van der Waals torque and force between anisotropic topological insulator slabs

We investigate the character of the van der Waals (vdW) torque and force between two coplanar and dielectrically anisotropic topological insulator (TI) slabs separated by a vacuum gap in the nonretardation regime, where the optic axes of the slabs are each perpendicular to the normal direction to the slab-gap interface and also generally differently oriented from each other. We find that in addition to the magnetoelectric coupling strength, the anisotropy can also influence the sign of the vdW force, viz., a repulsive vdW force can become attractive if the anisotropy is increased sufficiently. In addition, the vdW force oscillates as a function of the angular difference between the optic axes of the TI slabs, being most repulsive/least attractive (least repulsive/most attractive) for angular differences that are integer (half-integer) multiples of $\pi$. Our third finding is that the vdW torque for TI slabs is generally weaker than that for ordinary dielectric slabs. Our work provides an instance in which the vector potential appears in a calculation of the vdW interaction for which the limit is nonretarded or static.

Figure 1

A pair of flat, coplanar topological insulator slabs (labeled 1 and 3) separated by a vacuum gap (labeled 2) of width $\ell$. The optic axis of slab 1 (3) is colored cyan (green), and the angular difference between the orientations of the optic axes is $\theta$. In the figure we have only shown a finite slice of the slabs, which are assumed to have infinitely large thicknesses and cross-sectional areas.

Figure 2

Behavior of the van der Waals force (in units of $A/\beta {\ell }^3$, where $A$ is the cross-sectional area of the slabs, $\beta$ is the inverse temperature, and $\ell$ is the interslab separation) as a function of optic axis misalignment $\theta$, between a pair of dielectrically similar topological insulator slabs with the same dielectric permittivity, but different magnetoelectric polarizabilities $\overline{\alpha }$: (a) ${\overline{\alpha }}_1=-{\overline{\alpha }}_3=3/137$ and static dielectric permittivity ${\epsilon }_{1z}\left(0\right)={\epsilon }_{1y}\left(0\right)={\epsilon }_{3z}\left(0\right)={\epsilon }_{3y}\left(0\right)=4$; (b) ${\overline{\alpha }}_1=-{\overline{\alpha }}_3=21/137$ and static dielectric permittivity ${\epsilon }_{1z}\left(0\right)={\epsilon }_{1y}\left(0\right)={\epsilon }_{3z}\left(0\right)={\epsilon }_{3y}\left(0\right)=1.20179$. For both cases the intervening gap is a vacuum $\left({\epsilon }_2=1\right)$. In (a) the behavior is plotted for $\gamma =0.11$ (green, dot-dashed line), 0.1 (blue, dashed line), and 0.09 (red, dotted line). In (b) the behavior is plotted for $\gamma =0.0103$ (green, dot-dashed line), 0.01 (blue, dashed line), and 0.0097 (red, dotted line).

Figure 3

Behavior of the van der Waals force (in units of $A/\beta {\ell }^3)$ as a function of optic axis misalignment $\theta$, for a pair of dielectrically similar topological insulator slabs with ${\overline{\alpha }}_1=-{\overline{\alpha }}_3=21/137$ and $\gamma =0.01$, separated by a vacuum gap $\left({\epsilon }_2=1\right)$; the curves are plotted for static dielectric permittivity ${\epsilon }_{1z}\left(0\right)={\epsilon }_{1y}\left(0\right)={\epsilon }_{3z}\left(0\right)={\epsilon }_{3y}\left(0\right)$ equal to the value of (i) 1.2018 (green, dot-dashed line), (ii) 1.20179 (blue, dashed line), and (iii) 1.20178 (red, dotted line).

Figure 4

Behavior of the torque (multiplied by ${\ell }^2/A$, where $A$ is the cross-sectional area of each slab and $\ell$ is the interslab separation distance) for (i) ordinary dielectric slabs (black, thin line), (ii) topological insulator slabs with ${\overline{\alpha }}_1={\overline{\alpha }}_3=69/137$ (green, dashed line), (iii) topological insulator slabs with ${\overline{\alpha }}_1=-{\overline{\alpha }}_3=69/137$ (red, dot-dashed line), and (iv) topological insulator slabs with ${\overline{\alpha }}_1=-{\overline{\alpha }}_3=11/137$ (blue, dashed line). We take the slabs to be dielectrically uniaxial, with static dielectric permittivity ${\epsilon }_z\left(0\right)={\epsilon }_y\left(0\right)=4$ and $\gamma =0.2$, and the gap to be a vacuum $\left({\epsilon }_2=1\right)$.