Editorial Feature

Association Between Casimir Force and Material Characteristics of Mixed-State Superconductors

The Casimir effect is a macroscopic quantum phenomenon caused by electromagnetic field quantum fluctuations in a vacuum. Lifshitz created a generalized theory of the phenomena that may be applied to any dielectric body, including metals. The optical characteristics of objects and superconductivity of superconductors have a big impact on the Casimir force, which has been explored in the journal Quantum Reports. 

superconductivity, superconductors, Casimir force, Casimir, Lifshitz

Study: Casimir Effect between Superconducting Plates in the Mixed State. Image Credit: SeniMelihat/Shutterstock.com

The Casimir force is based on changes in Casimir energy caused by changes in superconductor optical characteristics. Recent advances in Casimir force measurements have shown that the Casimir force exists and obeys Lifshitz’s theory. Both transitions generated by temperature changes and transitions caused by an external magnetic field were examined in these studies.

The use of a nanophotonic sensing device on a chip to measure changes in the Casimir force between superconductive plates was presented as an experimental platform. As the external magnetic field rises, the conductivity of type-I superconductors varies suddenly. The magnetic field is initially expelled by the Meissner effect. The superconductivity of type-I superconductors, however, is reduced by a high magnetic field over a critical magnetic field Bc.

The normal phase of the magnetic flux is dictated by the velocity of the vortices. Between the lower critical magnetic field Bc1 and the upper critical magnetic field Bc2, there is a coexisting phase known as the mixed state or magnetic vortex state. In the superconductor, the magnetic flow is quantized.

The magnetic field decays exponentially with London penetration depth in the Meissner condition. The Casimir force between high-Tc superconductors in this condition has been theoretically studied by Villarreal and Caballero-Benitez.

Because the electric characteristics of type-I superconductors, such as aluminum, are well established, they are ideal for validating the Casimir energy transition. Many type-II superconductors, such as high-Tc superconductors, are anisotropic, in contrast. Measuring the Casimir force between superconductors can therefore reveal information on the electrical interplay between vacuum quantum fluctuation and quantum magnetic flux.

In this paper, researchers studied how the applied magnetic field affects the Casimir force between high-Tc superconductors in a mixed state. Because YBCO(YBa2Cu3O7) is a typical type-II superconductor, and the mixed state reflection data were measured, it was chosen as an example. Furthermore, the dissipation in terms of vortex motion helps explain the dependency of the reflectance on the external magnetic field.

The link between the Casimir force and material characteristics of superconductors in the mixed state is summarized in the conclusion section.


The angular frequency ω, wave vector k = (k, kz), where k is defined as the wave vector parallel to the surface, and polarization (transverse magnetic mode (TM) and transverse electric mode (TE) describe the electromagnetic field between the plates, as described in Figure 1.

Figure 1. Light propagation in (a) TM mode and (b) TE mode from a vacuum to a superconductor. The yz-plane is the plane of incidence. The external magnetic field B0 is perpendicular to the surface of the superconductor. The magnetic field of the light Blight in TM mode is perpendicular to the plane of incidence. Conversely, the electric field of the light is perpendicular to the plane of incidence, and the magnetic field exits at the yz plane. Image Credit: Inui, 2021


The YBCO’s parameters are presented in Table 1.

Table 1. Parameters for vortex dynamics. Source: Inui, 2021

. .
Tc 88.2 K
Tc2 87 K
λ0 140 nm
Bc2,0 112 T
κ0 2.1 × 104 N/m2
c0 2.0 × 10-7 Ω m
c1 9.7 × 10-7 Ω m/K


Figure 2 illustrates the superconductor’s mixed state permittivity along the imaginary axis at t = 0.1 for B = 0 and 10 T.

Permittivity parallel to the surface of YBCO in the mixed state at T = 8.82 K (t = 0.1) along imaginary axis in a log–log scale.

Figure 2. Permittivity parallel to the surface of YBCO in the mixed state at T = 8.82 K (t = 0.1) along imaginary axis in a log–log scale. Image Credit: Inui, 2021

The dependency of plasma frequencies ωp and ωD at t = 0.1 and 0.5, respectively, is shown in Figure 3(a, b).

Dependence of frequencies (a) ?p (plasma model) and (b) ?D (Drude model) on magnetic field at t = 0.1 (solid line) and 0.5 (dashed line).

Figure 3. Dependence of frequencies (a) ωp (plasma model) and (b) ωD (Drude model) on magnetic field at t = 0.1 (solid line) and 0.5 (dashed line). Image Credit: Inui, 2021

ωD decreases as the temperature rises, as seen in Figure 4.

Temperature dependence of frequency ?D (Drude model).

Figure 4. Temperature dependence of frequency ωD (Drude model). Image Credit: Inui, 2021

Figure 5(a) displays the normalized Casimir force ηC = P(a,T)/Pc (a) for a magnetic field between 1 and 60 T at a given separation distance a = 1 μm.

(Color online) Dependence of normalized Casimir force between YBCO plates in the mixed state on (a) magnetic field, (b) separation distance, and (c) temperature.

Figure 5. (Color online) Dependence of normalized Casimir force between YBCO plates in the mixed state on (a) magnetic field, (b) separation distance, and (c) temperature. Image Credit: Inui, 2021

Figure 6 shows a comparison of the temperature dependence of the YBCO plate’s Casimir force with and without the Lorentz component.

Comparison between normalized Casimir force acting on the superconductive plates with and without terms based on the Lorentz model.

Figure 6. Comparison between normalized Casimir force acting on the superconductive plates with and without terms based on the Lorentz model. Image Credit: Inui, 2021


One of the most essential concepts for determining the superconducting state is the magnetic field penetration depth. The Kramers–Kronig relation was used to determine the permittivity along with the imaginary frequency from the conductivity of superconductors stated in terms of penetration depth. Permittivity was calculated using a set of functions similar to those found in the plasma and Drude models.

The penetration depth was then calculated using the oscillator’s amplitude, and the material dependency of plasma frequencies was calculated using the spring constant. The electrical resistivity in the normal phase was used to calculate the material dependence of the relaxation parameter in the Drude model. A polycrystalline superconductor’s spring constant is stated to be 103 to 104 times lower than that of a single crystal.

The modification of the Casimir force by altering material characteristics has piqued interest. The Casimir force, which is created by irradiating laser pulses over a gold spherical and a Si membrane, is one example. The plasma frequency is related to the charge carrier concentration squared. As a result of the increase in electrons and holes caused by irradiation, the plasma frequency rises and the Casimir force increases.

Optical conductors have not been studied enough for electromagnetic waves with a frequency greater than microwaves in superconductors. As a result, the optical characteristics of the superconductor may be examined by measuring the Casimir force imparted on it (especially for short separation distances), which is dependent on conductivity at higher frequencies.

Journal Reference:

Inui, N (2021) Casimir Effect between Superconducting Plates in the Mixed State. Quantum Reports, 3(4), pp.731-745. Available Online: https://www.mdpi.com/2624-960X/3/4/46/htm

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Skyla Baily

Written by

Skyla Baily

Skyla graduated from the University of Manchester with a BSocSc Hons in Social Anthropology. During her studies, Skyla worked as a research assistant, collaborating with a team of academics, and won a social engagement prize for her dissertation. With prior experience in writing and editing, Skyla joined the editorial team at AZoNetwork in the year after her graduation. Outside of work, Skyla’s interests include snowboarding, in which she used to compete internationally, and spending time discovering the bars, restaurants and activities Manchester has to offer!


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