39-磁化反转Experimental evidence for an angular dependent transition of

Experimental evidence for an angular dependent transition of magnetization reversal modes in magnetic nanotubes
Ole Albrecht, Robert Zierold, Sebastián Allende, Juan Escrig, Christian Patzig et al.
Citation: J. Appl. Phys. 109, 093910 (2011); doi: 10.1063/1.3583666
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Experimental evidence for an angular dependent transition of magnetization reversal modes in magnetic nanotubes
Ole Albrecht,1Robert Zierold,1Sebastia´n Allende,2,3Juan Escrig,3,4Christian Patzig,5
Bernd Rauschenbach,5Kornelius Nielsch,1and Detlef Go¨rlitz1,a)
1Institute of Applied Physics,University of Hamburg,Jungiusstrasse11,D-20355Hamburg,Germany
2Departamento de Fı´sica,FCFM Universidad de Chile,Casilla487-3,8370415Santiago,Chile
3Center for the Development of Nanoscience and Nanotechnology(CEDENNA),Avda.Ecuador3493,
9170124Santiago,Chile
4Departamento de Fı´sica,Universidad de Santiago de Chile(USACH),Avda.Ecuador3493,9170124
Santiago,Chile
5Leibniz-Institute of Surface Modification,Permoserstrasse15,D-04318Leipzig,Germany
(Received7February2011;accepted29March2011;published online9May2011)
We report on the experimental and theoretical investigation of the magnetization reversal in
magnetic nanotubes that have been synthesized by a combination of glancing angle and atomic
layer ing superconducting quantum interference device magnetometry the angular
dependence of the coercivefields is determined and reveals a nonmonotonic behavior.Analytical
calculations predict the crossover between two magnetization reversal modes,namely,the
movement of different types of domain boundaries(vortex wall and transverse wall).This
transition,already known in the geometrical dependences of the magnetization reversal in various
nanotubes,is found within one type of tube in the angular dependence and is experimentally
confirmed in this work.V C2011American Institute of Physics.[doi:10.1063/1.3583666]
I.INTRODUCTION
Highly anisotropic magnetic nanostructures can be used to overcome the superparamagnetic limit found in spherical nanomagnets1,2such as magnetic nanowires and nanotubes. Several methods for the preparation3–6of wires and their magnetic properties7–11have been described in the last dec-ade.Whereas wirelike structures offer two geometric param-eters,namely length and radius,that can be independently adjusted to influence the magnetic properties,tubular struc-tures allow for the variation of an additional degree of free-dom:the wall thickness.To benefit from this fact a detailed understanding of the magnetization reversal processes in such structures is of great importance.A very convenient route for the fabrication of tubular nanostructures is the use of a porous alumina membrane12,13as a template for a subsequent covering with magnetic material.Atomic layer deposition14,15and electrodeposition16,17are among the most prominent techniques used for such template preparation. Recently,Albrecht et al.18synthesized arrays of magnetic nanotubes by using the combination of glancing angle depo-sition(GLAD)and atomic layer deposition(ALD).
The magnetic measurements on nanotubes have mostly been performed by superconducting quantum interference device(SQUID)or vibrating sample magnetometry(VSM) parallel or perpendicular to the long axis of the tube.19–22 Bachmann et al.23investigated the influence of the magnetic layer thickness in arrays of nanotubes on the reversal mode. For an applied magneticfield parallel to the long axis of the tubes they discovered a transition between two reversal modes that depends on the material and the geometrical parameters.
Allende et al.24theoretically predicted a nonmonotonic behavior of the angular dependent coercivefield in a mag-netic nanotube,and explained this feature by a transition between a domain wall of a vortex and a transverse domain wall at a specific angle between the long axis of the tube and the applied magneticfield.
In this work we present a combined experimental and analytical investigation of the angle dependent magnetiza-tion reversal in Fe3O4nanotubes.Glancing angle deposition was used to deposit Si columns on a Si substrate.The tech-nique allows for a deposition of nearly arbitrarily shaped structures by using self-shadowing effects.Subsequently,a magnetic tubular layer of Fe3O4was deposited by atomic layer deposition.The deposited layer thickness is close to a reported thickness dependent transition of the magnetization reversal modes.23The ALD technique is capable of produc-ing layers with a high thickness accuracy.As a simple exam-ple for the combined GLAD-ALD synthesis,here we present straight columns inclined to the substrate as a mechanical support structure for the Fe3O4nanotubes that are aligned neither parallel nor perpendicular to the substrate to avoid in-cidental substrate effects.A detailed description of the syn-thesis of the sample investigated here and the structural characterization of those nanotubes including TEM images has been given in a recent publication.18
The investigated sample,the experimental setup,and the main results of the angular dependent measurements are described in Sec.II.Models of different modes of magnet-ization reversal in nanotube modes are discussed in Sec.III. In Sec.IV the experimentally obtained values of the coercive field are compared to the theoretical predictions.
a)Author to whom correspondence should be addressed.Electronic mail:
goerlitz@physnet.uni-hamburg.de.
0021-8979/2011/109(9)/093910/5/$30.00V C2011American Institute of Physics
109,093910-1
JOURNAL OF APPLIED PHYSICS109,093910(2011)
II.EXPERIMENTAL
We investigate a sample with a mechanical support struc-ture consisting of inclined Si columns of b¼ð5762Þ grown on a Si substrate.The cylindrically shaped Si columns with a length ofð14006190Þnm have a mean radius of r¼ð6066Þnm and are covered with a Fe3O4layer of a thickness,RÀr¼ð1061Þnm.Thus,nanotubes of length ð14006190Þnm are produced by the ALD-process and the ratio of inner and outer radius of the Fe3O4tubes is g¼r=R%0:86.The layer thickness has been measured by x-ray reflectometry(XRR)on aflatfilm produced under the same conditions.As verified by XPS,electron diffraction,and high-resolution transmission electron microscopy in recent publications19,25,26the ALD synthesis yields a complete layer of magnetite,Fe3O4.It should be noted that the support struc-ture is needed to ensure the stability of the nanotube array and is retained for all subsequent investigations.Figure1(a)dis-plays a SEM micrograph of the investigated structure,and Fig.1(b)depicts a schematic drawing of the relevant direc-tions and angles in a single inclined nanotube synthesized on a substrate.The plane of the substrate is defined by the axes,x and y.The inclination of the structure to the substrate is described by b,i.e.,the angle between the surface normal,z,
and the direction of the long axis of the tube,z0.The rotation around the y axis is described by the angle,c.The rotation of the tube around this axis is characterized by two extreme con-figurations.At a certain angle the tube axis,i.e.,the easy axis, is parallel to the appliedfield,H,and,shifted further by90 , z0is perpendicular to H.
To investigate the angle dependent magnetization reversal processes of the nanotube array at a temperature of300K. magnetization isotherms were recorded,using a commercial SQUID-magnetometer(MPMS-XL,Quantum Design)equipped with a rotatable sample holder that allowed for a rotation of the sample in the range of360 with an adjustable increment down to0:1 .As displayed in Fig.2,where the two promi-nent angles H k z0and H?z0are selected exemplary,the angular dependent coercivefields were extracted from these recordings.
The full angular range of the coercivefields is displayed in Fig.3.Starting from point A,(c¼0 ),the coercivity slowly rises between c¼45 and c¼85 (Point B),fol-lowed by a sharp decrease with a prominent minimum at the angle,c¼125 (Point C).At this point,the tubes are perpen-dicular to the applied magneticfield,H,i.e.,the hard axis is aligned with H.By a further increase of the angle,c,the coercivity rises again until c¼153 (Point D),and decreases again.It should be noted that at points B and D the substrate is oriented in different angles with respect to the
applied FIG.1.(Color online)(a)Scanning electron micrographs of inclined Si col-
umns with an inclination angle with respect to the substrate normal
b¼ð5762Þ fabricated by glancing angle deposition.(b)Coordinate
system which is used in the discussion:The inclination of the columns with
respect to the xy plane of the substrate is described by the angle,b.The
angle,c,defines the rotation axis of the sample in respect of the applied
field,H.The inset in(b)displays a schematic top drawing of a tube with
both radii,namely the inner(r)and the outer(R)
one.
FIG.2.(Color online)Magnified part of magnetization isotherms of two
selected angles H k z0(~)and H?z0( ).From this recording the angular
dependent coercivity,H c,and squareness SQ¼l r=l s were extracted.For a
magneticfield applied parallel to the tube the isotherm reveals a squareness
of approximately73%.The squareness in the case of a perpendicular applied
field reaches a value of22%.Inset:Two magnetization isotherms for an
applied magneticfield parallel(!)or perpendicular(^)to the substrate
plane.
FIG.3.(Color online)(a)Angular dependence of the coercivefield of
inclined columns(b¼5762) for aðH k x!z!ÀxÞrotation by the
angle.c.Point A defines the starting angle of the measurement(c¼0 ).At
Point B the appliedfield is perpendicular to the substrate plane.At the mini-
mum of the coercivefield at c%125 (Point C),the columns are perpendic-
ular to the appliedfield.At Point D the magnitude of the magnetization
component parallel to the appliedfield is the same as at Point B.
field.The nonmonotonic behavior between A and C or C and D,respectively,cannot originate from a magnetization rever-sal due to the movement of a single domain wall type,as will be shown shortly.
III.MAGNETIZATION REVERSAL MODELS
In magnetic nanotubes the magnetization reversal can occur by a coherent rotation of all moments or by the nuclea-tion and movement of the domain walls.For tubes with a negligible impact of defects,three different reversal modes are possible.21,27These reversal modes are illustrated in Fig.4and correspond to the coherent rotation (left panel),where all magnetic moments simultaneously rotate;the vortex wall (center panel),where magnetic moments rotate progressively via propagation of a vortex domain wall (this mode is fre-quently called the curling reversal mode);and transverse wall (right scenario),where magnetic moments rotate pro-gressively via propagation of a transverse domain wall.As was pointed by Landeros et al.27the coherent reversal mode is only favorable for very short magnetic tubes where the length of the tube is in the same range as the wall width.21,27Since we are interested in high aspect ratio tubes we will focus our attention on the vortex and transverse reversal modes.Recently,Allende et al.24calculated the angular de-pendence of the transverse and vortex modes in magnetic nanotubes.We shortly review their approach and adapt it as a first model to describe the nonmonotonic angular behavior of the coercivity in our samples.
The nucleation theory with its analytic solutions allows the determination of the wall nucleation field,which describes for each reversal mode (transverse and vortex)the field and functional form at which the magnetization just starts to deviate from the saturated state.By using the correlation between the coercive field and the nucleation field mentioned later,it is possible to calculate the coerciv-
ity.From the experimental point of view,it is easier to extract the coercive field from the recorded measurements to compare it to the theoretically predicted values than to use the wall nucleation field.
A.Transverse reversal mode
Recently,Landeros et al.27calculated the total energy for the transverse reversal mode considering the sum of the exchange and dipolar contributions.Then,they minimized the energy with regard to the domain wall width,w T .How-ever,they only calculated energies and they could not calcu-late the nucleation or coercive field.To solve this problem,Escrig et al.21assumed that the nucleation field of a system that reverses its magnetization by means of the nucleation and propagation of a transverse wall is equivalent to the nucleation field of an equivalent system with an effective volume that reverses its magnetization by coherent rotation.Fortunately,Stoner and Wohlfarth 28discussed the angular dependence of a coherent magnetization reversal.Then,this model can be adapted 21,29to describe the angular depend-ence of a transverse reversal mode by replacing the length of the whole structure,in which the coherent rotation takes place,by the reduced length of the involved domain wall with width,w T .Thus we have introduced the effective vol-ume proposed by Escrig et al.21The geometry of the effec-tive volume is described by a demagnetization factor,N z ðw T Þ.27Using this value for the real nanotube in a first simple model leads to the following equation for the angular dependent nucleation field
H T
n
ðh Þ¼À2K ðw T ÞþK a ½ l 0M 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1Àt 2þt 4p 1þt M 0;(1)where t ¼tan 1=3h ðÞwith h being the angle between the long
axis of the tube and the applied field.Besides,
K ðw T Þ¼14l 0M 2
01À3N z ðw T Þ½ is the shape anisotropy con-stant.In the original work of Escrig et al.,21K a denotes the magneto-crystalline anisotropy (MCA)constant.Since we have polycrystalline nanotubes in our case,the MCA is aver-aged out and we generalize K a to be a universal anisotropy constant K a .Stoner and Wohlfarth have shown 28that the coercivity expression can be written as a function of the nucleation field as
j H T c
ðh Þj ¼j H T n ðh Þj 0 h p 42H T n p 4 Àj H T n
ðh Þj p 4 h p 28><
>::(2)Figure 5(black dashed curve)displays the angular depend-ence of the coercive field,using Eqs.(1)and (2),for a trans-verse reversal mode.The parameters used for the analytical
calculation are the geometric parameters described in Sec.II ;the saturation magnetization,M 0¼4:8Â105Am À1,at room temperature and a constant for the uniaxial anisotropy,K a ¼1:9Á104Jm À3,along the tube axis presumably caused by internal stress due to the production process.It should be noted that the used value corresponds to a stress induced uniaxial anisotropy in thin magnetite films on MgO
of
FIG.4.(Color online)Possible magnetization reversal modes in magnetic nanotubes of length,‘.The orientations of the local magnetic moments are displayed by the arrows.Left:coherent reversal mode.Center:vortex rever-sal mode,with a domain wall of thickness,w V .Right:transverse reversal mode,with a domain wall of thickness,w T .
approximately 104J =m 3found by Margulies et al.30Fol-lowing Landeros et al.27the domain wall width can be esti-mated to be w T %65nm.The obtained curve shows a monotonic decrease of the coercive field from an applied field aligned along the tube axis to a perpendicularly applied field.
B.Vortex mode magnetic reversal
The angular dependence of the curling nucleation field in a prolate spheroid has been calculated first by Aharoni 31in 1997.Escrig et al.29adapted the approach to calculate the switching field for finite nanotubes with demagnetization factors,N x ;y ;z ,describing the magnetic behavior in an applied field parallel to the tube axis.The authors used the exact geometry of hollow cylinders for the demagnetization and took into account the internal and external radii of the tubes.Their result,Eq.(5)in Ref.29,yields an angular de-pendence of the coercive field,which,however,does not include any other anisotropy term than the shape anisotropy.The effect of adding an anisotropy is essentially the same as that of changing the shape anisotropy by changing the aspect ratio.The effect is far from being negligible,and the anisot-ropy,along with the finite size,must be taken into account.In order to include the same uniaxial anisotropy along the tube axis as in the case of the transverse reversal mode,we adapted the calculations by Aharoni 31for the vortex nuclea-tion field in nanotubes with uniaxial anisotropy along the tube axis
H V
n
ðh Þcos ðh Àx Þ¼N x ð‘Þsin 2ðx ÞþN z ð‘Þcos 2ðx ÞÀÀc Àd 3cos 2ðx ÞÀ1ÂÃÁ
M 0;
(3a)H V
n ðh Þsin ðh Àx Þ¼N x ð‘ÞÀN z ð‘Þþd ½
sin ð2x Þ2
M 0;(3b)
where c ¼q 2L 2ex =R 2;d ¼K a =l 0M 2
0;and q satisfies
32qJ 0q ðÞÀJ 1q ðÞ01Àg qJ 0g q ðÞÀJ 1g q ðÞ
01¼0:
(4)
Here,J p z ðÞand Y p z ðÞare Bessel functions of the first and
second kind,respectively,L ex ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2A =l 0M 20
p is the exchange length,‘is the tube length,R is the external radius of the tube,g is the ratio of the inner and outer tube radius,and the angle,x ,is the angle 31where the nucleation starts with respect to the tube axis.Equation (4)has an infinite number of solutions,where only the one with the smallest nucleation field has to be considered.33To obtain the nucleation field,H V
n
ðh Þ,we simultaneously numerically solved Eqs.(3a)and (3b)for each applied field angle,h .As pointed out by Aharoni,31a jump of the magnetization for an isolated system occurs at or near the vortex nucleation field.Therefore,the coercivity is quite close to the absolute value of the nucleation field and we assume here,as in other studies,19,31,34that in the
V mode ÀH V n
is a good approximation to the coercivity,H V c .This assumption is supported by the high squareness value (73%)of the magnetization isotherm obtained for the H k z (vortex regime),displayed in Fig.2.
The angular dependence of the coercive field in the vor-tex regime was calculated with the same parameters as in the transverse one including the anisotropy term mentioned before and the exchange constant,A ¼1Â10À11Jm À1taken from Ref.35.Unlike the transverse reversal mode,it exhibits a monotonic increase of the coercive field up to 90 as shown in Fig.5(red solid curve).For symmetry reasons a mono-tonic decrease is obtained between 90 and 180 .
Neither the sole consideration of the movement of a transverse or a vortex wall,respectively,can explain the non-monotonic behavior of the experimentally obtained depend-ence of the coercive field.
IV.TRANSITION BETWEEN TRANSVERSE AND VORTEX REVERSAL MODE
By a combination of the angular dependences for the transverse and the vortex reversal modes considered above (the system will reverse its magnetization by whichever mode opens an energetically accessible route first,that is,by the mode that offers the lowest coercivity)we obtain a non-monotonic behavior of the angular dependence of the coer-cive field with distinct maxima and minima.As displayed in Fig.6,the calculation (solid lines)for the dominant majority of tubes with R ¼70nm and r ¼60nm exhibit crossing points at the same angles as the experimentally obtained maxima,which allows us to attribute them to a transition from a vortex to a transverse mode of magnetization rever-sal.21The prominent minimum is recovered by the calcula-tion for the transverse mode.It should be noted that for direct comparison with the theoretical results the experimen-tally derived curve is shifted in Fig.6with respect to Fig.3by the inclination angle,90 Àb ¼33 .We observe that the theoretical curves match the experimental data very well.The small shift of the experimental data points to higher val-ues,compared to the calculated curves at small angles,can possibly be explained by the deviation of the tubes’
cross-
FIG.5.(Color online)Angular dependence of the coercive field determined by analytical calculations for the vortex and transverse reversal mode between two prominent configurations.The applied field is parallel to the long axis of the tube at c ¼0 and perpendicular to the long axis of the tube at c ¼90 .The vortex mode (red solid line)exhibits a monotonic increase of the coercive field between the parallel and perpendicular configuration.In stark contrast to this,the coercive fields in the transverse mode (black dashed line)monotonically decrease.
section from perfect circularity (see the inset to Fig.6).We at-tribute the deviations for larger angles as stemming from the magnetic (dipolar)interaction in the tube ensemble.It has been pointed out by Escrig et al.21that the dipolar interaction of neighboring tubes results in a lowering of the coercive fields.
V.CONCLUSION
We have presented a combined experimental and theo-retical investigation of a novel type of magnetic nanostruc-tures obtained by a preparative approach in which ALD is used to coat a template obtained by GLAD.By the angular dependent magnetization measurements,namely the extrac-tion of the coercive field from the magnetization isotherms,for these tubular structures we find experimental evidence for an angular dependent transition of magnetization reversal modes which have been theoretically predicted.Analytical calculations allow us to identify the transition between the vortex and the transverse reversal mode.Shamaila et al.36also observed a nonmonotonic angular dependent coercive field in thick-walled nanotubes,but in contrast to our results,they explained it by a transition between the vortex and the coherent reversal mode.Despite the fact that they explained the nonmonotonic behavior for the angular dependence of the coercivity,it is just a qualitative explanation and not a quantitative understanding such as the one we propose in this paper.However,the coherent rotation should only apply to extremely small (monodomain)particles.For somewhat larger particles,such as magnetic nanotubes,we have to con-sider transverse and vortex reversal modes.
ACKNOWLEDGMENTS
The authors appreciate constructive discussions with J.Bachmann (Hamburg),and thank A.Zolotaryov (Hamburg)for the supporting XRR measurements.S.A.and J.E.acknowledge the support from Fondecyt under Grant Nos.3090047and 1110784,the Millennium Science Nucleus
“Basic and Applied Magnetism”(Project No.P06-022-F),and Financiamiento Basal para Centros Cientı´ficos y Tecnolo ´-gicos de Excelencia.Financial support by the Deutsche For-schunsgemeinschaft under Grant Nos.FOR 522(C.P.,B.R.,and K.N.),SPP 1165(R.Z.and K.N.),and SFB 668(K.N.,O.A.,and S.A.)as well as financial support from the Free and Hanseatic City of Hamburg in the context of the “Landesexzellenzinitiative Hamburg”(Project “NAME,”R.Z.and K.N.)is gratefully acknowledged.
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FIG.6.Direct comparison between the experimentally (open circles)deter-mined and the theoretically (solid lines)predicted angular dependence of the coercive field for Fe 3O 4nanotubes.At h ¼0 the tube axis z is parallel to the applied field and at h ¼90 it is perpendicular.Inset:Micrograph with a top view of Si rods determining the nanotube cross-section.。