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Aligia4 1Gerencia de Investigacio´n y Aplicaciones, GAIDI, CNEA, 1650 San Mart´ın, Buenos Aires, Argentina 2Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, 1025 CABA, Argentina 3Centro At´omico Bariloche, GAIDI, 8400 Bariloche, Argentina 4Instituto de Nanociencia y Nanotecnolog´ıa CNEA-CONICET, GAIDI, Centro At´omico Bariloche, 8400 Bariloche, Argentina We investigate the thermoelectric properties of a double quantum dot system coupled to two metallic reservoirs, focusing on two main effects: (i) the influence of coupling asymmetry between the quantum dot and the reservoirs on the Seebeck coefficient, and (ii) the impact of asymmetry in the energy levels of the dots on current rectification.Fromanexperimental the coupling to the leads [31, 34], andthat energyasym- perspective, DQDs have been realized in systems such metry in the dots can result in significant rectification as AlGaAs/GaAs heterostructures [52], Si-MOS devices effects [44], this workexploresthe impact of these asym- [53], InAs/InP nanowires[54] and Ge hut nanowires [55], metries on the DQD system.(inter-) dot Coulomb repulsion respectively, ν = L,R | − i L↓ R↓| i { } labels the left and right dot (and leads) and σ = , ↑ ↓ The energy of the degenerate triplet is given by E = stands for the spin projection.ie J = dω Tr ΓL f (ω) ΓR f (ω) G>(ω) A. Thermoelectric power C L R d h Z h(cid:0) − (cid:1) + ΓL f ( ω) ΓR f ( ω) G<(ω) , (7) In this section, we present the results for the ther- L R d (cid:0) − − − (cid:1) i moelectric power, also known as the Seebeck coefficient.cels in the expression for the rectification , f (ω) = ν 1+exp[(ω µ )/T ] −1 is the Fermi functR ion, and ν ν { − } f (E ) f (E )D ν ν ν¯ ν¯ ν n = − , ν(21) ν ν ν ν ν − To simplify the analysis, assume that the temperature at the right lead T = 0, E < 0 and E +U′ > 0 and R ν ν both Fermi levels are set at the origin of energies (µ = ν 0).6. resembling arcs or forks, similar to those previously ob- Note that in a different context, a “Kondo circle” has servedinthesimplerspinlesscase[44],whichcorresponds been reported in calculations of the Seebeck effect of a to our situation with the addition of a strong magnetic single dot as a function of the temperature of both leads field that projects out of the relevant Hilbert space the [37].When the visit the spinless case, where analytical expressions for temperature of both leads increase the effects add and the current have been derived in the regime where the this results in a circular shape for the change of sign of coupling to the leads is smaller than all other energy the Seebeck coefficient.[34] A. Manaparambil and I. Weymann, Nonequi- [16] R. S´wirkowicz, M. Wierzbicki, and J. Barna´s, Thermo- librium Seebeck effect and thermoelectric effi- electric effects in transport through quantum dots at- ciency of Kondo-correlated molecular junctions, tachedtoferromagneticleadswithnoncollinearmagnetic Phys.B. Rigo, and A. K. Mitchell, [17] T. A. Costi and V. Zlati´c, Thermoelectric trans- Linear response quantum transport through interact- port through strongly correlated quantum dots, ing multi-orbitalnanostructures, arXiv preprint (2022), Phys.Rev.B 81, 235127 (2010).Pen˜aloza., Universalityandthethermoelectric transport [19] J. Azema, A.-M. Dar´e, S. Sch¨afer, and P. Lom- properties of a double quantum dot system: Seeking bardo, Kondo physics and orbital degeneracy in- forconditionsthatimprovethethermoelectricefficiency, teract to boost thermoelectrics on the nanoscale, SciPost Phys.Core 7, 058 (2024).A. Aligia, and P. S. and I. Weymann, Nonequilibrium steady-state Cornaglia, Thermopower of an SU(4) Kondo res- thermoelectrics of Kondo-correlated quantum dots, onance under an SU(2) symmetry-breaking field, arXiv preprint (2024), 2409.03102.L. Arrachea, Nonlinear charge and energy dynamics [43] H. K. Yadalam and U. Harbola, Statistics of heat trans- of an adiabatically driven interacting quantum dot, port across a capacitively coupled double quantum dot Phys.Rev.B 95, 235117 (2017).Level anticrossing effect in single-level or multilevel [61] M. A. Romero, F. Flores, and E. C. Goldberg, double quantum dots: Electrical conductance, zero- Effective treatment of charge and spin fluctuations frequency charge susceptibility, and Seebeck coefficient, in dynamical and static atom-surface interactions, Phys.Rev.B 102, 115112 (2020).[46] P. Lombardo, R. Hayn, D. Zhuravel, and S. Sch¨afer, [62] A. Cr´epieux, S. Sahoo, T. Q. Duong, R. Zamoum, and Kondo-assisted switching between three conduc- M. Lavagna, Emission noise in an interacting quantum tion states in capacitively coupled quantum dots, dot: Roleofinelasticscatteringandasymmetriccoupling Phys.Rev.Res.Clerk, and [49] A. Manaparambil and W. I., Nonequilibrium Seebeck K. Le Hur, Theory of nonequilibrium transport in the effect and spin Seebeck effects in nanoscale junctions, SU(n) Kondo regime, Phys.Rev.B 80, 155322 (2009).A. Aligia, Ther- of asymmetric nanodevices in the Kondo regime, moelectric properties of a double quantum dot out of Phys.A. Saraiva, X. Hu, H. Jiang, M. Friesen, and S. N. [70] Y. Teratani, R. Sakano, T. Hata, T. Arakawa, Coppersmith, Lifting of spin blockade by charged M. Ferrier, K. Kobayashi, and A. Oguri, Field-induced impurities in si-mos double quantum dot devices, SU(4) to SU(2) Kondo crossover in a half-filling nan- Phys.Rev.B 101, 155411 (2020).M.Kumar,C.Thelander,A.Wacker,andA.Burke,Heat [71] A. Georges and Y. Meir, Electronic correla- driven transport in serial double quantum dot devices, tions in transport through coupled quantum dots, NanoLetters 21, 988 (2021).vices: A scattering-states numerical renormalization- [74] R. Aguado and D. C. Langreth, Out-of- group approach to open quantum systems, equilibrium Kondo effect in double quantum dots, Phys.Rev.Lett.[57] H. T. M. Nghiem and T. A. Costi, Time-dependent [75] R. L´opez, R. Aguado, and G. Platero, Nonequi- numerical renormalization group method for multiple librium transport through double quantum dots: quenches: Towards exact results for the long-time limit Kondo effect versus antiferromagnetic coupling, of thermodynamic observables and spectral functions, Phys.[76] N. S. Wingreen and Y. Meir, Anderson model [58] A. Manaparambil, A. Weichselbaum, J. von out of equilibrium: Noncrossing-approximation Delft, and I. Weymann, Nonequilibrium spin- approach to transport through a quantum dot, tronic transport through Kondo impurities, Phys.A. Aligia, Nonequilibrium ence effects in transport through a quantum dot, dynamics of a singlet–triplet Anderson im- 9 purity near the quantum phase transition, ing approximation, Phys.Rev.lar quantum dots: gate-tunable hund’s rule, under- [87] T. A. Costi, J. Kroha, and P. W¨olfle, Spectral prop- screened Kondo effect and quantum phase transitions, erties of the Anderson impurity model: Compari- J.A. Aligia, Width of the charge-transfer Aligia, Destructive quantum interference peak in the SU(n) impurity Anderson model in transport through molecules with elec- and its relevance to nonequilibrium transport, tron–electron and electron-vibration interactions, Phys.A. Aligia4 1Gerencia de Investigacio´n y Aplicaciones, GAIDI, CNEA, 1650 San Mart´ın, Buenos Aires, Argentina 2Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, 1025 CABA, Argentina 3Centro At´omico Bariloche, GAIDI, 8400 Bariloche, Argentina 4Instituto de Nanociencia y Nanotecnolog´ıa CNEA-CONICET, GAIDI, Centro At´omico Bariloche, 8400 Bariloche, Argentina We investigate the thermoelectric properties of a double quantum dot system coupled to two metallic reservoirs, focusing on two main effects: (i) the influence of coupling asymmetry between the quantum dot and the reservoirs on the Seebeck coefficient, and (ii) the impact of asymmetry in the energy levels of the dots on current rectification.Fromanexperimental the coupling to the leads [31, 34], andthat energyasym- perspective, DQDs have been realized in systems such metry in the dots can result in significant rectification as AlGaAs/GaAs heterostructures [52], Si-MOS devices effects [44], this workexploresthe impact of these asym- [53], InAs/InP nanowires[54] and Ge hut nanowires [55], metries on the DQD system.1.1 1.0 0.02 0.2 2 0.02 0.2 2 We have looked for large values of the thermoelectric δ δ power S for different asymmetry of the couplings α, voltage| V| , temperature T and splittings δ.When the visit the spinless case, where analytical expressions for temperature of both leads increase the effects add and the current have been derived in the regime where the this results in a circular shape for the change of sign of coupling to the leads is smaller than all other energy the Seebeck coefficient.[1] S. Guo, G. Zhou, and N. Tao, Single molecule hofer, P. Reddy,and H. Linke, Influenceof quantum in- conductance, thermopower, and transition voltage, terference on the thermoelectric properties of molecular NanoLett.[3] L. Rinc´on-Garc´ıa, C. Evangeli, G. Rubio-Bollinger, and [7] P. A. Erdman, J. T. Peltonen, B. Bhandari, B. Dutta, N. Agra¨ıt, Thermopower measurements in molecular H. Courtois, R. Fazio, F. Taddei, and J. P. Pekola, junctions, Chem.P. Reddy, Perspective: Thermal and ther- [8] A. Svilans, M. Josefsson, A. M. Burke, S. Fahlvik, moelectric transport in molecular junctions, C. Thelander, H. Linke, and M. Leijnse, Thermoelec- J. Chem.[34] A. Manaparambil and I. Weymann, Nonequi- [16] R. S´wirkowicz, M. Wierzbicki, and J. Barna´s, Thermo- librium Seebeck effect and thermoelectric effi- electric effects in transport through quantum dots at- ciency of Kondo-correlated molecular junctions, tachedtoferromagneticleadswithnoncollinearmagnetic Phys.B. Rigo, and A. K. Mitchell, [17] T. A. Costi and V. Zlati´c, Thermoelectric trans- Linear response quantum transport through interact- port through strongly correlated quantum dots, ing multi-orbitalnanostructures, arXiv preprint (2022), Phys.Rev.B 81, 235127 (2010).A. Aligia, and P. S. and I. Weymann, Nonequilibrium steady-state Cornaglia, Thermopower of an SU(4) Kondo res- thermoelectrics of Kondo-correlated quantum dots, onance under an SU(2) symmetry-breaking field, arXiv preprint (2024), 2409.03102.[38] S. E. Deghi and R. A. Bustos-Maru´n, Second-order [21] A.A.Aligia, Nonequilibriumself-energies, Ngapproach, adiabatic expansion of heat and charge currents and heat current of a nanodevice for small bias voltage within the nonequilibrium green’s function approach, and temperature, Phys.[22] J. Azema, P. Lombardo, and A.-M. Dar´e, Conditions [39] S. Donsa, S. Andergassen, and K. Held, Double for requiring nonlinear thermoelectric transport theory quantum dot as a minimal thermoelectric generator, in nanodevices, Phys.Rev.B 90, 205437 (2014).L. Arrachea, Nonlinear charge and energy dynamics [43] H. K. Yadalam and U. Harbola, Statistics of heat trans- of an adiabatically driven interacting quantum dot, port across a capacitively coupled double quantum dot Phys.Rev.B 95, 235117 (2017).Level anticrossing effect in single-level or multilevel [61] M. A. Romero, F. Flores, and E. C. Goldberg, double quantum dots: Electrical conductance, zero- Effective treatment of charge and spin fluctuations frequency charge susceptibility, and Seebeck coefficient, in dynamical and static atom-surface interactions, Phys.Rev.B 102, 115112 (2020).[46] P. Lombardo, R. Hayn, D. Zhuravel, and S. Sch¨afer, [62] A. Cr´epieux, S. Sahoo, T. Q. Duong, R. Zamoum, and Kondo-assisted switching between three conduc- M. Lavagna, Emission noise in an interacting quantum tion states in capacitively coupled quantum dots, dot: Roleofinelasticscatteringandasymmetriccoupling Phys.Rev.Res.[47] Y. Zhang and S. Su, Thermal rectification [63] A. Oguri, Fermi-liquid theory for the Anderson model and negative differential thermal conductance out of equilibrium, Phys.Clerk, and [49] A. Manaparambil and W. I., Nonequilibrium Seebeck K. Le Hur, Theory of nonequilibrium transport in the effect and spin Seebeck effects in nanoscale junctions, SU(n) Kondo regime, Phys.Rev.B 80, 155322 (2009).to thermoelectric transport in double quantum dots, [68] D. B. Karki, C. Mora, J. von Delft, and M. N. arXiv preprint (2024), 2409.17064.A. Saraiva, X. Hu, H. Jiang, M. Friesen, and S. N. [70] Y. Teratani, R. Sakano, T. Hata, T. Arakawa, Coppersmith, Lifting of spin blockade by charged M. Ferrier, K. Kobayashi, and A. Oguri, Field-induced impurities in si-mos double quantum dot devices, SU(4) to SU(2) Kondo crossover in a half-filling nan- Phys.Rev.B 101, 155411 (2020).M.Kumar,C.Thelander,A.Wacker,andA.Burke,Heat [71] A. Georges and Y. Meir, Electronic correla- driven transport in serial double quantum dot devices, tions in transport through coupled quantum dots, NanoLetters 21, 988 (2021).[57] H. T. M. Nghiem and T. A. Costi, Time-dependent [75] R. L´opez, R. Aguado, and G. Platero, Nonequi- numerical renormalization group method for multiple librium transport through double quantum dots: quenches: Towards exact results for the long-time limit Kondo effect versus antiferromagnetic coupling, of thermodynamic observables and spectral functions, Phys.[76] N. S. Wingreen and Y. Meir, Anderson model [58] A. Manaparambil, A. Weichselbaum, J. von out of equilibrium: Noncrossing-approximation Delft, and I. Weymann, Nonequilibrium spin- approach to transport through a quantum dot, tronic transport through Kondo impurities, Phys.[77] M. H. Hettler, J. Kroha, and S. Hershfield, Nonequi- [59] M. A. Romero, S. C. G´omez-Carrillo, P. G. Bol- librium dynamics of the Anderson impurity model, catto, and E. C. Goldberg, Spin fluctuation effects on Phys.A. Aligia, Repli- [89] L.Tosi,P.Roura-Bas,andA.A.Aligia,Non-equilibrium cas of the Kondo peak due to electron-vibration conductance through a benzene molecule in the Kondo interaction in molecular transport properties, regime, J.A. Aligia, Width of the charge-transfer Aligia, Destructive quantum interference peak in the SU(n) impurity Anderson model in transport through molecules with elec- and its relevance to nonequilibrium transport, tron–electron and electron-vibration interactions, Phys.".You have three tasks, which are:\n    1.to summarize the text I provided into a Summary .Please answer within 150-300 characters.\n    2.to summarize the text I provided, using up to seven Highlight.\n    3.to summarize the text I provided, using up to seven Key Insights. Each insight should include a brief in-depth analysis. Key Insight should not include timestamps.\n    Your output should use the following template strictly, provide the results for the three tasks:\n    ## Summary\n    ## Highlights\n    - Highlights\n    ## Key Insights\n    - Key Insights .\n  Importantly your output must use language "English"
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