Distance-Dependent Sign Reversal in the Casimir-Lifshitz Torque

The Casimir-Lifshitz torque between two biaxially polarizable anisotropic planar slabs is shown to exhibit a nontrivial sign reversal in its rotational sense. The critical distance $a_c$ between the slabs that marks this reversal is characterized by the frequency ${\omega }_c\sim c/2a_c$ at which the in-planar polarizabilities along the two principal axes are equal. The two materials seek to align their principal axes of polarizabilities in one direction below $a_c$, while above $a_c$ their axes try to align rotated perpendicular relative to their previous minimum energy orientation. The sign reversal disappears in the nonretarded limit. Our perturbative result, derived for the case when the differences in the relative polarizabilities are small, matches excellently with the exact theory for uniaxial materials. We illustrate our results for black phosphorus and phosphorene.

Figure 1

Preview: The torque between two biaxially polarizable planar slabs as a function of the separation distance $a$. Our perturbation theory predicts that, when the in-planar polarizabilities along the two principal axes of one of the materials are equal at a characteristic frequency ${\omega }_c$, the torque may change sign at a critical distance $a_c$. Black phosphorus, whose crystalline structure is shown in the inset, and its monolayer phosphorene are such materials used in our analysis. The solid curve is a preview of our central result, shown here for the torque between two semi-infinite slabs of black phosphorus when the relative angle $\theta =\pi /4$. The dashed curve is the corresponding nonretarded limit.

Figure 2

(Upper) Principal components of the dielectric tensors ${\mathit{ϵ}}_i=\mathrm{diag}({ϵ}_i^x,{ϵ}_i^y,{ϵ}_i^z)$ of BP and 2D-P, calculated using density functional theory (described in the text). The imaginary Matsubara frequency ${\zeta }_m$ is obtained after making the Euclidean rotation; ${\omega }_m=i{\zeta }_m=i2\pi m{k}_{B}T/\hslash$. Components ${ϵ}_i^x={ϵ}_i^y$ for both materials close to ${\zeta }_c\sim 4\times {10}^{15}\mathrm{rad}/\mathrm{s}$. (Lower) Perturbative parameter ${\beta }_i$ for BP and 2D-P (shown for all three faces of BP crystal).

Figure 3

Schematics for choices of interacting media for BP. We denote ${\mathrm{BP}}^{(XY)}$ for the configuration when the $X\text{−}Y$ face of the BP aligns perpendicular to $\widehat{\mathbf{n}}$. Similarly, ${\mathrm{BP}}^{(XZ)}$ and ${\mathrm{BP}}^{(ZY)}$ describe the configurations when $X\text{−}Z$ and $Y\text{−}Z$ faces of BP are set perpendicular to the $\widehat{\mathbf{n}}$, respectively.

Figure 4

Torque per unit area as a function of the separation distance $a$ between different combinations of BP and 2D-P for $\theta =\pi /4$. The torque is not only nonmonotonic, but also changes its sense of rotation as a function of $a$.

Figure 5

Torque per unit area as a function of the separation distance $a$ between two identical 2D-P layers (left) and two identical semi-infinite BP slabs (right). We show the TE, TM, and mixed mode contributions, which have different hierarchical order in the two cases.