Near-field thermal rectification devices using phase change periodic nanostructure

: We theoretically analyze two near-ﬁeld thermal rectiﬁcation devices: a radiative thermal diode and a thermal transistor that utilize a phase change material to achieve dynamic control over heat ﬂow by exploiting metal-insulator transition of VO 2 near 341 K. The thermal analogue of electronic diode allows high heat ﬂow in one direction while it restricts the heat ﬂow when the polarity of temperature gradient is reversed. We show that with the introduction of 1-D rectangular grating, thermal rectiﬁcation is dramatically enhanced in the near-ﬁeld due to reduced tunneling of surface waves across the interfaces for negative polarity. The radiative thermal transistor also works around phase transition temperature of VO 2 and controls heat ﬂow. We demonstrate a transistor-like behavior wherein heat ﬂow across the source and the drain can be greatly varied by making a small change in gate temperature. terms


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][7][8], while more recently radiative thermal rectification devices have gathered considerable attention [4,5,[9][10][11][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 0.7 Ca 0.15 Sr 0.15 MnO 3 (LCSMO) and vanadium dioxide (VO 2 ) [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 VO 2 based simple far-field radiative thermal diode, numerous studies on thermal rectification [16][17][18][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, VO 2 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 VO 2 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 VO 2 as phase-transition material. VO 2 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 VO 2 , 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 VO 2 .
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 VO 2 at temperature T 1 = 341 K + ∆T. On the passive side, structure 2 has its temperature T 2 = 341 K − ∆T. Mean temperature is chosen to be the phase transition temperature of VO 2 at 341 K. When T 1 > T 2 (referred to as forward bias), VO 2 layer is in metallic phase; when T 1 < T 2 (reverse bias), VO 2 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 VO 2 , while gate temperature is varied. Proposed configurations will be discussed in more detail along with results.
Although 341 K is said to be the phase transition temperature of VO 2 . 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 (∆T = ±10 K), as it reflects intrinsic properties of the proposed device around its intended temperature of use. Choice of these temperatures assure that VO 2 is in either completely metallic or completely insulator phase. For thermal transistor calculations, we considered a gradual change in optical properties of VO 2 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.

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.
where Θ(ω, T) = ( ω/2) coth( ω/2k B T) is the energy of harmonic oscillator at frequency ω and temperature T, is the reduced Planck constant, and k B is the Boltzmann constant. The function T 1→2 (ω, L) is known as the spectral transmissivity in radiative transfer between media 1 and 2 separated of distance L [18, 26, 27, 30]. The function T 1→2 (ω) corresponds to the spectral transmissivity in radiative transfer between media 1 and 2 separated by distance L and is expressed as Here, k ρ is the parallel component of wavevector and the integrand is known as energy transmission coefficient and can be defined as 2 are polarization dependent reflection coefficients of the two half spaces, µ = s (or p) refers to transverse electric (or magnetic) polarization, and k z is the z-component of wavevector in vacuum. k ρ ≤ ω/c and k ρ > ω/c correspond to propagating and evanescent modes, respectively.
As our proposed designs involve 1-D grating structure of VO 2 in vacuum, we use second order approximation of effective medium theory to obtain the dielectric properties given by the expressions [31-33] where ε A and ε B are dielectric functions of the two materials (VO 2 and vacuum) in surface gratings, λ is the wavelength, Λ is grating period and filling ratio φ = w/Λ where w is the width of VO 2 segment. The expressions for zeroth order effective dielectric functions ε T E,0 and ε T M,0 are given by [31,34] 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. The insulating state of VO 2 (below 341 K) is anisotropic and supports several phonon modes. Such anisotropic optical response can be described using separate classical oscillator formula i − jγ i ω − ω 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 . Here

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 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 VO 2 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 VO 2 and metallic VO 2 , 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 (∆T =10 K).
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 ρ c/ω is normalized parallel wavevector. In the forward bias configuration, the transmission coefficient is close to unity well beyond light line (k ρ c/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 VO 2 does not support surface phonon polariton in the infrared region [18], the symmetric and antisymmetric surface phonons supported by the BN layer (k ρ c/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 ρ c/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 VO 2 support surface phonon modes, they do not overlap and near-field radiative transfer is dominated by non-resonant surface waves. Note that although metallic VO 2 does not support surface phonons, surface phonons of BN exists in a range where metallic VO 2 has high extinction coefficient. On the contrary, surface phonons of insulator VO 2 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 VO 2 is much weaker than that between BN and metallic VO 2 , that leads to thermal rectification. Furthermore, when bulk VO 2 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 VO 2 and BN. Tunneling between BN and metallic VO 2 however is relatively unchanged. Consequently, a higher rectification is achieved. Resulting difference in spectral heat fluxes can also be observed. More detailed analysis 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 VO 2 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 VO 2 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 VO 2 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 VO 2 , 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 VO 2 . Transistor-like amplification can be characterized by amplification factor α defined as [24], 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, where, ∆Q G is the maximum change in Q G during phase transition and ∆Q D is the corresponding change in Q D . The points corresponding to ∆Q G and ∆Q D are highlighted in Fig. 4. For Fig.  4(a) and 4(b), α using this definition is 2.24 and 2.36, respectively. The fundamental difference between the thermal diode and the thermal transistor is that the transistor would behave the same way even when polarity of source and drain is reversed. If T S < T G < T D , transistor characteristics are essentially unchanged, meaning source and drain can be flipped.

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 VO 2 .
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 VO 2 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.