We have theoretically studied the thermoelectric properties of serially coupled quantum dots (SCQDs) embedded in an insulator connected to metallic electrodes. In the framework of Keldysh Green’s function technique, the Landauer formula of transmission factor is obtained using the equation of motion method. Based on such analytical expressions of charge and heat currents, we calculate the electrical conductance, Seebeck coefficient, electron thermal conductance, and figure of merit (ZT) of SCQDs in the linear response regime. The effects of interdot hopping and electron Coulomb interactions on ZT are analyzed. We demonstrate that ZT is not a monotonic increasing function of interdot electron hopping strength (tc). We also show that in the absence of phonon thermal conductance, SCQD can reach the Carnot efficiency as tcapproaches zero.
Recently, many considerable studies have been devoted to seeking efficient thermoelectric materials with the figure of merit (ZT) larger than 3 because there exist potential applications of solid-state thermal devices such as coolers and power generators[1–6]. Some theoretical efforts have pointed out that a single quantum dot (QD) junction system can have a very impressive ZT in the absence of phonon conductance[7–9]. However, in practice, it is difficult to maintain a large temperature gradient which is needed to produce sufficient temperature difference across the nanoscale junction. To reduce the temperature gradient across the QD junction, it is essential to consider many serially coupled quantum dots (SCQDs)[1, 5]. The transport property of a junction involving N serially coupled QDs with strong electron Coulomb interactions is one of the most challenging topics of condensed matter physics. To gain some insight, we investigate in the present paper the thermoelectric effect of serially coupled quantum dots (SCQDs) as shown in the inset of Figure1a.
It has been shown that the transport properties of the SCQD system exhibit several interesting features, including current rectification (due to the Pauli spin blockade), negative differential conductance, nonthermal broadening of tunneling current, and coherent tunneling in the Coulomb blockade regime. Although many theoretical investigations of the above phenomena have been reported, most of them did not investigate the thermoelectric properties of SCQDs[11–13]. This study investigates the ZT of a SCQD embedded in a semiconductor nanowire with small phonon thermal conductance. It is expected that the SCQD system has a potential to enhance the ZT of nanowires. Here, we consider nanoscale semiconductor QDs, in which the energy level separations are much larger than their on-site Coulomb interactions and thermal energies. Thus, only one energy level for each quantum dot needs to be considered. A two-level Anderson model  is employed to simulate the SCQD junction system.
Using Keldysh-Green’s function technique, the charge and heat currents of SCQD connected to metallic electrodes are given by
where is the transmission factor. denotes the Fermi distribution function for the left (right) electrode. The left (right) chemical potential is given byμL(μR).TL(R) denotes the equilibrium temperature of the left (right) electrode. e and h denote the electron charge and Planck’s constant, respectively. denotes the transmission function, which can be calculated by evaluating the on-site retarded Green’s function (GF) and lesser GF. The indices ℓ and j denote the ℓ th QD and the j th QD, respectively. Based on the equation of motion method, we can obtain analytical expressions of all GFs in the Coulomb blockade regime. Details are provided in. The transmission function in the weak interdot limit (tc/Uℓ≪ 1, where tc and Uℓ denote the electron interdot hopping strength and on-site Coulomb interaction, respectively) can be recast into the following form:
where Im means taking the imaginary part of the function that follows and
Γℓ=L(1),R(2)(ε) denotes the tunnel rate from the left electrode to dot A (E1) and the right electrode to dot B (E2), which is assumed to be energy- and bias-independent for simplicity. μℓ= ε−Eℓ + iΓℓ/2. We can assign the following physical meaning to Equation 3. The sum in Equation 3 is over eight possible configurations labeled by m. We consider an electron (of spin σ) entering level ℓ, which can be either occupied (with probability) or empty (with probability). For each case, the electron can hop to level j, which can be empty (with probability), singly occupied in a spin state (with probability) or spin σ state (with probabilitybj,σ= Nj,σ− cj), or a double-occupied state (with probabilitycj). Thus, the probability factors associated with the eight configurations appearing in Equation 4 become,,,,,,, and. Σm in the denominator of Equation 4 denotes the self-energy correction due to Coulomb interactions and coupling with level j (which couples with the other electrode) in configuration m. We have,,,,,,, and. Eℓ, Uℓ, and Uℓ,j denote, respectively, the energy levels of dots, intradot Coulomb interactions, and interdot Coulomb interactions. Here,Im Σjdenotes the effective tunneling rate from level l to the other electrode through level j in configuration m. For example,Im. It is noted that has a numeratorΓj for all configurations. Furthermore, is just the on-site single-particle retarded GF for level ℓ as given in Equation (A16) of, and corresponds to its partial GF in configuration m. The transmission function written this way has the same form as Landauer’s formula for a single QD with multiple energy levels including intralevel and interlevel electron Coulomb interactions[14, 15].
The probability factors of Equation 3 are determined by the thermally averaged one-particle occupation number and two-particle correlation functions, which can be obtained by solving the on-site lesser Green’s functions:
Note that ℓ ≠ j in Equations 3, 5, and 6. In the linear response regime, Equations 1 and 2 can be rewritten as follows:
where ΔV = μL− μRand ΔT = TL− TR are the voltage and temperature differences across the junction, respectively. Thermoelectric response functions in Equations 7 and 8 are given by
Here, and are evaluated in the equilibrium condition. It can be shown that the Onsager relation is preserved. These thermoelectric response functions can also be found in, where authors investigated the thermoelectric properties of a single QD.
If the system is in an open circuit, the electrochemical potential will form in response to a temperature gradient; this electrochemical potential is known as the Seebeck voltage (Seebeck effect). The Seebeck coefficient (amount of voltage generated per unit temperature gradient) is defined as. To judge whether the system is able to generate power or refrigerate efficiently, we need to consider the figure of merit, which is given by
Here, is the electrical conductance, and is the electron thermal conductance. (ZT)0 represents the ZT value in the absence of phonon thermal conductance, κph. For simplicity, we assume κph= κph,0Fs[16–18]. is the universal phonon thermal conductance arising from acoustic phonon confinement in a nanowire[16–18], which was confirmed in the phonon wave guide. The expression of κph= κph,0Fs with Fs= 0.1 can explain well the phonon thermal conductance of silicon nanowire with surface states calculated by the first-principles method. The dimensionless scattering factor Fsarises from phonon scattering with surface impurities or surface defects of quantum dots[1, 16]. Here, we adopt Fs= 0.02, which is smaller than Fs= 0.1 because QDs can enhance the phonon scattering rates and reduce phonon thermal conduction as pointed out in.
Results and discussion
Here, we consider the case of identical QDs in the optimization of ZT, although it is understood that the size fluctuation of QDs can suppress ZT. In Figure1a,b we plot (ZT)0 and ZT as a function of temperature for various electron hopping strengths. We adopt the following physical parameters: Eℓ=EF + 30Γ0, Uℓ= 30Γ0, Uℓ,j= 10Γ0, and ΓL= ΓR= Γ = 1Γ0. All energy scales are in the units of the characteristic energy, Γ0. In Figure1a, we see that (ZT)0 increases with decreasing tcand diverges as tc → 0. This behavior can be proved rigorously as we shall illustrate below. It implies that SCQD can reach the Carnot efficiency in the limit of extremely weak interdot coupling, if one can fully suppress κph, for example, by inserting a nanoscale vacuum layer to block the phonon heat current. Although it would be a challenging task to implement a vacuum layer between one of the electrodes and SCQD, it may be possible to test this idea out via a scanning tunneling microscopic experiment using a setup as shown in the inset of Figure1a. In Figure1b, we see that ZT is enhanced with increasing tc until tc reaches 3Γ0, and it becomes reduced for higher tc.
The diverging behavior of (ZT)0 with respect to tc is further illustrated in Figure2d. The maximum ZT is suppressed in the presence of κph, which is much larger than κefor small tc. The behaviors of ZT shown in Figure1b are mostly determined by the power factor (S2Ge). Once tc is larger than 3Γ0, the reduction of S2 is faster than the increase of Ge. This explains why the maximum ZT at tc= 4Γ0 is smaller than that at tc= 3Γ0. The location of ZTmax is nearly independent of tc, and it occurs near kBT = 8.8Γ0. For comparison, we also show the results (curves with triangle marks) for the case without electron Coulomb interactions in Figure1b. It is seen that the maximum ZT is enhanced when we turn off the electron Coulomb interactions. Such a behavior is similar to that of a single QD with multiple energy levels[7, 8]. The effect of electron Coulomb interactions is significant only for temperature between 6Γ0 and 50Γ0. Namely, the electron Coulomb interactions are negligible when U/(kBT)≫1or U/(kBT)≪1.
To further understand the behavior of ZT with respect to tc, we plot the electrical conductance (Ge), Seebeck coefficient (S), electrical conductance κe, and (ZT)0 as functions of tc in Figure2 for various detuning energies, Δ≡Eℓ−EF. When Eℓ is close to the Fermi energy, Ge and κe are enhanced, whereas S and (ZT)0 are suppressed. The behavior of (ZT)0 at Δ = 30Γ0 in the absence of Coulomb interactions is also shown by the curve with triangles, which has a similar trend as the solid line. Thus, it is instructive to analyze (ZT)0 in the absence of Coulomb interactions. Keeping the leading order of, we have,, and. Therefore,, S = − Δ/eT is independent on tc, and vanishes up to. Thus, the leading order of κeis. This indicates that (ZT) in the limit of weak interdot hopping.
Figure3 shows ZT as a function of Δ = Eℓ−EFfor various electron hoping strengths at kBT = 10Γ0. Other physical parameters are kept the same as those for Figure1. When tc= 0.1Γ0, the maximum ZT (ZTmax) occurs at near Δ = 27Γ0. The peak position only shifts slightly to higher Δ with increasing tc. We have ZTmax = 2.79 and 3.18 for tc= 1Γ0 and 3Γ0, respectively. However, at tc= 4Γ0, we have ZTmax = 3.07, which is smaller than ZTmax for tc= 3Γ0. Thus, it also illustrates that ZT is not a monotonically increasing function of tc. We further calculated ZT as a function of tcfor Δ = 10,20,30Γ0 and kBT = 10Γ0 in the presence of κphand found that again, ZT is not a monotonically increasing function of tc (not shown here). We conclude that as long as κphdominates over κe, the tc dependence of ZT is mainly determined by the power factor S2Ge, where the behaviors of Ge and S are similar to the results shown in Figure2a,b. When tc/Γ0≤1, Ge increases much faster than the reduction ofS2 for increasing tc, and the power factor slowly reaches the maximum when tcapproaches 3Γ0. When tc> 3Γ0, the power factor decreases due to the fast reduction ofS2 which prevails over the increase of Ge. The curve with triangle marks is for tc= 3Γ0 in the absence of Coulomb interaction. We see that ZTmax is larger when Uℓ=Uℓ,j= 0. Based on the results of Figure3, we conclude that it is important to control the detuning energy Δ for the optimization of ZT.
In Figures1,2, and3 we have considered the case with EFbelow QD energy levels. It would be interesting to investigate the case with EF above the energy levels of QDs. Figure4 shows Ge, S, κe, and ZT of an SCQD with tc= 3Γ0 as functions of applied gate voltage for various temperatures. Once tc>(ΓL + ΓR) = 2Γ0, the eight peaks for Ge can be resolved at kBT = 1Γ0. These eight peaks correspond to the following resonant channels: Eℓ−tc, Eℓ + tc, Eℓ + Uℓ,j−tc, Eℓ + Uℓ,j + tc, Eℓ + Uℓ,j + Uℓ− tc, Eℓ + Uℓ,j + Uℓ + tc, Eℓ + 2Uℓ,j + Uℓ− tc, and Eℓ + 2Uℓ,j + Uℓ + tc, which are tuned by the gate voltage to be aligned with EF. These eight channels result from the four configurations of p1, p3, p6, and p8 in Equation 4. Such a result implies that SCQD with identical QDs acts as a QD with effective two levels of Eℓ−tc and Eℓ + tc and satisfying Hund’s rule. These eight peaks are smeared out with increasing temperature. The sign changes of S with respect to the gate voltage result from the bipolar effect, i.e., the competition between electrons and holes, where holes are defined as the unoccupied states below EF. The electronic thermal conductance (κe) also exhibits eight peaks, and we noticed that the local maxima of the κe curve nearly coincide with the local minima of the Ge curve. We see that ZT values are still larger than 3 even when Eℓis deeply below EF(say, at eVg= 70Γ0). This is attributed to the electron Coulomb interaction. To illustrate that, we also show the results with Uℓ=Uℓ,j= 0 at kBT = 3Γ0(see the curve with triangle marks). The oscillation of ZT in the case of Uℓ=Uℓ,j= 0 is attributed to the sign change of S at Vg= 10Γ0. Note that S goes to zero at Vg= 10Γ0, which results from the electron-hole symmetry (with Eℓ + tcand Eℓ−tc straddling EF symmetrically). We see that ZT vanishes for eVg≥ 40Γ0in the absence of electron Coulomb interactions. Unlike the case of EF<Eℓ, where the finite U causes reduction of ZT, here, the electron Coulomb interaction leads to enhancement of ZT when EF>Eℓ.
In summary, the thermoelectric properties including Ge, S, κe, and ZT of the SCQD junction system are investigated theoretically. We demonstrate that the Carnot efficiency can be reached when tcapproaches zero in the absence of phonon thermal conductance. When the phonon contribution dominates the thermal conductance of the SCQD junction, the optimization of ZT can be obtained by the thermal power defined as S2Ge. We also found that the presence of electron Coulomb interactions can lead to either reduction or enhancement of ZT, depending on whether the Fermi level is below or above the QD level.
DMTK received his Ph.D. degree from National Taiwan University in 1996. He joined the Department of Electrical Engineering, National Central University in 2003 as an assistant professor, became an associate professor in 2005, and professor in 2008. His main research interests include nanodevices and quantum transport.
Y-CC received his Ph.D. degree from the California Institute of Technology in 1980. He joined the Physics Department, University of Illinois at Urbana-Champaign in 1980 as a visiting research assistant professor and became an assistant professor in 1982, associate professor in 1986, and professor in 1991. In 2005, he joined Academia Sinica, Taiwan as a Distinguished Research Fellow of the Research Center for Applied Sciences. His main research interests include condensed matter theory, semiconductor electronics, photonic materials, and optoelectronic devices.
This work was supported in part by the National Science Council, Taiwan under contract nos. NSC 99-2112-M-008-018-MY2 and NSC 98-2112-M-001-022-MY3.
Department of Electrical Engineering and Department of Physics, National Central University
Research Center for Applied Sciences, Academic Sinica
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