Delft University of Technology

Single-electron thermal devices coupled to a mesoscopic gate

Sánchez, Rafael; Thierschmann, Holger; Molenkamp, Laurens W.

DOI10.1088/1367-2630/aa8b94Publication date2017Document VersionFinal published versionPublished inNew Journal of Physics

Citation (APA)Sánchez, R., Thierschmann, H., & Molenkamp, L. W. (2017). Single-electron thermal devices coupled to amesoscopic gate. New Journal of Physics, 19(11), [113040]. https://doi.org/10.1088/1367-2630/aa8b94

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New J. Phys. 19 (2017) 113040 https://doi.org/10.1088/1367-2630/aa8b94

PAPER

Single-electron thermal devices coupled to amesoscopic gate

Rafael Sánchez1 , Holger Thierschmann2 and LaurensWMolenkamp3

1 InstitutoGregorioMillán,UniversidadCarlos III deMadrid, E-28911 Leganés,Madrid, Spain2 Kavli Institute ofNanoscience, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628CJDelft, The

Netherlands3 Experimentelle Physik 3, Physikalisches Institut, UniversitätWürzburg, AmHubland,D-97074Würzburg, Germany

Keywords: quantumdot, heat currents, thermal devices, single-electron tunneling

AbstractWe theoretically investigate the propagation of heat currents in a three-terminal quantumdot engine.Electron–electron interactions introduce state-dependent processes which can be resolved by energy-dependent tunneling rates.We identify the relevant transitionswhich define the operation of thesystem as a thermal transistor or a thermal diode. In the former case, thermal-induced chargefluctuations in the gate dotmodify the thermal currents in the conductor with suppressed heatinjection, resulting in huge amplification factors and the possible gatingwith arbitrarily low energycost. In the latter case, enhanced correlations of the state-selective tunneling transitions redistributeheatflows giving high rectification coefficients and the unexpected cooling of one conductor terminalby heating the other one.We propose quantumdot arrays as a possible way to achieve the extremetunneling asymmetries required for the different operations.

1. Introduction

The control of heatflows in electronic conductors is one of the present day technological challenges. Besides theconversion of heat currents into electrical powerwhich is themain focus of thermoelectrics, there is thepossibility ofmaking all-thermal circuits that work onlywith heat currents and temperature gradients. For thispurpose, all-thermal versions of electronic devices such as diodes and transistors need to be operative.

Mesoscopic conductors are good candidates [1, 2]: their energy spectrum can be easily designed by using lowdimensional structures showing quantum confinment (quantumdots, quantumwires...). Also, the presence ofdifferent gaps due to interaction effects (e.g. Coulomb blockade) or superconductor interfaces can be controlled.This possibility has been evident in the last years with the proposal and experimental realization of heat engines[3–17] and electronic refrigerators [1, 18–24] based on nanoscale systems, andwith the detection of heatcurrents [18, 25–29]. Thermal rectifiers and transistors have also been proposed [30–43] and experimentallyimplemented [44–47].

A crucial aspect for an electronic thermal device is how it interacts with its environment. Inelastic transitionsin the device can be due to the coupling tofluctuations in the electromagnetic environment [43, 48, 49], or tophononic [50, 51] ormagnetic [52] baths. Remarkably inmesoscopic conductors, the dominant interaction canbe engineered by introducing additional components whichmediate the coupling to the environment, e.g. aquantumpoint contact [53], quantumdots [11, 12, 54], or photonic cavities [55–57]. This allows one to definedifferent interfaces for the different operations. From a theoretical point of view, this permits the environmentto be described as a third terminal, and the auxiliary system (mediating the system-environment coupling) to betreated in equal footing as the conductor.

Here we propose amulti-terminal systemof two capacitively coupled quantumdots as a versatileconfiguration for the efficientmanipulation of electronic heat currents, as sketched infigure 1. It works as abipartite system: one of them is coupled to two terminals and is considered as the conductor whose heat currentsare to bemanipulated. The other dot is tunnel-coupled to a third terminal and serves as a gate system. Energyexchange between the two systems ismediated by electron–electron interactions [58–62]. Effects such asmesoscopic Coulomb drag [63–66], enhanced correlations [67–70], heat engines [11–14, 71–74], rectification of

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noise [54, 75], non-local power generation [76], relaxation time scales [77], or the relation of information andthermodynamics [78–80] have been investigated in similar configurations. Thermoelectric effects in threeterminal configurations have been describedwhere the coupling of a junction to the heat source ismediated byother kinds of interactions [15, 48–53, 57].

Energy exchangemediatedby electrostatic coupling has the additional advantageof allowing for definingthermal insulating system-gate interfaces. Thisway eventual heat currents leaking from the gate to the conductor,e.g. due to phonons, are suppressed.Also, phononicheat currents along the conductor are not affected by chargefluctuations in the gate andhence arenot expected to affect the thermal gating effects. Furthermore, in the lowtemperature regimediscussed here, the contribution of phonons in the conductor can be neglected.

In a recent experiment [81], the gating of voltage-induced electronic currents in the conductor by themodulation of the temperature of the gate was observed.We extend the investigation of such thermal gating tothe gating of thermally-induced heat currents. Afirst stepwas done in [82]where the gate dot acts as a switch. Aswe showhere, exploiting the energy dependence of tunneling effects in the conductor dot, the same devicebecomes a versatile thermal device.We connect the different operations of the device to different asymmetries.Energy-filtered couplings lead to a thermal transistor (section 4). Combined energy-dependent and left–rightasymmetries give a thermal diode aswell as the effect of cooling by heating (section 5).We discuss possibleimplementations of all these asymmetries and how to tune themwith quantumdot arrays in section 6.

In the remaining of the text, section 2 describes the theoreticalmodel and the heat currents and section 3introduces the thermal gating effect. Conclusions are discussed in section 7.

2.Heat currents

Ourmodel is based on the interaction of twomesoscopic systems: the conductor system and the thermal gate.For capacitively coupled quantumdots, the interaction is given in terms of the electrostatic repulsion ofelectrons occupying them. For simplicity, wewill assume that it is very large for electrons in the same dot, sowerestrict our considerations to single-electron occupations in each quantumdot. Interdot Coulomb interaction isdescribed by a constant EC given by the geometric capacitances of the system [11].

Through all this work, we consider non-equilibrium situations only due to temperature gradientsT T Ti iD = - applied to one terminal with respect to the reference temperature,T.We do not restrict to the

linear regimewhere gradients are small. No bias voltage is applied. Index l L, R= will always refer to theconductor terminals, while i labels any terminal. Gwill denote the gate. The internal potentialUim of eachquantumdot i is then shifted by ECwhen the charge of the other dotm fluctuates:U U Ei i C1 0= + . Hence everyenergy exchange between the system and the gate is given in terms of this quantity.

In an experimental realization, the internal potentials are tuned bymeans of gate voltages whichwe includewith their corresponding lever arms ia and U eV eV: S0 S S S Gb e a b= + + , andU eV eVG0 G S G Se a b= + + .This way, the charge configuration of the system can be externallymanipulated bymovingUimwith respect tothe Fermi energy of the leads, im . This is shown infigure 1: each region of this stability diagram is dominated byone of the occupations (N, n), withN, n=0, 1, whereN represents the occupation of the system and n denotesthe occuppation of the gate. Charge fluctuates around the intermediate regions.

Figure 1. Left: a quantumdot thermal gate. The heatflows in a two terminal electronic conductor aremodulated by the temperatureTG of a third reservoir. The coupling ismediated by theCoulomb interaction at two quantumdots. Energy-dependent tunneling ratesare affected by the charging of the other dot. Right: charge stability diagramof the system as the quantumdot gate voltages are tuned inthe region (N, n)with N n, =0, 1.We have considered 0.1ia = and 0.02b = .

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New J. Phys. 19 (2017) 113040 R Sánchez et al

Two triple points appearwhere charge fluctuations are present in both dots.We call the region betweenthem the stability vertex, whose size is given byEC. Thermoelectric heat engines based on the correlation of thesystem and the gate operate in this region [11, 12, 80].

Wewrite a rate equation for the occupation of the charge states N n,l = ( ):

P P P , 1ål l l= G ¢ - Gl

l l l l¢

¬ ¢ ¢¬˙ ( ) [ ( ) ( )] ( )

with the tunneling rates n n l ln1, 0,G = å G¬+

( ) ( ) (with l L, R= ), and N N N,1 ,0 GG = G¬+

( ) ( ) . For the reversedprocesses, we replace+-. They arewritten as f Uim im i imG = G D ( ), where the transparencies of the barrier

imG depend on the occupationm of the other dot, f E 1 ei

E k T 1iB= ++ -( ) [ ] is the Fermi function, andf E f E1i i

= -- -( ) ( ).We have defined U Uim im imD = - .We assumeweak dot-lead couplings, k Tim BG ,where transport is dominated by sequential tunneling events. A schematic representation of the possibletrajectories is represented infigure 2.

The dc heat currents are obtained by the steady state occupations P l¯ ( ) satisfying: P 0l =¯̇ ( ) . They are givenby [11]:

J U P n P n0, 1, , 2ln

ln ln lnå= D G - G+ -[ ¯ ( ) ¯ ( )] ( )

J U P N P N, 0 , 1 . 3N

N N NG G G Gå= D G - G+ -[ ¯ ( ) ¯ ( )] ( )

They are defined as positive when heatflows out the terminal. From the previous expressions, one canstraightforwardly write the state-resolved currents Jln and J NG such that: J Ji m im= å [71]. Aswe are applying nobias voltage, heat is conserved, and J 0i iå = . In the following, wewrite J Ti jD( ) as the heat current injected fromterminal i in response to a temperature gradient applied to terminal j, with all other terminals being (exceptwhen explicitly stated) at temperatureT.

The discrete level of the quantumdots filters the energy of the tunneling processes. Theflowof heat isrestricted to the regionwhere they are close to the Fermi energy, as shown infigure 3.When the temperaturegradient is applied longitudinally in the conductor (i.e. either in the left or right terminal), the conductor heatcurrents present a double peak around the crossing of the level with the Fermi energy. At that particular point farfrom the stability vertex (such thatfluctuations in the gate are frozen), Jl=0 because of particle-hole symmetry.This is a well-known feature of two-terminal quantumdots [83]. In our system, such a signal is broken at thestability vertex as a consequence of the charging of the gate dot.

Themechanism for heat transport in the gate is different. It relies on chargefluctuations in the twodotswhichinvolve the four charging states in a closed loop trajectory [11]. In a sequenceof the form0, 0 1, 0 1, 1 0, 1 0, 0« « « «( ) ( ) ( ) ( ) ( ), the events of tunneling in andout of the gate dot occur at differentenergies due to thedifferent occupations of the conductor dot. The energydifference, EC , is transferred betweenthe two-terminal conductor and the gate. The sign of the transfer depends on thedirectionof the above sequence.Hence J TjG D( ) showsup as a spot in themiddle of the stability vertex [80], as shown infigure 3.

In agreement with theClausius statement of the second law, heatflows out of the hot reservoirs, as shown inthe diagonal panels, starting from the top left, infigure 3.We note that in the symmetric configuration shown

Figure 2.Tunneling processes involved in the dynamics of the system. The rates imG for tunneling in (+) or out (−) of each quantumdot depend on the occupation of the other dot,m.

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New J. Phys. 19 (2017) 113040 R Sánchez et al

there, the absorption of this heat current is shared by the corresponding other (cold) terminals, where currentsare negative. Aswe discuss below, this is not necessarily the case in the presence of energy-dependent rates.

3. Thermal gating

The effect of the gate dot on the heatflow in the conductor is twofold: it exchanges heat with the conductor andaffects the heatflowwithin the conductor. The former onemodifies energy conservation. The second is due tothe shift in the position of the energy level of the conductor when the occupation of the gate dot changes. Thisbreaks the double peak structure shown infigure 3. The effect is evident in the stability vertexwhere transportfluctuations in the conductor coincide with charge fluctuations in the gate dot [70]. These occur at a higher ratewhen the gate reservoir is hot.Hence tuning the temperatureTG modifies the transport properties of theconductor.We call this the thermal gating effect.

In the electric response, itmanifests as a clover-leaf structure with positive and negative regions around thestability vertex [81, 84]. The sign can be understood in terms of dynamical channel blockade: it depends onwhether the fluctuations in the gate dot contributes to open or close the relevant transport channels[58, 59, 67, 69, 85].

Here we are interested in the thermal gating of heat currents.We quantify it by defining themodulation:

J T J T T J T, , 0 . 4i j i j i jGD D = D D - D( ) ( ) ( ) ( )

Weplot it infigure 4 for a symmetrical configurationwith lnG = G. For heat currents, we observe a clover-leafstructure centered at each triple point. Interestingly, J 0lD = along the conditions P N P N, 0 , 1=( ) ( ). There,the level of the gate dot is alignedwith the Fermi energy such that charging/uncharging the gate dot isindependent of the lead temperature.

The clover-leaf structures are the effect of thermal gating due tofluctuations in the gate.However, in thecenter of the stability vertexwe also observe signatures of the additional effect of energy exchangewith the gatesystem.Obviously, if we inject heat from a third terminal, the total heat flow in the conductor is affected. Theperformance of our system as an all-thermal device, e.g. a thermal transistor, requires the suppression of suchcontributions. For example, reducing the transparency NGG reduces JG, leaving the clover leaf structure almostunaffected [84]. This is an indication that based onfluctuations of the gate only one can have a thermal gatingeffect, in principle involving no heat injection into the conductor.

Figure 3.Heat currents in response to the different temperature gradients as functions of the quantumdot gate voltages. In panel (a)wemark the limits of the charge stability regions.Heat current in the gate dot is onlyfinite close to the two triple points. As a guide tothe eye, panel (e) includes a cut of J TR RD( ) at V 0.5G = - mVshowing the double peak structure. Parameters:EC=90meV,k T 0.243B = K, T k T 2i BD = , 10lNG = μeV, 1nGG = μeV.

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New J. Phys. 19 (2017) 113040 R Sánchez et al

4. Energyfiltering: a thermal transistor

An all-thermal transistormodifies the heat current in a conductor due to changes in the temperature of a third(gate) terminal. This is the thermal gating effect discussed in the previous section. In this case, the two conductorterminals act as the emitter and the collector, while the gate terminal is considered as the base. As for an electrictransistor, we can define the thermal amplification factor as [39, 42]:

J T

J T, 5lm

l m

mG

a =D DD D∣ ( )∣∣ ( )∣

( )

where J TmGD D( ) is the change of heat injected from the gate when tuningTG. For the system towork as athermal transistor, one needs 1.lma

The challenge is to achieve ameasurable thermal gating by injecting a small amount of heat from the gate. Inour system, heat is injected only via electron–electron interactions andwe can therefore control it by selectingthe charge of the system. As discussed above, themechanism for heat transfer in our system is based on theoccurance of charge fluctuations involving all the four charging states in a loop, as presented infigure 2. In thissectionwe discuss how to avoid such trajectories.Wewill see that this is themost effective way to suppress theenergy exchange between the system and the gate, while still having a gating effect. This can be achieved either byoperating at low tempreature, kT EC (i.e. eliminating one of the states), or by acting on the energy-resolvedtunneling rates (eliminating one of the transitions) [82]. Time-resolvedmeasurement offluctuations in theformer configuration have been recently reported in single-electron transistors [86].

We focus here on the second case, where the tunneling rates in the conductor are strongly energy dependent,for instance l l0 1G G . This allows us to consider the regime k T ECB ~ . An energydependence in tunneling rates isexpected to occurnaturally as in quantummechanics the tunneling probability depends exponentially on theenergy of the electronwith respect to the height of thebarrier. Furthermore, in semiconductor quantumdots, theenergy dependenceof thebarrier transparencies can bemodulated experimentally bymeansof gate voltages [12].Additionally, amore complicated quantumdot composition can beused,where only twodots provide the system-gate interface and theother ones are used as energyfilters.Wediscuss several such geometries in section 6.

In the limit 0l0G = , the transitions (0, 0)↔(1, 0) are avoided.Hence the state 1, 0( ) is only populatedthroughfluctuations of the form (1, 1)↔(1, 0) taking place in the gate dot. Currents in the conductor areconditioned on the occupation of the gate dot.When it is empty, the transport transitions are blocked.We canthenwrite: J Jl l1= and J 0l0 = . The gate dot becomes a switch.

In this case, there are no transitions that correlate fluctuations at the two dots. Then, state-dependentcurrents are conserved in each conductor: J 0l0 = , J J 0L1 R1+ = , and J 0NG = [71]. From the latest andequation (3), we get the relations:

P N P N, 0 e , 1 , 6U k TNG B G= D¯ ( ) ¯ ( ) ( )

emphasizing that the gate satisfies detailed balance. Using the level-resolved conservation laws given above, thismodel admits a simple analytical solutionwithout needing to solve themaster equation. The heat currents in theconductor are:

J U f f n T , 7l l l l1 seq1 1 1 G= D G - á ñ( ) ( ) ( )¯

wherewe havewritten l lseq11

11G = å G- - , f f Ul l l1 1= D( ), and l̄ is the opposite terminal to l in the conductor. The

effect of the gate enters only in the average occupation n T P P0, 1 1, 1Gá ñ = +( ) ( ) ( ) given by:

Figure 4.Thermal gating of heat currents when increasing the gate temperature T k T 2G BD = for the same symmetric configurationas infigure 3.Note that J 0l mD D =( ) along the conditions where the fluctuations between (N, 0) and (N, 1) are balanced.

5

New J. Phys. 19 (2017) 113040 R Sánchez et al

n Tf f

f f. 8l l

l l lG

1 G0 G1

1 G0 1 G1

åå

á ñ =G

G + G+ -( )( )

( )

Itmodifies the current expected for an isolated single channel at energy ε:

J f f . 9l l lsc, seqe e e e m e m= G - - -( ) ( )[ ( ) ( )] ( )¯

This effect can be clearly observed infigure 5, where all possible currents in response to a temperaturegradient applied to the different terminals of the conductor are plotted. By comparing it with the symmetric(energy independendent) rates configuration shown infigure 3, we observe that transport is suppressed in thelower region of the stability diagram,where n 0á ñ » as a result of the asymmetric tunneling rates.

We note that no heat current flows from the gate into the conductor. Nevertheless, the thermal gating effectexists (seefigure 6) due to the dynamical blockade of the heat currents:

J T J U n T T n T . 10l m l lsc, 1 G G GD D = D á + D ñ - á ñ( ) ( )[ ( ) ( ) ] ( )

In this case only the clover-leaf around the triple point close to (1, 1) appears. Also, J 0G = for any configuration,as expected, and J T 0G LD D =( ) . As a consequence, the thermal amplification factor diverges (within ourapproximations) and the systemoperates as an ideal transistor. Note that the leading order in an expansion ofthe thermal gating for this case is quadratic in the temperature gradient: J T T Tl j j GD D ~ D D( ) .

4.1. Leakage currentsOf course, in a real setup the transition rates cannot be exactly suppressed. In that case, heat leaks from the gateand the amplification coefficient becomes finite.We take this into account infigure 6, where LLa is calculated atthe twomaxima of J TL LD D( ) (labeled asA andB infigure 6(a)) forfinite values of l0 0G = G . For 1000 1G < G ,the amplification factor is several orders ofmagnitude larger than 1.Note that in the presence of a leakagecurrent, themaximumatB (which is further from the stability vertex and is hence less affected by the leakage)gives a larger amplification factor, even if J T J TA BL L L LD D > D D( )∣ ( )∣ for 00G = .

Being closer to the relevant triple point where sequential tunneling transitions are dominant,B is alsoexpected to be less sensitive to higher-order tunneling processes (neglected here). These involve energy transferbetween the two systems and hence avoid the divergence also in the ideal case 0l0G = .

4.2. Thermal gatingwithout heatingAswe discuss above, a thermal transitor canworkwithout any injection of heat in the conductor. This does notnecessarilymean that it workswithout any energy cost: in order to increase TG one in principle has to inject heatinto the gate terminal.

In this respect, single-level quantumdots are also beneficial: the state of the dots is highly non-thermal andno temperature of the dot can be defined [76]. The dynamics of the systemdepends on the non-equilibriumchargefluctuations in the quantumdots. The relevant gate temperature is that of the reservoir towhich thequantumdot is coupled.We can then think of a configuration such that the gate dot is tunnel-coupled to two

Figure 5.Heat currents in response to the different temperature gradients as functions of the quantumdot gate voltages in the energy-selected-tunneling configuration, with 0L0 R0G = G = . Currents are hence confined to the regionwith n 1G = . Note that there is noheat injected from the gate dot: J 0G = . Also, the gradient in the gate terminal does not lead to any heat current: J T 0l GD =( ) , sowedo not plot themhere. The rest of the parameters is as in figure 3.

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New J. Phys. 19 (2017) 113040 R Sánchez et al

terminals, one cold and one hot, as depicted infigure 7. By alternatively opening and closing one of the barriers,the chargefluctuations will adapt to the temperature of only one of them. Increasing TGD can then be done at anarbitrarily low energy cost (that of tuning the gates that control the barriers).

Furthermore, the cold and hot reservoirs in the gate can even be in thermal contact with the conductorterminals. This way, only one thermal gradient is needed: the one that drives the heat currents in the conductor.The system is then reduced to two terminals connected in parallel by two interacting quantumdots: onesupports a heat current. The other one is put in contact with only one of the reservoirs at a time. The current thusdepends on towhich side the gate dot is coupled.

5. Combined state-dependent andmirror asymmetric tunneling: thermal rectification

The energy-dependent tunneling rates introduced by the coupling to the gate dotmay also generate left–rightasymmetries in the propagation of heat in the conductor: the energy at which electronswillmore probablytunnel through the two barriers will be different. Transport would then depend onwhere the temperaturegradient is applied, leading to a thermal rectification effect, if J T J TL R R LD ¹ D( ) ( ).

A conductor that exploits this property by allowing the heat toflow for a forward, but not for a backwardgradient is a thermal diode. It is characterized by the thermal rectification coefficient, defined as:

J T J T

J T J T. 11L R R L

L R R L

=D - DD + D

∣ ( ) ( )∣∣ ( )∣ ∣ ( )∣

( )

Figure 6. (a)–(c)Thermal gating of heat currents when increasing the gate temperature T k T 2G BD = for the same configuration asin figure 5. (d)Thermal amplification coefficient as a function of the tunneling rate state-dependence. The two curves correspond tothemaximal thermal gating signal I Tl

LLD D( ) in (a). As the rates l0 0G = G (corresponding to an empty gate dot) vanish, the

amplification increases and the system behaves as an ideal thermal transistor.

Figure 7.Gate thermal transistor. The temperature that defines the gate fluctuations can be controlled by putting the gate quantumdot in contact with either a cold or a hot reservoir. This can be done by just opening or closing the corresponding tunneling barriers.

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New J. Phys. 19 (2017) 113040 R Sánchez et al

An ideal thermal diode operates at 1 » .We emphasize that, due to the presence of the gate terminal, heat isnot conserved in the conductor subsystem and therefore requires two currents to be defined. A symmetricconfiguration as the one considered infigure 3 obviously gives 0 = .

Finite rectification coefficients are obtained for a simple left–right asymmetric configuration, ln lG = G. Thiscase can be understood by considering for example the case L RG G . an electron residing in the left terminalwill easily get into the dot, but once there, it takes a longer time to tunnel through the right barrier. During thistime, it is in contact with the gate reservoir. On the other hand, leftmoving electrons hardly enter the dot, butthen they can rapidly tunnel to the left. Therefore, the time that left- and right-moving electrons expend incontact with the gate reservoir is different, and so is the probablility that they lose heat in it. As a consequence, theheat current arriving to the left terminal when heating the right onewill be larger. The role of the gate is totallypasive in this case, only acting as a heat sink.

A configuration that emphasizes the left–right asymmetric energy-dependent rates suppresses tunnelingprocesses through different barriers at different energies, for instance: L0 R1G = G = G, and xL1 R0G = G = G. Letusfirst consider for simplicity the case x=0. Then transitions n n0, 1,«( ) ( ) only occur through the leftbarrier for n=0 and through the right one, for n=1. In section 6we propose an implementation of thisconfiguration. It also corresponds to an optimal (Carnot efficient) heat engine [11], and provides amaximalcorrelation between the different flows [70]:

J

U

J

U

J

E, 12

C

L

S0

R

S1

G D

= -D

= = ( )

where

e e e e 13L0 R1 G0 G13

Uk T

Uk T

Uk T

ECk T

G0B G

s1B R

s0B L B G

g=

G G G G-

+ + + + D D D( ) ( )

is the particle current. The denominator 3g is determined by the normalization of the occupation probabilities.Note that if only one terminal is hot, 0 ¹ for finiteEC. An electron can only be transported accross theconductor if it exchanges an energyECwith the gate. This property, which is usually known as tight energy-matter coupling, here applies to all heat fluxes in the system.

Infigure 8, the different heat currents are displayed for the temperature gradient applied to the differentterminals. They occur only around the stability vertexwhere energy can be exchangedwith the gate. Eachcurrent is closer to a different triple point, corresponding to the charge configuration at which tunneling is

Figure 8.Heat currents in response to the different temperature gradients as functions of the quantumdot gate voltages in themaximally correlated configuration, with 0L1 R0G = G = . Note that the heat currents JL, JR change signwhen the gradient is applieedto the opposite terminal in the conductor. In that case also, they are centered at different triple points. The rest of the parameters is asin figure 3.

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New J. Phys. 19 (2017) 113040 R Sánchez et al

permitted. In agreementwith equations (12) and (13), when the temperature gradient is applied to one of theconductor terminals, l, the corresponding heat current Jl vanishes (but does not change sign)when U 0s0D =(l=L) orwhen U 0s1D = (l=R).

5.1. Cooling by (really)heatingVery remarkably, the current at theopposite terminal vanishes at twopoints, when U 0nsD = , where it changessign.As a consequence, wefind configurations in the center of the stability vertexwhere heat is extracted from thetwo terminals of the conductor, J J, 0L R > , i.e. byheating one of them, the other one gets cooled.The excess heat isabsorbed by the gate system.Hence only in the strictly hot terminal, our setup satisfies theClausius statement thatheatflows fromhot to cold in the absence ofwork doneonto the system.Wealsofindheatflowing between twocold reservoirs.Differently fromother proposals of cooling byheating [87–89], herewedonot couple to anincoherent source that drives a particular transition, but rather really heat onepart of the systemup.

Similar effects have been found for twofinite systems coupled by energy filters [90]. In our case, this is adynamical effect due to the coupling to the gate, which enables the transported electrons to tunnel at differentenergies depending on the involved lead. Related properties have been described for asymmetric quantumdotsin the dynamical Coulombblockade regime [43], suggesting the possibility to define thermal devices by properlyengineering the coupling to their environment.

5.2. Thermal rectificationLet us concentrate on the thermal rectification. Infigure 8, one can clearly observe an asymmetry of the heatcurrents due to an opposite gradient: J T J Tl l l lD ¹ D( ) ( )¯ ¯ . It is due to the heat currents being largest at differentdot potentials for different terminals. The thermal rectification coefficient will therefore befinite, except for thecase when the system is equidistant from the conditions U 0s0D = and U 0s1D = , as shown infigure 9 forx=0. It increases with the detuning of the conductor level with respect to the Fermi energy, where on the otherhand the currents are exponentially suppresed. Interestingly, increases with the applied temperaturegradient, TD , and gets closer to 1 = for larger gradients.

5.3. Leakage currents: thermal diodeRemarkable deviations from this behavior appear for x 0¹ . In this case, a small current will leak at undesiredenergies at each of the terminals. Thismodifies the conditions at which the heat currents vanish. In particular,

Figure 9.Thermal rectification, for the case with reversed energy-dependent tunneling rates: with xL1 R0G = G = G. (a)–(c)Thermalrectification coefficient as function of the quantumdot gate voltages, for different values of x. The lower panels present cuts along thedashedwhite line: (d)–(e) represent the involved heat currents and (f)–(i) the dependence of the rectification coefficients for differentvalues of the temperature gradient. Regions where 1LR = appear when the heat currents have opposite signs. The rest of theparameters are otherwise as infigure 8.

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J TL RD( ) and J TR LD( ) change sign at different points, seefigure 9 for x=0.1. As a consequence, two smallvoltagewindows openwhen they are opposite in sign, i.e. they have the same direction.Within thesewindows,the systembehaves as an ideal thermal diode, with 1 = . The size of thewindows increases with the appliedtemperature gradient.

For larger leakage currents, the cooling by heating effect disappears, so both J TL RD( ) and J TR LD( ) arenegative, see figure 9 for x=0.5. They are still different, in general, and hence afinite but small rectificactioncoefficient is obtained.

6.Quantumdot arrays for energyfiltering

Weend by analyzing the experimental feasibility of the different configurations discussed above. The energy-dependent rates of systems of two coupled quantumdots can bemodified bymeans of gate voltages [12].However, even if configurations with large enough asymmetries can be found [64], they are very difficult tocontrol.

One can however think ofmore complicated quantumdot arrays. The discrete level of the outermostquantumdots can be used as energy filters which can be tuned externally. The coupling between the conductorand gate subsystems ismediated by only two dots, as sketched infigure 10. This way, tunneling from the differentterminals can be put in resonancewith the required configurations at UimD , as long as thewidth of the filterlevels is smaller than EC [91].

The simplest case, as it only involves one additional barrier, is when the gate contains a double quantumdot.The one connected to the reservoir behaves as a zero-dimensional contact [92]. Its level can be chosen to injectelectrons only either at UG0D or UG1D , thus providing a configurationwith either 0G1G = , or with 0G0G = ,respectively. Theywouldwork as thermal transistors around different triple points in the stability diagram [82].

Controlling tunneling in the conductor (as discussed in this work)would require a triple quantumdot, seefigure 10. Fine tuning of such structures has been achieved in the last decade [93–96]. On the other hand, itpermits for amoreflexible operation of the system: if the twofilters are resonant with the same state, UlnD , therates with n n¢ ¹ will vanish ( 0lnG =¢ ), and the system is a thermal transistor, as discussed in section 4. If nowone of thefilters is put in resonancewith the other state, it works as a thermal diode or a refrigerator, seesection 5.We can therefore change the operation of the device from a transistor to a diode by just changing a gatevoltage.

7. Conclusions

Wehave investigated the properties of electronic heat transport in a systemof capacitively coupled quantumdots in a three-terminal configuration.One of the dots acts as amesoscopic thermal gate which can either serveas a heat source or sink, or control the occupation of the other dot. Heat exchangemediated by electron–electroninteractions introduces different ways to tune the energy-dependence of the relevant tunneling events.

Figure 10.Possible realizations of energy selective tunneling with quantumdot arrays. The coupling between the system and the gateis given by the capacitive coupling of only two dots. The additional dots filter the energy at which they are coupled to the reservoirs.Different operations as a thermal transistor or a thermal diode can be defined this way.

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New J. Phys. 19 (2017) 113040 R Sánchez et al

All-thermal control of heat currents can be achieved by controlling the appropriate system asymmetries. Itallows us to define different operations such as a thermal transistor, a thermal diode, and a refrigerator based onthe interactionwith the gate system.Onone hand, dynamical channel blockade of the conductor heat currentsvia charge fluctuations in the gate enable a thermal transistor with huge amplification factors. On the other hand,the correlation of state-dependent transitions in the three different contacts rectifies the heat currents andinduces the cooling of one terminal by the heating of the other one. These two effects combine in the operationof an ideal thermal diodewith 1 = .

The different configurations can be implemented nowadays, under experimental state of the art. In arrays ofquantumdots, the optimum level of performance can be achieved.Our results consider realistic parameterestimations from the experiment [12].

Ourwork uses amesoscopic structure tomediate the coupling of the conductor with the thermalenvironment.Working as a switch or inducing inelastic transitions, themediator defines the different behaviorsof the system. This strategy opens the possibility to use engineered system-environment couplings to improve ordefine new functionalities.

Acknowledgments

Weacknowledge financial support from the SpanishMinisterio de Economía yCompetitividad via grantsNo.MAT2014-58241-P andNo. FIS2015-74472-JIN (AEI/FEDER/UE), and the EuropeanResearchCouncilAdvancedGrantNo. 339306 (METIQUM).We also thank theCOSTActionMP1209 ‘Thermodynamics in thequantum regime’.

ORCID iDs

Rafael Sánchez https://orcid.org/0000-0002-8810-8811Holger Thierschmann https://orcid.org/0000-0002-6869-2723

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