## Abstract

Vanadium dioxide (VO_{2}) is a material that undergoes thermal phase transition resulting in drastic changes in its material properties. The phase change can also be brought on by optical pumping. Several experimental results have been presented in the literature dealing with such phase transitions brought on by optical pumping. In this manuscript we present a theoretical framework, which addresses this problem by self consistently solving the electromagnetic problem and the thermodynamic problem using a multiphysics approach when such transitions are thermally mediated, as is the case with continuous-wave optical pumps. Such an analysis provides us with insights into the transition process and also helps explain the conditions under which some of the observed experimental results like bistability takes place. Such optically induced phase transition materials also present the intriguing possibility of ultrahigh nonlinearity where the input optical signal essentially converts a dielectric into a plasmonic material. These materials can find significant applications in nonlinear metatronics.

© 2015 Optical Society of America

## 1. Introduction

Vanadium dioxide (VO_{2}) is a phase change material that has been observed to undergo a thermally induced insulator-metal phase transition around a temperature of 68 °C [1,2]. The transition occurs between a dielectric monoclinic phase at lower temperatures and a metallic rutile phase at the higher temperatures [3]. Although the VO_{2} transition is commonly controlled through thermal means, it has also been shown that the VO_{2} transition could be induced and controlled through optical means by using either continuous wave excitation [4,5] or pulsed excitation [6,7].

VO_{2} exists in two distinct phases, a low temperature insulator phase where it possesses a monoclinic structure, and a high temperature plasmonic phase with a tetragonal rutile structure. At intermediate temperature VO2 exists as a mixture of the two constituent phases and its properties could be described using the effective medium theory (EMT) [8–10]. Such a mixture could be parameterized using the filling fraction [8] (*f*) which represents the fraction of the material existing in the plasmonic phase, i.e. *f* is 0 and 1 for the insulator and plasmonic phase, respectively, and *f* is between 0 and 1 while the material is undergoing the transition. Figure 1(c)-1(d) shows the permittivity of VO_{2} for various values of the filling fraction. The transition values of permittivity were calculated using Bruggeman EMT [1,10]. As seen in the figure, the permittivity shows a gradual transition from dielectric to plasmonic behavior as *f* is increased from 0 to 1. The change is more significant at the longer wavelengths where the real part of permittivity becomes negative for higher values of *f*.

The phase transition of VO_{2} has been studied thoroughly when the phase transition is induced by thermal means [2,3,11]. There are also a number of studies dealing with phase transition induced through optical means [4–7]. The optically induced transition can be caused by electronic excitations (as is the case with pulsed-laser excitation) or optical heating (as is the case with continuous-wave excitation). In this manuscript we are dealing with continuous-wave excitations, which are largely thermally mediated. It is fairly straightforward to comprehend the phase transition when a sample of VO_{2} is heated to achieve the transition since the phase of VO_{2} is directly related to its temperature, but it is more complicated to understand the process when a VO_{2} sample is heated using an optical source. The complexity arises from the fact that the optical heating in the VO_{2} sample depends on the permittivity of VO_{2} and as the temperature changes the permittivity of VO_{2} changes so will the amount of incident optical energy that is converted to heat. This “nonlinearity” in the optical properties could give rise to interesting responses including bistability. The schematic of the structure studied in this manuscript is shown in Fig. 1(a). An optical input signal, which also acts as the pump, is normally incident on a VO_{2} film deposited on a glass substrate resulting in absorption of the pump by the VO_{2} film and consequent increase in temperature. We assume that the pump is in the form of a continuous wave such that at steady state the heat generated in the VO_{2} film is dissipated through the glass substrate by heat conduction and to the air by heat convection [12].

## 2. Theory and results

In the previous section we saw how the permittivity of VO_{2} can show significant variation while it is undergoing the dielectric-plasmonic transition. The phase of VO_{2} depends on the temperature and shows a hysteretic response that is typical of first order transitions [2]. The exact parameters of the transition depend on the fabrication of the sample, but the transition is usually centered around 341 K (68 °C) [2,3,5]. In the current work, we assume that the transition begins and ends at 335 K and 345 K, respectively, while the sample is being heated and conversely the transition back to the dielectric phase begins and ends at 335 K and 325 K, respectively, while it is being cooled. To model the phase transition we assume that the filling fraction (*f*) varies smoothly as a function of the temperature. This behavior is represented in Fig. 2. We have modeled the filling fraction in the transition region using a fifth order polynomial and ensuring that the first- and second-order derivatives of *f* with respect to the temperature stay continuous. Having continuous first- and second-order derivatives helps in the convergence of the numerical simulation, but the procedure we describe here does not need a particular polynomial fitting and we could use any reasonable dependence between *f* and the temperature.

In order to model the behavior of a VO_{2} film we need to solve the electromagnetic (EM) problem and the heat transfer (HT) problem simultaneously. The EM problem provides us with the heat generation in the VO_{2} film in the form of absorption. This heat source is then used to model the HT problem, which gives us the temperature distribution in the sample. The temperature further modifies the permittivity of VO_{2} thus affecting the solution of the EM problem. In order to find the response we need to find a self-consistent solution to the EM and HT problem. As seen in Fig. 1(a), the HT problem involves heat conduction in the VO_{2} film and the substrate, and heat convection in the air. Although air could also conduct heat the convection in air significantly dominates over conduction and we could safely ignore the heat conduction in air. The heat conduction process in solids is modeled using the heat equation [12].

*Q*represents the heat source and

*k*represents the thermal conductivity of the material. The thermal conductivity of VO

_{2}is 4 W/(m K) and 6 W/(m K) for the dielectric and metallic phase, respectively [13]. The thermal conductivity of the glass substrate is around 1 W/(m K) depending on the kind of glass [12]. On the side of air the VO

_{2}loses heat by convection into the air. Unlike conduction, heat convection is a fairly complicated process that strongly depends on a number of parameters and conditions including the orientation of the surface, air flow conditions and whether the flow is laminar or turbulent and there are detailed theories on how to model the convection process under various conditions [12]. As a simplification the interface experiencing the convective heat transfer can be modeled using Newton’s law of cooling as shown below [12].where

*q*represents the heat flux across the interface,

*T*is the temperature of the surface and

_{s}*T*is the temperature of the ambient fluid, i.e. air in this case. The parameter

_{a}*h*is referred to as the convective heat flux coefficient and it has the units W/(m

^{2}K). This parameter reflects the combined effect of all the conditions that affects the heat convection process and can change significantly depends on the surrounding conditions including air flow. In the present study, we will be assuming a value of 300 W/(m

^{2}K) for

*h*of air.

We also need to account for the heat transfer process occurring at the far end of the glass substrate. In case the sample is placed on a heat conductive surface with a fixed temperature we can assume that the far end of the glass substrate is held at a constant temperature and solve the heat transfer problem using a Dirichlet boundary condition. But more frequently in optical experiments the whole sample is held in an optical holder such that both sides are exposed to air. In this case, the far end of the glass substrate also experiences convective heat transfer to air. If we assume that the VO_{2} is at a temperature *T _{f}*, and the temperature drop across the glass substrate is given by Δ

*T*, the convective heat flux from the glass surface is given by

*h*(

*T*– Δ

_{f}*T*–

*T*). This rate should be equal to the conductive heat flux across the glass substrate given by

_{a}*k*Δ

*T*/Δ

*x*, where Δ

*x*is the thickness of the substrate. On equating the two terms we find that the temperature drop across the substrate is given by the following.

_{2}sample is exposed to air on both side. In case the glass substrate cannot be ignored we can still assume that the glass side of the VO

_{2}film experiences a convective heat flux with a modified convective heat flux coefficient given by $h/\left(1+\frac{h\Delta x}{k}\right)$.

As mentioned before, in order to model the behavior of the VO_{2} film the heat transfer problem has to be solved along with the EM problem in a self-consistent manner. In case of a thin VO_{2} film we can assume that the temperature stays constant within the VO_{2} film. Any temperature gradient within the sub-micron film would imply an unusually high heat flux. This allows us to analytically solve the problem as described below. Firstly, we describe the filling fraction *f*, as a function of the film temperature (*T _{f}*) with dependence as shown in Fig. 1(a). This relationship can be expressed in an equation form as $f=\theta \left({T}_{f}\right)$. The permittivity and consequently the refractive index (

*n*) of VO

_{2}can be expressed as a function of

*f*by using the effective medium theory as shown in Fig. 1(c)-1(d) [8–10], which can be further used to calculate the transmittance, reflectance and absorptance of the VO

_{2}film as a function of

*f*, i.e. $A=\varphi \left(f\right)$. The exact equations for the transmittance, reflectance and absorptance are shown below.

*n*, as a function of

*f*. Also,

*t*represents the thickness of the VO

_{f}_{2}film and

*n*represents the refractive index of the glass substrate. In case of a thin film we could use a Taylor series approximation in terms of the film thickness and express the absorptance in the following form.

_{g}*ε'*and

*ε”*represent the real and imaginary parts of the film permittivity. Under equilibrium the heat lost by the VO

_{2}film due to the convective heat flux is equal to the absorbed intensity and can be written as ${I}_{abs}={h}_{eff}\left({T}_{f}-{T}_{a}\right)$ where

*h*accounts for the heat convection on both sides of the sample. Since the absorbed intensity and the absorptance are related as ${I}_{abs}=A{I}_{inc}$, where

_{eff}*I*represents the incident intensity, we can express the relationship between the incident intensity and the film temperature as follows.

_{inc}_{2}film as a function of the incident intensity at two different wavelengths. The permittivity shows a hysteretic response similar to the hysteresis observed on thermally induced phase transition in VO

_{2}. When the incident pump is set to 500 nm wavelength the transition proceeds and the metal filling fraction of the film gradually increases from 0 to 1 as we gradually increase the incident intensity. The transition occurs when the incident intensity is around 3 to 4 W/cm

^{2}and is consistent with experimentally observed values in literature [4]. On the other hand when the incident pump is set to 900 nm wavelength the transition shows an unstable branch indicating a bistable behavior. In this case as we increase the pump intensity the film stays in the dielectric phase until the intensity is around 6.7 W/cm

^{2}when the film abruptly flips to ametallic state. On reducing the intensity the film stays in the metallic state until the intensity reaches a value of 4 W/cm

^{2}when it begins transitioning back to a dielectric phase. On further reducing the incident intensity to around 3.75 W/cm

^{2}, when the metal filling fraction is around 0.8, the film abruptly switches back to the dielectric phase. Figure 2(c) shows the wavelength and incident intensity combinations where the film exists in a bistable state. We can see that at shorter wavelengths the film does not exhibit any bistability and consequently we can tune the permittivity by gradually changing the incident intensity. The tunability is limited by the hysteresis in the permittivity and consequently we have two different tunability regimes corresponding to when the film is heating up or cooling down. But at longer wavelengths the film in inherently bistable and the region of bistability increases as we further increase the incident wavelength. This explains why a VO

_{2}film shows a gradual transition when excited by a 532 nm pump [4] but shows a strongly bistable response when excited by a 1.55

*µ*m pump [14].

To verify our theoretical results we performed numerical simulations of the same geometry. The simulations were performed on a commercial Finite-Element-Method (FEM) solver (COMSOL Multiphysics^{®}), which allows simultaneous simulation of the EM and HT problems with coupling between the two problems. The simulation results for the permittivity of the film are shown in Fig. 3(a)-3(b) along with the analytical results. We can see that there is a very good agreement between the theoretical and numerical results. Also as expected the numerical simulations cannot access the unstable branch when the wavelength is 900 nm. There is no such unstable branch when the wavelength is set to 450 nm. Figure 3(c) shows the norm of the electric field when the wavelength and the intensity are set to 900 nm and 5 W/cm^{2} respectively. We can see that the film behaves like a metallic film during the cooling phase and shows a stronger reflection and lower transmittance. But during the heating phase the same film behaves like a dielectric film even though the wavelength and intensity are at the same value.

## 3. Summary and conclusion

We have demonstrated that by simultaneously solving the heat transfer and electromagnetic problem in a self-consistent manner we can accurately describe the transition behavior of a VO_{2} film when the phase transition is induced by means of a continuous-wave optical pump. The continuous-wave-pump-induced transition is primarily a thermally mediated transition and can be adequately described using the heat transfer and electromagnetic model. The self-consistent solution shows good agreement with experiments reported in the literature, and also predicts the bistable states observed in experiments elsewhere when the pump is set to longer wavelengths. The analytical results were also confirmed with numerical solutions obtained using multiphysics simulations.

## Acknowledgments

This work was supported in part by the US Air Force Office of Scientific Research (AFOSR) grant number FA9550-10-1-0408.

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