Thermal Bias on the Pumped Spin-Current in a Single Quantum Dot

Commun. Theor. Phys. 62 (2014) 86 90 Vol. 62, No. 1, July 1, 2014 Thermal Bias on the Pumped Spin-Current in a Single Quantum Dot LIU Jia ( ) 1,2, and CHENG Jie ( ) 1 1 School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China 2 Key Laboratory of Integrated Exploitation of Bayan Obo Multi-Metal Resources IMUST, Baotou 014010, China (Received December 16, 2013; revised manuscript received February 26, 2014) Abstract Temperature effect on the spin pump in a single quantum dot (QD) connected to Normal (NM) and/or Ferromagnetic (FM) leads is investigated with the help of master equation method. Results show that the magnitude and the direction of the temperature difference between the source (L) and drain (R) leads have great impact on the spin current processes. In practical devices, the thermal bias is quite general and then our results may be useful in quantum information processing and spintronics. PACS numbers: 72.25.-b Key words: thermal bias, spin pump, quantum dot To create devices based on the spin of electrons is the main aim of spintronics [. [1] The most necessary element for any spin-based electronics is the ability to generate a spin current. Spin pumps, [2 6] in the past ten years, have attracted much attention since it can generate a net spin current flowing between unbiased leads. Experimentally, it has been demonstrated through the operation of a quantum spin pump, which is based on cyclic radiofrequency excitation of a GaAs quantum dot to pump pure spin. [7] Several proposals, such as nonmagnetic systems with the Zeeman field, ferromagnet with a rotating magnetization [8 11] and ferromagnet with time-dependent magnetization, [12] for generating a pure spin current have been identified. These architectures are the basis for recent spin-battery design. [13 15] In 2008, Uchida et al. [16] discovered the spin Seebeck effect that describes the relation between a temperature gradient and a spin bias voltage, and then increasing attention has been paid on the effect caused by temperature difference between the two leads coupled to a QD. [17 19] The thermal spin effect or spin Seebeck effect has been widely studied in the system of QD coupled with ferromagnetic leads during the past few years. [20 22] However, all these discussions about spin pumps have neglected the weak temperature gradient between the two leads. The heat dissipation and the environment temperature fluctuations are reflected at the temperature gradient (that is, thermal bias), which can not be avoided in the practical application. In this paper we discuss the effect of weak thermal bias on the NM-QD-NM\FM-QD-NM\FM-QD- FM architectures. When the temperature of the L lead is higher than that of the R lead, the thermal bias is positive ( T = T L T R > 0); otherwise the thermal bias is negative ( T = T L T R < 0). The device, [4] the spin Zeeman levels typical pump, can be described by the following Hamiltonian, [20] H = k (ε k µ )c + k c k + ε d + d + Un n + k (τ c + k d + H.c.) + H rf (t), (1) where c + k creates an electron in the lead with spin and energy ε k ; d + creates an electron in the dot with spin ; n = d + d is the number operator; U is the Coulomb charging energy; ε = ε d µ B B is the energy level in the dot. It is spin dependent due to a Zeeman splitting, which originates from the magnetic field induced by the FM leads or by an external field. It may also arise from the presence of spin-dipoles which are dynamically formed around a nanojunction. = ε ε = 2µ B B is the energy interval. τ is the coupling between the leads and the dot. H rf (t) describes the coupling between the spin states due to the rotating field B rf (t) and can be written as H rf (t) = R rf ( e iwt d + d + d + d e iwt ), (2) where the Rabi frequency R rf = g µ B B rf, with g and B rf (t) representing g-factor and the amplitude of the rotating field, respectively. The chemical potentials µ L and µ R are set to be the zero energy. The Coulomb repulsion inside the QD is so large that only single occupation (unoccupied or spin- state) is allowed 0 and describe the occupation probability in the QD being, respectively, so the density matrix Supported by the National Natural Science Foundation of China Project under Grant No. 11147010, the Natural Science Foundation of Inner Mongolia under Grant No. 2012MS0113, and the Research Program of Higher Education of Inner Mongolia Autonomous Region under Grant No. NJZY12111 Corresponding author, E-mail: jialiu@imust.cn

No. 1 Communications in Theoretical Physics 87 of the system can be written as: ρ 00 ρ 0 ρ 0 ρ = ρ 0 ρ ρ. (3) ρ 0 ρ ρ These matrix elements can be calculated by ρ ij = Tr ( i j ρ). The non diagonal elements represent the coherence among electronic occupation states, so ρ 0i = ρ i0 = 0. Studies indicated that two time scales that are introduced by Ref. [20] in a single QD was probed to be an order of microsecond via transport experiment, which is notably longer than other time scales. If the temperatures are higher than the Kondo temperature, in the sequential tunneling approximation, the time evolution of the probabilities obeys the following quantum rate equations: [23 24] ρ 00 =, P 0 ρ 00 +, P 0ρ, (4) ρ = P0 ρ 00 P 0ρ + ir rf (ρ ρ ), (5) ρ = 1 ( ) P 0 ρ + ir rf (ρ ρ ) + iδ ESR ρ.(6) 2, In these equations, δ ESR = w is the ESR detuning and P n m describes the lead s probability per unit time to transfer from state n to m. Assuming the coupling between the leads and the dot is energy independent (wide band approximation), the coupling of the dot to the lead can be written as Γ = 2π τ 2 ρ, where ρ is the spin-dependent density of states of lead. By introducing the definition of the spin polarization of lead, p = ρ ρ ρ + ρ, one can write Γ = γ (1 + p ) with γ = (1/2) Γ. P 0 = γ (1 + p )f + (ε, µ, T ), (7) P 0 = γ (1 + p )f (ε, µ, T ), (8) where f ± (ε, µ, T ) is the Fermi distribution of lead with the corresponding temperature and chemical potential, ± represent, respectively, the occupied and unoccupied states of two leads energy level. Under the weak coupling approximation, the particle current I QD flowing from the lead to the QD can be expressed as: [6,24 25] ( I QD = e P0 ρ 00 ) P 0ρ, (9) I QD = e(p0 P 0), (10) ( IQD s = P 0ρ ) P0 ρ 00. (11) In the following numerical calculations we choose = ε ε = 2µ B B as the energy unit and set B = 0.15 T. The system temperature in equilibrium state is fixed as T = 0.25, which is about 0.1 K for 35 µev in experiments. Under zero temperature gradient (T L = T R = T ) and symmetric coupling γ L = γ R = 0.5, the Fermi function of the two leads are the same. For simplicity, the spin decoherence effects are not considered in the following numerical calculations. Case 1 NM-QD-NM In this case, we consider the thermal bias that is neglected in Ref. [6] between the two NM leads connected to a QD. The polarization directions in these leads are desultory, so p L = p R = 0. For simplicity, we set the ESR detuning δ ESR = 0, the Zeeman splitting energy interval = 1, the initial temperature environment T = 0 and the dot-lead coupling parameter γ L = γ R = 0.5 in the following numerical calculations. Figure 1 shows the spincurrent I s (solid line) and charge-current I c (dotted line) changing with ESR Rabi frequency R rf and thermal bias T = 0 K (black online), 0.2 K (red online), 0.2 K (blue online). Fig. 1 (Color online) Spin-current I s (solid line) and charge-current I c (dotted line) as a function of ESR Rabi frequency R rf and thermal bias T = 0 K (black online), 0.2 K (red online), 0.2 K (blue online) with p L = p R = 0 and γ L = γ R = 0.5 for NM-QD-NM case. In the case of zero bias voltage, one can make the chemical potentials in both leads lie between the two energy levels by tuning the gate voltage, i.e., ε < µ L = µ R < ε. When the L lead has the same temperature with R lead, i.e., T = T L T R = 0 K. The spin-up electron has opportunity to tunnel into the QD and then is trapped in it. When the spin-flip mechanism is applied to QD, this spin-up electron can be lifted to the higher energy level and the electron becomes a spin-down one simultaneously. Because the higher QD level is above the chemical potentials in both leads, the electron with spin down tunnels out of the QD. In the presence of spin pump, the stationary charge current is exactly zero since the spin current in the two sides has the same size but opposite directions (see Fig. 1, the black solid line). The pure spin current

88 Communications in Theoretical Physics Vol. 62 does not increase with the increase of R rf any more but approaches to a saturated value I s = 0.25 (see Fig. 1, the black dotted line) that is determined by the coupling between the leads and the dot τ. When T = T L T R = 0.2 K > 0, thermal bias breaks the tunneling rates balance at the thermal equilibrium state. L lead has holes (electrons) under (up) the Fermi level. P0 L, the opportunity of the spin-down electron to tunnel from L lead into the QD and P 0 L, the opportunity of spin-up electron to tunnel from the QD into L lead increased. The QD level ε is above the chemical potentials in right lead µ R, and thus the spin-down electrons flow into the right lead, while the opportunity of spin-up electron on the ε to tunnel into the right lead shrinked. I s = 0.065 at the zero pumping intensity. The spin-down electron has high occupation probability at the low pumping intensity. With the pumping intensity increases, more extra spin-up electrons are pumped to the level of spin down, and thermal bias therefore becomes secondary factors. The flowing of extra spin-down electrons into the right lead and extra spin-up electrons into the QD results in spin current increase. In the presence of strong spin pumping R rf = 2, the pure spin current does not increase with the increase of R rf any more but approaches to a saturated value I s = 0.25 (see Fig. 1, the red solid line) which is the same with the case T = T L T R = 0 K (Fig. 1 the black solid line). The direction of the negative charges motion is set to be the positive direction of the electric current. In the presence of spin pump and positive thermal bias, the stationary charge current I c is negative (see Fig. 1, the red solid line) and tends to zero with the increase of R rf. If the thermal bias is negative, that is T = T L T R = 0.2 K < 0, R lead has holes (electrons) under (up) the Fermi level. The spin-down electron has opportunity to tunnel from R lead into the QD and spin-up electron has opportunity to tunnel out of the QD into R lead. ε is higher than µ L, and thus the spin-down electrons flow into the left lead at zero pumping intensity with I s = 0.065. With the pumping intensity increases, more extra spin-up electrons are pumped to the level of spin down, therefore, the pure spin current does not increase with the increase of R rf any more but approaches to a saturated value I s = 0.105 (see Fig. 1, the blue solid line) which is lower than positive thermal bias one. The charge current is just the opposite of positive one (see Fig. 1, the blue dotted line). Case 2 FM-QD-NM The effect of thermal bias between the FM (left) and NM (right) lead, both of which are connected to a QD is considered here. The FM lead has relatively consistent polarization direction, so we set p L = 0.5, p R = 0. Figure 2 shows the spin-current I s (solid line) and charge-current I c (dotted line) changing with ESR Rabi frequency R rf and thermal bias T = 0 K (black online), 0.2 K (red online), 0.2 K (blue online). For an FM lead with Γ L > Γ L, the spin-up electrons have the large possibility to tunnel into the QD. When the two leads have the same temperature T = T L T R = 0 K, a spin-up electron enters QD, and then the spin-flip mechanism lift it to the higher energy ε. Because the higher QD level is above the chemical potentials in both leads, the electron with spin down tunnels out of the QD with different intensity in two sides. In the presence of spin pump, the charge current approaches to a saturated value I c = 0.06 (see Fig. 2, the black dash line). The pure spin current increases nonlinearly with the increase of R rf, and finally approaches to a threshold I s = 0.235. When T = T L T R = 0.2 K > 0, the increase of both P0 L and P 0 L is greater than that of normal metal lead (case 1), so I s = 0.075 greater than NM-QD-NM case when R rf = 0. With the increase of pumping intensity, spin current and charge current have the same variation trend with case 1, but I s and I c reach the threshold of 0.245 and 0.06, respectively. At the strong spin pumping R rf = 2 (Fig. 2, the red lines). Thermal spin effect and thermoelectric effect can not be ignored in this structure. When the right lead has the high temperature T = T L T R = 0.2 K < 0, the change in the NM lead is the same with NM-QD-NM case. P0 R and P 0 R increase. The spin-down electrons flow into the left lead at zero pumping intensity with I s = 0.075. With the pumping intensity increasing, more extra spin-up electrons from L lead are pumped to the level of spin down, and thus temperature gradient also becomes secondary factors. The pure spin current approaches to a saturated value I s = 0.099 (see Fig. 2, the blue solid line) and the stationary charge current is I c = 0.06 (see Fig. 2, the blue dotted line). Fig. 2 (Color online) Spin-current I s (solid line) and charge-current I c (dotted line) as a function of ESR Rabi frequency R rf and thermal bias T = 0 K (black online), 0.2 K (red online), 0.2 K (blue online) with p L = 0.5, p R = 0 and γ L = γ R = 0.5 for FM-QD-NM case.

No. 1 Communications in Theoretical Physics 89 Case 3 FM-QD-FM In this case we consider two FM leads connected to a QD. Figures 3 and 4 show the change of spin-current I s (solid line) and charge-current I c (dotted line) changing with ESR Rabi frequency R rf and thermal bias T = 0 K (black online), 0.2 K (red online), 0.2 K (blue online) for parallel (p L = p R = 0.5) and antiparallel (p L = p R = 0.5) magnetization FM, respectively. In the parallel case, the symmetry of spin electron density distribution around the two leads leads to a similar situation occurred in the NM-QD-NM at zero thermal bias. They have the same spin pumping process so that the curves of I s and I c have the same trend. The saturated value I s = 0.165 is lower than that of case 1 because Γ L > Γ L and Γ R > Γ R (see Fig. 3, the black solid line). In the presence of spin pump, the stationary charge current is exactly zero since the spin current in the two sides has the same size but opposite directions (see Fig. 3, the black dotted line). When T = 0.2 K, I s = 0.085 at R rf = 0. With the increase of Rabi frequency, spin current and charge current trend to thresholds I s = 0.19 and I c = 0.035 at R rf = 2, respectively (see Fig. 3, the red lines). When T = 0.2 K, I s = 0.085 at R rf = 0. With the increase of Rabi frequency, I s = 0.19 and I c = 0.035 at R rf = 2 (see Fig. 3, the blue lines). Thermal bias has great effect in both situations due to the spin polarization in ferromagnetic leads. FM lead has relatively consistent polarization direction spin down, so ( ep0 R ρ 00) is the main contribution of charge current at low pumping intensity. Therefore, extra spin-down electrons flow into the QD, and meanwhile a positive charge current I c = 0.03 and a negative spin current I s = 0.05 are generated at R rf = 0. In spite of the strong pumping intensity, the charge current is not significantly enhanced since P 0 R is very small in Eq. (10) (see Fig. 4, the blue dotted line). With the increase of Rabi frequency, spin current and charge current tend to approach thresholds I s = 0.12 and I c = 0.085 at R rf = 2, respectively (see Fig. 3, the blue lines). When T = 0.2 K, I s = 0.05 and I c = 0.025 at R rf = 0; I s = 0.24 and I c = 0.08 at R rf = 2 (see Fig. 4, the red lines). The spin current and charge current in antiparallel FM-QD-FM has lower strength at R rf = 0 and higher strength at R rf = 2 than parallel case at positive or zero thermal bias. In the case of negative thermal bias, both I s and I c have weaker strength at R rf = 0 and 2 (comparing Fig. 3 with Fig. 4). These can be explained by Eqs. (7) (11). Fig. 4 (Color online) Spin-current I s (solid line) and charge-current I c (dotted line) as a function of ESR Rabi frequency R rf and thermal bias T = 0 K (black online), 0.2 K (red online), 0.2 K (blue online) with p L = p R = 0.5 and γ L = γ R = 0.5 for anti-parallel FM-QD-FM case. Fig. 3 (Color online) Spin-current I s (solid line) and charge-current I c (dotted line) as a function of ESR Rabi frequency R rf and thermal bias T = 0 K (black online), 0.2 K (red online), 0.2 K (blue online) with p L = p R = 0.5 and γ L = γ R = 0.5 for FM-QD-FM case. In the antiparallel case, when T = 0 K, I s = 0.25 and I c = 0.125 at R rf = 0; With the increase of pumping intensity, spin current and charge current tend to approach thresholds I s = 0.25, I c = 0.125 at R rf = 2, respectively (see Fig. 4, the black lines). Thermal bias breaks the tunneling rate balance at the thermal equilibrium state when T = 0.2 K, so P 0 R and P 0 R increased. The right In conclusion, we investigated the spin pumping process in a QD under the effect of thermal bias. Due to the combined effect of temperature gradient of the device and the spin polarization of the leads, thermal spin effect and thermoelectric effect can not be ignored even under weak temperature. The zero or positive temperature bias facilitates the production of spin current and the impact of positive temperature bias can be balanced to some extent by increasing the spin pump intensity. The generation of the charge current depends largely on the material structure, i.e., FM-QD-NM and FM-QD-FM have none zero charge current at strong pumping intensity, and meanwhile thermal bias becomes a secondary effects.

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