A normal metal tunnel-junction heat diode

We propose a low-temperature thermal rectifier consisting of a chain of three tunnel-coupled normal metal electrodes. We show that a large heat rectification is achievable if the thermal symmetry of the structure is broken and the central island can release energy to the phonon bath. The performance of the device is theoretically analyzed and, under the appropriate conditions, temperature differences up to $\sim$ 200 mK between the forward and reverse thermal bias configurations are obtained below 1 K, corresponding to a rectification ratio $\mathcal{R} \sim$ 2000. The simplicity intrinsic to its design joined with the insensitivity to magnetic fields make our device potentially attractive as a fundamental building block in solid-state thermal nanocircuits and in general-purpose cryogenic electronic applications requiring energy management.

be immediately exploited in low-temperature solid-state thermal circuits. Moreover, it would be virtually unaffected by magnetic fields ensuring high performance also in conditions where a hybrid superconducting thermal diode 30 could lose its effectiveness. We shall start, first of all, by defining two parameters that will help us to describe the diode's performance, i.e., the rectification effectiveness R and the thermal efficiency η. Toward this end, we analyze in more detail the structure of our system. As shown in Fig. 1(a) and 1(b), N 2 is coupled to N 1 and to N 3 by means of two tunnel junctions characterized by resistances R 1 and R 2 , respectively. For simplicity, from now on, we will set R 1 = 500 Ω and consider only the parameter r = (R 2 /R 1 ) ≥ 1, accounting for the asymmetry of the device. N 1 and N 3 act as thermal reservoirs and are used to establish a temperature gradient across the device. Electrode N 2 , instead, represents the core of the diode, since it controls the heat flow from a reservoir to the other by releasing energy to the phonon bath (residing at temperature T bath ). In the for-ward configuration, the electronic temperature of N 1 is set to T hot > T bath . This temperature bias leads to heat currents J in,fw and J fw flowing into N 2 and N 3 , respectively. On the other hand, in the reverse configuration the electronic temperature of N 3 is set to T hot , generating the heat currents J in,rev and J rev flowing into N 2 and N 1 , respectively. Under these assumptions, we can define the rectification effectiveness as: In general, a highly-effective thermal diode is characterized by R 1 or 1. In our case, since we have chosen r ≥ 1, the forward configuration results to be the most transmissive, as we will show. The thermal diode's response can be heuristically compared to that of the well-known electric diode by plotting the output current J out vs. the bias temperature T bias . In the forward configuration we define T bias = T hot − T bath and J out = J f w , whereas in the reverse configuration T bias = −(T hot − T bath ) and J out = −J rev . As shown in Fig. 1(c), if R 1 the behavior of J out vs. T bias is strongly asymmetrical, indicating a drastic mismatch of the diode's heat transport properties between the forward and the reverse configuration. Yet, another important parameter that has to be considered is the thermal efficiency. In the transmissive configuration, it can be defined as: indicating the fraction of power that is transferred from N 1 to N 3 . An ideal diode should exhibit J fw = J in,fw and J rev = 0, leading to R → ∞ and η = 1.
We describe now the equations governing heat transport in our device. First, if we consider two N electrodes residing at electronic temperatures T 1 and T 2 (with T 1 ≥ T 2 for definiteness) coupled by means of a tunnel junction, the stationary electronic thermal current flowing through the junction can be written as: 18 where R N is the contact resistance, e is the electron charge and k B is the Boltzmann's constant. Moreover, we must take into account the heat exchanged by electrons in the metal with lattice phonons: 6,31 Here Σ is the material-dependent electron-phonon coupling constant, V is the volume of the electrode and n is the characteristic exponent of the material. In this work we will consider two materials that are commonly exploited to realize N electrodes in nanostructures, i.e., copper (Cu) and manganesedoped aluminum (AlMn). The former is typically characterized by Σ Cu = 3 × 10 9 WK −5 m −3 and n Cu = 5, 1,5 while the latter exhibits Σ AlMn = 4 × 10 9 WK −6 m −3 and n AlMn = 6. 6,30 Furthermore, we assume that N 1 and N 3 are identical, with volumes V 1 = V 3 = 2 × 10 −20 m −3 . 30 Equations 3 and 4 can be used to formulate a thermal model accounting for heat transport through the device. The model is sketched in Fig  2(a) and describes the forward temperature bias configuration, in which the electrodes of the chain reside at temperatures T hot > T 2,fw > T fw > T bath . Here, T 2,fw and T fw represent the electronic temperatures of N 2 and N 3 , respectively. The terms J in,fw = J e (T hot , T 2,fw ) and J fw = J e (T 2,fw , T fw ) account for the heat transferred from N 1 to N 3 . The reservoirs can release energy to the phonon bath by means of J e−ph,1 and J e−ph,3 . Photon-mediated thermal transport, [32][33][34] owing to poor impedence matching, as well as pure phononic heat currents are neglected in our analysis. 5,6,30 We can now write a system of energy-balance equations that account for the detailed thermal budget in N 2 and N 3 by setting to zero the sum of all the incoming and outgoing heat currents: is the heat current that flows from N 2 to the phonon bath. Since the rectification effectiveness is defined under the condition of equal temperature bias in both the configurations, in Eqs. 5 we set T hot and T bath as independent variables and we calculate the resulting T 2,fw and T fw . Another system of energy-balance equations can be written and solved for the reverse configuration, 35 in which N 2 and N 1 reach electronic temperatures T 2,rev and T rev , respectively. Finally, we can extract the values of R and η.
We discuss in the following the crucial role of N 2 , which is the core of the proposed thermal diode. Figures 2(b), 2(c) and 2(d) display three possible designs of the central electrode. It is illustrative to start with the simplest one, at least conceptually, consisting of an island perfectly isolated from the phonon bath [see Fig. 2(b)]. In this case, the term J cool is null and from the energy-balance equations we obtain: Then, since by assumption we set Σ 1 = Σ 3 and V 1 = V 3 , we have T fw = T rev = T cold and we can easily show that the rectification efficiency becomes: for every value of r. This means that in this system no rectification occurs regardless the asymmetry of the coupling between N 2 and the reservoirs. As a confirmation of the latter result, the lower panel of Fig. 2(b) displays the symmetrical behavior of J out vs. T bias , calculated for r = 100.
In order to envision a device exhibiting a sizable thermal rectification, it is useful to rewrite R 1 as the following condition for temperatures: where δ T fw(rev) = T 2,fw(rev) − T fw(rev) and the mean temperatures T fw(rev) = (T 2,fw(rev) +T fw(rev) )/2. Expression 9 indicates a simple approach to pursue our goal: in the reverse configuration the electronic temperatures of N 1 and N 2 must be similar and close to the lowest temperature in the system, i.e., T bath . This condition can be satisfied by setting r > 1 and by coupling N 2 to the phonon bath. In this way, R 2 > R 1 reduces heat transfer to N 2 , which is able to release energy to the bath, thereby lowering both T rev and δ T rev . On the other hand, the coupling between N 2 and the phonon bath should have a limited impact on heat transport in the forward configuration. Figures 2(c) and 2(d) show two possible designs that allow to create a thermal link between N 2 and the phonon bath. The former just exploits the natural coupling between electrons and lattice phonons in metals, while the latter requires an additional N electrode labeled F, acting as a thermalizing cold finger. To this end, F must be tunnel-coupled to N 2 (with a resistance R F ) and must reside at T bath . We shall demonstrate that the second option provides the best rectification performance, as shown by the dependence of J out on T bias in the lower panels of Fig. 2(c) and 2(d). The former curve is calculated for r = 100 and V 2 = 1 × 10 −18 m −3 , while the latter is obtained for r = 100, V 2 = 1 × 10 −20 m −3 and R F = 500 Ω. We focus first on the design sketched in Fig. 2(c), which corresponds to setting the term J cool = J e−ph,2 in the energybalance equations. Figure 3(a) and 3(b) show respectively the behavior of R and η vs. T hot in an AlMn device for different values of V 2 at T bath = 50 mK and for r = 100. The rectification effectiveness exhibits a non-monotonic trend that reaches a maximum value (R max ) around T hot = 300 mK, whereas η has a maximum at T hot T bath and monotonically approaches zero at larger temperatures. The inset of Fig. 3(b) displays R as a function of T hot for different values of the resistance asymmetry r and for V 2 = 1 × 10 −18 m −3 . It appears evident how increasing r and V 2 generate larger values of R but gradually suppress the thermal efficiency. Furthermore, the latter parameter clearly indicates that the influence of the electronphonon coupling increases with temperature (see Eq. 4), causing strong energy losses in both the configurations. This can be noted also in Fig. 3(c), which displays the output temperatures T fw and T rev vs. T hot of a diode with V 2 = 1 × 10 −18 m −3 and r = 100 for three representative values of T bath . In particular, the derivatives of T fw and T rev strongly decrease as T hot and T bath increase, pinpointing a reduction in the thermal efficiency of the device. Moreover, we also notice that at low T bath the electron-phonon coupling is not much effective in lowering T rev and δ T rev . This effect is highlighted in Fig. 3(d), which compares the R max dependence on T bath for two devices made of AlMn and Cu. In the AlMn diode R max reaches a maximum value of 18 at T bath 200 mK, which indicates the temperature where the electron-phonon coupling attains the highest effectiveness in lowering T rev δ T rev , while having limited repercussions on T fw δ T fw . On the other hand, in Cu J e−ph follows a T 5 power law that leads to a shift of the R max peak to lower values of T bath . These results can be largely improved by tunnel-coupling the electrode F to N 2 , thereby creating an efficient channel through which the diode's core can release energy. As sketched in Fig. 2(d), this can be taken into account by setting J cool = J e−ph,2 + J e,2 in the energy-balance equations. Figure 4(a) shows R and η vs. T hot for different values of R F at T bath = 50 mK. Remarkably, R max ∼ 2000 is obtained for R F = 500 Ω, V 2 = 1 × 10 −20 m −3 and r = 100.
Similarly to the previous case, increasing R F and r leads to a suppression of the thermal efficiency, which appears to be lower than that obtained above. Nevertheless, η presents new important features: it has a non-monotonic behavior with a minimum at T hot T bath and a peak centered at a specific T hot which depends on the parameters R F , r and V 2 and can correspond to high values of R. The low-temperature dependence is dominated by the F channel, which exchanges heat following a T 2 dependence. The influence of the cold finger gradually vanishes at larger temperatures and the electron-phonon coupling starts playing the most important role reducing η. It is then clear how F is highly efficient in keeping T rev close to T bath and δ T rev close to zero, while attenuating its effect on T 2,fw that can reach relatively high temperatures. This mechanism allows us to decrease V 2 and consequently to reduce the influence of the electron-phonon coupling at high T hot . By defining the global efficiency of the thermal rectifier Rη, it is possible to determine the best parameters to optimize the diode's performance. The contour plot shown in Fig. 4(b) highlights the dependence of the maximum value of Rη on R F and r, indicating the optimal working region of the proposed device. Figure 4(c) displays the rectifier's output temperatures T fw and T rev vs. T hot for r = 100 and R F = 500 Ω at different values of T bath . The results point out a maximum difference of ∼ 200 mK between the forward and reverse configurations at T bath = 50 mK. Finally, Fig. 4(d) confirms the noxious effect of the electron-phonon coupling on the performance of the diode. As a matter of fact, the behavior of R max as a function of T bath shows that the T 5 dependence of J e−ph in Cu is extremely detrimental in the forward configuration and can reduce the rectification effectiveness up to a factor 10.
It is worth mentioning that adding a fourth electrode to the chain would allow us to obtain R = 1 even when r = 1, provided that the coupling between the central islands and the phonon bath is asymmetrical. However, it turns out that this effect cannot improve the performance obtained with the chain made of three electrodes. As a matter of fact, the additional element in the diode's core produces further energy losses and mitigates the temperature gradient δ T fw , thereby reducing J fw .
In summary, we have proposed and theoretically analyzed the performance of a thermal rectifier consisting of a simple NININ junction. We have demonstrated that a large rectification is achievable if the thermal symmetry of the system is broken and the central electrode is coupled to the phonon bath. Extremely high values of R ∼ 2000 can be obtained if a cold finger is connected to the core of the rectifier, thereby creating an efficient channel through which the diode can release energy in the non-transmissive temperature bias configuration. The device could be easily implemented by standard nanofabrication techniques 30 and, combined with heat current interferometers, 5,6 might become one of the building blocks of coherent caloritronic nanocircuits. [5][6][7] Moreover, its essential design and composition candidate our diode to become a promising tool for thermal management in general-purpose cryogenic electronic applications, even in presence of magnetic fields.
The Marie Curie Initial Training Action (ITN) Q-NET 264034 and the Italian Ministry of Defense through the PNRM project TERASUPER are acknowledged for partial financial support.