Magnetic Field Distribution Due to Domain Walls in Unconventional Superconductors

a r X i v :0710.3695v 1 [c o n d -m a t .s u p r -c o n ] 19 O c t 2007

Magnetic Field Distribution Due To Domain Walls

In Unconventional Superconductors

N. A.Logoboy

Racah Institute of Physics,The Hebrew University of Jerusalem,Jerusalem 91904,Israel

(Dated:February 2,2008)Steady-state properties of 1800Bloch domain wall (DW)in superconducting ferromagnet (SCFM)are studied.The distribution of magnetic ?eld above and below the surface of the SCFM due to the permanent magnetization supercurrent ?owing in the DW plane is calculated by solving Maxwell equations supplemented by London equation.It is shown that part of the magnetic ?ux of the two neighboring domains closures in the nearest vicinity of the surface of the sample giving rise to declination of the line of the force from being parallel to the DW plane.As a result,the value of the normal component of magnetic ?eld at the surface of the sample reaches only half of the value of the bulk magnetic ?ux.At the distances of the order of value of London penetration depth the magnetic ?eld decreases as inverse power law due to the long-range character of dipole-dipole interaction.The last two circumstances are important for comparison the calculated magnetic ?eld with the data obtained by the methods on measurement of normal component of magnetic ?eld,e.g.Hall probe technique,aimed to con?rm the existence of magnetic order parameter in unconventional superconductor.

PACS numbers:74.25.Ha,74.90.+n,75.60.-d

I.INTRODUCTION

Unconventional superconductors are a subject of in-tensive experimental and theoretical studies during past decade [1]-[8].Coexistence of superconductivity and magnetism results in a number of unusual phenomena,which have both fundamental and application interests.Unconventional superconductors,such as Sr 2RuO 4[3],ZrZn 2[4]and UGe 2[5],which are characterized by the state with broken time-reversal symmetry (TRSB),pos-sess the magnetic structure related to non-zero orbital magnetic moment due to spin-triplet p -wave state of multi-component SC order parameter.The properties of such superconductors are not trivial.In particular,the macroscopic magnetization is not well-de?ned qual-ity,and the equations of motion cannot be expressed in terms of local magnetization [7].

Starting from the pioneering work [10],[11]on inves-tigation of non-uniform states of superconducting order parameter (see,also review [12]),the planar defects such as domain walls (DW)and surfaces have been attracted recently a lot of interest [13],[14],[15],[16]due to new experiment possibilities,e.g.scanning superconduct-ing quantum interference device (SQUID)and Hall probe technique,on investigation of magnetic ?eld distribution resulting from spontaneously generated superconducting currents which can serve as a prove of TRSB origin of su-perconducting order parameter in unconventional chiral superconductors.

Although,the magnetostatic ?elds are screened by su-perconducting current,metastable domain walls (DWs),as topologically stable planar defects,may exist even in the Meissner state [6].Thus,the domain structure of SCFM cannot be ignored.The anomalies in the local magnetization loop for Sr 2RuO 4near B =0are con-sidered as a strong indication of the presence of chiral

domains [15].The strong coordinate dependence of mag-netization of such a 2D magnetic defect creates the intrin-sic magnetic ?eld which interacts with superconducting (SC)current.For unconventional superconductors the discontinuity of magnetic induction at the DW can be interpreted as an e?ective magnetization,and the con-tribution of DW current to the energy density can be transformed to an e?ective Zeeman term providing the possibility of excitation of orbital magnetization waves by incident electromagnetic ?eld [7].

In present publication we solve the Maxwell equations for distribution of magnetic ?eld inside the sample and above it due to the presence of planar defect,such as a DW,show rapid decrease of the magnetic ?eld in the vicinity of sample surface,discuss the long-range origin of this ?eld resulting in existence of the tails at the distance ～λ(London penetration depth)and compare our results with recent experimental data on measurements of the magnetic ?eld in Sr 2RuO 4by SQUID [13]and Hall probe technique [19].It is shown that for in?nitely thin DW the stray ?elds above the sample depend only the magnetic ?eld jump at the DW,but not on the details of DW structure.

II.BASIC EQUATIONS

Let us consider the 1800DW of Bloch type in a semi-in?nite sample of superconducting ferromagnet occupy-ing y ≤0.We assume,that the surface of a crystal is parallel to the x ?z plane at y =0,and the domain wall,being parallel to the y ?z plane,separates two domains with magnetization M (x )along the +y or ?y direction provided that the quality parameter α=H K /4πM 0>1,e.g.the ?eld of magnetic crystallographic anisotropy of the easy-axis (y -axis )type (H K )is high enough in or-

2

-10-50510

-4-2

2

4

b

y

x /

b

z

x /

FIG.1:Shown are the y -component (a )and z -component (b )of magnetic induction for Bloch DW at constant value of the domain wall width ?and variable London penetration depth λ.The indexes 1,2and 3correspond to δ=?/λ=0.1,0.3and 0.6respectively.The dashed line shows the y -component (a )and z -component (b )of magnetization which are the lim-iting values for corresponding components of magnetic induc-tion at λ?→∞,e.g.in neglecting of screening by supercon-ducting current.

der to stabilize the system against the ?ip of the domain magnetization into the x -z plane.We restrict ourselves to the simplest case when the London penetration length λexceeds the DW thickness ?.This means that at the spatial scales of the order of ?,the domain wall struc-ture is governed by the exchange and magnetic crystallo-graphic energies and is not a?ected by the kinetic energy associated with superconducting currents.Thus,we as-sumed that the next hierarchy of characteristic lengths for SCFM is ful?lled:?<λ

The free energy of the superconducting ferromagnet can be written as follows [8]:F =

d 3

x

2πα

M 2⊥

+?2

(?i M i )(?i M i )

+

1

2π

?φ?A 2

+

B 2

δq i

=0,(2)

where q i =θ,φ,A .Minimization of the free energy den-sity Eq.(1)with respect to azimuthal,φ,polar,θ,an-gles and vector potential A leads to the system of non-linear di?erential equations describing the ground state of SCFM with DW of Bloch type.In case of δ=?/λ<1

-10-50510

-10-50510

h

y

x /

h

z

x /

FIG.2:Shown are the y -component (a )and z -component (b )of magnetic ?eld for Bloch DW at constant value of the domain wall width ?and variable London penetration depth λ.The indexes 1,2and 3correspond to δ=?/λ=0.1,0.3and 0.6consequently.

-2-1012

-2-1012

h

y

x /

b

y

x /

FIG.3:Shown are the y -components of magnetic induction (a )and magnetic ?eld (b )for Bloch DW at constant value of the London penetration depth λand variable DW width ?.The indexes 1,2and 3correspond to δ=?/λ=0.1,0.3and 0.6consequently.The dash lines correspond the limit values for magnetic induction (a )and magnetic ?eld (b )in case of in?nitely thin DW [8].

the solution of these equations are as follows:θ0=2tan ?1e x/?,φ0=0,

b (0)y

=cos θ0e (x/λ)cos θ0,b (0)z =sin θ0e

(x/λ)cos θ0

,(3)where we used the reduced values for the components

of magnetic induction,b (0)y =B (0)y /4πM 0and b (0)

z =B (0)

z /4πM 0.The condition δ<1allows us to neglect the in?uence of Meissner current on the structure of the DW,which is de?ned entirely by the magnetic anisotropy and exchange energy.The screening action of the current re-sults in decreasing of magnetic induction at the distance of ～λfrom the center of the domain wall x =0.The graphical representation of the solutions (3)are shown in Fig.1for di?erent values of London penetration depth λat constant DW width ?.The distribution of magnetic ?eld H =B ?4πM ,created by Meissner current,are shown in Fig.2.It follows from Fig.1and Fig.2that the screening e?ect of superconducting current degreases with increasing of London penetration depth λ.

At constant λin limit case when the DW width ??→0,the jump of the tangential component of magnetiza-tion M y at the plane of the geometric domain boundary y ?z de?nes the current sheet responsible for the jump

of tangential component of magnetic induction B y [8].The distributions of tangential components of magnetic induction B y and magnetic ?eld H y of Meissner current at di?erent values of the DW width ?and constant Lon-don penetration depth λare shown in Fig.3.The results of our calculation based on Eq.(3)show the decrease of maximum of magnetic induction splitting with increase of DW width ?(see,Fig.3a )which con?rms the e?ect of smoothing due to ?nite DW width [20].

It follows from Maxwell equation curl H =j that there exists a two-component screening current j (x )=(0,j y (x ),j z (x )),which is the result of 2-dimensional structure of the DW Eq.(3).

III.

RESULTS AND DISCUSSION

To calculate the distribution of magnetic ?eld near the surface of SCFM we neglect the domain wall ?,assuming that ?<<λ.This assumption does not a?ect the results qualitatively,but essentially simpli?es the problem.In the end we shall discuss the e?ects of ?nite DW width ?=0.

A.

In?nitely thin DW (?→0)

The di?erence between a normal FM and a SCFM is important at distances larger than ?:while in normal FMs the magnetic ?eld H =B ?4πM vanishes and the magnetic induction B =4πM is constant inside domains,in SCFMs the magnetic induction B is con?ned in the Meisssner layers of width λ[8]:

B y =±

1

π

+∞

dk k exp (?ky )π

+∞

dk

k exp (?ky )

b

y

(x ,y )

FIG.4:(color online)Shown are the y -component of mag-netic ?eld in vacuum above the surface of SCFM (blue line)

and in the sample (red line)for one Bloch DW lying in y ?z plane.The presence of the identical tails in the distribution at large distances are due to long-range origin of dipole-dipole interaction in accordance to Eq.(9).

for the components of the reduced magnetic ?eld h i =2H i /δB i in vacuum,and

b x =?

2

k (k + k )

cos (kx ),b y =

2

k 2(k + k )

sin (kx )

(6)

for the components of the reduced magnetic induction b i =2B i /δB i in the sample.In Eqs.(5),(6)we used the notation k =(k 2+λ?2)1/2.The expression for h y (last equation in (5))in slightly di?erent form was calculated in [20].

The components of magnetic ?eld in vacuum Eqs.(5)can be calculated by introducing the complex potential ψ=ψ(w )which is analytical function of complex vari-able w =y ?ix .Thus,the complex magnetic ?eld

h (w )=h y (x,y )+ih x (x,y )

(7)

is derived from ψ(w )by h (w )=??w ψ(w ).The complex

potential ψ(w )can be expressed through special func-tions

ψ(w )=?i

2w

?i?w [H 0(w )?N 0(w )],

(8)

where H 0(w )and N 0(w )are zero-order Struve and Neu-mann functions correspondingly (see,e.g.[21]).For

λ,the di?erence H 0(w )?N 0(w )≈2/πw ,there-main contribution to the potential ψ(w )is due term in Eq.(8)which allows to calculate the distribution of the component of magnetic the sample (y ≥λ)

=

2

(x 2+y 2)2

,

h y =

4

(x 2+y 2)2

.

(9)

it follows from Eq.(9),that at |w |>>1

of magnetic ?eld decreases as inverse power g.|h |～|w |?2,due to long-range magnetostatic results of numerical calculations of normal to the surface component of magnetic ?eld in vacuum surface of the sample h y Eq.(5)and in the b y Eq.(6)are represented in Fig.4which shows the sample surface y =0,magnetic induction split-half of it bulk value δB y and are characterizes rapid decrease with the distance above the DW.the distance of y =0.1λabove the DW the mag-decays till the third of the bulk value δB y .At the magnetic ?eld splitting reaches about 0.05of it bulk value and decays with x as inverse power low Eq.(9)due to the long-range origin of dipole-dipole interaction.In a limit of in?nitely thin DW,?→0,the stray ?elds in vacuum above the superconducting ferromagnet depend only on the jump of magnetic induction at DW plane,but not on the details of DW structure.In this case,the tangential to the sample surface component of magnetic ?eld is non-analytic function of y and is char-acterized by the discontinuous jump at the DW position,y =0.In next subsection we take into account the e?ects of small (?<<λ),but ?nite DW width ?=0.

B.Finite DW width (?=0)

To consider the e?ects related to the ?nite DW width,?=0,we assume the linear distribution of magnetic induction in the DW deep in the bulk, e.g.B y ≈(1/2)δB y (x/?)at |x |≤?,which is not a?ected by superconducting currents,inasmuch as ?<<λ.This assumption signi?cantly simpli?es the calculation and al-lows to express the results in analytic form.It can be shown that for ?nite DW width the kernels in Eqs.(5)and (6)are modi?ed by the multiplier sin(k ?)/k ?.It results in slight suppression of y -component of magnetic ?eld and restores the analyticity of x -component of mag-netic ?eld.The discontinuous jump of tangential compo-nent of magnetic ?eld at the position of DW is replaced

by the value h max x

=(1/π)ln(γδ),where γ≈1.78107is Euler-Mascheroni constant.

Recent experiments on imaging of magnetic ?eld dis-tribution above the surface of the sample of Sr 2RuO 4by scanning SQUID and Hall probe microscopy has re-vealed no evidence for existence of DWs in this uncon-ventional superconductor [19].This negative result can

5

be understood in the framework of the developed the-ory.If the DWs exist,the maximum value of magnetic induction due to permanent current?owing in the plane of the DW does not exceed the lower critical?eld,e.g.δB y/2～H c1≈30G above which the domain structure is unstable due to formation of Abrikosov vortices.Thus, at the distance of y=λ≈190nm,the normal component of magnetic?eld due to the presence of DW is of order of3G which de?nitely improbable by the used methods.

Acknowledgments

The stimulated inspiring discussions with Prof.E.B. Sonin are highly appreciated.This work has been sup-ported by the grant of the Israel Academy of Sciences and Humanities.

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