1. Introduction

Thermal analogues of electronic rectification devices such as thermal diodes [1], thermal transistors [2] and memory elements [3] have been studied during past decade. During early years, conduction based devices were discussed [6–8], while more recently radiative thermal rectification devices have gathered considerable attention [4,5, 9–12]. In this work, we analyse radiative thermal rectification devices viz. thermal diode and thermal transistor. Thermal rectification has a wide variety of potential applications such as thermal management and energy storage, thermal circuits [13], thermal logic gates [14] and information processing [15].

Analogous to an electrical diode, a thermal diode has two terminals where a temperature bias is applied. The thermal diode exhibits a high degree of asymmetry in the magnitude of heat flow depending on the applied temperature bias. The goal is to increase this asymmetry as much as possible. Exploiting temperature dependent properties of phase changing materials such as La_0.7Ca_0.15Sr_0.15MnO₃ (LCSMO) and vanadium dioxide (VO₂) [10, 16, 17] is a common theme to achieve radiative thermal rectification. In the far-field, thermal rectification is achieved by change in emissivity/reflectivity of a phase change structure. In the near-field, difference in the degree of tunneling of surface waves between structures dictates thermal rectification. Several studies focus on modulation of radiative heat transfer in near-field regime, while many others deal with far-field thermal radiation. In general it has been observed that a higher rectification can be achieved in the near-field regime. Following Ben-Abdallah and Biehs’s work on a VO₂ based simple far-field radiative thermal diode, numerous studies on thermal rectification [16–19], thermal transistors [20], negative differential conductance [21] have been published. While materials such as SiC, AIST (an alloy of Ag, In, Sb, and Te) have been investigated, VO₂ is arguably the most commonly used material in the context of radiative thermal rectification. Modulation in near-field thermal radiative heat transfer upon phase change of VO₂ has also been experimentally demonstrated [22, 23].

A thermal transistor is a device that has three terminals: source, drain and gate. A temperature bias is applied across the the source and the drain while gate temperature is varied. By controlling the temperature of gate one can control the heat flow across the other two terminals. Li et al. [2] introduced a phonon based thermal analogue of electronic transistor. Later, concepts of thermal logic gates [14] and thermal memory devices [15] was also demonstrated. More recently, concepts of radiative thermal transistors in the near-field and far-field regimes were also presented. [5, 24].

Present study focuses on rectification in a near-field thermal diode and a near-field thermal transistor that employ VO₂ as phase-transition material. VO₂ can be reversibly switched in a very short amount of time (∼ 100 fs) from an insulating state to a metallic state [25]. To enhance the rectification, one may use several different materials. It has been observed that metamaterials such as dielectric mixtures [26] and grating structures [27] can manipulate radiative transfer in near-field and can be considered for enhancement of thermal rectification. Previous studies on radiative thermal rectification have been devoted mainly to bulk materials and thin films. Authors’ previous work [28] dealt with concepts of near-field thermal diode with a detailed discussion about effect of bulk VO₂, thin film, rectangular and triangular gratings on thermal rectification at various separations. In the present study, grating-enhanced near-field thermal diode introduced in [28] is revisited and the concept is extended to near-field thermal transistor. Our calculations indicate that thermal rectification in thermal diode and heat modulation in thermal transistor can be raised significantly by using 1-D surface gratings of VO₂.

A typical near-field thermal diode consists of two planar structures at a separation distance less than thermal wavelength [16]. One structure (hereafter referred to as active terminal) has a phase change material and its counterpart has fixed material properties (passive terminal). Figure 1(a) introduces the concepts of thermal diode proposed in this study that has two structures at a distance of L = 100 nm. The active terminal contains top layer of phase change material VO₂ at temperature T₁ = 341 K + ΔT. On the passive side, structure 2 has its temperature T₂ = 341 K − ΔT. Mean temperature is chosen to be the phase transition temperature of VO₂ at 341 K. When T₁ > T₂ (referred to as forward bias), VO₂ layer is in metallic phase; when T₁ < T₂ (reverse bias), VO₂ layer is in insulator phase with its optical axis aligned along the distance between them. This concept can be extended to thermal transistor as shown in Fig. 1(b). It has three terminals. The source and the drain are same as the passive side of thermal diode. The gate has same structure on both faces as the active side of thermal diode. In present study, source and drain temperatures are fixed above and below the phase transition temperature of VO₂, while gate temperature is varied. Proposed configurations will be discussed in more detail along with results.

 

Fig. 1 Schematics of (a) near-field thermal diode and (b) near-field thermal transistor. Active side of the diode and both the faces of the gate of the transistor have top layer of 1-D rectangular grating made of VO₂ of height h, width w, period Λ and filling ratio ϕ on a gold layer deposited on a substrate. The passive side of thermal diode and the source and the drain of the transistor has 1 μm layer of boron nitride (BN) on 1 μm layer of gold on a substrate.

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Although 341 K is said to be the phase transition temperature of VO₂. the phase transition is not homogeneous nor does it happen instantaneously [18, 25, 29]. We calculate thermal rectification values for the thermal diode at a minimal temperature difference of 20 K (ΔT = ±10 K), as it reflects intrinsic properties of the proposed device around its intended temperature of use. Choice of these temperatures assure that VO₂ is in either completely metallic or completely insulator phase. For thermal transistor calculations, we considered a gradual change in optical properties of VO₂ from 341 K to 345 K rather than modelling it as a sudden change. This provides a better insight into transistor-like behavior of proposed device.

2. Theoretical fundamentals

A well known formula derived from dyadic Green’s function formalism [30] has been employed to calculate near-field radiative transfer between planar surfaces of thermal diode and thermal transistor.

Here, k_ρ is the parallel component of wavevector and the integrand is known as energy transmission coefficient and can be defined as

As our proposed designs involve 1-D grating structure of VO₂ in vacuum, we use second order approximation of effective medium theory to obtain the dielectric properties given by the expressions [31–33]

In this study, temperatures involved are around 341 K and grating period (50 nm) is much less than the thermal wavelength (∼ 8.5 μm). Thus, the condition for effective medium approximation [33] is satisfied. For the near-field radiative transfer pertaining to periodic gratings, effective medium theory is valid for gaps larger than the grating period [35]. If the condition for EMT is not satisfied, more rigorous numerical methods such as discussed in Lussange et al. [36] and Guérout et al. [37] should be used.

The insulating state of VO₂ (below 341 K) is anisotropic and supports several phonon modes. Such anisotropic optical response can be described using separate classical oscillator formula $\epsilon (\omega )={\epsilon }_{\infty }+{\displaystyle \sum _{i=1}^N\frac{S_i{\omega }_i^2}{{\omega }_i^2-j{\gamma }_i\omega -{\omega }^2}}$ for ordinary and extraordinary mode of dielectric function. In our case, ordinary mode ε_O is along the plane perpendicular to the optical axis (x − y plane) and extraordinary mode ε_E is along the optical axis z axis. Note that, for both the thermal diode and thermal transistor configuration, the optical axis is assumed to be along the distance between the bodies. Barker et al. [38] provides experimental values of high-frequency constant ε_∞, phonon frequency ω_i, scattering rate γ_i and oscillator strength S_i. In the metallic state, VO₂ is isotropic and the dielectric function follows Drude model [38] given by $\epsilon (\omega )=\frac{-{\omega }_p^2{\epsilon }_{\infty }}{{\omega }^2-j\omega \mathrm{\Gamma }}$. The dielectric function for BN is taken from Palik [39] and is of the form $\epsilon (\omega )={\epsilon }_{\infty }\frac{\left({\omega }^2-{\omega }_{LO}^2+j\omega \gamma \right)}{\left({\omega }^2-{\omega }_{TO}^2+j\omega \gamma \right)}$. Here ω_TO and ω_LO are known as transverse and longitudinal optical phonon frequencies respectively, and γ is the damping constant. The values of ε_∞, ω_TO, ω_LO and γ for BN are 4.46, 0.1309 eV, 0.1616 eV and 6.55 × 10⁻⁴ eV respectively. Dielectric properties of gold can be found in Johnson and Christy [40].

3. Results

First, we will consider the thermal diode concept shown in Fig. 1(a). To quantify rectification, the rectification ratio R can be defined as R = (Q_f − Q_r)/Q_r where Q_f and Q_r refer to forward and reverse heat flux respectively [41]. Alternatively, the rectification coefficient can be defined as η = (Q_f − Q_r)/max(Q_r, Q_f). The goal is to increase R as much as possible so that Q_f ≫ Q_r and rectification is reasonably high to be utilized in a practical device. Fig. 2 highlights thermal rectification characteristics of thermal diodes. Heat flux is plotted against temperature difference between active and passive sides. Two cases are considered, the first is a simple diode with bulk VO₂ and bulk BN while the second is the configuration shown in Fig. 1. 1-D rectangular grating has a period (Λ) of 50 nm, filling ratio (ϕ) of 0.3 and height (h) of 0.5 μm. In both the cases, the separation between the two sides is 100 nm. Owing to different optical properties of insulator VO₂ and metallic VO₂, weak rectification exists in the bulk thermal diode device as can be seen from Fig. 2. For second case, when 1-D grating is used, significant enhancement in rectification is seen and rectification value is around 14 at the gap of 100 nm and temperature difference of 20 K (ΔT =10 K).

 

Fig. 2 Rectification characteristics of the proposed thermal diode highlighted by heal flux plot against temperature difference. Positive temperature difference corresponds to forward bias while negative temperature difference corresponds to reverse bias.

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In order to understand near-field thermal rectification across proposed thermal diode, we plot energy transmission coefficient ξ(ω, k_ρ) across the interfaces of the configuration considered (Figs. 3(a)) at a gap of 100 nm. Here k_ρc/ω is normalized parallel wavevector. In the forward bias configuration, the transmission coefficient is close to unity well beyond light line (k_ρc/ω = 1). The two prominent frequencies (highlighted by dashed lines) seen where transmission coefficient is high are close to characteristic wavelengths of BN (7.6 μm and 9.8 μm). Since metallic VO₂ does not support surface phonon polariton in the infrared region [18], the symmetric and antisymmetric surface phonons supported by the BN layer (k_ρc/ω ≫ 1 in Fig. 3) are primary modes of near-field radiative transfer in forward bias. In addition, there exists a secondary contribution due to Fabry-Perot modes and frustrated modes (k_ρc/ω ≈ 1) in the near-field regime. High energy transmission in forward bias is due to tunneling of surface waves across interfaces. In reverse bias, although both BN and insulator VO₂ support surface phonon modes, they do not overlap and near-field radiative transfer is dominated by non-resonant surface waves. Note that although metallic VO₂ does not support surface phonons, surface phonons of BN exists in a range where metallic VO₂ has high extinction coefficient. On the contrary, surface phonons of insulator VO₂ and BN occur in frequency range where the opposite side has low extinction coefficient (κ ≈ 0). This results in a mismatch of phonon frequencies. As a result, tunneling between BN and insulator VO₂ is much weaker than that between BN and metallic VO₂, that leads to thermal rectification. Furthermore, when bulk VO₂ is replaced by a 1-D rectangular grating, the transmission coefficient for reverse bias is reduced due to the presence of grating which suppresses the tunneling of surface waves supported by insulator VO₂ and BN. Tunneling between BN and metallic VO₂ however is relatively unchanged. Consequently, a higher rectification is achieved. Resulting difference in spectral heat fluxes can also be observed. More detailed analysis of effect of 1-D gratings on thermal rectification when compared to bulk and thin films can be found in Ghanekar et al. [19].

 

Fig. 3 Coefficient of energy transmission ξ(ω, k_ρ) across the the two interfaces of thermal diode with 1-D rectangular grating plotted against angular frequency ω and normalized parallel wavevector k_ρc/ω for (a) forward bias, (b) reverse bias cases.

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Next, we consider an extension of thermal diode concept that is, a thermal transistor. Only a specific case of thermal transistor is analysed. It has a structure similar to the thermal diode discussed earlier. The source and the drain has 1 μm layer of BN over 1 μm layer of gold. The gate has two faces and each face has a 0.5 μm thick 1-D rectangular grating of VO₂ with period Λ = 50 nm and filling ratio ϕ = 0.3. The gate is separated by 100 nm from the source and the drain. The optical axis of insulating VO₂ lies along the separation. Source temperature T_S and drain temperature T_D are fixed, while gate temperature is varied. To realistically model properties of VO₂ during phase transition, effective medium approximation is used. As the temperature is increased, metallic puddles are formed whose volume fraction gradually increased to 1 at 345 K [25]. Alternatively, Bruggeman approximation can also be used. As seen in Ben-Abdallah and Biehs [5], both EMT and Bruggeman approximation lead to very similar results.

Figure 4 displays transistor characteristics of the proposed thermal transistor for two different cases of source and drain temperature. Source and drain temperatures are kept constant while gate temperature is varied. Heat flux across the source (Q_S), the drain (Q_D) and the gate (Q_G) are plotted against gate temperature T_G. Under steady state operation, Q_S = Q_D + Q_G. In Fig. 4(a), source temperature T_S and drain temperature T_D are 317 K and 321 K, respectively. Upon phase transition of VO₂, a small change in gate heat flux Q_G and a large change in Q_D is seen. Before and after the phase transistor region, the heat transfer coefficient for Q_S and Q_D (e.g. Q_S/(T_S − T_G) for Q_S) is nearly constant, owing to linear temperature dependence of radiative heat transfer for small temperature differences. During phase transition however, the heat transfer coefficient varies with T_G. This is the transistor-like behavior. Note that earlier studies Ben-Abdallah and Biehs [5] and Prod’homme et al. [20] show similar transistor characteristics with exactly opposite trend. For instance, increase in gate temperature leads to decrease in heat flux during phase transition while in present study it increases. This transistor-like behavior is simply inherited from corresponding thermal diodes that would also show an opposite trend. Present transistor behaves more like an amplifier than a switch. It can be observed that, by changing the temperatures T_S and/or T_D, one can change the sensitivity of the transistor. In Fig. 4(a), Q_G is positive meaning heat is flowing out of the gate. This setting is not desirable as cooling is more complex than heating. Hence in Fig, 4(b), we consider a case where heat is flowing into the gate. This configuration is achieved simply because T_S − T_G < T_G − T_D. Here T_S and T_G are chosen to be 361 K and 311 K, respectively. As seen in Prod’homme et al. [20], for some combination of T_D and T_S, Q_G varies monotonically (no local maxima). However, for any combination of T_S and T_D, local maxima of Q_G if exists, must occur within the transition zone of VO₂. Transistor-like amplification can be characterized by amplification factor α defined as [24],

 

Fig. 4 Heat flux across source, drain and gate is plotted against gate temperature while source and drain temperatures are fixed. (a) T_S = 371 K and T_D = 321 K, (b) T_S = 361 K and T_D = 311 K.

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Note that, when Q_G reaches a maxima, δQ_G ≈ 0. This produces extremely high value of α which is not a true representative of the overall transistor behavior. Therefore we define an alternative way to quantify amplification,

4. Conclusion

Thus we have theoretically demonstrated that an enhanced thermal rectification can be achieved in a thermal diode and a thermal transistor by using 1-D grating of phase change material VO₂. For the near-field thermal diode considered in the study, we predict a reasonably high value of rectification ratio (∼ 14) that can be obtained at a gap of 100 nm. Improved rectification is attributed to reduced tunneling of surface waves across the interfaces for reverse bias. The concept of thermal diode was extended to thermal transistor with a source, a drain and a gate. Source and drain temperatures are fixed above and below the phase transition temperature of VO₂ respectively, while gate temperature is varied. The device displays a typical transistor-like behaviour wherein a small change in gate heat flux leads to a large change in drain heat flux. Essentially, the gate acts like a tap that controls heat flow across the source and the drain.