Enhancement of thermoelectric efficiency and violation of the Wiedemann-Franz law due to Fano effect
G. Gómez-Silva, O. Ávalos-Ovando, M. L. Ladrón de Guevara, P. A. Orellana
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Enhancement of thermoelectric efficiency and violation of the Wiedemann-Franz lawdue to Fano effect
G. G´omez-Silva, O. ´Avalos-Ovando, M. L. Ladr´on de Guevara, and P. A. Orellana
Departamento de F´ısica, Universidad Cat´olica del Norte, Casilla 1280, Antofagasta, Chile (Dated: August 19, 2018)We consider the thermoelectric properties of a double-quantum-dot molecule coupled in parallelto metal electrodes with a magnetic flux threading the ring. By means of the Sommerfeld expansionwe obtain analytical expressions for the electric and thermal conductances, thermopower and figureof merit for arbitrary values of the magnetic flux. We neglect electronic correlations. The Fanoantiresonances in transmission demand that terms usually discarded in the Sommerfeld expansionare taken into account. We also explore the behavior of the Lorenz ratio L = κ/σT , where κ and σ are the thermal and electrical conductances and T the absolute temperature, and we discuss thereasons why the Wiedemann-Franz law fails in presence of Fano antiresonances. Recently, there has been an increasing interest in thethermoelectric properties of low dimensional systems andnanostructured materials. Theoretical predictions aswell as experiments show that these structures ex-hibit higher efficiencies than bulk materials, makingthem very attractive for their potential application inenergy-conversion devices. The efficiency of a thermo-electric device is described by the dimensionless param-eter ZT , known as figure of merit, where T is the abso-lute temperature and Z characterizes the electrical andthermal transport properties of the device. The figureof merit is given by ZT = S σT /κ , where S , σ and T are, respectively, the thermopower (Seebeck coefficient),electronic conductance and absolute temperature, and κ = κ e + κ ph is the thermal conductance, where κ e isthe electron and κ ph the phonon thermal conductance. On the other hand, the thermal and electric conductancesfor most macroscopic metals at very low and room tem-peratures are constrained by the Wiedemann-Franz law, κ/σT = L , where L = ( π / k B /e ) is the Lorenznumber, with k B the Boltzmann constant and e the elec-tron charge. This relationship is a consequence of theFermi-liquid behavior of electrons in metals, and expressbasically the fact that free electrons support both chargeand heat transport.While the most efficient thermoelectric bulk materi-als have values of ZT not higher than 1, larger figuresof merit have been measured in nanostructured systems.Ref. reported ZT = 2 . ZT = 1 . More recently, two ex-periments showed that the thermoelectric efficiency ofSi nanowires can be substantially enhanced relative tothe bulk value for Si.
Different mechanisms explainthe improvement of thermoelectric efficiencies in systemsat the nanoscale. The figure of merit can be enhanceddue to the decrease of the thermal conductance pro-duced by the scattering of phonons off the structure, or because of the presence of an enhanced density ofstates at the Fermi level, which produces an increaseof the thermopower.
Fano resonances, a signatureof coherent transport of electrons, have been also pre- dicted to improve the thermoelectric efficiency in sys-tems as molecules or multiple-level quantum dots. A Fano resonance arises by the quantum interference oftwo transport pathways, a resonant and a non-resonantone, and manifests in a characteristic asymmetric line-shape in the transmission probability. On the otherhand, since in low dimensional structures electron trans-port is affected by mechanisms such as confinement, elec-tronic correlations, and others, the Wiedemann-Franzlaw is not necessarily fulfilled at the nanoscale. The vio-lation of this law in quantum dots has been predicted atdifferent regimes.
It has been directly associated tothe Coulomb interaction in a quantum dot connected tometal leads, nodes in transmission or to mesoscopicfluctuations in an open quantum dot. In this article we are concerned with the thermoelectricproperties of a double-quantum dot molecule embeddedin an Aharonov-Bohm ring. This system is described by atransmission amplitude with two components of differentspectral linewidths, the combination between them giv-ing rise to a convolution of a Breit-Wigner and a Fanoresonance in the transmission probability.
The ther-moelectricity of this system was studied numerically byLiu et al. both in the absence and in presence ofelectronic correlations, finding that the figure of merithave a significant increase in the Fano lineshape regime.Our work advances further on the findings of Refs. ,presenting an analytical work showing clearly that theenhancement of the thermoelectric efficiency comes fromthe Fano antiresonances, which are also responsible forthe failure of the Wiedemann-Franz law. In the frame-work of a noninteracting model, we use the Sommerfeldexpansion to derive analytical expressions of the ther-mopower, the electric and thermal conductances, and thefigure of merit. The Fano antiresonances in transmissiondemand that terms usually discarded in this expansionare taken into account. Other analytical approaches todescribe thermoelectric effects in similar systems havebeen developed. Nakanishi and Kato studied the ther-mopower of a multilevel quantum dot when zeros intransmission take place. Inasmuch as the Mott’s for-mula, widely used for analysis of thermopower in metals, e e L R t c G G G G F L L R R FIG. 1. Scheme of the double quantum dot molecule embed-ded in an Aharonov-Bohm ring. is not valid when Fano antiresonances occur, they derivedan “extended Mott’s formula” which allows an analyticalcalculation of this quantity. First principles calculationsof the thermoelectric efficiency of a nanojunction are de-veloped in Ref. .The system under consideration is shown in Fig. 1.Two quantum dots forming a molecule are coupled to left( L ) and right ( R ) leads. We assume that one energy levelis relevant in the dots 1 and 2, with ε and ε their respec-tive energies. The parameter t c is the coupling constantbetween dots, and Γ αi the line broadening of the energylevel of the dot i ( i = 1 ,
2) due to the coupling to thelead α ( α = L, R ), and Φ the total magnetic flux thread-ing the Aharonov-Bohm ring, assumed to be distributedevenly between the two sections. The interdot and in-tradot electron-electron interactions are neglected. Thisand similar interferometers have been carried out in two-dimensional electron gas (2DEG) systems in semiconduc-tor heterostructures (typically AlGaAs/GaAs).
Thisdevice has the exceptional property of being highly con-trollable through several of their parameters. The energyof the discrete levels of the quantum dots can be adjustedindependently via gate potentials, and so does the cou-pling between dots t c and the couplings between dots andleads Γ L,Ri ( i = 1 , φ is controlled throughthe magnetic field across the ring. The values of Γ inthese systems are of the order of a few meV. The stud-ied system also can be though as a double quantum dotmolecular junction, where Γ is of the order of 1 eV.
We model the system by means of a non-interactingAnderson Hamiltonian, as done in Ref. . We assume aneffective drop voltage ∆ V and a temperature difference∆ T between the left and right leads. In the linear tem-perature and bias regime, the charge current I e and theheat current I Q through the system are given by I e = − e K ∆ V + eT K ∆ T, (1a) I Q = eK ∆ V − T K ∆ T, (1b)where e is the charge of electron, T the absolute tem-perature, and K n ( µ, T ) = 2 h Z ∞ (cid:18) − ∂f∂E (cid:19) ( E − µ ) n τ ( E ) dE, (2) with h the Planck constant, µ the Fermi energy, f ( E − µ ) = [exp β ( E − µ ) + 1] − the Fermi distribution, and β = 1 /k B T , k B being the Boltzmann constant, and τ ( E )the transmission probability through the device. Thethermopower S is defined as the voltage drop induced bya difference of temperature when the charge current I e vanishes, and it is given by S = − ∆ V ∆ T = − eT K K , (3)as follows from Eqs. (1). From Eq. (1a) the electricconductance σ = − I e / ∆ V is obtained, which is measuredat zero temperature gradient, giving σ = e K , (4)and Eqs. (1) lead to the electron thermal conductance κ e , corresponding to the ratio between heat current andthe temperature gradient when the charge current is zero, κ e = − I Q ∆ T = 1 T (cid:20) K − K K (cid:21) . (5)The phonon contribution to the thermal conductance, κ ph , is neglected in this model. We assume that this hasbeen reduced by the choice of poor thermal contacts orby some mechanism of phonon confinement. The transmission τ ( µ ) was obtained through the equa-tion of motion approach for the Green’s function, andcan be written as τ ( µ ) = 4Γ [ t c − ( µ − ε ) cos( φ/ Ω( µ ) , (6a)with Ω( µ ) = [( µ − ε ) − t c − Γ ] +4Γ [ µ − ε − t c cos( φ/ , (6b) where φ = 2 π Φ / Φ is the Aharonov-Bohm phase, Φ = h/e the flux quantum, and we have assumed ε = ε ≡ ε and Γ Li = Γ Ri ≡ Γ ( i = 1 , τ ( µ ) is ingeneral a convolution of a Breit-Wigner and a Fano res-onance. These resonances develop around the molecularenergies ε − t c and ε + t c , of linewidths 2Γ cos ( φ/
4) and2Γ sin ( φ/ µ a = t c sec( φ/ t c determineshow close from each other are the resonances, and de-pending on how large it is as compared to Γ the dif-ferent peaks in transmission are visible or not: in thelimit t c ≪ Γ the two resonances are not resolved, when t c ≫ Γ they are perfectly resolved, and the case t c = Γrepresents the crossover between the two situations. Thephase φ determines not only the width but also the na-ture of each resonance, the Fano lineshape always beingthe narrower one, with a width ranging from 0 to Γ. TheBreit-Wigner resonance, in turn, has a linewidth betweenΓ and 2Γ. Special features occur in the transmissionspectrum when φ = 0 (or 2 nπ , n integer) where only theBreit-Wigner resonance exists, the other resonance beingabsent due to the localization of the associated molecu-lar state, and when φ = nπ ( n odd) the transmissionis a convolution of two Breit-Wigner lineshapes of equalwidths. The transmission τ as a function of φ has a 4 π period, but its basic structure is contained in an intervalof size π , where all the possible features of the resonancesare found. Given this, our analysis below only considersthe interval φ ∈ [0 , π ].When φ = 0, no Fano resonance occurs in transmis-sion, then the quantities σ , S , κ e and ZT can be obtainedresorting to the Sommerfeld expansion keeping the firstnon-zero term in each of the K , K and K . This re-sults in σ, κ e ∝ τ ( µ ), S ∝ τ (1) ( µ ) /τ ( µ ) (Mott’s formula),and ZT ∝ /τ ( µ ). When 0 < φ < π , the transmission isa convolution of a Breit-Wigner and a Fano resonance,with a zero occurring at µ = µ a . Then such approx-imations are not valid anymore, since both S and ZT diverge at that energy. Considering terms of higher or-ders in k B T ≡ ξ we have K = 2 h (cid:20) τ + π τ (2) ξ + 7 π τ (4) ξ + O ( ξ ) (cid:21) , (7a) K = 2 h (cid:20) π τ (1) ξ + 7 π τ (3) ξ + O ( ξ ) (cid:21) , (7b) K = 2 h (cid:20) π τ ξ + 7 π τ (2) ξ + O ( ξ ) (cid:21) , (7c)where τ ( n ) ≡ τ ( n ) ( µ ) = ( d n τ /dE n )( µ ). It follows fromEq. (7a) that at least the term of order O ( ξ ) must betaken into account in K , in order that the thermopower, S ∝ /K , does not diverge at the antiresonance. On theother hand, according to Eqs. (5) and (7), κ e at secondorder vanishes at µ a , since K ∝ τ (1) and K ∝ τ , makingthe figure of merit ZT ∝ /κ e diverge, then terms ofhigher orders in ξ are required. We show below thata good agreement between the numerical and analyticalcurves is obtained when terms up to order 4 are includedin the expansions of the K n .In the examples below we assume ε = 0, so that thetwo transmission resonances are located around µ = − t c and µ = t c , and we take t c = Γ. We focus our attention intwo values of φ ∈ (0 , π ) representing a narrow and a wideFano resonance. We consider two different values of k B T ,namely, k B T = 10 − Γ and 10 − Γ. For quantum dots(where Γ ∼ T = 10 mK and 100 mK, respectively, and for molecularjunctions (where Γ ∼ T = 10 K and 100 K,respectively.Figure 2 shows the transmission τ , the thermopower S , the figure of merit ZT and the Lorenz ratio L normal-ized by L as a function of the Fermi energy for φ = π/ K n (lines) and by using the Sommer-feld expansion keeping terms up to fourth order in ξ (circles). For this value of φ the transmission exhibitsa Fano resonance of a linewidth much smaller than thewidth of the Breit-Wigner resonance. For k B T = 10 − Γthe numerical and analytical results coincide, while when -3 -2 -1 0 1 2 30.00.51.0-2-10120.00.51.01.12 1.14 1.16 1.18 1.201234 t S / ( k B / e ) ZT L / L m / G FIG. 2. (Color online) Transmission probability τ , ther-mopower S , figure of merit ZT and ratio L/L versus Fermienergy µ in units of Γ, for k B T = 10 − Γ (solid line, blue cir-cles) and k B T = 10 − Γ (dash line, red circles), ε = 0, t c = Γ,and φ = π/
3. The lines stand for numerical curves, the circlesfor the analytical expressions via the fourth-order Sommerfeldexpansion. In the upper panel we have highlighted the regionof µ plotted in the lower panels. In the lowest panel we haveincluded L max /L = 4 .
19 (dotted line). K B T = 10 − Γ the analytical results of ZT and L/L dif-fer slightly from the numerical ones. We find in generalthat the closer to zero is φ , the smaller has to be the valueof k B T in order that the analytical and numerical curvesmatch. Figure 3 shows analogous graphs for φ = 3 π/ k B T , the curves obtainedanalytically for S , ZT and L fit exactly those obtainednumerically. For larger values of k B T the agreement be-tween both curves is better when the Fano resonance iswider, that is, when φ approaches to π , as evidenced byFig. 3. The approximate expressions in these cases arevalid even at room temperature in the case of molecularjunctions.Now, let us briefly discuss on the role that the parame-ter t c has on the graphs of the thermopower and figure ofmerit as a function of the Fermi energy. For t c = 0, thatis, the two dots not forming a molecule, S ( µ ) is symmet-ric with respect to the origin, and ZT ( µ ) shows peaks ofequal heights. The smallest values of ZT are found inthis case. If t c = 0 the symmetries of both S and ZT are affected, as observed in Figs. 2 and 3. If the Fanoresonance is wide enough (that is, φ is close to π ), t c doesnot have important effects of the shapes of both curves, -4 -2 0 2 40.00.51.0-2-10120.00.30.60.9 2.58 2.60 2.62 2.64 2.661234 t S / ( k B / e ) ZT L / L m / G FIG. 3. (Color online) Transmission probability τ , ther-mopower S , figure of merit ZT and ratio L/L versus Fermienergy µ in units of Γ, for k B T = 10 − Γ (solid line, blue cir-cles) and k B T = 10 − Γ (dash line, red circles) ε = 0, t c = Γ,and φ = 3 π/
4. The lines stand for numerical curves, the cir-cles for the analytical expressions via the fourth-order Som-merfeld expansion. In the upper panel we have highlightedthe region of µ plotted in the lower panels. In the lowest panelwe have included L max /L = 4 .
19 (dotted line). just limiting to shift them horizontally, the latter beingexpected by the fact that t c determines the position ofthe Fano antiresonance. The situation changes slightlywhen the Fano resonance gets thinner and φ get closer tozero. In this case the value of t c has more influence on theheights and symmetries of the two peaks present in bothcurves, the highest thermoelectric efficiencies occurringwhen t c is close to Γ.Last, let us pay attention to the Lorenz ratio L . For φ = 0 (and in general φ = 2 nπ , n integer), where noFano resonances take place, L = L for all values of µ at any value of k B T ∈ [0 , × − Γ]. Whenever φ = 0( φ = 2 nπ ), as is the case of Figs. 2 and 3 (lower panels), L departs from L in a small region around the antireso-nance energy. According to the expansion (7b), K con-tains only odd derivatives of the transmission τ , whichis dominated by the term proportional to τ (1) ( µ ), whichvanishes at µ = µ a . As consequence of this, K /K inEq. (5) falls to zero close to µ a , making the thermalconductance κ e have a small peak in this region, whilethe electric conductance σ presents a single minimum, asshown in Fig. 4. The shape difference of both curves re-sults in the violation of the Wiedemann-Franz law. Fur-thermore, we observe in the examples that L reaches the maximum L max = (7 k B π ) / (5 e ) = 4 . L , which cor-responds to the universal maximum given in Ref. . Al-though the Wiedemann-Franz law fails whenever Fanoantiresonances exist, the maximum L max is reached forall φ = 0 (2 nπ , n integer) only at very low tempera-tures ( k B T ∼ − Γ or smaller), and in general in aconstrained interval of values of φ around π , the size ofthis interval decreasing with temperature, when temper-atures are not very large ( k B T not exceeding 2 × − Γ). C ondu c t an c e ( a r b . un i t s ) m/G k e s FIG. 4. (Color online) Thermal and electric conductances κ e (solid line) and σ (dash line) versus Fermi energy µ in unitsof Γ, for ε = 0, t c = Γ, φ = π/
3, and k B T = 10 − Γ obtainedby integrating numerically Eqs. (2).
In summary, we have used the Sommerfeld expansionto describe analytically the thermoelectric properties of adouble quantum dot molecule embedded in an Aharonov-Bohm ring, which exhibits a Fano resonance in trans-mission. The existence of antiresonances demands thatusually discarded terms are taken into account, in or-der to avoid divergences in both the thermopower andfigure of merit. When the linewidth of the Fano res-onance is close to Γ, the obtained expressions for thethermopower, figure of merit, and Lorenz ratio are valideven at room temperature in the case of molecular junc-tions; for quantum dots they hold up to temperatures ofthe order of tenths of Kelvin degrees. If the Fano reso-nance is narrow, its linewidth being a small fraction ofΓ, these approximations hold only at very low tempera-tures, namely, k B T ∼ − Γ or less, which in quantumdots (molecular junctions) corresponds to temperaturesof a few mK (K). Our analysis shows clearly that theFano antiresonances are responsible in this system forthe enhancement of the thermopower magnitude and thethermoelectric efficiency, as well as for the violation ofthe Wiedemann-Franz law.
ACKNOWLEDGMENTS
The authors acknowledge financial support fromFONDECYT, under grants 1080660 and 1100560. G.G. and O. A. thank financial support from CONICYTMaster Scholarships. L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B ,16631(R) (1993). A. Khitun, A. Balandin, J. L. Liu, and K. L. Wang, J.Appl. Phys. , 696 (2000). A. A. Balandin and O. L. Lazarenkova, Appl. Phys. Lett. , 415 (2003). R. Venkatasubramanian, E. Siivola, T. Colpitts, and B.O’Quinn, Nature (London) , 597 (2001). T. C. Harman, P. J. Taylor, M. P. Walsh, and B. E.LaForge, Science , 2229 (2002). A. I. Hochbaum et al.
Nature (London) , 163 (2008). A. I. Boukai et al.
Nature (London) , 168 (2008). S. J. Thiagarajan, V. Jovovic, and J. P. Heremans, Phys.Stat. Sol. (RRL) , 256 (2007). G. Mahan, B. Sales, and J. Sharp, Phys. Today , 42(1997). G. D. Mahan and J. O. Sofo, Proc. Natl. Acad. Sci. USA , 7436 (1996). P. Murphy, S. Mukerjee, and J. Moore, Phys. Rev. B ,161406(R) (2008). J. P. Bergfield and C. A. Stafford, Nano Lett. , 3072(2009). C. M. Finch, V. M. Garc´ıa-Su´arez, and C. J. Lambert,Phys. Rev. B , 033405 (2009). O. Karlstr¨om, H. Linke, G. Karlstr¨om, and A. Wacker,Phys. Rev. B , 113415 (2011). T. Nakanishi and T. Kato, J. Phys. Soc. Jap. , 034715(2007). A. A. Clerk, X. Waintal, P. W. Brouwer, Phys. Rev. Lett. , 4636 (2001). D. Boese and R. Fazio, Europhys. Lett. , 576 (2001). X. Zianni, Phys. Rev. B , 045344 (2007). M. Tsaousidou and G. P. Triberis, J. Phys.: Condens. Mat-ter , 355304 (2010). Y. S. Liu, D. B. Zhang, X. F. Yang and J. F. Feng, Nan-otechnology , 225201 (2011). Y. Ahmadian, G. Catelani, I.L. Aleiner, Phys. Rev. B , 245315 (2005). B. Kubala, J. K¨onig, and J. Pekola, Phys. Rev. Lett. ,066801 (2008). M. G. Vavilov and A. D. Stone, Phys. Rev. B , 205107(2005). M. L. Ladr´on de Guevara, F. Claro, and P. A. Orellana,Phys. Rev. B , 195335 (2003). K. Kang and S. Y. Cho, J. Phys.: Condens. Matter ,117 (2004). Y. S. Liu and X. F. Yang, J. Appl. Phys. , 023710(2010). See, e.g., N. W. Ashcroft and N. D. Mermin,
Solid StatePhysics (Brooks Cole, Belmont, MA 1976). Y. S. Liu, Y. R. Chen, and Y. C. Chen, ACS Nano , 3497(2009). K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. Iye, Phys.Rev. Lett. , 256806 (2002). A. W. Holleitner, C. R. Decker, H. Qin, K. Eberl, andR. H. Blick, Phys. Rev. Lett. , 256802 (2001); A. W.Holleitner, R. H. Blick, A. K. H¨uttel, K. Eberl, and J. P.Kotthaus, Science , 70 (2002). J.C. Chen, A.M. Chang, and M.R. Melloch, Phys. Rev.Lett. , 176801 (2004). G. D. Mahan,
Many-Particle Physics (Plenum, New York2000). R. Swirkowicz, M. Wierzbicki, and J. Barnas, Phys. Rev.B , 195409 (2009). See, e.g., H. Bruus and K. Flensberg,
Many-body quan-tum theory in condensed matter physics (Oxford UniversityPress, Oxford 2004). P. A. Orellana, M. L. Ladr´on de Guevara, and F. Claro,Phys. Rev. B , 233315 (2004). See e. g. F. Chi, J. L. Liu, and L. L. Sun, J. Appl. Phys. , 093704 (2007). Yu. Boikov, B. M. Goltsman, and V. A. Danilov, Semicon-ductors , 464 (1995). A. Balandin and K. L. Wang, Phys. Rev. B58