JOURNAL OF APPLIED PHYSICS 109, 053712 (2011)
Pure spin thermoelectric generator based on a rashba quantum dot molecule Yu-Shen Liu,1,2,a) Feng Chi,3,b) Xi-Feng Yang,1,2 and Jin-Fu Feng1,2,c) 1
Jiangsu Laboratory of Advanced Functional Materials, Changshu 215500, China College of Physics and Engineering, Changshu Institute of Technology, Changshu 215500, China 3 Department of Physics, Bohai University, Jinzhou 121000, China 2
(Received 24 October 2010; accepted 31 January 2011; published online 11 March 2011) We propose a pure thermoelectric spin generator based on a Rashba quantum dot molecular junction by using the temperature difference instead of the usual voltage bias difference. A magnetic flux penetrating through the device is also considered. The spin Seebeck coefficient SS and the spin figure of merit ZS T of the molecular junction are calculated in terms of the Green’s function formalism and the equation of motion (EOM) technique. It is found that a pure spin-up (spin-down) Seebeck coefficient can be generated by the coaction of the magnetic flux and the Rashba spin-orbit C 2011 American Institute of Physics. [doi:10.1063/1.3560772] (RSO) interaction. V
I. INTRODUCTION
Recently, low-dimensional thermoelectric materials have been extensively investigated due to their high energy conversion efficiency.1–8 The efficiency of a material directly converting thermal energy into electrical energy or vice versa can be described by the dimensionless charge figure of merit ZC T ¼ S2C Ge T=ðjel þ jph Þ, where SC is the charge Seebeck coefficient, Ge is the electric conductance, jelðphÞ is the electric (phonon) thermal conductance, and T is the operating temperature. In bulk materials, the value of ZC T remains at about one mainly due to the Wiedemann-Franz law.9 ZC T in low-dimensional materials, however, is much larger than that of their three-dimensional (3D) bulk counterpart as demonstrated by recent experimental and theoretical works.10–14 For example, in zero-dimensional quantum dots (QDs) device, ZC T ’ 2 was reported experimentally by Harman et al,10 and predicted by theoretical calculations.11–14 More recently, Uchida et al reported an experimental observation of the spin Seebeck effect in a metallic magnet by using a spin detection technique, where the spin Seebeck effect denotes the generation of spin voltage arising from a temperature difference.15 From the application point of view, the study of the spin Seebeck effects in nanoscale QDs system will open a new door to design and fabricate thermospin quantum device to generate spin current by using temperature gradient rather than the usual electric bias. After that, there have been much academic works on spin-dependent thermoelectric effect in single QD attached to ferromagnetic leads with collinear or noncollinear magnetic moments.16–19 Up to now, we note that a large number of theoretical works on electron transport through double-quantum-dot (DQD) molecular junction have been reported,20–25 while less ones have addressed the thermoelectric effect in it.26 In this paper, we propose a pure spin-up (spin-down) Seebeck coefficient generator consisting of a Rashba DQD molecule sandwiched between two metallic electrodes by a)
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0021-8979/2011/109(5)/053712/5/$30.00
using temperature difference other than the usual voltage bias difference. It should be noted that the Rashba spin-orbit (RSO) effect in QDs, which arises from the inversion asymmetry of the interface electric field,27 may provide an all-electrical spin control method. Some theoretical and experimental works have shown that the current spin-polarization based on the RSO interaction can reach as high as 100%28,29 or infinite.30 The behind reason is that a spin-dependent RSOinduced phase factor appears in the tunnel coupling matrix between the QDs and the metallic electrodes. Until now, thermoelectric effects related to the RSO effect in the DQD molecular junction has never been investigated, which is the motivation of the present work. As will be seen later, the electron conductances of each spin components are modulated by the quantum interference effect arisen from the Rashbainduced phase factor. Under certain conditions, one spin component conductance is totally suppressed while that of the other spin component is finite, resulting the spin filter effect. From the definition of the Seebeck coefficient, it can be seen that now only one spin direction Seebeck coefficient is finite. As a result, we find that a pure spin-up or spin-down thermoelectric generator can be obtained at room temperature by tuning the magnetic flux and the RSO-induced phase factor. II. MODEL AND METHOD
The proposed spin-dependent thermoelectric generator consisting of two QDs in the presence of RSO interaction attached to two metallic electrode is depicted in Fig. 1, which can be described by the following total Hamiltonian, X X † eak a†akr aakr þ en dnr dnr Htotal ¼ n¼1; 2; r a¼L;X R; kr † þ Varn dnr aakr þ H:c: ; (1) k; a; r; n
where a†akr ðaakr Þ is the creation (annihilation) operator for an electron with energy eak , momentum k and spin index r in † ðdnr Þ creates (destroys) an electron with electrode a. dnr energy en and spin r in the nth QD. The tunnel matrix element Varn in a symmetric gauge is assumed to be independent
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equation of Gar ðeÞ ¼ ½Grr ðeÞ† . The heat current with spin index r flowing from electrode a is calculated by,33 Jra ¼
ð i de r a ðela ÞTr Car ðeÞfG< r ð EÞþfa ðeÞ½Gr ðeÞGr ðeÞg : h 2p (4)
At steady state, we arrive at the compact forms for the electric current and heat current through the DQD molecular junction as,34 ð e desr ðeÞ½fL ðeÞ fR ðeÞ; Ir ¼ (5) h FIG. 1. (Color online) Schematic plot of the spin thermoelectric generator consisting of double quantum dots (DQDs) sandwiched between two metallic electrodes in the presence of magnetic flux U. A spin-dependent thermoelectric voltage DVr is generated when an infinitesimal temperature difference DT is applied. l is the chemical potential of the metallic electrodes, and T is the electrode temperature.
of momentum k, and can be written as VLr1 ¼ jVLr1 j VLr2 ¼ jVLr2 jeiður/R Þ=4 , VRr1 ¼ jVRr1 j eiður/R Þ=4 , iður/R Þ=4 , VRr2 ¼ jVRr2 jeiður/R Þ=4 , with the phase e u ¼ 2pU=U0 and the flux quantum U0 ¼ h=e. /R denotes the difference between /R1 and /R2 , where /Rn is the phase factor induced by the RSO interaction inside the nth quantum dot. For simplicity, we have assumed that the RSO effect exists only inside the quantum dots, which often brings about two main results: One is the extra spin-dependent phase factor in the tunnel matrix elements; and the other is the interlevel spin-flip term.31 Here, we neglect the latter term since only one energy level inside each QD is taken into account. Using the Green’s function method and the Dyson’s equation technique, spin-dependent current in each lead a can be expressed as,32 ð ie de a r a Tr Cr ðeÞfG< Ira ¼ r ðEÞ þ fa ðeÞ½Gr ðeÞ Gr ðeÞg ; h 2p (2) where the spin-dependent line-width parameter Car is a 2 2 matrix as, 1 e6ið/ruR Þ=2 LðRÞ Cr ðeÞ ¼ C ið/ruR Þ=2 : (3) e 1 For simplification, we assume identical tunnel coupling between the QDs and the electrodes with Varn ¼ V and C ¼ 2pjVj2 qa , where qa is the density of states in the electrode a. G< r ðeÞ is the 2 2 matrix of the Fourier transform of Keldysh lesser QD Green’s function, whose matrix elements in the time space are r † defined as G< nr; mr ðtÞ ¼ i < dmr ð0Þdnr ðtÞ > . Gr ðeÞ is the 2 2 matrix of the Fourier transform of retarded QD Green’s function, and its matrix elements in the time space can be calcu† ð0Þg > , where lated by Grnr; mr ðtÞ ¼ iHðtÞ < fdnr ðtÞ; dmr HðtÞ is the step function. fa ¼ f1 þ exp½ðe la Þ=ðkB Ta Þg1 is the Fermi-Dirac distribution function in electrode a, where la and Ta are the chemical potential and the temperature in electrode a, respectively; kB is the Boltzmann constant. The advanced QD Green’s function can be obtained by the relation
and Jra
ð 1 deðe la Þsr ðeÞ½fL ðeÞ fR ðeÞ; ¼ h
(6)
respectively. sr ðeÞ is the transmission function of electron with spin index r, which is given by sr ðeÞ ¼ Tr CLr Grr CRr Gar . Since the two spin channels are independent of each other, we can introduce the spin-dependent Seebeck coefficient Sr ¼ limD!0 DVr =D T, where DVr is spin-dependent bias voltage induced by the temperature difference DT between the two leads under the condition of zero electric current DIr ¼ 0. DIr is calculated by the following equations,35 DIr ¼ Ir ðl; T; l þ eDVr ; T þ DT Þ Ir ðl; T; l; T Þ: (7) In the present paper, we consider the linear response case, which indicates that both the bias voltage and the temperature differences between the two leads tend to zero, i.e., lL ¼ lR ¼ l and TL ¼ TR ¼ T. After expanding the FermiDirac distribution function to the first order in DT and DVr , we obtain the spin-dependent Seebeck coefficient in terms of Eq. (7) as,35 Sr ðl; TÞ ¼
1 K1r ðl; TÞ ; eT K0r ðl; TÞ
(8)
and the electronic thermal conductance,35 1 K2r r ; (9) jel ðl; T Þ ¼ K1r eSr þ T h Ð m where Kmr ðl; T Þ ¼ de @f @e ðe lÞ sr ðeÞðm ¼ 0; 1; 2Þ. In the low temperature regime, using the Sommerfeld expansion,36 we can expand Kmr ðl; TÞ up to the lowest order in temperatures as K0r ’ sr , K1r ’ ½p2 kB2 s0r ðlÞ=3T 2 , and K2r ’ ½p2 kB2 sr ðlÞ=3T 2 , respectively. The differential conductance with spin index r may be expressed as Gr ðl; TÞ ¼ ðe2 =hÞK0r ðl; TÞ. Finally, we define the spin and the charge Seebeck coefficients as SS ¼ ð1=2ÞðS" S# Þ and SC ¼ ð1=2ÞðS" þ S# Þ, respectively.18 The charge and the spin figure of merits can be defined as ZC T ¼ ðS2c Ge T=jel þ jph Þ and ZS T ¼ ðS2S Gs T=jel þ jph Þ respectively. The corresponding conductances are given by Ge ¼ ðe2 =hÞ½K0" ðl; TÞ þ K0# ðl; TÞ and Gs ¼ ðe2 =hÞ½K0" ðl; TÞ K0# ðl; TÞ. jel is the electrical thermal conductance, which can be calculated by using the equation of jel ¼ j"el þ j#el . In the low
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temperature regime, jrel has a simple approximation expression of jrel ¼ brT T brV T 3 , where brT ¼ ½p2 kB2 sr ðlÞ=3h and brV ¼ fp4 kB4 ½s0r ðlÞ2 =9hsr ðlÞg. jph , the phonon thermal conductance of the junction, which is typically limited by the QD-electrode contact, is ignored in the present paper. This is reasonable in the case of poor link to phonon transport.35 III. RESULTS AND DISCUSSION
In the following numerical calculations, we set C ¼ 1 eV as the energy unit and fix the dots’ levels en ðn ¼ 1; 2Þ to be zero. The interdot tunnel coupling is neglected assuming that the distance between the two QDs is large enough. In Fig. 2, we show the spin-dependent thermoelectric effects in the presence of the RSO-induced phase factor ð/R ¼ 0:25pÞ and the magnetic flux threading through the molecular junction ðu ¼ 0:5pÞ. The RSO-induced phase factor /R arising from the Rashba effect is determined by the QDs’ parameters via the relationship of uR ¼ bm L=h2 , where b is the spin-orbit interaction strength, m is the electron effective mass, and L is the length of the dot, respectively. The magnitude of /R can be tuned in experiment by gate voltage or QDs’ configurations.37,38 When the spindependent phase factor is involved, the transmission function sr can be calculated by using the equation of sr ðlÞ ¼ Tr½CLr Grr CRr Gar , and it has an exact expression as,28 sr ¼
4C2 l2 cos2 ð/r =2Þ ½l2 C2 sin2 ð/r =2Þ2 þ 4C2 l2
;
(10)
where /r ¼ u r/R . The above equation shows that there are two resonant peaks in the transmission function sr appearing at the chemical potentials of l ¼ 6C sin /r =2, and a dip positioned at l ¼ 0 due to the quantum destructive interference effect. Equation (10) also shows that sr is modulated by the factor of cos2 ð/r =2Þ and, as a consequence, results in cos2 ð/" =2Þ > cos2 ð/# =2Þ in the case of u ¼ 0:5p and /R ¼ 0:25p. As a result, we note that the value of s" is
always larger than that of s# except for the zero energy point l ¼ 0. The numerical results for the spin-dependent transmission function sr as a function of the chemical potential is shown in Fig. 2(a), which are in good agreement with the Eq. (10). In Fig. 2(b), we plot the spin-dependent Seebeck coefficient Sr , the charge Seebeck coefficient SC , and the spin Seebeck coefficient SS as functions of the chemical potential l. Sr can be calculated by numerical integration for Eq. (8), then SC and SS are calculated by the equations of SC ¼ ð1=2ÞðS" þ S# Þ and SS ¼ ð1=2ÞðS" S# Þ, respectively. The numerical results show that the thermoelectric effects are enhanced obviously when the chemical potential is close to the dots’ energy levels of en ¼ 0. Since spin-up and spindown electrons have different phase factors, S" and S# become different from each other, and SS is thus introduced in the DQD molecular junction. The maximum value of jS# j has a larger value than the spin-up one, and exhibits a universal maximum value of about 156 lV=K at the two sides of the zero energy point, while the maximum value of jS" j has a smaller value of about 90 lV=K. The numerical results can be interpreted by the following descriptions. When the chemical potential l goes across the locations of the transmission resonances at 6C sin /r and approaches to the zero energy point, K1" become larger than K1# , while K0# is smaller than K0" [see the insets in Fig. 2(a)]. From Eq. (8), we can see that Sr is dependent on the ratio of K1r and K0r . Finally we obtain the relationship of jK1# =K0# j > jK1" =K0" j in such a case. Since the two transmission resonances of the spindown electron is far away from each other, s# has a simple approximate quadratic transmission node as,39 s# /
4l2 cos2 ð/# =2Þ C2 sin4 ð/# =2Þ
:
(11)
Thus jS# j has a larger value near the universal maximum value of 156 lV=K (Ref. 39). In Fig. 2(c), jrel as a function of the chemical potential l is shown, which can be calculated by the numerical integration for Eq. (9). Due to temperature
FIG. 2. (Color online) Behaviors of considered quantities as functions of the chemical potential: (a) Transmission function sr ; (b) Seebeck coefficients with K0r and K1r in the insets; (c) Electronic thermal conductance jrel ; (d) Charge ZC T and spin ZS T. The phase factors of u and /R are fixed as 0:5p and 0:25p, respectively. The electrode temperature T is chosen as 300 K.
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FIG. 3. (Color online) (a) charge conductance Ge and electronic thermal conductance jel (inset), (b) spin conductance GS , (c) charge coefficient SC , (d) spin Seebeck coefficient SS , (e) charge and (f) spin figure of merits as functions of the magnetic flux u for different values of /R . The chemical potential l and the temperature T of electrodes are fixed at 0:05 eV and 300 K, respectively. /R is chosen as 0 (black solid line), 0:25p (red dashed line), 0:5p (blue dotted line).
effects, jrel has a finite value at the zero energy point. We also find that j" is always bigger than j# due to the larger value of the spin-up transmission function. ZC T and ZS T as functions of the chemical potential are displayed in Fig. 2(d). The numerical results show that ZC T and ZS T have three dips and two peaks. These zero points of the charge and the spin figure of merits represent different physical mechanisms. When l ¼ 0, we have ZC T ¼ ZS T ¼ 0 due to the fact S" ¼ S# ¼ 0. Whereas, ZC T ¼ 0 and ZS T 6¼ 0 at the chemical potential of about 60:7 eV since S" ¼ S# . In this case, only spin current in the absence of charge current can be expected in the present DQD molecular junction. Around the chemical potential of 61:4 eV, we have ZS T ¼ 0 and ZC T 6¼ 0 because of S" ¼ S# . Now we study the oscillation behaviors of the charge and the spin thermoelectric effects as functions of the magnetic flux for different values of /R . In the low temperature regime, using Eq. (10) and the approximate equation Sr ’ p2 kB2 s0 ðlÞT=½3esðlÞ,35 we obtain an approximate result for the spin-dependent Seebeck coefficient as, Sr ’
p2 kB2 T ½l4 C4 sin4 ð/r =2Þ=l : 3e ½l2 C2 sin2 ð/r =2Þ2 þ 4C2 l2
(12)
When /R ¼ 0, we have /r ¼ u and SC ¼ Sr accordingly. From the above equation, it is evident that jSC j has a minimum (SC ’ 0) at u ¼ 2npðn ¼ 0; 1; 2; Þ and a maximum near the universal maximum value of 156 lV=K for u approaches to u ¼ ð2n þ 1Þpðn ¼ 0; 1; 2; Þ [see the black line in Fig. 3(a)]. The sign of SC is dependent on the factor ½l4 C4 sin2 ð/r =2Þ=l, and thus SC < 0 when l ¼ 0:05eV, which indicates that the DQD molecular junction is a N-type
conductor. In the case of /R 6¼ 0, S" and S# will be separated in the u space. As a result, we see that the resonant peaks around 2np split into two and the distance between the peaks in the u space equals to the phase difference of spin-up and spin-down electrons. The electronic thermal conductance jel as a function of the magnetic flux u under different Rashbainduced phase factors is plotted in the inset of Fig. 3(a), whose behavior is similar to that of the charge conductance. When /R ¼ p, the periods of GC , SC and ZC T vary from 2p to p as shown in Figs. 3(a), 3(c), and 3(e), respectively. The spin conductance GS , the spin Seebeck coefficient, and the spin figure of merit ZS T are plotted in Figs. 3(b), 3(d), and 3(f), respectively. When the Rashba effect is considered (/R 6¼ 0), SS and ZS T have a series of maximums around u ¼ /R þ 2np and minimums around u ¼ /R þ arcsin ðl=CÞ þ 2np ðn ¼ 0; 1; 2; Þ, respectively. From Eq. (12), we can clearly see that S" ¼ 0 when u ¼ /R þ arcsin ðl=CÞ þ arcsinðl=CÞ þ 2np ðn ¼ 0; 1; 2; Þ, while S# has a finite value. So a pure spin-down thermoelectric generator can be obtained in this case. When u ¼ /R þ arcsin ðl=CÞ þ 2np ðn ¼ 0; 1; 2; Þ, we have S# ¼ 0 with a finite value of S" . As a result, a pure spin-up thermoelectric generator can be obtained at room temperature. IV. SUMMARY
The authors study the spin dependent thermoelectric effects in a DQD molecular junction based on the Green’s function formalism and the EOM technique, in which a magnetic flux penetrates through the device. Due to the existence of the RSO interaction in the dots, the electrons flowing through different arms of the molecule junction will acquire
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a spin-dependent phase factor. By tuning the values of this RSO-induced phase factor and the magnetic flux, a pure spin thermoelectric generator can be fabricated at room temperature. ACKNOWLEDGMENTS
We thank support from the National Natural Science Foundation of China (NSFC) (Grant No. 10947130) and the Science Foundation of the Education Committee of Jiangsu Province (Grant No. 09KJB140001). We also thank support from the Foundation of Changshu Institute of Technology. Chi acknowledges support from SKLSM (Grant No. CHJG200901). 1
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