- Email: [email protected]
MicroSQUID magnetometry and magnetic imaging
Physica C 332 Ž2000. 140–147 www.elsevier.nlrlocaterphysc
MicroSQUID magnetometry and magnetic imaging K. Hasselbach a,) , C. Veauvy a , D. Mailly b a
CRTBT-CNRS, 25 aÕenue des Martyrs, BP166, 38042 Grenoble Cedex 9, France b LMM-CNRS, 196 H. RaÕera, 92220 Bagneux, France
Abstract We present a new instrument for local magnetization measurements: the micro-Superconducting Quantum Interference Device ŽmicroSQUID.. The microSQUID consists of a small Ž; 1 mm diameter. superconducting loop with two microbridge Josephson junctions. The SQUID loop itself serves as the input coil for the magnetic signal. In placing the SQUID very close to the sample, the magnetization measurement of signals as small as 10 000 m B has been demonstrated. In this limit of close proximity, the spatial resolution of the SQUID is a fraction of the width of the SQUID ring Ž0.2 mm.. We are now building a scanning SQUID microscope. The SQUID is placed by electron beam lithography at the apex of a silicon cantilever. The lever is attached to a force sensor, allowing us to image magnetically, as well as topographically. We expect to obtain a spatial resolution of 50 nm and a flux resolution of about 10y4 F 0 . The ensemble is mounted inside an inverted dilution refrigerator. We will discuss the application of this technique to the imaging of vortices in artificial networks. q 2000 Elsevier Science B.V. All rights reserved. Keywords: SQUID; Applications of high-Tc superconductors; Vortex; Magnetization; Magnetic imaging
1. Introduction Artificial superconducting networks are a model system for phase transitions w1x. The superconducting order parameter can be tuned by its dependence on the external field, temperature and the geometry of the system. New phase transitions are observed: for example, the formation of edge states in superconducting films with holes w2x and trapping of vortices in regular defects w3x. All these phenomena can be monitored at the superconducting transition via resistivity measurements w4x, statically by bitter decoration techniques w5x or scanning Hall probe microscopy w6x. In order to extend the domain of
)
Corresponding author. E-mail address: [email protected] ŽK. Hasselbach..
observability, we would like to employ microSQUID imaging to these systems.
2. What is a SQUID? Flux quantization in a superconducting loop is known as the Little Parks effect: the sum of the flux due to superconducting screening currents and the external flux are multiples of F 0 s hr2 e s 2.07 = 10y1 5 V s. A DC-SQUID consists of a superconducting loop interrupted by two Josephson junctions. The critical current, Ic , of this device is periodic in the magnetic flux passing through the loop, with period F 0 . To operate a SQUID, a current at least as high as the critical current has to be applied. The current induces flux due to the self-inductance, L, of
0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 6 5 7 - 7
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the SQUID. To obtain optimal modulation depth, Ic and L are related by the expression w7x b s 2p LIcrF 0 ; 1. As self-inductance is proportional to the diameter of the loop, this relation says that the critical current has to increase as the SQUID size is reduced. The Josephson junctions can be realized either by planar junctions Žmost commonly used are Superconducting–Insulator–Superconducting., point contacts or microbridge junctions. The planar junction SQUIDs are made by a Nb–Al 2 O 3 –Nb trilayer technology. Their critical currents can be adjusted by adjusting the area of the tunnel barrier and the critical current density. State of the art SQUIDs w8x are characterized by a SQUID inductance of about 50 pH, corresponding to a surface of about 800 mm2 , a critical current of 16 mA with a junction area of 3 mm2 . Their flux noise is of the order of Fn2 s 1 = 10y1 3 F 02rHz. A figure of merit exists for SQUIDs, called energy sensitivity per unit bandwidth, ´ s Fn2r2 L. It is the smallest measurable energy coupled into the SQUID. The abovementioned flux noise translates to an energy sensitivity w9x per unit bandwidth of 4 h at 4.2 K. These SQUIDs have to be shielded from external magnetic fields, otherwise their performance is reduced by vortex pinning and de-pinning. They can be exposed to a magnetic field as high as 280 G, if applied in the plane of the SQUID. Under these conditions w10x the flux sensitivity is reduced to Fn s 6 = 10y5F 0r 'Hz . Thus, these SQUIDs are placed far away from the zone of the applied field and the signal of the sample has to be coupled into the SQUID either by a low inductance line or by a flux transformer. The SQUID sensitivity may be limited only by the quantum limit Ž h.; but, when a magnetic signal has to be coupled to the SQUID, the coupled sensitivity is reduced depending on the position of the sample relative to the SQUID and the sample’s B field.
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As sample fabrication may be already very demanding, we opted for the simplest possible design of a SQUID: a superconducting loop with two microbridges as Josephson junctions. In order that the constrictions act as Josephson junctions, their dimensions must be comparable to the temperature dependent coherence length. We use SQUIDs, Fig. 1, having a typical area of 1–4 = 10y1 2 m2 , a linewidth of 0.2 mm, the microbridge is about 0.03-mm wide and 0.3-mm long. The self-inductance of such a ring Ž1-mm diameter. is about 1.5 pH w12x. All the SQUIDs are fabricated by electron beam lithography. Niobium SQUIDs are patterned with Reactive Ion Etching ŽRIE. using an aluminum mask. Aluminum SQUIDs are evaporated through a PMMA mask. The microSQUIDs have a hysteric V Ž I . characteristic, Fig. 2. Ramping the current up from zero, the SQUID transits from the superconducting to the normal state at Ic . A voltage step is generated as the normal state resistance of the junction appears and the dissipated energy heats the entire SQUID loop. When the current is lowered, the SQUID stays in the resistive state for currents much smaller than Ic . The thermal hysteresis excludes the usual current biasing schemes used as SQUID-readout. A new detection technique had to be implemented. A computer-con-
3. What is a microSQUID? In our group, the work concentrates on building SQUIDs that can be easily coupled to mesoscopic samples w11x.
Fig. 1. Electron microscope image of a Nb microSQUID: diameter, 2 mm; linewidth, 0.3 mm.
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supercurrent phase relation ships are valid. Nb SQUIDs are in the other limit: junction dimensions 4 coherence length, the supercurrent phase relationship w13x is changed and the modulation depth w14x decreased. The flux noise of the microSQUIDs is recorded in measuring the transfer function Ic Žflux.. Fig. 3a,b and in measuring the critical current noise Fn s InŽEFrEI .. The current noise is measured in accumulating the critical current values periodically in time and calculating the power spectral density per unit time by a Fast Fourier Transform. The current noise has been measured for an Al microbridge SQUID, Fig. 3c. The resulting flux noise, Fn ,
Fig. 2. Oscilloscope readings: Ža. the critical current, Ic is different from the current at which the superconductivity sets in again upon lowering the bias current; Žb. a computer controlled current source generates current ramps, that are stopped as soon as the critical current is attained.
trolled circuit triggers simultaneously a current ramp and a 40-MHz quartz clock. As soon as a EVrEt pulse of a preset height is detected at the SQUID, the clock stops and the current is set to zero. The clock reading is transferred to the computer, and the cycle begins again. The critical current is proportional to the duration of the current ramp. The fastest repetition rate is 10 kHz, limited by the time needed to settle the current. One single wire is sufficient to connect the SQUID, the EVrEt pulse is detected on the current biasing lead of the SQUID. The critical current of the Nb SQUIDs is five to 10 times higher than the critical current of Al SQUIDs. This is mainly due to the shorter coherence length of Nb compared to Al Ž j 0,Al s 1.6 mm and j 0,Nb s 0.04 mm.. In the case of Al SQUIDs, the dimensions of the microbridges are of the order, or less than, the coherence length, and the standard
Fig. 3. Ža. The flux-modulation of a Nb SQUID. Every point corresponds to one discreet Ic measurement. Žb. The flux-modulation of an Al SQUID and the current noise, In , of an Al SQUID.
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is 3.7 = 10y5 F 0r 'Hz . The flux noise of Nb is a factor two to three higher. This flux noise level is at least one order of magnitude higher than in state-of-the-art SQUIDs. This is due to the difference in the measuring technique. The usual SQUIDs are current biased, and the apparent voltage is actually a periodic function w15x of time with period V Ž t .2 erh, e.g., V Ž t . s 10 mV results in 4.8 GHz. This high frequency averaging is absent in the microSQUID technique as the first phase slip event triggers thermal dissipation, and the SQUID transits completely in the normal state. Thus, the highest sampling frequency is the repetition rate of the current ramp, 10 kHz. What is more important is the sensitivity for measuring a sample signal and the magnetic field range in which such measurements are possible. Small size, small thickness and narrow lines make these SQUIDs insensitive to flux trapping and field penetration.
4. What are microSQUIDS used for? The first application of microSQUIDs was for the detection of persistent currents. The prediction is that an orbital persistent current would circulate under application of a perpendicular magnetic field even in a normal metal ring, as long as the electron phase coherence is preserved. In order to measure the persistent current, it is necessary to record the magnetic response of the SQUID upon application of a magnetic field perpendicular to the SQUID loop. In order to subtract the direct SQUID response due to the magnetic field, it is necessary to build a gradiometer. A gradiometer consists of two connected loops wound in opposite directions, sensitive only to the flux difference between the two loops. One loop of the Al-microSQUID gradiometer is aligned on top of the 2DEG ring, etched from the AlGaAsrGaAs substrate, the other loop is on top of an calibration coil. The SQUID is about 0.2 mm above the actual 2DEG ring. A typical current of 4 nA is expected in such a ring of 2.7 mm diameter and 0.16 mm width. The mutual inductance between two such loops is abut 8 = 10y1 2 H, resulting in a flux of 1.5 = 10y5 F 0 at the SQUID. After the detection w16x of the persistent current in a single ring for the ballistic
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limit, the studies are currently extended to the e–e interaction effects in arrays of connected rings w17x. Thin Ž30 nm. Nb SQUIDs Žsee Fig. 1. have been used in the study of the underlying mechanisms responsible for magnetization reversal in ferromagnetic grains. The magnetization of the sample is usually parallel to the SQUID loop. For this geometry, the sample is placed on one branch of the SQUID. Half of the sample’s total flux is coupled into the SQUID loop. The external magnetic field is applied in the plane of the SQUID. During the in-plane field sweep, a feedback loop acting on the perpendicular magnetic field maintains the total flux through the SQUID constant. Calculations w31x show, that a magnetization reversal of a spherical 5 nm large Co sample would induce a signal of 4 = 10 4 F 0 if the sample is placed on a 0.2 mm wide SQUID arm. The magnetization of such a particle corresponds to 4000 m B . Hysteresis w18x loops of 20-nm diameter Co particles have been recorded, in order to follow the regime of thermal activation to low temperatures. The amplitude of the induced flux jump upon field sweeping was about 0.05 F 0 . The flux coupling is optimal as the particle is located next to one of the microbridges. A particle of such a volume contains about 2.5 = 10 5 m B . In order to observe the magnetization reversal, the magnetic field is swept in the plane of the SQUID from y0.3 to 0.3 T. MicroSQUIDs have been used to detect w19x the magnetic stray field of ferromagnetic micro-crystallites in the study of the macroscopic quantum tunneling in organic magnets.
5. Magnetic imaging with microSQUIDS There are many techniques for imaging magnetic fields at surfaces: decoration techniques w20x, magnetic force microscopy w21,22x, scanning Hall probe microscopy w23x, Lorentz Microscopy w24x, and scanning SQUID microscopy w25–28x. Every technique has its advantages and enormous progress has been achieved since their advent. SQUID microscopy has already resolved many questions concerning the symmetry w29x of the order parameter and the underlying mechanisms of high Tc superconductivity, but the spatial resolution w30x was limited to about 4 mm, and the SQUID could work only in very low applied
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magnetic fields. We are extending the microSQUID technique to scanning SQUID microscopy, hoping to combine high magnetic resolution with high spatial resolution. As a first example, a superconducting particle inducing flux into a microSQUID is shown in Fig. 4. Fig. 4 shows the result of a calculation of the flux coupling between a spherical Al grain of 50-nm diameter and a wire of square section and infinite extension w31x; this is appropriate as long as the particle is much smaller than the SQUID. The magnetization of the particle is oriented parallel to the SQUID plane and its amplitude is Bcrm 0 times the demagnetization factor. We neglect the fact that the particle size is of the order of the penetration depth, reducing the diamagnetic signal. This shows how flux is reduced as the particle is moved from the center of the wire, either sidewise or vertically. Spatial resolution of the SQUID depends on at least two parameters: the distance between sample and SQUID and the SQUID linewidth; flux linkage is inversely proportional to the linewidth. The shape of the sample’s B-field is a third ingredient. Flux coupling between a SQUID and a wire array can be calculated via the mutual inductance of two circuits. Fig. 5a shows the calculated image of a square loop of 8 mm a side, pierced by 1 F 0 , as it is scanned by a 2 mm diameter SQUID loop. The two loops are parallel and 1 mm apart. Fig. 5b shows line scans for different scanning heights for the same geometry. For a scanning height of 1 mm, the magnetic contrast is of about 0.1 F 0 . The scanning curve shows two opposite peaks as the SQUID loops pass
Fig. 4. The calculated flux induced into a SQUID by a Al sphere of 50-nm diameter, as function of lateral and vertical position.
Fig. 5. Ža. Plot of the mutual inductance between a square SQUID of 2 mm diameter scanned over an 8 mm diameter loop. Diameter, as function of lateral and vertical position.
over the ring. The sign change is due to the change of the direction of the flux between the inside and the outside of the ring, and the maxima evidence that the flux coupling is best when the wires are closest. The choice of SQUID sensor must take into account the back-action of the SQUID onto the sample. The SQUID receives the flux from the sample, but the sample receives also the flux from the SQUID. The mutual inductance, M, times the SQUID current is the flux piercing the sample. The 4 mm2 SQUID loop has an L of 4 = 10y1 2 H. In order to have optimal modulation depth, it should have a critical current of 80 mA. For a scan height of 0.1 mm, M in Fig. 5 is y5 = 10y1 3 H at the center. In consequence, the flux induced from the SQUID is 2 = 10y2 FrF 0 , 2% of the sample’s flux. The design of conventional Žtunnel-junction. SQUIDs allows the critical current to be adjusted. There is no evident manner to control the critical current of microbridges. A critical current of 80 mA
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can be obtained either by using Al SQUIDs, or Nb SQUIDs, if they are operated close to Tc . High resolution SQUID microscopy Žmagnetic spatial resolution better than 1 mm. is currently being developed. The first challenge is to build the SQUID probe. Possible approaches are: the use of a very soft magnetic needle w32x serving as flux guide to a SQUID, the fabrication of a small diameter pickup connected either directly w30x or via a flux transformer w28x to the SQUID, or the use of a microSQUID. The second challenge is common to all scanning probe devices: it is necessary to scan the device in close proximity to the sample in order to obtain the highest possible spatial and magnetic resolution. Our microscope is placed inside an inverted dilution refrigerator 1 ŽSionludi., custom made at the CRTBT. The refrigerator itself is vibration-isolated with a cutoff frequency of 2 Hz. The microscope is placed on top of the refrigerator above the mixing chamber. The microscope stage can be either mounted on the 4.2 K flange or the still 0.7 K flange. The design of the refrigerator allows the SQUID andror the sample to be thermalized at other temperatures than the stage. No further vibration-isolation is used. The coarse approach is done with a piezoelectric linear motor made of titanium Žlength: 30 mm, width: 40 mm, height: 10 mm.. The displacement range is "5 mm. In order to control the approach, the motor is equipped with a capacitive displacement sensor, having a resolution of 10 nm. We adapted a piezoelectric tip sample control w34x, as recently used in near field optical microscopy. In the original set-up a thin optical fiber is glued on a prong of a quartz tuning fork. The quartz tuning fork is excited mechanically at its resonance frequency. When the fork is resonating in a plane perpendicular to the sample, the interaction between tip and sample can be modeled as a viscous drag force using the differential equation of the harmonic oscillator. Due to the piezoelectricity of quartz, it is possible to deduce the oscillation amplitude from the voltage at the quartz. As the sample is approached, damping sets in and the ampli-
1
Licensed to Air Liquide, DTA, 38000 Sassenage, France.
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tude of oscillation is reduced, the phase begins to turn. The tuning fork technique has also been used for topographical imaging, a metallic STM tip w35x replaced the optical fiber. Instead of gluing an optical fiber on the prong, we attached a Si-chip carrying a microSQUID. The Sichip changes the weight of the prong, shifting the resonance frequency from 32 768 Hz Žempty. to about 29 000 Hz. We obtained resonances with a quality factor of 6000 at 4.2 K. The use of the tuning fork allows scans at constant height, and topographical imaging. The sample is scanned in front of the SQUID using a large range piezoelectric scanner w33x. The piezoelectric benders are 34-mm long, allowing a "60-mm scan-range in the xy direction at 4.2 K. The z control is realized with a fifth bender, Fig. 6. The last difficulty is to place the SQUID as close as possible to the edge of the scanning tip, in order to get the highest possible resolution. One technique is to build on a Si-wafer an AFM cantilever in SiC, containing the SQUID. All patterning and positioning can be done with electron beam lithography and RIE. The next process is to glue by ion interdiffusion a precut Pyrex wafer on the SiC. The grooves in the Pyrex wafer are aligned under the tips. The last step consists of dissolving the Si-wafer in Pyrocatecol and free standing tips on SiC membranes are obtained. This technique is elegant, but very difficult to carry out because many process steps are involved Žannealing of the membrane, . . . ..
Fig. 6. Scheme of the microscope set-up in the reversed dilution refrigerator.
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SQUIDs for the imaging of domains in hard ferromagnets or the imaging of Josephson vortices in superconductors with short coherence length. Due to their low critical current Al SQUIDs are better adapted for the exploration of vortex lattice transitions in artificial networks.
Acknowledgements
Fig. 7. Aluminum microSQUID on a SiC membrane.
An alternative technique w36x to fabricate tips is being explored: fabrication of the SQUID, masking it with an aluminum film and removing the unprotected Si by RIE. The SQUID layout can be modified according to the experiment. The design shown in Fig. 7 is not the best, as the loop is rather wide at the apex. The highest spatial resolution is obtained by inserting a narrow pickup loop in the inductance of the SQUID itself, placing a microbridge at the apex, and scanning as close as possible to the surface. This design is similar to what has been adopted by Ref. w27x. We are using microSQUIDs. Their fine linewidth results in high spatial resolution and makes them much less prone to flux trapping than conventional SQUIDs.
6. Conclusion Fine linewidth of microSQUIDs and close proximity enable the detection of flux of very small magnetic objects. We built a scanning SQUID microscope placed inside an inverted dilution refrigerator. All the displacements are piezoelectric. Topographical imaging is possible as a piezoelectric force sensor carries the SQUID. MicroSQUID imaging is a promising technique for the visualization of magnetic structures. According to the expected magnetization signal, different probes can be used: Nb
The authors acknowledge the encouragement of B. Pannetier and the fruitful discussions with A. Benoit and J.R. Kirtley’s practical and theoretical support. A. Madouri is thanked for his expertise in tip fabrication. The project is supported by the Ultimatech Program of the CNRS and a grant from DGA n 96 136.
References w1x J. Vidal, R. Mosseri, B. Doucot, Phys. Rev. Lett. 81 Ž1998. 5888. w2x A.I. Buzdin, Phys. Rev. B 47 Ž1993. 11416. w3x A. Bezryadin, B. Pannetier, J. Low Temp. Phys. 102 Ž1996. 73. w4x B. Pannetier, J. Chaussy, R. Rammal, J.C. Villegier, Phys. Rev. Lett. 53 Ž1984. 1845. w5x K. Runge, B. Pannetier, Europhys. Lett. 24 Ž1993. 737. w6x A.M. Chang, H.D. Hallen, L. Harriot, H.F. Hess, H.L. Kao, J. Kwo, R.E. Miller, R. Wolfe, J. van der Ziel, T.Y. Chang, Appl. Phys. Lett. 61 Ž1992. 1974. w7x C.D. Tesche, J. Clark, J. Low Temp. Phys. 29 Ž1977. 301. w8x M.B. Ketchen, IEEE Trans. Magn. 27 Ž1991. 2916. w9x M.B. Ketchen, M. Bhushan, S.B. Kaplan, W.J. Gallagher, IEEE Trans. Magn. 27 Ž1991. 3005. w10x L.R. Narasimhan, C.K.N. Patel, M.B. Ketchen, Appl. Phys. Lett. 73 Ž1998. 993. w11x C. Chapelier, M. El Khatib, P. Perrier, A. Benoit, D. Mailly, SQUID 91, in: H. Koch, H. Lbbig ŽEds.., Superconducting Devices and their Applications, Springer, Berlin, 1991, pp. 286–291. w12x J.M. Jaycox, M.B. Ketchen, IEEE Trans. Magn. 17 Ž1981. 400. w13x K.K. Likharev, L.A. Yacobson, Sov. Phys. Tech. Phys. 20 Ž1976. 950. w14x K. Hasselbach, J.R. Kirtley, D. Mailly, in preparation. w15x A.Th.A.M. De Waele, R. de Bruyn Ouboter, Physica 41 Ž1969. 225. w16x C. Chapelier, A. Benoit, D. Mailly, Phys. Rev. Lett. 70 Ž1993. 2020. w17x W. Rabaud, L. Saminadayar, D. Mailly, A. Benoit, in progress. w18x W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit,
K. Hasselbach et al.r Physica C 332 (2000) 140–147
w19x w20x w21x
w22x w23x w24x w25x
w26x w27x
B. Barbara, N. Demoncy, A. Loiseau, H. Pascard, D. Mailly, Phys. Rev. Lett. 70 Ž1997. 1791. W. Wernsdorfer, T. Ohm, C. Sangregario, R. Sessoli, D. Mailly, C. Paulsen, Phys. Rev. Lett. 82 Ž1999. 3903. B. Pannetier, A. Bezryadin, A. Eichenberger, Physica B ŽAmsterdam. 222 Ž1996. 253. A. Moser, H.J. Hug, I. Parashikov, B. Stiefel, O. Fritz, H. Thomas, A. Baratoff, H.-J. Guntherodt, P. Chaudhari, Phys. ¨ Rev. Lett. 74 Ž1995. 1874. A. Volodin, K. Temst, C. Van Haesendonck, Y. Bruynseraede, Appl. Phys. Lett. 73 Ž1998. 1134. A. Oral, S.J. Bending, H. Henini, Appl. Phys. Lett. 69 Ž1996. 1324. T. Matsuda, K. Harada, H. Kasai, O. Kamimura, A. Tonomura, Science 271 Ž1996. 1393. R.C. Black, A. Mathai, F.C. Wellstood, E. Dantsker, A.H. Miklich, D.T. Nemeth, J.J. Kingston, J. Clarke, Appl. Phys. Lett. 62 Ž1993. 2128. L.N. Vu, M.S. Wistrom, D.J. van Harlingen, Appl. Phys. Lett. 63 Ž1993. 1693. J.R. Kirtley, M.B. Ketchen, K.G. Stawiasz, J.Z. Sun, W.J.
w28x w29x
w30x
w31x
w32x w33x w34x w35x w36x
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Gallagher, S.H. Blanton, S.J. Wind, Appl. Phys. Lett. 66 Ž1995. 1138. T. Morooka, S. Nakayama, A. Odawara, K. Chinone, Jpn. J. Appl. Phys. 38 Ž1999. L119. C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, T. Shaw, J.Z. Sun, M.B. Ketchen, Phys. Rev. Lett. 73 Ž1994. 593. J.R. Kirtley, M.B. Ketchen, C.C. Tsuei, J.Z. Sun, W.J. Gallagher, L.S. Yu-Jahnes, A. Gupta, K.G. Stawiasz, S.J. Wind, IBM J. Res. Dev. 39 Ž1995. 655. W. Wernsdorfer, PhD, Joseph Fourier University, Grenoble Ž1996. and E. Durand Electrostatique et magnetostatique, ´ Paris: Masson 1953. P. Pitzius, V. Dworak, U. Hartmann, ISEC 97 Berlin. J. Siegel, J. Witt, N. Venturi, S. Field, Rev. Sci. Instrum. 66 Ž1995. 2520. K. Karrai, R.D. Grober, Appl. Phys. Lett. 66 Ž1995. 1842. J. Rychen, T. Ihn, P. Studerus, A. Herrmann, K. Ensslin, Rev. Sci. Instrum. 70 Ž1999. 2765. F. Ayela, J. Chaussy, T. Fournier, E. Menegaz, Sens. Actuators, A Ž1999. to appear.
MicroSQUID magnetometry and magnetic imaging K. Hasselbach a,) , C. Veauvy a , D. Mailly b a
CRTBT-CNRS, 25 aÕenue des Martyrs, BP166, 38042 Grenoble Cedex 9, France b LMM-CNRS, 196 H. RaÕera, 92220 Bagneux, France
Abstract We present a new instrument for local magnetization measurements: the micro-Superconducting Quantum Interference Device ŽmicroSQUID.. The microSQUID consists of a small Ž; 1 mm diameter. superconducting loop with two microbridge Josephson junctions. The SQUID loop itself serves as the input coil for the magnetic signal. In placing the SQUID very close to the sample, the magnetization measurement of signals as small as 10 000 m B has been demonstrated. In this limit of close proximity, the spatial resolution of the SQUID is a fraction of the width of the SQUID ring Ž0.2 mm.. We are now building a scanning SQUID microscope. The SQUID is placed by electron beam lithography at the apex of a silicon cantilever. The lever is attached to a force sensor, allowing us to image magnetically, as well as topographically. We expect to obtain a spatial resolution of 50 nm and a flux resolution of about 10y4 F 0 . The ensemble is mounted inside an inverted dilution refrigerator. We will discuss the application of this technique to the imaging of vortices in artificial networks. q 2000 Elsevier Science B.V. All rights reserved. Keywords: SQUID; Applications of high-Tc superconductors; Vortex; Magnetization; Magnetic imaging
1. Introduction Artificial superconducting networks are a model system for phase transitions w1x. The superconducting order parameter can be tuned by its dependence on the external field, temperature and the geometry of the system. New phase transitions are observed: for example, the formation of edge states in superconducting films with holes w2x and trapping of vortices in regular defects w3x. All these phenomena can be monitored at the superconducting transition via resistivity measurements w4x, statically by bitter decoration techniques w5x or scanning Hall probe microscopy w6x. In order to extend the domain of
)
Corresponding author. E-mail address: [email protected] ŽK. Hasselbach..
observability, we would like to employ microSQUID imaging to these systems.
2. What is a SQUID? Flux quantization in a superconducting loop is known as the Little Parks effect: the sum of the flux due to superconducting screening currents and the external flux are multiples of F 0 s hr2 e s 2.07 = 10y1 5 V s. A DC-SQUID consists of a superconducting loop interrupted by two Josephson junctions. The critical current, Ic , of this device is periodic in the magnetic flux passing through the loop, with period F 0 . To operate a SQUID, a current at least as high as the critical current has to be applied. The current induces flux due to the self-inductance, L, of
0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 6 5 7 - 7
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the SQUID. To obtain optimal modulation depth, Ic and L are related by the expression w7x b s 2p LIcrF 0 ; 1. As self-inductance is proportional to the diameter of the loop, this relation says that the critical current has to increase as the SQUID size is reduced. The Josephson junctions can be realized either by planar junctions Žmost commonly used are Superconducting–Insulator–Superconducting., point contacts or microbridge junctions. The planar junction SQUIDs are made by a Nb–Al 2 O 3 –Nb trilayer technology. Their critical currents can be adjusted by adjusting the area of the tunnel barrier and the critical current density. State of the art SQUIDs w8x are characterized by a SQUID inductance of about 50 pH, corresponding to a surface of about 800 mm2 , a critical current of 16 mA with a junction area of 3 mm2 . Their flux noise is of the order of Fn2 s 1 = 10y1 3 F 02rHz. A figure of merit exists for SQUIDs, called energy sensitivity per unit bandwidth, ´ s Fn2r2 L. It is the smallest measurable energy coupled into the SQUID. The abovementioned flux noise translates to an energy sensitivity w9x per unit bandwidth of 4 h at 4.2 K. These SQUIDs have to be shielded from external magnetic fields, otherwise their performance is reduced by vortex pinning and de-pinning. They can be exposed to a magnetic field as high as 280 G, if applied in the plane of the SQUID. Under these conditions w10x the flux sensitivity is reduced to Fn s 6 = 10y5F 0r 'Hz . Thus, these SQUIDs are placed far away from the zone of the applied field and the signal of the sample has to be coupled into the SQUID either by a low inductance line or by a flux transformer. The SQUID sensitivity may be limited only by the quantum limit Ž h.; but, when a magnetic signal has to be coupled to the SQUID, the coupled sensitivity is reduced depending on the position of the sample relative to the SQUID and the sample’s B field.
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As sample fabrication may be already very demanding, we opted for the simplest possible design of a SQUID: a superconducting loop with two microbridges as Josephson junctions. In order that the constrictions act as Josephson junctions, their dimensions must be comparable to the temperature dependent coherence length. We use SQUIDs, Fig. 1, having a typical area of 1–4 = 10y1 2 m2 , a linewidth of 0.2 mm, the microbridge is about 0.03-mm wide and 0.3-mm long. The self-inductance of such a ring Ž1-mm diameter. is about 1.5 pH w12x. All the SQUIDs are fabricated by electron beam lithography. Niobium SQUIDs are patterned with Reactive Ion Etching ŽRIE. using an aluminum mask. Aluminum SQUIDs are evaporated through a PMMA mask. The microSQUIDs have a hysteric V Ž I . characteristic, Fig. 2. Ramping the current up from zero, the SQUID transits from the superconducting to the normal state at Ic . A voltage step is generated as the normal state resistance of the junction appears and the dissipated energy heats the entire SQUID loop. When the current is lowered, the SQUID stays in the resistive state for currents much smaller than Ic . The thermal hysteresis excludes the usual current biasing schemes used as SQUID-readout. A new detection technique had to be implemented. A computer-con-
3. What is a microSQUID? In our group, the work concentrates on building SQUIDs that can be easily coupled to mesoscopic samples w11x.
Fig. 1. Electron microscope image of a Nb microSQUID: diameter, 2 mm; linewidth, 0.3 mm.
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supercurrent phase relation ships are valid. Nb SQUIDs are in the other limit: junction dimensions 4 coherence length, the supercurrent phase relationship w13x is changed and the modulation depth w14x decreased. The flux noise of the microSQUIDs is recorded in measuring the transfer function Ic Žflux.. Fig. 3a,b and in measuring the critical current noise Fn s InŽEFrEI .. The current noise is measured in accumulating the critical current values periodically in time and calculating the power spectral density per unit time by a Fast Fourier Transform. The current noise has been measured for an Al microbridge SQUID, Fig. 3c. The resulting flux noise, Fn ,
Fig. 2. Oscilloscope readings: Ža. the critical current, Ic is different from the current at which the superconductivity sets in again upon lowering the bias current; Žb. a computer controlled current source generates current ramps, that are stopped as soon as the critical current is attained.
trolled circuit triggers simultaneously a current ramp and a 40-MHz quartz clock. As soon as a EVrEt pulse of a preset height is detected at the SQUID, the clock stops and the current is set to zero. The clock reading is transferred to the computer, and the cycle begins again. The critical current is proportional to the duration of the current ramp. The fastest repetition rate is 10 kHz, limited by the time needed to settle the current. One single wire is sufficient to connect the SQUID, the EVrEt pulse is detected on the current biasing lead of the SQUID. The critical current of the Nb SQUIDs is five to 10 times higher than the critical current of Al SQUIDs. This is mainly due to the shorter coherence length of Nb compared to Al Ž j 0,Al s 1.6 mm and j 0,Nb s 0.04 mm.. In the case of Al SQUIDs, the dimensions of the microbridges are of the order, or less than, the coherence length, and the standard
Fig. 3. Ža. The flux-modulation of a Nb SQUID. Every point corresponds to one discreet Ic measurement. Žb. The flux-modulation of an Al SQUID and the current noise, In , of an Al SQUID.
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is 3.7 = 10y5 F 0r 'Hz . The flux noise of Nb is a factor two to three higher. This flux noise level is at least one order of magnitude higher than in state-of-the-art SQUIDs. This is due to the difference in the measuring technique. The usual SQUIDs are current biased, and the apparent voltage is actually a periodic function w15x of time with period V Ž t .2 erh, e.g., V Ž t . s 10 mV results in 4.8 GHz. This high frequency averaging is absent in the microSQUID technique as the first phase slip event triggers thermal dissipation, and the SQUID transits completely in the normal state. Thus, the highest sampling frequency is the repetition rate of the current ramp, 10 kHz. What is more important is the sensitivity for measuring a sample signal and the magnetic field range in which such measurements are possible. Small size, small thickness and narrow lines make these SQUIDs insensitive to flux trapping and field penetration.
4. What are microSQUIDS used for? The first application of microSQUIDs was for the detection of persistent currents. The prediction is that an orbital persistent current would circulate under application of a perpendicular magnetic field even in a normal metal ring, as long as the electron phase coherence is preserved. In order to measure the persistent current, it is necessary to record the magnetic response of the SQUID upon application of a magnetic field perpendicular to the SQUID loop. In order to subtract the direct SQUID response due to the magnetic field, it is necessary to build a gradiometer. A gradiometer consists of two connected loops wound in opposite directions, sensitive only to the flux difference between the two loops. One loop of the Al-microSQUID gradiometer is aligned on top of the 2DEG ring, etched from the AlGaAsrGaAs substrate, the other loop is on top of an calibration coil. The SQUID is about 0.2 mm above the actual 2DEG ring. A typical current of 4 nA is expected in such a ring of 2.7 mm diameter and 0.16 mm width. The mutual inductance between two such loops is abut 8 = 10y1 2 H, resulting in a flux of 1.5 = 10y5 F 0 at the SQUID. After the detection w16x of the persistent current in a single ring for the ballistic
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limit, the studies are currently extended to the e–e interaction effects in arrays of connected rings w17x. Thin Ž30 nm. Nb SQUIDs Žsee Fig. 1. have been used in the study of the underlying mechanisms responsible for magnetization reversal in ferromagnetic grains. The magnetization of the sample is usually parallel to the SQUID loop. For this geometry, the sample is placed on one branch of the SQUID. Half of the sample’s total flux is coupled into the SQUID loop. The external magnetic field is applied in the plane of the SQUID. During the in-plane field sweep, a feedback loop acting on the perpendicular magnetic field maintains the total flux through the SQUID constant. Calculations w31x show, that a magnetization reversal of a spherical 5 nm large Co sample would induce a signal of 4 = 10 4 F 0 if the sample is placed on a 0.2 mm wide SQUID arm. The magnetization of such a particle corresponds to 4000 m B . Hysteresis w18x loops of 20-nm diameter Co particles have been recorded, in order to follow the regime of thermal activation to low temperatures. The amplitude of the induced flux jump upon field sweeping was about 0.05 F 0 . The flux coupling is optimal as the particle is located next to one of the microbridges. A particle of such a volume contains about 2.5 = 10 5 m B . In order to observe the magnetization reversal, the magnetic field is swept in the plane of the SQUID from y0.3 to 0.3 T. MicroSQUIDs have been used to detect w19x the magnetic stray field of ferromagnetic micro-crystallites in the study of the macroscopic quantum tunneling in organic magnets.
5. Magnetic imaging with microSQUIDS There are many techniques for imaging magnetic fields at surfaces: decoration techniques w20x, magnetic force microscopy w21,22x, scanning Hall probe microscopy w23x, Lorentz Microscopy w24x, and scanning SQUID microscopy w25–28x. Every technique has its advantages and enormous progress has been achieved since their advent. SQUID microscopy has already resolved many questions concerning the symmetry w29x of the order parameter and the underlying mechanisms of high Tc superconductivity, but the spatial resolution w30x was limited to about 4 mm, and the SQUID could work only in very low applied
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magnetic fields. We are extending the microSQUID technique to scanning SQUID microscopy, hoping to combine high magnetic resolution with high spatial resolution. As a first example, a superconducting particle inducing flux into a microSQUID is shown in Fig. 4. Fig. 4 shows the result of a calculation of the flux coupling between a spherical Al grain of 50-nm diameter and a wire of square section and infinite extension w31x; this is appropriate as long as the particle is much smaller than the SQUID. The magnetization of the particle is oriented parallel to the SQUID plane and its amplitude is Bcrm 0 times the demagnetization factor. We neglect the fact that the particle size is of the order of the penetration depth, reducing the diamagnetic signal. This shows how flux is reduced as the particle is moved from the center of the wire, either sidewise or vertically. Spatial resolution of the SQUID depends on at least two parameters: the distance between sample and SQUID and the SQUID linewidth; flux linkage is inversely proportional to the linewidth. The shape of the sample’s B-field is a third ingredient. Flux coupling between a SQUID and a wire array can be calculated via the mutual inductance of two circuits. Fig. 5a shows the calculated image of a square loop of 8 mm a side, pierced by 1 F 0 , as it is scanned by a 2 mm diameter SQUID loop. The two loops are parallel and 1 mm apart. Fig. 5b shows line scans for different scanning heights for the same geometry. For a scanning height of 1 mm, the magnetic contrast is of about 0.1 F 0 . The scanning curve shows two opposite peaks as the SQUID loops pass
Fig. 4. The calculated flux induced into a SQUID by a Al sphere of 50-nm diameter, as function of lateral and vertical position.
Fig. 5. Ža. Plot of the mutual inductance between a square SQUID of 2 mm diameter scanned over an 8 mm diameter loop. Diameter, as function of lateral and vertical position.
over the ring. The sign change is due to the change of the direction of the flux between the inside and the outside of the ring, and the maxima evidence that the flux coupling is best when the wires are closest. The choice of SQUID sensor must take into account the back-action of the SQUID onto the sample. The SQUID receives the flux from the sample, but the sample receives also the flux from the SQUID. The mutual inductance, M, times the SQUID current is the flux piercing the sample. The 4 mm2 SQUID loop has an L of 4 = 10y1 2 H. In order to have optimal modulation depth, it should have a critical current of 80 mA. For a scan height of 0.1 mm, M in Fig. 5 is y5 = 10y1 3 H at the center. In consequence, the flux induced from the SQUID is 2 = 10y2 FrF 0 , 2% of the sample’s flux. The design of conventional Žtunnel-junction. SQUIDs allows the critical current to be adjusted. There is no evident manner to control the critical current of microbridges. A critical current of 80 mA
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can be obtained either by using Al SQUIDs, or Nb SQUIDs, if they are operated close to Tc . High resolution SQUID microscopy Žmagnetic spatial resolution better than 1 mm. is currently being developed. The first challenge is to build the SQUID probe. Possible approaches are: the use of a very soft magnetic needle w32x serving as flux guide to a SQUID, the fabrication of a small diameter pickup connected either directly w30x or via a flux transformer w28x to the SQUID, or the use of a microSQUID. The second challenge is common to all scanning probe devices: it is necessary to scan the device in close proximity to the sample in order to obtain the highest possible spatial and magnetic resolution. Our microscope is placed inside an inverted dilution refrigerator 1 ŽSionludi., custom made at the CRTBT. The refrigerator itself is vibration-isolated with a cutoff frequency of 2 Hz. The microscope is placed on top of the refrigerator above the mixing chamber. The microscope stage can be either mounted on the 4.2 K flange or the still 0.7 K flange. The design of the refrigerator allows the SQUID andror the sample to be thermalized at other temperatures than the stage. No further vibration-isolation is used. The coarse approach is done with a piezoelectric linear motor made of titanium Žlength: 30 mm, width: 40 mm, height: 10 mm.. The displacement range is "5 mm. In order to control the approach, the motor is equipped with a capacitive displacement sensor, having a resolution of 10 nm. We adapted a piezoelectric tip sample control w34x, as recently used in near field optical microscopy. In the original set-up a thin optical fiber is glued on a prong of a quartz tuning fork. The quartz tuning fork is excited mechanically at its resonance frequency. When the fork is resonating in a plane perpendicular to the sample, the interaction between tip and sample can be modeled as a viscous drag force using the differential equation of the harmonic oscillator. Due to the piezoelectricity of quartz, it is possible to deduce the oscillation amplitude from the voltage at the quartz. As the sample is approached, damping sets in and the ampli-
1
Licensed to Air Liquide, DTA, 38000 Sassenage, France.
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tude of oscillation is reduced, the phase begins to turn. The tuning fork technique has also been used for topographical imaging, a metallic STM tip w35x replaced the optical fiber. Instead of gluing an optical fiber on the prong, we attached a Si-chip carrying a microSQUID. The Sichip changes the weight of the prong, shifting the resonance frequency from 32 768 Hz Žempty. to about 29 000 Hz. We obtained resonances with a quality factor of 6000 at 4.2 K. The use of the tuning fork allows scans at constant height, and topographical imaging. The sample is scanned in front of the SQUID using a large range piezoelectric scanner w33x. The piezoelectric benders are 34-mm long, allowing a "60-mm scan-range in the xy direction at 4.2 K. The z control is realized with a fifth bender, Fig. 6. The last difficulty is to place the SQUID as close as possible to the edge of the scanning tip, in order to get the highest possible resolution. One technique is to build on a Si-wafer an AFM cantilever in SiC, containing the SQUID. All patterning and positioning can be done with electron beam lithography and RIE. The next process is to glue by ion interdiffusion a precut Pyrex wafer on the SiC. The grooves in the Pyrex wafer are aligned under the tips. The last step consists of dissolving the Si-wafer in Pyrocatecol and free standing tips on SiC membranes are obtained. This technique is elegant, but very difficult to carry out because many process steps are involved Žannealing of the membrane, . . . ..
Fig. 6. Scheme of the microscope set-up in the reversed dilution refrigerator.
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SQUIDs for the imaging of domains in hard ferromagnets or the imaging of Josephson vortices in superconductors with short coherence length. Due to their low critical current Al SQUIDs are better adapted for the exploration of vortex lattice transitions in artificial networks.
Acknowledgements
Fig. 7. Aluminum microSQUID on a SiC membrane.
An alternative technique w36x to fabricate tips is being explored: fabrication of the SQUID, masking it with an aluminum film and removing the unprotected Si by RIE. The SQUID layout can be modified according to the experiment. The design shown in Fig. 7 is not the best, as the loop is rather wide at the apex. The highest spatial resolution is obtained by inserting a narrow pickup loop in the inductance of the SQUID itself, placing a microbridge at the apex, and scanning as close as possible to the surface. This design is similar to what has been adopted by Ref. w27x. We are using microSQUIDs. Their fine linewidth results in high spatial resolution and makes them much less prone to flux trapping than conventional SQUIDs.
6. Conclusion Fine linewidth of microSQUIDs and close proximity enable the detection of flux of very small magnetic objects. We built a scanning SQUID microscope placed inside an inverted dilution refrigerator. All the displacements are piezoelectric. Topographical imaging is possible as a piezoelectric force sensor carries the SQUID. MicroSQUID imaging is a promising technique for the visualization of magnetic structures. According to the expected magnetization signal, different probes can be used: Nb
The authors acknowledge the encouragement of B. Pannetier and the fruitful discussions with A. Benoit and J.R. Kirtley’s practical and theoretical support. A. Madouri is thanked for his expertise in tip fabrication. The project is supported by the Ultimatech Program of the CNRS and a grant from DGA n 96 136.
References w1x J. Vidal, R. Mosseri, B. Doucot, Phys. Rev. Lett. 81 Ž1998. 5888. w2x A.I. Buzdin, Phys. Rev. B 47 Ž1993. 11416. w3x A. Bezryadin, B. Pannetier, J. Low Temp. Phys. 102 Ž1996. 73. w4x B. Pannetier, J. Chaussy, R. Rammal, J.C. Villegier, Phys. Rev. Lett. 53 Ž1984. 1845. w5x K. Runge, B. Pannetier, Europhys. Lett. 24 Ž1993. 737. w6x A.M. Chang, H.D. Hallen, L. Harriot, H.F. Hess, H.L. Kao, J. Kwo, R.E. Miller, R. Wolfe, J. van der Ziel, T.Y. Chang, Appl. Phys. Lett. 61 Ž1992. 1974. w7x C.D. Tesche, J. Clark, J. Low Temp. Phys. 29 Ž1977. 301. w8x M.B. Ketchen, IEEE Trans. Magn. 27 Ž1991. 2916. w9x M.B. Ketchen, M. Bhushan, S.B. Kaplan, W.J. Gallagher, IEEE Trans. Magn. 27 Ž1991. 3005. w10x L.R. Narasimhan, C.K.N. Patel, M.B. Ketchen, Appl. Phys. Lett. 73 Ž1998. 993. w11x C. Chapelier, M. El Khatib, P. Perrier, A. Benoit, D. Mailly, SQUID 91, in: H. Koch, H. Lbbig ŽEds.., Superconducting Devices and their Applications, Springer, Berlin, 1991, pp. 286–291. w12x J.M. Jaycox, M.B. Ketchen, IEEE Trans. Magn. 17 Ž1981. 400. w13x K.K. Likharev, L.A. Yacobson, Sov. Phys. Tech. Phys. 20 Ž1976. 950. w14x K. Hasselbach, J.R. Kirtley, D. Mailly, in preparation. w15x A.Th.A.M. De Waele, R. de Bruyn Ouboter, Physica 41 Ž1969. 225. w16x C. Chapelier, A. Benoit, D. Mailly, Phys. Rev. Lett. 70 Ž1993. 2020. w17x W. Rabaud, L. Saminadayar, D. Mailly, A. Benoit, in progress. w18x W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit,
K. Hasselbach et al.r Physica C 332 (2000) 140–147
w19x w20x w21x
w22x w23x w24x w25x
w26x w27x
B. Barbara, N. Demoncy, A. Loiseau, H. Pascard, D. Mailly, Phys. Rev. Lett. 70 Ž1997. 1791. W. Wernsdorfer, T. Ohm, C. Sangregario, R. Sessoli, D. Mailly, C. Paulsen, Phys. Rev. Lett. 82 Ž1999. 3903. B. Pannetier, A. Bezryadin, A. Eichenberger, Physica B ŽAmsterdam. 222 Ž1996. 253. A. Moser, H.J. Hug, I. Parashikov, B. Stiefel, O. Fritz, H. Thomas, A. Baratoff, H.-J. Guntherodt, P. Chaudhari, Phys. ¨ Rev. Lett. 74 Ž1995. 1874. A. Volodin, K. Temst, C. Van Haesendonck, Y. Bruynseraede, Appl. Phys. Lett. 73 Ž1998. 1134. A. Oral, S.J. Bending, H. Henini, Appl. Phys. Lett. 69 Ž1996. 1324. T. Matsuda, K. Harada, H. Kasai, O. Kamimura, A. Tonomura, Science 271 Ž1996. 1393. R.C. Black, A. Mathai, F.C. Wellstood, E. Dantsker, A.H. Miklich, D.T. Nemeth, J.J. Kingston, J. Clarke, Appl. Phys. Lett. 62 Ž1993. 2128. L.N. Vu, M.S. Wistrom, D.J. van Harlingen, Appl. Phys. Lett. 63 Ž1993. 1693. J.R. Kirtley, M.B. Ketchen, K.G. Stawiasz, J.Z. Sun, W.J.
w28x w29x
w30x
w31x
w32x w33x w34x w35x w36x
147
Gallagher, S.H. Blanton, S.J. Wind, Appl. Phys. Lett. 66 Ž1995. 1138. T. Morooka, S. Nakayama, A. Odawara, K. Chinone, Jpn. J. Appl. Phys. 38 Ž1999. L119. C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, T. Shaw, J.Z. Sun, M.B. Ketchen, Phys. Rev. Lett. 73 Ž1994. 593. J.R. Kirtley, M.B. Ketchen, C.C. Tsuei, J.Z. Sun, W.J. Gallagher, L.S. Yu-Jahnes, A. Gupta, K.G. Stawiasz, S.J. Wind, IBM J. Res. Dev. 39 Ž1995. 655. W. Wernsdorfer, PhD, Joseph Fourier University, Grenoble Ž1996. and E. Durand Electrostatique et magnetostatique, ´ Paris: Masson 1953. P. Pitzius, V. Dworak, U. Hartmann, ISEC 97 Berlin. J. Siegel, J. Witt, N. Venturi, S. Field, Rev. Sci. Instrum. 66 Ž1995. 2520. K. Karrai, R.D. Grober, Appl. Phys. Lett. 66 Ž1995. 1842. J. Rychen, T. Ihn, P. Studerus, A. Herrmann, K. Ensslin, Rev. Sci. Instrum. 70 Ž1999. 2765. F. Ayela, J. Chaussy, T. Fournier, E. Menegaz, Sens. Actuators, A Ž1999. to appear.