1 IntroductionDuring the last few years, a new subfield of spintronics named as thermospintronics[1–2] (or spin caloritronics) has emerged in condensed matter physics. To realize the functionalities of thermospintronics devices, a variety of efforts from both theoretical and experimental areas have been devoted to investigate how to generate a spin current by means of the temperature difference instead of the electric voltage bias. The findings of the spin-dependent thermoelectric effects in various materials and nanostructures have paved the way for the realization of thermally driven spin-polarized transport through solid state devices. A pioneering work about this was an experimental one[3] that demonstrated the possibility of generating a longitudinal spin current in the permalloy NiFe by using a temperature difference. After then, this phenomenon which is often referred to as the spin Seebeck effect was also observed in a ferromagnetic semiconductor system,[4] in thin films of the Heusler compound Co2MnSi[5] and even in magnetic tunneling junctions.[6–8] In addition to the spin Seebeck effect, the thermospin effect in quantum dot (QD) systems[9–16] and the spin-dependent Seebeck effect in Aharonov-Bohm interferometers[17–22] have also attracted a great amount of attention. For instance, Gong et al.[9] proposed that an apparent thermospin effect can occur in a quantum dot system under the action of circularly polarized light. This effect originates from the spin polarization induced by the polarized light. Up to now, the Aharonov-Bohm interferometers for research of the spin-dependent Seebeck effect are mostly two-terminal ones. Due to the phase locking effect,[23] driving a spin current to flow through such a system must require the help of magnetic materials or magnetic flux. However, a three-terminal interferometer can get rid of this effect and in fact it has been demonstrated that in three-terminal quantum interferometers a spin-polarized current can be produced purely by electric voltage or temperature bias.[24–28]
Inspired by the previous works about the spin-polarized electron transport in three-terminal mesoscopic interferometers,[24–28] the spin-dependent thermoelectric effect in a three-terminal double-dot interferometer, in which the Rashba spin-orbit interaction (RSOI) is considered in one of the QDs, is investigated in this work. Due to the spin-dependent quantum interference effect based on the RSOI, temperature differences applied to the system can also give rise to a spin current. By using the nonequilibrium Green’s function method, we studied the properties of the generated thermal spin current in detail and demonstrated the controllability of the thermoelectric and thermospin transport processes in the studied system. The rest of the paper is organized as follows. In Sec. 2, a model Hamiltonian for the studied system is given and the general formalism for the thermal spin current, the charge (spin) Seebeck coefficient and the charge (spin) figure of merit are derived. In Sec. 3, the numerical investigation is presented to show the properties of the thermoelectric and thermospin transport in the mesoscopic interferometer. Finally, a brief conclusion is drawn in Sec. 4.
2 Model and MethodIn this paper we investigate a thermospin device consisting of two single-level QDs, which are coupled to three normal leads as displayed schematically in Fig. 1. The left and right leads are coupled to both of the two QDs, but the middle lead is only coupled to QD2. A magnetic flux Φ may penetrate through the area enclosed by this device and the RSOI is only taken into account in QD1. Using the method of second quantization, the studied three-terminal mesoscopic interferometer can be described by the following Hamiltonian
The term
HC in Eq. (
1) is the Hamiltonian for electrons in the three normal leads. With the free electron gas model,
HC can be written as
where
(
Ckασ) stands for the creation(annihilation) of an electron with energy
εkα, momentum
k, and spin index
σ (
σ = ↑, ↓;±1) in lead
α. The term
HD describes two isolated QDs by
where
is the creation(annihilation) operator for an electron with spin index
σ in QD
i (
i = 1, 2). Here for simplicity we neglect the Coulomb interaction between electrons and
εi denotes the dot energy level in QD
i. Finally, the tunneling between the QDs and the leads is described by the third term on the right side of Eq. (
1)
In the above equation, the tunnel matrix element
tαiσ is assumed to be independent of momentum
k and it can be written as
tLiσ = |
tLiσ|,
tR1σ = |
tR1σ|e
iφσ,
tR2σ = |
tR2σ|,
tM2σ = |
tM2σ|. The spin-dependent phase factor
φσ can be described as
φσ =
ϕ −
σφ, in which the Aharonov-Bohm (AB) phase
ϕ = 2
πΦ/Φ
0 with Φ
0 being the flux quantum and the RSOI-induced spin precession phase
σφ =
σαLm*/
ħ2[29] with
α being the RSOI strength,
L being the length of QD1, and
m* being the electron effective mass.
In this work, we only focus on the electron transport in the middle lead. By employing the nonequilibrium Green’s function method, the spin-dependent electric current in the middle lead in stationary state can be expressed in the general Landauer-Büttiker formula form[30–31]
where
fα(
ε) = [e
(ε − μα)/kBTα + 1]
−1 is the Fermi distribution function for lead
α with chemical potential
μα and temperature
Tα. In this paper, we assume that
μL =
μR =
μM which means that no electric bias is applied to the device. Then when the leads hold different temperatures a spin-dependent electric current
Iσ will be driven to flow through the middle lead. In terms of
Iσ the charge and spin currents flowing through the middle lead can be defined as
Ic =
I↑ +
I↓ and
Is =
I↑ −
I↓ respectively. In Eq. (
5),
τMασ stands for the transmission coefficient for an electron with
σ-spin tunneling from the lead
α to the middle lead, which is given by
. Here
denote the spin-dependent line-width parameters and in the local basis they are 2 × 2 matrices as
For simplicity, the tunnel couplings between the QDs and the leads are assumed identical with |
tLiσ| = |
tR1σ| = |
tR2σ| = |
tM2σ| =
t and Γ =
πt2ρα where
ρα is the density of electron states in lead
α. In the expression of the transmission coefficient,
and
denote the spin-dependent retarded and advanced Green’s functions of the QDs in the spectrum space respectively which satisfy the relation
. Following from the Dyson equation, the retarded QDs Green’s function can be calculated as
where the bare QDs Green’s function
can be obtained by the equation of motion method
with
η being a positive infinitesimal and within the wide-band approximation the self-energy contributed by the three leads is written as
. After some straightforward algebra calculations, we get
To investigate the influence of the temperature distribution on the spin-dependent thermoelectric transport properties of the system qualitatively, we focus on the three cases for TL = TR ≠ TM, TL = TM ≠ TR, and TL ≠ TR ≠ TM. The case of TR = TM ≠ TL is similar with the case of TL = TM ≠ TR, so it is not specially discussed in this paper. For the case of TL = TR ≠ TM, due to fL(ε) = fR(ε) Eq. (5) can be reduced to
with
It is clear that only when
ϕ ≠ 0 and
ϕ ≠ 0, i.e., the magnetic flux and RSOI coexist, can
τσ(
ε) and thus
Iσ be dependent on electron spin. That is to say the coexistence of the magnetic flux and RSOI is essential for obtaining thermally driven spin current for the case of
TL =
TR ≠
TM. However, for the other two cases the term sin
φσ can not be canceled out in the expression of
Iσ and thus even with
ϕ = 0 the thermoelectric transport through the system is spin-resolved if the RSOI is present.
3 Results and DiscussionsFor numerical calculations, the dot-lead coupling strength Γ is assumed to be the energy unit, i.e., Γ = 1. The common chemical potential of the three leads is taken as the energy reference and the dot energy level in QD1 is set to be aligned with it, i.e., μL = μR = μM = ε1 = 0.
In Fig. 2 we show the thermally driven charge current Ic and spin current Is as functions of the dot energy level ε2 for the case of TL ≠ TR ≠ TM. As we have discussed above, in this situation only if the RSOI is present in the system a spin current can be generated to flow through the middle lead by temperature differences even without any magnetic flux. So as shown in Fig. 2(a) in addition to a charge current a spin current also varies with ε2 for ϕ = 0. But the charge current and spin current possess different symmetries. To be exact, the charge current satisfies the symmetry Ic(−ε2) = −Ic(ε2) while the spin current satisfies the symmetry Is(−ε2) = Is(ε2). To understand this difference, we define ταc, ταs and Δfα as ταc = τMα↑ + τMα↓, ταs = τMα↑ − τMα↓ and Δfα = fα − fM. Then according to Eq. (5) and the relevant definitions, the charge and spin currents have the forms
It is easy to demonstrate that
when
ϕ =
nπ(
n = 0, 1, 2, 3, . . .). These symmetries together with the fact that
lead to the integration on the right side of Eq. (
15) (and thus
Ic) and Eq. (
16) (and thus
Is) having centrosymmetry and mirror symmetry about
ε2 = 0 respectively for
ϕ = 0 (see Fig.
2(a)) and
ϕ =
π (see Fig.
2(d)). When the phase
ϕ lies between 0 and
π, the relations
ταc(−
ε, −
ε2) =
ταc(
ε,
ε2) and
ταs(−
ε, −
ε2) = −
ταs(
ε,
ε2) cannot hold ever so that the charge and spin currents loose their original symmetries (see Figs.
2(b) and
2(c)). The sign of the spin current can also be changed with the variation of
ϕ. Compared to the sign of the spin current in the case of
ϕ = 0, the sign of the spin current in the case of
ϕ =
π is completely reversed. This is because the sign of
ταs is reversed under the operation
ϕ →
ϕ +
π. But
ταc remains unchanged under this operation and thus so does the charge current. In Fig.
2, at some points (
A1,
A2,
A3,
A4) the charge current completely vanishes while the spin current is nonzero. Thus by tuning the system parameters appropriately a thermally driven pure spin current can be produced in the proposed device. Moreover, the formation of the pure spin current at point
A1 also implies that the proposed device can act as an all-electrical pure-spin-current thermal generator since the magnetic flux is absent in Fig.
2(a) and the dot energy level
ε2 can be tuned by a gate voltage. At some other points (
B1,
B2,
B3,
B4) the charge current is identical with the spin current indicating that a fully spin-polarized charge current, which is composed of only one spin component current, can also be thermally driven to flow in the middle lead.
The RSOI induced spin-dependent quantum interference effect should play a crucial role in the generation of the spin current. To show this clearly, in Fig. 3 we plot the spin current as a function of the RSOI-induced phase φ with different values of the dot-lead coupling strength Γ for TL ≠ TR ≠ TM and ϕ = 0. The numerical results show that the spin current oscillates as a function of φ with a period of 2π. For weak dot-lead coupling, the spin current is a perfect sinusoidal function. However, with the increase of the dot-lead coupling strength the spin current exhibits a non-sinusoidal oscillation. We can explain this as a result of the spin interference effect between different Feynman paths. For the studied device, there are two possible paths for an electron tunneling from the αth lead to the middle lead. One path is the αth lead → QD1 → the βth (β ≠ α) lead → QD2 → the middle lead and the other one is the αth lead → QD2 → the middle lead. As the dot-lead coupling is weak, one can only consider the lowest-order tunneling process, the spin-dependent effective coupling strength between the αth lead and the middle lead can be expressed as[27–28,32–33]
Obviously when
ϕ = 0, sin
φσ =
σsin
φ and thus
∝ sin
φ such that the thermally driven spin current flowing in the middle lead behaves like a sinusoidal function. On the contrary, for the strong coupling case the contribution from higher-order tunneling processes is not negligible, which causes the difference between the two spin-dependent effective coupling strength never to be proportional to sin
φ and consequently leads to the deviation of the spin current from a sinusoidal function of
φ.
For the two cases that TL = TR ≠ TM and TL = TM ≠ TR, the charge and spin currents are determined by the temperature difference ΔT, which is defined in detail in the following. So in Fig. 4, we show the dependences of the charge and spin currents on the temperature difference. Meanwhile, in Fig. 4 we also plot the charge and spin currents versus the dot energy level ε2 for the two cases since the dot energy level can be easily controlled by a applied gate voltage in experiments. As we have demonstrated earlier, for TL = TR ≠ TM a thermally driven spin current can never be produced in the proposed device without the help of a magnetic flux, hence in Figs. 4(a) and 4(c) the spin current is maintained at zero when ϕ = 0. But as the magnetic flux is switched on a spin current as well as a charge current can form in the middle lead. From Fig. 4(a) we can see that the values and positions of the peaks in the charge current curve are affected by the magnetic flux because the spin interference effect can be modulated by it. With the increase of the absolute value of the dot energy level ε2, the charge current approaches to zero gradually. This is due to that the increase of the absolute value of the dot energy level ε2 will cause the conductance of electron to be far away from the resonant points. For the case of TL = TM ≠ TR, the magnetic flux is not a key factor for the generation of a thermal spin current, thus in Figs. 4(b) and 4(d) a finite thermal spin current exists even in the absence of the magnetic flux. Following Eqs. (15) and (16) and limiting the αth lead to the right lead, one can easily find that the charge and spin currents here have the same characteristics of symmetry as possessed by the charge and spin currents for the case of TL ≠ TR ≠ TM. Therefore, as illustrated in Fig. 4(b), the charge and spin currents are strictly centrosymmetric and axisymmetric about ε2 = 0 respectively at zero magnetic flux and these symmetries can be destroyed by a finite magnetic flux. Under the condition that TL = TR ≠ TM, we can define the temperature difference ΔT as ΔT = TM − TL, then by expanding the Fermi distribution function in Eq. (13) to the first order in ΔT and employing the Sommerfeld expansion[34] Eq. (13) can be approximated with
Equation (
18) shows a linear relation between the spin-dependent current
Iσ and the temperature difference in the linear response regime, which induces the charge (spin) current to be linear functions of
kBΔ
T (see Fig.
4(c)). As
TL =
TM ≠
TR, the temperature difference is defined by Δ
T =
TR −
TM. From Eq. (
5) the spin-dependent current in the middle lead for this case can be expressed as
Utilizing the same approximation method as adopted above, Eq. (
19) yields
As a result, the charge and spin currents for the case of
TL =
TM ≠
TR are also linear with
kBΔ
T (see Fig.
4(d)) when the temperature difference is considerably small (
kBΔ
T ≤ 0.5).
Since in the two cases of TL = TR ≠ TM and TL = TM ≠ TR the spin-dependent current in the middle lead can always be written into a two-terminal current formula (see Eqs. (13) and (19)), we can introduce the spin-dependent Seebeck coefficient Sσ to measure the ability of generating the spin-dependent current by the temperature difference ΔT in the proposed device. With the zero current condition Iσ = 0,[35–36] the spin-dependent Seebeck coefficient is defined as Sσ = ΔVσ/ΔT, where Vσ is the spin-dependent thermoelctric voltage. In the linear response regime, ΔVσ and ΔT tend to be zero, the spin-dependent Seebeck coefficient can be computed as Sσ = −L1σ/eTL0σ with T the common temperature in the absence of temperature difference and the intermediate quantities Lnσ(n = 0,1,2) determined by . Here μ and f(ε) denote the common chemical potential and Fermi distribution function without any temperature difference respectively, the spin-dependent electron transmission function corresponds to τσ(ε)(τMR σ(ε)) for the case of TL = TR ≠ TM(TL = TM ≠ TR). Additionally, the spin-dependent conductance Gσ and the spin-dependent thermal conductance κσ can also be obtained in the terms of Lnσ as Gσ = e2L0σ and . Then we can arrive at the charge (spin) thermopower Sc(s) = (1/2)(S↑ ± S↓) as well as the charge (spin) figure of merit with Gc(s) = G↑ ± G↓ and κ = κ↑ + κ↓. We first consider the case of TL = TR ≠ TM at ϕ = 0.25π. By the coaction of the magnetic flux and the RSOI, thermoelctric as well as thermospin transport can be realized in this case. Hence as shown in Fig. 5(a), both the charge and spin thermopowers can be nonzero and they vary with the dot energy level ε2. For the parameters chosen in Fig. 5(a), Sc/Ss is lesser than ten at some values of ε2, i.e., the spin thermopower is comparable to the charge one, which means that a considerable spin current can be generated thermally in the studied system. Around ε2 = 0 the two thermopowers are both centrosymmetric. With increasing ε2 from negative to positive, the sign of the two thermopowers changes from positive to negative. In order to better understand the change of the sign of the thermopowers, we plot the ratio at ε = μL = 0 as a function of the dot energy level ε2 in Fig. 5(c). As is shown, and are both negative for ε2 ˂ 0 and positive for , always holds regardless of the sign of ε2. Combining these features of the thermopowers with the approximative equation ,[37] it can be readily concluded that Sc(s) = (1/2)(S↑ ± S↓) ˃ 0 for ε2 ˂ 0 while Sc(s) = (1/2)(S↑ ± S↓) ˂ 0 for ε2 ˃ 0. In Fig. 5(b), the charge and spin figure merits as functions of ε2 are depicted. Figure 5(b) shows that the two figure of merits are both symmetric with respect to the axis ε2 = 0. The spin figure of merit owns three zero points located at ε2 = −3.3, 0, 3.3 and the charge figure of merit has only one zero point centered at ε2 = 0. It is interesting that at ε2 = −3.3, 3.3 a pure charge figure of merit without accompanying the spin counterpart is obtained and at these two points the charge figure of merit attains its maximal value which exceeds 1 suggesting the proposed device has a value of practical application.
When TL = TM ≠ TR, the thermoelectric transport can become spin-resolved for both the cases of ϕ = 0 and ϕ ≠ 0. To compare these two cases, we display the dependences of Sc(s) and Zc(s) on the dot energy level ε2 at ϕ = 0 and ϕ = 0.25π in Fig. 6. As Figs. 6(a) and 6(b) display, the charge thermopower (spin thermopower) is a perfect centrosymmetric (axisymmetric) function in the absence of the magnetic flux while the centrosymmety (axisymmetry) can not be preserved when the area enclosed by the device is threaded by a magnetic flux. This result is consistent with the behaviors of the thermally driven charge and spin currents in the case of TL = TM ≠ TR, which originates from the spin-dependent quantum interference effect that can be strongly influenced by the magnetic flux. It is also shown that the charge thermopower is enhanced but the spin counterpart is suppressed by the magnetic flux. For the spin thermopower, another distinct influence brought by the magnetic flux is that its original single maximal value is split into a maximal and a local maximal values. As the magnetic flux is absent, both of the charge and spin figure of merits are perfectly axisymmetric about ε2 = 0 and possess two peaks as well as a dip (see Fig. 6(c)). Similar to the thermopowers, the symmetry possessed by the two figure of merits can be broken by the magnetic flux and when the magnetic flux is introduced the peak values of the charge figure of merit are increased while those of the spin counterpart are decreased (see Fig. 6(d)). In addition, one of the peak values of the charge figure of merit can even be increased to be over 1 after introducing the magnetic flux. More interestingly, no matter whether the magnetic flux is present or not, the dip of the charge figure of merit locates at ε2 = 0 and the dip value remains zero. In the absence of the magnetic flux, the dip of the spin figure of merit also locates at ε2 = 0 and the dip value is zero. However, for a nonzero magnetic flux the value of the dip at ε2 = 0 can become finite and an additional dip emerges around ε2 = 7.3 whose value is exactly zero.
4 ConclusionIn conclusion, we have explored the thermoelectric and thermospin transport in a three-terminal double-dot interferometer with one dot containing RSOI. The results evaluated by the non-equilibrium Green’s function technique show that some particular temperature distribution makes the generation of the thermal spin current demand the help of a magnetic flux. However, at other temperature distributions the magnetic flux is not indispensable for producing the spin current thermally. By appropriately adjusting the system parameters, the proposed device can work as a pure-spin-current or fully-spin-polarized-current thermal generator. For other two cases that only one terminal is kept at a temperature different from the others, the system under study resembles a two-terminal device, the defined spin thermopower is shown to be comparable to the charge one and the defined charge figure of merit is expected to exceed 1. These characteristics of the currents, thermopowers and figure of merits endow the considered setup with potential application values. Furthermore, some other interesting properties of the currents, thermopowers and figure of merits are also revealed in this paper.