Casimir interaction with black phosphorus sheets

Lei Wang; Haiqing Sun; YongLei Jia; Lixin Ge; Zhichao Ji; Ke Gong

doi:10.1364/OE.489635

1. Introduction

The Casimir force is a macroscopic quantum effect due to the fluctuations of the electromagnetic field [1]. This interaction plays an important role in fabricating and operating nano- and micro-electromechanical systems (NEMS and MEMS) [2]. According to Casimir’s theory, the attractive force per unit area between two electrically neutral and perfect metals is $\frac {F_0}{A}=-\frac {\hbar c\pi ^2}{240 d^4}$[1], where the negative sign indicates attraction, $A$ is the in-plane area, $\hbar$ is the reduced Planck’s constant, $c$ is the speed of light in vacuum, and $d$ is the distance between the plates. Subsequently, the Casimir force based on isotropic two-dimensional (2D) materials such as graphene has been widely reported [3,4]. Drosdoff et al. found the Casimir force per unit area between pristine graphene sheet with a constant conductivity is $\frac {F_g}{A}=-\frac {3e^2}{128\pi ^2\varepsilon _0 d^4}$, which show the same distance dependence as the one for perfect metals [5]. Recently, anisotropic 2D materials, such as black phosphorus (BP) [6,7], ReS$_2$ [8,9] and TiS$_3$ [10,11] have received enormous attentions. Particularly, mono-layer or few-layer black phosphorus, showing direct bandgap, high mobility and large current on/off ratios [12], are promising 2D material for applications in optoelectronic devices [6,13]. Compared with the well-known Casimir interactions in graphene sheets [14,15] and strained graphene sheets [16,17], highly natural in-plane anisotropic is one of the most intriguing properties for BP sheets [12,18,19], which make it very interesting in nano- or micro-electromechanical systems.

In this work, we investigated the Casimir attractive force between parallel isotropic materials and anisotropic 2D sheets. Based on the fluctuation-dissipation theory, the Casimir force between isotropic thin plates (gold plates or graphene sheets) and BP sheets is calculated. We found that the Casimir interaction can be tuned by the carrier doping of BP sheets and graphene sheets. In addition, we also show that the Casimir force can be affected by ambient temperature and semi-infinite substrates.

2. Simulation and theory

The system in the study is depicted in Figs. 1(a) and 1(b), consisting of a few-layer BP sheet and a gold nanoplate with thickness $L_g$ or a single-layer graphene sheet brought into close proximity with a vacuum gap $d$. The in-plane dimension of the gold nanoplate is much larger than $L_g$ and $d$, thus it is considered as a slab during our calculations.

Fig. 1. (a) Schematic view of a gold nanoplate suspended above a few-layer black phosphates sheet. (b) Schematic view of a doped graphene sheet suspended above a few-layer BP sheet. The separation between the gold nanoplate or graphene sheet and the BP sheet is $d$.

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The Casimir free energy per unit area between two parallel flat bodies characterized by their reflection matrices $\mathbf {R}$, is defined by the Lifshitz formula [20,21]

(1)$$\frac{E_c(d)}{A}=\hbar\int_0^\infty\frac{d\xi}{2\pi}\int\frac{d^2\mathbf{k}_{\|}}{(2\pi)^2}\log\det[1-\mathbf{R_1}\cdot\mathbf{R_2}e^{{-}2\kappa_0d}],$$

where $\xi$ is the imaginary frequency with $\omega = i\xi$, $\kappa _0=\sqrt {k_{\|}^2+\xi ^2/c^2}$ represents the imaginary vertical wave vector in vacuum, $k_{\|}=\sqrt {k_x^2+k_y^2}$ is the parallel wave vector, $\mathbf {R_1}$ and $\mathbf {R_2}$ are the reflection coefficient matrices of object 1 and 2, respectively. Then the Casimir force can be obtained by $F_c=-\partial E_c(d)/\partial d$. In the study, we choose few-layer black phosphorus (BP) sheets as object 1, isotropic gold plates or single layer graphene sheets as object 2, vacuum as the intervening medium, and also take the zero-temperature limit of the expression. Considering an incident plane wave with in-plane wavevector $\mathbf {k}_{\|} = k_x\,\hat {x} + k_y\,\hat {y}$, the reflection matrix of a BP sheet is given by [22,23]

(2)$$\begin{aligned} \mathbf{R_1} & =\frac{1}{\Delta}\begin{pmatrix} \tilde{r}_{ss} & \tilde{r}_{sp}\\ \tilde{r}_{ps} & \tilde{r}_{pp} \end{pmatrix},\\ \Delta & =(\frac{\varepsilon \kappa_0}{\kappa}+\frac{\sigma_{xx}\kappa_0}{\varepsilon_0\xi}+1)(1+\frac{\kappa}{\kappa_0}+\frac{\mu_0\sigma_{yy}\xi}{\kappa_0})-\frac{\mu_0}{\varepsilon_0}\sigma_{xy}^2,\\ \tilde{r}_{ss} & =(\frac{\varepsilon \kappa_0}{\kappa}+\frac{\sigma_{xx}\kappa_0}{\varepsilon_0\xi}+1)(1-\frac{\kappa}{\kappa_0}-\frac{\mu_0\sigma_{yy}\xi}{\kappa_0})+\frac{\mu_0}{\varepsilon_0}\sigma_{xy}^2,\\ \tilde{r}_{pp} & =(\frac{\varepsilon \kappa_0}{\kappa}+\frac{\sigma_{xx}\kappa_0}{\varepsilon_0\xi}-1)(1+\frac{\kappa}{\kappa_0}+\frac{\mu_0\sigma_{yy}\xi}{\kappa_0})-\frac{\mu_0}{\varepsilon_0}\sigma_{xy}^2,\\ \tilde{r}_{sp} & =\tilde{r}_{ps}=2\sqrt{\frac{\mu_0}{\varepsilon_0}}\sigma_{xy}, \end{aligned}$$

where $\mu _0$ and $\varepsilon _0$ are the permeability and permittivity in vacuum, $\kappa =\sqrt {k_{\|}^2+\varepsilon \xi ^2/c^2}$ is the imaginary vertical wave vector in substrate, $\varepsilon =\varepsilon (i\xi )$ is the relative permittivity of the substrate evaluated with imaginary frequency, the subscripts $s$ and $p$ denote transverse electric (TE) and transverse magnetic(TM) mode, respectively, and

(3)$$\sigma=\begin{pmatrix} \sigma_{xx} & \sigma_{xy}\\ \sigma_{yx} & \sigma_{yy} \end{pmatrix}=\begin{pmatrix} \sigma_{\text{AC}}k_x^2/k_\|^2+\sigma_{\text{ZZ}}k_y^2/k_\|^2 & (\sigma_{\text{AC}}-\sigma_{\text{ZZ}})k_xk_y/k_\|^2\\ (\sigma_{\text{AC}}-\sigma_{\text{ZZ}})k_xk_y/k_\|^2 & \sigma_{\text{AC}}k_y^2/k_\|^2+\sigma_{\text{ZZ}}k_x^2/k_\|^2 \end{pmatrix}$$

are optical conductivity’s tensor of BP for the rotation with respect to unit vector of $\mathbf {k}_{\|}$. Incidentally, the off-diagonal elements of reflection matrix $r_{sp} = r_{ps} = 0$ for isotropic 2D materials as expected because of $\sigma _{xy} = \sigma _{yx} = 0$. However, the off-diagonal terms of the reflection matrix may nonzero for anisotropic 2D materials due to the anisotropic conductivity. The photonic properties of few layer BP sheets can be described by [24,25]:

(4)$$\sigma_{j}(i\xi)=\sigma_{\text{intra}}+\sigma_{\text{inter}}=\frac{D_j}{\pi(\xi+\eta)}+s_j\frac{2}{\pi}\tan^{{-}1}(\frac{\xi}{\omega_j}),$$

where $j\in \{\text {AC, ZZ}\}$ denotes the armchair (AC) or zigzag (ZZ) direction of the BP sheets, and $D_j=\frac {\pi e^2n}{m_j}$ is the related Drude weight, which is determined by anisotropic effective electron mass $m_j$ and carrier density $n$, $\eta$ corresponds to the relaxation rate, $s_j$ represents the strength of interband transitions, and $\omega _j$ is the frequency of the related interband transitions. We adopt the electron doping $n=5\times 10^{13}\text {cm}^{-2}$ as well as $\hbar \eta =10$ meV to describe the relaxation rate, which are within the range reported in ab initio studies [12].

With medium 2 being isotropic, the reflection matrix of a gold nanoplate or a graphene sheet is given by

(5)$$\mathbf{R_2}=\begin{pmatrix} r^{s} & 0\\ 0 & r^{p} \end{pmatrix},$$

where the reflection coefficients can be given analytically as follows

(6)$$\begin{aligned} r^{s,p}=&\frac{r^{s,p}_{u}+r^{s,p}_{d}e^{{-}2\kappa_gL_g}}{1+r^{s,p}_{u}r^{s,p}_{d}e^{{-}2\kappa_gL_g}}\\ r^s_{u}=&\frac{\kappa_0-\kappa_g}{\kappa_0+\kappa_g},\quad r^p_{u}=\frac{\varepsilon_g(i\xi)\kappa_0-\kappa_g}{\varepsilon_g(i\xi)\kappa_0+\kappa_g}\\ r^s_{d}=&\frac{\kappa_g-\kappa}{\kappa_g+\kappa},\quad r^p_{d}=\frac{\kappa_g/\varepsilon_g(i\xi)-\kappa/\varepsilon(i\xi)}{\kappa_g/\varepsilon_g(i\xi)+\kappa/\varepsilon(i\xi)} \end{aligned}$$

where $\kappa _g=\sqrt {k_{\|}^2+\varepsilon _g(i\xi )\xi ^2/c^2}$ is the imaginary vertical wave vector in medium 2 with $\varepsilon _g(i\xi )$ and $L_g$ being the relative permittivity and thickness of the nanoplates or nanosheets, respectively. The generalized Drude-Lorentz model is applied for gold, where four pairs of Lorentz poles are taken into account [26,27]

(7)$$\varepsilon(i\xi)=\varepsilon_\infty+\frac{\gamma\sigma}{\xi(\xi+\gamma)}+\sum_{j=1}^4(\frac{i\sigma_j}{i\xi-\Omega_j}+\frac{i\sigma_j^*}{i\xi+\Omega_j^*}).$$

The parameters shown in Ref. [27] are fitted from the experimental data in a wide range of frequency. Moreover, the complex surface conductivity of the graphene mono-layer is governed by the Kubo formula with the random phase approximations [4,28,29]

(8)$$\sigma(i\xi)=\frac{e^2E_F}{\pi\hbar^2(\xi+1/\tau)}+\frac{e^2}{2\hbar\pi}\tan^{{-}1}(\frac{\xi}{2E_F}),$$

where $E_F$ denotes the Fermi energy and $\tau$ is the relaxation time, then one can treat the graphene sheet as a thin layer of material with permittivity $\varepsilon (i\xi )=1+\sigma (i\xi )/(\varepsilon _0\xi L)$, where $L\approx 0.3$ nm is the thickness of single layer graphene sheets. The permittivity for different materials are shown in Fig. 2.

Fig. 2. The permittivity of gold, graphene (Gr), Teflon, and BP in the armchair ($\varepsilon _{AC}$) and zigzag ($\varepsilon _{ZZ}$) direction as a function of imaginary frequency. The Fermi energy and relaxation time of graphene are set as 1 eV and 1 ps, respectively. Regarding to the BP, we set $\hbar \omega _{\text {AC}}=1.0$ eV, $\hbar \omega _{\text {ZZ}}=0.35$ eV, $\hbar \eta =0.01$ eV, $m_{\text {AC}} =0.2\,m_0, m_{\text {ZZ}}=m_0, s_{\text {AC}}=1.7s_0, s_{\text {ZZ}}=3.7s_0, s_0=\sigma _0=e^2/4\hbar$ and $n=5.0\times 10^{13}\,\text {cm}^{-2}$. $m_0$ is the mass of a free electron. Here the thickness of BP sheets is set as 1 nm for comparison.

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The Casimir force per unit area between a freestanding BP sheet and a suspended gold nanoplate(BP-Gold) or a freestanding graphene nanosheet(BP-Gr), normalized by perfect metal limit, is shown in Fig. 3. Remarkably, they are of the order of $\sim 10^{-2}$ (i.e., $\sim 1/\alpha$) times perfect metal limit, where $\alpha =e^2/(4\pi \varepsilon _0c\hbar )\approx 1/137$ is the fine structure constant. It has the same order of magnitude as the Casimir force in graphene system [5]. Moreover, when the separation $d$ is larger than 50 nm, the Casimir force between the BP sheet and the gold nanoplate is about twice as much as it between the BP sheet and the graphene nanosheet.

Fig. 3. Relative Casimir force between a BP sheet and a gold plate (solid line), and between a BP sheet and a graphene sheet (broken line), normalized by perfect metal limit $F_0$. Here $L_g$=40 nm.

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The spectral Casimir force contribution between a BP sheet and a graphene sheet, defined as $\int d^2\mathbf {k}_{\|}\log \det [1-\mathbf {R_1}\cdot \mathbf {R_2}e^{-2\kappa _0d}]$, are shown in Fig. 4. Clearly, one can find that the contribution mainly comes from low frequency components, i.e., $\hbar \omega <0.5\,\text {eV}$. Besides, one can find that the spectra become narrower as the separation $d$ decrease. To understand the mainly contribution mechanism in Fig. 4, the contours plots of at $k$-space are shown in Fig. 5 for three different photon energies. For low photon energy $\hbar \omega =0.1\,\text {eV}$, $\varepsilon _{\text {AC}}>\varepsilon _{\text {ZZ}}$ is satisfied, the most effective coupling appears at $k_x$ axis with $k_xd\approx \pm 0.45$. For photon energy $\hbar \omega =0.382\,\text {eV}$, $\varepsilon _{\text {AC}}\approx \varepsilon _{\text {ZZ}}$, the contribution pattern is isotropic. While for photon energy $\hbar \omega =1\,\text {eV}$, $\varepsilon _{\text {AC}}<\varepsilon _{\text {ZZ}}$ is fulfilled, the most effective coupling appears at $k_y$ axis with $k_yd\approx \pm 1.14$ right now. Considering that the intensity of spectral Casimir force at low frequency is one order of magnitude higher than that of high frequency, one can conclude that the whole Casimir force contribution mainly comes from low frequency components, and the most effective coupling appears at $k_x$ direction.

Fig. 4. Contours plots of Casimir force contributed from different frequencies and different separation $d$.

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Fig. 5. Contours plots of Casimir force contributed from parallel wave vectors for different photon energies. (a) 0.1 eV; (b) 0.382 eV; (c) 1 eV. Here, $d$ = 10 nm.

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The Casimir force as a function of BP concentration $n$ are shown in Fig. 6. To our surprise, the Casimir force increases as $n$ varies from $1\times 10^{12}\,\text {cm}^{-2}$ to $1\times 10^{14}\,\text {cm}^{-2}$, which show the opposite dependence with the near-field radiative heat transfer [30,31]. This may be due to the near-field radiative heat transfer is mainly originated from the surface plasmon polaritions resonance while there is no resonance in imaginary frequency space [32]. Similarly, raising the Fermi energy, i.e., increasing the doping concentration in graphene, can also enhance the Casimir force. However, the thickness of gold plates has very little impact on the Casimir force between gold plates and BP sheets.

Fig. 6. (a) Relative Casimir force between a BP sheet and a gold plate as a function of carrier concentration $n$ in the BP sheet with different $L_g$. (b) Relative Casimir force between a BP sheet and a gold plate as a function of carrier concentration $n$ in the BP sheet with different Fermi energy in graphene.

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3. Finite temperature effect

The temperature of the system needs to be considered in the actual configuration. To consider the thermal excitation in graphene and BP sheets, one can replace the E$_F$ in Eq. (6) by $2k_bT\ln [2\cosh (E_F/2k_bT)]$, and replace the doping carrier density $n$ in Eq. (4) by

(9)$$n\approx\frac{\sqrt{m_{\text{AC}}m_{\text{ZZ}}}k_BT}{\pi\hbar^2}\log\bigg[1+\exp\big(\frac{\mu_c-E_1}{k_BT}\big)\bigg],$$

where $\mu _c$ is the chemical potential, E$_1$ is the minimum energy of the first conduction subband [33,34]. Moreover, the integral over frequency $\xi$ for 0 K approximation in Eq. (1) should be rewritten as a discrete summation [20,35]

(10)$$\frac{\hbar}{2\pi}\int_0^\infty d\xi\rightarrow k_bT {\sum_{n=0}^{\infty}}',$$

where $\xi$ is replaced by discrete Matsubara frequencies $\xi _n =2\pi \frac {k_bT}{\hbar }n(n=0,1,2,3,\ldots )$, k$_B$ is the Boltzmann’s constant, $T$ is the temperature of the system and the prime denotes a prefactor 1/2 for the term $n = 0$. Actually, regardless of suspenended BP sheets, graphene sheets or gold plates, one can conclude $\mathbf {R_1}(\xi _0)\approx \mathbf {R_2}(\xi _0)\rightarrow \begin {pmatrix} 0 & 0\\ 0 & 1 \end {pmatrix}$. The derivation process is shown in the Appendix. By the way, the reflection matrix of pefect metal is $\begin {pmatrix} -1 & 0\\ 0 & 1 \end {pmatrix}$. The Casimir force under different temperatures is shown in Fig. 7. It is found that the Casimir force increases as the temperature. Moreover, when $d\lesssim 50\,$nm, the deviation between 0 and 300 K is so small that one can adopt the 0 K approximation for room temperature.

Fig. 7. Relative Casimir forces between a BP sheet and a gold plate, and between a BP sheet (solid lines) and a graphene sheet (broken lines) calculated for finite temperatures and 0 K approximation from Eq. (1).

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4. Substrate effect

To investigate the impact of the substrate on the Casimir force between semi-infinite substrates, we consider the commonly used substrates made of Teflon. For Teflon substrate, the dielectric function is described by the Lorentz oscillator model, where eight pairs of Lorentz poles are taken into account [27,36]

(11)$$\varepsilon(i\xi)=1+\sum_{j=1}^8\frac{C_j}{1+(\xi/\omega_j)^2}.$$

The values of $C_j$ and $\omega _j$ shown in Ref. [27] are fitted from the experimental data in a wide range of frequency. Figure 8(a) shows the Casimir force between BP sheets and graphene or gold plate located on Teflon substrate. It was found that the features of Fig. 8(a) are similar to those of Fig. 3, except for the fact that Teflon substrates have enhanced the Casimir interaction. Actually, the Casimir force has almost doubled after introducing the Teflon substrate. To further study of the properties of substrate on the Casimir force, the dependence of relative Casimir force on the relative permittivity $\varepsilon$ of the substrates are shown in Fig. 8(b). One can find that the relative Casimir force monotonically increases with the increase of $\varepsilon$. When $\varepsilon$ is less than 10, the Casimir force approximately scale with $F\propto \lg (\epsilon )$, as the permittivity continues to increase, the Casimir force no longer increases significantly.

Fig. 8. (a) Relative Casimir force between a BP sheet and a gold plate (solid lines) on top of Teflon substrate, and between a BP sheet and a graphene sheet (broken lines) on top of Teflon substrate, normalized by perfect metal limit. Here $L_g$=40 nm. (b) The dependence of relative Casimir force on the relative permittivity $\varepsilon$ of the substrates. Here $d$=10 nm.

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5. Conclusion

In summary, the Casimir interaction between gold plates/graphene sheets and BP sheets has been theoretically investigated using the Lifshitz theory. Our model shows that the Casimir force with BP sheet is of the order of $\alpha$ times the perfect metal limit. The contribution of the Casimir force mainly comes from low frequency components, and the most effective coupling appears at $k_x$ direction. Besides, introducing dielectric substrate and increasing temperature can also enhance the Casimir force. Moreover, active modulation of Casimir interaction can also be realized by tuning the doping concentrations in BP and graphene sheet though external gating. Our work may pave the way for applications of anisotropic 2D materials in non-contact mechanical modulators, resulting from the quantum fluctuations of the electromagnetic field.

Appendix

When $\xi \rightarrow 0$, $\sigma _{xx},\sigma _{xy}$ and $\sigma _{yy}$ are finite, and $\lim \kappa =\lim \kappa _0=k_\parallel$. For simplify, we only consider the freestanding BP sheet, Eq. (2) can be read as

\begin{aligned} \lim_{\xi\rightarrow0}\Delta & =\lim_{\xi\rightarrow0}[(2+\frac{\sigma_{xx}\kappa_0}{\varepsilon_0\xi})(2+\frac{\mu_0\sigma_{yy}\xi}{\kappa_0})-\frac{\mu_0}{\varepsilon_0}\sigma_{xy}^2]=\frac{2\sigma_{xx}k_\|}{\varepsilon_0\xi}+4+\frac{\mu_0}{\varepsilon_0}(\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2)+o(\xi),\\ \lim_{\xi\rightarrow0}\tilde{r}_{ss} & =\lim_{\xi\rightarrow0}[(2+\frac{\sigma_{xx}\kappa_0}{\varepsilon_0\xi})(-\frac{\mu_0\sigma_{yy}\xi}{\kappa_0})+\frac{\mu_0}{\varepsilon_0}\sigma_{xy}^2]={-}\frac{\mu_0}{\varepsilon_0}(\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2)+o(\xi),\\ \lim_{\xi\rightarrow0}\tilde{r}_{pp} & =\lim_{\xi\rightarrow0}[(\frac{\sigma_{xx}\kappa_0}{\varepsilon_0\xi})(2+\frac{\mu_0\sigma_{yy}\xi}{\kappa_0})-\frac{\mu_0}{\varepsilon_0}\sigma_{xy}^2]=\frac{2\sigma_{xx}k_\|}{\varepsilon_0\xi}+\frac{\mu_0}{\varepsilon_0}(\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2)+o(\xi),\\ \lim_{\xi\rightarrow0}\frac{\tilde{r}_{ss}}{\Delta} & =\lim_{\xi\rightarrow0}\frac{-\frac{\mu_0}{\varepsilon_0}(\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2)+o(\xi)}{\frac{2\sigma_{xx}k_\|}{\varepsilon_0\xi}+4+\frac{\mu_0}{\varepsilon_0}(\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2)+o(\xi)}=0,\\ \lim_{\xi\rightarrow0}\frac{\tilde{r}_{pp}}{\Delta} & =\lim_{\xi\rightarrow0}\frac{\frac{2\sigma_{xx}k_\|}{\varepsilon_0\xi}+\frac{\mu_0}{\varepsilon_0}(\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2)+o(\xi)}{\frac{2\sigma_{xx}k_\|}{\varepsilon_0\xi}+4+\frac{\mu_0}{\varepsilon_0}(\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2)+o(\xi)}=1,\\ \lim_{\xi\rightarrow0}\frac{\tilde{r}_{sp}}{\Delta} & =\lim_{\xi\rightarrow0}\frac{\tilde{r}_{ps}}{\Delta}=\lim_{\xi\rightarrow0}\frac{2\sqrt{\frac{\mu_0}{\varepsilon_0}}\sigma_{xy}}{\frac{2\sigma_{xx}k_\|}{\varepsilon_0\xi}+4+\frac{\mu_0}{\varepsilon_0}(\sigma_{xx}\sigma_{yy}-\sigma_{xy}^2)+o(\xi)}=0, \end{aligned}

one can obtain that $\mathbf {R_2}(\xi _0)=\begin {pmatrix} 0 & 0\\ 0 & 1 \end {pmatrix}$.

Similarly, for graphene sheets or gold plates, when $\xi \rightarrow 0$, $\lim \kappa _g=\lim \kappa _0=\lim \kappa =k_\parallel$, and $\lim \varepsilon _g=\sigma /\xi$. For suspended graphene sheets or gold plates,

\begin{aligned} \lim_{\xi\rightarrow0}r^s_{u} & =\lim_{\xi\rightarrow0}r^s_{d}=0,\\ \lim_{\xi\rightarrow0}r^p_{u} & =\lim_{\xi\rightarrow0}\frac{\varepsilon_g-1}{\varepsilon_g+1}=1,\\ \lim_{\xi\rightarrow0}r^p_{d} & =\lim_{\xi\rightarrow0}\frac{1/\varepsilon_g-1}{1/\varepsilon_g+1}={-}1,\\ \lim_{\xi\rightarrow0}r^s & =\lim_{\xi\rightarrow0}\frac{r^{s}_{u}+r^{s}_{d}e^{{-}2\kappa_gL_g}}{1+r^{s}_{u}r^{s}_{d}e^{{-}2\kappa_gL_g}}=0,\\ \lim_{\xi\rightarrow0}r^p & =\lim_{\xi\rightarrow0}\frac{r^{p}_{u}+r^{p}_{d}e^{{-}2\kappa_gL_g}}{1+r^{p}_{u}r^{p}_{d}e^{{-}2\kappa_gL_g}}=\frac{1-e^{{-}2k_\|L_g}}{1-e^{{-}2k_\|L_g}}=1, \end{aligned}

one can conclude that $\mathbf {R_1}(\xi _0)=\mathbf {R_2}(\xi _0)=\begin {pmatrix} 0 & 0\\ 0 & 1 \end {pmatrix}$

Funding

National Natural Science Foundation of China (11604282, 11604283, 61571386, 61974127); Natural Science Foundation of Henan Province (232300420120); Nanhu Scholars Program for Young Scholars of Xinyang Normal University.

Disclosures

The authors declare no conflicts of interest.

Data availability

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

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Casimir interaction with black phosphorus sheets

Abstract

1. Introduction

2. Simulation and theory

3. Finite temperature effect

4. Substrate effect

5. Conclusion

Appendix

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (13)

Optics Express