Thermoelectric Phenomena in a Quantum Dot Attached to Ferromagnetic Leads in Kondo Regime

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1 Commun. Theor. Phys. 62 (2014) Vol. 62, No. 3, September 1, 2014 Thermoelectric Phenomena in a Quantum Dot Attached to Ferromagnetic Leads in Kondo Regime CHEN Qiao (íü) 1,2, and ZHAO Li-Li ( ÛÛ) 3, 1 Department of Maths and Physics, Hunan Institute of Engineering, Xiangtan , China 2 The Cooperative Innovation Center of Wind Power Equipment and Energy Conversion, Xiangtan , China 3 Department of Fundamental Courses, Academy of Armored Force Engineering, Beijing , China (Received March 10, 2014; revised manuscript received May 19, 2014) Abstract We have studied the thermoelectric properties through ferromagnetic leads-qd coupled system (F-QD-F) in the Kondo regime by nonequilibrium Green s functions method. The spin-flip effect induced by ferromagnetic leads and Kondo effect influence the thermoelectric properties significantly. The peak-valley structure emerges at the low temperature due to Kondo resonance, and the peak-valley structure also relies on the polarization angle θ, the spindependent linewidth function Γ γσ and the energy level of QD ǫ d. Novel resonant peak also emerges in the curve of ZT c versus polarization angle θ. The Kondo effect suppresses the figure of merit ZT c and the spin-dependent figure of merit ZT s. In addition, the spin-dependent figure of merit ZT s is relate with the gap between Γ γ and Γ γ. PACS numbers: r, c, b, Bg Key words: thermoelectric properties, Kondo effect, ferromagnetic leads, nonequilibrium Green s function 1 Introduction The observation of Kondo effect in quantum dots [1 2] has led to an increased interest in this many-body phenomenon. The unambiguous many-body information can be obtained in transport through quantum dot devices due to the fact that the QD parameters can be well controlled. [3 4] The Kondo effect is caused by correlated cotunneling events resulting in enhanced conductance when sequential tunneling is Coulomb blocked. [5] The underlying mechanism is due to the formation of a coherent spin state between electrons in the leads and on the quantum dot. The Kondo effect can be used to investigate the interaction between spins on a double-dot system [6 7] and the exchange interaction with ferromagnetic leads. [8] The Kondo effect is important for the understanding of strongly correlated electrons. For example, ferromagnetic and antiferromagnetic correlations and Fermi and non-fermi liquid behaviors are believed to result from the competition between Kondo and RKKY interaction. In recent years, thermoelectric effect and spinthermoelectric effect in nano devices have been widely studied both experimentally and theoretically. [9 15] The efficiency of thermoelectric device measured by a dimensionless parameter ZT while ZT = GS 2 T/κ. [16] S and G denote Seebeck coefficient and electrical conductance, respectively, and T is the operating temperature. A large value of ZT should own a large value of Seebeck coefficient S, a large value of electric conductance G, and a low value of the thermal conductance κ. The value of ZT in bulk materials remains at about one due to the Wiedemann Franz law. [17] Due to level quantization and Coulomb blockade effects, the situation of in nano devices is different from that in bulk system. Recently, Harman et al. [18] reported ZT 2 in quantum dots system,the results also predicted by theoretical calculations. [14,19] Spin-polarized transport between ferromagnetic leads-qd coupled system has become the subject of extensive research because of the potential device applications. [20 21] The transport properties in quantum dot attached ferromagnetic leads can be controlled with the aid of the electron spin degree of freedom. Spin Seebeck effect was observed for ferromagnetic slab and spin voltage generated by temperature gradient was measured. [22] In the present study the theoretical analysis of thermoelectric and spin-thermoelectric properties of F-QD-F system in Kondo regime is addressed. The Kondo effect and the spin-flip effect affect the thermoelectric properties significantly. Novel properties emerge due to Kondo resonance and these properties are relate with the polarization angle θ, energy level of center QD ǫ d and the gap between Γ γ and Γ γ. This paper is organized as follows. In Sec. 2, we Supported by the National Natural Science Foundation of China under Grant No , the Hunan Provincial Natural Science Foundation of China under Grant No. 11JJ4005 and Innovative Fund Project of the Academy of Armored Forces Engineering under Grant No. 2013CJXS11 cqhy1127@aliyun.com ponydoll@163.com c 2014 Chinese Physical Society and IOP Publishing Ltd

2 418 Communications in Theoretical Physics Vol. 62 present the Hamiltonian of our system and detailed algebraic expressions by nonequilibrium Green s functions (NGF). The numerical results are given in Sec. 3 with analyses. Finally, a summary is presented in Sec Model and Formalism Let us consider a single-level quantum dot with a charging energy U coupled to two ferromagnetic leads. The magnetic moment M γ of the γ-th lead is tilted at angle θ γ to the z axis. We take θ L as the reference of the angle by setting θ L = 0, i.e., the magnetic moment M L of the left lead is assumed to be parallel to the z axis and θ R = θ. The Hamiltonian of the system can be written as [23 24] H = γkσ{[ǫ γ,kσ σm γ cosθ γ ]a γ,kσ a γ,kσ ( a θ ) ( γ,kσ = α γ,kσ cos σα θ ) γ,k σ 2 sin. (2) 2 With this unitary transformation, the Hamiltonian of the system becomes diagonal, and the tunnelling part of the Hamiltonian becomes spin-dependent. The interaction terms between the ferromagnetic leads and the QD are now written as the elements T γk,σσ of the interaction strength matrix T γk = T γk R(θ) in the spin-space,where the rotation matrix is employed R(θ) = cos θ sin θ 2 2 sin θ cos θ. (3) 2 2 Herein, the Hamiltonian of the system is transformated to H = γkσ[ǫ γk σm γ cosθ γ ]α γ,kσ α γ,kσ M γ sin θ γ a γ,kσ a γ,k σ} + σ ǫ d d σ d σ Un σn σ + σ ǫ d d σ d σ Un σn σ + γkσ(t γk a γ,kσ d σ + H.c.). (1) Here a γ,kσ (a γ,kσ) is the creation (annihilation) electron operators of the γ-th lead. d σ(d σ ) is creation (annihilation) electron operator of the central QD. n σ = d σd σ is electron occupation operator. The magnitude of magnetic moment M γ = (1/2)gµ B h γ is associated with the Bohr magneton µ B, Landé factor g, and the molecular field strength h γ of the γ th lead. ǫ γ,kσ are the isolated energies of leads and ǫ d is the isolated energy level of QD. T γk is the coupling strength of central QD with the γ th ferromagnetic lead. In order to make the Hamiltonian diagonalization, we make the transform over the Hamiltonian of ferromagnetic leads ( θ ( θ a γ,kσ = α γ,kσ cos σα γ,k σ sin, 2) 2) + (α γ,kσ T γkσσ d σ + H.c.). (4) γkσσ In order to obtain the formulas for electronic current and heat current through F-QD-F system, we follow the same way described in Refs. [25 28] and express the electronic current and the heat current with Keldysh Green s functions. Then, we can obtain the formulas for the electronic current J cσ = e dω[γ L G r (ω)γ R G a (ω)] σσ [f L (ω) f R (ω)], (5) h and the heat current J hσ = e dω(ω ev )[Γ L G r (ω)γ R G a (ω)] σσ h [f L (ω) f R (ω)]. (6) The matrix Γ γ is given by θ Γ γ = cos2 2 Γ γ + sin 2 θ 2 Γ γ cos θ 2 sin θ 2 (Γ γ Γ γ ) cos θ 2 sin θ 2 (Γ γ Γ γ ) cos 2 θ 2 Γ γ + sin 2 θ. (7) 2 Γ γ The linewidth function of the γ-th lead is defined by Γ γσ (ω) = 2π k T γk 2 δ(ω ǫ γ,kσ ). We can obtain the Green s function matrices G r of the central QD by employing EOM method, [26 27] in which the spin-flip effect makes important contribution. The Fourier transformed retarded Green s functions are given by G r σσ (ω) = (ω ǫ d χ r σ σ(ω))[1 n σ ]δ σσ + [1 n σ ]χ r σ σ(ω)δ σσ (ω ǫ d χ r σσ(ω))(ω ǫ d χ r σ σ(ω)) χ r, (8) σ σ(ω)χ r σσ(ω) where χ r σσ (ω) = Σr σσ (ω) + Σ(1)r (ω). The self-energy matrixes are given by Σ r (t, t ) = γk σσ T γk gr γk(t, t )T γk, Σ (1)r (t, t ) = γk T γk gr γk(t, t )T γk f(ǫ γ,kσ σm γ cosθ γ ), respectively. The diagonal Green s function of the γ-th lead is defined by the matrix elements gγk,σσ r (t, t ) = i [ θ(t t )exp i ] (ω ǫ γk + σm γ cosθ γ )(t t ) δ σσ.

3 No. 3 Communications in Theoretical Physics 419 The results obtained by EOM approach are quantitatively valid at the high temperature T > T K, and it is qualitatively valid for T T K. We can obtain the proper Kondo peaks by this method. [29 30] Therefore, our results are at least trustable to the whole qualitative behaviors at chosen temperature. The occupation number n σ should be calculated self-consistently through the equation n σ = 1 2π Im dǫg < σσ. The Keldysh Green s function matrix is expressed by G < (ω) = i G r (ω)γ γ (ω)g a (ω)f γ (ω). (9) γ In the linear regime, for infinitesimally small sourcedrain bias V and temperature gradients T, we can expand the electronic and heat currents as J cσ = e 2 I 0σ V + e T I 1σ T, J hσ = ei 1σ V 1 T I 2σ T. (10) The coefficient I nσ is defined by I nσ = 1 dω[γ L G r (ω)γ R G a f(ω) (ω)] σσ h ω. (11) The spin-dependent electrical conductance, the thermal conductance and Seebeck coefficient can be given by G σ = e 2 I 0σ, κ σ = (1/T)[I 2σ I 2 1σ /I 0σ] and S σ = I 1σ /eti 0σ by using the notification I nσ. The conductance are given by G e = G + G and G s = G G. Finally, we define the charge and spin Seebeck coefficients as S c = (1/2)(S +S ) and S s = (1/2)(S S ), respectively. The charge and spin figure of merit can be defined as ZT c = (S 2 cg e T/κ) and ZT s = (S 2 sg s T/κ). The thermal conductance κ is defined by κ = κ + κ. 3 Results and Analysis We now perform numerical calculations to examine the thermoelectrical properties of F-QD-F system in the Kondo regime. Without loss of generality, we consider only the case of single level QD by setting ǫ d = 0. We choose = 0.1 mev as the energy unit and consider the symmetric barriers with Γ Lσ = Γ Rσ in numerical calculations. The spin-dependent linewidth function Γ γ = pγ γ. We set µ L = µ R = 0 in the linear region. The conductance, the Seebeck coefficient, and the thermal conductance are scaled by 2e 2 /h, k B /e and 2k B /h, respectively. Fig. 1 The Seebeck coefficient S c and the figure of merit ZT c versus the temperature T for different polarization angle θ with U = 0 and U =. The linewidth functions Γ γ = 1 and Γ γ = 0.1 and the energy level of QD ǫ d = 1.2. Figure 1 shows the temperature dependence of Seebeck coefficient S c and ZT c for different polarization angle θ. In order to compare the contribution of Kondo effect in our system with their noninteracting systems, we also depict the Seebeck coefficient S c and the figure of merit ZT c as the charging energy U = 0. As charging energy U = 0, the Seebeck coefficient is larger than zero regardless of the tilted angle θ. However, the Seebeck coefficient S c is less than zero at low temperature in the Kondo regime. The Seebeck coefficient S c also shows a peak-valley structure at low temperature, and the magnitude of the peakvalley structure is closely related to the polarization angle θ. The magnitude of the peak-valley structure becomes small with the increasing of the polarization angle. The Seebeck coefficient is sensitive to the density of state at femi energy E f. As we know, Kondo resonance emerges at femi energy E f due to strong Coulomb interaction in the Kondo regime. The ferromagnetic leads split the Kondo peak to form two Kondo peaks around femi energy. The locations of the two Kondo peaks are associate with the

4 420 Communications in Theoretical Physics Vol. 62 polarization angle intimately. In addition, the Seebeck coefficient S c is suppressed by Kondo effect. The figure of merit ZT c shows a single peak in noninteracting QD system. However,the figure of merit ZT displays small peaks at low temperature in the Kondo regime. The small peaks are related with the valley in the curve of Seebeck coefficient due to the Kondo effect. The magnitude of peak also decreases with polarization angle θ. The figure of merit ZT c in Kondo regime is less than in noninteracting system. The energy level ǫ d of central QD is an important parameter in Kondo regime, which is always tuned by the gate voltage. We give the Seebeck coefficient S c and figure of merit ZT c versus temperature T with different ǫ d. We found that the thermoelectric properties are sensitive to the energy level of QD. The peak-valley structure caused by Kondo effect is very remarkable as ǫ d = 1.2. The figure of merit ZT c increases with ǫ d at low temperature and decreases with ǫ d at high temperature. The linewidth function is also very important in tunneling process. [31 32] In Figs. 2(c) and 2(d) we discuss the influence of linewidth function on thermoelectric properties. The linewidth functions Γ γ and Γ γ are different in ferromagnetic lead- QD coupled system due to spin-flip effect. We denote Γ γ = pγ γ. The peak-valley structure disappears gradually as the parameter p increases. As the parameter p decreases, the gap between the spin-up linewidth function Γ γ and the spin-down linewidth function Γ γ due to the spin-flip effect becomes large. The spin-flip effect has huge influence on Kondo effect. At high temperature, the Seebeck coefficient S c has nothing to do with the parameter p. Because the temperature is higher than the Kondo temperature T K. The figure of merit ZT c increases with the parameter p due the linewith function increases. Fig. 2 The Seebeck coefficient S c and the figure of merit ZT c versus the temperature T for different ǫ d (Diagram (a)) and different p (Diagram (b)). The other parameters are θ = π/6, Γ γ = 1.0, Γ γ = 0.1 for Diagram (a) and ǫ d = 1.2 for Diagram (b). Figure 3 represents the variation of the figure of merit ZT c with respect to the polarization angle θ in the absence and presence of Coulomb interaction. The figure of merit ZT c oscillates with the polarization angle θ with period 2π in both conditions. In the absence of Coulomb interaction, there is only one resonant peak in each period. However, a novel resonant peak emerges in the presence of Coulomb interaction due to Kondo effect. In addition, the magnitude of resonant peaks in the Kondo regime is smaller than in the absence of Coulomb interaction, i.e., the Kondo effect suppresses the figure of merit ZT c. It is also found in Figs. 1 and 2. The spin Seebeck effect denotes the generation of spin voltage arising from a temperature difference, and we can generate spin current by using temperature gradient. Figure 4 represents the spin Seebeck coefficient and the spin dependent figure of merit with respect to the polarization angle θ in the absence and presence of Coulomb interaction with different parameter p. The spin Seebeck coefficient and the spin dependent figure of merit are influenced by the parameter p significantly. As the parameter p increases, i.e., the difference Γ γ Γ γ is smaller, the spin Seebeck coefficient and the spin dependent figure of merit decrease in both conditions (U = 0 and U = ). In the Kondo regime, the resonant peak becomes blunt and a small shoulder emerges due to Kondo resonance. The spin dependent figure of merit ZT s also decreases as the Kondo resonance occurs.

5 No. 3 Communications in Theoretical Physics 421 Fig. 3 The figure of merit ZT c versus the polarization angle theta θ with U = 0 and U =. The other parameters are chosen as ǫ d = 1.2, Γ γ = 1.0 and Γ γ = 0.1. Fig. 4 The spin-dependent Seebeck coefficient S s and the spin-dependent figure of merit ZT s versus the polarization angle theta θ with U = 0 and U =. The other parameters are chosen as ǫ d = 1.2, k BT = 0.1 and Γ γ = Summary In summary, we have investigated thermoelectric properties through quantum dot attached to ferromagnetic leads in the Kondo regime by nonequilibrium Green s function technique. We also have calculated the thermoelectric properties in the absence of Coulomb interaction for comparison. The information of spin-flip effect caused by ferromagnetic leads is included in the Green s functions of QD and the linewidth functions. The Kondo effect and spin-flip effect have huge influence on thermoelectric

6 422 Communications in Theoretical Physics Vol. 62 properties of the QD system. The peak-valley structure emerges at the low temperature due to Kondo resonance and the peak-valley structure is relate with the polarization angle θ, spin-dependent linewidth function and the energy level of QD intimately. The Kondo effect suppresses the the figure of merit ZT c and the spin-dependent figure of merit ZT s. The spin-dependent figure of merit ZT s also is dependent on the gap between Γ γ and Γ γ. The ZT s increases with the magnitude of this gap. References [1] D. Goldhaber-Gordon, et al., Nature (London) 391 (1998) 156. [2] S.M. Cronenwett, et al., Science 281 (1998) 540. [3] J. Martinek, et al., Phys. Rev. Lett. 91 (2003) [4] H.B. Heersche, et al., Phys. Rev. Lett. 96 (2006) [5] S. Sasaki, S.De Franceschi, J.M. Elzerman, W.G. van der Wiel, M. Eto, S. Tarucha, and L.P. Kouwenhoven, Nature (London) 405 (2000) 764. [6] A.M.C.H. Jeong and M.R. Melloch, Science 293 (2001) [7] J.C. Chen, A.M. Chang, and M.R. Melloch, Phys. Rev. Lett. 92 (2004) [8] A.N. Pasupathy, et al., Science 306 (2004) 86. [9] Y. Dubi and M.Di Ventra, Rev. Mod. Phys. 83 (2011) 131. [10] C.W.J. Beenakker and A.A.M. Starling, Phys. Rev. B 46 (1992) [11] B. Dong and X.L. Lei, J. Phys.: Condens. Matter 14 (2002) [12] J. Koch, F.von Oppen, Y. Oreg, and E. Sela, Phys. Rev. B 79 (2009) [13] B.C. Hsu, Y.S. Liu, S.H. Liu, and Y.C. Chen, Phys. Rev. B 83 (2011) [14] J. Liu, Q.F. Sun, and X.C. Xie, Phys. Rev. B 81 (2010) [15] R. Świrkowicz, M. Wierzbicki, and J. Barnaś, Phys. Rev. B 80 (2009) [16] G.S. Nolas, J. Sharp, and H. Goldsmid, Thermoelectrics: Basic Principles and New Materials Developments, Springer, New York (2001). [17] A. Majumdar, Science 303 (2004) 777. [18] T.C. Harman, P.J. Taylor, M.P. Walsh, and B.E. LaForge, Science 297 (2002) [19] D.M.T. Kuo and Y.C. Chang, Phys. Rev. B 81 (2010) [20] Mahn-Soo Choi, D. Sánchez, and R. López, Phys. Rev. Lett. 92 (2004) [21] P. Zhang, Q.K. Xue, Y.P. Wang, and X.C. Xie, Phys. Rev. Lett. 89 (2002) [22] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature (London) 455 (2008) 778. [23] N. Sergueev, Q.F. Sun, H. Guo, B.G. Wang, and J. Wang, Phys. Rev. B 65 (2002) [24] Z.G. Zhu, G. Su, Q.R. Zheng, and B. Jin, Phys. Rev. B 70 (2004) [25] A.P. Jauho, N.S. Wingreen, and Y. Meir, Phys. Rev. B 50 (1994) [26] H.K. Zhao, L.L. Zhao, and J. Wang, Eur. Phys. J. B 77 (2010) 441. [27] H.K. Zhao and L.L. Zhao, Europhys. Lett (2011). [28] Y.S. Liu and X.F. Yang, J. Appl. Phys. 108 (2010) [29] Y. Meir, N.S. Wingreen, and P.A. Lee, Phys. Rev. Lett. 70 (1993) [30] T.K. Ng, Phys. Rev. Lett. 76 (1996) 487. [31] S.K. Maiti, Physica B 394 (2007) 33. [32] S.K. Maiti, Org. Electron. 8 (2007) 575.