Effect of finite temperature and uniaxial anisotropy on the Casimir effect with three-dimensional topological insulators

In this work we study the Casimir effect with three-dimensional topological insulators including the effects of temperature and uniaxial anisotropy. Although precise experimental values for the optical properties of these materials are yet to be established, a qualitative analysis is still possible. We find qualitatively that the reported repulsive behavior and the equilibrium point are robust features of the system, and are favored by low temperatures and the enhancement of the optical response parallel to the optical axis. The dependence of the equilibrium point with temperature and with the topological magnetoelectric polarizability characteristic of three-dimensional topological insulators is also discussed.

Figure 1

Topological insulating plates separated by a distance $d$ covered with a thin ferromagnetic layer of thickness $l\ll d$ (not to scale). Changing the sign of the magnetization of one of the plates (parallel or antiparallel to the surface normal) is equivalent to changing the sign of (either) ${\theta }_{1,2}$.

Figure 2

(a) ${\mathrm{d̄}}_{\text{eq}}$ as a function of ${\theta }_1$ for different values of ${\theta }_2$ ranging from $\pi$ to $5\pi$. The solid blue line represents ${\mathrm{d̄}}_{\text{eq}}\propto \left|{\theta }_1{\theta }_2\right|$. (b) ${\mathrm{d̄}}_{\text{eq}}$ as a function of $\theta \equiv {\theta }_1=-{\theta }_2$ (black curve with dots). The behavior deviates from ${\theta }^2$ (bottom curve red) reasonable for small $\theta$. Beyond $\theta =17\pi$ the equilibrium point is lost (see text). For both figures the remaining parameters of the model were fixed to $\frac{{\omega }_e}{{\omega }_R}=0.45$.

Figure 3

Attraction vs repulsion in the classical limit ($T\to \infty$) as a function of $\frac{{\omega }_e}{{\omega }_R}$ and $\frac{\theta }{\pi }$, where ${\theta }_1=-{\theta }_2\equiv \theta$. The white upper region shows attraction while the lower green region shows repulsion. The black region is a forbidden region for the parameters due to the positive energy condition $\frac{\alpha \left|\theta \right|}{\pi }<\sqrt{\epsilon (0)}$ (see text).

Figure 4

(a) Casimir energy density [in units of $E_0=A\hslash c/(2\pi ){}^2({\omega }_R/c){}^3$] as a function of the dimensionless distance $\mathrm{d̄}$ for $\frac{{\omega }_e}{{\omega }_R}=0.45$ for temperatures ranging from $\mathrm{T̄}=0$ to 10 and ${\theta }_1=-{\theta }_2=3\pi$. Temperature pushes the repulsive behavior to shorter distances. (b) Equilibrium distance as a function of temperature. (c) Casimir energy density [in units of $E_0=A\hslash c/(2\pi ){}^2({\omega }_R/c){}^3$] as a function of $\mathrm{d̄}$ for $\frac{{\omega }_e}{{\omega }_R}=0.16$ for temperatures ranging from $\mathrm{T̄}=0$ to 10. For values where there is repulsion, decreasing the temperature shifts the curve toward smaller distances, as shown inside the inset figure.

Figure 5

Effect of changing the relative oscillator strengths between parallel and perpendicular components of the dielectric function on the Casimir energy density [in units of $E_0=A\hslash c/(2\pi ){}^2({\omega }_{R_z}/c){}^3$] as a function of the dimensionless distance $\mathrm{d̄}$ with (a) ${\theta }_{1,2}=0$, (b) ${\theta }_1={\theta }_2=\pi$, and (c) ${\theta }_1=-{\theta }_2=\pi$. (a) Attraction is obtained as no magnetoelectric term is included. (b) Both magnetoelectric couplings have the same sign and so no repulsion occurs. (c) Repulsion appears at short distances. Increasing ${\omega }_{e_{\perp }}$ makes the minimum shift to lower distances while increasing its depth. Therefore, repulsion is favored by the condition ${\omega }_{e_z}>{\omega }_{e_{\perp }}$.

Figure 6

Effect of changing the relative positions of the oscillator resonances between parallel and perpendicular components of the dielectric function on the Casimir energy density [in units of $E_0=A\hslash c/(2\pi ){}^2({\omega }_{R_z}/c){}^3$] as a function of the dimensionless distance $\mathrm{d̄}$ with (a) ${\theta }_{1,2}=0$, (b) ${\theta }_1={\theta }_2=\pi$, and (c) ${\theta }_1=-{\theta }_2=\pi$. (a) Attraction is obtained as no magnetoelectric term is included. (b) Both magnetoelectric couplings have the same sign and so no repulsion occurs. (c) Increasing ${\omega }_{R_{\perp }}$ makes the minimum shift to higher distances while decreasing its depth. Therefore, repulsion is favored by increasing $\beta$. (See text.)

Figure 7

Casimir energy density [in units of $E_0=A\hslash c/(2\pi ){}^2({\omega }_{R,z}/c){}^3$] and ${\theta }_1=-{\theta }_2=\pi$ including anisotropy and temperature effects: (a) Effect of changing the temperature from $\mathrm{T̄}=0$ to 7 with ${\omega }_{e,z}=0.45$, ${\omega }_{e,\perp }=0.3$, and $\beta =5$. (b) Effect of varying ${\omega }_{e,\perp }=0$–1 with $\mathrm{T̄}=5$, ${\omega }_{e,z}=0.45$, and $\beta =5$.