Tunable Casimir equilibria with phase change materials: from quantum trapping to its release

Now we consider the Casimir forces as the substrate is composed of a VO₂ film and Teflon (see Fig. 1(a)). The Casimir pressure ($P_{c}=F_{c}/A$) for the thick film of VO₂ is given in Fig. 2(a). The results show that the curves are almost identical for $L_{\mathrm{V}}$=200, 500 and 1000 nm, indicating the weak impact of the thickness for thick-film configurations. The pressure is repulsive at small separation (e.g., $d<60$ nm), making the nanoplate stay away from the substrate. As the separation increases further, the Casimir equilibria (zero pressure) occur and quantum traps can be realized for both metallic (solid lines) and insulating phases (dashed lines). In addition, the equilibrium distance $d_{c}$ is shifted under the phase transition of VO₂. On the other hand, the thin-film thickness and the phase transition of VO₂ can play an important role in Casimir pressure as shown in Fig. 2(b). For the thickness $L_{\mathrm{V}}$ =10 and 20 nm, quantum traps can be realized for the metallic phase, whereas no trap is found for the insulating phase. Under such configurations, a switch from quantum trapping of the nanoplate(“on” state) to its release (“off” state) can be triggered by the metal-insulator transition of VO₂. However, the quantum trapping occurs for both metallic and insulating phases as the thickness $L_{\mathrm{V}}$ increases to 30 nm, and the “off” state disappears. Compared with the vacuum-separated configuration Castillo-Garza et al. (2007), not only the magnitude of Casimir forces can be modified in a liquid environment, but also the sign could be switched (e.g., from attraction to repulsion for $d$=100 nm, $L_{V}$=30 nm), due to the phase-transition of VO₂.

To understand the switch transition from the “on” to the “off” state, the contour plots of Casimir pressure are shown in Fig. 3 under different separations. The sign of pressure is determined by the competition of VO₂ film (attraction) and low-refractive-index Teflon (repulsion). For small separation $d$=30 nm, the pressure is dominant by the repulsive component as shown in Figs. 3(a) and 3(b). For the metallic phase, the attractive component increases and it compensates the repulsive one as the separation becomes 85 nm ($d\approx d_{c}$), resulting in Casimir equilibrium (see Fig. 3(c)). While the repulsion is still dominant for the insulating phase as shown in Fig. 3(d). As $d$ increases further to 150 nm, the Casimir pressure turns out to be dominantly attractive in Fig. 3(e) for the metallic phase, resulting in a restoring force for stable trapping. By contrast, the pressure is still dominant by repulsion for the insulating phase as shown in Fig. 3(f). The pressure maps between the metallic and insulating phases are almost identical for large energy (e.g., $>$2 eV), whereas the discrepancy manifests at low energy. The results indicate that the attractive component appears only at low frequency and small $k$ vector for metallic VO₂, where the field cannot penetrate the metalZhao et al. (2019). Conversely, the field can penetrate the thin-film of insulating VO₂ easily, leading to repulsive Casimir forces.

Practically, the influences of gravitation and buoyancy on the force balances should be taken into account. The condition for the force equilibrium is written as $\vec{n}\cdot(\mathbf{F}_{c}+\mathbf{F}_{\mathrm{GB}})$=0, where $\vec{n}$ is the unit vector normal to the surface, $F_{\mathrm{GB}}=(\rho_{g}-\rho_{liq})gL_{g}A$ is the sum of gravity and buoyancy, $g$ is the gravitational acceleration, $\rho_{g}\approx$19.3 g/cm³ and $\rho_{liq}\approx$1.50 g/cm³ is the density of gold and liquid bromobenzene, respectively. The magnitude of $F_{\mathrm{GB}}/A$ is about 7.0 mPa as the thickness $L_{g}$=40 nm. Three types of configurations are depicted in the inset of Fig. 4(a) for the cross-section views. The type I configuration corresponds to a zero-projection (or weightlessness in aerospace), where the switching from quantum trapping (metallic state) to its release (insulating state) can be obtained as $L_{\mathrm{V}}$ in a proper range, from about 2 to 22 nm. For type II configuration, the attractive $F_{\mathrm{GB}}$ can compensate the long-range repulsive Casimir force at large $d$, leading to stable suspensions for both $T>T_{c}$ and $T<T_{c}$. However, the equilibrium distances are different, and it can be inferred that the stiffness of trapping for metallic phase is stronger than that of the insulating phase. For type III configuration (a flipped down system), the switching between trapping and its release can also be realized. Interestingly, there are two equilibrium distances for this configuration. It is not difficult to know that the smaller equilibrium distance (solid lines) is stable, whereas the other one (dashed lines) with larger distance is unstable to small perturbations in position. For both type II and III configurations, the deviations from Type I become strong as $d_{c}$ is large.

In addition to the thickness of VO₂ film, the top-layer Teflon can also play a significant role in the Casimir effect. The plots of Casimir pressure via the thicknesses of the coating Teflon $L_{T}$ are shown in Figs. 4(b) and 4(c), where $L_{\mathrm{V}}$=20 nm is fixed. The results show that the switching between quantum trapping and it release occurs only when $L_{T}$ is larger than about 42 nm (no gravity). The larger the $L_{T}$, the larger of the position for the Casimir equilibrium. As $L_{T}$ is smaller than 42 nm, the equilibrium distance is also small, and quantum trappings can be realized for both metallic and insulating phases. For comparison, the gravitation and buoyancy are taken into account. Again, strong discrepancies among three configurations occur as the equilibrium positions larger than about 150 nm, resulting from the comparable magnitude of $F_{GB}$ and the Casimir force. The impact of $F_{GB}$ can be further reduced by decreasing the thickness $L_{g}$ near the skin depth (about 22 nm) Lisanti et al. (2005).

The active control of the low-refractive-index nanoplates can also be significant in many applications. Inspiring by the work Zhao et al. (2019), a complementary design is schematically shown in the inset of Fig. 5(a). A Teflon nanoplate is suspended in a liquid of bromobenzene, and the substrate is a semi-infinite plate of Teflon coated by a VO₂ film (high refractive index). Under such design, the Casimir force is repulsive at very short separation, due to the dominant interaction between Teflon/bromobenzene/VO₂. As the separation increases, the attractive interaction from Teflon/bromobenzene/Teflon can be dominant instead, resulting in a stable Casimir trapping. To verify the design, the Casimir pressure is given quantitatively in Figs. 5(a) and 5(b) as a function of separation. Interestingly, the Casimir pressure shows a long-range repulsive behavior for the metallic VO₂, which corresponds to the “off” state. The repulsion pressure becomes stronger as the thickness $L_{\mathrm{V}}$ enlarges from 2 to 6 nm. For $L_{\mathrm{V}}$= 2 nm, a Casimir equilibria and strong restoring forces can be found when VO₂ is in the insulating phase. Therefore, the quantum trapping and release of a Teflon nanoplate can be achieved under the insulator-to-metal transition of VO₂. As the thickness is 4 nm, the restoring force decreases and the trapping stiffness drops considerably. The calculation results indicate that the Casimir pressure is quite sensitive to the thickness of VO₂. Due to the low density of Teflon (2.1 g/cm³), the pressure $F_{GB}/A$ for the Teflon nanoplate is about 0.6 mPa, which is reduced significantly compared with those of gold nanoplates.

To achieve the phase transition of VO₂, the temperatures of the devices need to be changed. We assume that the dielectric functions of the gold and Teflon are temperature-independent. For organic liquids, the change of refractive index due to the temperature Li et al. (1994) is an order of $10^{-4}/$ K, and the permittivity of bromobenzene is also treated as temperature-independent. Nonetheless, it is interesting to check the finite temperature effect on Casimir forces. The integral over frequency $\xi$ in Eq. (1) now is replaced by a discrete summation Rahi et al. (2009):

where $\xi$ is replaced by discrete Matsubara frequencies $\xi_{n}=2\pi\frac{k_{b}T}{\hbar}n(n=0,1,2,3\ldots),$ $k_{B}$ is the Boltzmann’s constant and the prime denotes a prefactor 1/2 for the term $n$=0. The Casimir pressures under different temperatures are shown in Figs. 6(a) and 6(b), where two different designs are demonstrated. It is found that the curves for temperature 320 K (insulating phase) overlap with those calculated from Eq. (1). For the temperature of 360 K, there is only a small deviation between 0 K and 360 K. Overall, the calculation results from 320 and 360 K confirm the accuracy of the 0 K approximation. Recently, the switching between repulsive and attractive Casimir forces based on PCM has also been reported Boström et al. (2018), where the equilibrium distances for switching occur only at several nanometers. The equilibrium distances in our work are more accessible to experiments, and it can be tuned by designing the geometric thickness of VO₂ and the Teflon.

In a real configuration, the position of a nanoplate has a fluctuation around the equilibrium distances due to the Brownian motion. To evaluate the effect of Brownian motion, the total energy of the suspended nanoplate should be known, which are written as $U(d)=E_{c}+\Lambda\times(E_{g}+E_{b})$, where $E_{c}$ is the Casimir energy given by Eq. [1], $E_{g}=\rho_{p}gL_{p}Ad$ and $E_{b}=-\rho_{liq}gL_{p}Ad$ are respectively the energies caused by the gravity and buoyancy Phan et al. (2012), $\rho_{p}$ and $L_{p}$ represent the density and thickness of the suspended nanoplate. The coefficient $\Lambda$ is the parameter depending on the gravity projection. For type I configuration (see the inset of Fig. 4), $\Lambda$=0. While we have $\Lambda$=1 and -1 for type II and type III configurations. The total energy of a gold and Teflon nanoplate are shown in Figs. 7(a) and 7(b), respectively. The minimum of $U(d)/k_{B}T$ corresponds to the equilibrium distance $d_{c}$. Clearly, stable quantum trapping can be realized for a gold (Teflon) nanoplate when VO₂ is in the metallic (insulating) phase. Due to the balance of repulsive Casimir force and gravity, stable trapping can also be realized for type II configuration. Theoretically, the transition rate from the equilibrium distance to another position due to the Brownian motion is proportional to $\exp(-\triangle U/k_{B}T)$ Phan et al. (2012); Rodriguez et al. (2010b), where $\triangle U$ represents the energy barrier between these two positions. The calculated results indicate that the transition rates from Casimir equilibria to stiction are negligible since the energy barriers $\triangle U//k_{B}T$ are quite large (e.g., over $10^{4}$) for the gold and Teflon nanoplates. For a flipped-down system (type III), quantum trapping can be realized for gold (Teflon) nanoplate when VO₂ is in the metallic (insulating) phase. However, there is a nonzero possibility that the nanoplates can escape from the equilibrium distances to the free-liquid regime ($d\rightarrow\infty$). Fortunately, the energy barrier $\triangle U/k_{B}T$ for such a transition is the order of $10^{2}$ as shown in Figs. 7(a) and 7(b), and the transition rate of the escape is also negligible.