Large Negative Differential of Heat Generation in a Two-Level Quantum Dot Coupled to Ferromagnetic Leads

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1 Commun. Theor. Phys. 64 (2015) Vol. 64, No. 6, December 1, 2015 Large Negative Differential of Heat Generation in a Two-Level Quantum Dot Coupled to Ferromagnetic Leads PENG Ya-Jing ( ), 1 ZHENG Jun (Ü ), 2 and CHI Feng (ðõ) 3, 1 School of Mathematics and Physics, Bohai University, Jinzhou , China 2 College of New Energy, Bohai University, Jinzhou , China 3 College of Engineering, Bohai University, Jinzhou , China (Received June 16, 2015; revised manuscript received August 20, 2015) Abstract Heat current exchanged between a two-level quantum dot (QD) and a phonon reservoir coupled to it is studied within the nonequilibrium Green s function method. We consider that the QD is connected to the left and right ferromagnetic leads. It is found that the negative differential of the heat generation (NDHG) phenomenon, i.e., the intensity of the heat generation decreases with increasing bias voltage, is obviously enhanced as compared to that in single-level QD system. The NDHG can emerge in the absence of the negative differential conductance of the electric current, and occurs in different bias voltage regions when the magnetic moments of the two leads are arranged in parallel or antiparallel configurations. The characteristics of the found phenomena can be understood by examining the change of the electron number on the dot. PACS numbers: b, k, Gk Key words: quantum dot, heat generation, ferromagnetic lead 1 Introduction Over the past more than two decades, electronic transport through quantum dot (QD), which has controllable size, shape, energy levels and various interactions in it, has been intensively studied both theoretically and experimentally in mesoscopic physics. [1 2] Many attractive transport features in single QDs, including the Coulomb blockade effect, [3 4] the Kondo effect, [5 6] the spin blockade effect, [7 8] and the Fano effect [9] have been found. Since these effects traditionally occur only in atom or molecule, QD or QD matrix were then called as artificial atoms or molecules. The small size of a QD results in fully quantized energy levels in it, and the transport properties will become more complex and interesting when more than one localized dot level enter into the bias window. [10] One of the most important phenomena found in such systems with strong Coulomb interactions is the negative differential conductance (NDC), [11 12] which means that the current intensity decreases with rasing bias voltage. On the other hand, with the rapid development of mesoscopic and nanoscale structures, the problem of current-induced heat generation has been paid much attention in recent years. Some previous works have demonstrated that the Joule law in macroscopic systems no longer holds true in these small devices, i.e., the heat intensity is no longer directly proportional to the product of the current intensity and the bias voltage. Currentinduced local heating in single molecules was experimentally observed by Huang et al. by measuring the rupture force of the electrode-molecule bonds. [13] It is found that the force is enhanced with reduced device dimensions. Hot phonon generation from a current bias was recently measured in experiment by Oron Carl and Krupke. [14] Theoretically, local heating in nanoscale junctions and heat generation in nanotube were studied individually by Chen et al. [15] and Lazzeri et al. [16] by the first-principles calculations. It has been shown that, similar to the case in macroscopic systems, the dominant factors for the cause of heating in nanostructures are the inelastic electronelectron and electron-phonon scattering. But in these miniaturized devices, the energy is carried by phonons in a quantized way, which has been theoretically predicted [17] and observed in the following experiment, [18] which become a concrete evidence for the microscopic theory. Very recently, heat generation by electric current in QD systems was studied by Sun and Xie by the nonequilibrium Keldysh Green s function technique. [19] They found that in the resonant tunneling region, small heat current is generated while a large electric current flows through the device. Such a phenomenon is quite favorable in device design. [20] Under some particular conditions, negative differential of the heat generation (NDHG), Supported by the National Natural Science Foundation of China under Grant No and the Liaoning Excellent Talents Programand (LJQ ), the Foundation of State Key Laboratory of Explosion Science and Technology of Beijing Institute of Technology (KFJJ14-08M) Corresponding author, chifeng@semi.ac.cn c 2015 Chinese Physical Society and IOP Publishing Ltd

2 748 Communications in Theoretical Physics Vol. 64 i.e., the intensity of the heat generation decreases with rasing bias voltage, emerges regardless of the monotonously increased electric current with the bias voltage. [20] Transient heat generation induced by a step-like or a squareshaped pulse bias was also theoretically studied and a heat absorbtion from the zero-temperature phonon bath was predicted. [21] In our earlier work, it has been shown that the resonant tunneling electrons can absorb heat from the phonon bath to a QD in the central region when a temperature gradient across the system is applied. Such a device can be used as a nano-refrigerator to cool down the host material of the QD. [22] Until now, works about heat generation in QD-based systems mainly focused on the single-level case. [19 27] Investigation on how the heat flow is affected by multiple energy levels in the dot is a nature extension. Here we study the heat generation by electric current in a two-level QD that is coupled to the left and right ferromagnetic leads, and restrict our considerations to the Coulomb blockade regime. Such a system is realizable in experiments. [28] Moreover, heat generation by electric current in singlelevel QD coupled to ferromagnetic leads has been studied in our previous works. [29 30] It is shown that the intensity of the heat generation and the NDHG effect will be obviously influenced by the ferromagnetism on the leads. 2 Model and Method In the present paper, we assume that the two energy levels on the dot are coupled to the same single-mode phonon reservoir and to the leads with the same coupling strengths. The system s Hamiltonian with ferromagnetic leads can be divided into four parts ( = 1): H = H leads + H D + H p + H T, in which the first term H leads = k,β,σ ε kβσc kβσ c kβσ describes the left (β = L) and right β = R leads 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 term H D corresponds to the two-level dot and is of the following Anderson-like form [19 20,30 31] H D = iσ [ε i0 + λ q (a + a)]n iσ i,j,σ,σ (iσ jσ ) U 0 ijn iσ n jσ, (1) where n iσ = d iσ d iσ is the electron number operator with the creation (annihilation) operator of an electron with on the dot level ε i0 and spin σ. The quantity U 0 ij describes either the intralevel or the interlevel Coulomb correlations. [11 12] 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 single phonon mode is described by the Hamiltonian H p = ω q a a, and H T = i,k,β,σ (V ikβσc kβσ d iσ + V ikβσ d iσ c kβσ) accounts for the tunneling effects between the dot and the leads. The quantity V ikβσ are elements of the tunneling matrix of the dot-lead tunneling. To eliminate the EPI, we make a canonical transformation to the Hamiltonian, i.e., Refs. [19 20, 30 31] H = ˆXH ˆX with ˆX = exp[(λ q /ω q ) iσ (a a)n iσ ]. The transformed Hamiltonian reads H = k,βσ ε kβσ c kβσ c kβσ + iσ i,j,σ,σ (iσ jσ ) i,k,β,σ ε i n iσ Ũ ij n iσ n jσ + ω q a a (Ṽikβσc kβσ d σ + H.c.), (2) where the renormalized dot level and the Coulomb interactions are respectively given by ε i = ε i0 gω q and U ij = Uij 0 2gω q with g = (λ q /ω q ) 2. The tunneling matrix element is also renormalized to Ṽi,kασ = V i,kασ X, where the operator X = exp[ g(a a)]. Next we replace the operator X 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. [19 20] The heat generation Q(t) due to the electric current can be calculated by the time evolution of the phonon energy E ph = ω q a a, i.e., and Q(t) = de ph /dt. Since there is no time-dependent external field in the present paper, the Fourier transform of Q(t) is [19 20] Q = Re dω ω q λ 2 { q G< 2π iiσ (ω) G > iiσ ( ω) iσ 2N ph [ G > iiσ (ω) G a iiσ( ω) + G r iiσ(ω) G > iiσ ( ω)]}, (3) where ω = ω ω q. The dot single-electron Green s functions G iiσ (ω) are the Fourier transforms of G iiσ (t) r,a,<,> r,a,<,> defined in terms of Hamiltonian (2), i.e., the retarded r(a) (advanced) one G ijσ (t) = iθ( t) {d iσ(t), d jσ (0)}, the lesser Green s function G < ijσ (t) = i d jσ (0)d iσ(t), and the greater Green s function G > ijσ (t) = i d jσ (t)c iσ(0). Since the e-p interaction is decoupled in Hamiltonian, the Green s functions G r σ and G < σ can be easily calculated respectively by the Dyson and Keldysh equations, [32] [ ] 1 G r Gr σ = 11σ Gr 12σ = [ g r 1 σ Σ r σ ] 1, (4) G r 21σ G r 22σ [ G < G< σ = 11σ G< 12σ G < 21σ G < 22σ ] 1 = G r σ Σ < σ G a σ, (5) where the matrix g r σ is the Green function of the QD free from interaction between the leads, which can be calculated by the equation of motion technique as g r iiσ = (1 n i σ)(1 n ī ) A iσ + (1 n i σ)n ī A iσ U 12

3 No. 6 Communications in Theoretical Physics n i σ(1 n ī ) n i σ n + ī, (6) A iσ U i A iσ U i U 12 and g 12 r = g 21 r = 0. The subscript indexes are ī = 2 for i = 1 and ī = 1 for i = 2, σ = for σ = and σ = for σ = ; A iσ = ω ε i [n ī ]U 12 + i0 +, {n i } = n i [n i ] with [n i ] is the integer part of the electron total occupation number in dot i n i = n i + n i. n iσ is self-consistently calculated by the equation of n iσ = Im dω G < iiσ (ω)/2π. Σ r,< σ The self-energies represents the retarded and lesser self-energies and are given by, [32] [ Σ r i( ΓL σ = 11σ + Γ R 11σ)/2 i( Γ L 12σ + Γ R ] 1 12σ)/2 i( Γ L 21σ + Γ R 21σ)/2 i( Γ L 22σ + Γ R, 22σ)/2 [ Σ < i( ΓL σ = 11σf L + Γ R ] 1 11σf R ) 0 0 i( Γ L 22σf L + Γ R, (8) 22σf R ) with the Fermi distribution function in Eq. (8) is f β=l,r (ω) = 1/{exp[ω µ β ]/k B T e + 1} with µ β being the chemical potential in lead β. The line-width function Γ β ijσ = k 2πṼikβσṼ jkβσ δ(ε ε kβσ). In the present paper we assume that the two levels are coupled to the leads with equal strength, and then Γ β 11σ = Γ β 22σ = Γ β 12σ = Γ β σ. The spin-dependent line-width function Γ β σ is defined in terms of a spin-polarization parameter p β in each lead as p β = ( Γ β Γ β )/( Γ β + Γ β ), or Γ β σ = Γ β 0 (1 + σp β) with Γ β 0 = ( Γ β + Γ β )/2. After solving G r σ and G < σ, the advanced and greater Green s functions are obtained via the relationships of G a σ = [ G r σ ] and G > σ = G < σ + G r σ G a σ. The electric current flowing through the system can also be obtained with the help of the above Green s functions by using the same method as in Refs. [30 32]. Finally, it should be clarified that the heat generation in the present paper actually is the rate of heat generation. In addition, the present system is a net resistive one, and then the total dissipation is equal to the input power IV, where I is the current intensity and V the bias voltage. The remaining dissipation Q r = IV Q occurs in the leads. [19 20] 3 Numerical Results In the following numerical investigation we choose the phonon frequency ω q 1 as the unit of energy ( 1). The chemical potentials of the two leads tuned by the the bias voltage V as µ L = µ R = V/2. We also assume that the QD and the phonon bath are held at the same temperature, i.e., T ph = T e. We first present the electric current and the heat generation in Fig. 1 without either intra-level or inter-level Coulomb interactions. The tunneling current shows stair-like plateaus when the dot levels and the phonon-induced sub-levels enter into the bias window. The increase in the heat generation, however, depends on the value of the dot levels. When the dot level ε 1 = 0, the current starts to increase at V = 0 in Fig. 1(a), whereas the heat generation in Fig. 1(b) emerges at about V/2 = ω q. When the dot level is ε 1 = 0.5ω q, the first plateau of the current happens at V/2 = ε 1 in Fig. 1(c), where the heat generation in Fig. 1(d) also develops into a plateau. This phenomenon is quite different from that in single-level QD system, [19] in which the heat generation always has a delay of ω q with respect to the electric current. The reason is that the phonon emitting process in the present system may occur between the two dot levels, which is different from the case of single-level case. We observe there are several plateaus induced by the phonon-assisted electron tunneling process in the current in Figs. 1(a) and 1(c). But there are only two plateaus in the heat generation started respectively at V/2 = ε 1 + ω q and V/2 = ε 2 +ω q in Fig. 1(b). For ε 1 = 0.5ω q in Fig. 1(d), there are three plateaus happens at about V/2 = ε 1, V/2 = ε 1 +ω q, and V/2 = ε 2 +ω q, respectively. There are no phonon-induced plateaus in the heat generation. Here it can be seen that the heat generation is not proportional to the electric current, [19 20,29] indicating the breakdown of the Joule law in this nanoscale system. When the leads become ferromagnetic, the electric current of the parallel configuration (dashed red line) overlaps with that of the nonmagnetic leads (solid black line). The current of antiparallel configuration, however, is suppressed. As shown in Figs. 1(b) and 1(d), the magnitude of the heat generation in both parallel and antiparallel configurations are suppressed as compared to the case of nonmagnetic leads. This behavior is similar to the case of singe-level QD, in which detailed explains can be found. [29] The Coulomb interactions will drastically influence the electric current and the heat generation as shown in Fig. 2. As compared to the case of zero Coulomb interactions, many current plateaus emerge in Figs. 2(a) and 2(c) due to the Coulomb blockade effect. Generally, the first eight plateaus due to the electron tunneling start at about the states of ε 1, ε 1 + U 1, ε 2, ε 1 + U 1 + U 12, ε 2 + U 2, ε 2 + U 12, ε 1 + U 1 + 2U 12, ε 2 + U 2 + U 12, where the electric differential conductance shows sharp peaks of Lorentzian shape. [15,32] Meanwhile, there are also some substeps due to the phonon-assisted tunneling. [19 20] The influence of the ferromagnetism on the leads resembles that in Fig. 1. The behavior of the heat generation in the presence of the Coulomb interactions is quite complicate as compared to the case in Fig. 1. Firstly, although the heat generation in the parallel configuration of the ferromagnetism on the leads (the dashed lines in Figs. 2(b) and 2(d)) is suppressed as compared to the case of nonmagnetic leads (the solid lines), the heat generation in the antiparallel configuration is no more monotonously suppressed by the ferromagnetism on the leads (the dotted lines). It is shown that in the lower bias regimes (ev < 5U 12 ), the heat generation of antiparallel configuration is smaller than those of the parallel configuration and the nonmagnetic leads expect at the bias regime for the NDHG. In the higher

4 750 Communications in Theoretical Physics Vol. 64 bias regimes, the heat generation in the antiparallel configuration is larger than those of the other two cases. Such a phenomenon has been found in single-level QD structure in our previous work. [29] Moreover, there are two bias regions where NDHG emerges in the parallel configuration and the nonmagnetic leads in both Figs. 2(b) and 2(d). But the NDHG phenomenon disappears in the antiparallel configuration under the condition of ε 1 = 0 in Fig. 2(b). For ε 1 = 0.5ω q as shown in Fig. 2(d), there is one NDHG bias regime in the antiparallel configuration. It should be noted that in the single-level QD structure, there is only one bias regime for the NDHG regardless of the arrangement of the ferromagnetism on the leads. [20,29] We observe that in the cases of nonmagnetic leads and parallel configuration, the first NDHG occurs when V is between 2(U 1 + ε 1 + ω q ) and 2(U 1 + ε 1 ), and the other bias regime for the NDHG happens about at 2(ε 2 +U 2 +U 12 +ω q ) and 2(ε 2 + U 2 + U 12 + ω q ). The nature for the NDHG in this two-level QD is the same as that in single-level one, [19 20] but additional work region for the phonon absorbing process is brought about in the present system. Fig. 1 Electric current J and heat generation Q as functions of the bias voltage for different values of the ferromagnetism on the leads. In (a) and (b) the dot level is fixed at ε 1 = 0, and in (c) and (d) ε 1 = 0.5 in the absence of both the intra-level and inter-level Coulomb interactions. Other parameters are Γ L 0 = Γ R 0 = 0.01ω q, T e = T ph = 0.02ω q, λ = 0.8ω q, and ε = ω q. Fig. 2 Electric current J in (a) and (c), and heat generation Q in (b) and (d) as functions of the bias voltage for different values of the dot level and ferromagnetism on the leads. The values of Coulomb interactions are set as U 1 = U 2 = U 12 = 3ω q. Other parameters are as in Fig. 1.

5 No. 6 Communications in Theoretical Physics 751 Fig. 3 Electric current and heat generation in (a) and the electron numbers on the dot in (b) versus the bias voltage. The spin-polarization on the lead is p L = 0.95 and p R = 0. Under such a condition the system shows spin Pauli blockade effect as demonstrated by the behavior of the electric current. Other parameters are as in Fig. 2. We study in Fig. 3(a) the electric current and the heat generation in an asymmetrical system induced by the ferromagnetism on the two leads. In the calculation we fixed ε 1 = 0 and p L = 0.95 (nearly a half-metallic lead) and p R = 0 (a nonmagnetic lead), respectively. This asymmetry between the left and right electrodes gives rise to asymmetrical transport characteristics of the current and heat generation with respective to the bias reversal. [29] As compared to the case in Fig. 2, both the magnitude of the current and the heat generation in this asymmetric structure is suppressed due to the mismatching of the tunneling rates of the spin-up and spin-down electrons. It is shown that the degree of the asymmetry with respective to the bias reversal in the heat is much pronounced than that of the current. The width and the position of the bias regime for the NDHG phenomenon can be effectively tuned by the ferromagnetism on the leads. By comparing Figs. 3(a) with 3(b), one can observe that the abrupt changes in the heat generation can be attributed to the change of the electron number on the QD, [29] which is shown in Fig. 3(b). This can be understood that the heat generation arises from the phonon emitting and absorb processes of the electron number on the dot, and then their changes must be synchronous. From the above discussions we can see that the ferromagnetism servers as an effective way for adjusting the heat generation through changing the dot occupation number. 4 Conclusion In conclusion, we have investigated the heat generation in a two-level QD molecular sandwiched between two ferromagnetic leads and interacts with a phonon bath. It is found that multiple NDHG phenomenon occurs in the present system, whereas there is only bias regime for this phenomenon in single-level dot structure. The heat generation may also monotonously or non-monotonously depend on the leads spin-polarization depending on the strength of the Coulomb interaction in the dot. Moreover, both the width and the position of the bias regimes for the NDHG phenomenon can be changed by the ferromagnetism on the leads. References [1] I. Zutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76 (2004) 323. [2] W.G. van der Wiel, S. De. Franceschi, J.M. Elzerman, T. Fujisawa, S. Tarucha, and L.P. Kouwenhoven, Rev. Mod. Phys. 75 (2002) 1. [3] C.W.J. Beenakker, Phys. Rev. B 44 (1991) 1646; D.V. Averin, A.N. Korotkov, and K.K. Likharev, Phys. Rev. B 44 (1991) [4] L.P. Kouwenhoven, D.G. Austing, and S. Tarucha, Rep. Prog. Phys. 64 (2001) 701. [5] T.K. Ng and P.A. Lee, Phys. Rev. Lett. 61 (1988) 1768; Y. Meir, N.S. Wingreen, and P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601; L. Kouwenhoven and L. Glazman, Physics World 14 (2001) 33. [6] N. Sergueev, Q.F. Sun, H. Guo, B.G. Wang, and J. Wang, Phys. Rev. B 65 (2002) [7] D. Weinmann, W. Häusler, and B. Kramer, Phys. Rev. Lett. 74 (1995) 984. [8] M. Braun, J. König, and J. Martinek, Phys. Rev. B 70 (2004) ; W. Rudziński, J. Barnaś, R. Świrkowicz, and M. Wilczyński, Phys. Rev. B 71 (2005) [9] J. König, and Y. Gefen, Phys. Rev. B 65 (2002) ; R. Franco, M.S. Figueira, and E.V. Anda, Phys. Rev. B 67 (2003) [10] W. Belzig, Phys. Rev. B 71 (2004) (R); A. Thielmann, M.H. Hettler, J. König, and G. Schön, Phys. Rev. B 71 (2005) ; F. Elste and C. Timm, Phys. Rev. B 73 (2006) [11] I. Weymann and J. Barnaś, J. Phys.: Condens. Matter 19 (2007) [12] Y.C. Chang and D.M.T. Kuo, Phys. Rev. B 77 (2008) [13] Z. Huang, B.Q. Xu, Y.C. Chen, M. Di Ventra, and N.J. Tao, Nano Lett. 6 (2006) 1240; Z. Huang, F. Chen, R. D Agosta, P.A. Bennett, M. Di Ventra, and N.J. Tao, Nat. Nanotechnol. 2 (2007) 698.

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