Quantum Thermal Rectification to Design Thermal Diodes and Transistors

2.1 Thermal Diode

The system under consideration consists of two coupled TLS, each of them related to a thermal bath, as depicted in Figure 1.

Figure 1:

Quantum thermal diode made up of two TLS coupled with each other and connected to a thermal bath.

The two TLS are labelled with the letters L (left) and R (right), which is also the case of the temperature of the thermal baths related to TLS. We use the strong-coupling formalism developed by Werlang et al. [45]. Each of the TLS is characterised by an angular frequency ω_L or ω_R. The coupling between the two TLS has the typical angular frequency ω_LR. The Hamiltonian of the system is (in ћ=1 units)

where σzP (P=L, R) is the Pauli matrix z, whose eigenstates for the system P are the states ↑ and ↓. Therefore, angular frequencies ω have the dimension of energy. H_S eigenstates are given by the tensorial product of the individual TLS states, so that we have four eigenstates labelled as |1〉=|↑↑〉, |2〉=|↑↓〉, |3〉=|↓↑〉, |4〉=|↓↓〉. The coupling between the TLS and the thermal bath P constituted of harmonic oscillators [47] is based on the spin-boson model in the x component HTLS−bathP=σxP∑kgk(akP+akP†). The two reservoirs P have their Hamiltonians equal to HbathP=∑kωkakP†akP. This modelling implies that baths can only flip one spin at a time. There are therefore four authorised transitions. The transitions 1↔3 and 2↔4 are induced by the thermal bath L, whereas the transitions 1↔2 and 3↔4 are induced by the thermal bath R. The transitions 1↔4 and 2↔3 are forbidden.

The system state is described by a density matrix ρ, which obeys a master equation. In the Born–Markov approximation, it reads

As in [45], [48], the Lindbladians ℒ_P[ρ] are written for an Ohmic bath according to classical textbooks [48], [49], so that we take the expression

of [45], where

and ℐ(ω)=κω in the case of a Ohmic bath with a coupling constant κ. We now consider a steady state situation. We define

the heat current injected by the bath P into the system. Averaging the master equation, we find J_L+J_R=0, in accordance with the energy conservation.

The master equation is a system of four equations on the diagonal elements ρ_ii. If we introduce the net decaying rate from the state |i〉 to the state |j〉, due to the coupling with bath P, with the help of the Bose–Einstein distribution nωP=(eω/TP−1) − 1 (in k_b=1 units): ΓijP=κωij[(1+nωP)ρii−nωPρjj]=− ΓjiP, the master equation yields

from which it can be deduced that

The definition of the thermal currents (6) gives then the final expression of the thermal currents

As an example, let us consider the example of a system where ω_L=1, ω_R=0, and ω_LR=0.1. The energy levels and the authorised transitions are depicted in Figure 2. Note that we took in this example, as well as in the rest of the paper, κ=1. This means that the currents have to be multiplied by κ and divided by ћ in order to be expressed in W. When T_L>T_R, the left reservoir populates level |1〉 from level |3〉 through the transition 1↔3. Level |1〉 de-excitates through level |2〉 by transferring energy to reservoir R. Level |2〉 de-excitates through level |4〉 by transferring energy to reservoir L and finally level |4〉 de-excitates through level |3〉 by transferring energy to reservoir R. If T_L<T_R, the energy transfers are reversed. Now imagine that T_L is of the order of the transition energies, whereas T_R is much lower. Then, energy will easily flow from reservoir L to reservoir R according to the process described above. On the contrary, if T_R is much lower than the transition energies and T_L<T_R, then the energy transfer is poor since excitation by reservoir R through the transitions 4↔3 and 2↔1 is low. Hence, if we study the flux J_L(T_L, T_R) with T_R fixed at a value lower than the transition energies (e.g. T_R=0.1, Fig. 3), we see that the flux is close to 0 when T_L<T_R. When T_L is increased to values larger than T_R, the current increases until it reaches saturation at high temperatures. The calculation of Γ, which gives the current, can be achieved by solving the system of equations on the populations (7). Note that the system of equations are not totally independent since the fourth equation is actually the sum of the three others. One has to use the fact that the trace of the density matrix is equal to 1 (Tr[ρ]=1). The exact expression of Γ can be found in [45]. In the case studied here, this expression can be simplified and the current reads

Figure 2:

Energy levels and transitions in the case ω_L=1, ω_R=0, and ω_LR=0.1. The arrow direction shows the balance of the authorised transition between levels. Left: T_L>T_R. Right: T_L<T_R.

Figure 3:

J_L(T_L, T_R) in the case ω_L=1, ω_R=0, and ω_LR=0.1 with T_R=0.1.

where the transition from low current values, at low T_L, to high current values, at higher T_L, can be seen.

Let us note that the system proposed here constitutes a passive thermal switch at low temperature. As long as T_L is larger than T_R, the current in the structure is important and the thermal contact is good between the reservoirs L and R. However, when the temperature T_L reduces to values below T_R, the thermal current is drastically lowered, so that it can be seen as switched off. This system could therefore be used to isolate objects from a cold environment while it would be thermally linked to a hot environment. In a case of an environment with temperatures oscillating between high and low values, this simple quantum system can be seen as a passive heater and a thermal rectifier, that is, that heat flow through it depends on the direction of the heat flux.

There is actually another way to quantify the rectification of a system. This is the ratio between the sum of the fluxes through the system when the temperatures are reversed and the maximum of these two fluxes:

The rectification ratio R(T_L, T_R) variations with T_L for different T_R are represented in Figure 4. When T_R is small enough (T_R<1), rectification is strong except for values of T_L very close to those of T_R. When T_R is larger, rectification is smaller, even for T_L values that are greatly different from T_R. We note, in particular, that rectification is low for high T_R temperature. In this latter case, there is no rectification, because heat transfer can occur with both reservoirs with the help of the energy transitions presented above. However, when T_R is fixed, and T_L goes to 0, then J_L(T_R, T_L) tends to 0. Also, rectification rises to 1. This kind of device can thus be seen as a thermal diode, since the heat current through the system is nonzero when the heat flux is in a given direction and 0 when it is in the opposite one.

Figure 4:

Rectification ratio R(T_L, T_R) variations with T_L for different values of T_R in the case ω_L=1, ω_R=0, and ω_LR=0.1. (a) T_R=0.1. (b) T_R=1. (c) T_R=10. (d) T_R=100.

This type of system paves the way to develop more complicated ones. For example, it is well known that electronic transistors as the bipolar ones can be made up of NPN and PNP junctions whereas the PN junction constitutes a diode. One can therefore wonder if it is also possible to conceive a transistor with the elementary quantum system that constitutes the thermal diode that we have just studied in this section. This is the subject of the next section.

2.2 Thermal Transistor

We consider in this part a system constituted of three TLS coupled with each other. Each of the three TLS is also coupled to a thermal bath at thermal connection (Fig. 5). This system is therefore similar to the previous one with one supplementary TLS and reservoir. The three TLS are now indexed with the letters L (left), M (medium), and R (right). The thermal bath temperatures are labelled in the same way. As in the previous part, we use the strong-coupling formalism developed by Werlang et al. [45]. Similarly, TLS can be in the up state ↑ or in the down one ↓. Let us write the Hamiltonian of the system (in ћ=1 units)

Figure 5:

Quantum system made up of three TLS coupled with each other and connected to a thermal bath.

where ω_P denotes the energy difference between the two spin states, whereas ω_PQ stands for the interaction between the spin P and the spin Q. Following the preceding part on the quantum thermal diode, we have eight eigenstates labelled as |1〉=|↑↑↑〉, |2〉=|↑↑↓〉, |3〉=|↑↓↑〉, |4〉=|↑↓↓〉, |5〉=|↓↑↑〉, |6〉=|↓↑↓〉, |7〉=|↓↓↑〉, and |8〉=|↓↓↓〉. There are now 12 authorised transitions. The left bath (L) induces the transitions 1↔5, 2↔6, 3↔7, and 4↔8, and the middle one (M) drives the transitions 1↔3, 2↔4, 5↔7, and 6↔8. The right bath (R) triggers the transitions 1↔2, 3↔4, 5↔6, and 7↔8. All other transitions flipping more than one spin are forbidden.

The master equation fulfilling the density matrix, in the Born–Markov approximation, reads

we now go to the steady state situation. Averaging the master equation, we find J_L+J_M+J_R=0, in accordance with the energy conservation.

The master equation is a system of eight equations on the diagonal elements ρ_ii. Introducing the net decaying rate from the state |i〉 to the state |j〉 due to the coupling with bath P, the master equation becomes

The sum of these eight equations is 0 and therefore they are not independent. The condition Tr[ρ]=1 is added to the system in ρ_ii whose resolution provides all state occupation probabilities as well as the currents J_P.

Let us now show that such a device makes a thermal transistor with a close analogy to an electronic one. In an electronic bipolar transistor, such as a PNP or an NPN transistor, a driven current at the base modulates, switches, or amplifies the collector and emitter currents. Therefore, switching, modulation, and amplification have to be exhibited in order to have a transistor. We are going to show here that by slightly changing J_M it is possible to control J_L or J_R. Let us consider the following situation: the left and right TLS are both connected to thermal baths at respective temperatures T_L and T_R. The third bath at temperature T_M changes the fluxes J_L and J_R by means of the current J_M injected into the system. The dynamical amplification factor α, defined as

is a measure of the transistor ability to amplify a small heat flux variation at the base (M). If a small change in J_M makes a large one in J_L or J_R, that is, |α_L,R |>1, then the thermal transistor effect can be identified. The system presented here exhibits many parameters: the frequencies ω_P and ω_PQ and the temperatures T_L and T_R. The last temperature T_M, which is taken here between T_L and T_R, controls the transistor properties and is related to the current J_M. The number of parameters involved can be reduced by choosing a situation that will not change the physics of the system but will allow a good understanding of the physical phenomena involved. We therefore restrict our analysis to a case for which ω_LM=ω_MR=Δ, whereas ω_RL and the three TLS energies are equal to 0. As shown below, this configuration provides a good transistor effect, easy to handle with simple calculations. The transistor effect disappears when the three couplings are equal (symmetric configuration), but it still occurs and can even be optimised if the three TLS energies are nonzero [50]. The operating temperature T_L is taken so that e−Δ/TL≪1 (T_L/Δ≪0.25), whereas e−Δ/TR≪e−Δ/TL (T_R/Δ≪0.0625).

Under these conditions, the system states are degenerated 2 by 2. There are now only four states and three energy levels (see Fig. 6).The states |1〉 and |8〉 are now state |I〉, |2〉 and |7〉 state |II〉, |3〉 and |6〉 state |III〉, and |4〉 and |5〉 state |IV〉. One can define the new density matrix elements ρ_I=ρ₁₁+ρ₈₈, ρ_II=ρ₂₂+ρ₇₇, ρ_III=ρ₃₃+ρ₆₆, and ρ_IV=ρ₄₄+ρ₅₅. Using the net decaying rates between the states, the three currents read

Figure 6:

Energy levels for ω_L=ω_M=ω_R=0, ω_RL=0, and ω_LM=ω_MR=Δ. There are four states (|I〉, |II〉, |III〉, and |IV〉) but three energy levels since E_II=E_IV=0. The arrows indicate the net decaying rate between the states due to bath L (red), bath M (green), and bath R (blue) for T_L=0.1Δ, T_R=0.01Δ, and T_M=0.05Δ.

Transitions between the different states are plotted in Figure 6, for T_L/Δ=0.1, T_R/Δ=0.01, and T_M/Δ=0.05. The direction of the arrows shows the transition direction, whereas its width is proportional to the decay time. Energy exchanges are mainly dominated by the III–II and IV–III transitions. J_M is expected to be larger than J_R and J_L. This is shown in Figure 7, in which J_L, J_M, and J_R are plotted versus T_M, for T_L/Δ=0.1 and T_R/Δ=0.01. J_L and J_R increase linearly with T_M, at low temperature, and behave sublinearly as T_M gets close to T_L. Note that over the whole range, J_M remains lower than J_L and J_R, as expected. Thus, T_M will be controlled by changing a little bit the current J_M: a tiny change of J_M can modify J_L and J_R in a larger proportion. Moreover, J_L and J_R are switched off when J_M approaches 0, for small temperatures T_M: the three TLS exhibit the transistor switching property. One also remarks that the J_M slope is larger than the ones of J_L and J_R over a large part of the temperature range. Given the definition of the amplification factor α, the thermal current slopes are essential to figure out amplification.

Figure 7:

Upper: thermal currents J_L, J_M, and J_R versus T_M for ω_L=ω_M=ω_R=0, ω_RL=0, ω_LM=ω_MR=Δ, T_L=0.1Δ, and T_R=0.01Δ. Lower: thermal current J_M versus T_M.

In Figure 8, the two amplification coefficients α_L and α_R are plotted versus temperature T_M. We see that α remains much larger than 1 (around 2.2×10⁴) at low T_M. One also notes that α diverges for a certain value of the temperature for which J_M has a minimum. This occurs for T_M0.07444Δ. In these conditions, an infinitely small change in J_M makes a change in J_L and J_R. As T_M approaches T_L, the amplification factor drastically decreases to reach values below 1: the transistor effect does not exist anymore. Note also that, in between, there exists a temperature for which J_M=0. This temperature is the one at which the bath M is at thermal equilibrium with the system since it does not put any thermal current in it. At this temperature (T_M0.08581Δ), J_L=−J_R=3.325×10⁻⁶. Amplification still occurs since α_L=831 and α_R=−832.

Figure 8:

Amplification factors α_L (red) and α_R (dashed blue) versus T_M for ω_L=ω_M=ω_R=0, ω_RL=0, ω_LM=ω_MR=Δ, T_L=0.1Δ, and T_R=0.01Δ.

Populations and current expressions explain these observations well. In the present case, if we limit the calculation to first order of approximations on e− Δ/TL and e− Δ/TM, populations can be estimated by

ρ_III remains very close to 1 and ρ_II to 10⁻². ρ_I and ρ_IV change by one to two orders of magnitude with temperature and are much smaller than the two preceding ones.

We now explicitly present the three thermal current expressions and their dependence on temperature which is the core of our study.

These formulas are in accordance with the linear dependence of the thermal currents for small values of T_M. Note also that J_L and J_R seem to be driven by ρ_IV, the state population at the intermediate energy (E_IV=0) since their expressions (21) and (20) are very similar. Examining the authorised transitions, one expects J_M to be driven by the population of the most energetic state, that is, ρ_I. The main difference between ρ_IV and ρ_I is the temperature dependence, which is linear in one case and exponential (e^−2Δ/T ) in the other one. The result is that even when T_M is close to T_L, ρ_I remains low. Therefore, J_M keeps low values in the whole temperature range due to the low values of ρ_I. A careful look at J_M shows that it is the sum of two terms. The first one is roughly linear on T_M and is similar to the one that appears in ρ_IV. J_M depends on the population of the state IV, which also changes the population of the state I with the transition IV–I. Increasing ρ_IV with T_M facilitates the IV–I transition, and raises ρ_I. This increases the state I decay through the I–III transition. This term is negative and decreases as T_M increases. This can be seen as a negative differential resistance since a decrease in J_M (cooling in M) corresponds to an increase in the temperature T_M. In this temperature range, the amplification factor |αL| ≈ |αR| ≈eΔ/TL(e¹⁰=22026.5). The second term in J_M is the classical e − Δ/TM Boltzmann factor, which makes the population of the state I increase with T_M. J_M is a compromise between these two terms. At low temperature, the linear term is leading. As T_M increases, the term e − Δ/TM dominates. Consequently, there exists a value where the increase in ρ_I reverses the I–IV transition, so that the I–III transition bids with both the I–V and I–II transitions. I–III is then reversed. With these two terms competing, there is a temperature for which J_M reaches a minimum and a second temperature where J_M=0, as described above.

One can wonder what are the conditions on the parameters to obtain the best transistor effect in the conditions studied here. There are two criteria that will quantify a good transistor. One is the amplification factor and the other one is the intensity of the heat currents at the emitter and the collector (J_L and J_R). Note that the amplification factor depends on eΔ/TL and that the currents depend on e − Δ/TL. Let us also recall that we have assumed up to now that e − Δ/TL≪1. Therefore, the best choice to have a transistor with a sufficiently collector or emitter current is to take the lowest Δ/T_L with the condition e − Δ/TL≪1 and the criteria chosen to fulfil this last condition (here Δ/T_L≈5).

One can summarise the conditions needed for the system to undergo a thermal transistor effect. Two baths (here L and R) induce transitions between two highly separated states with an intermediate energy level, whereas the third one (M) makes only a transition between the two extremes. This will first make J_M much smaller than J_L and J_R, and second, it will set a competition between a direct decay of the highest level to the ground level and a decay via the intermediate one. This competition between the two terms makes the thermal dependence of J_M on T_M slow enough to obtain a high amplification.

Finally, one can wonder what could be the type of system that could be used in order to make a thermal diode or a thermal transistor. Electrons in quantum dots that are used to form qubits [51] and in particular electron qubits are good candidates if they can be coupled to a thermal reservoir that can be controlled in temperature. The idea would be to place these qubits on nano objects the temperature of which would be controlled by electrical currents. The thermal currents calculated in this article would correspond to the heat deposited by the electrical currents applied. Using spin qubits, one could naturally study the influence of the qubit quantum coherence on the thermal objects presented here as well as the time scale at which decoherence would kill coherence effects and retrieve the thermal behaviour presented in this article.