1. Introduction

The Casimir effect is a macroscopic quantum effect existing between neutral macroscopic bodies arising from electromagnetic zero-point energy fluctuations in the vacuum due to the existence of the boundaries. In 1948, Casimir theoretically predicted the presence of an attractive force between two neutral, perfectly conducting parallel planes in the vacuum [1]. A few years later, Lifshitz [2] generalized a method to study the attractive force between two semi-infinite dielectric plates containing dispersion and absorption at finite temperatures. A large number of papers have been devoted to obtaining a theoretical understanding of the Casimir effect [3–5] and acquiring experimental measurements of the attractive force [6–8] in recent decades. Generally, the attractive Casimir force between two usual nonmagnetic bodies at small separations is always overwhelming [9]. However, this attractive force may cause irreversible adhesion between neighboring elements in micro- and nanoelectromechanical systems (MEMS and NEMS) [10–12]. Therefore, changing the sign of the Casimir force has become a popular research topic. Repulsive Casimir forces can arise in the systems containing nanowire materials [13,14], or graphene sheets [15,16], or materials with a high magnetic permeability [17,18]. Therefore, repulsive Casimir force systems usually contain magnetically responsive materials, such as saturated ferrite materials [19], or magnetoelectric materials, such as topological insulators [20–23]. In fact, metamaterials [24–29] are extensively used to investigate the repulsive Casimir force; metamaterials are desirable for this purpose because they are artificial, and thus, their electromagnetic properties are controllable in the optical and near-infrared regimes [30]. In addition, the repulsive Casimir force has been observed experimentally between a large plate and a polystyrene sphere coated with a thick gold film, which is separated from the plate by a fluid [31].

Recently, a new kind of artificial material with an unusual permittivity tensor has attracted a great deal of attention [32]. Specifically, the diagonal elements of the permittivity tensor are different and have opposite signs [33]. As a result, the isofrequency contour for TM polarization, which depends on the permittivity, is a hyperboloid. Therefore, this material has been labeled a hyperbolic metamaterial (HMM). Generally, most HMMs are electrically anisotropic; we call this type of HMM an electric hyperbolic metamaterial (ε-HMM). Many potential applications of ε-HMMs, such as sub-diffraction imaging [34,35], negative refraction [36], efficient single-photon sources [37,38], and Casimir interaction torque [39], have been proposed. We previously studied the attractive Casimir force between two ε-HMM slabs, and we demonstrated that hyperbolic dispersion is essential to achieving an enormous increase in the Casimir force [40]. In experiments, ε-HMMs are easily fabricated by metallic nanowire arrays embedded in a dielectric host [35] or by metal-dielectric layered structures [37].

To obtain full flexibility for arbitrarily polarized waves, the magnetic responses of metamaterials that define the dispersion characteristics for TE polarization should be developed to produce magnetic hyperbolic metamaterials (μ-HMMs). Similar to ε-HMMs, μ-HMMs are anisotropic materials with a permeability tensor whose diagonal elements possess opposite signs. Therefore, the resulting isofrequency surface for TE polarization is a hyperboloid. A popular microstructure employed to manipulate the magnetic response is the multilayer fishnet structure, which was predicted theoretically to possess a magnetic hyperbolic dispersion [41]; subsequently, the magnetic hyperbolic dispersion of this structure was confirmed experimentally in a metal-dielectric multilayer fishnet system [42]. Additionally, a theoretical study demonstrated that a wire medium consisting of high-index ε-positive nanowires could also be treated as a μ-HMM at a certain frequency [43]. Therefore, μ-HMMs provide a new platform for modifying the electromagnetic field.

It is well known that the Casimir force can be modified due to the electromagnetic properties of the boundary material [4]. Here, we investigate the formation of the repulsive Casimir effect between HMMs. In particular, we discuss the conditions that are necessary for achieving repulsive and restoring Casimir forces between an ε-HMM slab and a μ-HMM slab in the light of their particular hyperbolic dispersion characteristics. This study indeed is different from our previous research on the influence of hyperbolic dispersion on the attractive Casimir force between two ε-HMM slabs [40]. The remainder of this paper is organized as follows. In Sec. II, we introduce a model considering the Casimir force between two slabs and the electromagnetic responses of HMMs. In Sec. III, we calculate the Casimir force; the repulsive and restoring forces as well as multiple equilibria are obtained, and the influence of the temperature is also considered. In Sec. IV, we draw our conclusions.

2. Model and theoretical framework

The scheme of the system considered here is depicted in Fig. 1. A represents the ε-HMM, while B is the μ-HMM. They are parallel to each other and separated by a distance $a$. The $x\text{-}y$ plane is set to be parallel to the surfaces of the slabs, while the $z$ axis is the out-of-plane optical axis of the anisotropic slabs.

 

Fig. 1 Scheme of the considered system. A and B are an ε-HMM and a μ-HMM, respectively.

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2.1 Effective medium theory for hyperbolic metamaterials (HMMs)

HMMs are a particular kind of composite medium. It is important to choose a suitable effective medium theory for HMMs since the calculation of the Casimir force is very sensitive to the choice of which effective medium model is used [44]. In our previous work [40], we demonstrated that the anisotropic Maxwell Garnett mixing formula [45] is suitable for multilayered HMMs when the thicknesses of the layers are much smaller than the operating wavelength. Therefore, based on our research [40] and previous experimental investigations [37,42], we focus here on the case in which both ε-HMM and μ-HMM slabs are constructed from multilayer systems.

The ε-HMM slab A is a multilayer structure with alternating metal and dielectric layers. The metal is nonmagnetic and described by ${\epsilon }_m^{\text{A}}$. Similarly, the dielectric layer is described by ${\epsilon }_d^{\text{A}}$. Based on the anisotropic Maxwell Garnett mixing formula [45], the permittivity of slab A is a tensor and can be expressed as

The μ-HMM slab B is a combination of nonmagnetic dielectric and magnetic metamaterials with an effective ${\mu }_{eff}^{\text{B}}$. Similar to the expression for the permittivity of the ε-HMM slab, the permeability of the μ-HMM slab is described by the tensor

2.2 Casimir force between two parallel slabs

Based on the Maxwell electromagnetic stress tensor method with the properties of macroscopic field operators, the Casimir force at zero temperature can eventually be written as [5]

Since slabs A and B have been considered as bulk anisotropic metamaterials by using the effective medium theory, in this work, we suppose that A and B are both semi-infinite slabs. Then, the reflection coefficients between the vacuum and the ε-HMM (A) described by Eq. (1) can be expressed as [40]

Meanwhile, the reflection coefficients between the vacuum and the μ-HMM (B) described by Eq. (4) are obtained as follows:

3. Repulsive and restoring Casimir forces

In this section, we discuss the repulsive Casimir force between an ε-HMM slab and a μ-HMM slab. Since the Casimir force is calculated over all frequencies, all ingredient materials should be considered frequency dependent without any loss of generality. These ingredient materials are then reasonably described by the Drude-Lorentz model [24,28,29,46]. For an ε-HMM, the ingredient materials can be described as

3.1 The hyperbolic dispersion regions

The real parts of ${\epsilon }_{xx}^{\text{A}}$, ${\epsilon }_{zz}^{\text{A}}$, ${\mu }_{xx}^{\text{B}}$ and ${\mu }_{zz}^{\text{B}}$ are plotted as functions of $\omega$ in Fig. 2 for different filling factors; in the corresponding calculations, these characteristic frequencies are chosen as ${\Omega }_m={\omega }_0$, ${\Omega }_d=0.1{\omega }_0$, ${\Omega }_{eff}=0.3{\omega }_0$, ${\omega }_d={\omega }_{eff}=0.25{\omega }_0$, ${\gamma }_m=0.01{\omega }_0$ and ${\gamma }_d={\gamma }_{eff}=0.006{\omega }_0$. The frequencies are scaled with a reference frequency ${\omega }_0$. The frequencies that satisfy $\mathrm{Re}[{\epsilon }_{xx}^{\text{A}}(\omega )]\mathrm{Re}[{\epsilon }_{zz}^{\text{A}}(\omega )]<0$ or $\mathrm{Re}[{\mu }_{xx}^{\text{B}}(\omega )]\mathrm{Re}[{\mu }_{zz}^{\text{B}}(\omega )]<0$ are called regions of electric or magnetic hyperbolic dispersion, which are illustrated by the shaded areas in Fig. 2. It is clear that an increase in the filling factor can expand the bandwidth of hyperbolic dispersion and enhance the electromagnetic response. In this work, we mainly discuss the influence of this region on the Casimir force.

 

Fig. 2 Real parts of ${\epsilon }_{xx}^{\text{A}}$ (solid blue lines) and ${\epsilon }_{zz}^{\text{A}}$ (dotted red lines) as functions of $\omega$ with different filling factors: (a) $f_{\text{A}}=0.2$, and (b) $f_{\text{A}}=0.5$. Real parts of ${\mu }_{xx}^{\text{B}}$ (solid blue lines) and ${\mu }_{zz}^{\text{B}}$ (dotted red lines) with different filling factors: (c) $f_{\text{B}}=0.2$, and (d) $f_{\text{B}}=0.5$. Other parameters are mentioned in the text. Shaded areas indicate regions of hyperbolic dispersion.

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3.2 The repulsive Casimir force

To investigate the role of the hyperbolic relation on the Casimir force, we first assume that ${\mu }_{\text{A}}=1$ and ${\epsilon }_{\text{B}}=1.37$, indicating that only the electric (magnetic) response is frequency dependent for the ε-HMM (μ-HMM). The relative Casimir force $F_r=F_C/F_0$ as a function of the separation $a$ is plotted in Fig. 3(a) for different values of $f_{\text{A}}$ and fixed $f_{\text{B}}=0.5$, while Fig. 3(b) shows the results for the opposite case for fixed $f_{\text{A}}=0.5$ but different $f_{\text{B}}$, and the other parameters are the same as in Fig. 2. Here, $F_0=\hslash c{\pi }^2/240a^4$ is the well-known formula for the Casimir force between two perfectly conducting plates with a separation $a$.

 

Fig. 3 The relative Casimir force $F_r=F_C/F_0$ between an ε-HMM and a μ-HMM as a function of the separation $a$ with different filling factors. (a) The filling factor of the ε-HMM $f_{\text{A}}$ varies, but the filling factor of the μ-HMM $f_{\text{B}}$ is fixed as $f_{\text{B}}=0.5$. The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency when $f_{\text{B}}=0.5$. (b) The filling factor of the μ-HMM $f_{\text{B}}$ varies, but the filling factor of the ε-HMM $f_{\text{A}}$ is fixed as $f_{\text{A}}=0.5$. The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency when $f_{\text{B}}=0.2$. Other parameters are mentioned in the text.

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The separation $a$ is scaled by the unit ${\lambda }_0=2\pi c/{\omega }_0$. Repulsive Casimir forces, denoted by negative values of $F_r$, are evident in the results.

It is well known that the repulsive Casimir force can be obtained between an electric plate and a magnetic plate [17,18]. In the present work, slab A is an ε-HMM, for which the electric response dominates across the whole frequency range. From Fig. 3(a), the separation distance of the attraction-repulsion transition decreases with the growth of $f_{\text{A}}$, indicating that the separation region of the repulsive force becomes larger. The force at a small separation is related to the electromagnetic modes of the high frequency region [28], where both slabs are mainly electric. Therefore, the force at a small separation is attractive. Furthermore, when the separation increases, the frequency range of the main contribution shifts to lower frequencies. From the inset of Fig. 3(a), the electromagnetic properties of the μ-HMM are mainly magnetic in these hyperbolic dispersion regions. Consequently, the attractive Casimir forces gradually decrease and finally become repulsive. In addition, as shown in Fig. 2, the increased $f_{\text{A}}$ of the ε-HMM leads to a wider hyperbolic dispersion region and a stronger electric response. Thus, the attractive and repulsive forces are both strengthened.

The influence of $f_{\text{B}}$ on the force is shown in Fig. 3(b). When $f_{\text{B}}=0.2$, although the μ-HMM has a magnetic hyperbolic response in some frequencies, its electric response is stronger across the whole frequency range, as shown in the inset of Fig. 3(b); thus, the force is attractive. However, the permeability of the μ-HMM is higher than its permittivity ${\epsilon }_{\text{B}}$ at lower frequencies for $f_{\text{B}}=0.5$, as shown in the inset of Fig. 3(a). For larger separations, the Casimir forces mainly originate from contributions at lower frequencies, where the A slab (ε-HMM) is mainly electric, and the B slab (μ-HMM) is mainly magnetic. Thus, repulsive forces are found at large separations. This means that the repulsive force will emerge between the ε-HMM and the μ-HMM only if the magnetic response of the μ-HMM is stronger than its electric response.

3.3 The restoring Casimir force and multiequilibrium

From the above discussion, we note that forces can transform from attractive to repulsive with an increase in the separation. However, the forces do not become attractive again with a further increase in the separation. These conditions do not represent a stable equilibrium; with the initial separation possessing $F_r=0$, they will be either attracted together or repulsed away after a small perturbation. A stable equilibrium can be established when the force becomes attractive with an increasing separation. The Casimir forces at different separations originate from different frequency ranges [28]. Thus, when the electromagnetic response of the μ-HMM changes between electric and magnetic several times across the whole frequency, the polarity of the force converts correspondingly. Consequently, a stable equilibrium may emerge. Therefore, we suppose that the μ-HMM possesses not only the magnetic dispersion described in Eq. (17) but also the electric dispersion whose ${\epsilon }_{\text{B}}$ is described by the Drude-Lorentz model as

For the μ-HMM with this frequency-dependent electric dispersion, we plot the Casimir force as a function of the separation in Fig. 4. The restoring forces, defined as the transition from repulsive to attractive with increasing separation, appear in the cases of $f_{\text{B}}=0.2$, $0.5$ and $1$. As mentioned above, increasing $f_{\text{B}}$ can expand the hyperbolic region and enhance its magnetic response. Thus, the repulsion effect is stronger. In the meantime, for low frequencies, the μ-HMM has a strong electric response, as shown by the inset in Fig. 4; therefore, the corresponding repulsive forces at large separations are weakened and finally become attractive.

 

Fig. 4 The relative Casimir force $F_r$ between an ε-HMM and a μ-HMM as a function of the separation $a$ with different $f_{\text{B}}$. The restoring forces are shown in the curves for $f_{\text{B}}=0.2$ (dashed blue line), $0.5$ (dotted magenta line), and $1$ (dash-dotted red line). The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency with $f_{\text{B}}=0.2$.

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Because the permittivity of an ε-HMM and the permeability of a μ-HMM are anisotropic, the electromagnetic properties of two slabs are complex across the whole frequency spectrum. Here, we focus on the possibility of multiequilibrium, i.e., the existence of more than one restoring Casimir force with different separations. As an example, the adjusted parameters are chosen as follows: ${\Omega }_m=0.01{\omega }_0$, ${\gamma }_m=0.001{\omega }_0$, ${\Omega }_{eff}=1.7{\omega }_0$, ${\omega }_{eff}=0.1{\omega }_0$, and ${\gamma }_{\text{B}}=0.0006{\omega }_0$; the other parameters are the same as those in Fig. 4. The resulting Casimir forces varying with the separation are shown in Fig. 5. It is evident that for $f_{\text{B}}=0.5$, there exist four polarity transitions with an increase in the separation, two of which correspond to stable restoring forces and equilibria, as shown by the squares in Fig. 5. This result can be easily explained as follows. First, along the real frequencies for the μ-HMM, there are two resonances corresponding to ${\mu }_{xx}^{\text{B}}$ and ${\mu }_{zz}^{\text{B}}$, which are the main contributors to the Casimir repulsion at different separations; thus, there are two different separation regions corresponding to the repulsive force. Second, along the imaginary frequencies (see the inset in Fig. 5), the electromagnetic responses are different according to different imaginary frequencies. The attractive forces at very long separations can be explained via the dispersive behavior within the band where ${\epsilon }_{\text{B}}(i\xi )>{\mu }_{xx}^{\text{B}}(i\xi ),{\mu }_{zz}^{\text{B}}(i\xi )$. When the frequencies in the band ${\mu }_{xx}^{\text{B}}(i\xi )>{\epsilon }_{\text{B}}(i\xi )>{\mu }_{zz}^{\text{B}}(i\xi )$ provide the main contribution, these electromagnetic response properties are involved. Thus, the Casimir force converts its polarity from attractive to repulsive and then to attractive again with decreasing separation. Subsequently, due to the frequency band where ${\mu }_{xx}^{\text{B}}(i\xi ),{\mu }_{zz}^{\text{B}}(i\xi )>{\epsilon }_{\text{B}}(i\xi )$, the force becomes repulsive again at relatively short separations. Because the ε-HMM and μ-HMM are both mainly electric across the entire range of frequencies as mentioned above, the force finally becomes attractive at very short separations. Due to the frequent switching between attraction and repulsion, the magnitude of the force is small. However, the corresponding effect may still be enhanced by carefully adjusting the parameters and filling factors.

 

Fig. 5 The relative Casimir force between an ε-HMM and a μ-HMM as a function of the separation $a$ with different $f_{\text{B}}$. Multiequilibrium appears when $f_{\text{B}}=0.5$ (dash-dotted magenta line). Circles and squares are the transition points of the force polarity. Squares indicate stable equilibria. The inset shows the permeability and permittivity of the μ-HMM as a function of the imaginary frequency with $f_{\text{B}}=0.5$. Other parameters are mentioned in the text.

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3.4 The Casimir force at finite temperatures

Here, we consider the Casimir force at finite temperatures. Equation (8) is rewritten in the form of a summation over the Matsubara frequencies ${\xi }_m=2\pi m{k}_{B}T/\hslash$ as

 

Fig. 6 The relative Casimir force between an ε-HMM and a μ-HMM as a function of the separation $a$ with different $f_{\text{B}}$ at room temperature $T=300K$. Circles and squares are the transition points of the force polarity. Squares indicate a stable equilibrium. The parameters are the same as those in Fig. 5.

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3.5 Discussion

It is known that, all materials are mainly electric at high frequencies. Therefore, the Casimir force at a small separation is attractive since it is related to the electromagnetic modes in the high frequency region. In order to get the repulsive Casimir force, one of the slabs must be magnetic at low frequencies, which is guaranteed by Eq. (17) in the present work. Previous works [47,48] analyzed the Casimir interaction between metal-dielectric metamaterials, and concluded that the Casimir force is always attractive in such systems. However, one still may not eliminate the possibility of fabricating HMMs with the magnetic materials. Recently, a theoretical work provides a new approach to realize hyperbolic metamaterial [49]. They demonstrated that a hyperbolic dispersion relation in diamond with NV centers can be engineered and dynamically tuned by applying a magnetic field. Here we suggest that one can utilize ferrimagnets or saturated ferrite materials as an ingredient material to fabricate the required magnetic metamaterial. The effects of HMM model beyond the Drude-Lorentz expression are left as a subject for further work. The present effects can be measurable in the micron range, thus the reference frequency ${\omega }_0$ should be chosen to be about ${10}^{14}$ rad/s, which can be achieved in recent experiments. Take $a={\lambda }_0=1\mu m$ for example, the force is about ${10}^{-7}~{10}^{-5}$ N/m². Furthermore, our purpose is to achieve multi-stable equilibria with the Casimir force. As we demonstrate above, to realize the multi-equilibrium, the permeability of μ-HMM should be anisotropic, and the dispersion of permittivity must be taken into account. Correspondingly, the electromagnetic response of the μ-HMM changes between electric and magnetic several times across the whole frequency, then the multiequilibrium will emerge. Therefore, the different resonant frequencies for ${\mu }_{xx}^{\text{B}}$ and ${\mu }_{zz}^{\text{B}}$ of μ-HMM are essential to the multi-equilibrium. From these standpoints, the present results do not contradict the conclusions of Refs [47,48].

4. Conclusion

In conclusion, we study the Casimir effect between two types of HMMs, i.e., electric and magnetic hyperbolic metamaterials, both of which exhibit multilayer structures. Due to the effective medium theory, a region of hyperbolic dispersion will emerge, thereby offering a different contribution to the Casimir effect. Repulsive Casimir forces emerge at the zero-temperature limit, and they can be enhanced by adjusting the hyperbolic dispersion. Additionally, we find that the restoring Casimir force occurs between two slabs by manipulating the permittivity dispersion of the μ-HMM. Furthermore, multiple stable equilibria can be established due to the extreme anisotropy of the hyperbolic metamaterials. This phenomenon is understood by analyzing the properties of the electromagnetic responses within the corresponding frequency range at certain separations. Finally, the forces between two types of HMMs at room temperature are also calculated. The formation of multiple stable equilibria is maintained, and they appear at larger separations when one chooses suitable parameters. This work could be helpful for overcoming adhesion problems in MEMS and NEMS in the future.