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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 26 — Dec. 25, 2006
  • pp: 13007–13013
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Improved longitudinal magneto-optic Kerr effect signal contrast from nanomagnets with dielectric coatings

L. F. Holiday and U. J. Gibson  »View Author Affiliations


Optics Express, Vol. 14, Issue 26, pp. 13007-13013 (2006)
http://dx.doi.org/10.1364/OE.14.013007


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Abstract

We report on the use of dielectric coatings to improve the contrast of longitudinal magneto-optic Kerr effect signals from submicron magnetic structures. Electron-beam lithography was used to define disks in 22 nm thick Ni films deposited on Si substrates. The structures were measured in four configurations: as-deposited, through a fused silica prism using index-matching fluid, coated with ZnS, and using a prism on top of the ZnS layer. The modified samples show up to 20 times improvement in the MOKE contrast due to admittance matching to the magnetic material and suppression of the substrate reflectance. The behavior is successfully predicted by a model that includes the magneto-optic response of the nickel layer and accounts for the fraction of the beam intercepted by the magnetic structure.

© 2006 Optical Society of America

1. Introduction

Longitudinal magneto-optic Kerr effect (MOKE) magnetometry is an important characterization technique for studies of domain wall motion and in-plane magnetization of planar structures. With increasing research into the use of nanomagnets for applications such as MRAM[1

1. A. Ney, C. Pampuch, R. Koch, and K. H. Ploog, “Programmable computing with a single magnetoresistive element,” Nature. 425, 485–487 (2003). [CrossRef] [PubMed]

] and magnetic logic gates[2–5

2. D.A. Allwood, G. Xiong, M.D. Cooke, C.C. Faulkner, D. Atkinson, and R.P. Cowburn, “Characterization of submicrometer ferromagnetic NOT gates,” J. Appl. Phys. 95, 8264–8270 (2004). [CrossRef]

], there is great interest in extending the utility of this technique to smaller structures. Focused MOKE systems for longitudinal measurements operate with spot sizes of a few microns (limited by the need for non-normal incidence), but some of the most interesting effects, such as vortex ring formation[6

6. T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, “Magnetic vortex core observation in circular dots of permalloy,” Science. 289, 930–932 (2000). [CrossRef] [PubMed]

] and single-domain remanent states[5

5. A.-V. Jausovec, G. Xiong, and R.P. Cowburn, “Stability of magnetization states in submicron Permalloy disks,” J. Appl. Phys. 99, 08B103-1-3 (2006). [CrossRef]

], occur within submicron structures. While magnetic force microscopy is invaluable for detailed studies of domains, MOKE remains an attractive method for non-contact sample measurement, and the characterization of structures over a wafer scale. Measurements of the average response of arrays of nanostructures have been made[7–9

7. S.U. Jen and K.C. Chen, “Enhancement of polar Kerr effect by forming Au nanoparticles on Ni surface,” J. Appl. Phys. 97, 10M311-1-3 (2005). [CrossRef]

], but information on individual structures is also of interest.

In the limit of small structures, most of the light reflected to the detector derives from the substrate, and the difficulty of separating the small Kerr component from the larger Fresnel reflection becomes extremely challenging. Optical coatings can be used to both increase the Kerr component of the sample reflectivity, and suppress the substrate and sample Fresnel reflections. Coatings have been used extensively in improving the response of polar Kerr structures for magneto-optical recording systems[10–12

10. R. Atkinson, I.W. Salter, and J. Xu, “Quadrilayer Magnetooptic Enhancement with Zero Kerr Ellipticity,” J. Magn. Magn. Mat. 102, 357–364 (1991). [CrossRef]

], and have been recently applied to polar Kerr measurements of nanomagnets[13

13. N. Qureshi, H. Schmidt, and A.R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004). [CrossRef]

, 14

14. N. Qureshi, S. Wang, M.A. Lowther, A.R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005). [CrossRef] [PubMed]

]. In this paper we report on the use of dielectric coatings for enhancing longitudinal Kerr contrast (the height of the magnetization loop, (dI), divided by the background signal (Iavg)) on nanomagnets with both experimental data and modeling results.

Dramatic improvements in the MOKE contrast (dI/I) were observed when a dielectric coating was applied to the small structures and the surrounding substrate. Additional gains were made by increasing the index of refraction of the incident medium through coupling to the coated structure using a prism and index-matching oil.

2. Theory and modeling

2.1 Kerr contrast

We describe the origin of the signal contrast in MOKE, following the development of Allwood[15

15. D.A. Allwood, G. Xiong, M.D. Cooke, and R.P. Cowburn, “Magneto-optical Kerr effect analysis of magnetic nanostructures,” J. Phys. D-Appl. Phys. 36, 2175–2182 (2003). [CrossRef]

]. P-polarized light reflected from a magnetized metallic film has two amplitude components, Epp and Eps (where the first subscript indicates the incident polarization, and the second, the polarization of the reflected light); the Fresnel and Kerr contributions. For longitudinal MOKE in Ni, the ratio of the Kerr to Fresnel components is of order 10-3. In general, these components are out of phase, but a quarter-wave plate can be used to correct for any coherent shift, yielding plane polarized light rotated by a small angle relative to the input light. Figure 1 shows the orientation of these components and a polarization analyzer, set at an angle, ϕ, to the s-polarized component. Since the detector measures intensity, the sum of the projections of Epp and Eps onto the analyzer is squared, yielding (for+x magnetization)

I+=(Eppsin(ϕ)+Epscos(ϕ))2=Epp2sin2(ϕ)+2EppEpssin(ϕ)cos(ϕ)+Eps2cos2(ϕ)
(1)

A similar result is found for the opposite magnetization, but with a negative cross-term. In the measurement of a magnetization loop, the height of the loop, dI, is given by the difference of these terms, and the average intensity, Iavg, by their mean. The detection of the signal is dependent on the ratio, dI/Iavg, which we identify as the Kerr contrast[16

16. P.R. Cantwell, U.J. Gibson, D.A. Allwood, and H.A.M. Macleod, “Optical coatings for improved contrast in longitudinal magneto-optic Kerr effect measurements,” J. Appl. Phys. 100, 093910 (2006). [CrossRef]

].

Fig. 1. Electric field components in MOKE magnetometry; Eps is enlarged for clarity. θk. is the Kerr angle, and ϕ is the analyzer angle
dIIavg=4EpsEppcos(ϕ)sin(ϕ)Eps2cos2(ϕ)+Epp2sin2(ϕ)+γ
(2)

An additional intensity term, γ, (of order 10-4) appears in the denominator to represent the experimental case, where non-idealities in the optical system lead to leakage of a small amount of light, in addition to the Kerr component, through crossed polarizers (ϕ=0). Determination of Kerr contrast for a system with magnetic and dielectric layers therefore reduces to a problem of calculating the reflectance coefficients for the thin film stack, and determining the combination of layers and analyzer angle that gives the optimum Kerr contrast. While a well-chosen antireflection coating will increase both the Kerr angle and contrast, overenthusiastic suppression of the Fresnel component, Epp, will lead to a huge Kerr angle, and no measurable signal. The general concepts of thin film antireflection coating design can be applied to the problem, but design of optimized coatings requires consideration of the interrelated effects of the coating(s) on the Kerr and Fresnel components, for which we used numerical modeling.

2.2 Multilayer response

There are many approaches to calculating the performance of a multilayer stack which includes magneto-optic layers[17–21

17. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A-Pure and Applied Optics. 1, 646–653 (1999). [CrossRef]

]. Our modeling was performed using a MATLAB® routine that calculated the optical admittance of the film stack including the magnetic materials, using the matrix multiplication formulation of Abdulhalim[18

18. I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A-Pure and Applied Optics. 2, 557–564 (2000). [CrossRef]

]. For small spots, we calculated the Kerr and Fresnel reflectivity components for the magnetic region and the substrate independently, and a linear average was performed, based on the fraction of the beam intensity intercepted by the magnetic region. The appropriate fraction was determined by the relative diameter of the dot and the measured x- and y- profiles of the beam. We have modeled and measured the increase in Kerr contrast for Ni magnets on Si substrates for circular structures down to 400 nm in diameter.

We then calculated the MOKE contrast based on equation (2), suitably weighted for the magnetic and non-magnetic regions, for comparison with experiments. Note that the value for γ (0.8×10 -4 for uncoated samples) in Equation 1 is taken from a fit to a freshly deposited Ni film, where the entire beam contributes to the signal.

3. Experimental

The substrates used were Si(100) wafers, cleaned by ultrasonication in acetone and isopropanol. E-beam lithography was used to define micron and submicron features in PMMA resist, isolated from, but located between, larger fiducial marks to allow alignment of the beam with the smaller spots. A thin layer of Ni (22 nm) was deposited onto the PMMA, and the structures were defined by liftoff (see Fig. 2). In addition to a sample with disks having diameters of 10, 5, 2, 0.4, 0.2 and 0.1 microns, a substrate was completely covered with Ni. This sample was used to verify agreement of the model with the measured effects of the dielectric coating and the prism, independent of beam size effects. Liftoff of the smallest disks was imperfect, and thus these were not measured.

Fig. 2 Scanning Electron Micrograph (SEM) of the Ni nanomagnets and fiducial marks. The triangles and squares were used to align the measurement beam.

All samples were measured at a wavelength of 635 nm, with an incidence angle of 45 degrees, using a NanoMOKE® system from Durham Nanomagnetics, Inc. A simplified layout of the system is shown in Fig. 3. The sample is mounted on a computer-controlled X-Y-Theta stage with submicron motion control, and a diffuse white light source allows imaging of the sample for alignment purposes.

Fig. 3. Simplified diagram of MOKE measurement system. Inset shows the mounting of the quartz prism.

For each sample, the as-deposited Ni on Si disk was measured. A second set of measurements was made by mounting the hypotenuse of a fused silica 45-90 prism on the sample using a small quantity of index matching oil. The inset of Figure 3 shows the geometry relative to the incoming beam. Using this prism corresponds to increasing the index of refraction of the incident medium to 1.46, without changing the incident angle. In addition to the index change, the focal length and spot size of the beam are slightly altered, leading to a reduction in γ. The prism was then removed, the sample was cleaned, overcoated with 58 nm of thermally evaporated ZnS, and re-measured. ZnS was chosen because it is a near-perfect antireflection coating for Si at 45 degrees, and also gives ~3× increase in the Kerr contrast for a Ni film [16

16. P.R. Cantwell, U.J. Gibson, D.A. Allwood, and H.A.M. Macleod, “Optical coatings for improved contrast in longitudinal magneto-optic Kerr effect measurements,” J. Appl. Phys. 100, 093910 (2006). [CrossRef]

]. A final set of measurements was made with both the coating and the prism.

The beam diameter of our setup was measured by scanning a magnetic square in front of the beam, and measuring the magnitude of the MOKE signal (dI) as a function of position. These curves were fit to an integrated Gaussian, yielding a FWHM of 7 microns in the y-direction and 10 in the x-direction (stretched by the 45 degree incidence angle).

4. Results and discussion

We first modeled the effect of the coating and prism coupling on the continuous Ni film, and compared our results with experiment, as shown in Fig. 4. The agreement between the model and the experiment is very good, although the calculated contrast for the prism-covered sample is a bit higher, and the (film+prism) prediction, a little lower, than observed. A smaller value of γ was used for the prism cases, proportional to the index change.

Fig. 4 Model and experimental values of the Kerr contrast for Ni film on silicon. Magenta (◊) - bare; black (▫) - prism covered, red (∆) - ZnS coated and blue (O), ZnS coated and prism covered. γ=8×10-5 without, and 6×10-5 with the prism.

In Fig. 5, we show the modeled effect of the coating as a function of ZnS thickness and analyzer angle for 10% and 0.5% of the beam intensity being intercepted. The Kerr contrast for the uncoated film (curve at film thickness=0) is reduced from that of the continuous Ni film almost linearly with the beam intensity fraction, but the peak value with the coating (thickness 58nm) decreases more slowly. This occurs because the ZnS film is acting as an antireflection coating for the Si substrate, reducing the overall reflectivity (Iavg)of the sample.

Fig. 5. Predicted Kerr contrast for Ni disks on Si with ZnS coatings as a function of analyzer angle and coating thickness for a) 10%, and b) 0.5 % of the beam intercepted by the Ni.; γ=0.9×10-4 was used for both calculations. The dielectric tensor elements used were (εxx=εyy=εzz=5.52 for ZnS, 15.3+0.08i for Si -13.2+16.5i, 3.91+.01i for Ni, with εxy=0.24-0.02i).

Thus, more than an area-proportional amount of the light received by the detector comes from the Ni. Because the overall amount of light is reduced, the best contrast is expected at higher analyzer angles than for unpatterned films. In our experiments we found that the amount of depolarization, represented by γ in the model, was also significantly reduced for small spots. This leads to a shift of the peak of dI/Iavg to smaller angles, and an increase in the maximum value of dI/Iavg, as shown in Fig. 6. There are several factors that contribute to the reduction; the overall reflected intensity is lower, and the disk only samples the central region of the laser spot, minimizing any effects due to a multiplicity of incidence planes or deviations from a planar wavefront. Additional improvement is expected when contacting the sample through the prism, due to the reduced spot size, and refractive effects increasing the focal length in the fused silica. The values of γ were based on the measured leakage through the system at extinction, scaled to the value used for the continuous nickel film, but because of the difficulty of determining the exact value of γ, we made small adjustments to the parameter to fit the data for small disks.

Fig. 6. Modeled Kerr contrast for a Ni disk on Si, intercepting 0.04 of the spot intensity. The solid curves are for γ=0.9×10-4, and the dotted curves are for γ=0.25×10-4. The blue curves are for an overcoat of 55nm of ZnS, and the red curves are bare Ni on Si.

Measurements and modeled values of the Kerr contrast, dI/Iavg (as a function of analyzer angle) for disks with diameters of 2 and 0.4 microns are shown in Fig. 7. The percentage of the intensity intercepted by each disk was based on the measured beam size, SEM images of the disks, and the magnitude of the Kerr signal, dI, relative to that of the Ni film. The contrast improvement over the bare Ni on Si is 13× for the 2 micron disk, and 20× for the 400nm disk. dI increases by a factor of ~3; values for γ are shown on the graphs. As observed for the Ni film, the model overestimates the improvement in performance for the prism alone(adjusting γ to smaller values worsens the agreement), and the value of γ decreases as the size of the disk goes down.

Fig. 7 Experimental and modeling results for (left) 2 micron and (right) 400 nm disks. Magenta (◊) — bare disk, black (▫) — prism covered, red (∆) — ZnS coated, and blue (O) — ZnS coated and prism covered.6.

5. Summary and conclusions

Optical coatings were applied to micron and submicron magnetic structures, and resulted in a dramatic increase in the predicted and observed Kerr contrast of the structures. Using a prism to couple the light into the structure improved the performance further. A multilayer model was used to predict the coating performance, with slight variation of the experimental depolarization used as a fitting parameter. By reducing the substrate contribution to the Fresnel reflectance, and optimizing the ratio of Kerr and Fresnel contributions from the magnetic film, the Kerr contrast, dI/Iavg could be increased by a factor of 20 for a 400 nm Ni disk on Si. A ZnS coating increases the slope of dI/(cos(ϕ)sin(ϕ)), and using an index-matched prism reduces the depolarization, γ, increasing the contrast. Proper choice of substrate, coating and measurement conditions will thus permit ready measurement of the longitudinal Kerr effect in individual submicron structures.

Acknowledgments

We gratefully acknowledge the assistance of J. Cui, T, Davis, and support from NIST.

References and links

1.

A. Ney, C. Pampuch, R. Koch, and K. H. Ploog, “Programmable computing with a single magnetoresistive element,” Nature. 425, 485–487 (2003). [CrossRef] [PubMed]

2.

D.A. Allwood, G. Xiong, M.D. Cooke, C.C. Faulkner, D. Atkinson, and R.P. Cowburn, “Characterization of submicrometer ferromagnetic NOT gates,” J. Appl. Phys. 95, 8264–8270 (2004). [CrossRef]

3.

D.A. Allwood, G. Xiong, C.C. Faulkner, D. Atkinson, D. Petit, and R.P. Cowburn, “Magnetic domain-wall logic,” Science. 309, 1688–1692 (2005). [CrossRef] [PubMed]

4.

A. Imre, G. Csaba, L. Ji, A. Orlov, G.H. Bernstein, and W. Porod, “Majority logic gate for magnetic quantum-dot cellular automata,” Science. 311, 205–208 (2006). [CrossRef] [PubMed]

5.

A.-V. Jausovec, G. Xiong, and R.P. Cowburn, “Stability of magnetization states in submicron Permalloy disks,” J. Appl. Phys. 99, 08B103-1-3 (2006). [CrossRef]

6.

T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, “Magnetic vortex core observation in circular dots of permalloy,” Science. 289, 930–932 (2000). [CrossRef] [PubMed]

7.

S.U. Jen and K.C. Chen, “Enhancement of polar Kerr effect by forming Au nanoparticles on Ni surface,” J. Appl. Phys. 97, 10M311-1-3 (2005). [CrossRef]

8.

M. Klaui, C.A.F. Vaz, J.A.C. Bland, and L.J. Heyderman, “Domain wall coupling and collective switching in interacting mesoscopic ring magnet arrays,” Appl. Phys. Lett. 86, 032504-1-3 (2005). [CrossRef]

9.

P. Vavassori, V. Metlushko, M. Grimsditch, B. Ilic, P. Neuzil, and R. Kumar, “Magneto-optical studies of superlattice dot arrays,” Phys. Rev. B. 61, 5895–5898 (2000). [CrossRef]

10.

R. Atkinson, I.W. Salter, and J. Xu, “Quadrilayer Magnetooptic Enhancement with Zero Kerr Ellipticity,” J. Magn. Magn. Mat. 102, 357–364 (1991). [CrossRef]

11.

K. Balasubramanian, A.S. Marathay, and H.A. Macleod, “Modeling Magneto-Optical Thin-Film Media for Optical-Data Storage,” Thin Solid Films. 164, 391–403 (1988). [CrossRef]

12.

M. Mansuripur, G.A.N. Connell, and J.W. Goodman, “Signal and Noise in Magneto-Optical Readout,” J. Appl. Phys. 53, 4485–4494 (1982). [CrossRef]

13.

N. Qureshi, H. Schmidt, and A.R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004). [CrossRef]

14.

N. Qureshi, S. Wang, M.A. Lowther, A.R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005). [CrossRef] [PubMed]

15.

D.A. Allwood, G. Xiong, M.D. Cooke, and R.P. Cowburn, “Magneto-optical Kerr effect analysis of magnetic nanostructures,” J. Phys. D-Appl. Phys. 36, 2175–2182 (2003). [CrossRef]

16.

P.R. Cantwell, U.J. Gibson, D.A. Allwood, and H.A.M. Macleod, “Optical coatings for improved contrast in longitudinal magneto-optic Kerr effect measurements,” J. Appl. Phys. 100, 093910 (2006). [CrossRef]

17.

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A-Pure and Applied Optics. 1, 646–653 (1999). [CrossRef]

18.

I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A-Pure and Applied Optics. 2, 557–564 (2000). [CrossRef]

19.

M. Mansuripur, “Analysis of Multilayer Thin-Film Structures Containing Magnetooptic and Anisotropic Media at Oblique-Incidence Using 2X2 Matrices,” J. Appl. Phys. 67, 6466–6475 (1990). [CrossRef]

20.

S. Visnovsky, K. Postava, and T. Yamaguchi, “Magneto-optic polar Kerr and Faraday effects in periodic multilayers,” Opt. Express. 9, 158–171 (2001). [CrossRef] [PubMed]

21.

S. Višnovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B36, 625–650 (1986). [CrossRef]

OCIS Codes
(210.3820) Optical data storage : Magneto-optical materials
(240.0310) Optics at surfaces : Thin films
(260.2130) Physical optics : Ellipsometry and polarimetry

ToC Category:
Optical Data Storage

History
Original Manuscript: October 4, 2006
Revised Manuscript: December 1, 2006
Manuscript Accepted: December 7, 2006
Published: December 22, 2006

Citation
L. F. Holiday and U. J. Gibson, "Improved longitudinal magneto-optic Kerr effect signal contrast from nanomagnets with dielectric coatings," Opt. Express 14, 13007-13013 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-13007


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References

  1. A. Ney, C. Pampuch, R. Koch, and K. H. Ploog, "Programmable computing with a single magnetoresistive element," Nature. 425,485-487 (2003). [CrossRef] [PubMed]
  2. D.A. Allwood, G. Xiong, M. D. Cooke, C. C. Faulkner, D. Atkinson, and R. P. Cowburn, "Characterization of submicrometer ferromagnetic NOT gates," J. Appl. Phys. 95, 8264-8270 (2004). [CrossRef]
  3. D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, "Magnetic domain-wall logic," Science. 309, 1688-1692 (2005). [CrossRef] [PubMed]
  4. A. Imre, G. Csaba, L. Ji, A. Orlov, G. H. Bernstein, and W. Porod, "Majority logic gate for magnetic quantum-dot cellular automata," Science. 311, 205-208 (2006). [CrossRef] [PubMed]
  5. A.-V. Jausovec, G. Xiong, and R. P. Cowburn, "Stability of magnetization states in submicron Permalloy disks," J. Appl. Phys. 99, 08B103-1-3 (2006). [CrossRef]
  6. T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, "Magnetic vortex core observation in circular dots of permalloy," Science. 289, 930-932 (2000). [CrossRef] [PubMed]
  7. S. U. Jen and K. C. Chen, "Enhancement of polar Kerr effect by forming Au nanoparticles on Ni surface," J. Appl. Phys. 97, 10M311-1-3 (2005). [CrossRef]
  8. M. Klaui, C. A. F. Vaz, J. A. C. Bland, and L. J. Heyderman, "Domain wall coupling and collective switching in interacting mesoscopic ring magnet arrays," Appl. Phys. Lett. 86, 032504-1-3 (2005). [CrossRef]
  9. P. Vavassori, V. Metlushko, M. Grimsditch, B. Ilic, P. Neuzil, and R. Kumar, "Magneto-optical studies of superlattice dot arrays," Phys. Rev. B. 61, 5895-5898 (2000). [CrossRef]
  10. R. Atkinson, I. W. Salter, and J. Xu, "Quadrilayer magneto-optic enhancement with Zero Kerr Ellipticity," J. Magn. Magn. Mater 102, 357-364 (1991). [CrossRef]
  11. K. Balasubramanian, A. S. Marathay, and H. A. Macleod, "Modeling magneto-optical thin-film media for optical-data storage," Thin Solid Films. 164, 391-403 (1988). [CrossRef]
  12. M. Mansuripur, G.A.N. Connell, and J. W. Goodman, "Signal and Noise in Magneto-Optical Readout," J. Appl. Phys. 53, 4485-4494 (1982). [CrossRef]
  13. N. Qureshi, H. Schmidt, and A. R. Hawkins, "Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures," Appl. Phys. Lett. 85, 431-433 (2004). [CrossRef]
  14. N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, "Cavity-enhanced magneto optical observation of magnetization reversal in individual single-domain nanomagnets," Nano Lett. 5, 1413-1417 (2005). [CrossRef] [PubMed]
  15. D. A. Allwood, G. Xiong, M. D. Cooke, and R. P. Cowburn, "Magneto-optical Kerr effect analysis of magnetic nanostructures," J. Phys. D-Appl. Phys. 36, 2175-2182 (2003). [CrossRef]
  16. P. R. Cantwell, U. J. Gibson, D. A. Allwood, and H. A. M. Macleod, "Optical coatings for improved contrast in longitudinal magneto-optic Kerr effect measurements," J. Appl. Phys. 100, 093910 (2006). [CrossRef]
  17. I. Abdulhalim, "Analytic propagation matrix method for linear optics of arbitrary biaxial layered media," J. Opt. A-Pure and Appl. Opt. 1, 646-653 (1999). [CrossRef]
  18. I. Abdulhalim, "Analytic propagation matrix method for anisotropic magneto-optic layered media," J. Opt. A-Pure and Appl. Opt. 2, 557-564 (2000). [CrossRef]
  19. M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto optic and anisotropic media at oblique-incidence using 2X2 matrices," J. Appl. Phys. 67, 6466-6475 (1990). [CrossRef]
  20. S. Visnovsky, K. Postava, and T. Yamaguchi, "Magneto-optic polar Kerr and Faraday effects in periodic multilayers," Opt. Express. 9, 158-171 (2001). [CrossRef] [PubMed]
  21. S. Višňovský, "Magneto-optical ellipsometry," Czech. J. Phys. B36, 625-650 (1986). [CrossRef]

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