Stabilizing Diamagnetic Levitation of a Graphene Flake through the Casimir Effect

: Graphene exhibits diamagnetism, enabling it to be lifted by the repulsive force produced in an inhomogeneous magnetic ﬁeld. However, the stable levitation of a graphene ﬂake perpendicular to the magnetic ﬁeld is impeded by its strong anisotropic of magnetic susceptibility that induces rotation. A method to suppress this rotation by applying the Casimir force to the graphene ﬂake is presented in this paper. As a result, the graphene ﬂake can archive stable levitation on a silicon plate when the gravitational force is small.


Introduction
One of the promising applications of the Casimir effect [1,2] is the actuation of microelectromechanical systems (MEMS) [3][4][5][6].Unlike conventional actuation methods that rely on electric power sources, the Casimir force, which arises from vacuum fluctuations, does not require an external energy source such as a battery.This advantage makes it a desirable option for MEMS actuation.However, the Casimir force can also lead to unexpected adhesion [7] between different parts of MEMS, causing their function to cease due to the omnipresence of vacuum fluctuations.To prevent this adhesion, a straightforward approach is to generate a repulsive force that levitates the MEMS parts.While the repulsive Casimir force [8] has been studied and observed in liquid environments [9], its utilization in atmospheric conditions has not been explored [10].
Levitating MEMS parts without them adhering to substrates can reduce friction and enhance the sensitivity sensors.Numerous methods for levitating objects in the atmosphere have been proposed, with magnetic levitation being a well-known technique.According to Earnshaw's theorem, the stable levitation of magnets cannot be achieved through a combination of static magnetic or gravitational force alone; additional control is required.However, the unique property of diamagnetism enables levitation in the atmosphere without the need for control [11,12].
Graphite, a highly diamagnetic material [13], can be stably levitated above neodymium magnets, and more recently, multi-layer graphene flakes have also been successfully levitated [14,15].However, achieving stable levitation of a single-layer graphene flake above a magnet in an atmosphere setting remains an unmet challenge.When a magnetic field is applied perpendicularly to the surface of graphene, a strong magnetic moment is induced.In contrast, applying a magnetic field parallel to the surface results in minimal magnet moment induction.Consequently, the graphene rotates to align itself parallel to the magnetic field, leading to a loss of the levitation force.
This study investigates the levitation of a single graphene flake above a substrate, considering the interplay between the diamagnetic force and the Casimir force within the framework of the proximity force approximation (PFA) [16].Normally, the Casimir force between a graphene sheet and a substrate is attractive, thereby reducing the levitation force.However, it is demonstrated that the Casimir effect can counteract rotation and enable the stable levitation of graphene.
For a configuration involving a small square plate and an infinitely large substrate arranged in parallel, the Casimir energy between them is negative and proportional to the area of the small plate.When the small plate tilts around an axis passing through its center and parallel to its sides, the projection area onto the substrate decreases due to the inclination.This decrease in the projection area contributes to an increase in the Casimir energy.Additionally, the distance between one side of the small plate and the substrate increases, further increasing the Casimir energy.However, the distance between the other side of the small plate and the substrate decreases, leading to a decrease in the Casimir energy.The stability of the parallel state relative to the substrate is determined by the summation of these contributions.For large separations, the last contribution is smaller than the others, resulting in a stable parallel state.If the torque induced by magnetic interaction can be canceled out by the Casimir effect, a graphene flake can be levitated stably in a vacuum.
The remainder of this paper is organized as follows: In Section 2, the magnetic properties of a graphene flake based on the tight-binding model is explained.Section 3 focuses on the levitation of a graphene flake, considering the balance between the force induced by a magnetic field and gravity.In Section 4, the approximation of the Casimir energy between a single-layer graphene and a silicon substrate as a summation of power functions is given.In Section 5, the change in the Casimir energy which is expressed as a power function, due to the rotation being is calculated.This also explores the relationship between the stability of the parallel state and the separation distance.In Section 6, the stability of a levitated graphene flake above a silicon substrate through the application of diamagnetic force is examined.In Section 7, the optimizing the applied magnetic field can lead to stabilization through the Casimir effect on Earth is demonstrated.Additionally, the calculation method beyond PFA is discussed.Finally, in the conclusion, the essential conditions required to achieve stable levitation are summarized.

Magnetic Properties of Graphene
The potential energy of a graphene flake in the presence of an external magnetic field is examined.In this study, the magnetic field is generated by passing an electric current through a coil.When the electric current is sufficiently large, the graphene flake can be levitated above the coil, as depicted in Figure 1a.The coil, with a radius R, is positioned above a silicon substrate at a separation distance d as illustrated in Figure 1b.It is important to note that the discussion focuses solely on the levitation achieved through the magnetic force, while the influence of the Casimir effect on the levitation is explored in subsequent Sections.The magnetic flux density along the z-axis is expressed as follows: where I is the electric current through the coil, and µ 0 = 1.257 × 10 −6 NA −2 is the permeability in the vacuum.The direction of the electric current is counterclockwise when observed from above the ring.On the z-axis, the component of the magnetic flux density parallel to the substrate is zero.The diamagnetic characteristics of graphene stem from the interaction between the induced electric current and the magnetic field.This study neglects the spin interactions and focus on the calculation of the energy eigenvalues of graphene, denoted as i (B), which depend on the magnetic field.These eigenvalues are typically determined using the tight-binding model and the Peierls substitution method [17][18][19].The Hamiltonian can be expressed as follows: where γ 0 (=3 eV) is the transfer energy; ĉ † n and ĉn are the annihilation and creation operators of an electron at site n, respectively.The magnetic dependence of the Hamiltonian is represented by Peierls phase, φ nm , defined by where A(r) is the vector potential, and r s is the position of site s.The symbols e and h denote the elemantary charge and the reduced Planck constant, respectively.If the applied uniform magnetic field, B, is perpendicular to the graphene surface, the vector potential can be expressed by (0, Bx, 0) in the Landau gauge.
By calculating eigenvalues i (B) of the Hamiltonian, the free energy of graphene including the contribution from the orbital current induced by applying the magnetic field at temperature T is expressed as where k B and µ denote the Boltzmann constant and the chemical potential, respectively.In Equation ( 4), the factor of 2 is the spin degeneracy of the levels.The present sudy considers only the case of µ = 0.When considering small magnetic fields, the induced magnetic moment of the graphene flake is directly proportional to the applied magnetic field.Consequently, the magnetic potential can be described as proportional to the inner product of the magnetic moment and the magnetic field.Thus, the magnetic potential for the small magnetic field is expressed as c m B 2 .The coefficient c m depends on the size and edge type of graphene and is expressed as follows:

Levitation by Diamagnetic Force
In the case when the angle of inclination is zero and the change in the magnetic field within the graphene flake is disregarded, the total energy of the graphene system can be expressed as the sum of the magnetic energy and the gravitational energy: where m is the mass of graphene and g is gravitational acceleration.If d = 0 and the normalized position, ζ ≡ z/R, is introduced, the total energy is expressed as where Figure 2a shows the dependence of U mg on ζ for a different γ.For a small γ, there is a local minimum of U mg , at which the magnetic force balances.Furthermore, this equilibrium point is stable.Thus, the graphene can be levitated stably along the z-axis.If γ > γ c ≡ 1.329, no local minimum exists and the graphene falls to the substrate.Figure 2b shows the dimensionless levitation height, ζ m , where U mg takes a minimum valueas a function of γ.The dimensionless levitation height approaches 0.378 in the limit of γ → γ c .The magnetic susceptibility of graphite vertical to the surface per mass, χ ⊥ , is −2.7 × 10 −7 m 3 /kg and one parallel to the surface per mass, χ , is −6.3 × 10 −9 m 3 /kg.Similarly, the orbital magnetic susceptibility of graphene when aligned parallel to the surface is expected to be significantly smaller than that when aligned perpendicularly to the surface.Thus, it is assumed that χ is zero in the following calculations.Accordingly, the magnetic energy of the inclined graphene with the angle θ of inclination is given by The gravitational energy of a graphene flake remains unaffected by its rotation.Consequently, the flake will rotate until it aligns as parallel to the magnetic field, causing the levitation force to diminish.As a result, the graphene flake descends and eventually comes in contact with the substrate.

Casimir Force between a Graphene Flake and a Silicon Plate
To address the issue of rotation, the utilization of the Casimir effect becomes crucial.According to the Lifshitz theory, the Casimir free energy per unit area between a single-layer graphene sheet and a dielectric plate, separated by a separation a and at a temperature T, can be expressed as the sum of contributions for different polarizations η of the electromagnetic field, namely transverse magnetic (TM) and transverse electric (TE): where Here, ζ l with nonnegative integer variable l is the dimensionless Matsubara frequencies defined by 4πak B Tl/hc, with c denoting the speed of light, and r η are the reflection coefficients on graphene and on a plate for the polarization η, respectively [20][21][22][23][24][25].
The reflection coefficients on a silicon plate are expressed as where l is the dielectric permittivity of silicon at the imaginary frequency, 2πik B Tl/h, and calculated from the optical data [26] based on the Kramers-Kronig relation, The reflection coefficients on graphene using the Dirac model are expressed as follows: where Π 00 is the 00-component of the polarization tensor, Π, and The polarization tensor is determined by the temperature [27], mass gap parameter, δ g , chemical potential, µ, and the Fermi velocity, v F = c/300.The polarization tensor is described in detail in Ref. [21].
The circles in Figure 3 show the dependence of the Casimir energy per area between graphene with δ g = µ = 0 and a silicon plate on the separation distance at temperature 300 K.The line represents a fitting function, where c 3 = 1.11 × 10 −12 Jm and c 2 = 7.96 × 10 −11 J.

Change in the Casimir Energy by Inclining
The calculation of the Casimir energy between an inclined plate and a flat plate can be a computationally intensive task [28][29][30].As an alternative approach, the PFA is employed.Within the PFA framework, the Casmir energy between an inclined square plate, with one side measuring 2L, and an infinite substrate can be approximated by the following expression: where u(z) is the Casimir energy per area.If the Casimir energy per area obeys the power function, u(z) = −c β z −β , where c β is a constant and β > 1, its dependence on θ is expressed as By introducing a dimensionless coordinate α ≡ a/L, the normalized energy, Ũ, with the absolute value at θ = 0 is expressed as Figure 4a shows Ũ(α, θ, 2) for α = 1.2, √ 2, and 3.For the small values of α, representing small separation distances, a local minimum is present at the nonzero inclination angle and the flat state (θ = 0) becomes unstable.As the separation increases above a threshold, α c = √ 2, the Casimir energy takes the minimum value at θ = 0, and the flat state becomes stable.Figure 4b

Levitation of a Graphene Flake above a Silicon Plate in a Magnetic Field
Let us proceed to investigate whether the suppression of rotation, discussed in Section 5, can contribute to stabilizing the levitation of a graphene flake in the presence of a magnetic field.If the graphene flake is levitated on a silicon plate, as depicted in Figure 1b, the total potential energy can be approximated given the following expression: where the Casimir energy between a graphene flake and a coil is neglected.Figure 5a shows the relationship between the total energy of a flat graphene flake and a silicon plate, and the separation distance, considering the gravitational acceleration g = 0 and 0.1 m/s 2 .The parameters employed in the calculation are summarized in Table 1.Notably, one of the parameters, c m , plays an important role in determining the magnetic force.Its value is specific to a graphene flake with hexagonal armchair edges at a temperature of 300 K (see Ref. [17] for details).The levitation heights are measured to be 7.3 and 6.9 µm for the gravitational acceleration of 0 and 0.1 m/s 2 , respectively.Figure 5b displays the change in the total energy due to inclination, with the minimum value of ∆U set to zero.The total energy monotonously increases as the graphene flake tilts away from θ = 0, indicating stable levitation when the gravitational acceleration is small.As the gravitational acceleration increases, the levitation height decreases.The results for g = 0.3 m/s 2 and 0.4 m/s 2 are presented in Figure 6a,b and illustrate the dependence of total energy on the position and inclination angle, respectively.When the graphene flake derivates from θ = 0, the total energy decreases, reaching its minimum at θ = 0.52 rad for g = 0.3 m/s 2 .The inclined angle increases with an increase in gravitational acceleration.For g = 0.4 m/s 2 , the angle of inclination, at which the total energy is minimized, is 0.67 rad. Figure 6c shows the total energy of inclined graphene flakes at θ = 0.52 rad for g = 0.3 and θ = 0.67 rad for g = 0.4 m/s 2 .In the case of g = 0.3 m/s 2 , a local minimum exists in the total energy, enabling the levitation of the graphene flake in the inclined state.However, for g = 0.4 m/s 2 , no local minimum exists, causing the graphene flake to fall onto the substrate after tilting.

Role of a Diamagnetic Force and the Casimir Torque
A diamagnetic force and the Casimir torque must be effectively combined to achieve stable levitation.In previous calculations, the magnetic field is generated by a circular current and the levitation was unsuccessful for large gravitational accelerations.However, if an appropriate magnetic field is generated, then stable levitation can be realized.For example, when the magnetic flux density, which is expressed by 0.5 − 0.35 z + 0.08z 2 T, (where z is in µm) near z = 2 µm is generated, the total energy of the flat graphene flake for g = 9.8 m/s 2 takes a minimum at 2 µm as shown in Figure 7a.Furthermore, Figure 7b shows the total energy at the levitation height and takes the minimum in the flat state (θ = 0), and indicates that stable levitation can be achieved on Earth if the appropriate magnetic field is generated.
(a)  The stabilization of levitation results from the feature that the Casimir energy takes a minimum value when the plates are parallel for large separations.PFA worsens as the separation distance increases.However, the predicted stability of parallel configurations may be correct.For perfectly conductive plates, an analytical formula of the Casimir energy between non-parallel plates was presented in Ref.
[30] using the optical approximation, which is one of the calculation methods beyond PFA [31][32][33].Figure 8 shows the inclination angle of a square plate with side of size L above an infinite plate in a stable configuration, calculated using the optics approximation; θ * is a function of the ratio of the separation distance between the square center and the infinite plate (δ).The parallel configuration with a zero inclination angle stabilizes when δ > 0.81.Although sophisticated calculation methods are necessary to determine the magnetic field accurately to realize levitation, the guideline indicating that stabilization through the Casimir effect is highly effective for large separations can be useful.

Conclusions
The coexistence of attractive and repulsive forces is crucial for achieving the levitation of objects.Furthermore, the behavior of these forces near the equilibrium point is of great importance.If they follow the power functions with the same exponent, the resulting function becomes monotonic, rendering stable levitation impossible.
The Casimir energy between objects is influenced by various physical parameters, such as the objects' shapes, distance between objects, permittivity, and temperature [27,34].It is often represented by a power function.In the case of perfectly conductive plates, Casimir energy is inversely proportional to the cube of the separation distance, i.e., a −3 , regardless of the distance.However, if the permittivity is finite, Casimir energy varies as a −2 for small separations.This implies that the force between objects and its derivative can be manipulated by selecting appropriate materials and separation distances from an engineering perspective.
To counteract the rotation of the graphene flake and achieve stable levitation, a diamagnetic force was employed as a repulsive force.In order for the flake's surface to remain perpendicular to the magnetic field, it is necessary to fixate the surface and ensure that the product of the applied magnetic field and its derivative is sufficiently large.However, in a vacuum environment, the graphene flake tends to rotate, causing its surface to become parallel to the magnetic field.To address this issue, the Casimir torque was utilized as one of the methods for stabilization.
The numerical analysis revealed that the Casimir effect can provide some degree of stabilization for the diamagnetic levitation of a graphene flake.However, its effectiveness is limited, and it is insufficient to fully suppress the rotation in Earth's gravity by the magnetic field generated through the circular electric current.This limitation arises from the feature that the suppression of the rotation by the Casimir effect is most effective at larger separation distances.As the separation distance increases, the Casimir energy rapidly diminishes, resulting in a weaker restoring torque.Therefore, in order to achieve magnetic levitation in a strong gravitational field, furthering the optimization of the magnetic field is necessary as shown in Section 7.
As the separation distance approaches zero, the magnetic energy remains finite.However, the Casimir energy diverges to negative infinity.This implies that the levitation state is metastable [35][36][37], and a potential barrier is present near the surface.The height of this potential barrier is not significant.Therefore, to maintain the levitation state, it is necessary to maintain a high vacuum and low temperature.These conditions help to stabilize the system and prevent the graphene flake from falling onto the substrate.

Figure 1 .
Figure 1.(a) Illustration of the levitation of a graphene flake in a magnetic field generated by an electric current flowing through a coil positioned above a silicon plate.(b) Positional relationship between the inclined graphene, coil, and silicon plate.

J/m 2 )Figure 3 .
Figure 3. Casimir energy per area between a graphene sheet and a silicon plate calculated using the Lifshitz formula (circles) and the fitting function (line).

Figure 5 .
Figure 5. (a) Dependence of the total Casimir energy, magnetic energy, and gravitational energy on the separation distance for gravitational acceleration g = 0 and 0.1 m/s 2 .(b) Dependence of the total energy at equilibrium heights on the inclined angle.

Figure 6 .
Figure 6.(a) Dependence of the total energy of the flat graphene flake for g = 0.4 and 0.3 m/s 2 on the vertical position.(b) Dependence of the total energy at equilibrium heights on the inclined angle for g = 0.4 and 0.3 m/s 2 .The angle at which takes the minimum potential energy exists at non-zero inclined angle.(c) Dependence of the total energy of the graphene flakes that incline with the equilibrium angle on the separation distance.

Figure 7 .
Figure 7. (a) Dependence of the total energy of the flat graphene flake for g = 9.8 m/s 2 on the vertical position.(b) Dependence of the total energy at equilibrium heights of 2 µm on the inclined angle for g = 9.8 m/s 2 .

Figure 8 .
Figure 8. Dependence of the equilibrium of inclination angle, θ * , obtained by the optical approximation on nomalized separation distance with the size of a plate, δ.The parallel configuration is stable for δ > 0.81.

Table 1 .
Parameters used in the calculations.See text for details.