The Casimir force, one of the most relevant causes of stiction problems, gives rise to critical impediments in the fabrication and operation of nano/micro‐electromechanical systems (NEMS/MEMS). In almost all cases in which the scale is of hundreds of nanometers, Casimir interactions produce such significant amount of friction as to attract the attention of scientists from a wide variety of research fields [1]-[3]. The theoretical understanding and measurements of Casimir interactions have advanced substantially in the last ten years, allowing physicists to have a more detailed understanding of the fundamental physics, not only in nanophysics, but particle physics and cosmology as well.

The simplest Casimir system was generated theoretically by Casimir in 1948 when he developed a model describing the interaction between two parallel conducting plates [4]. Since then, the Casimir forces between real materials such as metals [5], semiconductors [6], semimetals [7] and high‐T c superconductors [8] have been extensively studied theoretically and experimentally. It has been found that the presence of liquids between two objects allows the sign of the Casimir force to switch [1]-[3]. These repulsive Casimir forces appear when the dielectric functions of object 1 and object 2 immersed in a medium 3 satisfy the relation over a wide imaginary frequency range It is also possible to achieve the repulsion using arrays of gold nanopillars on two plates [9]. In addition, metamaterials are promising candidates for creating these repulsive interactions [10]. The combination between experimental measurements and theoretical calculations has provided essential information to help researchers design nanoscale devices.

The well‐known Lifshitz theory developed a generalization of the Casimir force [3], [11], [13]. In the theory, the force between uncharged objects made of real materials is given by an analytical formula with frequency‐dependent dielectric permittivity Variation of the dielectric functions causes the change of the Casimir interactions. The Casimir–Lifshitz force has been studied for systems at the thermal equilibrium in atom–atom, plane–plane and atom–plane configurations [11], [12].

Cuprate superconductors, high‐T c superconductors and anisotropic materials are widely used in various devices. It has been shown theoretically that the Casimir force in the BSCCO–air–gold system is significantly affected by the anisotropy in the dielectric functions. In the present Letter, the Casimir–Lifshitz force is calculated in the case of a perpendicular cleave between cuprate superconductor and silica with bromobenzene in between. The calculation of the force takes into account the thermal effect and the influence of thickness on the dispersion force. The force is repulsive and is associated with the sticking process in nanodevices.

The general expression describing the Casimir interaction between two infinite parallel plates is the Lifshitz formula. At a given separation a and given temperature T, the Casimir pressure between two plates is given [11], [13] by

((1))

Here is the Boltzmann constant, is the wave vector component perpendicular to the plate, and denote the reflection coefficients of the transverse magnetic (TM) and transverse electric (TE) field, respectively. The superscripts (1) and (2) correspond to the first body (silica) and the second body (BSCCO). In addition,

are the Matsubara frequencies, is an integer, and is the dielectric function of the medium between the two objects. In the calculation, bromobenzene is the medium. The dielectric function of the liquid can be described using the oscillator model [2]:

((2))

where parameters and were obtained by fitting with experimental data in a large frequency range [2].

It is important to note that for the prefactor of the integration is instead of for other values of n. In the case of silicon dioxide, the reflection coefficients are given [2], [3] as

((3))

((4))

in which is the dielectric function of silica. The dielectric function still has the form of an oscillator model as in Eq. (2), with parameters as generated in Ref. [2]. Considering the role of the thickness D of the silica slab on the Casimir interaction, the TM and TE reflection coefficients in Eqs. (3) and (4) become [14], [15]

((5))

((6))

Because of the uniaxial property of BSCCO, its permittivity in the perpendicular cleave is represented in the form of a tensor [8]:

((7))

where and are the dielectric components along the optical axis and perpendicular to the optical axis, respectively. Therefore, the expression of the reflection coefficients for BSCCO must be modified. The TM and TE coefficients on the liquid–BSCCO interface are [8], [16]

((8))

((9))

For BSCCO, the dielectric responses and are modeled based on the damped‐multioscillator model. Parameters corresponding to the resonance frequency, damping and oscillator strength are given in Ref. [8].

As shown in Fig. 1, the Casimir forces are significantly influenced by the thickness of the slab. The Casimir interactions in the real system are much smaller than the force in the ideal case. It is clear that the presence of bromobenzene makes the Casimir force in this case repulsive. For nm, the effect of thickness nearly vanishes and the Casimir force in the system can be modeled as an interaction between two plates. For metals such as gold, when the thickness nm, a gold thin film can be treated as a gold plate [17], [18]. The reason for the discrepancy between the two cases is that the conductivity of metals is much larger than that of silicon dioxide. When the thickness of a metal slab is reduced, it appears as if the skin‐depth effect occurs. This effect, however, is not present in the case of silica.

Figure 1 Open in figure viewer PowerPoint Relative Casimir pressures between a BSCCO plate and a silica slab in the presence of bromobenzene. Here is the Casimir force between two ideal metal plates.

At small distances, there is not much change in the force with different thicknesses. The reason is that in this range so The influence of thickness on the interaction disappears.

Bromobenzene molecules exist in liquid form over an important temperature range from 242 K to 429 K. Figure 2 shows the Casimir force in this temperature range. It is evident that the interaction depends notably on temperature. There are variations in the Casimir force at different thicknesses. It is now possible to design a non‐touching system because the Casimir force is repulsive for the en‐ tire range of distances. The Casimir force and the gravitational forces between the two bodies lead to a repulsive–attractive transition in our system and result in our system reaching an equilibrium distance [19]. Obviously, the stable position can be varied by changing the size of the bodies, the thickness and the temperature. Vidal et al. [20] proposed the measurement of Casimir forces using a tiny spring in order to get a balanced position and oscillation frequency. However, in our case, the equilibrium positions exist naturally. It is not necessary to attach a spring to the system to measure the Casimir force. The force can be found via observation of the oscillation frequencies. Because of anisotropy, the expressions for the TE and TM reflection coefficients in the case of parallel cleave orientation are different from Eqs. (8) and (9). This discrepancy is proof of a difference between the Casimir forces in the two orientations.

Figure 2 Open in figure viewer PowerPoint Relative Casimir pressures taking into account the thermal effect with different values of thickness.

The thermal effect in the Casimir interaction plays an important role at long distances [13]. For short distances, this effect can be ignored. One can use double integration, instead of summation and single integration as in Eq. (1), in the calculations at short distances. The expression of double integration provides a good agreement with experiment.

This work presents a reliable anti‐stiction method for NEMS/MEMS structures. The presence of a liquid can address the stiction issue causing catastrophic failure in nanoscale devices. The temperature and thickness dependence of the Casimir force allows for controlling the adhesion force between two surfaces. Currently, it is difficult to

measure properties in fluidic environments. However, using liquid films in NEMS/MEMS, devices with a thickness range of the liquid layers from 2 nm to 70 nm have been intensively investigated [21], [22]. Moreover, the behavior of tiny devices in liquids was described [23]. This research makes it possible to design nanostructures in microfluidic environments, and our study gives an interesting view of what happens physically in systems submerged in liquids.