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Non-stationary spin-filtering effects in correlated quantum dot
Physica E 93 (2017) 224–229
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Non-stationary spin-filtering effects in correlated quantum dot a,⁎
a
MARK
b
V.N. Mantsevich , N.S. Maslova , P.I. Arseyev a b
Lomonosov Moscow State University, Moscow 119991, Russia P.N. Lebedev Physical Institute RAS, Moscow 119991, Russia
A R T I C L E I N F O
A BS T RAC T
Keywords: Spin polarized transport Correlated quantum dots Relaxation times and mean free paths Spin filtering
The influence of external magnetic field switching “on” and “off” on the non-stationary spin-polarized currents in the system of correlated single-level quantum dot coupled to non-magnetic electronic reservoirs has been analyzed. It was shown that considered system can be used for the effective spin filtering by analyzing its nonstationary characteristics in particular range of applied bias voltage.
1. Introduction One of the key issues of spintronics is the control and generation of spin-polarized currents. Nowadays generation and detection of spinpolarized currents in semiconductor nanostructures has attracted great attention since this is the key problem in developing semiconductor spintronic devices [1–4]. To generate tunable highly spin-polarized stationary currents the variety of systems has been already proposed ranging from semiconductor heterostructures to low-dimensional mesoscopic samples [5–8]. Significant progress has been achieved in experimental and theoretical investigation of stationary spin-polarized transport in magnetic tunneling junctions [9–12]. Nevertheless spinpolarized current sources based on the non-magnetic materials are attractable as one could avoid the presence of accidental magnetic fields that may result in the existence of undesirable effects on the spin currents. It was demonstrated recently that stationary tunneling current could be spin dependent in the case of non-magnetic leads [13,14]. There have been several proposals to generate stationary spinpolarized currents using non-magnetic materials: small quantum dots [15,16] and coupled quantum dots [5,17] built in semiconducting nanostructures in the presence of external magnetic field. Moreover, quantum dots systems based on the non-magnetic materials were proposed as a spin filter prototypes [18,19]. Effective spin filtering in such systems requires to have many quantum dots with the Coulomb correlations inside each dot [20–22] and also between the dots [5]. To the best of our knowledge usually stationary spin-polarized currents are analyzed. However, creation, diagnostics and controllable manipulation of charge and spin states in the single and coupled quantum dots (QDs), applicable for ultra small size electronic devices design requires analysis of non-stationary effects and transient processes [23–28]. Consequently, non-stationary evolution of initially ⁎
Corresponding author. E-mail address: [email protected] (V.N. Mantsevich).
http://dx.doi.org/10.1016/j.physe.2017.06.027 Received 29 May 2017; Received in revised form 21 June 2017; Accepted 23 June 2017 Available online 24 June 2017 1386-9477/ © 2017 Elsevier B.V. All rights reserved.
prepared spin and charge configurations in correlated quantum dots is of great interest both from fundamental and technological point of view. In this paper we analyze non-stationary spin polarized currents through the correlated single-level QD localized in the tunnel junction in the presence of applied bias voltage and external magnetic field, which can be switched “on” or “off” at a particular time moment. We demonstrate that single biased QD in the external magnetic field can be considered as an effective spin filter based on the analysis of nonstationary spin-polarized currents, which can flow in the both leads. Currents direction can be tuned by the external magnetic field switching “on” or “off”. 2. Theoretical model We consider non-stationary processes in the single-level quantum dot with Coulomb correlations of localized electrons situated between two non-magnetic electronic reservoirs in the presence of external magnetic field B switched “on”/“off” at t = t0 . We are interested in the influence of magnetic field directly on the quantum dot [5]. Modern experimental scanning probe technique provides possibility of applying magnetic field to various subsystems: directly to the dot, to the reservoirs or both to the dot and reservoir with the controllable field direction [29–32]. Moreover, when the uniform magnetic field in the dot and in the lead is applied to the system the situation when g-factor in the leads is much smaller than in the dots can be realized due to the materials properties (strongly different effective masses of electrons in the reservoir and in the dot). So only the effect of external magnetic field on the dot states was considered. The Hamiltonian of the system
l=H lQD + H lR + H lT H
(1)
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V.N. Mantsevich et al.
can be written as a sum of the single-level quantum dot part
lQD = ∑ ε1 nl1σ + Unl1σ nl1−σ , H
i (2)
σ
i
non-magnetic electronic reservoirs Hamiltonian
lR = H
∑ εk clkσ+ clkσ + ∑ (εp − eV ) clpσ+ clpσ kσ
k ′≠ k
i
∑ tk (clkσ+ cl1σ + cl1+σ clkσ ) + ∑ tp (clpσ+ cl1σ + cl1+σ clpσ ). kσ
pσ
(5)
∂nl1σk = − (ε1σ − εk ) · nl1σk − U · nl1−σ nl1σk ∂t + tk · (nl1σ − nlkσ ) − ∑ tk ′ · clk+′ σ clkσ ,
(3)
pσ
and the tunneling part
lT = H
∂nl1σ = − ∑ tk · (nlkσ1 − nl1σk ), ∂t k,σ
(4)
∂clk+′ σ clkσ = − (εk ′ − εk ) · clk+′ σ clkσ ∂t − tk ′ · cl1+σ clkσ + tk · clk+′ σ cl1σ .
(6)
(7)
and
Here index k(p) labels continuous spectrum states in the leads, tk ( p ) is the tunneling transfer amplitude between continuous spectrum states and quantum dot with the energy level ε1 which is considered to be independent on the momentum and spin. Operators clk+( p ) σ / clk ( p ) σ are the creation/annihilation operators for the electrons in the continuous spectrum states k(p). nl1σ (−σ ) = cl1+σ (−σ )cl1σ (−σ ) -localized state electron occupation numbers, where operator cl1σ (−σ ) destroys electron with spin σ (−σ ) on the energy level ε1. U is the on-site Coulomb repulsion for the double occupation of the quantum dot. External magnetic field B leads to the Zeeman splitting of the impurity single level ε1 proportional to the atomic g factor. In the absence of external magnetic field direct exchange and spin-orbit interaction can often lead to the appearance of effective magnetic field acting on electron spins. In the simple mean-field approach electron spin is affected by a mean nuclear spin polarization similar to effective magnetic field and thus hyperfine level splitting appears. The non-negligible value of effective magnetic field averaged value appears due to the finite number of nuclei within the quantum dot. For example, in GaAs this field leads to effective level splitting about 135μeV [36,37]. Consequently, for deep QD's electron levels (about 10 − 100 meV ) effective level splitting can be omitted as it doesn't influence opposite spin electron occupation numbers. When quantum dot's energy levels position is comparable to the hyperfine levels splitting two different cases can be distinguished. The first one deals with the value of level splitting can be of the order or smaller then the relaxation rate caused by the coupling to reservoir. In this situation level splitting due to the hyperfine interaction is not important. The second possible situation is when the value of level splitting exceeds the relaxation rate. In this case if the effective magnetic field has the same sign for each spin direction the symmetry between opposite spin states for localized electrons is broken even without external magnetic field. The effects of external magnetic field switching “on” and “off” now should drastically depend on its direction. Another interesting effect can appear in the presence of strong spinorbit interaction, when the direction of the effective magnetic field is opposite for the two different electron spin projections. This situation is typical for spin-Hall systems and one should expect the appearance of a pair of counter-propagating currents for each spin projection. Further results deal with the case of deep energy levels or correspond to the situation when relaxation rate exceeds hyperfine level splitting (as typical values of relaxation rate can be of order of 1 − 5 meV [38,39], while hyperfine level splitting is about 135μeV [36,37]). Further analysis deals with the low temperature regime when the Fermi level is well defined and the temperature is much lower than all the typical energy scales in the system. Consequently, the distribution function of electrons in the leads (band electrons) is close to the Fermi step.
i
+ ∂clpσ clkσ
∂t
+ = (εk − εp ) · clpσ clkσ + + + tk · clpσ cl1σ − tp · cl1+σ clkσ ,
(8)
+ clkσ clkσ
nlkσ
is an occupation operator for the electrons in the where = reservoir and εσ = ε1 + σ μB where σ = ± 1. Equations of motion for the electron operators products cl1+σ clpσ and clp+′ σ clpσ can be obtained from Eqs. (6) and (7) correspondingly by the indexes substitution k ↔ p and k′ ↔ p′. Following the logic of Ref. [33] one can get kinetic equations for the electron occupation numbers operators time evolution in the case of external magnetic field B switching “on” at the time moment t = t0 > 0 :
∂n1σ = − 2Θ (t0 − t ) · γ × [n1σ − (1 − n1−σ ) · ΦεT (t ) − n1−σ · ΦεT+ U (t )] ∂t − 2Θ (t − t0 ) · γ × [n1σ − (1 − n1−σ ) · Φε+T (t ) − n1−σ · Φε++TU (t )], ∂n1−σ = − 2Θ (t0 − t ) · γ × [n1−σ − (1 − n1σ ) · ΦεT (t ) − n1σ · ΦεT+ U (t )] ∂t − 2Θ (t − t0 ) · γ × [n1−σ − (1 − n1σ ) · Φε−T (t ) − n1σ · Φε−+TU (t )], (9) where γ = γk + γp and γk ( p ) = states in the leads and
πν0 tk2( p ) .
ν0 – is the unperturbed density of
γ lε± T (t ) = γk · Φ lkε± (t ) + p · Φ lpε± (t ), Φ γ γ γ lε±+ UT (t ) = γk · Φ lkε±+ U (t ) + p · Φ lpε±+ U (t ), Φ γ γ
(10)
where
lε± (t ) = 1 i · dεk · f σ (εk ) Φ k 2 ⎡ 1 − ei (ε1 ± μB + iΓ − εk ) t 1 − e−i (ε1 ± μB − iΓ − εk ) t ⎤ ×⎢ − ⎥, ⎣ ε1 ± μB + iΓ − εk ε1 ± μB − iΓ − εk ⎦ lε±+ U (t )= 1 i · dεk · f σ (εk ) Φ k 2 ⎡ 1 − ei (ε1 ± μB + U + iΓ − εk ) t 1 − e−i (ε1 ± μB + U − iΓ − εk ) t ⎤ ×⎢ − ⎥. ⎣ ε1 ± μB + U + iΓ − εk ε1 ± μB + U − iΓ − εk ⎦
∫
∫
(11)
Initially (t < t0) magnetic field B is absent [μB = 0 in Eqs. (9)–(11)] lε± (t ) = Φ lε (t ). To and, consequently, the following relation is valid Φ analyze system kinetics in the situation when magnetic field was initially present in the system and switched “off” at t = t0 one can easily generalize Eq. (9) by substitution t ↔ t0 . Equations for the localized electrons occupation numbers n1σ ± (t ) can be obtained by averaging Eqs. (9)–(11) for the operators and by decoupling electrons occupation numbers in the leads. Such decoupling procedure is reasonable if one considers that electrons in the macroscopic leads are in the thermal equilibrium [34,35]. After decoupling one has to replace electron occupation numbers operators in the reservoir nlkσ in Eqs. (9)–(11) by the Fermi distribution functions f kσ .
3. Non-stationary electronic transport formalism Let us further consider = = 1 and e = 1 elsewhere, so the motion equations for the electron operators products nl1σ , nl1σk = cl1+σ clkσ and clk+′ σ clkσ can be written as: 225
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Fig. 1. Occupation numbers time evolution. Panels a)-c) correspond to the magnetic field switching “on”, panels d)-f) correspond to the magnetic field switching “off”. a),d) ε1/2γ = −1.25 , eV /2γ = −2.5; b),e) ε1/2γ = −1.25 , eV /2γ = −7.5; c),f) ε1/2γ = −5, eV /2γ = −6.25. Parameters U /2γ = 10 , μB /2γ = −3.25, γk = γp = γ = 1 and initial conditions n1σ (0) = 0.6 ,
n1−σ (0) = 0.4 are the same for all the figures.
their direction and polarization can be tuned by magnetic field B switching “on”/“off”. Non-stationary spin-polarized currents Ik±( p )(t ) behavior for magnetic field switching “on”/“off” is shown in Figs. 2 and 3. Corresponding electron occupation numbers behavior is depicted in Fig. 1. Schemes of the QD energy levels both in the presence (ε+ and ε-) and in the absence (ε1) of magnetic field are shown in Fig. 4. Let us first focus on the situation when magnetic field is present at the initial time moment and switched “off” at t = t0 (see Fig. 1a,c,e and Fig. 2). In the presence of magnetic field when condition ε+ < EF − eV occurs (energy level ε- can be localized higher or lower than EF) (see Fig. 4a,c), nonstationary spin-polarized currents Ik−(t ) and Ik+(t ) in the lead with EF = 0 are flowing in the same direction (see Fig. 2a,c), contrary to the currents I p−(t ) and I p+(t ) flowing in the opposite directions in the lead with the Fermi level shifted by the applied bias voltage (lead p) (see Fig. 2d,f). In the
4. Non-stationary spin-polarized currents If the initial state is a “magnetic” one, non-stationary spin-polarized currents Ik ( p ) (t ) ± flow in the each contact lead:
Ik± (t ) = − 2γk · [n1± σ − (1 − n1∓σ ) · Φkε± (t ) − n1∓σ · Φkε±+ U (t )] − 2γk · [n1± σ − (1 − n1∓σ ) · Φkε (t ) − n1∓σ · Φkε + U (t )], I p± (t )
= − 2γp · [n1± σ − (1 − n1∓σ ) · Φpε± (t ) − n1∓σ · Φpε±+ U (t )] − 2γp · [n1± σ − (1 − n1∓σ ) · Φpε (t ) − n1∓σ · Φpε + U (t )],
(12)
where electron occupation numbers n1± σ are determined from the system of Eq. (9) with the magnetic initial conditions. Non-stationary spin-polarized currents can flow in the both leads and 226
Physica E 93 (2017) 224–229
V.N. Mantsevich et al.
Fig. 2. Normalized non-stationary spin-polarized tunneling currents Ik ( p ) ± (t ) /2γ in the case of magnetic field switching “off” at t = t0 . Panels a)-c) demonstrate Ik ± (t ) /2γ , panels d)-f) demonstrate I p ± (t ) /2γ . a),d) ε1/2γ = −1.25 , eV /2γ = −2.5; b),e) ε1/2γ = −1.25, eV /2γ = −7.5; c),f) ε1/2γ = −5, eV /2γ = −6.25. Parameters U /2γ = 10 , μB /2γ = −3.25, γk = γp = γ = 1 and initial conditions n1σ (0) = 0.6 , n1−σ (0) = 0.4 are the same for all the figures.
stationary state all currents Ik±( p )(t ) values turn to zero. Magnetic field switching “off” results in the appearance of non-zero spin-polarized currents in both leads. Non-stationary spin-polarized currents Ik−(t ) and Ik+(t ) in the lead with EF = 0 continue flowing in the same direction with the same non-zero amplitude (see Fig. 2a,c). Currents I p−(t ) and I p+(t ) are also flowing in the same direction but magnetic field switching “off” results in the appearance of total current strong spin polarization at the initial stage of relaxation as the amplitude of current I p+(t ) strongly exceeds the amplitude of non-stationary current I p−(t ) (see Fig. 2d,f). Similar behavior of electron occupation numbers and non-stationary spinpolarized currents for two different positions of ε- (see Fig. 4a,c) is the result of the Coulomb correlations presence in the system. In both cases energy level ε+ is occupied and energy level ε- is unoccupied (even in the case depicted in Fig. 4c) due to the strong Coulomb repulsion. Non-
stationary current I p+(t ) changes direction with the magnetic field switching “off”, while currents Ik± (t ) and I p−(t ) are flowing in the same direction both in presence and in the absence of magnetic field. This effect can be applied for the effective spin-filtering in the single QD system alternatively to the previously proposed spin-filtering mechanisms based on the analysis of multiple QDs stationary characteristics [5]. Fig. 4b demonstrates that in the presence of magnetic field at the initial stage of relaxation non-stationary spin-polarized currents Ik± (t ) can flow in the opposite directions and currents I p± (t ) in the same directions (see Fig. 2b,e). Stationary state reveals the presence of only one non-stationary current flowing in each lead (Ik+(t ) and I p+(t ) correspondingly). Magnetic field switching “off” causes the appearance of both spin-polarized currents Ik± (t ) and I p± (t ) flowing in each lead in the same direction. In the stationary state spin currents values in each lead become equal.
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Fig. 3. Normalized non-stationary spin-polarized tunneling currents Ik ( p ) ± (t ) /2γ in the case of magnetic field switching “on” at t = t0 . Panels a)-c) demonstrate Ik ± (t ) /2γ , panels d)-f) demonstrate I p ± (t ) /2γ . a),d) ε1/2γ = −1.25 , eV /2γ = −2.5; b),e) ε1/2γ = −1.25, eV /2γ = −7.5; c),f) ε1/2γ = −5, eV /2γ = −6.25. Parameters U /2γ = 10 , μB /2γ = −3.25, γk = γp = γ = 1 and initial conditions n1σ (0) = 0.6 , n1−σ (0) = 0.4 are the same for all the figures.
netic field switching “on” allows to consider single QD as an effective spin-filter based on the analysis of its non-stationary characteristics. In the absence of magnetic field non-stationary spin-polarized currents in each lead Ik± (t ) and I p± (t ) are flowing in the same direction and
Electron occupation numbers and non-stationary spin-polarized currents behavior in the case when magnetic field is absent at the initial time moment and switched “on” at t = t0 is shown in Fig. 1b,d,f and Fig. 3 correspondingly. Obtained results demonstrate that mag-
Fig. 4. Sketch of the correlated QD energy levels coupled to non-magnetic leads both in the presence (ε+ and ε-) and in the absence (ε1) of magnetic field.
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[4] P. Chuang, S.C. Ho, L.W. Smith, F. Sfigakis, M. Pepper, C.H. Chen, J.C. Fan, J.P. Griffits, I. Farrer, H.E. Beer, et al., Nat. Nanotechnol. 10 (2015) 35. [5] M.Yu Kagan, V.V. Val’kov, S.V. Aksenov, Phys. Rev. B 95 (2017) 035411. [6] I. Zutic, J. Fabian, S.D. Sarma, Rev. Mod. Phys. 76 (2004) 323. [7] M.E. Torio, K. Hallberg, S. Flach, A.E. Miroshnichenko, M. Titov, Eur. Phys. J. B 37 (2004) 399. [8] A.M. Shakirov, Yu.E. Shchadilova, A.N. Rubtsov, P. Ribeiro, Phys. Rev. B 94 (2016) 224425. [9] E.Y. Tsymbal, O. Mryasov, P.R. LeClair, J. Phys, Condens. Matter 15 (2003) R109. [10] H.J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier, H.-P. Schonherr, K.H. Ploog, Phys. Rev. Lett. 87 (2001) 116601. [11] Y. Ohno, D.K. Young, B. Beschoten, F. Matsukura, H. Ohno, D.D. Awschalom, Nature 402 (1999) 790. [12] R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag, L.W. Molenkamp, Nature 402 (1999) 787. [13] V.I. Perel', S.A. Tarasenko, I.N. Yassievich, S.D. Ganichev, V.V. Bel'kov, W. Prettl, Phys. Rev. B 67 (2003) 201304. [14] M.M. Glazov, P.S. Alekseev, M.A. Odnoblyudov, V.M. Chistyakov, S.A. Tarasenko, I.N. Yassievich, Phys. Rev. B 71 (2005) 155313. [15] P. Recher, E.V. Sukhorukov, D. Loss, Phys. Rev. Lett. 85 (2000) 1962. [16] R.M. Potok, J.A. Folk, C.M. Marcus, V. Umansky, M. Hanson, A.C. Gossard, Phys. Rev. Lett. 91 (2003) 016802. [17] J.A. Andrade, P.S. Cornaglia, Phys. Rev. B 94 (2016) 235112. [18] T. Koga, J. Nitta, H. Takayanagi, S. Datta, Phys. Rev. Lett. 88 (2002) 126601. [19] A. Voskoboynikov, S.S. Liu, C.P. Lee, Phys. Rev. B 58 (1998) 15397. [20] W. Gong, Y. Zheng, Y. Liu, T. Lu, Phys. Rev. B 73 (2006) 245329. [21] J.H. Ojeda, M. Pacheco, P.A. Orellana, Nanotechnology 20 (2009) 434013. [22] H.-H. Fu, K.-L. Yao, Appl. Phys. Lett. 100 (2012) 013502. [23] I. Bar-Joseph, S.A. Gurvitz, Phys. Rev. B 44 (1991) 3332. [24] S.A. Gurvitz, M.S. Marinov, Phys. Rev. A 40 (1989) 2166. [25] P.I. Arseyev, N.S. Maslova, V.N. Mantsevich, Eur. Phys. J. B 85 (7) (2012) 249. [26] C.A. Stafford, N.S. Wingreen, Phys. Rev. Lett. 76 (1996) 1916. [27] B.L. Hazelzet, M.R. Wegewijs, T.H. Stoof, Y.V. Nazarov, Phys. Rev. B 63 (2001) 165313. [28] E. Cota, R. Aguado, G. Platero, Phys. Rev. Lett. 94 (2005) 107202. [29] C.F. Hirjibehedin, C.P. Lutz, A.J. Heinrich, Science 312 (2006) 1021. [30] C.F. Hirjibehedin, C.-Y. Lin, A.F. Otte, M. Ternes, C.P. Lutz, B.A. Jones, A.J. Heinrich, Science 317 (2007) 1199. [31] A.F. Otte, M. Ternes, K. von Bergmann, S. Loth, H. Brune, C.P. Lutz, C.F. Hirjibehedin, A.J. Heinrich, Nat. Phys. 4 (2008) 847. [32] D. Iaia, A. Kubetzka, K. von Bergmann, R. Wiesendanger, Phys. Rev. B 93 (2016) 134409. [33] N.S. Maslova, P.I. Arseyev, V.N. Mantsevich, Solid State Commun. 48 (2016) 21. [34] J.Q. You, H.-Z. Zheng, Phys. Rev. B 60 (1999) 8727. [35] J.Q. You, H.-Z. Zheng, Phys. Rev. B 60 (1999) 13314. [36] D. Paget, G. Lampel, B. Sapoval, V.I. Safronov, Phys. Rev. B 15 (1977) 5780. [37] B. Urbaszek, X. Marie, T. Amand, O. Krebs, P. Voisin, P. Maletinsky, A. Hogele, A. Imamoglu, Rev. Mod. Phys. 85 (2013) 79. [38] E. Nielsen, Phys. Rev. B 88 (2013) 195131. [39] J. Fransson, Phys. Rev. B 69 (2004) 201304. [40] S. Amaha, W. Izumida, T. Hatano, S. Teraoka, S. Tarucha, J.A. Gupta, D.G. Austing, Phys. Rev. Lett. 110 (2013) 016803.
demonstrate equal non-zero stationary values (see Fig. 3). Magnetic field switching “on” results in the direction changing of one of the spinpolarized currents in the leads (see Fig. 3a,d,f). Another possible situation deals with fast switching “off” of one of the spin-polarized currents in each lead when magnetic field is switched “on” (see Fig. 3b,e). Consequently, only non-stationary current with a certain spin orientation continue flowing in each lead in the presence of magnetic field. To observe these effects the switching times of magnetic field must be smaller than the lifetime of the initially prepared magnetic states. Modern scanning tunneling microscopy/spectroscopy experiments provide possibility to achieve typical spin-polarized current values of the order of 10 pA÷10 nA (1 nA ≃ 6 × 109e / sec ) [40,39], which corresponds to the relaxation time scales 1/ γ ≃ 1 ÷ 100 ns for the system parameters depicted in Figs. 2 and 3. It gives the possibility to realize proposed protocols for the state of art experiments by means of pico- or femtosecond switchers, for example by means of laser pulses. 5. Conclusion We have analyzed the behavior of spin-polarized non-stationary currents in the system of single-level quantum dot situated between two non-magnetic electronic reservoirs with Coulomb correlations of localized electrons in the presence of external magnetic field switched “on” or “off” at particular time moment. It was demonstrated that single-level correlated quantum dot can be considered as an effective spin filter depending on the ration between the values of magnetic field induced energy level splitting and applied bias voltage. Acknowledgements This work was supported by RFBR Grant 16-32-60024 mol-a-dk and by the RF President Grant for young scientists MD-4550.2016.2. References [1] Semiconductor spintronics and quantum computation, in: D.D. Awschalom, D. Loss, N. Samarth (Eds.), Nanoscience and Technology, Springer, Berlin, 2002. [2] G.A. Prinz, Science 82, 1660, 1998, [3] R. Crook, J. Prance, K.J. Thomas, S.J. Chorley, I. Farrer, D.A. Ritchie, M. Pepper, C.G. Smith, Science 312 (2006) 1359.
229
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Non-stationary spin-filtering effects in correlated quantum dot a,⁎
a
MARK
b
V.N. Mantsevich , N.S. Maslova , P.I. Arseyev a b
Lomonosov Moscow State University, Moscow 119991, Russia P.N. Lebedev Physical Institute RAS, Moscow 119991, Russia
A R T I C L E I N F O
A BS T RAC T
Keywords: Spin polarized transport Correlated quantum dots Relaxation times and mean free paths Spin filtering
The influence of external magnetic field switching “on” and “off” on the non-stationary spin-polarized currents in the system of correlated single-level quantum dot coupled to non-magnetic electronic reservoirs has been analyzed. It was shown that considered system can be used for the effective spin filtering by analyzing its nonstationary characteristics in particular range of applied bias voltage.
1. Introduction One of the key issues of spintronics is the control and generation of spin-polarized currents. Nowadays generation and detection of spinpolarized currents in semiconductor nanostructures has attracted great attention since this is the key problem in developing semiconductor spintronic devices [1–4]. To generate tunable highly spin-polarized stationary currents the variety of systems has been already proposed ranging from semiconductor heterostructures to low-dimensional mesoscopic samples [5–8]. Significant progress has been achieved in experimental and theoretical investigation of stationary spin-polarized transport in magnetic tunneling junctions [9–12]. Nevertheless spinpolarized current sources based on the non-magnetic materials are attractable as one could avoid the presence of accidental magnetic fields that may result in the existence of undesirable effects on the spin currents. It was demonstrated recently that stationary tunneling current could be spin dependent in the case of non-magnetic leads [13,14]. There have been several proposals to generate stationary spinpolarized currents using non-magnetic materials: small quantum dots [15,16] and coupled quantum dots [5,17] built in semiconducting nanostructures in the presence of external magnetic field. Moreover, quantum dots systems based on the non-magnetic materials were proposed as a spin filter prototypes [18,19]. Effective spin filtering in such systems requires to have many quantum dots with the Coulomb correlations inside each dot [20–22] and also between the dots [5]. To the best of our knowledge usually stationary spin-polarized currents are analyzed. However, creation, diagnostics and controllable manipulation of charge and spin states in the single and coupled quantum dots (QDs), applicable for ultra small size electronic devices design requires analysis of non-stationary effects and transient processes [23–28]. Consequently, non-stationary evolution of initially ⁎
Corresponding author. E-mail address: [email protected] (V.N. Mantsevich).
http://dx.doi.org/10.1016/j.physe.2017.06.027 Received 29 May 2017; Received in revised form 21 June 2017; Accepted 23 June 2017 Available online 24 June 2017 1386-9477/ © 2017 Elsevier B.V. All rights reserved.
prepared spin and charge configurations in correlated quantum dots is of great interest both from fundamental and technological point of view. In this paper we analyze non-stationary spin polarized currents through the correlated single-level QD localized in the tunnel junction in the presence of applied bias voltage and external magnetic field, which can be switched “on” or “off” at a particular time moment. We demonstrate that single biased QD in the external magnetic field can be considered as an effective spin filter based on the analysis of nonstationary spin-polarized currents, which can flow in the both leads. Currents direction can be tuned by the external magnetic field switching “on” or “off”. 2. Theoretical model We consider non-stationary processes in the single-level quantum dot with Coulomb correlations of localized electrons situated between two non-magnetic electronic reservoirs in the presence of external magnetic field B switched “on”/“off” at t = t0 . We are interested in the influence of magnetic field directly on the quantum dot [5]. Modern experimental scanning probe technique provides possibility of applying magnetic field to various subsystems: directly to the dot, to the reservoirs or both to the dot and reservoir with the controllable field direction [29–32]. Moreover, when the uniform magnetic field in the dot and in the lead is applied to the system the situation when g-factor in the leads is much smaller than in the dots can be realized due to the materials properties (strongly different effective masses of electrons in the reservoir and in the dot). So only the effect of external magnetic field on the dot states was considered. The Hamiltonian of the system
l=H lQD + H lR + H lT H
(1)
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can be written as a sum of the single-level quantum dot part
lQD = ∑ ε1 nl1σ + Unl1σ nl1−σ , H
i (2)
σ
i
non-magnetic electronic reservoirs Hamiltonian
lR = H
∑ εk clkσ+ clkσ + ∑ (εp − eV ) clpσ+ clpσ kσ
k ′≠ k
i
∑ tk (clkσ+ cl1σ + cl1+σ clkσ ) + ∑ tp (clpσ+ cl1σ + cl1+σ clpσ ). kσ
pσ
(5)
∂nl1σk = − (ε1σ − εk ) · nl1σk − U · nl1−σ nl1σk ∂t + tk · (nl1σ − nlkσ ) − ∑ tk ′ · clk+′ σ clkσ ,
(3)
pσ
and the tunneling part
lT = H
∂nl1σ = − ∑ tk · (nlkσ1 − nl1σk ), ∂t k,σ
(4)
∂clk+′ σ clkσ = − (εk ′ − εk ) · clk+′ σ clkσ ∂t − tk ′ · cl1+σ clkσ + tk · clk+′ σ cl1σ .
(6)
(7)
and
Here index k(p) labels continuous spectrum states in the leads, tk ( p ) is the tunneling transfer amplitude between continuous spectrum states and quantum dot with the energy level ε1 which is considered to be independent on the momentum and spin. Operators clk+( p ) σ / clk ( p ) σ are the creation/annihilation operators for the electrons in the continuous spectrum states k(p). nl1σ (−σ ) = cl1+σ (−σ )cl1σ (−σ ) -localized state electron occupation numbers, where operator cl1σ (−σ ) destroys electron with spin σ (−σ ) on the energy level ε1. U is the on-site Coulomb repulsion for the double occupation of the quantum dot. External magnetic field B leads to the Zeeman splitting of the impurity single level ε1 proportional to the atomic g factor. In the absence of external magnetic field direct exchange and spin-orbit interaction can often lead to the appearance of effective magnetic field acting on electron spins. In the simple mean-field approach electron spin is affected by a mean nuclear spin polarization similar to effective magnetic field and thus hyperfine level splitting appears. The non-negligible value of effective magnetic field averaged value appears due to the finite number of nuclei within the quantum dot. For example, in GaAs this field leads to effective level splitting about 135μeV [36,37]. Consequently, for deep QD's electron levels (about 10 − 100 meV ) effective level splitting can be omitted as it doesn't influence opposite spin electron occupation numbers. When quantum dot's energy levels position is comparable to the hyperfine levels splitting two different cases can be distinguished. The first one deals with the value of level splitting can be of the order or smaller then the relaxation rate caused by the coupling to reservoir. In this situation level splitting due to the hyperfine interaction is not important. The second possible situation is when the value of level splitting exceeds the relaxation rate. In this case if the effective magnetic field has the same sign for each spin direction the symmetry between opposite spin states for localized electrons is broken even without external magnetic field. The effects of external magnetic field switching “on” and “off” now should drastically depend on its direction. Another interesting effect can appear in the presence of strong spinorbit interaction, when the direction of the effective magnetic field is opposite for the two different electron spin projections. This situation is typical for spin-Hall systems and one should expect the appearance of a pair of counter-propagating currents for each spin projection. Further results deal with the case of deep energy levels or correspond to the situation when relaxation rate exceeds hyperfine level splitting (as typical values of relaxation rate can be of order of 1 − 5 meV [38,39], while hyperfine level splitting is about 135μeV [36,37]). Further analysis deals with the low temperature regime when the Fermi level is well defined and the temperature is much lower than all the typical energy scales in the system. Consequently, the distribution function of electrons in the leads (band electrons) is close to the Fermi step.
i
+ ∂clpσ clkσ
∂t
+ = (εk − εp ) · clpσ clkσ + + + tk · clpσ cl1σ − tp · cl1+σ clkσ ,
(8)
+ clkσ clkσ
nlkσ
is an occupation operator for the electrons in the where = reservoir and εσ = ε1 + σ μB where σ = ± 1. Equations of motion for the electron operators products cl1+σ clpσ and clp+′ σ clpσ can be obtained from Eqs. (6) and (7) correspondingly by the indexes substitution k ↔ p and k′ ↔ p′. Following the logic of Ref. [33] one can get kinetic equations for the electron occupation numbers operators time evolution in the case of external magnetic field B switching “on” at the time moment t = t0 > 0 :
∂n1σ = − 2Θ (t0 − t ) · γ × [n1σ − (1 − n1−σ ) · ΦεT (t ) − n1−σ · ΦεT+ U (t )] ∂t − 2Θ (t − t0 ) · γ × [n1σ − (1 − n1−σ ) · Φε+T (t ) − n1−σ · Φε++TU (t )], ∂n1−σ = − 2Θ (t0 − t ) · γ × [n1−σ − (1 − n1σ ) · ΦεT (t ) − n1σ · ΦεT+ U (t )] ∂t − 2Θ (t − t0 ) · γ × [n1−σ − (1 − n1σ ) · Φε−T (t ) − n1σ · Φε−+TU (t )], (9) where γ = γk + γp and γk ( p ) = states in the leads and
πν0 tk2( p ) .
ν0 – is the unperturbed density of
γ lε± T (t ) = γk · Φ lkε± (t ) + p · Φ lpε± (t ), Φ γ γ γ lε±+ UT (t ) = γk · Φ lkε±+ U (t ) + p · Φ lpε±+ U (t ), Φ γ γ
(10)
where
lε± (t ) = 1 i · dεk · f σ (εk ) Φ k 2 ⎡ 1 − ei (ε1 ± μB + iΓ − εk ) t 1 − e−i (ε1 ± μB − iΓ − εk ) t ⎤ ×⎢ − ⎥, ⎣ ε1 ± μB + iΓ − εk ε1 ± μB − iΓ − εk ⎦ lε±+ U (t )= 1 i · dεk · f σ (εk ) Φ k 2 ⎡ 1 − ei (ε1 ± μB + U + iΓ − εk ) t 1 − e−i (ε1 ± μB + U − iΓ − εk ) t ⎤ ×⎢ − ⎥. ⎣ ε1 ± μB + U + iΓ − εk ε1 ± μB + U − iΓ − εk ⎦
∫
∫
(11)
Initially (t < t0) magnetic field B is absent [μB = 0 in Eqs. (9)–(11)] lε± (t ) = Φ lε (t ). To and, consequently, the following relation is valid Φ analyze system kinetics in the situation when magnetic field was initially present in the system and switched “off” at t = t0 one can easily generalize Eq. (9) by substitution t ↔ t0 . Equations for the localized electrons occupation numbers n1σ ± (t ) can be obtained by averaging Eqs. (9)–(11) for the operators and by decoupling electrons occupation numbers in the leads. Such decoupling procedure is reasonable if one considers that electrons in the macroscopic leads are in the thermal equilibrium [34,35]. After decoupling one has to replace electron occupation numbers operators in the reservoir nlkσ in Eqs. (9)–(11) by the Fermi distribution functions f kσ .
3. Non-stationary electronic transport formalism Let us further consider = = 1 and e = 1 elsewhere, so the motion equations for the electron operators products nl1σ , nl1σk = cl1+σ clkσ and clk+′ σ clkσ can be written as: 225
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Fig. 1. Occupation numbers time evolution. Panels a)-c) correspond to the magnetic field switching “on”, panels d)-f) correspond to the magnetic field switching “off”. a),d) ε1/2γ = −1.25 , eV /2γ = −2.5; b),e) ε1/2γ = −1.25 , eV /2γ = −7.5; c),f) ε1/2γ = −5, eV /2γ = −6.25. Parameters U /2γ = 10 , μB /2γ = −3.25, γk = γp = γ = 1 and initial conditions n1σ (0) = 0.6 ,
n1−σ (0) = 0.4 are the same for all the figures.
their direction and polarization can be tuned by magnetic field B switching “on”/“off”. Non-stationary spin-polarized currents Ik±( p )(t ) behavior for magnetic field switching “on”/“off” is shown in Figs. 2 and 3. Corresponding electron occupation numbers behavior is depicted in Fig. 1. Schemes of the QD energy levels both in the presence (ε+ and ε-) and in the absence (ε1) of magnetic field are shown in Fig. 4. Let us first focus on the situation when magnetic field is present at the initial time moment and switched “off” at t = t0 (see Fig. 1a,c,e and Fig. 2). In the presence of magnetic field when condition ε+ < EF − eV occurs (energy level ε- can be localized higher or lower than EF) (see Fig. 4a,c), nonstationary spin-polarized currents Ik−(t ) and Ik+(t ) in the lead with EF = 0 are flowing in the same direction (see Fig. 2a,c), contrary to the currents I p−(t ) and I p+(t ) flowing in the opposite directions in the lead with the Fermi level shifted by the applied bias voltage (lead p) (see Fig. 2d,f). In the
4. Non-stationary spin-polarized currents If the initial state is a “magnetic” one, non-stationary spin-polarized currents Ik ( p ) (t ) ± flow in the each contact lead:
Ik± (t ) = − 2γk · [n1± σ − (1 − n1∓σ ) · Φkε± (t ) − n1∓σ · Φkε±+ U (t )] − 2γk · [n1± σ − (1 − n1∓σ ) · Φkε (t ) − n1∓σ · Φkε + U (t )], I p± (t )
= − 2γp · [n1± σ − (1 − n1∓σ ) · Φpε± (t ) − n1∓σ · Φpε±+ U (t )] − 2γp · [n1± σ − (1 − n1∓σ ) · Φpε (t ) − n1∓σ · Φpε + U (t )],
(12)
where electron occupation numbers n1± σ are determined from the system of Eq. (9) with the magnetic initial conditions. Non-stationary spin-polarized currents can flow in the both leads and 226
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Fig. 2. Normalized non-stationary spin-polarized tunneling currents Ik ( p ) ± (t ) /2γ in the case of magnetic field switching “off” at t = t0 . Panels a)-c) demonstrate Ik ± (t ) /2γ , panels d)-f) demonstrate I p ± (t ) /2γ . a),d) ε1/2γ = −1.25 , eV /2γ = −2.5; b),e) ε1/2γ = −1.25, eV /2γ = −7.5; c),f) ε1/2γ = −5, eV /2γ = −6.25. Parameters U /2γ = 10 , μB /2γ = −3.25, γk = γp = γ = 1 and initial conditions n1σ (0) = 0.6 , n1−σ (0) = 0.4 are the same for all the figures.
stationary state all currents Ik±( p )(t ) values turn to zero. Magnetic field switching “off” results in the appearance of non-zero spin-polarized currents in both leads. Non-stationary spin-polarized currents Ik−(t ) and Ik+(t ) in the lead with EF = 0 continue flowing in the same direction with the same non-zero amplitude (see Fig. 2a,c). Currents I p−(t ) and I p+(t ) are also flowing in the same direction but magnetic field switching “off” results in the appearance of total current strong spin polarization at the initial stage of relaxation as the amplitude of current I p+(t ) strongly exceeds the amplitude of non-stationary current I p−(t ) (see Fig. 2d,f). Similar behavior of electron occupation numbers and non-stationary spinpolarized currents for two different positions of ε- (see Fig. 4a,c) is the result of the Coulomb correlations presence in the system. In both cases energy level ε+ is occupied and energy level ε- is unoccupied (even in the case depicted in Fig. 4c) due to the strong Coulomb repulsion. Non-
stationary current I p+(t ) changes direction with the magnetic field switching “off”, while currents Ik± (t ) and I p−(t ) are flowing in the same direction both in presence and in the absence of magnetic field. This effect can be applied for the effective spin-filtering in the single QD system alternatively to the previously proposed spin-filtering mechanisms based on the analysis of multiple QDs stationary characteristics [5]. Fig. 4b demonstrates that in the presence of magnetic field at the initial stage of relaxation non-stationary spin-polarized currents Ik± (t ) can flow in the opposite directions and currents I p± (t ) in the same directions (see Fig. 2b,e). Stationary state reveals the presence of only one non-stationary current flowing in each lead (Ik+(t ) and I p+(t ) correspondingly). Magnetic field switching “off” causes the appearance of both spin-polarized currents Ik± (t ) and I p± (t ) flowing in each lead in the same direction. In the stationary state spin currents values in each lead become equal.
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Fig. 3. Normalized non-stationary spin-polarized tunneling currents Ik ( p ) ± (t ) /2γ in the case of magnetic field switching “on” at t = t0 . Panels a)-c) demonstrate Ik ± (t ) /2γ , panels d)-f) demonstrate I p ± (t ) /2γ . a),d) ε1/2γ = −1.25 , eV /2γ = −2.5; b),e) ε1/2γ = −1.25, eV /2γ = −7.5; c),f) ε1/2γ = −5, eV /2γ = −6.25. Parameters U /2γ = 10 , μB /2γ = −3.25, γk = γp = γ = 1 and initial conditions n1σ (0) = 0.6 , n1−σ (0) = 0.4 are the same for all the figures.
netic field switching “on” allows to consider single QD as an effective spin-filter based on the analysis of its non-stationary characteristics. In the absence of magnetic field non-stationary spin-polarized currents in each lead Ik± (t ) and I p± (t ) are flowing in the same direction and
Electron occupation numbers and non-stationary spin-polarized currents behavior in the case when magnetic field is absent at the initial time moment and switched “on” at t = t0 is shown in Fig. 1b,d,f and Fig. 3 correspondingly. Obtained results demonstrate that mag-
Fig. 4. Sketch of the correlated QD energy levels coupled to non-magnetic leads both in the presence (ε+ and ε-) and in the absence (ε1) of magnetic field.
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[4] P. Chuang, S.C. Ho, L.W. Smith, F. Sfigakis, M. Pepper, C.H. Chen, J.C. Fan, J.P. Griffits, I. Farrer, H.E. Beer, et al., Nat. Nanotechnol. 10 (2015) 35. [5] M.Yu Kagan, V.V. Val’kov, S.V. Aksenov, Phys. Rev. B 95 (2017) 035411. [6] I. Zutic, J. Fabian, S.D. Sarma, Rev. Mod. Phys. 76 (2004) 323. [7] M.E. Torio, K. Hallberg, S. Flach, A.E. Miroshnichenko, M. Titov, Eur. Phys. J. B 37 (2004) 399. [8] A.M. Shakirov, Yu.E. Shchadilova, A.N. Rubtsov, P. Ribeiro, Phys. Rev. B 94 (2016) 224425. [9] E.Y. Tsymbal, O. Mryasov, P.R. LeClair, J. Phys, Condens. Matter 15 (2003) R109. [10] H.J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier, H.-P. Schonherr, K.H. Ploog, Phys. Rev. Lett. 87 (2001) 116601. [11] Y. Ohno, D.K. Young, B. Beschoten, F. Matsukura, H. Ohno, D.D. Awschalom, Nature 402 (1999) 790. [12] R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag, L.W. Molenkamp, Nature 402 (1999) 787. [13] V.I. Perel', S.A. Tarasenko, I.N. Yassievich, S.D. Ganichev, V.V. Bel'kov, W. Prettl, Phys. Rev. B 67 (2003) 201304. [14] M.M. Glazov, P.S. Alekseev, M.A. Odnoblyudov, V.M. Chistyakov, S.A. Tarasenko, I.N. Yassievich, Phys. Rev. B 71 (2005) 155313. [15] P. Recher, E.V. Sukhorukov, D. Loss, Phys. Rev. Lett. 85 (2000) 1962. [16] R.M. Potok, J.A. Folk, C.M. Marcus, V. Umansky, M. Hanson, A.C. Gossard, Phys. Rev. Lett. 91 (2003) 016802. [17] J.A. Andrade, P.S. Cornaglia, Phys. Rev. B 94 (2016) 235112. [18] T. Koga, J. Nitta, H. Takayanagi, S. Datta, Phys. Rev. Lett. 88 (2002) 126601. [19] A. Voskoboynikov, S.S. Liu, C.P. Lee, Phys. Rev. B 58 (1998) 15397. [20] W. Gong, Y. Zheng, Y. Liu, T. Lu, Phys. Rev. B 73 (2006) 245329. [21] J.H. Ojeda, M. Pacheco, P.A. Orellana, Nanotechnology 20 (2009) 434013. [22] H.-H. Fu, K.-L. Yao, Appl. Phys. Lett. 100 (2012) 013502. [23] I. Bar-Joseph, S.A. Gurvitz, Phys. Rev. B 44 (1991) 3332. [24] S.A. Gurvitz, M.S. Marinov, Phys. Rev. A 40 (1989) 2166. [25] P.I. Arseyev, N.S. Maslova, V.N. Mantsevich, Eur. Phys. J. B 85 (7) (2012) 249. [26] C.A. Stafford, N.S. Wingreen, Phys. Rev. Lett. 76 (1996) 1916. [27] B.L. Hazelzet, M.R. Wegewijs, T.H. Stoof, Y.V. Nazarov, Phys. Rev. B 63 (2001) 165313. [28] E. Cota, R. Aguado, G. Platero, Phys. Rev. Lett. 94 (2005) 107202. [29] C.F. Hirjibehedin, C.P. Lutz, A.J. Heinrich, Science 312 (2006) 1021. [30] C.F. Hirjibehedin, C.-Y. Lin, A.F. Otte, M. Ternes, C.P. Lutz, B.A. Jones, A.J. Heinrich, Science 317 (2007) 1199. [31] A.F. Otte, M. Ternes, K. von Bergmann, S. Loth, H. Brune, C.P. Lutz, C.F. Hirjibehedin, A.J. Heinrich, Nat. Phys. 4 (2008) 847. [32] D. Iaia, A. Kubetzka, K. von Bergmann, R. Wiesendanger, Phys. Rev. B 93 (2016) 134409. [33] N.S. Maslova, P.I. Arseyev, V.N. Mantsevich, Solid State Commun. 48 (2016) 21. [34] J.Q. You, H.-Z. Zheng, Phys. Rev. B 60 (1999) 8727. [35] J.Q. You, H.-Z. Zheng, Phys. Rev. B 60 (1999) 13314. [36] D. Paget, G. Lampel, B. Sapoval, V.I. Safronov, Phys. Rev. B 15 (1977) 5780. [37] B. Urbaszek, X. Marie, T. Amand, O. Krebs, P. Voisin, P. Maletinsky, A. Hogele, A. Imamoglu, Rev. Mod. Phys. 85 (2013) 79. [38] E. Nielsen, Phys. Rev. B 88 (2013) 195131. [39] J. Fransson, Phys. Rev. B 69 (2004) 201304. [40] S. Amaha, W. Izumida, T. Hatano, S. Teraoka, S. Tarucha, J.A. Gupta, D.G. Austing, Phys. Rev. Lett. 110 (2013) 016803.
demonstrate equal non-zero stationary values (see Fig. 3). Magnetic field switching “on” results in the direction changing of one of the spinpolarized currents in the leads (see Fig. 3a,d,f). Another possible situation deals with fast switching “off” of one of the spin-polarized currents in each lead when magnetic field is switched “on” (see Fig. 3b,e). Consequently, only non-stationary current with a certain spin orientation continue flowing in each lead in the presence of magnetic field. To observe these effects the switching times of magnetic field must be smaller than the lifetime of the initially prepared magnetic states. Modern scanning tunneling microscopy/spectroscopy experiments provide possibility to achieve typical spin-polarized current values of the order of 10 pA÷10 nA (1 nA ≃ 6 × 109e / sec ) [40,39], which corresponds to the relaxation time scales 1/ γ ≃ 1 ÷ 100 ns for the system parameters depicted in Figs. 2 and 3. It gives the possibility to realize proposed protocols for the state of art experiments by means of pico- or femtosecond switchers, for example by means of laser pulses. 5. Conclusion We have analyzed the behavior of spin-polarized non-stationary currents in the system of single-level quantum dot situated between two non-magnetic electronic reservoirs with Coulomb correlations of localized electrons in the presence of external magnetic field switched “on” or “off” at particular time moment. It was demonstrated that single-level correlated quantum dot can be considered as an effective spin filter depending on the ration between the values of magnetic field induced energy level splitting and applied bias voltage. Acknowledgements This work was supported by RFBR Grant 16-32-60024 mol-a-dk and by the RF President Grant for young scientists MD-4550.2016.2. References [1] Semiconductor spintronics and quantum computation, in: D.D. Awschalom, D. Loss, N. Samarth (Eds.), Nanoscience and Technology, Springer, Berlin, 2002. [2] G.A. Prinz, Science 82, 1660, 1998, [3] R. Crook, J. Prance, K.J. Thomas, S.J. Chorley, I. Farrer, D.A. Ritchie, M. Pepper, C.G. Smith, Science 312 (2006) 1359.
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