Nonanalytic behavior of the Casimir force across a Lifshitz transition in a spin-orbit-coupled material

The Casimir effect is a fascinating phenomenon where quantum fluctuations of the electromagnetic field give rise to measurable forces between macroscopic systems. Here we propose that the Casimir effect can be used as a tool to detect changes in electronic structures. In particular, we focus here on the Lifshitz transition—a topological change in the Fermi surface—in a planar spin-orbit-coupled semiconductor in a magnetic field and calculate the Casimir force between the semiconductor and another probe system across the magnetic-field-tuned transition. We show that the Casimir force experiences a sharp kink at the topological transition and provide numerical estimates indicating that the effect is well within experimental reach. The simplest experimental realization of the proposed effect would involve a metal-coated sphere suspended from a microcantilever above a thin layer of InSb (or another semiconductor with a large $g$ factor).

Figure 1

The geometry typically used in experimental measurements of the Casimir force is a gold-coated sphere suspended above a planar plate from a cantilever. We consider a lower plate of indium antimonide with an applied magnetic field.

Figure 2

The Casimir force $F_c$ normalized by the ideal conductor value between one semiconductor plate and one metallic plate separated by $a=50$ nm as a function of applied magnetic field. The red plot (left axis) corresponds to $\mu >0$, and the blue plot (right axis) corresponds to $\mu <0$. The upper plot uses $\mu =\pm 6$ meV and the lower uses $\mu =\pm 10$ meV. The insets show the band structure above and below the transition point (marked with a dashed line) along with the two fixed values of the Fermi energy.

Figure 3

The Casimir force $F_c$ normalized by the ideal conductor value between two metallic plates at a fixed separation as a function of the applied magnetic field. The insets show the band structure above and below the transition along with the fixed value of the Fermi energy. With a large Fermi energy and small electronic $g$ factor ($\approx 2$), a prohibitively large magnetic field is needed to reach the transition.

Figure 4

The Casimir force $F_c$ normalized by the ideal conductor value $F_0$ between two semiconductor plates separated by $a=50$ nm as a function of applied magnetic field. The red plot (left axis) corresponds to $\mu >0$, and the blue plot (right axis) corresponds to $\mu <0$. The upper plot uses $\mu =\pm 6$ meV and the lower plot uses $\mu =\pm 10$ meV. The insets show the band structure above and below the transition point along with the two fixed values of the Fermi energy.

Figure 5

The imaginary frequency longitudinal conductivity of the InSb plate at $i\omega =2i\mu$ for $\mu =10$ meV as a function of the applied magnetic field. The Lifshitz transition point is indicated with a dashed line.

Figure 6

Examples of the two types of diagrams we are left with. In the first type, all photon lines connect one plate (in the form of ${\widehat{\Pi }}_1$, labeled here by A) to the other plate (in the form of ${\widehat{\Pi }}_2$, labeled by B). In the second type, there is at least a single photon line connecting a plate to itself.