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Curr. Opt. Photon. 2022; 6(6): 583-589

Published online December 25, 2022 https://doi.org/10.3807/COPP.2022.6.6.583

A Theoretical Study on the Low Transition Temperature of VO2 Metamaterials in the THz Regime

Jisoo Kyoung

Department of Physics, Dankook University, Chungnam 31116, Korea

Corresponding author: *kyoungjs@dankook.ac.kr, ORCID 0000-0001-6736-9118

Received: July 11, 2022; Revised: October 4, 2022; Accepted: November 2, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Vanadium dioxide (VO2) is a well-known material that undergoes insulator-to-metal phase transition near room temperature. Since the conductivity of VO2 changes several orders of magnitude in the terahertz (THz) spectral range during the phase transition, VO2-based active metamaterials have been extensively studied. Experimentally, it is reported that the metal nanostructures on the VO2 thin film lowers the critical temperature significantly compared to the bare film. Here, we theoretically studied such early transition phenomena by developing an analytical model. Unlike experimental work that only measures transmission, we calculate the reflection and absorption and demonstrate that the role of absorption is quite different for bare and patterned samples; the absorption gradually increases for bare film during the phase transition, while an absorption peak is observed at the critical temperature for the metamaterials. In addition, we also discuss the gap width and VO2 thickness effects on the transition temperatures.

Keywords: Critical temperature, Insulator-to-metal phase transition, Slit array, Terahertz, Thin film

OCIS codes: (050.1220) Apertures; (050.6624) Subwavelength structures; (160.3900) Metals; (160.3918) Metamaterials; (310.6860) Thin films, optical properties

Vanadium dioxide (VO2) is a classical transition metal oxide material that undergoes a reversible insulator-to-metal phase transition (IMT) near room temperature (Tc = 340 K) [1]. Due to the huge change in conductivity (four to five orders of magnitude) during the phase transition, VO2 has been intensively studied with respect to its fundamental physics [2–4] as well as its potential applications [5, 6]. This transition can be triggered by thermal [7], optical [4, 8, 9], electrical [10–12], and stress excitation [13]. The metal-insulator transition time ranges from few minutes to sub-picosecond depending on the excitation method. In particular, in the terahertz spectral range (0.1–3 THz), the giant change in conductivity enables large transmission modulation up to 50–80 % with only a few hundred nanometers thick VO2 film (~λ/10,000) [14]. This is because the THz wave is extremely sensitive to the free carrier dynamics in the thin film [15, 16]. Owing to these unique functionalities, VO2 materials have been widely used to realized active metamaterials in the THz spectral range. For example, a polarization tunable multiband resonator [17], controllable terahertz active absorber [18], switchable plasmonic metasurface with more than 130-degree phase shift [19], and tunable optical activity [20] have been recently demonstrated. Additionally, some early studies dramatically enhanced the transmission modulation depth by patterning negative type metallic nanostructures, slot antennas, or slit arrays on VO2 thin film [7, 8]. It is also reported that these metallic patterns significantly lower the transition (critical) temperature of VO2. Metal nanostructures made of gold can be fabricated on the VO2 thin film by standard photo or e-beam lithography and lift-off process. A titanium or chromium adhesive layer a few nanometers thick is widely used to improve the adhesion between gold and VO2. In particular, Jeong et al. [21] showed that the transition temperature could be lowered by 20 degrees with 5-nm-width gaps compared to the bare VO2 film. In this work, we theoretically study the phase transition characteristics of VO2 metamaterials (the periodic metal nanogap fabricated on the VO2 thin film) to understand the low transition temperature. By developing an analytical model of transmission and reflection through slit arrays on VO2 thin films, we demonstrate that the large modulation depth and shift of the transition temperature originate from the rapidly increasing reflection during the phase transition. Unlike the metamaterials, absorption plays a significant role in reducing transmission for bare VO2 thin film. In addition, we investigate the slit width and VO2 film thickness dependence on transmission modulation.

2.1. Theoretical Model

Figures 1(a) and 1(b) show the schematic diagrams of the bare VO2 thin film and the metamaterial (slit arrays on VO2), respectively. Both were placed on sapphire, one of the most widely used substrates to grow high quality VO2 thin film. We assumed that the thickness of the sapphire substrate was infinite with a refractive index of 3.1 in the THz spectral range. Ordinary simulation tools (finite difference time domain or finite element method) are hard to use for our system because the thicknesses of the metal and the VO2 thin film are too thin compared to the wavelength. Therefore, the development of analytical models is necessary. To calculate transmission and reflection by the bare VO2 film, a well-known transfer matrix method was used, while the modal expansion method [22, 23] was employed for metamaterials. Since the transfer matrix method has been introduced in the most optics textbooks, here we only discuss the modal expansion method. The metal film making up the slit can be assumed to be a perfect electric conductor (PEC), because most metals in the THz spectral range have a very high refractive index [24, 25]. The width, thickness, and period of the slit array are denoted by w, h, and p respectively, while d represents the VO2 thin film thickness. nt and ns are the refractive indices of the VO2 thin film and the sapphire substrate.

Figure 1.Schematic diagrams for (a) bare vanadium dioxide (VO2) film and (b) VO2 metamaterials. (c) Transmission versus temperature for the bare film (red dots) and metamaterials (blue triangles). (d) Normalized transmission of (c). The maximum and minimum transmission are adjusted to be 1 and 0, respectively, to easily compare the critical temperature.

The electromagnetic waves polarized along the x direction are incident from the air towards the slit (traveling in the z axis). The incident and reflected waves in region I and the transmitted waves in region IV can be expressed by the modal expansion method:

HyI=HyI,inc+HyI,ref=ε0μ0eikz+h+ε0μ0mR meik mxw/2+κ m1z+h

HyIV=ε0μ0mT meik mxw/2κ mnszd

where k=2πλ,km=2πpm,κmni=nj2k2km2, and nj is the refractive index of j’s region (j = I, II, III, or IV). Rm and Tm are reflection and transmission coefficients, respectively. The common time dependence term eiωt is omitted and the corresponding electric field can be derived from the Maxwell equation:

×H=iωεE

where ε is the permittivity of the matter. Inside the slit, the single mode approximation can be applied [26]:

HyII=Feikz+h/2+Beikz+h/2

where F and B are forward and backward propagation coefficients, respectively. Likewise, in the thin film, both the forward and backward propagations exist, which can be written as:

HyIII=ε0μ0mC me iκ mntz+D me iκ mntzeik m xw/2

where Cm and Dm are the coefficient of the forward and backward radiation, respectively, at each mode inside the thin film. After applying the boundary conditions at the three interfaces (superstrate/slit array, slit array/thin film, thin film/substrate), the zeroth order transmission coefficient T0 and reflection coefficient R0 can be expressed as

T0=wp2D11 nscos ntkdi1 ntsin ntkd

R0=1+wp2Dcoskh+iWnt,nssinkh

where

D=i1+W1Wnt,nssinkhW1+Wnt,nscoskh

W1=kwpm1κ m1sinc2wk m2

Wnt,ns=kwpmnt2κ m ntsinc2 wkm2G mnt,nsd

Gmnt,nsd= κm nt nt2cos κm ntdi κm ns ns2sin κm ntdi κm nt nt2sin κm ntd+ κm ns ns2cos κm ntd

The transmission T and reflection R are given by:

T= T02ns,R= R0 2

The absorption A was estimated from the energy conservation:

T+R+A=1

Based on our theoretical models (transfer matrix and modal expansion), we calculated the transmission, reflection, and absorption at 0.6 THz as a function of temperature for both bare and metamaterials. The temperature-dependent refractive indices of VO2 thin film used in the models will be explained in the next section.

2.2. Dielectric Constants of VO2

In our simulation, we assumed that the dielectric permittivity of VO2 in the terahertz range can be described by the Drude model:

ϵω=ϵωp2σω2+iγω

where ϵ = 12 is the permittivity at the infinite frequency, ωp(σ) is the conductivity-dependent plasma frequency and γ is the collision frequency. According to [27], the plasma frequency at σ can be approximately expressed as ωp2(σ) = σσ0ωp2(σ0) with σ0 = 3 × 105 S/m, ωp(σ0) = 1.4 × 1015 rad/s, and γ = 5.75 × 1013 rad/s, which is assumed to be independent of σ. The conductivity during the phase transition at 0.6 THz was extracted from Fig. 1(c) in [14] and listed in Table 1. By substituting conductivities into Eq. (14), the temperature-dependent dielectric constants were calculated and then the corresponding refractive indices for transfer matrix and modal expansion were figured out through the relation: ϵ(ω) = nt2 (ω).

TABLE 1 Temperature-dependent conductivity of vanadium dioxide (VO2) taken from [14]

Temperature (K)Conductivity (Ω · cm)−1
310.011.0
313.414.2
317.017.0
320.120.5
322.825.2
325.731.2
327.442.5
328.679.3
329.3127.5
330.0204.0
331.4311.6
332.6430.6
333.7538.2
335.0637.4
337.5759.2
340.0858.4
342.6915.0
345.0937.7
347.5966.0
350.1980.2
352.9994.3
356.21001.8
359.81002.8
363.11011.3
366.61016.0
370.11017.0

Figure 1(c) shows the calculated transmission through the bare film (red dots) and the VO2 metamaterials (blue triangles) during the phase transition at 0.6 THz. The thickness of the VO2 film was d = 100 nm for both the bare and patterned samples. The metal thickness h, period p, and width w of the slit array were 100 nm, 2 μm, and 100 nm, respectively. The transmission through the bare film was about 72% in the insulating state (310 K), while 20% transmission was recorded in the metallic state of VO2 (370 K). We would like to emphasize that our dielectric model of VO2 in Table 1 with the transfer matrix method reproduced the experimental results very well including the abrupt change of the transmission curve near the critical temperature [for example, see Fig. 1(c) in [14]. With the slit arrays on the VO2 thin film, the transmission through the sample was slightly reduced to 62% in the insulating state. Considering the extremely small opening ratio [(empty space between the metals versus total area) = wp] of the slits, 5%, this is a very surprising result. That is, compared to the bare film, the opening area of the surface is reduced by 95%, but the transmission is reduced by only 10%. This unexpectedly high transmission originates from the giant field enhancement at the gap, which is inversely proportional to wp [28].

When the VO2 metamaterial transitions to a metallic state, the transmission becomes almost zero; the transmission is further reduced by 19% compared to the bare film. The largely enhanced dynamic range or modulation depth (ratio between the maximum and minimum transmission) has also been observed in previous experimental work [7–10]. In addition to the modulation depth, there are two major differences between the phase transition characteristics of the bare film and the metamaterial. One is the slope of the transmittance decrease in the temperature range of 310 K to 330 K. For the bare film, the slope was almost zero even though the conductivity had increased by 20 times due to the thinness of the VO2 film (100 nm) compared to the wavelength (500 μm). On the contrary, the metamaterial had a distinct negative slope with the same conductivity values. The other major difference is the position of the critical temperature. In order to compare the critical temperature clearly, we normalized the transmission curves: the maximum and minimum transmission were set to 1 and 0, respectively [Fig. 1(d)]. As is clearly seen, the critical temperature had been shifted to a low temperature by several kelvin. These two characteristics of VO2 metamaterial during the phase transition were demonstrated by the experiment [21]. This “early phase transition” of the metamaterial originates from the fact that the metallic nanostructures can confine the long wavelength light in a very small region, which is very sensitive to changes in the dielectric environment [29, 30].

Experimentally, the early transition of VO2 metamaterial has been investigated only in the transmission geometry [21]. In this work, we calculated not only transmission but also reflection and absorption for both bare films and metamaterials to elucidate the main causes of low critical temperature. Figures 2(a) and 2(b) show the stack plot of absorption (blue area with ‘+’ hatch), reflection (orange area with ‘\’ hatch), and transmission (green area with ‘x’ hatch) for bare and metamaterial, respectively, as a function of temperature. For the bare film, absorption at room temperature was negligible even though large imaginary part of refractive index of VO2 because the film thickness was 5,000 times smaller than the wavelength. Instead, reflection was about 27%, so that the transmission reached 72% as discussed previously. Interestingly, absorption suddenly increased near the critical temperature (~335 K) and eventually exceeded 20% in the metallic state. The reflection also rose up to 56%. Therefore, unlike the insulating state, both reflection and absorption play an important role in the metallic state.

Figure 2.Stack plots of transmission, reflection, and absorption for (a) bare vanadium dioxide (VO2) film and (b) VO2 metamaterial.

In contrast to the bare film case, the absorption of the VO2 metamaterial at room temperature was not negligible; it was about 7% at 310 K, seven times greater than the bare film case. However, reflection (~31%) was still much larger than absorption. Meanwhile, in the temperature range of 310 K to 330 K, reflection as well as absorption clearly increased with increasing temperature, unlike the bare film, which shows flat reflection and absorption curves. This phenomenon appears as an early transition in transmission for the VO2 metamaterials. The dramatic difference between bare films and metamaterials occurs near the transition temperature and in the metallic state. For the bare film, the reflection and absorption gradually increased near the transition temperature and approximately half of the incident light was reflected and 20% light was transmitted with 24% absorption in the metallic state. That is, the maximum absorption appeared in the metallic state of VO2. In stark contrast, a strong absorption peak was observed at the critical temperature for the metamaterial and then the absorption decreased to a level similar to the insulator state. In other words, the maximum absorption did not occur in the metallic state of the metamaterial. Instead, the reflection increased much more rapidly compared to bare film as it passed the transition temperature. Finally, in the metallic state, reflection overwhelmed absorption and transmission. This is because the metallic VO2 blocks the slit apertures; incident light sees the flat metal plate without the aperture and most of the energy is reflected at the metallic surface. Therefore, we can conclude that the enhanced transmission modulation depth of the VO2 metamaterial is mainly due to the large reflection rather than the absorption. We would like to further point out that metal nanostructures do not increase the maximum absorption level; the maximum absorption of the bare film is about 24% in the metallic state, which is close to the maximum absorption value for the metamaterials observed at the transition temperature.

Figure 3(a) and 3(b) show the slit width and the VO2 film thickness dependence of the transition temperature of the metamaterials. The critical temperature (Tc) was defined as a temperature corresponding to a half value of the difference between the maximum and the minimum transmission. According to our definition, the critical temperature of the bare VO2 film [red dots in Fig. 1(c)] was about 331.1 K. For the metamaterials, the slit width varied from 1 μm to 50 nm with a fixed period of 2 μm. The transmission versus temperature curves were calculated at each width and the critical temperatures were extracted. The infinity mark (∞) on the x axis in Fig. 3(a) represents the bare film case. Even a relatively wide slit on the order of 1 um was shown to lower the critical temperature. In addition, the critical temperature gradually decreased with decreasing slit width, which was consistent with the previous experimental results [21]. We also investigated the VO2 thickness effect on the critical temperatures as displayed in Fig. 3(b). In this calculation, the slit width, metal thickness, and slit period remained constant, while the VO2 thickness changed from 50 nm to 1,000 nm. A gradual decrease in critical temperature was observed with increasing thickness [Fig. 3 (b)].

Figure 3.Shifts of the critical temperatures. (a) Width and (b) vanadium dioxide (VO2) thickness dependence of the critical temperature of the VO2 metamaterials.

In conclusion, we have theoretically studied the low transition temperature of VO2 metamaterial in the THz spectral range. The temperature-dependent refractive index was calculated by applying the Drude model from the experimentally measured conductivity. VO2 metamaterial consists of slit arrays on a VO2 thin film/sapphire substrate. To calculate transmission through the bare film, the transfer matrix method was used, while for the VO2 metamaterial, a modal expansion method was developed in this study. Our calculation results show that the critical temperature of the VO2 metamaterial is lowered and the modulation depth (ratio between the maximum and the minimum transmission) is largely enhanced compared to the bare film, which is consistent with earlier experimental work. We further calculated the reflection and absorption during the phase transition, and demonstrated that the absorption behaviors are significantly different between the bare and metamaterial. The absorption of the bare film is negligible in the dielectric state, and gradually increases during the phase transition. In contrast, the absorption peak is observed at the critical temperature and the absorption level in the dielectric state and the metallic state are similar for the metamaterial. Finally, we show that the narrower the slit width and the thicker the VO2 film, the lower the transition temperature. Recently, VO2-based active metamaterial or metasurfaces have been extensively studied because the conductivity of VO2 changes several orders of magnitude during the phase transition [3134]. Therefore, we believe that this theoretical work could help to design and understand the working principles of active VO2 metamaterials and metasurfaces. In addition, our results will provide an opportunity to develop near-room-temperature phase transition devices.

The author declares no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the author upon reasonable request.

The present research was supported by the research fund of Dankook University in 2020.

Research fund of Dankook University in 2020.

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Article

Article

Curr. Opt. Photon. 2022; 6(6): 583-589

Published online December 25, 2022 https://doi.org/10.3807/COPP.2022.6.6.583

A Theoretical Study on the Low Transition Temperature of VO2 Metamaterials in the THz Regime

Jisoo Kyoung

Department of Physics, Dankook University, Chungnam 31116, Korea

Correspondence to:*kyoungjs@dankook.ac.kr, ORCID 0000-0001-6736-9118

Received: July 11, 2022; Revised: October 4, 2022; Accepted: November 2, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Vanadium dioxide (VO2) is a well-known material that undergoes insulator-to-metal phase transition near room temperature. Since the conductivity of VO2 changes several orders of magnitude in the terahertz (THz) spectral range during the phase transition, VO2-based active metamaterials have been extensively studied. Experimentally, it is reported that the metal nanostructures on the VO2 thin film lowers the critical temperature significantly compared to the bare film. Here, we theoretically studied such early transition phenomena by developing an analytical model. Unlike experimental work that only measures transmission, we calculate the reflection and absorption and demonstrate that the role of absorption is quite different for bare and patterned samples; the absorption gradually increases for bare film during the phase transition, while an absorption peak is observed at the critical temperature for the metamaterials. In addition, we also discuss the gap width and VO2 thickness effects on the transition temperatures.

Keywords: Critical temperature, Insulator-to-metal phase transition, Slit array, Terahertz, Thin film

I. INTRODUCTION

Vanadium dioxide (VO2) is a classical transition metal oxide material that undergoes a reversible insulator-to-metal phase transition (IMT) near room temperature (Tc = 340 K) [1]. Due to the huge change in conductivity (four to five orders of magnitude) during the phase transition, VO2 has been intensively studied with respect to its fundamental physics [2–4] as well as its potential applications [5, 6]. This transition can be triggered by thermal [7], optical [4, 8, 9], electrical [10–12], and stress excitation [13]. The metal-insulator transition time ranges from few minutes to sub-picosecond depending on the excitation method. In particular, in the terahertz spectral range (0.1–3 THz), the giant change in conductivity enables large transmission modulation up to 50–80 % with only a few hundred nanometers thick VO2 film (~λ/10,000) [14]. This is because the THz wave is extremely sensitive to the free carrier dynamics in the thin film [15, 16]. Owing to these unique functionalities, VO2 materials have been widely used to realized active metamaterials in the THz spectral range. For example, a polarization tunable multiband resonator [17], controllable terahertz active absorber [18], switchable plasmonic metasurface with more than 130-degree phase shift [19], and tunable optical activity [20] have been recently demonstrated. Additionally, some early studies dramatically enhanced the transmission modulation depth by patterning negative type metallic nanostructures, slot antennas, or slit arrays on VO2 thin film [7, 8]. It is also reported that these metallic patterns significantly lower the transition (critical) temperature of VO2. Metal nanostructures made of gold can be fabricated on the VO2 thin film by standard photo or e-beam lithography and lift-off process. A titanium or chromium adhesive layer a few nanometers thick is widely used to improve the adhesion between gold and VO2. In particular, Jeong et al. [21] showed that the transition temperature could be lowered by 20 degrees with 5-nm-width gaps compared to the bare VO2 film. In this work, we theoretically study the phase transition characteristics of VO2 metamaterials (the periodic metal nanogap fabricated on the VO2 thin film) to understand the low transition temperature. By developing an analytical model of transmission and reflection through slit arrays on VO2 thin films, we demonstrate that the large modulation depth and shift of the transition temperature originate from the rapidly increasing reflection during the phase transition. Unlike the metamaterials, absorption plays a significant role in reducing transmission for bare VO2 thin film. In addition, we investigate the slit width and VO2 film thickness dependence on transmission modulation.

2.1. Theoretical Model

Figures 1(a) and 1(b) show the schematic diagrams of the bare VO2 thin film and the metamaterial (slit arrays on VO2), respectively. Both were placed on sapphire, one of the most widely used substrates to grow high quality VO2 thin film. We assumed that the thickness of the sapphire substrate was infinite with a refractive index of 3.1 in the THz spectral range. Ordinary simulation tools (finite difference time domain or finite element method) are hard to use for our system because the thicknesses of the metal and the VO2 thin film are too thin compared to the wavelength. Therefore, the development of analytical models is necessary. To calculate transmission and reflection by the bare VO2 film, a well-known transfer matrix method was used, while the modal expansion method [22, 23] was employed for metamaterials. Since the transfer matrix method has been introduced in the most optics textbooks, here we only discuss the modal expansion method. The metal film making up the slit can be assumed to be a perfect electric conductor (PEC), because most metals in the THz spectral range have a very high refractive index [24, 25]. The width, thickness, and period of the slit array are denoted by w, h, and p respectively, while d represents the VO2 thin film thickness. nt and ns are the refractive indices of the VO2 thin film and the sapphire substrate.

Figure 1. Schematic diagrams for (a) bare vanadium dioxide (VO2) film and (b) VO2 metamaterials. (c) Transmission versus temperature for the bare film (red dots) and metamaterials (blue triangles). (d) Normalized transmission of (c). The maximum and minimum transmission are adjusted to be 1 and 0, respectively, to easily compare the critical temperature.

The electromagnetic waves polarized along the x direction are incident from the air towards the slit (traveling in the z axis). The incident and reflected waves in region I and the transmitted waves in region IV can be expressed by the modal expansion method:

$HyI=HyI, inc+HyI, ref=ε0μ0eikz+h+ε0μ0∑mR me−ik mx−w/2+κ m1z+h$

$HyIV=ε0μ0∑mT me−ik mx−w/2−κ mnsz−d$

where $k=2πλ, km=2πpm, κmni=nj2k2−km2$, and nj is the refractive index of j’s region (j = I, II, III, or IV). Rm and Tm are reflection and transmission coefficients, respectively. The common time dependence term eiωt is omitted and the corresponding electric field can be derived from the Maxwell equation:

$∇→×H→=−iωεE→$

where ε is the permittivity of the matter. Inside the slit, the single mode approximation can be applied [26]:

$HyII=Fe−ikz+h/2+Beikz+h/2$

where F and B are forward and backward propagation coefficients, respectively. Likewise, in the thin film, both the forward and backward propagations exist, which can be written as:

$HyIII=ε0μ0∑mC me iκ mntz+D me −iκ mntze−ik m x−w/2$

where Cm and Dm are the coefficient of the forward and backward radiation, respectively, at each mode inside the thin film. After applying the boundary conditions at the three interfaces (superstrate/slit array, slit array/thin film, thin film/substrate), the zeroth order transmission coefficient T0 and reflection coefficient R0 can be expressed as

$T0=wp−2D11 nscos ntkd−i1 ntsin ntkd$

$R0=1+wp2Dcoskh+iWnt, nssinkh$

where

$D=i1+W1Wnt, nssinkh−W1+Wnt, nscoskh$

$W1=kwp∑m1κ m1sinc2wk m2$

$Wnt, ns=kwp∑mnt2κ m ntsinc2 wkm2G mnt, nsd$

$Gmnt, nsd= κm nt nt2cos κm ntd−i κm ns ns2sin κm ntd−i κm nt nt2sin κm ntd+ κm ns ns2cos κm ntd$

The transmission T and reflection R are given by:

$T= T02ns, R= R0 2$

The absorption A was estimated from the energy conservation:

$T+R+A=1$

Based on our theoretical models (transfer matrix and modal expansion), we calculated the transmission, reflection, and absorption at 0.6 THz as a function of temperature for both bare and metamaterials. The temperature-dependent refractive indices of VO2 thin film used in the models will be explained in the next section.

2.2. Dielectric Constants of VO2

In our simulation, we assumed that the dielectric permittivity of VO2 in the terahertz range can be described by the Drude model:

$ϵω=ϵ∞−ωp2σω2+iγω$

where ϵ = 12 is the permittivity at the infinite frequency, ωp(σ) is the conductivity-dependent plasma frequency and γ is the collision frequency. According to [27], the plasma frequency at σ can be approximately expressed as ωp2(σ) = $σσ0$ωp2(σ0) with σ0 = 3 × 105 S/m, ωp(σ0) = 1.4 × 1015 rad/s, and γ = 5.75 × 1013 rad/s, which is assumed to be independent of σ. The conductivity during the phase transition at 0.6 THz was extracted from Fig. 1(c) in [14] and listed in Table 1. By substituting conductivities into Eq. (14), the temperature-dependent dielectric constants were calculated and then the corresponding refractive indices for transfer matrix and modal expansion were figured out through the relation: ϵ(ω) = nt2 (ω).

TABLE 1. Temperature-dependent conductivity of vanadium dioxide (VO2) taken from [14].

Temperature (K)Conductivity (Ω · cm)−1
310.011.0
313.414.2
317.017.0
320.120.5
322.825.2
325.731.2
327.442.5
328.679.3
329.3127.5
330.0204.0
331.4311.6
332.6430.6
333.7538.2
335.0637.4
337.5759.2
340.0858.4
342.6915.0
345.0937.7
347.5966.0
350.1980.2
352.9994.3
356.21001.8
359.81002.8
363.11011.3
366.61016.0
370.11017.0

III. RESULTS AND DISCUSSION

Figure 1(c) shows the calculated transmission through the bare film (red dots) and the VO2 metamaterials (blue triangles) during the phase transition at 0.6 THz. The thickness of the VO2 film was d = 100 nm for both the bare and patterned samples. The metal thickness h, period p, and width w of the slit array were 100 nm, 2 μm, and 100 nm, respectively. The transmission through the bare film was about 72% in the insulating state (310 K), while 20% transmission was recorded in the metallic state of VO2 (370 K). We would like to emphasize that our dielectric model of VO2 in Table 1 with the transfer matrix method reproduced the experimental results very well including the abrupt change of the transmission curve near the critical temperature [for example, see Fig. 1(c) in [14]. With the slit arrays on the VO2 thin film, the transmission through the sample was slightly reduced to 62% in the insulating state. Considering the extremely small opening ratio [(empty space between the metals versus total area) = $wp$] of the slits, 5%, this is a very surprising result. That is, compared to the bare film, the opening area of the surface is reduced by 95%, but the transmission is reduced by only 10%. This unexpectedly high transmission originates from the giant field enhancement at the gap, which is inversely proportional to $wp$ [28].

When the VO2 metamaterial transitions to a metallic state, the transmission becomes almost zero; the transmission is further reduced by 19% compared to the bare film. The largely enhanced dynamic range or modulation depth (ratio between the maximum and minimum transmission) has also been observed in previous experimental work [7–10]. In addition to the modulation depth, there are two major differences between the phase transition characteristics of the bare film and the metamaterial. One is the slope of the transmittance decrease in the temperature range of 310 K to 330 K. For the bare film, the slope was almost zero even though the conductivity had increased by 20 times due to the thinness of the VO2 film (100 nm) compared to the wavelength (500 μm). On the contrary, the metamaterial had a distinct negative slope with the same conductivity values. The other major difference is the position of the critical temperature. In order to compare the critical temperature clearly, we normalized the transmission curves: the maximum and minimum transmission were set to 1 and 0, respectively [Fig. 1(d)]. As is clearly seen, the critical temperature had been shifted to a low temperature by several kelvin. These two characteristics of VO2 metamaterial during the phase transition were demonstrated by the experiment [21]. This “early phase transition” of the metamaterial originates from the fact that the metallic nanostructures can confine the long wavelength light in a very small region, which is very sensitive to changes in the dielectric environment [29, 30].

Experimentally, the early transition of VO2 metamaterial has been investigated only in the transmission geometry [21]. In this work, we calculated not only transmission but also reflection and absorption for both bare films and metamaterials to elucidate the main causes of low critical temperature. Figures 2(a) and 2(b) show the stack plot of absorption (blue area with ‘+’ hatch), reflection (orange area with ‘\’ hatch), and transmission (green area with ‘x’ hatch) for bare and metamaterial, respectively, as a function of temperature. For the bare film, absorption at room temperature was negligible even though large imaginary part of refractive index of VO2 because the film thickness was 5,000 times smaller than the wavelength. Instead, reflection was about 27%, so that the transmission reached 72% as discussed previously. Interestingly, absorption suddenly increased near the critical temperature (~335 K) and eventually exceeded 20% in the metallic state. The reflection also rose up to 56%. Therefore, unlike the insulating state, both reflection and absorption play an important role in the metallic state.

Figure 2. Stack plots of transmission, reflection, and absorption for (a) bare vanadium dioxide (VO2) film and (b) VO2 metamaterial.

In contrast to the bare film case, the absorption of the VO2 metamaterial at room temperature was not negligible; it was about 7% at 310 K, seven times greater than the bare film case. However, reflection (~31%) was still much larger than absorption. Meanwhile, in the temperature range of 310 K to 330 K, reflection as well as absorption clearly increased with increasing temperature, unlike the bare film, which shows flat reflection and absorption curves. This phenomenon appears as an early transition in transmission for the VO2 metamaterials. The dramatic difference between bare films and metamaterials occurs near the transition temperature and in the metallic state. For the bare film, the reflection and absorption gradually increased near the transition temperature and approximately half of the incident light was reflected and 20% light was transmitted with 24% absorption in the metallic state. That is, the maximum absorption appeared in the metallic state of VO2. In stark contrast, a strong absorption peak was observed at the critical temperature for the metamaterial and then the absorption decreased to a level similar to the insulator state. In other words, the maximum absorption did not occur in the metallic state of the metamaterial. Instead, the reflection increased much more rapidly compared to bare film as it passed the transition temperature. Finally, in the metallic state, reflection overwhelmed absorption and transmission. This is because the metallic VO2 blocks the slit apertures; incident light sees the flat metal plate without the aperture and most of the energy is reflected at the metallic surface. Therefore, we can conclude that the enhanced transmission modulation depth of the VO2 metamaterial is mainly due to the large reflection rather than the absorption. We would like to further point out that metal nanostructures do not increase the maximum absorption level; the maximum absorption of the bare film is about 24% in the metallic state, which is close to the maximum absorption value for the metamaterials observed at the transition temperature.

Figure 3(a) and 3(b) show the slit width and the VO2 film thickness dependence of the transition temperature of the metamaterials. The critical temperature (Tc) was defined as a temperature corresponding to a half value of the difference between the maximum and the minimum transmission. According to our definition, the critical temperature of the bare VO2 film [red dots in Fig. 1(c)] was about 331.1 K. For the metamaterials, the slit width varied from 1 μm to 50 nm with a fixed period of 2 μm. The transmission versus temperature curves were calculated at each width and the critical temperatures were extracted. The infinity mark (∞) on the x axis in Fig. 3(a) represents the bare film case. Even a relatively wide slit on the order of 1 um was shown to lower the critical temperature. In addition, the critical temperature gradually decreased with decreasing slit width, which was consistent with the previous experimental results [21]. We also investigated the VO2 thickness effect on the critical temperatures as displayed in Fig. 3(b). In this calculation, the slit width, metal thickness, and slit period remained constant, while the VO2 thickness changed from 50 nm to 1,000 nm. A gradual decrease in critical temperature was observed with increasing thickness [Fig. 3 (b)].

Figure 3. Shifts of the critical temperatures. (a) Width and (b) vanadium dioxide (VO2) thickness dependence of the critical temperature of the VO2 metamaterials.

IV. CONCLUSION

In conclusion, we have theoretically studied the low transition temperature of VO2 metamaterial in the THz spectral range. The temperature-dependent refractive index was calculated by applying the Drude model from the experimentally measured conductivity. VO2 metamaterial consists of slit arrays on a VO2 thin film/sapphire substrate. To calculate transmission through the bare film, the transfer matrix method was used, while for the VO2 metamaterial, a modal expansion method was developed in this study. Our calculation results show that the critical temperature of the VO2 metamaterial is lowered and the modulation depth (ratio between the maximum and the minimum transmission) is largely enhanced compared to the bare film, which is consistent with earlier experimental work. We further calculated the reflection and absorption during the phase transition, and demonstrated that the absorption behaviors are significantly different between the bare and metamaterial. The absorption of the bare film is negligible in the dielectric state, and gradually increases during the phase transition. In contrast, the absorption peak is observed at the critical temperature and the absorption level in the dielectric state and the metallic state are similar for the metamaterial. Finally, we show that the narrower the slit width and the thicker the VO2 film, the lower the transition temperature. Recently, VO2-based active metamaterial or metasurfaces have been extensively studied because the conductivity of VO2 changes several orders of magnitude during the phase transition [3134]. Therefore, we believe that this theoretical work could help to design and understand the working principles of active VO2 metamaterials and metasurfaces. In addition, our results will provide an opportunity to develop near-room-temperature phase transition devices.

DISCLOSURES

The author declares no conflicts of interest.

DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the author upon reasonable request.

ACKNOWLEDGMENT

The present research was supported by the research fund of Dankook University in 2020.

FUNDING

Research fund of Dankook University in 2020.

Fig 1.

Figure 1.Schematic diagrams for (a) bare vanadium dioxide (VO2) film and (b) VO2 metamaterials. (c) Transmission versus temperature for the bare film (red dots) and metamaterials (blue triangles). (d) Normalized transmission of (c). The maximum and minimum transmission are adjusted to be 1 and 0, respectively, to easily compare the critical temperature.
Current Optics and Photonics 2022; 6: 583-589https://doi.org/10.3807/COPP.2022.6.6.583

Fig 2.

Figure 2.Stack plots of transmission, reflection, and absorption for (a) bare vanadium dioxide (VO2) film and (b) VO2 metamaterial.
Current Optics and Photonics 2022; 6: 583-589https://doi.org/10.3807/COPP.2022.6.6.583

Fig 3.

Figure 3.Shifts of the critical temperatures. (a) Width and (b) vanadium dioxide (VO2) thickness dependence of the critical temperature of the VO2 metamaterials.
Current Optics and Photonics 2022; 6: 583-589https://doi.org/10.3807/COPP.2022.6.6.583

TABLE 1 Temperature-dependent conductivity of vanadium dioxide (VO2) taken from [14]

Temperature (K)Conductivity (Ω · cm)−1
310.011.0
313.414.2
317.017.0
320.120.5
322.825.2
325.731.2
327.442.5
328.679.3
329.3127.5
330.0204.0
331.4311.6
332.6430.6
333.7538.2
335.0637.4
337.5759.2
340.0858.4
342.6915.0
345.0937.7
347.5966.0
350.1980.2
352.9994.3
356.21001.8
359.81002.8
363.11011.3
366.61016.0
370.11017.0

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Wonshik Choi,
Editor-in-chief