I 1 8 0 IEEE TRANSACTIONS ON APPLLED SUPERCONDUCTIVITY, VOL. I I , NO. I , MARCH 200 I

Determination of Magnetic Properties Using a Room-Temperature Scanning SQUID

Microscope E. F. Fleet, S. Chatraphorn, F. C. Wellstood, and C. Eylem

Absrruct-We have used a YBCO de SQUID at 77 K to image room-temperature magnetic thin film samples. Samples imaged include Fe30r, rare earth magnets such as samarium cobalt, and CMR materials. We typically saturate the magnetization of the sample in fields up to 8.5 Tesla, and then image the remanent (zero applied field) state. To help quickly interpret and quantify the SQUID microscope data, we have developed several analytical techniques. These techniques yield quantifiable results of magnetic properties, including magnetization, total dipole moment, and the demagnetizing field. We will present our results and discuss applications and limits of SQUID microscopy to the characterization of bulk and thin-film magnetic materials.

Index Term-deconvolution, Fourier analysis, magnetism, SQUID.

1. INTRODUCTION he Scanning SQUID Microscope has a wide variety of possible applications. One possible area involves research on magnetic materials, including analyzing new

magnetic materials, tracing local resistive-magnetization dependencies in materials, and inspection of thin-film quality. To perform such tasks, it is necessary to develop analysis tools for quantifying and interpreting the data from the SQUID microscope. In this paper, we describe techniques we developed for extracting t F magnetic pole d2nsity pM, total magnetic dip05 moment m, magnetization M, and internal magnetic field H from our scanning SQUID microscope data.

We note that a unique solution for the case of out-of-plane magnetization has already been found [l]. This problem is solvable because there is only one component of the magnetization. In [l], however, Wikswo et al. note that the general two-dimensional in-plane magnetization problem cannot be solved given only an externally measured magnetic induction B,. While this is technically true, the following analysis will show that useful quantitative information can be obtained using reasonable assumptions about the sample's magnetization.

T

Manuscript received September 18, 2000. This work was supported in part by Lynntech, Inc. and the State of Maryland through the Center for Superconductivity Research.

E. F. Fleet, S. Chatraphom, and F. C. Wellstood are with the Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, MD 20742 USA (telephone: 301-405-7581).

C. Eylem was with Lynntech, Inc., College Station, TX 77840 USA. He is now with Duracell, Inc., Global Science Center, Berkshire Corporate Park, Bethel, CT 06801 USA (telephone: 203-731-6578, email: jeylem@duracell.com).

11. ANALYSIS

A . Magnetic Pole Density The magnetization M(x', y', z') of a sample relates to the

z-component of the magnetic induction B,(x, y, z) through the following equations (see [2], pp. 193-194):

and

+

+ + pu (x ' , y ' , z' ) = - V'. M ( x ' , y' , z ' ) (1)

Plo v, (X'>Y', 4 &'&dZ', (2) Bz = qx- y>' + G -$y + (. - yyy

where p,, = 4 ~ x 1 0 ~ Wm is the permeability of$ee space and pM(x',y',z') is the magnetic pole density. If M(x', y', z') is in-plane and uniform over the thickness t of the material, pM will have the following form:

(3)

B, is the induction measured by the SQUID at a SQUID sample separation of G. If ~0 >> t, we can set z' equal to 0 in the denominator integrate over z' to obtain

(4)

The integral is a convolution integral, and can be solved in Fourier space to find [3]

Finally, we can solve for pM:

p,(x,y) = F T - I p, = FT-' -eekzoFFTIB(x, y ) ] ] . [ A 1 [P:t (6)

The exponential factor in (6) is the most interesting part of the analysis for pM. This factor exponentially increases the large k (small x) features in the data. This is why the algorithm improves the spatial resolution of the resulting image, analogous to the calculation of 2-d current density from B, [l], [3]-[5]. To understand the maximum resolution obtainable, co9ider the relationship between pM and B,. A magnetization M produces a pole density pM whenever

V.MrO. (7) pM then produces a magnetic induction B, which the SQUID detects. Equation ( 5 ) shows how the large k (small x) features in €3, become exponentially damped in k-space. This means these features are blurred in x-y space as the SQUID moves farther away (increasing q).

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If the measurement apparatus were noiseless, every small feature could be recovered by applying (6). Of course, every measurement has noise, and typically this noise is independent of or weakly dependent on x, y, and z, and therefore is significant at all k. In other words, in k-space, the signal is exponentially damped while the noise is not. At some large enough k, the signal will drop below the noise of the measurement and be obscured. This means the smaller features, which have more information at higher k values, will be obscured more quickly than the larger features. Trying to recover those features whose k-space signal is weaker than the noise will simply introduce large noise in the resulting real-space pM. Therefore, the solution obtained with (6) must be windowed such that the data above some chosen cut-off k, is damped or discarded.

Ultimately, the SQUID-sample separation ~0 and the signal-to-noise (SNR) ratio determine k,. In turn, the choice of k, determines the spatial resolution s in the final image. Therefore, the maximum spatial resolution obtainable is a function of the SNR of the measurement and ~ 0 .

The exact relationship between k, and s depends upon what type of windowing function is used. The details of the effects of different window functions on the final solution have been covered elsewhere [ 5 ] . In this reference, the Gaussian window

where k = (k: + k?)'", was shown to be the best window for this analysis, particularly for calculating total dipole moment (discussed below).

To apply (6), we use Fortner's TransformTM image processing software and use the built-in Fast Fourier Transform (FFT) algorithms to quickly calculate everything needed. First, we take the FFT of the data B,(x,y,q) from the microscope. We then calculate the exponential and constant factor, and multiply these with the FFT of B,. We employ the noise cut-off discussed above, and inverse FFT back to (x,y) space. We note the FFT algorithm uses a definition of k,=l/x, whereas our defmition is k,=2nlx.

B. Magnetic Field In addition to pM, we would like to find M and H. To find

+ + + H from pMr we start with:

We note that, in the absence of electrical currents,

This allows the introduction of a magnetic scalar potential QM, such that

(9)

V x H = O . (10)

H = - V a M . (1 1)

G2@ M - -- PM' (12)

- - - - V a H = -V a M = pM . 4 -

Putting (1 1) into (9) yields

This can be solved for QM (see p. 193 of [2]), yielding

S2bstituting this into ( I I), we find the following equation for H:

+

Considering the x-component of (14) and taking the derivative. we find:

I (X - .')PM (x' ,Y', z ' ) m,dy,dZl (1 5) ((x - X1)2 + ( y - y y + (z - z')2

= III Now, assuming PM as in (3) and integrating over z', we find

I 4n

H =-

In (16), we have set z = 0, so that we are solving for the field inside the sample.

The formula for H, is now a convolution integral, and can be solved in Fourier space. A similar expression can be solved for Hy, yielding the final result [5]

Note (14) is for z = 0, the mid-plane of the sample, and therefore has no z-component.

C. Total Dipole Moment and Magnetization While the above analysis yields improved visualization of

magnetic structures, additional analysis is required to find usable qu5ntitative information. For example, the total dipole moment m is related to pM through the following equation (p. 2 17 of [6]):

where 2 and 9 are unit vectors in the x and y direction. Note that this equation assumes that the limits of integration lie outside the relevant area of magnetization, so there is no surface term. Assuming pM is uniform in z, we can integrate (1 8) to fmd:

(18) +

= JJf P M (x,y, z)d '

(19) A A

+ ' = '+ Y Y)pM (x, Y)&& *

If the material is homogeneously magnet+ized, dividing m by the volume V yields the magnetizatp M. In the case of non-uniform magnetization, this M is an average magnetization.

111. THE SQUID MICROSCOPE The principle of operation of the scanning SQUID

microscope is relatively simple. In our system, the SQUID is current biased and we use a flux-locked feedback loop (FLL) to linearize the output [7] . The SQUID is maintained at a fixed height above the sample during scanning, with the plane of the SQUID loop parallel to the sample so the SQUID measures the z-component of the magnetic field (see Fig. 1).

(13) A computer controls the scanning stage and records the voltage output of the FLL as the sample is scanned.

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sample

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x-y stage *

Fig. 1 Picture of SQUID microscope. The window manipulating mechanism is indicated at the top.

To obtain the best possible spatial resolution, the microscope has a custom vacuum housing which allows us to bring room-temperature samples close to the SQUID. The housing has a 25 pm thick sapphire window epoxied to the end of a G-IO nose cone. A window manipulating mechanism allows the window to be brought very close to the SQUID. Once the SQUID is positioned behind the window, we use a z-stage to move the sample up to the window. By using an optical microscope to observe the process, the final SQUID- sample separation can be reduced to about 30 pm under optimal conditions [ 5 ] , [SI, [9].

IV. DATA AND ANALYSIS To illustrate the analysis algorithm, we examined a thin-

film of Samarium Cobalt. The 35 nm thick film was sputter deposited on a glass substrate by Dr. Samuel Jiang at Argonne National Laboratory [see insert to Fig. 2 (a)]. Before scanning, we magnetized the sample in an 8.5 T magnetic field, applied in-plane in the direction indicated by the dashed arrow in Fig. 2 (a). From independent measurements, the saturation field is 7 T for this sample, so we should have fully saturated the magnetization.

x (") Fig. 2 (a) Gray-scale image of B, from SmCo sample. The dashed arrow indicates the direction of magnetization. The dashed line indicates the linecut shown in (b). The insert to (a) is a picture ofthe sample.

After magnetizing the sample, we scanned it with the SQUID microscope at a SQUID-sample separation of 150 pm. Fig. 2 (a) shows a gray-scale image of the 7 mm x 5 mm scan with a pixel size of 14 pn x 10 pm. Strong magnetic fields are present only at the edges of the sample that are perpendicular to the magnetization direction. This indicates the magnetization is very uniform over the area of the sample. Further, the orientation of the magnet's poles indicates that the direction of magnetization coincides with the applied magnetic field direction. Thus, the sample is either isotropic, or it is anisotropic with an easy axis in the direction of the applied field. The linecut shown in Fig. 2 (b) shows the fields range from f 18 pT.

We next applied (6) to the dataset to find pM (see Fig. 3). For this image, we used a Gaussian window with k, = 5 mm". Comparing Fig. 2 to Fig. 3, one readily sees that Fig. 3 displays the enhanced sharpness produced by the algorithm. Since, to date, no straight calculation for the right k, exists for pM, we rely on trial and error to find the optimum value for a given set of data. Usually the maximum value for k, can be estimated from experience with dealing with datasets that have a similar zo.

A gray scale image of H (the magnitude of H) is given in Fig. 4. In this image, the strongest fields are about 100 A/m, and are indicated by the bright areas. In comparison, the coercive field for this samplejs approximately 4 x lo6 A/m. The image clearly shows that H is strongest at the bright areas

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around the pole faces of the magnet and quickly falls off in the interior. Further, these fields are very weak everywhere in the magnet’s interior, indicating that the bulk of the sample has little demagnetizing field. Essentially, the film is in it> remanent state, or at the point on the hysteresis curve where H is roughly 0.

We applied (19) to the p M calculated for the Sn$o saFple. Using a film thickness of 35 nm, we calculated m and M for the three pM datasets at three different values of k,. The resulting values are shown in Table I. Also included in the table are the estim3ted area of the sample, the thickness, and the magnitude of m in the remanent state, as found from the hysteresis loop.

We first note that the value of m is independent of our choice of k,. In [ 5 ] , this is shown to be true provided the region of integration of (19) extends a distance r = 4/(2nk,) =

127 ym beyond the sample’s edges. In our calculations, we typically extended the integration at least 500 pm beyond the sample’s edges.

Second, the calculated value of 6 is 30% smaller than that found from the hysteresis data+ provided by Argonnne. However, the value we obtain for M critically depends on the value of the area and thickness of the sample, which are somewhat difficult to determine exactly. This causes at least some of the discrepancy.

Finally, we note that the techniques outlined above are possibly useful in a variety of ways. The analysis of pM helps isolate magnetic structures in the microscope data, and would be useful whe+n examining magnetic films for any reason. The analysis for H helps detennine+the local state of the magnet. Finally, the calculation for m, traditionally considered a “bulk” property, can quantify individual structures within a sample, or entire samples.

+

ACKNOWLEDGMENT We thank Dr. Samuel Jiang for the samarium cobalt sample

and magnetization data.

0 1 2 3 4 5 6 7

x (“1 Fig. 4 Gray-scale image of the magnitude of H for the SmCo sample.

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REFERENCES

B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, Jr., “Using a magnetometer to image a two-dimensional current distribution”, J. Appl. Phys., vol. 65, pp. 361-372, January 1989. J. D. Jackson, Classical Electrodvuamics 2”d Ed., John Wiley & Sons, Inc., New York, NY, 01975. E. F. Fleet, S. Chatraphorn, F. C. Wellstood, and L. A. Knauss, “HTS scanning SQUID microscopy of active circuits”, IEEE Trans. on Appl. Supercond., vol. 9, pp. 4103-6, June 1999. S. Chatraphorn, E. F. Fleet, F. C. Wellstood, L. A. Knauss, and T. M. Eiles, “Scanning SQUID microscopy of integrated circuits”, Appl. Phys. Lett., vol. 76, pp. 2304-6, April 2000. E. F. Fleet, “Design and applications of a cryo-cooled scanning SQUID microscope”, Ph. D. dissertation, University of Maryland, College Park (2000). J. R. Keitz, F. J. Milford, and R. W. Christy, Foundations of Electromannetic Theorv 3rd Ed., Addison-Wesley Publishing Co., Reading, MA, 01979. See, for example, J. Clarke, “SQUID Fundamentals”, in SQUID Sensors: Fundamentals, Fabrication and Applications, M. Weinstock, Ed., Kluwer Academic Publishers, pp. 1-62, 1996. F. C. Wellstood, Y. Gim, A. Amar, R. C. Black, and A. Mathai, “Magnetic microscopy using SQIJIDs”, IEEE Trans. Appl. Supercond., vol. 7, pp. 3134-3138, June 1997. E. F. Fleet, S. Chatraphom, F. C. Wellstood, S. M. Green, and L. A. Knauss, “HTS scanning SQUID microscope cooled by a closed-cycle refrigerator”, IEEE Trans. on Appl. Supercond., vol. 9, pp. 3704-3707, June 1999.