Non-Linear Thermovoltage in a Single-Electron Transistor
Paolo A. Erdman, Joonas T. Peltonen, Bibek Bhandari, Bivas Dutta, Hervé Courtois, Rosario Fazio, Fabio Taddei, Jukka P. Pekola
NNon-Linear Thermovoltage in a Single-Electron Transistor
P. A. Erdman, J. T. Peltonen, B. Bhandari, B. Dutta, H. Courtois, R. Fazio,
4, 1
F. Taddei, and J. P. Pekola NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy ∗ QTF Centre of Excellence, Department of Applied Physics,Aalto University School of Science, P.O. Box 13500, 00076 Aalto, Finland Univ. Grenoble Alpes, CNRS, Institut N´eel, 25 Avenue des Martyrs, 38042 Grenoble, France ICTP, Strada Costiera 11, I-34151 Trieste, Italy NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
We perform direct thermovoltage measurements in a single-electron transistor, using on-chip localthermometers, both in the linear and non-linear regimes. Using a model which accounts for co-tunneling, we find excellent agreement with the experimental data with no free parameters evenwhen the temperature difference is larger than the average temperature (far-from-linear regime).This allows us to confirm the sensitivity of the thermovoltage on co-tunneling and to find that inthe non-linear regime the temperature of the metallic island is a crucial parameter. Surprisingly,the metallic island tends to overheat even at zero net charge current, resulting in a reduction of thethermovoltage.
Introduction. —The use of nano-devices has emergedas one of the key technologies in the quest to establisha sustainable energy system, allowing at the same timethe control of heat flow in small circuits [1]. So far, mostof the investigations of thermal properties in nanostruc-tures have focused on the thermal conductance [2–11].Conversely the thermovoltage, which describes the elec-trical response to a temperature difference and is directlyrelated to both the power and efficiency of thermal ma-chines [1], is much less studied. This is due to the dif-ficulty in coupling local sensitive electron thermometersand heaters/coolers to the sample under study in order tohave a well-defined, known temperature difference acrossthe device. The thermovoltage has been measured in de-vices based on nanowires [12, 13] and on quantum dots[14–26]. In these experiments, however, the temperatureof the electrodes were typically not measured directly,but rather determined as fitting parameters, and thereare no experiments where the temperature of the elec-trodes and the thermovoltage are measured simultane-ously. Furthermore, there are no experiments probingthe thermovoltage in devices based on metallic islands,while theoretical works for these systems have focusedonly on the linear response regime [27–34].In this paper, we report for the first time on themeasurement of the thermovoltage in a metallic single-electron transistor (SET) using on-chip, local tunnel-junction-based thermometers and electron temperaturecontrol. This system allows us to perform thermoelec-tric measurements with an unprecedented control, bothwithin the linear and non-linear response regimes, impos-ing temperature differences exceeding the average tem-perature. Using a theoretical model which accounts fornon-linear effects and co-tunneling processes, we find anexcellent agreement with the experimental data with nofree parameters. On one hand, this allows us to nail downquantitatively the role of co-tunneling processes on thethermovoltage. On the other hand, we find that in the (a)
N I N I N (b) L , V L I , n R , V Rel − ph tun (c) th n g V b [ m V ] |I| [nA] −0.1−0.05 FIG. 1. Representation and characterization of the single-electron transistor. a) False-colored SEM image of the fulldevice and a zoomed in view around the metallic island (yel-low) tunnel coupled to two normal leads (red and green). b)Schematic representation of the system with the same color-ing as in the SEM image. The heat balance in the metallicisland is represented by red arrows. c) Absolute value of thecurrent through the SET as a function of the applied sourcedrain voltage V b and of the gate-induced charge n g . non-linear regime the temperature of the island emergesas a crucial parameter. Surprisingly, although the ther-movoltage is measured at zero net charge current, withinthe non-linear response the island tends to overheat to atemperature greater than the average lead temperature,which results in a suppression of the thermovoltage. Weshow, however, that the non-linear thermovoltage can beoptimized up to a factor two with respect to the experi-mentally observed value by lowering the temperature ofthe island to the temperature of the cold lead. This couldbe achieved by exploiting the phonons in the island whichact as a third thermal bath coupled to our system. The experimental setup. — Fig. 1a) is a colored scan- a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec ning electron micrograph of the device and Fig. 1b) isa schematic representation of the experiment with thesame colors highlighting the main elements of the fullynormal-conducting SET. The left lead L (red) and rightlead R (green) are tunnel and capacitively coupled to acentral metallic island I (yellow), which is under the in-fluence of a tunable gate electric field (orange). A voltagebias V b = V L − V R can be applied to the SET electrodesand the corresponding current I can be measured for aninitial characterization of the device. The temperature T R of the electrons in R is fixed to the bath tempera-ture, given the strong electron-phonon coupling in thelarge and “bulky” lead. On the other hand, the elec-tronic temperature T L in the left lead (red) can bothbe varied and measured using the superconducting tun-nel probes (blue). The tunability of the temperature ispossible thanks to the superconducting wire (purple) inclean contact with the left lead through which there isno heat conduction, and thanks to the limited size of thenormal (red) part of the lead that reduces the electron-phonon heat flux. Electrons within the island are inlocal equilibrium at temperature T I since the electron-electron interaction is much faster than the tunnelingrates [45]. The experiment is performed in a dilution re-frigerator at bath temperatures typically between 50 and400 mK. For the thermovoltage measurements, the SETvoltage bias source and current preamplifier (sketched inred in Fig. 1a) are disconnected. Crucially, the ther-movoltage V th is probed directly across the SET using aroom-temperature voltage preamplifier with ultralow in-put bias current below 20 fA. Fabrication details can befound in Ref. [9] where “sample B” is the device used forthis experiment.Figure 1c) shows the absolute value of the current I across the device at 65 mK as a function of the po-tential bias V b and of the gate-induced charge n g =( C L V L + C R V R + C g V g ) /e , where C L , C R and C g are,respectively, the capacitances of the island to L, R and tothe gate electrode, and V g is the gate voltage. In the darkblue regions, Coulomb diamonds, single electron tunnel-ing between the leads and the island is not allowed, andthe current is very small. At half integer values of n g ,“degeneracy points”, there are conductance peaks at zerobias since single electron tunneling is allowed for any fi-nite voltage bias. The model. —The state of the SET is characterized bythe probability P ( n ) to have n excess charges on the is-land. The electrostatic energy necessary for this is U ( n ) = E C ( n − n g ) , (1)where E C = e / (2 C ) is the charging energy with C = C L + C R + C g . Electron tunneling between the leadsand the island induces transitions between charge states.The leading order process in a perturbative expansionin the tunnel coupling between the island and the leadscorresponds to a single electron transfer between the leads and the island (sequential tunnelling) [46, 47]. Thesequential-tunneling rates for transferring electrons from α = L,R (I) to β = I (L,R), with the island initially hav-ing n charges, is denoted by Γ αβ ( n ) (see SupplementalMaterial for details [48]).Higher order processes can become dominant if allsequential-tunneling processes are energetically unfavor-able [in the Coulomb diamond region in Fig. 1c)]. In par-ticular, co-tunneling (second order process) refers to thetransfer of an electron from one lead to another, withoutchanging the charge state of the island but going througha virtual state. The dominant contribution of this kindis inelastic co-tunneling, i.e. the electron which tunnelsfrom lead L, say, to I via a virtual state has a differentenergy with respect to the electron tunneling from I toR [49]. We denote the rate of inelastic co-tunneling thattransfers a charge from α = L (R) to β = R (L), when n electrons are on the island before the process occurs, by γ αβ ( n ).The probabilities P ( n ) can be computed by solving amaster equation (see Supplemental Material for details[48]). The charge current can then be written as I ( V b ) = I seq + I cot , where I seq = e (cid:88) n P ( n ) [Γ LI ( n ) − Γ IL ( n )] (2)is the sequential-tunneling contribution, given by elec-trons tunneling between lead L and I, and I cot = e (cid:88) n P ( n ) [ γ LR ( n ) − γ RL ( n )] (3)is the inelastic co-tunneling contribution [29, 30, 47, 50,51]. We compute the sequential and co-tunneling ratesexactly, without linearizing in the voltage bias and tem-perature difference (see Supplemental Material for details[48]).In the presence of a fixed temperature bias ( T R (cid:54) = T L ),the thermovoltage V th is the solution to I ( V th ) = 0 . (4)Notice that the charge current also depends on the tem-perature of the island T I . By imposing that the chargecurrent and the net energy entering the island throughelectron tunneling are zero, we find that T I = T L R R + T R R L R L + R R , (5)where R L and R R are respectively the resistance ofthe left and right tunnel junctions. Eq. (5), whichis found performing a simple sequential tunneling cal-culation within linear response and in the two chargestate approximation (valid for E C (cid:29) k B T ), reduces to T I = ¯ T ≡ ( T L + T R ) / R L = R R . We will thus initially assume that T I . . . . . n g − − V t h [ µ V ] Sawtooth Seq. Cot. Exp.
FIG. 2. Experimental and theoretical thermovoltage as afunction of n g . The red thin curve represents the sawtoothbehavior predicted with a sequential-tunneling calculation inlinear response and accounting for two charge states. Thedashed red curve is found by solving Eq. (4) including onlysequential contributions, while the green curve includes alsoco-tunneling contributions. The temperatures of the leads are T L = 134 mK and T R = 190 mK and, according to Eq. (5),we assume that T I = ¯ T . is given by the average lead temperature ¯ T . However,as we will soon discuss in detail, we find that this as-sumption gives quantitatively wrong results beyond thelinear response regime, leading us to the exploration ofthe impact of T I on the thermovoltage. Results. —We focus on two data sets which representtwo different regimes: linear response (Fig. 2), i.e. whenthe modulus of the temperature difference ∆ T = T L − T R is smaller than the average lead temperature ¯ T = ( T L + T R ) /
2, and non-linear response (Fig. 3). In both cases,using the model detailed above, we could accurately re-produce the experimental data without any free parame-ter. The system parameters E C = 100 µ eV ≈ k B × .
16 Kand R L = R R = 26 kΩ are independently extracted fromcharge current measurements. Figures 2 and 3a) presentthe same qualitative behavior, namely a periodic oscilla-tion of the thermovoltage with the gate-induced charge n g and a linear dependence around degeneracy points,but they exhibit different amplitudes (note that the signof V th is opposite in the two cases since the temperaturebiases are opposite).We first analyze the linear response regime by choos-ing the set of data obtained when the temperature ofthe leads is T L = 134 mK and T R = 190 mK, suchthat | ∆ T | < ¯ T . In Fig. 2 we compare the measured V th (blue dots) as a function of n g with different theo-retical models. The red thin curve represents the typicalsawtooth behavior which is predicted within linear re-sponse accounting only for sequential tunneling and twocharge states. This is characterized by a linear function . . . . . n g − − V t h [ µ V ] (a)(b) Lin.Non-Lin. & T I = ¯ T Non-Lin. & Heat Bal.Exp. T I [ m K ] n g FIG. 3. a) Experimental and theoretical thermovoltage as afunction of n g . All theoretical curves include co-tunneling.The red dashed-dotted curve corresponds to a linear responsecalculation around ¯ T . The green dashed curve corresponds toa non-linear calculation where we fix T I = ¯ T , while the blackcurve corresponds to a non-linear calculation where T I , shownin b) as a function of n g , is calculated solving the heat balancecondition in Eq. (6) together with Eq. (4). The temperaturesof the leads are T L = 342 mK and T R = 63 mK. of n g , crossing zero at the degeneracy points with slope E C ∆ T / ¯ T [27]. The other two curves (red dashed andgreen solid) are instead determined by computing V th using Eq. (4) and assuming that T I = ¯ T [see Eq. (5)].The red dashed curve, which only accounts for sequen-tial tunneling, shows a smoothened sawtooth behavioras a consequence of including multiple charge states inthe master equation and of a finite temperature. How-ever, both models based on sequential tunneling (thinand dashed red curves) approximately fit the experimen-tal data only near the degeneracy points (near half in-teger values of n g ). In this case, indeed, sequential tun-neling is allowed and thus dominates over co-tunneling[29]. On the other hand the green solid curve, computedincluding co-tunneling contributions, shows a strong sup-pression of the thermovoltage as we move away from de-generacy points. The excellent agreement between thismodel and the experimental measurements pinpoints thecritical dependence of the thermovoltage on inelastic co-tunneling processes.We now move to the non-linear regime. In Fig. 3a)we show the measured thermovoltage as a function of n g (blue dots) compared to theoretical calculations, all ofwhich include co-tunneling contributions. The lead tem-peratures are T L = 342 mK and T R = 63 mK, such that | ∆ T | > ¯ T . The red dashed-dotted curve is computedwithin the linear response regime choosing the averagelead temperature ¯ T as the characteristic temperature.More precisely, we solve Eq. (4) setting T I = ¯ T and choos-ing a small temperature difference of the leads δT around¯ T to find the thermopower S ≡ V th /δT for δT →
0. Wethen calculate the thermovoltage as V th = S ( T L − T R ),where now T L = 342 mK and T R = 63 mK are the ac-tual lead temperatures. As we can see from Fig. 3a), thislinear response model overestimates the thermovoltagealmost by a factor two. A non-linear calculation (greendashed curve) improves the agreement with the exper-imental data. This calculation is performed by solvingEq. (4) using the actual lead temperatures and, as be-fore, we fix the island temperature at T I = ¯ T . The dif-ference between the red dashed-dotted and green dashedcurves proves that we are indeed in the non-linear re-sponse regime, and it shows that the main effect of thenonlinear response is to decrease the amplitude of thethermovoltage. However, we still do not obtain a goodagreement with the experimental data.We find that we can get a perfect agreement with theexperimental data if we further improve the model by de-termining also the island temperature T I through a heatbalance equation, rather than fixing it at ¯ T . More pre-cisely [see Fig. 1b)], we denote by ˙ Q tun the heat cur-rent entering the island from sequential and co-tunnelingevents (see Supplemental Material for details [48]) and by Q el-ph = Σ V ( T − T ) the heat current flowing from elec-trons in the island to the phonons (we assume that theelectronic temperature T R in the bulky right electrode isequal to the temperature of the phonons). V is the islandvolume and Σ is the electron-phonon coupling constantwhich only depends on the material. The temperature ofthe island can thus be determined by the following heatbalance equation ˙ Q tun = ˙ Q el-ph . (6)The values of the parameters entering Q el-ph that we useare determined independently: V = 225 × ×
29 nm is estimated from SEM images and Σ is obtained fromRef. [9] for this device (sample B). The value, Σ =2 . − m − , is close to the standard literature valuefor copper [45] and in agreement with measurements ofother samples fabricated using the same Cu target.The black curve in Fig. 3a) is thus determined bycomputing both V th and T I simultaneously by solvingEqs. (4) and (6) without any free parameters for eachvalue of n g . As we can see, the non-linear model, com-plemented with the heat balance equation, is in excellentagreement with the experimental measurements, demon-strating that T I is indeed an important parameter in thenon-linear regime. Conversely we have verified that, us-ing the parameters of Fig. 2 which are within the linearresponse regime, V th only weakly depends on the par-ticular choice of T I between T L and T R . In Fig. 3b) weplot the island temperature T I , as a function of n g over
50 100 150 200 250 300 350 T I [mK]10152025 V m a x t h [ µ V ] T R T L Non-Lin. & T I = ¯ T Non-Lin. & Heat Bal.
FIG. 4. The maximum amplitude of the thermovoltage V maxth is plotted as a function of the island temperature, for T R ≤ T I ≤ T L . The green dashed lines point to the values of V maxth and T I found in the non-linear calculation at fixed T I = ¯ T (seethe green dashed curve of Fig. 3a) while the black solid linesand the gray area refer to the non-linear calculation includingthe heat balance equation (see the black solid curve of Fig. 3). a single period, determined in the same calculation thatleads to the black curve in Fig. 3a). Remarkably, despitethe very low phonon temperature (63 mK), the calculated T I ≈
250 mK is much larger than the average lead tem-perature ¯ T = 202 . n g modulation of approximately 10 mK,but this prediction cannot be confirmed in the presentexperiment.Finally we discuss how the thermovoltage depends on T I . In Fig. 4 we plot V maxth , the maximum amplitude of V th , computed by solving Eq. (4) at fixed lead temper-atures T L = 342 mK and T R = 63 mK and varying T I between the lead temperatures. The black solid lines andthe gray area point to the actual experimental value of V maxth and to the corresponding computed T I which dif-fers from ¯ T [see black curves in Figs. 3a) and 3b)], whilethe dashed green lines point to V maxth calculated setting T I = ¯ T [see the green dashed curve in Fig. 3a)]. We findthat V maxth strongly depends on the choice of T I and thatit increases as T I is lowered. Indeed, at T I = T R = 63 mK,the amplitude of the thermovoltage reaches 27 µ eV, twicethe experimental value [see blue dots in Fig. 3a)]. Thus,by increasing the energy exchange between the electronsand phonons in the island, for example by increasing theisland’s volume, we can lower the temperature of the is-land which in turn results in an increase of V th . Conclusions. —We performed measurements of ther-movoltage in a metallic island tunnel coupled to nor-mal leads. Within the linear regime we nail down therole of co-tunneling in determining the thermovoltage.Within the non-linear response regime we explore tem-perature biases, determined with on-chip thermometers,even larger than the average lead temperature. Usinga theoretical model which accounts for co-tunneling andnon-linear effects, we find an accurate agreement withthe experimental data without any free parameters. Inparticular, we find that the temperature of the metallicisland becomes an important parameter which must bedetermined by solving a heat balance equation for the is-land. Surprisingly, even if the net charge current throughthe system is vanishing and the coupling to the leads issymmetric, the metallic island overheats to a tempera-ture larger than the average lead temperature. As a con-sequence, the amplitude of the thermovoltage oscillationsdecreases.
Acknowledgments. —This work has been supported bythe Academy of Finland (grant 312057), by the Euro-pean Union’s Horizon 2020 research and innovation pro-gramme under the European Research Council (ERC)programme (grant agreement 742559), by SNS-WIS jointlab “QUANTRA”, by the SNS internal projects “Ther-moelectricity in nano-devices”, by the CNR-CONICETcooperation programme “Energy conversion in quan-tum, nanoscale, hybrid devices”, and by the COST Ac-tionMP1209 “Thermodynamics in the quantum regime”.BD acknowledges support from the Nanosciences Funda-tion under the auspices of the Universite Grenoble AlpesFoundation. ∗ [email protected][1] G. Benenti, G. Casati, K. Saito, and R. S. Whitney, Sci.Rep. , 1 (2017).[2] L. W. Molenkamp, T. Gravier, H. van Houten, O. J. A.Buijk, M. A. A. Mabesoone, and C. T. Foxon, Phys.Rev. Lett. , 3765 (1992).[3] K. Schwab, E. Henriksen, J. Worlock, and M. L. Roukes,Nature , 974 (2000).[4] O. Chiatti, J. T. Nicholls, Y. Y. Proskuryakov, N. Lump-kin, I. 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The system is described by the following Hamiltonianˆ H = (cid:88) α = L,R ˆ H α + ˆ H I + ˆ H t , (7)where ˆ H α = (cid:80) kσ ( (cid:15) k + eV α ) a † kσα a kσα is the Hamiltonian of the free electrons in lead α = L,R, ˆ H I = (cid:80) kσ (cid:15) k c † kσ c kσ + E C (ˆ n − n g ) is the Hamiltonian of the electrons in the metallic island and ˆ H t = (cid:80) kpσα t ( α ) kp c † pσ a kσα + h.c. is theusual tunneling Hamiltonian between the leads and the island. a kσα ( a † kσα ) is the destruction (creation) operator ofelectrons in lead α with energy (cid:15) k + eV α and spin σ , c kσ ( c † kσ ) is the destruction (creation) operator of electrons inthe metallic island with energy (cid:15) k and spin σ , and ˆ n is the operator for the number of excess electrons on the island.In order to describe charge and heat transport in the system, we employ a master equation approach to computethe probabilities P ( n ) in terms of all processes that can induce transitions between charges states (the tunnelingrates). Sequential tunneling of electrons between the island and the leads changes the charge state by one, so it entersthe master equation. Co-tunneling processes instead transfer an electron from one lead to another one via a virtualstate in the island, but the overall process does not change the number of electrons in the island; consequently, themaster equation does not depend on co-tunneling. Second order processes that transfer two electrons from/to theleads to/from the island can be safely neglected as the charging energy E C is much larger than the thermal energy k B T and than the voltage bias range considered in this work. The master equation reads ∂P ( n ) ∂t = (cid:88) α = L,R {− P ( n ) [Γ αI ( n ) + Γ Iα ( n )] + P ( n − αI ( n −
1) + P ( n + 1)Γ Iα ( n + 1) } , (8)and we solve it by setting ∂P ( n ) /∂t = 0 for every n . Eq. (8) states that the probability of being in charge state n candecrease (first r.h.s. term) if the island has n excess charge states and an electron tunnels into or out of the island,while it can increase (second and third r.h.s. terms) if, after a sequential tunneling process, the number of excesscharges on the island is n .Given the probabilities, the charge current can be computed using Eqs. (2) and (3). The energy entering themetallic island ˙ Q tun can be computed as˙ Q tun ≡ I EL + I ER = I hL + I hR + e ( V L − V R ) I, (9)where I E α and I hα are respectively the energy (measured respect to the common voltage ground) and heat currentsleaving reservoir α , and we used the fact that I EL = I hL + eV L I and I EL = I hR − eV R I . We can simply interpret the r.h.s.of Eq. (9) by noticing that the heat entering the metallic island is given by the sum of the heat leaving the leads andthe heat generated by Joule effect. We notice that a shift of the energy reference shifts V L and V R , but it does notchange I hL and I hR , so ˙ Q tun , as defined in Eq. (9), does not depend on the un-physical energy reference.The heat currents can be calculated in terms of “heat rates”. We thus define Γ hα I ( n ) as the rate of heat leavingreservoir α when electrons tunnel sequentially from lead α to the island with n initial electrons, and Γ h I α ( n ) as therate of heat entering lead α when electrons tunnel sequentially from the island to lead α with n initial electrons.Analogously, we define γ h/ out αβ ( n ) as the rate of heat leaving lead α when a co-tunneling process transfers one electronfrom lead α to lead β with n electrons in the island, and γ h/ in αβ ( n ) as the rate of heat entering lead β when aco-tunneling process transfers one electron from lead α to lead β with n electrons in the island. Notice that alsoco-tunneling processes where α = β must be considered in the heat currents, since the electron leaving and the oneentering the same lead can have different energies. Also the heat currents can be written as I hα = I h/ seq α + I h/ cot α ,where I h/ seq = (cid:88) n P ( n ) (cid:2) Γ hα I ( n ) − Γ h I α ( n ) (cid:3) (10)is the sequential-tunneling contribution, given by electrons tunneling between lead α and I, and I h/ cot α = (cid:88) n,β =L,R P ( n ) (cid:104) γ h/ out αβ ( n ) − γ h/ in βα ( n ) (cid:105) (11)is the inelastic co-tunneling contribution.Using the T matrix theory (or generalized Fermi golden rule) [50–52], we can compute sequential and co-tunnelingrates. The transition rate from a given initial state | i (cid:105) to a final state | f (cid:105) is given byΓ i → f = 2 π (cid:126) p i (1 − p f ) |(cid:104) f | T | i (cid:105)| δ ( E f − E i ) , (12)where p i and p f are the probabilities of finding the system in state i and f , E i and E f are the energies of states i and f and T = ˆ H t + ˆ H t G ˆ H t + . . . is the T matrix with G = 1 / ( E i − ˆ H + iη ) denoting the Green function in theabsence of the ˆ H t , i.e. ˆ H = ˆ H L + ˆ H R + ˆ H I . We compute sequential rates by taking T at first order in ˆ H t . We thustake T = ˆ H t in Eq. (12) and sum over all states in the lead and in the island, yieldingΓ αI ( n ) = 2 π (cid:126) (cid:88) k σ ,k σ f α ( (cid:15) k ) f − I ( (cid:15) k ) (cid:12)(cid:12)(cid:12) (cid:104) | c k σ H t a † k σ α | (cid:105) (cid:12)(cid:12)(cid:12) δ [ (cid:15) k − (cid:15) k + ∆ E α ( n )] , (13)where ∆ E α ( n ) = U ( n + 1) − U ( n ) − eV α is the electrostatic energy difference to move an electron from lead α tothe island, f α/ I ( (cid:15) ) = [1 + exp ( (cid:15)/ ( k B T α/I ))] − is the Fermi distribution of lead α at temperature T α or of the islandat temperature T I , f − α/ I ( (cid:15) ) = f α/ I ( − (cid:15) ) = 1 − f α/ I ( (cid:15) ), and k B is the Boltzmann constant. An analogous expressionholds for Γ Iα ( n ). The heat rates are computed in the same way, taking into account that an amount of heat (cid:15) k isremoved(injected) from(into) a lead if an electron with momentum k tunnels from(into) the lead. We thus have thatΓ hαI ( n ) = 2 π (cid:126) (cid:88) k σ ,k σ (cid:15) k f α ( (cid:15) k ) f − I ( (cid:15) k ) (cid:12)(cid:12)(cid:12) (cid:104) | c k σ H t a † k σ α | (cid:105) (cid:12)(cid:12)(cid:12) δ [ (cid:15) k − (cid:15) k + ∆ E α ( n )] , Γ hIα ( n ) = 2 π (cid:126) (cid:88) k σ ,k σ (cid:15) k f I ( (cid:15) k ) f − α ( (cid:15) k ) (cid:12)(cid:12)(cid:12) (cid:104) | a k σ α H t c † k σ | (cid:105) (cid:12)(cid:12)(cid:12) δ [ (cid:15) k − (cid:15) k − ∆ E α ( n − . (14)By assuming that the energy levels in the leads and in the island form a continuum, by taking a constant density ofstates around the Fermi energy and by replacing the hopping parameters t ( α ) kp with their averaged value over k and p ,we can write the sequential rates and heat rates in terms of the functionsΥ α (∆ E ) ≡ e R α + ∞ (cid:90) −∞ d(cid:15)f α ( (cid:15) ) f − I ( (cid:15) − ∆ E ) , Υ hα (∆ E ) ≡ e R α + ∞ (cid:90) −∞ d(cid:15) (cid:15)f α ( (cid:15) ) f − I ( (cid:15) − ∆ E ) , (15)where R α is the tunnel resistance between lead α and the island, as follows:Γ α I ( n ) = Υ α [∆ E α ( n )] , Γ I α ( n + 1) = Υ α [ − ∆ E α ( n )] , Γ hα I ( n ) = Υ hα [∆ E α ( n )] , Γ h I α ( n + 1) = − Υ hα [ − ∆ E α ( n )] . (16)Co-tunneling rates are second order processes that involve initial and final states with two electrons, so we nowconsider T = ˆ H t G ˆ H t . We thus take | i (cid:105) = a † k σ α c † q τ | (cid:105) and | f (cid:105) = a † q τ β c † k σ | (cid:105) , which corresponds to consideringthe process where an electron in state k σ tunnels from lead α to the island into state k σ , and another onecoherently tunnels from the island in state q τ to lead β into state q τ . From Eq. (12) we have that γ αβ ( n ) = 2 π (cid:126) (cid:88) k σ ,k σ q τ ,q τ f α ( (cid:15) k ) f I ( (cid:15) q ) f − β ( (cid:15) q ) f − I ( (cid:15) k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) ν (cid:104) f | H t | ν (cid:105) (cid:104) ν | H t | i (cid:105) E i − E ν + iη (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ [ (cid:15) q + (cid:15) k − (cid:15) q − (cid:15) k + e ( V β − V α )] , (17)where E i = (cid:15) q + (cid:15) k + eV α + U ( n ) is the energy of state | i (cid:105) , the sum over | ν (cid:105) runs over a complete set of eigenstates {| ν (cid:105)} of H , and E ν is the energy, evaluated with H , of state | ν (cid:105) . As we did for the sequential rates, we notice thatin the processes described in Eq. (17), the heat leaving reservoir α is (cid:15) k , while the heat entering reservoir β is (cid:15) q .The co-tunneling heat rate leaving reservoir α , γ h/ out αβ ( n ), is thus given by Eq. (17) adding an (cid:15) k inside the sum overthe initial and final states, while the co-tunneling heat rate entering reservoir β , γ h/ in αβ ( n ), is also given by Eq. (17)adding an (cid:15) q inside the sum over the initial and final states. Manipulating Eq. (17) using the same approximationsmentioned for the sequential rates, we find that by defining υ αβ (∆ E, ∆ E , ∆ E ) = (cid:126) π + ∞ (cid:90) −∞ d(cid:15) Υ α ( − (cid:15) )Υ β ( (cid:15) + ∆ E ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) + ∆ E − iη − (cid:15) − ∆ E + ∆ E + iη (cid:12)(cid:12)(cid:12)(cid:12) ,υ h/ out αβ (∆ E, ∆ E , ∆ E ) = (cid:126) π + ∞ (cid:90) −∞ d(cid:15) Υ hα ( − (cid:15) )Υ β ( (cid:15) + ∆ E ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) + ∆ E − iη − (cid:15) − ∆ E + ∆ E + iη (cid:12)(cid:12)(cid:12)(cid:12) ,υ h/ in αβ (∆ E, ∆ E , ∆ E ) = − (cid:126) π + ∞ (cid:90) −∞ d(cid:15) Υ α ( − (cid:15) )Υ hβ ( (cid:15) + ∆ E ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) + ∆ E − iη − (cid:15) − ∆ E + ∆ E + iη (cid:12)(cid:12)(cid:12)(cid:12) , (18)we can write the co-tunneling rates and heat rates as γ αβ ( n ) = υ αβ [ e ( V β − V α ) , ∆ U α ( n ) , − ∆ U β ( n − , (19) γ h/ out αβ ( n ) = υ h/ out αβ [ e ( V β − V α ) , ∆ U α ( n ) , − ∆ U β ( n − , (20) γ h/ in αβ ( n ) = υ h/ in αβ [ e ( V β − V α ) , ∆ U α ( n ) , − ∆ U β ( n − . (21)At last, we notice that the integrals in Eq. (18) are divergent in the limit η → + . In order to overcome this problem,we adopt a commonly used “regularization scheme” [29, 30, 47, 50, 51]. All three integrals can be written in the form I = + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) − α − iη − (cid:15) − α + iη (cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) i =1 , + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) − α i − iη (cid:12)(cid:12)(cid:12)(cid:12) − + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) Re (cid:26) (cid:15) − α + iη )( (cid:15) − α + iη ) (cid:27) = (cid:88) i =1 , (cid:110) I (1) i (cid:111) − I (2) (22)where g ( (cid:15) ) is a suitable function and α and α are suitable constants. We now analyze each integral: I (1) i = + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) − α i − iη (cid:12)(cid:12)(cid:12)(cid:12) = + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) − g ( α i ) + g ( α i )( (cid:15) − α i ) + η = g ( α i ) + ∞ (cid:90) −∞ d(cid:15) (cid:15) − α i ) + η + + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) − g ( α i )( (cid:15) − α i ) + η = πg ( α i ) η + P + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) − g ( α i )( (cid:15) − α i ) + O ( η ) , (23)where P denotes a principal value integration. We notice that the last step of Eq. (23) is an expansion for small η .In particular, the first term diverges as 1 /η , the second one is finite and independent of η , while the third one goes tozero if η →
0. The regularization scheme consists of dropping the divergent term and retaining only the second term,which is finite and independent of η : I (1) i → P + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) − g ( α i )( (cid:15) − α i ) . (24)Let’s now turn to I (2) = + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) ) Re (cid:26) (cid:15) − α + iη )( (cid:15) − α + iη ) (cid:27) = + ∞ (cid:90) −∞ d(cid:15)g ( (cid:15) ) ( (cid:15) − α )( (cid:15) − α ) − η [( (cid:15) − α )( (cid:15) − α ) − η ] + η [( (cid:15) − α ) + ( (cid:15) − α )] = + ∞ (cid:90) −∞ d(cid:15)g ( (cid:15) ) ( (cid:15) − α )( (cid:15) − α ) − η ( (cid:15) − α ) ( (cid:15) − α ) + η [( (cid:15) − α ) + ( (cid:15) − α ) + η ] . (25)0We notice that the denominator in Eq. (25) is always positive and non-zero. In the limit η →
0, the term proportionalto ( (cid:15) − α )( (cid:15) − α ) turns into a principal value integration, while the term proportional to − η vanishes. Theregularization scheme thus consists of I (2) → P + ∞ (cid:90) −∞ d(cid:15) g ( (cid:15) )( (cid:15) − α )( (cid:15) − α ) , (26)which is now finite and independent of ηη