Drastic change of the Casimir force at the metal-insulator transition

The dependence of the Casimir force on material properties is important for both future applications and to gain further insight on its fundamental aspects. Here we apply the general Lifshitz theory of the Casimir force to low-conducting compounds, or poor metals. For distances in the micrometer range, the Casimir force for a large variety of such materials is described by universal equations containing a few parameters: the effective plasma frequency ${\omega }_p$, dissipation rate $\gamma$ of the free carriers, and electric permittivity ${\epsilon }_{\infty }$ for $\omega \ge {\omega }_p$ (in the infrared range). This theory of the Casimir force for poor metals can also describe inhomogeneous composite materials containing small regions with different conductivity. The Casimir force for systems involving samples made with compounds that have a metal-insulator transition shows a drastic change of the Casimir force within the transition region, where the metallic and dielectric phases coexist. Indeed, the Casimir force can increase by a factor of 2 near this transition.

Figure 1

The normalized Casimir force $\Pi =F/F_{\text{C}}$ versus $l/l_p$ for some values of parameters (shown near curves). This plot compares the results of numerical calculations of the two-dimensional integral Eq. (11) for the prefactor $\Pi$ (symbols ◻, ◇, and ○) with the results within the approximate approach based on Eq. (13) (solid curves). This plot shows a very good agreement between both.

Figure 2

(Color online) The normalized Casimir force $\Pi =F/F_{\text{C}}$ versus the parameter $\tilde{l}=l/l_p$, for $\nu =1,2$, using the typical value ${\epsilon }_{\infty }=8$, as well as the smaller ${\epsilon }_{\infty }=1.5$, and different values of the dissipation parameter $\alpha =\gamma /{\omega }_p$. The horizontal line on top gives the asymptotic value, $F/F_{\text{C}}=\Pi =1$, for an ideal metal.

Figure 3

Integrals $J_1$ and $J_2$, defined by Eq. (13), describing the Casimir force versus $l/l_p$ for ${\epsilon }_{\infty }=8$ and different values of the dissipation parameter $\alpha =\gamma /{\omega }_p$. The symbols indicated inside the square depict the results of numerical calculations.

Figure 4

The integral $J_1$ versus $l/l_p$ for ${\epsilon }_{\infty }=8$ and different values of the dissipation parameter $\alpha =\gamma /{\omega }_p$ (indicated by the symbols inside the square). The symbols depict the results of numerical calculations. The detailed behavior of $J_1$ for small $\tilde{l}$ is present in the inset. The dotted line describes the analytical result Eq. (15) for $\alpha =0$, solid lines are drawn in accordance with the fitting formula (18).

Figure 5

The integral $J_2$ versus $l/l_p$, calculated numerically for ${\epsilon }_{\infty }=8$ and ${\epsilon }_{\infty }=1.5$ and different values of $\alpha$ (indicated by the symbols inside the square). The shapes of the curves $J_2\left(l/l_p\right)$ depend very weakly on $\alpha$.

Figure 6

The integral $J_2$ versus $l/l_p$, calculated numerically for $\alpha =0.1$ and very different values of the dielectric permittivity, ${\epsilon }_{\infty }=8$ (▲) and ${\epsilon }_{\infty }=1.5$ (▼). The curves describe the approximating function Eq. (19). Note the very good agreement between numerical data and approximating functions. The inset shows the magnified curves for small values of $l/l_p$.

Figure 7

(Color online) The effective permittivity ${\epsilon }_{\text{eff}}\left(i\zeta \right)$ (in units of ${\epsilon }_{\infty }$), for $d=3$ and different concentrations $f$ of the metal phase, as a function of the dimensionless variable $X^2=\zeta \left(\zeta +\gamma \right)/{\omega }_p^2$. Inset: effective plasma frequency ${\omega }_{p,\text{eff}}$ (in units of ${\omega }_p$) versus $f$ in the coexistence region.

Figure 8

Casimir force for the interaction between two equivalent poor metals $\left(\nu =2\right)$ (with ${\epsilon }_{\infty }=8$ and $\alpha =0.3$) versus the concentration $f$ of the metallic phase in the coexistence region. The force is normalized by values of the force $F_{\text{C}}$ for ideal metals in the same geometry and for different values of $l/l_p=1$ (▼), 3(◼), and 5(▲), where $l_p$ is determined by the plasma frequency in the pure metallic phase.

Figure 9

Predicted dependence of the Casimir force on temperature for the interaction between a poor metal and an ideal metal, calculated for (a) ${\text{VO}}_2$ and for (b) ${\text{La}}_{0.7}{\text{Ca}}_{0.3}{\text{MnO}}_3$. The force is normalized by its value $F_{\text{C}}$ for an ideal metal for different values of $l/l_p=1$ (▼), 3(◼), and 5(▲), where $l_p$ is determined by the plasma frequency in the pure metallic phase. The horizontal dashed lines indicate the limit values of the force for the pure insulating phase and (for ${\text{VO}}_2$ only) its metallic phase. The corresponding values of $F_{\text{C}}$ are described by Eq. (1); for ${\lambda }_p\simeq 1.2\mu \text{m}$ these are $F_{\text{C}}=0.9$, $1.15\times {10}^{-2}$, and $1.5\times {10}^{-3}\text{dyn}/{\text{cm}}^2$ for $l/l_p=1$, 3, and 5, respectively. Note that the Casimir force can increase by about a factor of 2.