Electronic shot noise in the absence of currents
Jakob Eriksson, Matteo Acciai, Ludovico Tesser, Janine Splettstoesser
EElectronic shot noise in the absence of currents
Jakob Eriksson,
1, 2
Matteo Acciai, and Janine Splettstoesser Department of Microtechnology and Nanoscience (MC2),Chalmers University of Technology, S-412 96 G¨oteborg, Sweden University of Gothenburg, S-412 96 G¨oteborg, Sweden (Dated: March 3, 2021)Shot noise is typically associated with the random partitioning of a current. Recently, chargecurrent shot noise due to a temperature bias, in the absence of an average current, has attractedinterest and was dubbed delta- T noise. Here, we show that this concept is much more general, oc-curring in different nonequilibrium constellations and for different types of currents. We specificallyanalyze and derive bounds for the zero-current charge shot noise at thermovoltage, and for heatshot noise in the absence of a heat current. Introduction.—
In mesoscopic devices, noise arisesboth at equilibrium, due to thermal excitations leading tothermal noise [1, 2], and out of equilibrium, where parti-tioning of electrons at a scatterer generates shot noise [3].Shot noise measurements have proven extremely usefulto gain insights on charge carriers [4] and have lead, forinstance, to the experimental observation of fractionalcharges [5, 6]. Very recently, a distinct type of shotnoise has been measured: this so-called delta- T noise isof purely thermal origin—it arises due to a temperaturebias, without any voltage bias—but it is distinct fromthermal noise and it is finite despite the average cur-rent vanishes [7–9]. This previously overlooked sourceof nonequilibrium noise was first introduced for diffusiveconductors [10] and is currently attracting a lot of at-tention both from a theoretical [11–13] and from an ex-perimental perspective [7–9]. Until now, the analysis ofthis phenomenon has however mostly been restricted bythe following constraints: (1) conductors are assumed tohave energy-independent transmission probabilities, (2)the nonequilibrium state is induced by a pure temper-ature bias, (3) only charge-current shot noise has beenconsidered.In this Letter, we generalize this broadly by lifting allthree constraints and thereby provide concrete results fora wider class of noise phenomena, of which the delta- T noise is a specific manifestation. Allowing the conduc-tor to have an energy-dependent transmission probabil-ity, we open up the analysis for two important aspects.Finite shot noise arises at zero average current undermore generic nonequilibrium conditions, e.g., combina-tions of temperature and voltage biases. Moreover, notonly the charge current, but also other types of currentsare subject to shot noise persisting at zero current underappropriate nonequilibrium conditions: as an example,we study the heat shot noise in the absence of an averageheat current. We demonstrate that, under the conditionof zero current, the energy dependence of the transmis-sion probability enhances the contribution of both chargeshot noise and heat shot noise relative to their thermalcounterparts.More concretely, we study a two-terminal conductor as sketched at the bottom of Fig. 1. The energy dependenceof the conductor’s transmission probability goes alongwith a combination of ∆ µ and ∆ T fulfilling the require-ment of a vanishing current. Considering a temperaturebias ∆ T , the voltage required to have zero charge currentis the thermovoltage ∆ µ = ∆ µ I , which naturally devel-ops in a thermoelectric system under open-circuit condi-tions. Alternatively, it is possible to completely suppressthe heat current into one of the terminals, despite heatis generally generated at nonequilibrium. For a giventemperature bias, this suppression always requires thepresence of a non-vanishing heat thermovoltage ∆ µ J . Inboth situations, we find the corresponding shot noise topersist; we refer to this as the zero-current shot noise inthe following. This zero-current shot noise arises becauseopposite energy-resolved currents are present in the sys- (b)(a) I J T , , I = 0 T T , , J L = 0 T Figure 1. Schematic representation of non-equilibrium con-ditions leading to vanishing (a) charge current and (b) heatcurrent into contact L of the setup sketched at the bottomof the panels. Fermi functions f L and f R of left and rightcontact with T L > T R and with µ L and µ R are shown as redand blue lines. The purple dashed line shows a Lorentziantransmission probability of the scatterer inside the conduc-tor (grey). In order to fulfill the zero-current condition, wehave (a) µ R − µ L = ∆ µ I and (b) µ R − µ L = ∆ µ J . Arrowsshow the magnitude and direction (emphasized by the back-ground color) of the resulting energy-resolved fluxes addingup to zero-average charge or heat current. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r tem (see Fig. 1). We find analytical bounds for both thezero-current charge and heat shot noise in conductorswith constant transmission, which are reached at largetemperature bias. These bounds are shown to be brokenwhen the transmission is energy-dependent, leading tozero-current charge and heat shot noise which can be ofthe same magnitude as the corresponding thermal noise.The analysis of zero-current charge shot noise at thethermovoltage and of zero-current heat shot noise areparticularly relevant also for the topical field of quan-tum thermal transport and thermoelectrics [14–16]. Innonequilibrium thermal engines, average currents be-tween source and working substance might indeed van-ish [17–20], while their noise remains relevant [21, 22].Also the minimization of noise as a performance goalof small-scale heat engines has recently been high-lighted [23–26], and correlations of heat and charge cur-rents have been linked to their efficiency [27, 28]. More-over, out-of-equilibrium fluctuation relations at vanishingaverage current have been studied recently [29–31]. Formalism and model.—
We consider a quantum con-ductor connected to two reservoirs ( α = L , R) character-ized by Fermi distributions f α ( E ) = { β α ( E − µ α )] } − , where µ α are the chemical potentials and β α =( k B T α ) − the inverse temperatures. In the following, wefix µ L ≡
0. In the framework of scattering theory [4, 32],the conductor is described by a transmission probabil-ity, D ( E ), that an electron at energy E is transmittedfrom one reservoir to the other. In the recent measure-ments of delta- T noise, scattering theory has been shownto be in very good agreement with experimental data inmolecular junctions [7], quantum point contacts [8] andtunnel junctions [9]. In the following, we investigate themore general situation of zero-current charge and heatshot noise. Thus, we require vanishing charge currentbetween the contacts and heat current into contact α .These are found from the expectation values of the op-erators ˆ X = ˆ I L = − ˆ I R ≡ ˆ I and ˆ X = ˆ J α = ˆ I Eα − µ α ˆ I α ,the heat current operator being the energy current in ex-cess to the energy flow of particles at the electrochemicalpotential. The average currents are X = 1 h Z dE xD ( E )[ f L ( E ) − f R ( E )] , (1)with x → {− e, E − µ α } for X → { I, J α } and the ele-mentary charge e >
0. The zero-frequency noise of thesecurrents is defined as S X = 2 R h δ ˆ X ( t ) δ ˆ X (0) i dt , where δ ˆ X = ˆ X − X is the fluctuation of the operator ˆ X aroundits average value X . These noise expressions can be di-vided into two contributions S X = S X th + S X sh , S X th = 4 h X γ =R , L Z dE x D ( E ) f γ ( E )[1 − f γ ( E )] S X sh = 4 h Z dE x D ( E )[1 − D ( E )][ f L ( E ) − f R ( E )] (2) The first term, S X th , is thermal-like noise. It contains in-dependent contributions from the two contacts γ = R , Land is therefore present also at equilibrium. The chargecurrent noise at equilibrium is the famous Johnson-Nyquist noise [1, 2]. In contrast, what is important isthat the shot noise term, S X sh , is only present out of equi-librium , when f L = f R and it is usually associated withthe partitioning of a current flowing into the system.The standard situation is indeed when a pure voltagebias ∆ µ = µ L − µ R is applied, resulting in a net current I = 0. Then S I sh reduces to conventional shot noise [3].However, as it is clear from Eqs. (1) and (2), a finite shotnoise does not require a finite average current, I = 0!Indeed, another possibility to obtain a measurable shotnoise, largely unexplored until recently, is to impose apure temperature bias ∆ T = T L − T R . Then, it has beenfound for energy-independent transmission probabilitiesthat S I sh = 0 even though I = 0. This is the recentlyinvestigated delta- T noise [7–9].In order to study the more general situation of zero-current charge and heat shot noise, we allow for anenergy-dependent transmission probability D ( E ). Whilea number of analytical results can be found for rathergeneric energy dependencies of D ( E ), we require a con-crete expression of D ( E ) for the plots displayed in Figs. 2and 3. In this case, we use a Lorentzian profile, D Lor ( E ) = D Γ Γ + ( E − (cid:15) ) , (3)with 0 < D <
1, peak energy (cid:15) , and width Γ, as indi-cated by the purple dashed line in Fig 1. This is a con-venient choice to describe different regimes. Indeed, theinterplay of the transmission probability and the Fermifunctions in Eq. (1) is such that D Lor ( E ) → π Γ D δ ( E − (cid:15) ), when Γ (cid:28) k B max( T L , T R ) and D Lor ( E ) → D , asstudied in Refs. [7–9], when Γ (cid:29) k B max( T L , T R ). Charge current noise.—
We consider two differentlimiting regimes for the temperature bias ∆ T , whichwere previously investigated in experiments on delta- T noise [7–9]. The first one is determined by a small tem-perature difference ∆ T (cid:28) ¯ T ≡ ( T L + T R ) /
2. It wasfound in Ref. [7] for an energy-independent transmissionthat, while S I th | I =0 simply becomes the Johnson-Nyquistnoise at temperature ¯ T , the delta- T noise S I sh | I =0 is pro-portional to (∆ T / ¯ T ) . It is hence typically very smallcompared to the thermal noise (about 3% in Ref. [7]). Al-though an energy-dependent transmission slightly modi-fies the result for the zero-current charge noise [33], theimpact on the total noise remains small.This is different in the second limiting case, where onetemperature dominates over the other, say T L (cid:29) T R .Then, the zero-current charge shot noise can be of thesame order as the thermal noise. We hence mainly focuson this regime in the rest of the paper. For a generic,weakly energy-dependent transmission function, we findfor the ratio between the two noise contributions [33] R I ≡ S I sh S I th (cid:12)(cid:12)(cid:12)(cid:12) I =0 = [1 − D (0)] (cid:20) ln (cid:18) µ I k B T L (cid:19) − (cid:21) . (4)The thermovoltage ∆ µ I , found by imposing that thecharge current I vanishes, reads∆ µ I = Γ D (0) D (0) − s π (cid:18) k B T L Γ (cid:19) (cid:18) D (0) D (0) (cid:19) . (5)These results are valid when k B T L (cid:28) Γ and, here, Γ isa typical energy scale set by the transmission functionsuch that D (0) / Γ := dD/dE [hence we use the samesymbol as for the Lorentzian profile, Eq. (3)]. Equa-tion (4) generalizes a previous result for delta- T noise [9]to arbitrary, weakly-dependent transmission probabili-ties; it also shows that the zero-current charge shot noiseis comparable to the thermal one, unless D (0) is closeto unity. In the limiting case of a constant transmission D ( E ) = D , where the thermovoltage is ∆ µ I = 0, Eq. (4)reduces to R I | D ( E )= D = (1 − D )(2 ln 2 − . (6)Interestingly, in the case of a constant transmission prob-ability, a direct numerical evaluation of Eq. (2) for arbi-trary temperatures T L and T R [33] shows that Eq. (6) isan upper bound for R I . In contrast, we here find thatalready a weak energy-dependence in the transmissionprobability breaks this bound! The presence of the ther-movoltage ∆ µ I = 0 in Eq. (4) always leads to an in-crease with respect to Eq. (6), which is however smallsince Eq. (5) yields β L ∆ µ I ∼ k B T L / Γ (cid:28) D Lor ( E ), Eq. (3), with arbitrarily sharp energy depen-dence. As a first step, we find the thermovoltage ∆ µ I bynumerically solving the equation I = 0. Then, plugging∆ µ I into Eq. (2) yields the resulting zero-current chargeshot noise and thermal noise. The charge noise ratio, R I normalized to 2 ln 2 −
1, together with the charge ther-movoltage, ∆ µ I , are shown in Fig. 2 as function of thepeak energy (cid:15) and the width Γ of the Lorentzian, for T L (cid:29) T R . Since ∆ µ I ( R I ) is an odd (even) function of (cid:15) , we consider (cid:15) ≥ R I = 2 ln 2 − D Lor ( E ) is close toconstant [see Eq. (6)]. This is the case for β L Γ (cid:29) β L (cid:15) (cid:29)
1. In the limit β L Γ (cid:29)
1, the transmission proba-bility reaches the value D , which we choose to be smallcompared to 1, D = 10 − , in Fig. 2. When (cid:15) is thelargest energy scale, only the tail of the Lorentzian over-laps with the transport window (i.e., the region where Figure 2. (a) Charge noise ratio R I , normalized to the charac-teristic factor 2 ln 2 −
1, see Eq. (6). (b) Corresponding chargethermovoltage ∆ µ I . (c) Cuts of the density plot in (a) for dif-ferent values of Γ. For all plots, we use D ( E ) = D Lor ( E ), with D = 10 − , and T R /T L = 0 at fixed, finite T L . the difference of Fermi functions in Eq. (1) is large) andthe transmission function can be approximated by a con-stant much smaller than D . Correspondingly, we find∆ µ I → R I exceeds thevalue 2 ln 2 − R I ≈ (cid:15) →
0, where the ∆ µ I = 0 due to electron-hole symme-try. We conclude that even the charge noise ratio R I fordelta- T noise, namely for zero-current charge shot noiseat ∆ µ I = 0, can be increased by more than a factor 2compared to the case of constant transmission. This in-creased ratio stems from an overall suppression of thethermal noise when decreasing Γ which is more promi-nent than the decrease of the shot noise [33]. Heat current noise.—
Importantly, the concept of zero-current shot noise is not limited to charge currents, butcan greatly be extended to other transport quantities aswe show in this Letter. Here, as an example, we explic-itly introduce the zero-current heat shot noise, which toour knowledge has not been studied before. This choiceis motivated by the current interest in mesoscopic heat / / Figure 3. (a) Ratio R J between the zero-current heat shotnoise and heat thermal noise (normalized to the characteristicfactor R J , see Eq. (10)). (b) Corresponding heat thermovolt-age. (c) Cuts of the density plot in (a) for different values ofΓ. For all plots, we use D ( E ) = D Lor ( E ), with D = 10 − ,and T R /T L = 0 at fixed, finite T L . engines, where fluctuations in heat and power can playan important role [14]. Unlike conserved currents, theheat current in nonequilibrium conductors can in gen-eral be nullified only in one of the contacts at a time.We hence investigate the heat current noise in contact L, S J L ≡ S J , at vanishing heat current into the same con-tact [34]. This situation of non-vanishing zero-currentheat shot noise can only be obtained when both a tem-perature and a voltage difference across the conductorare present.First, we consider the small temperature bias regime,∆ T (cid:28) ¯ T . For a constant transmission probability, wefind the ratio between zero-current heat shot noise andheat thermal noise to be R J ≡ S J sh S J th (cid:12)(cid:12)(cid:12)(cid:12) J L =0 = π − " − (cid:18) T R T L (cid:19) (1 − D ) . (7)Similar to what was found for the zero-current chargeshot noise, this contribution is small compared to thecorresponding thermal noise. Expressions for more gen-eral transmission probabilities in this regime are shownin [33].Next, we now focus on the opposite regime, T L (cid:29) T R . Considering first a generic, weakly energy-dependenttransmission probability, with k B T L (cid:28) Γ, we find R J = 3 π [1 − D (0)] A ( x J ) , (8) where x J = ∆ µ J / ( k B T L ) is the dimensionless heat ther-movoltage and A ( x ) = 2 x ln(1 + e x ) − ( π + x ) / x Li ( − e x ) − ( − e x ), with the polylogarithm functionLi n . Solving J L = 0, one finds two solutions [35] x J = ± π √ (cid:20) ± π √ k B T L Γ D (0) D (0) (cid:21) . (9)It is instructive to analyze Eq. (8) in the limit of a con-stant transmission function D ( E ) = D . In this case, x J = ± π/ √ R J | D ( E )= D = 3 π A (cid:18) π √ (cid:19) (1 − D ) ≡ R J (1 − D ) . (10)Since R J ≈ .
45, this shows that also the zero-currentheat shot noise can be of the same order of magnitudeas the corresponding thermal noise! As for the chargenoise, we find that Eq. (10) provides an upper bound for R J when D ( E ) = D [33], which can already be brokenfor a weakly energy-dependent transmission function if D (0) >
0, as A ( x J ) is an increasing function of | x J | .We finally move to the Lorentzian-shaped transmissionprobability as an example for a specific conductor, with-out a weak energy dependence. By solving J L = 0, wefind ∆ µ J , which we plug into Eq. (2) to obtain the zero-current heat shot noise and heat thermal noise; their ratiofor T L (cid:29) T R is plotted in Fig. 3. The density plot for R J displayed in Fig. 3(a) shows extended regions (green),where the limit R J = R J is reached. As for the chargenoise ratio, this occurs when the transmission probabilityis close to constant and the heat thermovoltage [Fig. 3(b)]approaches the value − β L ∆ µ J = π/ √
3. Interestingly,also the zero-current heat shot noise can be brought tobe of the same order as the heat thermal noise, by prop-erly tuning the energy-dependence of the transmissionprobability. This limit is found in the light-green andyellow regions in Fig. 3(a) and is clearly visible also fromFig. 3(c). Note, however, that in contrast to the behav-ior of the charge noise ratio R I , we find extended regionswhere R J for the Lorentzian-shaped transmission proba-bility is lower than the bound for constant transmissionat large temperature biases. Finally, we also observe that R J does not saturate when (cid:15) →
0, in contrast to the caseof R I . This is due to a non-monotonic behavior of boththe zero-current heat shot noise and the thermal noise asfunction of (cid:15) , see [33] for details. Conclusions.—
We have investigated both the elec-tronic charge and heat noise in a quantum conductor withan arbitrary transmission probability. When the systemis out of equilibrium due to a temperature and/or volt-age bias, shot noise is generated even in absence of anet charge or heat current flow. We have provided gen-eral expressions for these zero-current shot noises andhave characterized them for a prototypical example of aLorentzian transmission probability. We have shown thatsuch energy-dependent transmission can enhance the rel-ative contribution of the shot noise with respect to thethermal noise, compared to a conductor with a constanttransmission. In this latter case, we have shown that theratio between shot and thermal noise have bounds givenby a closed simple form, as given in Eqs. (6) and (10).This work greatly extends previous studies on delta- T noise, showing that the concept of zero-current shot noisecan be explored under more general nonequilibrium con-ditions and for different types of currents. While we havehere focused on charge and heat transport, it would beinteresting to investigate this concept for other kinds ofcurrents such as spin currents and their noise in the fu-ture. Finally, the possible use of delta- T noise for shotnoise spectroscopy of thermal gradients has been pointedout [7] and we expect this scope to become more am-ple with extensions to different types of shot noise undermore general nonequilibrium conditions.We thank Rafael S´anchez, Jens Schulenborg, ChristianSp˚ansl¨att, and Robert Whitney for helpful comments onthe manuscript. We acknowledge financial support fromthe Swedish VR (J.S.) and the Knut and Alice Wallen-berg Foundation (M.A. and J.S.). [1] J. B. Johnson, “Thermal agitation of electricity in con-ductors,” Nature , 50–51 (1927).[2] H. Nyquist, “Thermal Agitation of Electric Charge inConductors,” Phys. Rev. , 110–113 (1928).[3] W. Schottky, “ ¨Uber spontane Stromschwankungen inverschiedenen Elektrizit¨atsleitern,” Annalen der Physik , 541–567 (1918).[4] Ya. M. Blanter and M. B¨uttiker, “Shot noise in meso-scopic conductors,” Phys. Rep. , 1–166 (2000).[5] R. de Picciotto, M. Reznikov, M. Heiblum, V. Uman-sky, G. Bunin, and D. Mahalu, “Direct observation of afractional charge,” Nature , 162–164 (1997).[6] L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne,“Observation of the e / , 2526–2529 (1997).[7] Ofir Shein Lumbroso, Lena Simine, Abraham Nitzan,Dvira Segal, and Oren Tal, “Electronic noise due totemperature differences in atomic-scale junctions,” Na-ture , 240–244 (2018).[8] E. Sivre, H. Duprez, A. Anthore, A. Aassime, F. D.Parmentier, A. Cavanna, A. Ouerghi, U. Gennser, andF. Pierre, “Electronic heat flow and thermal shot noisein quantum circuits,” Nature Communications , 5638(2019).[9] Samuel Larocque, Edouard Pinsolle, Christian Lupien,and Bertrand Reulet, “Shot Noise of a Temperature-Biased Tunnel Junction,” Phys. Rev. Lett. , 106801(2020).[10] Eugene V. Sukhorukov and Daniel Loss, “Noise in multi-terminal diffusive conductors: Universality, nonlocality,and exchange effects,” Phys. Rev. B , 13054–13066(1999).[11] Elena Zhitlukhina, Mikhail Belogolovskii, and Paul Sei-del, “Electronic noise generated by a temperature gradi-ent across a hybrid normal metal–superconductor nano- junction,” Applied Nanoscience (2020), 10.1007/s13204-020-01329-7.[12] J. Rech, T. Jonckheere, B. Gr´emaud, and T. Martin,“Negative Delta- T Noise in the Fractional Quantum HallEffect,” Phys. Rev. Lett. , 086801 (2020).[13] Masahiro Hasegawa and Keiji Saito, “Delta- T noise inthe Kondo regime,” Phys. Rev. B , 045409 (2021).[14] Giuliano Benenti, Giulio Casati, Keiji Saito, andRobert S. Whitney, “Fundamental aspects of steady-state conversion of heat to work at the nanoscale,”Physics Reports , 1 – 124 (2017).[15] Robert S. Whitney, Rafael S´anchez, and JanineSplettstoesser, “Quantum thermodynamics of nanoscalethermoelectrics and electronic devices,” in Thermody-namics in the Quantum Regime: Fundamental Aspectsand New Directions , edited by Felix Binder, Luis A.Correa, Christian Gogolin, Janet Anders, and GerardoAdesso (Springer, Cham, 2018) pp. 175–206.[16] Felix Binder, Luis A. Correa, Christian Gogolin, JanetAnders, and Gerardo Adesso,
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0, the heat current into the leftcontact equals the energy current across the conductor.The following discussion is hence also valid for the energycurrent and the corresponding zero-current energy shotnoise.[35] The existence of two solutions is most easily understoodin the constant-transmission limit, where the heat cur-rent can be obtained exactly and depends on x J . upplemental material: Electronic shot noise in the absence of currents Jakob Eriksson,
1, 2
Matteo Acciai, and Janine Splettstoesser Department of Microtechnology and Nanoscience (MC2),Chalmers University of Technology, S-412 96 G¨oteborg, Sweden University of Gothenburg, S-412 96 G¨oteborg, Sweden (Dated: March 3, 2021)
S1. CHARGE NOISE FOR A GENERAL TRANSMISSION
In this section, we evaluate the charge shot noise at the thermovoltage for a general transmission function D ( E/ Γ).Here, Γ is a typical energy scale in the system. As a concrete example, one can think of a Lorentzian transmissionfunction D Lor (cid:18) E Γ (cid:19) = D (cid:0) E Γ − (cid:15) Γ (cid:1) . (S1) A. Large temperature gradient
We are interested in evaluating the shot noise in the limit where one of the two temperatures is much smaller thanthe other; without loss of generality, we then take T L = T and T R = 0. We also consider µ L = 0 and µ R = − ∆ µ ,where ∆ µ is an up to now arbitrary voltage bias. Later on, we will set ∆ µ = ∆ µ I , where ∆ µ I is the thermovoltagewhich develops in the system in response to the temperature bias in such a way that the charge current vanishes.With the above assumptions, and defining the function F ( E/ Γ) = D ( E/ Γ)[1 − D ( E/ Γ)], the charge shot noise reads S I sh = 4 e h Z + ∞−∞ dE F ( E/ Γ)[ f L ( E ) − f R ( E )] = 4 e h Z + ∞−∞ dE F ( E/ Γ) [ f E ( E ) + k B T f L ( E ) − f L ( E )Θ( − E − ∆ µ ) + Θ( − E − ∆ µ )]= 4 e h (cid:20)Z + ∞− ∆ µ dE F ( E/ Γ) f L ( E ) + Z + ∞ ∆ µ dE F ( − E/ Γ) f L ( E ) + k B T Z + ∞−∞ dE F ( E/ Γ) f L ( E ) (cid:21) (S2)where Θ( x ) is the Heaviside step function. In order to proceed analytically, we assume that the temperature T inthe left reservoir is small compared to the typical energy scale Γ determined by the transmission function. In otherwords, this physically means that the transmission function is weakly energy-dependent on the scale of the relevanttransport window. To summarize, the approximations we are considering are k B T R (cid:28) k B T L (cid:28) Γ.Consider now the integral I = Z + ∞− ∆ µ dE F ( E/ Γ) f L ( E ) = k B T Z + ∞− ∆ µk B T dω F (cid:18) k B T Γ ω (cid:19)
11 + e ω . (S3)For k B T (cid:28) Γ, it can be approximated as I = k B T (cid:26) F (0) (cid:2) x + ln (cid:0) e − x (cid:1)(cid:3) + k B T Γ F (0) [ − x ln (1 + e x ) − Li ( − e x )]+ F (0)2 (cid:18) k B T Γ (cid:19) (cid:2) x ln (1 + e x ) + 2 x Li ( − e x ) − ( − e x ) (cid:3) + O "(cid:18) k B T Γ (cid:19) (S4)where x = ∆ µ/ ( k B T ), Li n ( · ) is the polylogarithm function and denotes the differentiation with respect to E/ Γ. Theother integrals in the last line of Eq. (S2) can be handled in the same way. Eventually, one finds, keeping terms up a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r to second order in k B T / Γ, S I sh = 4 e h k B T ( F (0) [ln (2 + 2 cosh x ) − − k B T Γ F (0) [ x ln (2 + 2 cosh x ) + L ( x )]+ (cid:18) k B T Γ (cid:19) F (0) (cid:20) x ln (2 + 2 cosh x ) + x L ( x ) − L ( x ) − π (cid:21) + O "(cid:18) k B T Γ (cid:19) , (S5)where L n ( x ) = Li n ( − e x ) + ( − n +1 Li n ( − e − x ). In a similar way, we evaluate the thermal noise as follows: S I th = 4 e h Z + ∞−∞ D ( E/ Γ) f L ( E )[1 − f L ( E )] ≈ e h k B T " D (0) + D (0) (cid:18) k B T Γ (cid:19) π . (S6)The only thing left is to find the thermovoltage ∆ µ I to be plugged in Eq. (S5) in order to have the shot noise at zerocurrent. To this aim, we have to impose the condition I = eh Z + ∞−∞ D ( E/ Γ)[ f L ( E ) − f R ( E )] = 0 . (S7)By performing the same expansion as in the previous calculations, we find∆ µ I D (0) − µ I D (0) − π ( k B T ) D (0) = 0 = ⇒ ∆ µ I = Γ D (0) D (0) − s π (cid:18) k B T Γ (cid:19) (cid:18) D (0) D (0) (cid:19) . (S8)The sign of the square root is chosen in order to have the physically relevant solution, which requires that when D (0) → µ I →
0, meaning that no thermoelectric effect is present to this order of approximation.From Eqs. (S5) and (S6) we can compute an approximation for the ratio R I in terms of the thermovoltage givenby Eq. (S8) and the transmission function and its derivatives evaluated at the Fermi level. As we show below, theleading terms in Eqs. (S5)-(S6) already provide a good approximation; we then find the simplified expression R I = S I sh S I th (cid:12)(cid:12)(cid:12)(cid:12) I =0 ≈ [1 − D (0)] (cid:20) ln (cid:18) µ I k B T (cid:19) − (cid:21) , (S9)which is the result reported in the main text.For a constant transmission D ( E ) = D the thermovoltage is ∆ µ I = 0 and all derivatives of D ( E ) are zero.Therefore the previous expression at ∆ µ I = 0 actually gives the exact result in this case: R I | D ( E )= D = (1 − D )(2 ln 2 − . (S10)Similar expressions have been previously obtained in this regime [1].We now illustrate the approximation (S9) for the Lorentzian transmission function in Eq. (S1). In Fig. S1(a-b) wecompare the ratio R I given in Eq. (S9) with the exact result for the same quantity, computed by numerical integration.As we can see, the agreement between the two is very good for k B T / Γ < − . Fig. S1(c) shows two different cutsof the density plots in the top panel and illustrates that the approximation is increasingly good the smaller k B T /
Γ.Finally, as discussed in the main text, notice how the ratio tends to the value 2 ln 2 − (cid:15) is the largest energyscale. This is because in this situation, the transmission function for the relevant energies in the transport windowappears as a constant D (cid:28) − D ≈ B. Small temperature gradient
We now consider the regime where the temperatures in the reservoirs are close to each other. This is the situationwhich has been investigated in the first measurement of the delta- T noise, see Ref. [2]. Explicitly, consider that T L , R = ¯ T ± ∆ T / µ L = 0, µ R = − ∆ µ . For a constant transmission D , one gets for the shot noise S I sh = 4 e h k B ¯ T D (1 − D ) "(cid:16) y coth y − (cid:17) + (cid:18) ∆ T ¯ T (cid:19) y ( y + 4 π ) coth( y/ − y + 4 π )48 sinh ( y/ + O "(cid:18) ∆ T ¯ T (cid:19) , (S11) (a) (b) (c) Figure S1: (a) Exact ratio, computed by numeric integration. (b) Ratio R I calculated with the approximation. (c)Comparison between the approximation (dashed lines) and the exact result (solid lines) for two representative valuesof Γ /k B T . In all the plots a Lorentzian transmission function centered at energy (cid:15) and with D = 0 . y = ∆ µ/ ( k B ¯ T ). In the limit y →
0, when no bias is applied, one recovers S I sh = 4 e h k B ¯ T D (1 − D ) (cid:18) ∆ T ¯ T (cid:19) π − , (S12)in agreement with previous results [2]. For a generic transmission function, in the limit k B ¯ T (cid:28) Γ, we instead find S I sh = 4 e h k B ¯ T (cid:20) S I + S I (cid:18) ∆ T ¯ T (cid:19)(cid:21) + O "(cid:18) ∆ T ¯ T (cid:19) , (S13)with S I = F (0) (cid:16) y coth y − (cid:17) + k B ¯ T Γ F (0) y (cid:18) −
12 coth y (cid:19) + O "(cid:18) k B ¯ T Γ (cid:19) ,S I = k B ¯ T Γ F (0) 4 π sinh y − y ( y + 4 π − y )24 sinh ( y/
2) + O "(cid:18) k B ¯ T Γ (cid:19) . (S14)These results hold for an arbitrary bias ∆ µ . Notice that in the small bias limit, y (cid:28)
1, the coefficient S I reduces to S I → k B ¯ T Γ F (0) y π − , (S15)in agreement with Ref. [2] (Supplementary Information, Eq. S9), and vanishes for y → µ I , i.e. the voltage for which I = 0. It is found to be∆ µ I k B ¯ T = Γ D (0) D (0) − s π (cid:18) k B ¯ T Γ (cid:19) (cid:18) D (0) D (0) (cid:19) ∆ T ¯ T → − π k B ¯ T Γ D (0) D (0) ∆ T ¯ T . (S16)By using this into Eq. (S14), we eventually can write S I (cid:12)(cid:12) I =0 = π (cid:18) k B ¯ T Γ (cid:19) D (0) D (0) " F (0) ∆ T ¯ T + π (cid:18) F (0)6 + F (0) D (0) D (0) k B ¯ T Γ (cid:19) (cid:18) ∆ T ¯ T (cid:19) ,S I (cid:12)(cid:12) I =0 = π (6 − π )9 (cid:18) k B ¯ T Γ (cid:19) D (0) D (0) ∆ T ¯ T , (S17)from which we conclude that the zero-current charge shot noise S I sh | I =0 behaves as ∆ T / ¯ T for small temperaturebiases. This is only true when D (0) = 0.Finally, a similar calculation for the thermal noise shows that, provided D (0) = 0, the leading order approximationin ∆ T / ¯ T is independent of the thermovoltage and reads S I th (cid:12)(cid:12) I =0 = 4 e h k B ¯ T D (0) ( π D (0) D (0) (cid:18) k B ¯ T Γ (cid:19) + ∆ T T " π D (0) D (0) (cid:18) k B ¯ T Γ (cid:19) . (S18)Together with the previous expressions for the shot noise contribution, this result shows that the thermal noise is thedominant term in the small temperature bias regime ∆ T (cid:28) ¯ T . S2. HEAT NOISE FOR A GENERAL TRANSMISSIONA. Large temperature gradient
Following the lines of the calculation in Sec. S1 A, it is possible to obtain an approximate expression for the heatnoise too. The calculation being very similar to the one presented above, here we simply report the results. Weassume again the conditions k B T R (cid:28) k B T L ≡ k B T (cid:28) Γ and we consider the heat noise in the left reservoir. The heatshot noise can be expressed as S J sh = 4 h ( k B T ) ( A ( x ) F (0) + (cid:18) k B T Γ (cid:19) B ( x ) F (0) + (cid:18) k B T Γ (cid:19) C ( x ) F (0) + O "(cid:18) k B T Γ (cid:19) , (S19)where x = ∆ µ/ ( k B T ) and A ( x ) = − π + x x ln(1 + e x ) + 4 x Li ( − e x ) − ( − e x ) , (S20) B ( x ) = x − π − x ln(1 + e x ) − x Li ( − e x ) + 12 x Li ( − e x ) − ( − e x ) , (S21) C ( x ) = − π + 3 x
30 + x ln(1 + e x ) + 4 x Li ( − e x ) − x Li ( − e x ) + 24 x Li ( − e x ) − ( − e x ) . (S22)Similarly, for the thermal heat noise the result is S J th = 4 h ( k B T ) π " D (0) + (cid:18) k B T Γ (cid:19) π D (0) . (S23)By imposing that the heat current J L in the left reservoir be zero, one finds the value of the (dimensionless) heatthermovoltage x J = ∆ µ J / ( k B T ). Keeping terms up to the first order in ( k B T /
Γ), we find a cubic equation for x J ,with the three real solutions ( k = 0 , , x ( k ) J = Γ k B T D (0)2 D (0) + Γ k B T (cid:12)(cid:12)(cid:12)(cid:12) D (0) D (0) (cid:12)(cid:12)(cid:12)(cid:12) cos
13 arccos ( sgn( D (0)) " − π (cid:18) k B T Γ (cid:19) (cid:18) D (0) D (0) (cid:19) − πk ! . (S24)Only two of them are physically relevant, namely x (1) J and x (2) J when D (0) > x (0) J and x (1) J when D (0) < k B T (cid:28) Γ, the previous equation may be simplified, yielding x J = ± π √ (cid:20) ± π √ k B T L Γ D (0) D (0) (cid:21) . (S25)The leading order approximation to the ratio between thermal and shot noise then reads R J = S J sh S J th (cid:12)(cid:12)(cid:12)(cid:12) J =0 ≈ π [1 − D (0)] A ( x J ) , (S26)which is the result in the main text.For a constant transmission D ( E ) = D , all the derivatives vanish and therefore the first terms in Eqs. (S19) and(S23) give the exact results for the heat noise (shot and thermal, respectively) at arbitrary voltage. Moreover, thesolutions for the heat thermovoltage are x J = ± π/ √
3, which can be obtained from Eq. (S24) or (S25) by taking thelimit D (0) →
0. As for the charge noise, one finally finds the exact expression shown in the main text. R J | D ( E )= D = (1 − D ) 3 π A (cid:18) π √ (cid:19) = R J (1 − D ) ≈ . − D ) . (S27) B. Small temperature gradient
Using the same notations (and under the same assumptions) as in Sec. S1 B, the heat shot noise for small temperaturegradient can be written as (up to first order in ∆
T / ¯ T and k B ¯ T / Γ) S J sh = 4 h ( k B ¯ T ) (cid:20) S J + S J (cid:18) ∆ T ¯ T (cid:19)(cid:21) , (S28)with S J = F (0) (cid:20) y (cid:0) y + π (cid:1) coth y − y − π (cid:21) + k B ¯ T Γ F (0) (cid:20) y (cid:0) y + π (cid:1) − y (cid:0) y + 2 π (cid:1) coth y (cid:21) ,S J = F (0) (cid:20) y − π y coth y y (cid:18) y π (cid:19) csch y (cid:21) + k B ¯ T Γ F (0) (cid:20) π (cid:18) y π (cid:19) coth y − y (cid:0) π + y (cid:1) − (cid:18) y
80 + 5 π y
24 + 7 π y (cid:19) csch y (cid:21) . (S29)In the case of a constant transmission F (0) vanishes and the remaining terms provide exact results for the heat shotnoise in this regime ∆ T (cid:28) ¯ T .The previous expressions are valid for an arbitrary bias y = ∆ µ/ ( k B ¯ T ), with µ L = 0 and µ R = − ∆ µ . As a nextstep, we find the heat thermovoltage ∆ µ J such that the heat current J L vanishes. For a constant transmission, thisis obtained exactly as ∆ µ J = ± k B π √ q T − T = ± k B ¯ T π √ "r T ¯ T + O (cid:18) ∆ T ¯ T (cid:19) , (S30)and thus the leading order approximation for the zero-current heat shot noise reads S J sh (cid:12)(cid:12)(cid:12)(cid:12) J =0 D ( E )= D = 4 h ( k B ¯ T ) D (1 − D ) π ( π − T ¯ T + O "(cid:18) ∆ T ¯ T (cid:19) . (S31)Figure S2: Noise ratios R I and R J at arbitrary temperatures for a constant transmission function D ( E ) = D .These plots are obtained with a numerical evaluation of the second integral in Eq. (2) of the main text, with avoltage bias such that the charge or the heat current vanishes. The dashed lines show the position of the upperbounds.For a generic transmission function we find instead∆ µ J = ± k B ¯ T π √ r T ¯ T " ± π √ r T ¯ T D (0) D (0) k B ¯ T Γ , (S32)which leads to a correction of order (∆ T / ¯ T ) / : S J sh (cid:12)(cid:12) J =0 = 4 h ( k B ¯ T ) " F (0) π ( π − T ¯ T ± K (cid:18) ∆ T ¯ T (cid:19) / + O "(cid:18) ∆ T ¯ T (cid:19) , (S33)with K = k B ¯ T Γ √ π (cid:20) F (0) 2( π − D (0) D (0) + F (0) π − (cid:21) . (S34) S3. NOISE RATIO AT ARBITRARY TEMPERATURES FOR CONSTANT TRANSMISSION
In this section, we finally show that the exact results in Eqs. (S10) and (S27) are actually upper bounds for thecharge and heat noise ratios R I and R J , when the transmission probability does not depend on energy, namely D ( E ) = D . The validity of this statement is shown in Fig. S2, where we show the ratios R I and R J for an arbitrarychoice of the temperatures T R and T L . We clearly see that the R I approaches the upper limit (1 − D )(2 ln 2 − T R (cid:28) T L , as considered in the main text, or T R (cid:29) T L . For all intermediate temperatures, the R I isalways smaller than this value and it vanishes when T R = T L because the zero-current shot noise is zero under thiscondition.Concerning the heat noise ratio R J , we observe the same behavior: it approaches the upper bound R J when T R (cid:28) T L and it decreases when T R increases. As a difference compared to the charge noise, notice that the plot for R J is only shown for T R ≤ T L . This is because the heat current into the left reservoir never vanishes when T R > T L . S4. CHARGE AND HEAT CURRENTS FOR A LORENTZIAN TRANSMISSION
In this section, we present analytical expressions for charge and heat currents for a conductor with an aribitrarilysharp Lorentzian-shaped transmission probability. Starting from Eqs. (1) and (3) in the main text, we find for thecharge current at any temperature and voltage bias I = − eD Γ h Im X α =R , L ϑ α (Ψ α ( z ) + 2 πif α ( − iz )) . (S35)Figure S3: Zero-current charge thermal and shot noise, S I th and S I sh , as a function of (cid:15) and for different values ofthe Lorentzian width Γ. The plots refer to the large temperature bias scenario, T R /T L = 0. Noises are normalized tothe value they take for a constant transmission, S I th | D ( E )= D and S I sh | D ( E )= D . The plots show that the reduction at (cid:15) = 0 is more relevant for the thermal contribution and this leads to an increased ratio R I exceeding theconstant-transmission bound in Eq. (S10).Here, Ψ α ( x ) = ψ ( + β α ( z − iµ α )2 π ) − ψ ( − β α ( z − iµ α ) ∗ π ), with ψ the digamma function, ϑ R / L = ∓
1, and z = Γ + i(cid:15) .For the heat current in contact α = L , R, we find for arbitrary ∆ T and ∆ µ , J L = D Γ2 h (cid:20)
2Γ ln T L T R + Re X α =R , L zϑ α ( ˜Ψ α ( z ) − πif α ( − iz )) (cid:21) (S36)with ˜Ψ α ( z ) = ψ ( + β α ( z − iµ α )2 π ) + ψ ( − β α ( z − iµ α )2 π ).These analytic expressions are the ideal starting point for the evaluation of the charge and heat thermovoltages,since they substantially simplify the finding of numerical solutions for I ≡ J L ≡ R I when (cid:15) →
0. In this case, due toparticle-hole symmetry, the thermovoltage is ∆ µ I = 0 and therefore only a temperature bias is present, leading toa purely delta- T noise, as in the case of a constant transmission. Still, we have shown in the main text that theenergy-dependence of D ( E ) results in a ratio R I exceeding the bound in Eq. (S10). In Fig. S3 we show that thishappens as, while both the thermal and shot contributions are reduced compared to the constant transmission case,the reduction is more pronounced for S I th , resulting in an increased ratio R I when (cid:15) →
0, as observed in Fig. 2(c) inthe main text. To show this effect more clearly, in Fig. S3 we normalized each noise contribution to the value it takesat constant transmission, namely S I th | D ( E )= D = 4 e D / ( hβ L ) and S I sh | D ( E )) D = 4 e D (1 − D ) / ( hβ L )(2 ln 2 − S5. ADDITIONAL PLOTS FOR NOISE RATIOS WITH A LORENTZIAN TRANSMISSION
Next, we consider the same particle-hole symmetric situation in the case of the heat noise. Here, a similarlyincreased ratio R J was observed [Fig. 3(c) in the main text], but its behavior as (cid:15) → R I . This is because, while both S I th and S I sh are monotonically increasing when (cid:15) →
0, their heat counterpartsexhibit a maximum at a finite (cid:15) , after which they decrease as (cid:15) →
0. This behavior is shown in Fig. S4 The way S J th and S J sh decrease as (cid:15) → R J are observed when Γ is varied. (a) (b) Figure S4: (a) Zero-current thermal and shot heat noise, S J th | J =0 and S J sh | J =0 , as functions of (cid:15) for different valuesof Γ. The two types of noise are normalized to the value they take at constant transmission. The thermal heat noisedecreases more rapidly compared to the shot heat noise when Γ is decreased. This results in regions where the heatnoise ratio is greater than in the case of constant transmission. (b) Heat noise ratio as a function of (cid:15) , normalizedto the value at constant transmission. The region 10 − < β L (cid:15) < − shows a non-trivial behaviour, which is notvisible with a linear scale for (cid:15) . [1] Samuel Larocque, Edouard Pinsolle, Christian Lupien, and Bertrand Reulet, “Shot Noise of a Temperature-Biased TunnelJunction,” Phys. Rev. Lett. , 106801 (2020).[2] Ofir Shein Lumbroso, Lena Simine, Abraham Nitzan, Dvira Segal, and Oren Tal, “Electronic noise due to temperaturedifferences in atomic-scale junctions,” Nature562