SUPPLEMENTARY FIGURES

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1 1 SUPPLEMENTARY FIGURES Supplementary Figure 1: Schematic representation of the experimental set up. The PC of the hot line being biased, the temperature raises. The temperature is extracted from noise measurements. Shot noise measurements are done by converting the current fluctuations into the voltage fluctuations (noted δv 1 and δv ) across a RLC circuit, cooled at 300 mk. The harvested current of the cold line is sent into two 5 kω resistors and converted into voltage before being amplified.

2 (a) (b) D c,1 D c, U 1 C C D h,1 D h, U Supplementary Figure : Comparison between our device and the model proposed by Sothmann et al. []. (a) Schematic representation of the experimental set up described in Ref.[]. A hot cavity is coupled to a cold line via a coupling capacitance C. (b) Scanning electron microscope view of the sample used in our experiment. The lower line defines the hot cavity. The temperature is raised by Joule heating. The harvested current is measured in the cold line. The two lines are coupled by the coupling capacitance C C.

3 3 PC Contact T f = R PCI T e = R PCI Contact T f 0 x Supplementary Figure 3: Schematic representation of the heat dissipated by one PC. This configuration corresponds to the case where the PC of the hot line is kept open and does not play any role. The injected power by the PC on each side of the mesa is equal to. We note T e the electronic temperature under the PC and T f the temperature of the contacts assumed to be at fridge temperature. The heat due to mesa resistance and contact resistances is not shown here. The PC is at position 0 in the x-axis. = R PCI

4 4 PC serie PC Det Contact T f = R PC,LI T e,left = R PC,LI = R PC,RI T e,right = R PC,RI Contact T f T 0 Inner ohmic contact T in Supplementary Figure 4: Schematic representation of the heat dissipated by two series PC. We note the left electronic temperature of the PC T e,left, the right one T e,right, in between the two PCs T 0 and the inner ohmic contact at T in (assumed to be at fridge temperature).

5 5 SUPPLEMENTARY NOTE Supplementary Note 1: Theoretical Model Semiclassical Limit The experimental work described in this letter is inspired from a theoretical study by Sothmann et al. []. Calculations are performed in the semiclassical limit, which assume that the electronic phase in the cavity is destroyed, but electrons energy is conserved. This assumption can be justified if the number of channels N through each PC is large. In our experiment, N is not large since transmissions are typically tuned between 0 and 1. But the size of the cavity between the two PCs is long enough 0 µm to ensure that coherence is lost after several scattering. Indeed, the chaotic nature of electronic trajectory will mix up the different phases. At low temperature, collisions are dominated by electron-electron scattering and the coherence length l ϕ is of the order of the electron-electron scattering length l e e. In our system, l e e is typically 00 µm [3]. Linear Regime The main approximation performed in [] is a linearization of the kinetic equation with respect to U, the voltage fluctuations induced in the cold cavity. It is interesting to question whether we stay in this linear regime, despite the large relative temperature difference between the two cavities T/T.5. In a first step, the temperature of the hot cavity is elevated based on Joule heating. At high bias, the Wiedemann- Franz law gives T δu. Fluctuations are transmitted to the cold line via C C 1 pf. This small value should ensure us that we remain in a linear regime also in the cold line. A quick evaluation of the amplitude of the voltage fluctuations in the cold line δu cold can be performed. We keep the notation of Fig. 1 in the main text. The output current of the device is simply given by I = D c,1 G 0 δu cold D c, G 0 δu cold, G 0 being the quantum of conductance. We introduce PCs non-linearity: D c,i = D 0 c,i ed c,i δu, And by averaging I = G 0 (D 0 c,1 D 0 c,)δu eg 0 (D c,1 D c,)δu, < I >= eg 0 (D c, D c,1) < δu >. The experiment gives < I >= 1 pa (Figure 4 in main text, for D c,1 = 0.5 and D c, = 1). We find < δu > =.4 µv which is small as compared to k B T/e ( 6 µv at 300 mk) and the linear approximation is still valid for the experiment. Model for PCs Following Büttiker saddle-point theory [1], transmission D n of the PC can be written: 1 D n (V g) = ( ), 1 + exp π e(vg V0,n) V g,n where V g,n is related to the negative curvature of the saddle point potential. We also introduce = ɛ/ V g the level-arm factor. We extract by measuring a differential conductance map of each PC versus gate voltage and source-drain bias. We find 0.0e. Therefore D = D/ ɛ can be evaluated:. D = 1 π e D n (1 D n ) V g,n n

6 6 Notations We keep the notations used in Ref. []. The device described in Ref. [] is similar to our sample, with the minor difference that in Ref. [] the hot cavity is connected to the thermal bath via only one constriction whereas we have two constrictions connecting our cavity to the lead. These two constrictions are necessary since the warm up is based on Joule heating, and a current has to be transmitted from one lead to the other. As described in the main text, the heat to current conversion is controlled by the rectification parameter Λ. The rectified current is then calculated using the two quantities introduced in Ref. [], C eff and G eff, defined as followed: C eff = C Σ(C C + C Σ )(CΣ + C CC Σ C C C µ ) CC C, µ G eff = G c,σg h,σ G c,σ + G h,σ. C µ is the cavities mesoscopic capacitance, defined as Cµ 1 = (e ν F ) 1 + C 1 Σ, with ν F the density of states at the Fermi level. C Σ is given by C Σ = C 1,Σ + C,Σ where C 1,Σ is the total capacitance of the upper cavity and C,Σ of the lower one. G 1,Σ = r G 1,r, G,Σ = r G text,r and G Σ = G 1,Σ + C,Σ. The parameter C eff is not monotonic with C C and thus does not describe the coupling between the two cavities. Indeed it grows as C C. Instead, it shows a minimum for a particular value of C C (other parasitic capacitances being fixed), which is the optimal value for the heat engine. The calculations performed in Ref. [] give the output current of the device and its maximal efficiency η max as a function of Λ and τ RC = C eff /G eff : I = Λ τ RC k B ( T ) η max = with T the temperature difference between the two cavities. Λ 4G 1 τ RC k B ( T ) On the number of channel in the PC Theoretical results in Ref. [] are discussed for a large number of channels since they consider the semiclassical limit. Nevertheless it is shown that the device is a better heat to electrical power converter for a small number of channels. Indeed the maximal power scales inversely with the number of channel, and maximal efficiency decreases with the number of channels squared. In the experimental work presented in this letter, we chose to work with a minimum number of channels (one or less) in order to maximize the device efficiency. Some other theoretical works have consider single-channel systems as thermoelectric devices. In Ref. [4], energy harvesting is discussed with a slightly different device. The cold part is a quantum dot with large charging effects set in the Coulomb blockade regime. It would correspond to our device with a much smaller cavity in order to have charging energy E C much greater than thermal excitation k B T. Reducing the size of the cavity reduces the total capacitance of the cavity C Σ (including capacitance of the gates and leads, and self-capacitance to the ground), which leads to an increase of the charging energy E C = e C Σ. Such a device can be operated in principle as close to Carnot efficiency as possible, that is to say with an efficiency η = 1 Tc T h (T c temperature of the cold cavity and T h of the hot one). Such a good efficiency can only be reached when current delivered by the device is small, which is not an asset for a practical realization of a heat-to-current engine. Supplementary Note : Heating effect single PC configuration At low temperature, electrons in the two dimensional electron gas are hardly thermalized by the phonons since typical sample lengths are very small compared to electron-phonon temperature relaxation length. A temperature

7 gradient between the PC and the contacts (assumed to be thermalized at the base temperature of the fridge T fridge ) will therefore appear. Combining the Wiedemann-Franz law and the Joule heating, the problem can be exactly solved. We note T (x) the electronic temperature at position x. The injected power by the PC on each side of the mesa is equal to RPCI (x = 0)= (see Fig. 3). The heat equation can be written: and T (x) j (x) = κ x (x) x = ρ mesa I () with κ the thermal conductivity of the mesa, R PC the resistance of the PC and ρ mesa the linear resistance of the mesa. The Wiedemann-Franz law enables us to relate κ, the conductivity of the mesa σ and T : κ σt = π 3 [k B e ] (3) We introduce T e the electronic temperature at the PC. Integrating over the length of the right mesa we get: T e T f = 4 G PC R m π 7 (1) VDS[1 + R mg PC ] (4) where R m is the total resistance of the mesa + contact resistance. We assume the contact resistance to be the same on the right and left of the PC. Therefore the total conductance of the right and left mesa + contact in parallel is equal to G m = 4/R m. We finally obtain: Since G PC G m, we have: with L = π k B 3e. T e = T f + 4 π G PC T e = T f G m [1 + G PC ]( ev DS ) (5) G m k B + G PC VDS (6) L G m Series PC configuration We now discuss the electron heating in our experiment, where two PCs are in series with in between an inner ohmic contact. As depicted in Fig. 4, because of the series PC, we introduce the left electronic temperature of the PC noted T e,left, the right one T e,right, in between the two PCs T 0 and the inner ohmic contact at T f. A similar approach to the one described in the single PC configuration gives: L (Te,right T f ) = G PC,RVDS (R m,r ) with G PC,R the conductance of the detector PC and R m,r the resistance of the right mesa+contact resistance. For T e,left : L (Te,left T f ) = G PC,LVDS (R m,l ) with G PC,L the conductance of the serie PC (tuned on a plateau)and R m,l the resistance of the left mesa+contact resistance. For T 0 the temperature in between the PCs, we write: L (T0 Tin ) = G PC,LVR + G PC,RVL (R c,in ) with R c,in the contact resistance of the inner ohmic contact. The electronic temperature of the left PC is given by the average of the left temperature T e,left and the temperature in between the two PCs T 0, (T e,left +T 0 )/. From our measurements we extract R m,l =R m,r = 400 Ω and R c,in = 1500 Ω. The inner ohmic contact resistance being much smaller this difference is not surprising. (7) (8) (9)

8 8 Supplementary Note 3: Energy harvesting efficiency Figure 3 in main text gives the output current of the device when harvesting energy. For G c,1 = G 0, G c, = 0.5 G 0 and V ds = 1.8 mv the device delivers pa in a 5 kω load, thus producing W = fw of electrical power. Ref. [] gives a theoretical frame to evaluate the total energy transferred from the hot cavity to the cold one, called, which yield to a total transmitted power of 0.04 pw for the previous working point. The efficiency which is the ratio between these two values η = W/J H is equal to η = According to Ref. [], the maximum efficiency scales with Λ /τ RC. While the rectification parameter Λ depends on the energy dependence of the transmission and therefore is subject to the randomness of PCs characteristics, modifying the design of the sample to decrease τ RC is more straightforward and reproducible on a large scale. The ideal situation is to lower the parasitic capacitances of the system as much as possible (self-capacitances and couplings to the leads), and to choose the optimal coupling capacitance C C which minimizes the parameter C eff. In our experiment, the self-capacitance could be reduced by decreasing the width of the mesa and the size of the cavity, which requires a precise control over fabrication techniques. J H = kb T τ RC SUPPLEMENTARY METHODS Sample Emitter and detector lines are two-dimensional electron gas (DEG) with cm V 1 s 1 mobility and cm density. The DEG is buried 100 nm under the surface of a GaAs/Ga(Al)As heterojunction and patterned using wet-etching and lithography techniques. Low resistance AuNiGe ohmic contacts connect both lines to the external circuit. On the middle of each line, two set of split gates are designed to create a cavity. Applying negative gate voltages deplete the DEG and form two PCs separating the cavity to its leads. Each cavity is electrically connected to one side of a metallic capacitor through an inner ohmic contact. The capacitor is made of 100 nm thick Ti/Au evaporated layer, patterned on top of the substrate by an electron beam lithography technique. Current and current noise measurements The experimental set up is depicted in Fig. 1. The PC of the hot line being biased, the temperature raises. The temperature is extracted from noise measurements. Shot noise measurements are done by converting the current fluctuations into the voltage fluctuations (noted δv 1 and δv ) across a RLC circuit, cooled at 300 mk using 3 MHz resonant frequency and 300 khz typical bandwidth. Home made cryogenic amplifiers, with ultra low input voltage noise (0. nv/ Hz) and located on the 3 K stage amplify the voltage fluctuations. Using fast acquisition card and Fast Fourier Transform, the voltage noise cross-correlation δv 1 δv is computed in real time. Then δv 1 δv is converted into a current shot noise S I. The harvested current of the cold line is sent into two 5 kω resistors and converted into voltage before being amplified. SUPPLEMENTARY REFERENCES [1] Büttiker, M. uantized transmission of a saddle-point constriction Phys. Rev. B, , [] Sothmann, B., Sánchez, R., Jordan, A.N. and Büttiker, M. Rectification of thermal fluctuations in a chaotic cavity heat engine Phys. Rev. B, , 01. [3] Yacoby, A., Sivan, U., Umbach, C. P. and Hong, J. M., Interference and dephasing by electron-electron interaction on length scales shorter than the elastic mean free path Phys. Rev. Lett., , [4] Sánchez, R. and Büttiker, M., Optimal energy quanta to current conversion Phys. Rev. B, , 011.

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