J Low Temp Phys (2012) 166:208–217DOI 10.1007/s10909-011-0417-2

Fano Effect on the Thermoelectric Efficiencyin Parallel-Coupled Double Quantum Dots

Jun Zheng · Min-Jun Zhu · Feng Chi

Received: 21 August 2011 / Accepted: 5 November 2011 / Published online: 17 November 2011© Springer Science+Business Media, LLC 2011

Abstract With the help of the Green function technique and the equation of motionapproach, we study the thermoelectric efficiency (characterized by the figure of merit)in parallel-coupled double quantum dots (QDs) system. It is found that the configu-rations of the dot-lead coupling strengths exert a significant impact on the figure ofmerit through the Fano effect, due to which electrons experience an abrupt changefrom resonant to antiresonant tunneling. The absolute value of the thermopower isincreased by the Fano resonance and antiresonance, while the electrical conductanceis separately enhanced and suppressed, leading to the enhancement of the thermalefficiency of the device.

Keywords Thermoelectric effect · Quantum dots · Fano effect

1 Introduction

Thermoelectric phenomena, which involves the conversion between thermal and elec-trical energies, has attracted extensive theoretical and experimental interest sinceSeebeck and Peltier discovered two different thermoelectric effects in the early 19thcentury: Seebeck discovered that temperature gradient can be converted into electric

J. ZhengState Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, ChineseAcademy of Sciences, Beijing 100083, Chinae-mail: junzheng@semi.ac.cn

M.-J. ZhuDepartment of Physics, Bohai University, Jinzhou 121013, China

F. Chi (�)Faculty of Engineering, Bohai University, Jinzhou 121013, Chinae-mail: chifeng@semi.ac.cn

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voltage in a conductor, while Peltier found that electrons moving through a materialcarry both charge and heat. Thermoelectric energy conversion technology based onthermoelectric effect has been recognized as the most desirable energy-conversiontechnology that show promise for thermoelectric applications. It is because thermo-electric generators (or coolers) are solid-state devices with no moving parts, they aresilent, reliable and scalable. However, except for specialized applications, such asmedical applications, laboratory equipment, and space missions, the vision of largescale implementation of thermoelectric energy-conversion devices has remained elu-sive due to its low thermoelectric efficiency.

The thermoelectric efficiency is usually characterized by the figure of meritZT = S2GT/(κel + κph), where S, G, T , κel , and κph are the thermopower, elec-trical conductance, system temperature, electron thermal conductance, and phononthermal conductance, respectively. To obtain high value figure of merit ZT , a largethermopower (absolute value of the Seebeck coefficient), high electrical conductance,and low thermal conductance are required. Until 2001, the room-temperature ZT val-ues of bulk semiconductors rarely exceeded 1 [1]. The challenge lies in the fact thataccording to Wiedemann-Franz law [2] and Mott relation [3, 4] S, G, κel , and κph areinterdependent, changing one of these physical quantities will unavoidably alter oth-ers: increase in the electrical conductance usually leads to a corresponding increase inthe thermal conductance, and always accompanied by a decrease in the thermopower.

Nevertheless, these classical laws can be violated in nanostructures due to thequantum effects [4]. Some higher values of S and ZT were found in low-dimensionalsystems due to its sharp change in the density of states near the Fermi level [5, 6].Mahan and Sofo [7] predicted that a narrow distribution of the electronic energyis needed for maximum thermoelectric efficiency. This indicates that quantum dots(QDs) and molecular junctions may offer higher thermoelectric efficiency than bulkmaterials, which make them promising candidates for solid-state thermoelectricenergy-conversion applications. Since then, the thermoelectric properties have beenextensively investigated in QDs [8–15] and molecular junction [16–19] systems cou-pled to normal metal or ferromagnetic leads in the Coulomb-blockade or Kondoregime.

In the present paper, we study the thermoelectric efficiency of an Aharonov-Bohmring with parallel-coupled noninteracting double QDs. As shown in Fig. 1, three typesof configuration are considered according to the spatial symmetry of the dot-leadcoupling strengths. Different from most previous works centered on the effect ofCoulomb interaction or magnetic flux, we focus our attention on how the energy-conversion efficiency can be influenced by the configurations of the dot-lead couplingstrengths. We found that by optimizing a number of parameters, ZT can be quite highdue to the existence of Fano effect.

2 Model and Calculation

The second-quantized form of the Hamiltonian describing the double-dot interferom-eter can be written as [22, 23]

H =∑

k,α

εkαc†kαckα +

∑

i

εid†i di − tc(d

†1d2 + H.c) +

∑

k,i,α

(tαic†kαdi + H.c), (1)

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Fig. 1 (Color online) System ofa double QDs interferometersconnected to the left and theright leads with different typesof dot-lead coupling.(a), (b) and (c) respectivelydenote structures 1, 2 and 3

where c†kα (ckα) is the creation (annihilation) operator of an electron with momentum

k and energy εkα in the αth (α = L,R) lead; d†i (di) creates (annihilates) an electron

in dot i (i = 1,2) with energy εi ; tc and tαi respectively describe the dot-dot and thedot-lead tunneling coupling. Here, for the sake of simplicity, the matrix elements tαi

are assumed to be independent of k.By using nonequilibrium Green’s function methods, the electric current and elec-

tron thermal current in each leads can be derived as [12–14, 24, 25](

Je

Q

)= 2

h

∫dε

(−e

ε − μ

)[fL(ε) − fR(ε)]T (ε), (2)

where fβ(ε) = {1 + e(ε−μβ)/kBT }−1 is the Fermi distribution function of lead β withchemical potential μβ , temperature T and Boltzmann constant kB . As we are in-terested in the linear-response regime, the chemical potentials and the temperaturesof the two leads are set to be μL = μR = μ and TL = TR = T . The transmissioncoefficient T (ε) can be expressed in terms of the Green function as [23]

T (ε) = Tr{Ga(ε)�RGr (ε)�L}, (3)

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where the 2 × 2 matrices Gr(a)(ε) and �β are the QDs’ retarded (advanced) Greenfunction and the line-width function, respectively. The matrix elements of �β aregiven by �α

ij = ∑k tαi t

∗αj 2πδ(ε−εkα). The electrical conductance, the thermopower,

and the electron thermal conductance can be expressed as [11–14]

G = 2e2

hK0(μ,T ), (4)

S = − 1

|e|K1(μ,T )

K0(μ,T ), (5)

κel = 1

h

[K1(μ,T )eS + K2(μ,T )

T

], (6)

where Kn(μ,T ) = ∫dε(−∂f/∂ε)(ε − μ)nT (ε). According to the structure types

shown in Fig. 1, the line-width functions are set to be: �L11 = �R

22 = �, �L22 = �R

11 =λ� for Fig. 1(a), �L

11 = �R11 = λ�, �L

22 = �R22 = �, for Fig. 1(b), and �L

11 = �L22 = λ�,

�R11 = �R

22 = � for Fig. 1(c), respectively. Wherein λ is asymmetric factor. In exper-iments the coupling strengths of the QDs to the source (drain) reservoirs �α

ij can betuned by adjusting the thickness and height of the tunneling barriers through gatevoltages, which has been realized in previous experiments [20, 21]. Accordingly, themagnitude of λ can be changed.

By employing the equation of motion method, one can obtain the matrix Green’sfunction as [14, 23]

Gr (ε) =⎡

⎢⎣ε − ε1 + i

2(�L

11 + �R11) tc + i

2(�L

12 + �R12)

tc + i

2(�L

21 + �R21) ε − ε2 + i

2(�L

22 + �R22)

⎤

⎥⎦

−1

. (7)

The advanced Green’s function Ga(ε) is the hermitian conjugate of Gr (ε).

3 Results and Discussion

In numerical calculations, we only consider the case in which the energy levels ofthe QDs are identical (ε1 = ε2 = ε0), since the difference between them does notchange the essential results in the present paper. As the energy levels in the QDs canbe modulated by gate voltages, we set ε0 = −Vg . As usual, [26] we also set � = 1 asthe energy unit and μL = μR = 0 throughout the paper. In experiment, the value of� varies from several meV up to the magnitude of eV. In the following we separatelyconsider the three kinds of dot-lead coupling configurations.

Figure 2 shows the relative quantities in the structure depicted in Fig. 1(a) for dif-ferent values of asymmetric factor λ. For 0 < λ < 1, the electrical conductance G inFig. 2(a) is composed of a wider Lorentzian resonance centered at the bonding state,and a narrower Fano resonance at the antibonding energy. The latter stems from theinterference between a discrete state in the QDs and a continuum in the lead, and isa good evidence of retention of the quantum phase coherence [26]. Its line shape inthe conductance spectrum is characterized by a sharp transition from resonance to

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Fig. 2 (a) Electrical conductance G, (b) thermopower S, (c) electric thermal conductance κel , and (d) fig-ure of merit ZT in structure 1 as functions of gate voltage Vg for different values of asymmetric factorλ = 0.3,0.6 and 1. The interdot tunneling coupling tc = 1, and the temperature is chosen as T = 300 K

antiresonance. Due to the existence of the antiresonance state in the electrical con-ductance, the absolute value of the thermopower in Fig. 2(b) is obviously increasedwhen the QDs levels are tuned to the antibonding state ε0 + tc (Vg is negative aswe set ε0 = −Vg). As for the thermal conductance, it resembles the configuration ofthe electrical conductance (see Fig. 2(c)). Due to the combined effect of these threephysical quantities, the figure of merit is enhanced around the antibonding state asshown in Fig. 2(d). When the gate voltage is tuned away from the antibonding state,the Fano resonance in the electrical conductance vanishes, leading to the disappear-ance of the resonance in S, κel and ZT . For λ = 0, the two dots are serially connectedand the electrons have only one transport channel. Now Fano effect can never happendue to the absence of the interference phenomenon. λ = 1 corresponds to a totallysymmetrical configuration in parallel, and the antibonding state is totally decoupledfrom the leads. Now electrons pass through the upper and lower arms with the samephase factor without interference effect, leading to the disappearance of Fano reso-nance (dotted lines in Fig. 2). As a result of it, the resonances in the thermal power,thermal conductance and the figure of merit all vanish.

We now qualitatively discuss the impacts of phonon and Coulomb interactions inthe QDs on the figure of merit. As was indicated by previous work, electron-phonon

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Fig. 3 (Color online) Figure ofmerit in structure 1 as a functionof gate voltage Vg andasymmetric factor λ withT = 300 K and tc = �

interaction influences both the dot level position and dot-lead coupling strength[27, 28] and has negative contribution to the magnitude of the figure of merit as isseen from its definition. Nevertheless, the Fano effect is robust against the presenceof phonon [29], and we expect the enhanced thermoelectric efficiency will hold qual-itatively true with such an interaction but with shifted dot levels’ position and weak-ened magnitude. For the Coulomb interaction, it has been demonstrated that besidesthe two peaks in the electrical conductance positioned at ±tc, there are two extraones at ±tc + U , where U is the intradot Coulomb interaction [30]. Then two reso-nances in the figure of merit may arise approximately at the other antibonding statetc + U .

Figure 3 shows the figure of merit as a function of asymmetric factor λ andgate voltage Vg . ZT shows two resonances for a large range of asymmetric factor(0.2 < λ < 0.8) with a main peak positioned around the antibonding state, whichhas been shown in Fig. 2. If the value of λ approaches to zero or unit, the Fano ef-fect vanishes and the magnitude of ZT is weakened accordingly. Dependences of theelectrical conductance, thermal conductance and the figure of merit around the anti-bonding state on system temperature is given in Fig. 4. The magnitude of the electricalconductance G is increased by the increase of the temperature in the energy regimeof antiresonance sate, but is suppressed when the gate voltage is tuned to the reso-nance state. The crossover energy point is exactly the antibonding energy Vg = −tcat which the magnitude of the electrical conductance equals to half of its quantumlimit (2e/�) (see Fig. 4(a)). The thermal conductance κel in Fig. 4(b), however, in-creases monotonously with increasing temperature, which resembles the behavior inbulk materials. Its antiresonance and resonance peaks in higher temperature regimeare less distinguished. As for the figure of merit shown in Fig. 4(c), the magnitude ofthe resonance peak in lower gate voltage regime tends to be weakened by increasingtemperature, while that of higher voltage gate regime is enhanced with rising temper-ature. The optimal ZT is found to be around the room temperature, i.e., T ≈ 300 K,and will be decreased with further increasing T , due to the combined effect of de-creasing electrical conductance in Fig. 4(a) and the enhanced thermal conductance inFig. 4(b). This behavior is consistent with the results reported in [12, 13].

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Fig. 4 (a) Electricalconductance, (b) electricthermal conductance, and(c) figure of merit in structure 1as functions of gate voltage fordifferent values of temperatureand fixed asymmetric factorλ = 0.5, tc = �

Figures 5 and 6 show the results in structures 2 and 3 depicted in Fig. 1. The elec-trical conductance in structure 2 exhibit Fano effect as shown in Fig. 5(a), while Fanoeffect in structure 3 vanishes (see Fig. 6(a)). Such a result has been systematicallydiscussed in [31]. As was indicated in it, the disappearance of the Fano resonance instructure 3 is due to the line-width function of the antibonding state tends to zero inthe absence of external magnetic flux, and then this state is totally decoupled fromthe leads, resulting in infinite lifetime [31]. The conductance spectra of structure 2(Fig. 5(a)) is quite different from that of structure 1 (Fig. 1(a)) because of the phasechange of the discrete level in the reference channel. As a result of the difference of

J Low Temp Phys (2012) 166:208–217 215

Fig. 5 (a) Electrical conductance, (b) thermopower, (c) electric thermal conductance, and (d) figure ofmerit in structure 2 as functions of gate voltage for different values of asymmetric factors. The otherparameters are the same as in Fig. 2

the Fano lineshape between structures 1 and 2, the thermopower S (Fig. 5(b)), thermalconductance κel (Fig. 5(c)) and figure of merit (Fig. 5(d)) show different behaviorsas compared to those of Fig. 2. The properties of these quantities can be understoodby exactly the same reason as in Fig. 2. As there is no Fano effect in structure 3,the properties of relevant quantities are quite normal. They behave like those of thenoninteracting case studied in [12], which can be explained in terms of the bipolareffect.

Finally, it should be pointed out that Dicke effect was also studied in such parallel-coupled QDs structure in recent years [32]. The Dicke effect, which originates fromspontaneous emission of two atoms radiating a photon, was first predicted and ob-served in atom systems. It is characterized in the luminescence spectrum by a nar-row and a broad peak, due to the existence of fast- and low-decaying modes orin other words, short- and long-lived states, respectively [33, 34]. This effect wasthen studied in electron through mesoscopic system [35–38]. Since the Dicke ef-fect has similar physical origin and line-shape as those of Fano effect, it is worthbeing investigated in detail in the future, which is beyond the scope of the presentpaper.

216 J Low Temp Phys (2012) 166:208–217

Fig. 6 (a) Electrical conductance, (b) thermopower, (c) electric thermal conductance, and (d) figure ofmerit in structure 3 as functions of gate voltage for different values of asymmetric factors. The otherparameters are the same as in Fig. 2

4 Conclusions

In conclusion, we have studied the thermoelectric efficiency of a two-terminalparallel-coupled double QDs device up to room temperature. Our numerical resultsindicate that the figure of merit ZT can be significantly enhanced by the Fano effectarose from configurations of the dot-lead coupling. The thermoelectric efficiency ofstructure 1 is obviously greater than that of mirror symmetry with respect to verti-cal axis (structure 2) and perfect symmetric (structure 3) coupling configuration. Byproperly choosing the asymmetric factor, dot-lead and interdot coupling strengths,quite large thermoelectric efficiency can be obtained, therefore can be used as highefficiency thermoelectric energy-conversion device.

Acknowledgements This work was supported by the Education Department of Liaoning Provinceunder Grants No. 2009A031 and 2009R01. Chi acknowledge support from SKLSM under GrantNo. CHJG200901.

References

1. A. Majumdar, Science 303, 777 (2004)

J Low Temp Phys (2012) 166:208–217 217

2. G. Jeffrey Snyder, E.S. Toberer, Nat. Mater. 7, 105 (2008)3. M. Cutler, N.F. Mott, Phys. Rev. 181, 1336 (1969)4. B. Kubala, J. Konig, J. Pekola, Phys. Rev. Lett. 100, 066801 (2008)5. L.D. Hicks, M.S. Dresselhaus, Phys. Rev. B 47, 12727 (1993)6. L.D. Hicks, M.S. Dresselhaus, Phys. Rev. B 47, 16631 (1993)7. G.D. Mahan, J.O. Sofo, Proc. Natl. Acad. Sci. USA 93, 7436 (1996)8. T.S. Kim, S. Hershfield, Phys. Rev. B 67, 165313 (2003)9. R. Franco, J. Silva-Valencia, M.S. Figueira, J. Appl. Phys. 103, 070726 (2008)

10. R. Swirkowicz, M. Wierzbicki, J. Barnas, Phys. Rev. B 80, 195409 (2009)11. T.A. Costi, V. Zlatic, Phys. Rev. B 81, 235127 (2010)12. J. Liu, Q.F. Sun, X.C. Xie, Phys. Rev. B 81, 245323 (2010)13. D.M.T. Kuo, Y.C. Chang, Phys. Rev. B 81, 205321 (2010)14. Y.S. Liu, X.F. Yang, J. Appl. Phys. 108, 023710 (2010)15. M. Tsaousidou, G.P. Triberis, J. Phys., Condens. Matter 22, 355304 (2010)16. M. Paulsson, S. Datta, Phys. Rev. B 67, 241403 (2003)17. J. Koch, F.V. Oppen, Y. Oreg, E. Sela, Phys. Rev. B 70, 195107 (2004)18. C.M. Finch, V.M. García-Suárez, C.J. Lambert, Phys. Rev. B 79, 033405 (2009)19. M. Leijnse, M.R. Wegewijs, K. Flensberg, Phys. Rev. B 82, 045412 (2010)20. A.W. Holleitner, C.R. Decker, H. Qin, K. Eberl, R.H. Blick, Phys. Rev. Lett. 87, 256802 (2001)21. A.W. Holleitner, R.H. Blick, A.K. Hüttel, K. Eberl, J.P. Kotthaus, Science 297, 70 (2002)22. Q.F. Sun, J. Wang, H. Guo, Phys. Rev. B 71, 165310 (2005)23. F. Chi, J.L. Liu, L.L. Sun, J. Appl. Phys. 101, 093704 (2007)24. A.P. Jauho, N.S. Wingreen, Y. Meri, Phys. Rev. B 50, 5528 (1994)25. Y. Meir, N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992)26. M.L. Ladrón de Guevara, F. Claro, P.A. Orellana, Phys. Rev. B 67, 195335 (2003)27. D.M.T. Kuo, Y.C. Chang, Phys. Rev. B 66, 085311 (2002)28. Z.Z. Chen, R. Lü, B. F Zhu, Phys. Rev. B 71, 165324 (2005)29. A. Ueda, M. Eto, Phys. Rev. B 73, 235353 (2006)30. H.Z. Lu, R. Lü, B.F. Zhu, J. Phys., Condens. Matter 18, 8961 (2006)31. H.Z. Lu, R. Lü, B.F. Zhu, Phys. Rev. B 71, 235320 (2005)32. P.A. Orellana, M.L. Ladron de Guevara, F. Claro, Phys. Rev. B 70, 233315 (2004)33. R.H. Dicke, Phys. Rev. 89, 472 (1953)34. P.R. Berman, Appl. Phys. 6, 283 (1975)35. T.V. Shahbazyan, M.E. Raikh, Phys. Rev. B 49, 17123 (1994)36. T.V. Shahbazyan, S.E. Ulloa, Phys. Rev. Lett. 79, 3478 (1997)37. T.V. Shahbazyan, S.E. Ulloa, Phys. Rev. B 57, 6642 (1998)38. A.L. Chudnovskiy, S.E. Ulloa, Phys. Rev. B 63, 165316 (2001)