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C 185189 (1991) 1671 I ~ NcaahHolland
MAGNETIC
FORCE
MICROSCOPY
AND
SUPERCONDUCTOILS
Chri~toph BRUDER Department of P h y M ~ , University ~ Pennsyl~ania, Philadelp~a~ PA 191~
Unconventional superconductors with a nontrivial complex order parameter (OP) (e.g., a 2dim~sionM O P in which the two components have a relative phase) break timereversal symmetry and are predicted to show a ~ r i e t y of magnetic phenomena, even in the absence of an external magnetic tleld [15I. Ex.a~nples include a) domain walls separating regmns ot mner~, t O P orientation that are dressed by supercurrents [13] b) OP suppression at ~_m_ple boundaries leading to supercurrents along the boundaries [2,3] c) (if an ~ ternal field is applied) unconventional vortices at tlae center of which the OP fails to x~nish [2,4]. Since all of these effects leave a signature in the magnetic fields just outside the superconductor (SC), I p m ~ e to u ~ a magnetic force microscope to try" to observe these fields and thereby investigate timereversal breaking superconductivity. Magnetic force microscopy [6,7] is one of ~veral scanning probe microscopies developed in the wake of the atomic force microscope. The principle of the magnetic force microscope ~ M . ) is that magnetic acting on a ferromagnetm (or higMy paramagnetm) Up lead to a deflection of a tiny cantilever that can be measured by electron tunneling or by an optical probe. Most of the work on M F M ' s has been concerned ~ t h imaging of ferromagnetic domains (see, e.g., I8] and references therein), but recently" it has also been proposed to study the di_amagnetic respon~ of a superconductor using an M F M [9]. A carehfl analysis by Schhnenberger and Alwdrado [8] led to the conclusion that the tip should be modelled as a magnetic charge (because the tipapex do:_ main is aligne(i with the tip axis and large compared to the tipsample distance). Therefore, the response of the MFM is directly proportional to the {coarsegrained) microfields produced by the sample at the location of the tip. To make these ideas more quantitative, I will now use realistic parameters and look in detail at case (a) described above, viz. a domain wall between two superconducting phases and the supercurrents connected with it. Following [2l and [3], I will consider a theory with a 2dimensional OP (r/i, q2) belonging to the represen
ration r e d the tet~go~M point gr0~tp D,~. ~ di~mc~siont~s Git=burgLandau {GL } free c~er~v i~ pro~. portionM to the integrM M (T/Tc

~)(b, ff 
+
b.ff)
+
(I,,ff
+
l'~ff) ~
* )2
+Kt(ID~:Thl ~ + 1D~,~l:) + K2(ID~,~t ~ + lD~,:i:)
+ K~( ( D,~I )'( D~rt: ) + ( D ~ ) ' {
D,V: ) + c c )
+K~(ID:,hl ~ + ID:~I :) + n~(B:  2 B
H~.~)
where D = V  2 i c A / h c . ~ is the GL p a r a m ~ and the 3~. K, are phenomenologicat coefficients. As a result, the supercurrents are mc~e involved than in conventional SC's. and i: t~urns out [3] that gradica~ of ~hc OP m the xdir~tion can bad to a sup~nt in the ydirection (and a magnetic fietd in the z  d i c t i o n ) . Con~,der now the fo~owing d ~ wall which exist for 0 > & >>  1 / 4 : ( , l , r / 2 )  , ( x } ( I , e imp}) where ,(x * +co) = l,el = V/(I  T/Tc}/(4 + & } and 7(z ~  c o ) = ~,7(z * oc) = O. U~ng ~b~e boundary conditions, the authors of [3] o b t ~ an approximate solution for ~ that leads to a supercurrent in the ydirection, i.e., p ~ e I to the domain w ~ :
a
tamh(ez)
where a new bngth scale (~~ = ~/K~K2,'(432iUoi: ) has been introduced. The corresponding field (including the diamagnetic curr~,nts){s
B: 
4~.2
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09214534/91/$03.5C © 1991  Elsevier Science Publishers B.V. All rights r~crved.
C. Bruder [
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Magneticforce microscopyand superconductors
11
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(:)
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of He1 = ],012 log ~/(vf2~2), which is 46 G in UBe13. Since images with a contrast of about 100 G between dark and light have been reported in [8], observation of a domain wall between two superconducting phases seems to be possible with existing technology. In conclusion, I have proposed to apply a magnetic force microscope to a domain wall between two superconducting phases as they are predicted for superconductors with vector order parameters. The length scales that appear in a domain watl are accesssible with a MFM; the forces depend on parameters in the GLfunctional that are difficult to estimate.
Acknowledgements F i g . 1. Domain wall (double line) together with the currents produced by the bending of the order parameter (dashed lines) and screening cw'rents (dasheddotted lines). where 6 = t~2/(21~012) is the London penetration depth. To observe this structure, assume that a domain wall is given as described above and that an M F M is used to scan a boundary of the sample lying in the xy plane. To avoid a distortion of the order parameter by the presence of the SCvacuum boundary (effect (b) described in the introduction) it is best to cover the SC with a thin layer of normal metal that should be as close as possible to the properties (i.e., Fermi surface and effective masses) of the SC in its normal state. This will soften the OPdiscontinuity. The current pattern to be scanned by the MFM will then resemble that in Fig. 1. To obtain numerical values, I will use the parameters of UBel3 which is a typical candidate for unconventional superconductivity, albeit not a tetragonal one: ~0 = 14nm, AL = 390nm, n = 40, and Hci = 46G for the critical field [10]. Unfortunately, this is not enough to determine all the parameters that occur in the expression for B z, and we have to make an educated guess for a  1 . Since it is a length connected to the bending of the order parameter close to the domain wall, the only reasonable choice is to identify c~1 and ~0, i.e., (x = 1 in dimensionless units, although the two quantities are different a priori. Assuming that K1 = K2, what is still required is the dimensionless constant K4, and I will set it equal to 1 as well. The force that the field distribution will exert on the magnetic tip will depend on the finite size of the tip. Assuming that the tip is a truncated cone with diameter 2H at its end, SchSnenberger and Alvarado [8] determined its transfer function in Fourier space to be /~0MtioTrR 2 exo((kR/2)2), here M,;, is the magnetization" of the tii3 • Another cutoff t~at I will neglect is given by the distance of the tip from the current distribution. A typical tip radius [8] is R = 40nm which is of the same order as ~0. The force on the tip has been calculated by a convolution of B,(x) and the transfer function just mentioned. In Fig. 2 I plot the magnetic field together with the forces for R = 40nm and R = 80nm. No units on the field or force axis are given, but by a rough estimate the maxima of the two quantities can be determined as follows: the maximum of B~ in the center of the domain wall is of the order
I would like to thank Manfred Sigrist for discussions. This work was supported by NSF Grant No. D M R 8815469.
References [1] G.E. Volovik and L.P. Gor'kov, Soy.Phys. J E T P 39, 674 (1984). [2] M. Sigrist, T.M. Rice, and K. Ueda, Phys.Rev. Lett. 63, 1727 (1989). [3] M. Sigrist, N. Ogawa, and K. Ueda, unpublished. T.A. Tokuyasu, D.W. Hess and 3.A. Sauls, Phys. Rev. B41, 8891 (1990). [5] M. Palumbo, P. Muzikar, and J.A. Sauls, Phys. Rev. B 4 2 , 2681 (1990). [6] Y. Martin and H.K. Wickramasinghe, Appl.Phys. Lett. 50, 1455 (1987). [8] C. SchSnenberger and S.F. Alvarado, Z.Phys.B 80, 373 (1990). [9] H.J. Hug, Th. Jung, H.3. Giinterodt, and H. Thomas, Physica C175, 357 (1991). [10] R.N. Kleinman, Experimental Evidence for HigherI Pairing in Heavy Fermion Superconductors, unpublished. ""
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x/~0 2. Magnetic field B:(x) (solid line) and force F(x) on the tip of the MFM as a function of x/~0. Longdasiied line: R/~o = 2, shortdashed line: R/~o =
Fig.
4 where R is the tip radius. The units on the yaxis are arbitrary and discussed in the main text.