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Measurement of the gradient of the Casimir force between a nonmagnetic sphere and a magnetic plate

We measured the gradient of the Casimir force between an Au sphere and a plate made of ferromagnetic metal (Ni). It is demonstrated that the magnetic properties influence the force magnitude. This opens prospective opportunities for the control of the Casimir force in nanotechnology and for obtaining Casimir repulsion by using ferromagnetic dielectrics.


I. INTRODUCTION
In the last few years the Casimir effect 1 has attracted much experimental and theoretical attention as a fluctuation-induced quantum phenomenon with diverse applications ranging from nanotechnology to elementary particle physics, gravitation and cosmology. 2,3 Many experiments on measuring the Casimir force in configurations with material boundaries made of nonmagnetic metals, semiconductors and dielectrics have been performed (see reviews [4][5][6] ).
On the theoretical side, the Lifshitz theory 7 of the van der Waals and Casimir forces between plane-parallel plates described by the frequency-dependent dielectric permittivity ε(ω) was generalized for the boundary surfaces of arbitrary shape [8][9][10] . Nevertheless, application of this theory to materials with free charge carriers is under much discussion. 3,4,6,[11][12][13][14][15][16][17][18][19][20][21][22][23][24] The Lifshitz theory has long been generalized 25 for the case of magnetic materials described by the frequency-dependent magnetic permeability µ(ω). The Casimir force between magnetodielectric plates was studied theoretically by many authors (see, e.g., [26][27][28][29][30][31][32][33], especially in connection with the possibility of Casimir repulsion, which has promise for nanotechnology. A large contribution of magnetic properties to the Casimir force (including the Casimir repulsion through a vacuum gap) was predicted 30 for constant ε and µ. For real magnetic metals described by the Drude model at nonzero temperature the effect of repulsion was not confirmed. 27,31 In Refs. 27,34 it was shown that for real magnetic materials at room temperature the influence of magnetic properties on the Casimir force can occur solely through the zero-frequency contribution to the Lifshitz formula (the reason is that µ(iξ) drops to unity at ξ several orders of magnitude lower than the first Matsubara frequency). It was also predicted 32,34 that in the interaction of a nonmagnetic metal with a ferromagnetic dielectric the Casimir repulsion is possible if the relaxation properties of free electrons in a metal and the dc conductivity of the dielectric are omitted. Keeping in mind that some measurements [15][16][17][20][21][22][23][24] have raised questions on whether the relaxation properties of free charge carriers and dc conductivity influence the Casimir force, there is a question pointed out in Ref. 34 if the magnetic properties of real materials will influence the Casimir force.
This paper starts the experimental investigation of the Casimir effect in the presence of magnetic boundaries. Using the dynamic atomic force microscope (AFM), operated in a frequency shift technique, we have measured the gradient of the Casimir force between an 2 Au-coated sphere and a plate covered with the ferromagnetic metal Ni. The experimental data were compared with predictions of the Lifshitz theory extrapolated to zero frequency both including 11,12 and omitting 13,14 the relaxation properties of free electrons (i.e., using the Drude and plasma model approaches discussed in the literature. 3,4 ) It was found that the measurement data are in excellent agreement with the predictions of the plasma model approach taking the magnetic properties into account and with the Drude model approach (which is not sensitive to magnetic properties for a nonmagnetic metal interacting with the magnetic one). Predictions of both these approaches are shown to be very close within the range of experimental separations 220 nm ≤ a ≤ 500 nm. If to combine the obtained results with an exclusion of the Drude model approach for nonmagnetic metals found in several experiments 15,16 , one can consider our experiment as a confirmation of the influence of magnetic properties on the Casimir force. This opens outstanding opportunities for the tuning of the Casimir force in nanotechnology by means of depositing ferromagnetic films on movable parts of microdevices and even obtaining repulsive Casimir forces (other means are considered in Refs. 4,5,35,36 ).
The paper is organized as follows. In Sec. II we describe the measurement scheme and the experimental setup using a dynamic AFM. Section III is devoted to the calibration procedures and measurement results. Section IV contains comparison between experiment and theory. In Sec. V the reader will find our conclusions and discussion.

II. MEASUREMENT SCHEME AND EXPERIMENTAL SETUP
We have applied different voltages V i to the plate and measured the gradient of the total force, electric plus Casimir, between an Au-coated sphere of radius R and Ni-coated plate using dynamic force microscopy, more specifically, frequency modulation AFM. 37 where ω r is the resonance frequency in the presence of external force F tot and ω 0 is the natural resonance frequency, is given by Here, C ≡ C(k, ω 0 ) = ω 0 /(2k), β ≡ β(a, z 0 , C, R) = C∂X(a, R)/∂a, k is the spring constant of the cantilever, absolute separations a = z piezo + z 0 , where z piezo is the plate movement due to the piezoelectric actuator and z 0 is the point of the closest approach between the Au sphere and Ni plate, and X(a, R) is a known function. 3,4,15,22 The residual potential difference V 0 in (2) can be nonzero even for a grounded sphere due to the different work functions of the sphere and plate materials or contaminants on the interacting surfaces.
Expression (2) is the basis of the technique used in this experimental study to explore the Casimir effect in magnetic materials. The electrostatic force gradient and therefore the frequency shift has a parabolic dependence on the voltage V i applied to the plate.
From Eq.(2), the minimum in the frequency shift corresponds to V 0 . The curvature of the parabola, which includes the spatial dependence of the electrostatic force and the cantilever parameters, is related to β and depends on z 0 and C. Thus these parameters can be extracted from this dependence. In order to test for systematic errors in the fitting parameters, the fitting is repeated at many different distance ranges.
The instrument was designed for a precision measurement of the Casimir force gradient.
The primary pieces of equipment necessary for frequency modulation measurements consist of a microfabricated cantilever, the cantilever motion controllers (piezoelectric transducers), plate motion controller (piezoelectric tube), optical detection system, and a phase locked loop (PLL) to measure the frequency shift. The cantilever was placed inside a high vacuum chamber. All these allowed us to reach subnanometer separation distance resolution over a distance range larger than two micrometers.
The sphere was attached to the cantilever. The chosen sphere needs to have a large R and a smooth clean surface. To attach it to the cantilever we used a dot of Ag epoxy at the tip of the free-end of the rectangular cantilever. To achieve high resonance frequencies and high signal to noise, hollow glass microspheres (3M Scotchlite) and stiff rectangular conductive Si cantilevers were used. The amount of epoxy has to be minimal to obtain the high frequencies. We used commercial monocrystalline cantilevers that are n-type and have a specific resistance of 0.01 to 0.05 Ω cm. We chosen them because they have very low amount of internal stresses, in comparison to other materials like silicon nitride (SiN 3 ). This leads to low energy dissipation, resulting in a high quality factor Q. We chosen ones with the smallest spring constant (∼ 0.03 N/m). The cantilevers were cleaned with high-purity acetone and then rinsed with distilled and deionized water. After that, the silicon dioxide (SiO 2 ) layer on the surface of the cantilever was etched with a solution of HF for 1 min.
Finally, they were double rinsed in a solution of deionized water. The Si cantilevers are conductive, which is necessary for good electrical contact to the sphere.
The surfaces of the hollow glass spheres are smooth as they are made from liquid phase.
We used the special procedure for cleaning the spheres before attachment to the cantilever to remove organic contaminants and debris from the surface. Details of this procedure are described in Ref. 16 .
The next step was to thermally evaporate a uniform coating of 280 ± 1 nm of Au on the sphere and at the top of the free end of the cantilever. The Au on the top of the cantilever has to be evaporated only at the tip, about 50 µm from the free end. The radius of the sphere was measured to be R = 64.1 ± 0.1 µm using a SEM. The tip of the cantilever and sphere were coated with Au using a specially designed mask. Coating only the cantilever tip, preserves the large oscillation Q-factor leading to high sensitivity. We used an oil-free thermal evaporator for the Au coating. This instrument is equipped with a scroll pump ("Varian", SH-100) and a turbomolecular pump ("Varian", TV301 Navigator) that permits us to coat at a pressure of 10 −6 Torr. To achieve uniformity the cantilever sphere system was rotated using a motor during evaporation. To obtain a smooth coating, the coating rate has to be low and done with cooler atoms. The Au coating was done over 4 hours, at a boat-cantilever distance of about 2.5 ′′ to achieve a uniform 280 nm thick layer. To obtain smooth coatings, it was found that an equilibrium pressure be reached after melting of the Au wire, before start of the coating process.
We used a Si plate as substrate for the Ni coating. The plate was first cleaned in acetone for 15 min using a sonicator, then rinsed with deionized water. This was repeated in IPA For frequency demodulation we used a PLL. The PLL frequency demodulator system combines a controller module to maintain the resonant frequency and a detector module to measure the force gradient induced resonant frequency shift. A piezoelectric actuator connected to the output of the PLL drives the cantilever at its resonant frequency with 7 a constant amplitude. For all separations the oscillation amplitude of the cantilever was fixed at < 10 nm. To control the amplitude, the fluctuation spectrum of the cantilever was measured with a spectrum analyzer. The latter was calibrated using the thermal noise oscillation spectrum which has an average amplitude of ∼ 0.4 nm. During experiment the signal on the spectrum analyzer was about 15 times higher than for the thermal noise. The output of the low pass filter was used to form a closed loop PID controller using a piezoelectric actuator, to maintain constant separation distance between the fiber end and the cantilever.
LabView software was used to control and monitor PID function and to acquire the data.
The program also sets the voltages applied to the Ni plate using a low-noise voltage supply.

III. CALIBRATION PROCEDURES AND MEASUREMENT RESULTS
The frequency shift of our oscillator due to the total force (electrostatic and Casimir) between the Au sphere and the Ni plate was measured as a function of sphere-plate separation distance. For this purpose, the Ni-coated plate was connected to a voltage supply (33220A, "Agilent Inc.") operating with 1 µV resolution. 11 different voltages in the range from -87.1 to 25.9 mV were applied to the Ni plate, while the sphere remained grounded.
The plate was moved towards the sphere starting at the maximum separation, and the corresponding frequency shift was recorded at every 0.14 nm. This measurement was repeated four times. Any mechanical drift was subtracted from the separation distance. To do so we took into account that at a > 2 µm, the total force between the sphere and plate is below the instrumental sensitivity. At these separations, the noise is far greater than the signal and in the absence of systematic errors the signal should average to zero. However, the drift caused the separation distance to increase by around 1 nm in 1000 s, where 100 s correspond to time taken to make the one measurement (note that positional precision much better than 1 nm was achieved in this experiment). The drift rate was calculated from the change in position at one frequency shift signal plotted as a function of time. This procedure was repeated for 15 different frequency shift signals to calculate the average drift rate, which was found to be 0.002 nm/s. The separation distance in all measurements was corrected for this drift rate. Note that the temperature was controlled through excellent thermal contact to the heat bath, which is the large thermal mass of the vacuum chamber and optical table.
The experiment was equilibriated for several hours before data were taken. The drift was experimentally found to be linear during 40 min, which is the time scale of the measurements (see Ref. 16 for details where the same setup was used).
After applying the drift correction the residual potential V 0 between the sphere and plate was determined using the following procedure. For every 1 nm separation the frequency shift signals were found by interpolation. Then these signals were plotted as a function of the applied voltage V i at every separation and the corresponding V 0 identified at the position of the parabola maxima. The curvature of the parabola β was also found. These V 0 from all four measurement sets are plotted in Fig. 1(a) versus separation distance. In Fig. 1(b) we show the systematic error of each individual V 0 , as determined from the fit, at different separations. The mean value is V 0 = −34.1 ± 1 mV, where the total error is determined at a 67% confidence level, and can be observed to be independent of separation. This is an indirect confirmation of the fact that the interacting regions of the surfaces are clean or the adsorbed impurities are randomly distributed with a sub-micrometer length scales and do not contribute to the total force. 16 The next step was to determine the separation distance at closest approach z 0 and the coefficient C in Eq. (2). As discussed above, these parameters associated with the cantilever can be found from the dependence of the parabola curvature β on distance. The corresponding theoretical expression for parabola curvature was fit to β as function of the separation We now turn to the determination of the experimental errors. The random error in the gradient of the Casimir force calculated from 44 repetitions (4 measurement sets with 11 applied voltages each) at a 67% confidence level is shown by the short-dashed line in Fig. 3.
The systematic error is determined by the instrumental noise including the background noise level, by the errors in calibration and by the error in the gradient of electrostatic force subtracted in accordance with Eq. (2) to get F ′ (a). It is shown by the long-dashed line in Fig. 3. The total experimental error found at a 67% confidence level is shown by the solid line in Fig. 3. The error in separation ∆a = 0.4 nm coincides with the error in the determination of z 0 (details of error analysis can be found in Ref. 16 ).
In Fig. 4(a-d) the measured gradients of the Casimir force are indicated as crosses, where the arms of the crosses are determined by the total experimental errors. It can be seen that the total relative experimental error in the measurements of F ′ (a) at a = 220, 250, 300, 400, and 500 nm is equal to 0.6%, 0.94%, 1.8%, 5.4%, and 12.8%, respectively.

IV. COMPARISON BETWEEN EXPERIMENT AND THEORY
Now we compare the experimental data for the gradient of the Casimir force in the configuration containing nonmagnetic (Au) and magnetic (Ni) metals with the predictions of the Lifshitz theory. Computations were performed using the following Lifshitz-type formula 34 for F ′ (a) obtained in the sphere-plate geometry with the help of the proximity force approximation (using the exact theory, it was recently shown 40,41 that the error in this case is less than a/R, i.e., less than 0.3% at the shortest separation): Here, k B is the Boltzmann constant, T = 300 K is the temperature at the laboratory, q 2 l = k 2 ⊥ + ξ 2 l /c 2 , and ξ l = 2πk B T l/ with l = 0, 1, 2, . . . are the Matsubara frequencies. The prime near the summation sign multiplies the term with l = 0 by 1/2. The reflection coefficients for transverse magnetic (TM) and transverse electric (TE) polarizations of the electromagnetic field on a Ni plate are given by where The reflection coefficients on Au are obtained from Eqs. (4) and (5) Fig. 4(a-d). The widths of the bands indicate the theoretical error. The difference between the two bands is explained by the fact that the reflection coefficient with TE polarization on a Ni plate described by the plasma model at zero Matsubara frequency depends on µ (Ni) (0): where k ⊥ is the projection of the wave vector on the plate. At the same time r (Au) TE,p (0, k ⊥ ) = 0. As can be seen in Fig. 4, the plasma model approach with magnetic properties included is in excellent agreement with the data over the entire measurement range. The plasma model approach with the magnetic properties omitted is excluded by the data at a 67% confidence level over the interaction range from 220 to 420 nm.
The Drude model approach is not sensitive to the presence of magnetic properties in this experiment because r (we note that these coefficients enter the Lifshitz formula (3) as a product). By coincidence, over the separation region from 220 to 500 nm the predictions of the Drude model approach almost coincide with the solid line in Fig. 4 (the magnitudes of relative differences at separations of 220, 300, 400, and 500 nm are only 0.5%, 0.2%, 0.4%, and 1.2%, respectively). Thus, for a nonmagnetic metal interacting with a magnetic one at short separations, the predictions of both approaches are much closer than for two Au test bodies where the respective differences were resolved experimentally. 15,16 Note that at large separations the relative differences between the Drude model approach and the plasma model approach with magnetic properties included are much larger (31.1% and 42.8% at separations 3 and 5 µm, respectively).

V. CONCLUSIONS AND DISCUSSION
In this work we have measured the gradient of the Casimir force between an Au-coated sphere and a Ni-coated plate using an AFM operating in the dynamic regime. We have