Vanadium dioxide (VO₂) is a classical transition metal oxide material that undergoes a reversible insulator-to-metal phase transition (IMT) near room temperature (T_c = 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. n_t and n_s 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:

(1)$$H_y^I=H_y^{I,inc}+H_y^{I,ref}=\sqrt{\frac{{\epsilon }_0}{{\mu }_0}}{e}^{ik\left(z+h\right)}+\sqrt{\frac{{\epsilon }_0}{{\mu }_0}}{\displaystyle \sum _m{R}_m{e}^{-i\left(k_m\left(x-w/2\right)+{\kappa }_m^1\left(z+h\right)\right)}}$$

(2)$$H_y^{IV}=\sqrt{\frac{{\epsilon }_0}{{\mu }_0}}{\displaystyle \sum _m{T}_m{e}^{-i\left(k_m\left(x-w/2\right)-{\kappa }_m^{n_s}\left(z-d\right)\right)}}$$

where $k=\frac{2\pi }{\lambda },k_m=\frac{2\pi }{p}m,{\kappa }_m^{n_i}=\sqrt{n_j^2{k}^2-k_m^2}$, and n_j is the refractive index of j’s region (j = I, II, III, or IV). R_m and T_m are reflection and transmission coefficients, respectively. The common time dependence term e^−iωt is omitted and the corresponding electric field can be derived from the Maxwell equation:

(3)$$\overrightarrow{\nabla }\times \overrightarrow{H}=-i\omega \epsilon \overrightarrow{E}$$

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

(4)$$H_y^{II}=F{e}^{-ik\left(z+h/2\right)}+B{e}^{ik\left(z+h/2\right)}$$

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:

(5)$$H_y^{III}=\sqrt{\frac{{\epsilon }_0}{{\mu }_0}}{\displaystyle \sum _m\left(C_m{e}^{i{\kappa }_m^{n_t}z}+D_m{e}^{-i{\kappa }_m^{n_t}z}\right)e^{-i{k}_m\left(x-w/2\right)}}$$

where C_m and D_m 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

(6)$$T_0=\left(\frac{w}{p}\right)\frac{-2}{D}\frac{1}{\left(\frac{1}{n_s}\cos \left(n_{t}kd\right)-i\frac{1}{n_t}\sin \left(n_{t}kd\right)\right)}$$

(7)$$R_0=1+\left(\frac{w}{p}\right)\frac{2}{D}\left(\cos \left(kh\right)+i{W}^{n_t,n_s}\sin \left(kh\right)\right)$$

where

(8)$$D=i\left(1+W^1{W}^{n_t,n_s}\sin \left(kh\right)\right)-\left(W^1+W^{n_t,n_s}\right)\cos \left(kh\right)$$

(9)$$W^1=k\frac{w}{p}{\displaystyle \sum _m\frac{1}{{\kappa }_m^1}{\text{sinc}}^2\left(\frac{w{k}_m}{2}\right)}$$

(10)$$W^{n_t,n_s}=k\frac{w}{p}{\displaystyle \sum _m\frac{n_t^2}{{\kappa }_m^{n_t}}{\text{sinc}}^2\left(\frac{w{k}_m}{2}\right)G_m^{n_t,n_s}\left(d\right)}$$

(11)$$G_m^{n_t,n_s}\left(d\right)=\frac{\left(\frac{{\kappa }_m^{n_t}}{n_t^2}\cos \left({\kappa }_m^{n_t}d\right)-i\frac{{\kappa }_m^{n_s}}{n_s^2}\sin \left({\kappa }_m^{n_t}d\right)\right)}{\left(-i\frac{{\kappa }_m^{n_t}}{n_t^2}\sin \left({\kappa }_m^{n_t}d\right)+\frac{{\kappa }_m^{n_s}}{n_s^2}\cos \left({\kappa }_m^{n_t}d\right)\right)}$$

The transmission T and reflection R are given by:

(12)$$T=\frac{{\left|T_0\right|}^2}{n_s},R={\left|R_0\right|}^2$$

The absorption A was estimated from the energy conservation:

(13)$$T+R+A=1$$

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:

(14)$$ϵ\left(\omega \right)={ϵ}_{\infty }-\frac{{\omega }_p^2\left(\sigma \right)}{{\omega }^2+i\gamma \omega }$$

where ϵ_∞ = 12 is the permittivity at the infinite frequency, ω_p(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 ω_p²(σ) = $\frac{\sigma }{{\sigma }_0}$ω_p²(σ₀) with σ₀ = 3 × 10⁵ S/m, ω_p(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) = n_t² (ω).

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

Temperature (K)	Conductivity (Ω · cm)−1
310.0	11.0
313.4	14.2
317.0	17.0
320.1	20.5
322.8	25.2
325.7	31.2
327.4	42.5
328.6	79.3
329.3	127.5
330.0	204.0
331.4	311.6
332.6	430.6
333.7	538.2
335.0	637.4
337.5	759.2
340.0	858.4
342.6	915.0
345.0	937.7
347.5	966.0
350.1	980.2
352.9	994.3
356.2	1001.8
359.8	1002.8
363.1	1011.3
366.6	1016.0
370.1	1017.0

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) = $\frac{w}{p}$] 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 $\frac{w}{p}$ [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 (T_c) 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 [31–34]. 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.