Heat Generation by Electric Current in a Quantum Dot Molecular Coupled to Ferromagnetic Leads

Transcription

1 Commun. Theor. Phys. 62 (2014) Vol. 62, No. 3, September 1, 2014 Heat Generation by Electric Current in a Quantum Dot Molecular Coupled to Ferromagnetic Leads GUO Yu (À ), 1 SUN Lian-Liang (ê ), 2 and CHI Feng (ðõ) 3, 1 School of Mathematics and Physics, Bohai University, Jinzhou , China 2 College of Science, North China University of Technology, Beijing , China 3 College of Engineering, Bohai University, Jinzhou , China (Received Feburary 19, 2014; revised manuscript received May 6, 2014) Abstract We study the properties of the heat flow generated by electric current in a quantum dot (QD) molecular sandwiched between two ferromagnetic leads. The heat is exchanged between the QD and the phonon reservoir coupled to it. We find that when the leads magnetic moments are in parallel configuration, the total heat generation is independent on the leads spin-polarization regardless of the magnitude of the intradot Coulomb interaction. This behavior is similar to that of the electronic current. In the antiparallel configuration, however, the influences of the leads ferromagnetism on the heat generation are quite different from those on the electric current. Under the conditions of weak intradot Coulomb interaction and small bias voltage, the heat generation is monotonously suppressed by increasing leads spin-polarization. Whereas for sufficient large intradot Coulomb interaction and bias voltage, the heat generation shows non-monotonous behavior due to the electron-phonon interaction and the spin accumulation induced on the dot. Furthermore, the magnitude of the negative differential of the heat generation previously found in a QD connected to nonmagnetic leads can be weakened by the increase of the spin-polarization of the ferromagnetic leads. PACS numbers: b, k, Gk Key words: quantum dot, heat generation, ferromagnetic lead 1 Introduction Heat generated by electric current flowing through a conductor is a very old problem in physics, which is known as the Joule heat early in In macroscopic conductor, the heat generation is directly proportional to the product of the current and the bias voltage, i.e., the Joule law. In nanoscale structures, however, previous theoretical and experimental works have shown that the behaviors of the heat generation are quite different from those in the usual macroscopic ones and then the Joule law is no longer hold true. [1 9] Sun and Xie have calculated the heat generated by electric current by using the non-equilibrium Keldysh Green s function and some behaviors unique to nanodevies were predicted. [10] For example, the heat generation can be very small despite a large current in the resonant tunneling region, which is favorable in device design. [10 11] It can even decrease with increasing bias voltage for some parameter regions, exhibiting negative differential phenomenon. [11] If the electrons are driven by external ac bias voltage, the time-dependent heat generation is not zero even when the current is zero. [10,12 13] Under some particular conditions, electrons can absorb energy (phonons) from the surrounding material held at zero temperature. [12] In the presence of the temperature difference between the two ends of the device, we have shown that the resonant tunneling electrons can absorb heat from the phonon bath to the central region that are held at the same temperature, behaving as a nano-refrigerator to cool down the host material of the central region. [14] From a microscopic point of view, the heat in solid-state device originates mainly from the inelastic electron-electron and electron-phonon scattering, [4,15] and is exchanged with environment through the terminals connected to the electron reservoirs and the phonon bath. Quantized energy transport carried by phonons has been theoretically predicted in a one-dimensional dielectric quantum well coupled to electron reservoirs, [1] and then was observed in subsequent experiment, [3] becoming a direct evidence for the microscopic theory. Experimentally, current-induced local heating effects have been demonstrated by Huang et al. in a single molecule. [5 6] Phonon generation by electric bias have been demonstrated by Oron Carl and Krupke in terms of the ratio of anti-stokes to Stokes lines. [7] Such findings are useful in the emerging phonon engineering techniques. [16] During various nanodevices, zero-dimensional quantum dot (QD) or QD moleculars has become an extensively studied object due to the existence of discrete energy levels and tunable Coulomb interaction between electrons. [17] It has been proven as a promising candidate Supported by National Natural Science Foundation of China under Grant No Corresponding author, chifeng@semi.ac.cn c 2014 Chinese Physical Society and IOP Publishing Ltd

2 424 Communications in Theoretical Physics Vol. 62 in photoelectric and quantum information devices. On the other hand, QD-based spintronic devices have also attracted much attention in recent years due to the attractive advantages as compared with their conventional charge-based counterparts, such as faster data-processing speed, less electric power consumption, and increased integration densities, etc. [18 19] Currently, the simplest way of generating and manipulating electron spin in QD is to introduce ferromagnetic leads. [20] Experimentally, semiconductor spacers of InAs QD with controllable size and energy spectrum have been inserted in between nickel or cobalt leads. [21 23] In such a device, the spin polarization of the current injected from the ferromagnetic leads and the tunnel magnetoresistance (TMR) can be effectively tuned by a gate nearby the QD, and has enabled some new possible applications. Moreover, spin-polarized transport through a QD coupled to two ferromagnetic leads in the presence of electron-phonon interaction (EPI) has been studied in recent years. [24 25] They found that the EPI will arise additional satellite resonances in the conductance and the oscillation of the TMR. Previous theoretical works about the heat generation have mainly focused on the structure of a QD coupled to two normal metal leads. [8 12,26 38] In the present paper, we study the heat generation in a system composed of a QD connected to ferromagnetic leads, which has been extensively investigated in the usual electron transport subject, [20] but the heat generation in it remains untouched yet. It should be noted that the heat generation has been studied very recently in a tunnel contact between electrodes of different temperatures with at least one of it is a ferromagnet. [39] It shows that the heat results from the spin-dependent band structure of the leads which is very different from that of the usual charge-based electric devices. This spin-dependent band structure or density of states (DOS) is also responsible for the ferromagnetism and thus is expected to induce some new phenomena in the heat generation of the present structure, which is the motivation of the present paper. As shown in Fig. 1(a), the QD is connected to the left and right leads via tunnel junctions and contains both intradot Coulomb interaction and EPI. The magnetic moments of the left and right leads are in either parallel or antiparallel configuration. Due to the EPI, the bare energy level in the QD ε 0 is renormalized to ε d. Based on the non-equilibrium Keldysh Green s function technique and the general formulae derived in Ref. [10], we calculate the heat generation and the electric current. Our numerical results show that the ferromagnetism of the external leads will influence the heat generation and the electric current in different way, especially in the case of antiparallel configuration with large spin-imbalance induced on the QD. For example, the heat generation can vary non-monotonously with the spin-polarization of the leads, and the intradot Coulomb interaction can either enhance or suppress the heat generation depending on the magnitude of the bias voltage. 2 Model and Method The system under consideration with a QD molecular connected to the left and right ferromagnetic leads is described by the following Hamiltonian ( = 1): [10,24 25] H = ε kβσ c kβσ c kβσ k,β,σ + σ [ε 0 + λ q (a + a)]d σ d σ + U 0 d d d d + ω q a a + (V kβσ c kβσ d σ + H.c.), (1) k,β,σ where the first term describes the left (β = L) and right (β = R) electron reservoir (lead) in the non-interacting quasi-particle approximation with c kβσ (c kβσ) being the creation (annihilation) operator of an electron with momentum k, energy ε kβσ and spin σ. The second and the third terms are for the electron on the QD, and d σ (d σ ) creates (annihilates) an electron with energy ε 0 and intradot Coulomb interaction U 0. a (a) is the creation (annihilation) operator of a phonon having frequency ω q ; the quantity λ q is the electron-phonon coupling strength. The fourth term accounts for the single phonon mode, and the last term describes the conduction electron hopping between the QD and the leads with V kβσ being the spin-dependent tunnelling matrix element. Following previous works, we make a canonical transformation to eliminate the electron-phonon coupling terms, i.e., [10,24 25] H = ˆXH ˆX with ˆX = exp[(λ q /ω q ) σ (a q a q )d σd σ ]. The transformed Hamiltonian reads H = ε kβ c kβσ c kβσ + ε d d σ d σ + Ud d d d k,β σ + k,α(ṽkβσc kασ d σ + H.c.), (2) where the renormalized dot level and the intradot Coulomb interaction are respectively given by ε d = ε 0 gω q and U = U 0 2gω q with g = (λ q /ω q ) 2. The tunnelling matrix element is also renormalized to Ṽkασ = V kασ X, where the phonon operator X = exp[ g(a a)]. When the conduction electron hopping V kασ is small compared to λ q, X can be replaced by its expectation value X = exp[ g(n ph + 1/2)], where N ph = 1/[exp(ω q /k B T ph ) 1] being the phonon distribution function with T ph the temperature of the phonon bath. [12 13] Note that in the above unitary transformation, the operators a q a q and d σ d σ remain unchanged, and then the heat flow between the electrons and the phonons Q σ (t) = ω q da q (t)a q(t)/dt [8,26] can be calculated from the transformed Hamiltonian in Eq. (2). Since there is no time-dependent fields, the Fourier transform of Q σ (t) can be obtained as, [10,14] Q σ = ω qλ 2 q dω { 2 Γσ (N e N q ) Γ Lσ ΓRσ 2π Γ Lσ ΓRσ

3 No. 3 Communications in Theoretical Physics 425 Γ βσ [f β ( ω) f β (ω)] β } + f LR (ω)f LR ( ω) T σ (ω)t σ ( ω), (3) where ω = ω ω q and N e = 1/[exp(ω q /k B T e ) 1]. The abbreviations f LR (ω) and f LR ( ω) in the above equation are individually f L (ω) f R (ω) and f L ( ω) f R ( ω), in which the Fermi distribution function is f L(R) (ω) = 1/{exp[ω µ L(R) ]/k B T e + 1}. µ β and T e are the chemical potential and the temperature of the electrons, respectively. The total renormalized line-width function is Γ σ = Γ Lσ + Γ Rσ, where Γ βσ = exp[ g(2n ph +1)]Γ βσ, with Γ βσ = 2π V kβσ 2 ρ βσ and ρ βσ being the spin-dependent local DOS in lead β. In real case, the ferromagnetism on the lead results from the spin-dependent band structure or the DOS. Here as an equivalent choice, we assume square bands of both spin-up and spin-down electrons (wide band approximation) and assume spin-dependent tunneling matrix V kβσ. The spin-dependent line-width function Γ βσ is defined in terms of a spin-polarization parameter P β in each lead. One can define P β as P β = (Γ β Γ β )/(Γ β + Γ β ), or Γ βσ = Γ β0 (1 + δ σ P β ) with δ = 1, δ = 1, and Γ β0 = (Γ β +Γ β )/2. [24 25] We consider the parallel and antiparallel configurations of the two leads. For parallel configuration, we have P L = P R = P and P L = P R = P for antiparallel configuration. The electron transmission coefficient T σ (ω) in Eq. (3) is calculated with the help of the Keldysh nonequilibrium technique, [10 14] and is given by T σ (ω) = Γ Lσ ΓRσ Im G r σ (ω)/ Γ σ. The retarded Green s function G r σ (ω) is the Fourier transforms of the single-particle retarded Green s function G r σ (t) defined in terms of the Hamiltonian (2). By employing the equation of motion technique, the retarded Green s function is obtained as, [11] G r σ (ω) = 1 g σ r 1 (ω) + i( Γ Lσ + Γ Rσ )/2, (4) where the dressed Green s function without coupling to the leads is g σ r (ω) = ω ε d Ũ(1 n σ), (5) (ω ε d )(ω ε d Ũ) in which the average occupation number n σ needs to be self-consistently calculated by the following equation dω Γ Lσ f L (ω) + Γ Rσ f R (ω) n σ = ImG r π Γ Lσ + Γ σ(ω). (6) Rσ Once the retarded Green s function is obtained, all the other Green s functions to be used can be determined, i.e., Ga σ (ω) = [ G r σ(ω)], G<(>) σ (ω) = G r σ(ω) Σ <(>) σ Ga σ (ω), in which the lesser and greater self-energies are respectively given by Σ < σ = i[ Γ Lσ f L (ω)+ Γ Rσ f R (ω)], and Σ > σ = i{ Γ Lσ [1 f L (ω)] + Γ Rσ [1 f R (ω)]}. [15] To calculate the spin-dependent electric current J σ and the intradot electron occupation number n σ, we must know the Green s functions defined in terms of Hamiltonian (1), i.e., G r,a,, σ (ω). These Green s functions have G r,a,, σ similar relationship as those between (ω), and then we need only the knowledge of G r σ (ω). Based on the above canonical transformation between Eqs. (1) and (2), G r σ (ε) is related to G r σ (ε) by[24 25] n= G r σ (ω) = { L n Gr σ (ω nω q ) n= [ G < σ (ω nω q) G } < σ (ω + nω q)], (7) where L n in equation is given by L n = exp[ g(2n ph + 1)] exp(nω q /2k B T)I n [ 2g 2N ph (N ph + 1) ], with I n (x) the modified n-th Bessel function. The electric current of each spin component flowing through the structure can also be calculated with the help of the Green s function defined in terms of Hamiltonian given in Eq. (1), and reads [24 25] J σ = e dω[f L (ω) f R (ω)]t σ (ω). (8) h 3 Numerical Results In the following numerical investigation we set the phonon energy ω q as the unit of energy and assume symmetric barriers with Γ L0 = Γ R0 = 0.01 ω q. The chemical potential of the right lead is chosen as the energy zero point µ R = 0, and then the bias voltage is ev = µ L. We also assume that the QD and the phonon bath are held at the same temperature T ph = T e so as N ph = N e. The expression of the spin-dependent heat generation given in Eq. (3) then is reduced to the following simple form, Q σ = ω qλ 2 q dω Γ Lσ ΓRσ 2π f LR(ω)f LR ( ω)t σ (ω)t σ ( ω), (9) which shows that heat generation is directly proportional to product of the bias windows at ev and ev ω q time the electron transmissions within them. We first study the electric current and the heat generation in the absence of the Coulomb interaction (U = 0). Figure 1(b) shows the total electric current J = J + J and the heat generation Q = Q + Q when the leads magnetic moments are arranged in parallel configuration. Now both the electric current and the heat generation become spin-polarized, i.e., J J and Q Q, but the total electric current and the heat generation are independent on the spinpolarization of the lead. As is expected, both the electric current and the heat generation increase with increasing bias voltage. The increase of the heat generation has a delay of ω q with respect to the electric current. This is because the heat generation originates from the process of phonon emitting when an electron at state of ω jumping to an empty state of ω ω q. If ev < ω q, the electrons can not absorb enough energy for this process and then the heat generation is zero. This delay effect will become more obvious at lower temperature regime. Another difference between the heat generation and the electric current is their

4 426 Communications in Theoretical Physics Vol. 62 rapid jumps happening at different bias voltage. The rapid jump of the electric current happens when the dot level ε d enters into the bias window. Due to the variation of the dot level with the bias voltage, the jump is at ev = ω q as indicated by the dashed line in Fig. 1(b). Whereas the jump of the heat generation happens at ev = 2ω q (the solid line). At last, one can see that there are many small substeps emerge in the current curves because of the phonon-assisted tunneling processes, but no phononassisted substeps in the heat generation curves. Mathematically, this can be understood that the Green s functions for the the current and heat are calculated respectively from Eqs. (1) and (2). In other words, they are obtained respectively in the electron and electron-phonon interaction representations. Figures 1(b) and 1(c) show the behavior of the electric current and the heat generation in the antiparallel configuration with different leads spinpolarization. Now the transmission coefficient can be obtained as T (ω) = T (ω) = Γ 2 0 (1 P 2 )/[(ω ε d ) 2 +Γ 2 0 ], and the current and the heat generation are spin-unpolarized with their magnitudes are monotonously suppressed by the increasing of the leads spin-polarization, which can be directly seen from Eqs. (8) and (9). Other behaviors resemble those in the parallel configuration. Fig. 1 (Color online) (a) Schematic plot of a quantum dot coupled to the electron and phonon reservoirs. Due to the interaction with the phonon bath, the energy level in the quantum dot is renormalized to ε d. The magnetic moments of the left and right leads are in either parallel or antiparallel configuration. (b) The total heat generation Q = Q + Q (the solid line) and the total electric current J = J + J (the dashed line) versus the bias voltage for the parallel configuration of the ferromagnetism of the leads. (c) and (d) are respectively the total electric current and the total heat generation versus the bias voltage for antiparallel configuration of the leads ferromagnetism and different values of the lead s spin polarization P L = P R = P. The dot level is allowed to vary with the bias voltage as ε d = ε 0 d + xev, where the bare dot level ε 0 d is fixed to be 0.5 and the asymmetry factor x = 0.5. The other parameters are U = 0, Γ L = Γ R = 0.02, T e = T ph = 0.15, λ q = 0.6ω q. We now study how the intradot Coulomb interaction influence the current and the heat flow in Fig. 2. Since the total current and the heat are still independent on the leads spin-polarization in the parallel configuration, we only present the quantities for the antiparallel case. As usual, the electric current in Fig. 2(a) has a step-like profile with two main steps appearing at threshold bias voltages at which the dot energy levels ε d and ε d + U cross the Fermi level of the left source electrode. Tunneling processes are mediated by the EPI, which results in additional phonon steps in the current. The current is monotonously suppressed in magnitude by the leads spin-polarization, and is antisymmetric with respective to the zero bias point. Different from the zero Coulomb interaction case, the current becomes spin-polarized in the Coulomb blockade regime (only ε d is within the bias window) now, with the spin-polarization of the current P J = (J J )/(J + J ) presented in Fig. 1(b). It is shown that the current polarization is negative in the negative bias regime and reverses its sign when the bias direction is reversed. The spin-polarization of the current originates from the spin accumulation induced on the dot.

5 No. 3 Communications in Theoretical Physics 427 Taking P = 0.2 for example, the line-width functions (tunneling rates) of the left and right leads are respectively Γ L = Γ R = ω q and Γ L = Γ R = ω q. In the negative bias regime where electrons tunneling from the right lead to the left one via the dot, Γ Rσ is the ingoing tunneling rate and Γ Lσ the out-going one. In the deep level case, i.e., both ε d and ε d +U are in the bias window (ev < 5), the spin-up and spin-down current are the same and P J = 0 as seen from Fig. 2(b). This is because the two levels are equally occupied by electrons of different spin directions and then they can transport through the dot with equal speed. For further increasing bias and only ε d is within the bias window ( 5 < ev < 1), spindown current is larger than the spin-up one because of inequality of the tunneling rates. Specifically, spin-down electrons come into the dot faster than they leave it because of Γ R > Γ L whereas the spin-up ones leave the dot faster than they come into it because of Γ R > Γ L. So on average the spin-down electrons spend more time in the dot than the spin-up ones and induce negative spinimbalance m = n n < 0. As a result of it, the spinup electrons tend to be more blocked than the spin-down ones and the magnitude of the corresponding current then is reduced, i.e., J > J. For the positive bias voltage, the in-going and out-going tunneling rates are interchanged and the signs of the spin-imbalance and the current polarization is interchanged correspondingly which is shown in Fig. 2(b). Fig. 2 (Color online) (a) and (b): Total electric current J and the spin-polarization P J; (c) and (d): Total heat generation Q and the spin-polarization P Q as functions of the bias voltage in the antiparallel configuration for different values of the leads spin polarization. The renormalized intradot Coulomb interaction is fixed as U = 2. Other parameters are as in Fig. 1. The behavior of the heat generation in Fig. 2(c) is quite complicate as compared to that of the current. First of all, negative differential of the heat generation emerges for P 0.8 whose origin was explained in Ref. [11], and the magnitude of it is weakened when the leads spinpolarization is large enough. Secondly, for small bias voltage, heat generation is monotonously suppressed by the leads spin-polarization whereas it is firstly enhanced and then suppressed by the leads spin-polarization in larger bias voltage regime. Thirdly, the heat generation is symmetric with respective to the zero bias point even its spinpolarization is antisymmetric with respective to this point as shown in Fig. 2(d). This is because that the heat generation is not directly proportional to the current as was demonstrated by Sun et al. [10 11] To show in more detail the dependence of the heat generation on the bias voltage and the Coulomb interaction, we present its behaviors in Fig. 3 for different values of the leads spinpolarization. For small bias in Fig. 3(a), we find that the heat generation is monotonously suppressed by increasing leads spin-polarization, which is consistent with the result found in Fig. 2. Moreover, the heat generation has a huge peak at U = ω q because of the phonon emitting process happening between real electron levels ε d and ε d + U. [11]

6 428 Communications in Theoretical Physics Vol. 62 Apart from this point, the magnitude of the heat generation tends to be suppressed by increasing Coulomb interaction. The behavior of the heat generation is quite different for larger bias as given in Figs. 3(b) and 3(c) expect for the existence of the huge peak located in U = ω q presented in the inset of Fig. 3(b). The heat generation depends monotonously (Fig. 3(b)) and non-monotonously (Fig. 3(c)) on the Coulomb interaction at the left and right parts of the point of U = ω q, respectively. In small Coulomb interaction regime, we find that the heat generation may be enhanced by increasing U as shown in Fig. 3(b), which is different from the case of Fig. 3(a). These complicate behaviors may be contributed to the spin-blockade effect originated from the spin-imbalance induced in the dot discussed above. Fig. 3 (Color online) Total heat generation as a function of the renormalized intradot Coulomb interaction in the antiparallel configuration for different values of the leads spin-polarization. The bias voltage in (a) is fixed as 1.5ω q. (b), (c) and the inset in (b) present the heat generation in different regimes of the renormalized intradot Coulomb with bias voltage ev = 6ω q. Other parameters are as in Fig Conclusion In conclusion, we investigate the heat generation in a quantum dot molecular sandwiched between two ferromagnetic leads and interacts with a phonon bath. We find some complicate phenomena emerge due to the ferromagnetism of the leads as compared to the structure of a dot coupled to normal metal leads. The magnitude of the negative differential of the heat generation previously found may be weakened at large leads ferromagnetism due to the spin-blockade effect originated from the spin-imbalance in the dot. The heat generation may also monotonously or non-monotonously depend on the leads spin-polarization because of the spin-imbalance in the dot. References [1] L.G.C. Rego and G. Kirczenow, Phys. Rev. Lett. 81 (1998) 232. [2] M.P. Blencowe, Phys. Rev. B 59 (1999) [3] K. Schwab, E.A. Henriksen, J.M. Worlock, and M.L. Roules, Nature (London) 404 ( 2000) 974. [4] A. Balandin, Nat. Mater. 10 (2011) 569. [5] Z. Huang, B.Q. Xu, Y.C. Chen, M. Di Ventra, and N.J. Tao, Nano Lett. 6 (2006) [6] Z. Huang, F. Chen, R. D Agosta, P.A. Bennett, M. Di Ventra, and N.J. Tao, Nat. Nanotechnol. 2 (2007) 698. [7] M. Oron-Carl and R. Krupke, Phys. Rev. Lett. 100 (2008) [8] J.T. Lü and J.S. Wang, Phys. Rev. B 76 (2007)

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