As outlined in the introduction, a purely photonic technology has been proposed as an alternative to the phononic one. An important step forward in this direction has been carried out by Otey et al. [12] by introducing the first radiative thermal rectifier by exploiting the thermal dependence of optical properties of materials in interaction. During the next 5 years, the rectification performances have been improved [13], [14], [15], [16], [17], [18] until the development of a phase-change radiative thermal diode able to rectify 66% of heat flux in far-field [19], [20] and even more than 99% in the near-field regime [21], respectively. Contrary to the relatively weak dependence with respect to the temperature of optical properties of materials used in the thermal rectifier developed so far, the phase-change thermal diode exploites a sudden change of these properties around a critical temperature, the transition temperature.

Following this idea, a general concept of radiative transistor has been introduced in 2014 [22] and applied in 2015 [23] in the particular situation where only propagating photons participate to the transfer. Before introducing it, let us briefly describe the classical electronic transistor. This last is sketched in Figure 1a. It is composed by three solid elements, the drain, the source, and the gate. It is basically used to control the flux of electrons exchanged in the channel between the drain and the source by changing the voltage bias applied on the gate. While the physical diameter of this channel is fixed, its effective electrical diameter can be tuned by application of a voltage on the gate. A tiny change in this voltage can cause a large variation in the current from the source to the drain.

Figure 1: (a) Electronic transistor made of three terminals, the source (S), the gate (G), and the drain (D). The gate is used to actively control (by applying a voltage bias *V* on it) the apparent electric conductivity of the channel between the source and the drain. (b) Radiative thermal transistor. A membrane of an MIT material (VO_{2}) acts as the gate between two silica (SiO_{2}) thermal reservoirs (source and drain). The temperature (*T*_{G}) of the gate is chosen between the temperatures of the source and the drain (*T*_{S} and *T*_{D}). Φ_{D} and Φ_{S} are the radiative heat fluxes received by the drain and emitted by the source, respectively.

The radiative analogue of an electronic transistor [22] is depicted in Figure 1b. It basically consists in a source and a drain, labelled by the indices S and D, which are maintained at temperatures *T*_{S} and *T*_{D} (which play an analogue role as the voltage) using thermostats where *T*_{S}>*T*_{D} so that a net heat flux is transferred from the source towards the drain. A thin layer of a metal-isulator transition material (MIT) labelled by *G* of width *δ* is placed between the source and the drain at a distance *d* from both media and operates as a gate. This configuration coincides with two heat-radiation diodes [19] which are connected in series, so that the heat radiation transistor corresponds to a bipolar transistor. In an MIT material, a small change in the temperature around its critical temperature *T*_{c} causes a sudden qualitative and quantitative changes in its optical properties. Vanadium dioxide (VO_{2}) is one of such materials (see Fig. 2) which undergoes a first-order transition (Mott transition [24]) from a high-temperature metallic phase to a low-temperature insulating phase [25] close to room-temperature (*T*_{c}=340 K). Different works have shown [21], [26], [27] that the heat-flux exchanged at close separation distances (i.e. in the near-field regime) between an MIT material and another medium can be modulated by several orders of magnitude across the phase transition of MIT materials.

Figure 2: Permittivity (real part) of VO_{2} in its metallic phase (*ε*_{metal} when *T*>*T*_{c}) and in its cristalline phase (*ε*_{||} and *ε*_{⊥} when *T*<*T*_{c}), *T*_{c} being the critical temperature of VO_{2}.

Without external excitation, the system will reach its steady state for which the net flux Φ_{G} received by the intermediate medium, the gate, is zero by heating or cooling the gate until, it reaches its steady state or equilibrium temperature ${T}_{\text{G}}^{\text{eq}}.$ In this case, the gate temperature *T*_{G} is set by the temperature of the surrounding media, that is, the drain and the source. When a certain amount of heat is added to or removed from the gate for example by applying a voltage difference through a couple of electrodes as illustrated in Figure 1b or by extracting heat using Peltier elements, its temperature can be either increased or reduced around its equilibrium temperature ${T}_{G}^{eq}.$ This external action on the gate allows to tailor the heat flux Φ_{S} between the source and the gate and the heat flux Φ_{D} between the gate and the drain.

To show that this system operates as a transistor let us examinate how the radiative heat flux evolves in this system with respect to the gate temperature. In a three-body system, the radiative flux received by the drain takes the form [28], [29]

$${\Phi}_{D}={\displaystyle {\int}_{0}^{\infty}\frac{d\omega}{2\pi}{\varphi}_{\text{D}}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}d\mathrm{)}},$$(1)

where the spectral heat flux is given by

$$\begin{array}{c}{\varphi}_{\text{D}}={\displaystyle \sum _{j\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{\{}\text{s}\mathrm{,}\text{p}\mathrm{\}}}{\displaystyle \int \frac{{\text{d}}^{2}k}{{\mathrm{(}2\pi \mathrm{)}}^{2}}\mathrm{[}{\Theta}_{\text{SG}}\mathrm{(}\omega \mathrm{)}{\mathcal{T}}_{j}^{\text{S}\mathrm{/}\text{G}}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}k\mathrm{;}\text{\hspace{0.17em}}d\mathrm{)}}}\\ \text{\hspace{0.17em}}+{\Theta}_{\text{GD}}\mathrm{(}\omega \mathrm{)}{\mathcal{T}}_{j}^{\text{G}\mathrm{/}\text{D}}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}k\mathrm{;}\text{\hspace{0.17em}}d\mathrm{)}]\mathrm{.}\end{array}$$(2)

Here, ${\mathcal{T}}_{j}^{\text{S}\mathrm{/}\text{G}}$ and ${\mathcal{T}}_{j}^{\text{G}\mathrm{/}\text{D}}$ denote the transmission coefficients of each mode (*ω*, *k*) between the source and the gate and between the gate and the drain for both polarisation states *j*=s, p. In the above relation, Θ_{ij} denotes the difference of functions Θ(*ω*, *T*_{i} ) and Θ(*ω*, *T*_{j} ), $\Theta \mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}T\mathrm{)}=\frac{\hslash \omega}{\mathrm{(}\mathrm{exp}\mathrm{(}\frac{\hslash \omega}{{k}_{\text{B}}T}\mathrm{)}-1\mathrm{)}}$ being the mean energy of a Planck oscillator at temperature *T*. According to the N-body near-field heat transfer theory [28], [29], the transmission coefficients ${\mathcal{T}}_{j}^{\text{S}\mathrm{/}\text{G}}$ and ${\mathcal{T}}_{j}^{\text{G}\mathrm{/}\text{D}}$ of the energy carried by each mode written in terms of optical reflection coefficients *ρ*_{E,j} (E>S, D, G) and transmission coefficients *τ*_{E,j} of each basic element of the system and in terms of reflection coefficients *ρ*_{EF,j} (E=S, D, G and *F*=S, D, G) of couples of elementary elements is given by (*κ*>*ω*/*c*)

$$\begin{array}{l}{\mathcal{T}}_{j}^{\text{S}\mathrm{/}\text{G}}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}k\mathrm{,}\text{\hspace{0.17em}}d\mathrm{)}=\frac{4|{\tau}_{\text{G}\mathrm{,}j}{|}^{2}\text{Im}\mathrm{(}{\rho}_{\text{S}\mathrm{,}j}\mathrm{)}\text{Im}\mathrm{(}{\rho}_{\text{D}\mathrm{,}j}\mathrm{)}{e}^{-4\gamma d}}{|1-{\rho}_{\text{SG}\mathrm{,}j}{\rho}_{\text{D}\mathrm{,}j}{e}^{-2\gamma d}{|}^{2}|1-{\rho}_{\text{S}\mathrm{,}j}{\rho}_{\text{G}\mathrm{,}j}{e}^{-2\gamma d}{|}^{2}}\mathrm{,}\\ {\mathcal{T}}_{j}^{\text{G}\mathrm{/}\text{D}}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}k\mathrm{,}\text{\hspace{0.17em}}d\mathrm{)}=\frac{4\text{Im}\mathrm{(}{\rho}_{\text{SG}\mathrm{,}j}\mathrm{)}\text{Im}\mathrm{(}{\rho}_{\text{D}\mathrm{,}j}\mathrm{)}{e}^{-2\gamma d}}{|1-{\rho}_{\text{SG}\mathrm{,}j}{\rho}_{\text{D}\mathrm{,}j}{e}^{-2\gamma d}{|}^{2}}\end{array}$$(3)

introducing the imaginary part of the wavevector normal to the surfaces in the multilayer structure *γ*=Im(*k*_{z0}). Similarly, the heat flux from the source towards the gate reads

$$\begin{array}{c}{\varphi}_{\text{S}}={\displaystyle \sum _{j\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{\{}\text{s}\mathrm{,}\text{p}\mathrm{\}}}{\displaystyle \int \frac{{\text{d}}^{2}k}{{\mathrm{(}2\pi \mathrm{)}}^{2}}\mathrm{[}{\Theta}_{\text{DG}}\mathrm{(}\omega \mathrm{)}{\mathcal{T}}_{j}^{\text{D}\mathrm{/}\text{G}}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}k\mathrm{;}\text{\hspace{0.17em}}d\mathrm{)}}}\\ \text{\hspace{0.17em}}+{\Theta}_{\text{GS}}\mathrm{(}\omega \mathrm{)}{\mathcal{T}}_{j}^{\text{G}\mathrm{/}\text{S}}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}k\mathrm{;}\text{\hspace{0.17em}}d\mathrm{)}]\end{array}$$(4)

where the transmission coefficients are analogue to those defined in (3) and can be obtained making the substitution S↔D. Note that here we have neglected the contribution of the propagating waves which is vanishingly small compared to the contribution of the evanescent waves for distances *d* much smaller than the thermal wavelength. At steady state, the net heat flux received/emitted by the gate which is just given by the heat flux from the source to the gate minus the heat flux from the gate to the drain vanishes so that

$${\Phi}_{\text{G}}={\Phi}_{\text{S}}-{\Phi}_{\text{D}}=0.$$(5)

This relation allows us to identify the gate equilibrium temperature ${T}_{\text{G}}^{\text{eq}}$ (which is not necessarily unique because of the presence of bistability mechanisms [22]) for given temperatures *T*_{S} and *T*_{D}. Note that out of steady state, the heat flux received/emitted by the gate is Φ_{G}=Φ_{S}−Φ_{D}≠0. If Φ_{G}<0 (Φ_{G}>0) an external flux is added to (removed from) the gate by heating (cooling).

The different operating modes of the transistor can be analysed from the evolution of flux curves with respect to the gate temperature. These curves are plotted in Figure 3 in the case of a silica source and a silica drain with a VO_{2} gate in between. We set *T*_{S}=360 K and *T*_{D}=300 K and choose a separation distance between the source and the gate and between the gate and the drain to *d*=100 nm. The thickness of the gate layer is set to *δ*=50 nm.

Figure 3: Heat fluxes inside the near-field thermal transistor with a VO_{2} gate of thickness *δ*=50 nm located at a distance *d*=100 nm between two massive silica samples maintained at temperatures *T*_{S}=360 K and *T*_{D}=300 K. By changing the flux Φ_{G} supplied to the gate, the temperature *T*_{G} is changed so that the transistor operates either as thermal switch or as thermal amplifier.

The equilibrium temperature of the gate, obtained by solving the transcendental equation (5), is for this configuration uniquely given by ${T}_{G}^{\text{eq}}=332\text{\hspace{0.17em}K}$ which is close to the critical temperature *T*_{c}≈340 K of VO_{2}. Hence, in the steady-state situation, the gate is in its insulating phase. In this situation, it supports surface phonon polaritons [21], [22] in the mid-infrared range and the source and the drain so that these surface waves can couple together making the heat transfer very efficient between the source and the drain which can be seen by inspection of the transmission coefficients in Figure 4. On the contrary, when the temperature of the gate is increased by external heating to values larger than *T*_{c} then VO_{2} undergoes a phase transition towards its metallic phase. In this case, the gate does not support surface wave resonances anymore so that the surface mode coupling between each solid element is suppressed (see Fig. 4) and the heat transfer drastically drops (Fig. 3). This drastic change in the transfer of energy can be used to modulate the heat flux received by the drain by changing the gate temperature around its critical value. The thermal inertia of the gate as well as its phase transition delay defines the timescale at which the switch can operate. Usually, the thermal inertia limits the speed to some microseconds [30], [31] or even less [32].

Figure 4: Energy transmission coefficient of modes in the (*ω*, *k*) plane for a SiO_{2}–VO_{2}–SiO_{2} transistor with a *δ*=50-nm-thick gate and a separation distance of *d*=100 nm between the source and the gate (respectively the drain and the gate). Wien’s frequency (where the transfer is maximum) at *T*=340 K is *ω*_{Wien}~1.3×10^{14} rad/s.

But more interesting is the possibility offered by this system to amplify the heat flux received by the drain. This effect is the thermal analogue of the classical transitor effect. To highlight this effect, let us focus our attention on the operating mode in the region of phase transition around *T*_{c}. As we see in Figure 3, a small increase in *T*_{G} leads to a drastic reduction of flux received by the drain. As described by Kats et al. in [33], this behaviour can be associated in far-field to a reduction of the thermal emission. This anomalous behaviour corresponds to the so-called negative differential thermal conductance as described in [34]. The presence of a negative differential thermal conductance is a necessairy condition (but not sufficient) for observing a transistor effect. Indeed, the amplification coefficient of a transistor is defined as (see for example [4])

$$\alpha \equiv \left|\frac{\partial {\Phi}_{\text{D}}}{\partial {\Phi}_{\text{G}}}\right|=\frac{1}{\left|1-\frac{{{\Phi}^{\prime}}_{\text{S}}}{{{\Phi}^{\prime}}_{\text{D}}}\right|}$$(6)

where

$${{\Phi}^{\prime}}_{\text{S}\mathrm{/}\text{D}}\equiv \frac{\partial {\Phi}_{\text{S}\mathrm{/}\text{D}}}{\partial {T}_{\text{G}}}\mathrm{.}$$(7)

By introducing the thermal resistances

$${R}_{D}={{\Phi}^{\prime}}_{D}^{-1}$$(8)

and

$${R}_{S}=-{{\Phi}^{\prime}}_{\text{S}}^{-1}$$(9)

associated to the drain and the source, this coefficient can be recasted under the form

$$\alpha =\left|\frac{{R}_{S}}{{R}_{S}+{R}_{D}}\right|\mathrm{.}$$(10)

It immediately follows from this expression that when both resistances are positive, then *α*<1. This, precisely happens outside the phase transition region (see Fig. 3) where ${{\Phi}^{\prime}}_{S}=-{{\Phi}^{\prime}}_{D}$ so that *α*=1/2 (Fig. 5). On the contrary, in the region where the phase transition occurs, this situation changes. A direct inspection of flux in this region (Fig. 3) clearly shows that ${{\Phi}^{\prime}}_{S}$ and ${{\Phi}^{\prime}}_{D}$ have the same (negative) sign so that *R*_{S}>0 and *R*_{D}<0. It turns out that *α*>1 (Fig. 5).

Figure 5: Amplification coefficient of a SiO_{2}–VO_{2}–SiO_{2} transistor. When *T*_{G}<<*T*_{c} or *T*_{G}>>*T*_{c}, the slopes of Φ_{S} and Φ_{D} are identical (modulo the sign) so that *α*=1/2. On the contrary, in the close neighborhood of *T*_{c}, we have *α*>1 owing to the negative differential thermal resistance in the transition region. The large *α* at the beginning and the end of the phase-change region are artefacts due to the model used to describe the permittivity in this region. Here, the parameters of the transistor are the same as in Figure 4.

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