocking). In fact, logical spinfiltering is not perfect because the value of electron transmission T is not equal to 1. The spinpolarization can be described as
(T
T )⁄(T
T
) where
. If P = -1, the
spin of transmitted electrons leftovers without changed but if P=1, the spin of transmitted electrons changes from up to down and vice versa. As we see in Fig. 2(c), the spin-polarization is positive and approximately about +1 for outgoing B, and negative and hardly reaches to -1 for outgoing C. It has seen that for the arbitrary Rashba constant, spin-polarization shifts from -1 (transmission without spin change) to +1 (transmission with mostly spin-inversion). In order to obtain an optimum logical spin-filtering, the contour plot of the spin-polarization versus α and
⁄
is depicted in Figs. 3(a) and (b) respectively. The values of E are assumed
about E=1.53 and E=1.38 for both outgoing leads. The spin polarization versus α is varying between negative and positive values related to spin-up and spin-down transmission probability, 9
while the sign in the two leads are opposite. The spin properties for some values of Rashba strength has optimized (for example α = 2.5). According to the Figs. 3(a) and (b), spin polarization in the lead B is exactly equal to the lead C, with the opposite signs. Adjustment of the energy of incident electron remained in RSOI strength, provided suitable conditions; in that, not only 100 percent of the incoming spin-down current transmitted into one of the outgoings leads, but also 100 percent of the incoming spin-down current could be directed toward the other one simultaneously, so proper spintronic properties observed clearly. In addition, perfect spininverting and spin-filtering are observed in outgoing leads B and C, respectively. The RSOI behaves like an effective in-plane momentum-dependent magnetic field, and the fully spin-down polarized electron transmission in the incoming lead will be changed to the spinup in the outgoing leads at specific RSOI. For the moving electrons along with the ring in different paths, their phase difference should be affected using the alteration of the Rashba constant. Therefore, it has affected the interferences of electron waves and consequently by changing RSOI constant, the transmission of the network has modified. By moderating the gate voltage, we can obtain the reliable value of Rashba constant for the state that the spin-inversion is perfectly present. Close quantum ring eigenvalue can affect the electron spin transition passing through the ring that means, when energy of incoming electron is equal to one of the quantum ring eigenvalues, the electron can pass through the ring. In the open quantum ring and network, the presence of RSOI causes that quantum ring eigenvalue is dependent on the electron spin. As we know, the RSOI acts similar to the effective magnetic field oriented along the Zdirection. There is a characteristic length scale called spin precession length. For normal incoming electrons when the Rashba barrier length is equal to one-half of spin precession length, 10
the spin of electrons rotates 1 0 , hence the spins along +z (up) invert to -z (down) and contrariwise [33]. Although we have obtained a reasonable value for spin polarization, the low value of transmission causes a reduction in the efficiency of that system. Therefore, in order to get an appropriate output, we have plotted T ⁄
T and T
T as a function of RSOI strength and
in Fig. 4 (a) and (b) respectively. According to the Fig. 4(a) and (b) there are large
differences between spin-up and spin-down transmission probability of two outgoings leads, leads us to have the high transmission probability in each outgoing lead. The value of +1(-1) in Fig. 4(a) (4(b)) demonstrates 100 percent spin-up (spin-down) state and indicates that the system operate as a perfect filtering of spin-down (spin-up). According to the contour plot in Figs. 3 and 4, we plotted transmission coefficients as a function of energy. Fig. 5 shows transmission coefficients T and T for an appropriate value of Rashba constant (i.e.
. ) and the normalized magnetic flux
⁄
0.0. Fig
5(a) and (b) denotes 100 percent spin transition for spin-up (spin-down) in E=1.43 and E=1.53, indicating perfect filtering of spin-down and spin-up respectively. For further clarification, the perfect spin-polarization is demonstrated in Fig. 5(c). Therefore, by choosing the appropriate Rashba constant, controlled by a gate voltage, the direction of the spin-up and spin-down transmission probability in two outgoing leads is reversed simultaneously in the appropriate range of energy spectra. From the theoretical point of view, spin transport dependence on magnetic flux and RSOI through the ring, so the changing in magnetic flux and Rashba strength affected the spin current of the ring. The RSOI breaks the degeneracy of up and down spin states and makes them
11
circulate in opposite directions in the quantum ring, thereby producing a pure persistent spin current in the system. In addition, the numerical calculations show that, the transmission coefficients with/without spin-inversion for incoming with spin-up electrons are the same as those of the incoming with spin down electrons (i.e. T
T or T
T ). As a significant result, the triangular network
of the quantum rings can act as a logical spin-filtering. Additionally, this network of the quantum ring can be used as a perfect spin-splitting based on the energy of incoming electron (E=1.53 and E=1.38). The point is that, when the energy of incoming electron is equal to 1.53, electron passes thorough lead B and when this is 1.38, they pass thorough the lead C. Moreover, this array of quantum rings with one input and two outgoing leads in the presence of RSOI has remarkable similarities with a Stern-Gerlach apparatus because we can separately direct each of the up and down spins toward each of the output leads simultaneously. 4. Conclusions In conclusion, we have calculated the Spin-dependent electron transmission and spinpolarization of a triangular network in presence of RSOI and external magnetic field by using the wave guide theory. We demonstrated that, a triangular network of the quantum ring with RSOI can serve as a logical spin-filtering for electron spins. In addition, it has been shown that efficiency of spin-filter has been improved and trended up to a high value (very close to 100%). It is important that, this network is more efficient than a single ring with fewer controllable parameters, so it can be seen as a multipurpose spintronic device. The spin-polarization of transmitted electrons can be changed from -1 (merely spin-down) to +1 (merely spin-up) for two outgoings and contrariwise by altering the Rashba strength and magnetic flux. Also by altering the input spin to spin-up, all the results will be inverted. To obtain perfect spin polarization and 12
spin transition, the contour maps of spin polarization and spin transition are plotted in term of the magnetic flux and Rashba constant respectively. We have also obtained the optimum values by varying the RSOI constant and the magnetic flux, and the result is that, a better spin-inverting can still be achieved. We believe that, the aforementioned study can be used to design efficient spintronic nano-devices. Besides, we have already found that, in case of symmetric geometry, spin-splitting is dependent on the energy of incoming electron, and the presence of RSOI causes the Stern-Gerlach type spin separation. References [1] S. Datta and M. Das, Electronic analog of the electro‐optic modulator, Appl. Phys. Lett. 56 (1990) 665-667. [2] V. Moldovaenu, and B. Tanatar, Tunable spin currents in a biased Rashba ring, Phys. Rev. B. 81, (2010) 035326-035329. [3] L. Diago-Cisneros, F. Mireles, Quantum-ring spin interference device tuned by quantum point contacts, Appl. Phys. 114 (19) (2013) 193706-193713. [4] P. D. Hodgson, GaSb quantum rings in GaAs/AlxGa1− xAs quantum wells, Appl. Phys. 119 (4) (2016) 044305-044307. [5] E.Cota, R.Aguado, and G.Platero, AC-driven double quantum dots as spin pumps and spin filters, Phys. Rev. Let. 94 (10) (2005) 107202-107206. [6] R. Sánchez, Spin-filtering through excited states in double-quantum-dot pumps, Phys. Rev .B. 74 (3) (2006) 035326-035335.
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[7] L. Eslami, E. Faizabadi, and S. Ahmadi, Quantum nano ring composed of quantum dots as a source of pure persistent spin or charge current, Phys. Lett. A. 380 (45) (2016) 3854-3860. [8] M. Molavi, and E. Faizabadi, Spin-polarization and spin-flip in a triple-quantum-dot ring by using tunable lateral bias voltage and Rashba spin-orbit interaction, Magn. Magn. Mater. 428 (2017) 488-492. [9] J. Nitta, F. E. Meijer, and H. Takayanagi, Spin-interference device, Appl. Phys. Lett. 75 (695) (1999) 143-144. [10] J. Splettstoesser, M. Governale, and U. Zu licke, Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling, Phys. Rev. B. 68 (16) (2003) 165341-165350. [11] P. Foldi, O. Ka ´lma ´n, and F. M. Peeters, tability of spintronic devices based on quantum ring networks, Phys. Rev. B. 80 (12) (2009) 125324-125332. [12] L. Minchul, and C. Bruder, Spin filter using a semiconductor quantum ring side coupled to a quantum wire, Phys. Rev .B, 73 (8) (2006) 085315-085319. [13] R. Citro, F. Romeo, and M. Marinaro, Zero-conductance resonances and spin filtering effects in ring conductors subject to Rashba coupling, Phys. Rev .B. 74 (11) (2006) 115329115339. [14] P. Földi, Quantum rings as electron spin beam splitters, Phys. Rev .B. 73 (15) (2006) 155325-155329. [15] O. Kálmán, Magnetoconductance of rectangular arrays of quantum rings, Phys. Rev .B. 78 (12) (2008) 125306-
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[16] A. Naeimi, M. Esmaeilzadeh, A wide range of energy spin-filtering in a Rashba quantum ring using S-matrix method, Appl. Phys. 113 (4) (2013) 044316-044322. [17] A. Naeimi, Spin transport properties in a double quantum ring with Rashba spin-orbit interaction, Appl. Phys. 113 (2013) 014303-014310. [18] E. Faizabadi, M. Molavi, Radius effect on the spintronic properties of a triangular network of quantum nanorings in the presence of Rashba spin-orbit interaction, Current Appl. Phys. 17 (2) (2017) 207-213. [19] S.Souma, and B.K. Nikolić, Spin Hall current driven by quantum interferences in mesoscopic rashba rings, Phys. Rev. Lett. 94 (10) (2005) 106602-106606. [20] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Electron spin interferometry using a semiconductor ring structure, Appl. Phys. Lett. 86 (5) (2005) 162107162110. [21] J. Cserti, A. Csord´as, U. Z¨ulicke, Electronic and spin properties of Rashba billiards, Phys. Rev. B. 70 (23) (2004) 233307-233311. [22] H. B. Heersche, Z. deGroot, J. A. Folk, L. P. Kouwenhoven, H. S. J. van der Zant, A. A. Houck, J. Labaziewicz, and I. L. Chuang, Kondo effect in the presence of magnetic impurities, Phys. Rev. Lett. 96 (1) (2006) 017205-017209. [23] F.E. Meijer, A.F.Morpugo, T. M. Klapwijk, One-dimensional ring in the presence of Rashba spin-orbit interaction: derivation of the correct Hamiltonian, Phys. Rev. B. 66 (3) (2002) 033107033110.
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[24] D. Grundler, Large Rashba splitting in InAs quantum wells due to electron wave function penetration into the barrier layers, Phys. Rev. Lett. 84 (26) (2000) 6074-6077. [25] P. F¨oldi, B. Moln´ar, M. G. Benedict, and F. M. Peeters, Spintronic single-qubit gate based on a quantum ring with spin-orbit interaction, Phys. Rev. B. 71 (3) (2005) 033309-033313. [26] A. Tsuneya, et al., eds, Mesoscopic physics and electronics. Springer Science & Business Media, (2012). [27] G. A. Prinz, Magnetoelectronics, Science 282 (5394) (1998) 1660-1663. [28] J. B. Xia, Quantum waveguide theory for mesoscopic structures, Phys. Rev. B. 45 (7) (1992) 3593-3599. [29] H. C. Wu, Y. Guo, X. Y. Chen, and B. L. Gu, Rashba spin-orbit effect on traversal time in ferromagnetic/semiconductor/ferromagnetic heterojunction, J. Appl. Phys. 93 (9) (2003) 53165320. [30] J. R. Shi, and B. Y. Gu, Quantum waveguide transport with side-branch structures: A recursive algorithm, Phys. Rev. B. 55 (7) (1997) 4703-4715. [31] S. Griffith, Trans. Faraday Soc. 49, 345 (1953); ibid. 49, 4825 (1997). [32] G. H. Golub and C. F. Van Loan, Matrix Computations (The Johns Hopkins University Press, Baltimore, 1996). [33] S. Ahmadi, Spin-inversion in nanoscale graphene sheets with a Rashba spin-orbit barrier, AIP Adv. 2 (1) (2012) 012130-012139.
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17
Figure 1: Schematic representation of a triangular array of quantum rings coupled to three leads in the presence of RSOI and magnetic flux. The arrows illustrate the possible directions of the currents.
(a)
18
(b)
(c)
19
Figure 2: Spin-dependent electron transmission coefficients T and T as a function of electron 1.
energy for an arbitrary Rashba constant (i.e.
) (a) The B outgoing (b) the C
outgoing (c) the Spin-polarization P as a function of electron energy corresponding to Figs. 2(a) and 2(b).
(a)
20
(b)
Figure 3: The contour plot of spin-polarization versus α, ϕ/ϕ0 (a) B outgoing lead and E=1.53 (b) C outgoing lead and E=1.38.
(a)
21
(b)
Figure 4: The contour plot of transition coefficient versus α, ⁄ .
22
(a)
and (b)
(a)
(b)
23
(c)
Figure 5: Spin-dependent electron transmission coefficients T and T and Spin-polarization as a function of electron energy for an arbitrary Rashba constant.(i.e. normalized magnetic flux
⁄
. ) and the
0.0 (a) The B outgoing (b) the C outgoing (c) Spin-
polarization (P) as a function of electron energy corresponding to Figs. 5(a) and 5(b).
24
PII: DOI: Reference:
S0921-4526(17)30671-3 http://dx.doi.org/10.1016/j.physb.2017.09.076 PHYSB310302
To appear in: Physica B: Physics of Condensed Matter Received date: 2 May 2017 Revised date: 16 September 2017 Accepted date: 18 September 2017 Cite this article as: E. Dehghan, D. Sanavi Khoshnoud and A.S. Naeimi, Logical spin-filtering in a triangular network of quantum nanorings with a Rashba spinorbit interaction, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.09.076 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Logical spin-filtering in a triangular network of quantum nanorings with a Rashba spin-orbit interaction E. Dehghan1, D. Sanavi Khoshnoud1,*, A.S. Naeimi2 1
2
Department of Physics, Semnan University, Semnan 35195-363, Iran
Department of Physics, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran
Abstract The spin-resolved electron transport through a triangular network of quantum nanorings is studied in the presence of Rashba spin-orbit interaction (RSOI) and a magnetic flux using quantum waveguide theory. This study illustrates that, by tuning Rashba constant, magnetic flux and incoming electron energy, the triangular network of quantum rings can act as a perfect logical spin-filtering with high efficiency. By changing in the energy of incoming electron, at a proper value of the Rashba constant and magnetic flux, a reverse in the direction of spin can take place in the triangular network of quantum nanorings. Furthermore, the triangular network of quantum nanorings can be designed as a device and shows several simultaneous spintronic properties such as spin-splitter and spin-inverter. This spin-splitting is dependent on the energy of the incoming electron. Additionally, different polarizations can be achieved in the two outgoing leads from an originally incoming spin state that simulates a Stern-Gerlach apparatus.
1
* Corresponding author. Tel.: +982331533254; fax: +982333654081 E-mail addresses: [email protected]
Keywords: Spintronics, Quantum nanoring, Rashba spin-orbit interaction, logical spin-filtering, Stern-Gerlach apparatus 1. Introduction Nowadays, improvement in technology, has led to a growing interest in a new product of lowdimensional quantum nanostructure [1-3]. Semiconductor-based devices are reliable for processing application and data storage, thereby as well as electronic, have a large contribution in the future of spintronics [4-6]. Low-dimensional quantum nanostructures such as confined zero-dimensional quantum dot (QD), quantum well (QW) and quantum ring (QR) have attracted enormous research interests in recent years and play an important role in microelectronic [7, 8]. In 1999, for the first-time quantum ring of the RSOI were indicated as a confining potential for spin interferometer device [9]. Afterward, specification of a quantum ring has been estimated in several articles [10, 11]. In recent years, some of the theoretical researchers posed quantum rings as a multipurpose and flexible spintronic device. For instance, Lee et al. introduced a new spin filter based on spinresolved Fano resonances coupled with a quantum wire (QW), in accordance with spin-split levels in a quantum ring (QR) [12]. In 2006, Citro et al investigated the effect of RSOI on the spin-dependent electron transport in a one-dimensional open semiconductor quantum ring. They explored a feasible spin-filtering model in the ring with tunable spin-orbit interaction. The proposed spin-filtering of Citro et al. is not perfect, and its efficiency dose not exceeding 70% 2
[13]. In another study, Foldi et al. analyzed the spin-orbit effect in an open quantum ring with one incoming and two outgoing leads. They also showed that such system can operate as a spinsplitter [14]. Later, a rectangular array of quantum rings exposed to the RSOI of a perpendicular magnetic field was explored by Kalman et al. (2008). They showed that, by changing various Rashba spin-orbit couplings (RSOC) in each individual ring [15], the system can control the spin-dependent electronic transmission. As a new study, Naeimi et al. explored spin-filtering properties of transmitted electrons through a quantum ring in the presence of RSOI and magnetic flux [16]. Additionally, in another work, they have studied spin-dependent electron transport in an open nonmagnetic double quantum ring, in the presence of RSOI; and showed that, the double quantum ring can act as a perfect electron spin-inverter with very high efficiency and spin-switch [17]. Newly, Faizabadi et al. have studied the radius effect of a triangular network of quantum nanorings in the presence of RSOI on the spintronic properties. They examined the effect of changing radius of the rings in the network and concluded that, such system can be considered as a multipurpose spintronic device [18]. As we know, any change in the radius of the ring, is not the most proper way for the network, because the geometry of a network will change as the radius of the rings experience a variation. An appropriate way to manipulate and control the electron spin is the Rashba spin-orbit interaction (RSOI) and magnetic flux. The strength of RSOI and magnetic flux can be easily controlled by an external electric field and a gate voltage. In this paper, we have studied the spin-dependent electron transport in a triangular network of quantum nanorings in the presence of RSOI and magnetic flux using quantum waveguide theory. It is shown that, for proper values of the RSOI and energy of incoming electrons, the designed system will indicate several spintronic properties such as logical spin-filtering, spin-inverter, and spin-splitter simultaneously. Moreover, we have introduced a new device which is able to be
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considered as a spintronic analog of the Stern-Gerlach apparatus. The incoming electrons, dependent on their energy, are forced to split into two different spatial parts with the geometrical construction of the semiconductor device [19-21]. This paper is as follows: The theoretical framework the model and formalism with RSOI is described in Section 2, the two-step numerical calculation is presented in Section 3, and the paper is concluded by a summary of the main results in Section 4. 2. Theoretical model As shown in Fig.1, we have considered a triangular quantum network which consists of three quantum ring attached to one incoming and two outgoing leads. Typically, in comparison with the radial dimensions, the thickness of the ring is small enough. Thus, the array is considered as a one-dimensional system, without transverse modes; so, with a pretty good approximation, one can consider only the lowest level of electron sub-band [22]. Furthermore, we ignore electron–electron and electron–phonon interactions, while the temperature of the system is considered to be zero. In addition, there are tunable magnetic fluxes through the center of each ring such that there is no magnetic field on them. At the first step, we will solve the problem for a single narrow quantum ring, and then will generalize the results for the triangular network of quantum rings. The relevant dimensionless Hamiltonian can be written as [23]
̂
[(
where
)
is the azimuthal angle,
]
(1) ⁄ m a is the dimensionless kinetic energy,
reduced Planck constant, m* is the effective mass of electron, a is the radius of the ring, 4
is the is the
r-component of the Pauli spin matrix in cylindrical coordinates (i.e., is the magnetic flux passing through the ring,
),
is flux quantum, and
≡
⁄ a
is the
normalized Rashba constant. The Rashba constant is assumed to (2) be tunable with the average electric field perpendicularly applied to the ring [24]. The Eigen values of the Hamiltonian [Eq. (1)] are given by [25] ψ(K, )
e
χ( ) ( )
where
1,2 referred to the Eigen-basis
〉 and
〉 z component of the spin,
respectively. The orthogonal spinors, χ( ) ( ) in [Eq. (2)], can be expressed in terms of the 1 0 eigenvectors ( ) and ( ) of Pauli matrix 0 1
χ( ) ( )
as
θ
e 1 ( √ π e
θ
(3)
)
And
χ( ) ( )
1 √ π
(
θ
e
θ
e ar ta (
where θ
(4)
)
). By use of Eq. (2), the corresponding energy eigenvalues are
obtained as follow [26]
E
[(K
1
Φ π
π
)
]
(5)
5
where Φ
( 1) √1
π[1
and K ≡ ( m E⁄
] is the so-called Aharonov-Casher (AC) phase [26],
) is electron wavenumber. Using Eq. (5), the possible values of electron
wavenumber k can be calculated as [27] ( 1)
K
q
(6) [( ⁄ )
where q
E⁄
]
⁄
and j=1,2 are correspond to the clockwise and
counterclockwise motions of an electron through the ring, respectively. We can write the state of the electron using of quantum waveguide theory in the upper and lower arms of one ring as [2830]. ψ
( )
,
ψ ( )
(7)
χ( ) ( )
∑ ∑a e ,
,
(8)
χ( ) ( )
∑ ∑b e ,
where the signs up and low denote the upper and lower arms of the ring, and a and b are unknown coefficients, which should be determined. In this paper, we apply Eqs. (7) And (8) for upper and lower arms of the three similar rings. According to Fig. 1, the electron wave function in the incoming, middle and outgoing leads, are as follows: f ( )e f
ψ (x)
ψ (x)
(
t t
t t
(
)e
r r
r )e r
(
r r
r r
(9)
)e
(10)
6
t t
t t
)e
(
t t
t t
)e
(
ψ (x)
(
ψ (x)
(
r r
r r
)e
(11)
r r
r r
)e
(12)
And
(13) ψ
(x)
(
t t
t )e t
ψ
(x)
(
t t
t )e t
where f (
(14)
, ) is the amplitude of the injected electron with spin , t and
transmission amplitudes of the electron with incoming and outgoing spins from the B (C) ring, and t
(t
,t
,)
and outgoing spin
) are the
respectively
is the transmission amplitude from the A (A, B) ring
forward to the B (C, C) ring, and similarly r incoming spin
(t
is the reflection amplitude of electron with
from the (A) ring and r
(r
,r
)is the reflection
amplitude from the A (A, B) ring backward to the B (C, C) ring. We have applied the boundary conditions to explain that wave function must be continues and the spin current density must be conserved at the junctions of the rings and the leads. Using Griffith’s boundary conditions [31], and employing the Gaussian elimination method [32], according to Eqs. (7) And (8) we can obtain
all
a , b , a , b , a , b , t
the ,t
, t
, t
unknown , t
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, r
,r
,r
, and
coefficients r
,
for three rings A, B
and C. Finally, the transmission coefficient for outgoing leads B and C can be determined by T
t
and T
t
.
3. Results and discussion A numerical study on spin-resolved properties of a triangular network of quantum nanorings with RSOI is presented using derived equations in Sec. 2. Here, the radius and the Rashba constant of the three rings are considered to be equal (i.e. aA=aB=aC and
). As the
Fig.1 shows, the angle between the leads in the system is considered to be constant. We choose the angles π⁄
and and
π⁄ and
of the rings as follows: for ring {A}, π⁄ , and ring {C},
π⁄
and
π⁄ , ring {B},
π⁄ . The effects of the
magnetic flux , the strength of the RSOI and energy of the incoming electron on spindependent electron transport are studied for different conditions. The advantage of the chosen triangular network allowed to derive simple analytical results and get their clear physical explanation. Comparing our results to the case of a single ring with three leads in Ref [14] it can be seen that for appropriate values of tunable parameters such as Rashba strength and magnetic flux, the system only can act as a perfect spin-splitter and it was separated incoming spin direction. While the main characteristics of triangular network are that, it acts as a multipurpose spintronic device such as logical spin-filtering, spin-inverting, spin-splitting and Stern-Gerlach apparatus at the same time. This network can be polarized spin direction in each output with high efficiency. As well as one of the most important advantages of this system is that, we can increase the system efficiency by changing the strength of Rashba spin-orbit intraction and the magnetic flux, that can be easily controlled by an external magnetic field and a gate voltage. In Fig. 2, the spin-resolved electron transmission coefficients T and T and the electron spinpolarization P are shown as a function of the energy of incoming electron, where T 8
is the
electron transmission coefficient without spin-inversion, and T is the electron transmission coefficient with spin-inversion from down to up. Here, the spin of incoming electrons are considered to be down and the normalized magnetic flux is illustrates the transmission coefficients T a
T
⁄
0. . Fig.2 (a) and (b)
as a function of the normalized electron
energy E for outgoing leads B and C, for an arbitrary value of Rashba constant ( . e. 1.
), respectively. As Fig. 2(a) shows, in a wide range of normalized electron energy (i.e.
0 < E < 1), the transmission coefficient for outgoing lead B, T (transmission without spininversion) is zero, while for all values of normalized electron energy the value (transmission with spin-inversion) is about 0.7, proving that spin-inversion can take place in presence of RSOI. According to Fig. 2(b) in a wide range of normalized electron energy (i.e. 0 < E < 1), the transmission coefficient T
is 0.7 (which is not a perfect transmission), as long as, for all values
of normalized electron energy T is zero (which shows a totally blocking). In fact, logical spinfiltering is not perfect because the value of electron transmission T is not equal to 1. The spinpolarization can be described as
(T
T )⁄(T
T
) where
. If P = -1, the
spin of transmitted electrons leftovers without changed but if P=1, the spin of transmitted electrons changes from up to down and vice versa. As we see in Fig. 2(c), the spin-polarization is positive and approximately about +1 for outgoing B, and negative and hardly reaches to -1 for outgoing C. It has seen that for the arbitrary Rashba constant, spin-polarization shifts from -1 (transmission without spin change) to +1 (transmission with mostly spin-inversion). In order to obtain an optimum logical spin-filtering, the contour plot of the spin-polarization versus α and
⁄
is depicted in Figs. 3(a) and (b) respectively. The values of E are assumed
about E=1.53 and E=1.38 for both outgoing leads. The spin polarization versus α is varying between negative and positive values related to spin-up and spin-down transmission probability, 9
while the sign in the two leads are opposite. The spin properties for some values of Rashba strength has optimized (for example α = 2.5). According to the Figs. 3(a) and (b), spin polarization in the lead B is exactly equal to the lead C, with the opposite signs. Adjustment of the energy of incident electron remained in RSOI strength, provided suitable conditions; in that, not only 100 percent of the incoming spin-down current transmitted into one of the outgoings leads, but also 100 percent of the incoming spin-down current could be directed toward the other one simultaneously, so proper spintronic properties observed clearly. In addition, perfect spininverting and spin-filtering are observed in outgoing leads B and C, respectively. The RSOI behaves like an effective in-plane momentum-dependent magnetic field, and the fully spin-down polarized electron transmission in the incoming lead will be changed to the spinup in the outgoing leads at specific RSOI. For the moving electrons along with the ring in different paths, their phase difference should be affected using the alteration of the Rashba constant. Therefore, it has affected the interferences of electron waves and consequently by changing RSOI constant, the transmission of the network has modified. By moderating the gate voltage, we can obtain the reliable value of Rashba constant for the state that the spin-inversion is perfectly present. Close quantum ring eigenvalue can affect the electron spin transition passing through the ring that means, when energy of incoming electron is equal to one of the quantum ring eigenvalues, the electron can pass through the ring. In the open quantum ring and network, the presence of RSOI causes that quantum ring eigenvalue is dependent on the electron spin. As we know, the RSOI acts similar to the effective magnetic field oriented along the Zdirection. There is a characteristic length scale called spin precession length. For normal incoming electrons when the Rashba barrier length is equal to one-half of spin precession length, 10
the spin of electrons rotates 1 0 , hence the spins along +z (up) invert to -z (down) and contrariwise [33]. Although we have obtained a reasonable value for spin polarization, the low value of transmission causes a reduction in the efficiency of that system. Therefore, in order to get an appropriate output, we have plotted T ⁄
T and T
T as a function of RSOI strength and
in Fig. 4 (a) and (b) respectively. According to the Fig. 4(a) and (b) there are large
differences between spin-up and spin-down transmission probability of two outgoings leads, leads us to have the high transmission probability in each outgoing lead. The value of +1(-1) in Fig. 4(a) (4(b)) demonstrates 100 percent spin-up (spin-down) state and indicates that the system operate as a perfect filtering of spin-down (spin-up). According to the contour plot in Figs. 3 and 4, we plotted transmission coefficients as a function of energy. Fig. 5 shows transmission coefficients T and T for an appropriate value of Rashba constant (i.e.
. ) and the normalized magnetic flux
⁄
0.0. Fig
5(a) and (b) denotes 100 percent spin transition for spin-up (spin-down) in E=1.43 and E=1.53, indicating perfect filtering of spin-down and spin-up respectively. For further clarification, the perfect spin-polarization is demonstrated in Fig. 5(c). Therefore, by choosing the appropriate Rashba constant, controlled by a gate voltage, the direction of the spin-up and spin-down transmission probability in two outgoing leads is reversed simultaneously in the appropriate range of energy spectra. From the theoretical point of view, spin transport dependence on magnetic flux and RSOI through the ring, so the changing in magnetic flux and Rashba strength affected the spin current of the ring. The RSOI breaks the degeneracy of up and down spin states and makes them
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circulate in opposite directions in the quantum ring, thereby producing a pure persistent spin current in the system. In addition, the numerical calculations show that, the transmission coefficients with/without spin-inversion for incoming with spin-up electrons are the same as those of the incoming with spin down electrons (i.e. T
T or T
T ). As a significant result, the triangular network
of the quantum rings can act as a logical spin-filtering. Additionally, this network of the quantum ring can be used as a perfect spin-splitting based on the energy of incoming electron (E=1.53 and E=1.38). The point is that, when the energy of incoming electron is equal to 1.53, electron passes thorough lead B and when this is 1.38, they pass thorough the lead C. Moreover, this array of quantum rings with one input and two outgoing leads in the presence of RSOI has remarkable similarities with a Stern-Gerlach apparatus because we can separately direct each of the up and down spins toward each of the output leads simultaneously. 4. Conclusions In conclusion, we have calculated the Spin-dependent electron transmission and spinpolarization of a triangular network in presence of RSOI and external magnetic field by using the wave guide theory. We demonstrated that, a triangular network of the quantum ring with RSOI can serve as a logical spin-filtering for electron spins. In addition, it has been shown that efficiency of spin-filter has been improved and trended up to a high value (very close to 100%). It is important that, this network is more efficient than a single ring with fewer controllable parameters, so it can be seen as a multipurpose spintronic device. The spin-polarization of transmitted electrons can be changed from -1 (merely spin-down) to +1 (merely spin-up) for two outgoings and contrariwise by altering the Rashba strength and magnetic flux. Also by altering the input spin to spin-up, all the results will be inverted. To obtain perfect spin polarization and 12
spin transition, the contour maps of spin polarization and spin transition are plotted in term of the magnetic flux and Rashba constant respectively. We have also obtained the optimum values by varying the RSOI constant and the magnetic flux, and the result is that, a better spin-inverting can still be achieved. We believe that, the aforementioned study can be used to design efficient spintronic nano-devices. Besides, we have already found that, in case of symmetric geometry, spin-splitting is dependent on the energy of incoming electron, and the presence of RSOI causes the Stern-Gerlach type spin separation. References [1] S. Datta and M. Das, Electronic analog of the electro‐optic modulator, Appl. Phys. Lett. 56 (1990) 665-667. [2] V. Moldovaenu, and B. Tanatar, Tunable spin currents in a biased Rashba ring, Phys. Rev. B. 81, (2010) 035326-035329. [3] L. Diago-Cisneros, F. Mireles, Quantum-ring spin interference device tuned by quantum point contacts, Appl. Phys. 114 (19) (2013) 193706-193713. [4] P. D. Hodgson, GaSb quantum rings in GaAs/AlxGa1− xAs quantum wells, Appl. Phys. 119 (4) (2016) 044305-044307. [5] E.Cota, R.Aguado, and G.Platero, AC-driven double quantum dots as spin pumps and spin filters, Phys. Rev. Let. 94 (10) (2005) 107202-107206. [6] R. Sánchez, Spin-filtering through excited states in double-quantum-dot pumps, Phys. Rev .B. 74 (3) (2006) 035326-035335.
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[7] L. Eslami, E. Faizabadi, and S. Ahmadi, Quantum nano ring composed of quantum dots as a source of pure persistent spin or charge current, Phys. Lett. A. 380 (45) (2016) 3854-3860. [8] M. Molavi, and E. Faizabadi, Spin-polarization and spin-flip in a triple-quantum-dot ring by using tunable lateral bias voltage and Rashba spin-orbit interaction, Magn. Magn. Mater. 428 (2017) 488-492. [9] J. Nitta, F. E. Meijer, and H. Takayanagi, Spin-interference device, Appl. Phys. Lett. 75 (695) (1999) 143-144. [10] J. Splettstoesser, M. Governale, and U. Zu licke, Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling, Phys. Rev. B. 68 (16) (2003) 165341-165350. [11] P. Foldi, O. Ka ´lma ´n, and F. M. Peeters, tability of spintronic devices based on quantum ring networks, Phys. Rev. B. 80 (12) (2009) 125324-125332. [12] L. Minchul, and C. Bruder, Spin filter using a semiconductor quantum ring side coupled to a quantum wire, Phys. Rev .B, 73 (8) (2006) 085315-085319. [13] R. Citro, F. Romeo, and M. Marinaro, Zero-conductance resonances and spin filtering effects in ring conductors subject to Rashba coupling, Phys. Rev .B. 74 (11) (2006) 115329115339. [14] P. Földi, Quantum rings as electron spin beam splitters, Phys. Rev .B. 73 (15) (2006) 155325-155329. [15] O. Kálmán, Magnetoconductance of rectangular arrays of quantum rings, Phys. Rev .B. 78 (12) (2008) 125306-
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[16] A. Naeimi, M. Esmaeilzadeh, A wide range of energy spin-filtering in a Rashba quantum ring using S-matrix method, Appl. Phys. 113 (4) (2013) 044316-044322. [17] A. Naeimi, Spin transport properties in a double quantum ring with Rashba spin-orbit interaction, Appl. Phys. 113 (2013) 014303-014310. [18] E. Faizabadi, M. Molavi, Radius effect on the spintronic properties of a triangular network of quantum nanorings in the presence of Rashba spin-orbit interaction, Current Appl. Phys. 17 (2) (2017) 207-213. [19] S.Souma, and B.K. Nikolić, Spin Hall current driven by quantum interferences in mesoscopic rashba rings, Phys. Rev. Lett. 94 (10) (2005) 106602-106606. [20] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Electron spin interferometry using a semiconductor ring structure, Appl. Phys. Lett. 86 (5) (2005) 162107162110. [21] J. Cserti, A. Csord´as, U. Z¨ulicke, Electronic and spin properties of Rashba billiards, Phys. Rev. B. 70 (23) (2004) 233307-233311. [22] H. B. Heersche, Z. deGroot, J. A. Folk, L. P. Kouwenhoven, H. S. J. van der Zant, A. A. Houck, J. Labaziewicz, and I. L. Chuang, Kondo effect in the presence of magnetic impurities, Phys. Rev. Lett. 96 (1) (2006) 017205-017209. [23] F.E. Meijer, A.F.Morpugo, T. M. Klapwijk, One-dimensional ring in the presence of Rashba spin-orbit interaction: derivation of the correct Hamiltonian, Phys. Rev. B. 66 (3) (2002) 033107033110.
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[24] D. Grundler, Large Rashba splitting in InAs quantum wells due to electron wave function penetration into the barrier layers, Phys. Rev. Lett. 84 (26) (2000) 6074-6077. [25] P. F¨oldi, B. Moln´ar, M. G. Benedict, and F. M. Peeters, Spintronic single-qubit gate based on a quantum ring with spin-orbit interaction, Phys. Rev. B. 71 (3) (2005) 033309-033313. [26] A. Tsuneya, et al., eds, Mesoscopic physics and electronics. Springer Science & Business Media, (2012). [27] G. A. Prinz, Magnetoelectronics, Science 282 (5394) (1998) 1660-1663. [28] J. B. Xia, Quantum waveguide theory for mesoscopic structures, Phys. Rev. B. 45 (7) (1992) 3593-3599. [29] H. C. Wu, Y. Guo, X. Y. Chen, and B. L. Gu, Rashba spin-orbit effect on traversal time in ferromagnetic/semiconductor/ferromagnetic heterojunction, J. Appl. Phys. 93 (9) (2003) 53165320. [30] J. R. Shi, and B. Y. Gu, Quantum waveguide transport with side-branch structures: A recursive algorithm, Phys. Rev. B. 55 (7) (1997) 4703-4715. [31] S. Griffith, Trans. Faraday Soc. 49, 345 (1953); ibid. 49, 4825 (1997). [32] G. H. Golub and C. F. Van Loan, Matrix Computations (The Johns Hopkins University Press, Baltimore, 1996). [33] S. Ahmadi, Spin-inversion in nanoscale graphene sheets with a Rashba spin-orbit barrier, AIP Adv. 2 (1) (2012) 012130-012139.
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Figure 1: Schematic representation of a triangular array of quantum rings coupled to three leads in the presence of RSOI and magnetic flux. The arrows illustrate the possible directions of the currents.
(a)
18
(b)
(c)
19
Figure 2: Spin-dependent electron transmission coefficients T and T as a function of electron 1.
energy for an arbitrary Rashba constant (i.e.
) (a) The B outgoing (b) the C
outgoing (c) the Spin-polarization P as a function of electron energy corresponding to Figs. 2(a) and 2(b).
(a)
20
(b)
Figure 3: The contour plot of spin-polarization versus α, ϕ/ϕ0 (a) B outgoing lead and E=1.53 (b) C outgoing lead and E=1.38.
(a)
21
(b)
Figure 4: The contour plot of transition coefficient versus α, ⁄ .
22
(a)
and (b)
(a)
(b)
23
(c)
Figure 5: Spin-dependent electron transmission coefficients T and T and Spin-polarization as a function of electron energy for an arbitrary Rashba constant.(i.e. normalized magnetic flux
⁄
. ) and the
0.0 (a) The B outgoing (b) the C outgoing (c) Spin-
polarization (P) as a function of electron energy corresponding to Figs. 5(a) and 5(b).
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