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Journal of Magnetism and Magnetic Materials 324 (2012) 3912–3917
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Journal of Magnetism and Magnetic Materials
0304-88
http://d
n Corr
Univers
E-m
journal homepage: www.elsevier.com/locate/jmmm
Domain walls confined in magnetic nanotubes with uniaxial anisotropy
A.P. Chen a, J. Gonzalez a, K.Y. Guslienko a,b,n
a Depto. Fisica de Materiales, Facultad Quımicas, Universidad del Pais Vasco, UPV/EHU, 20018 San Sebastian, Spainb IKERBASQUE, The Basque Foundation for Science, 48011 Bilbao, Spain
a r t i c l e i n f o
Article history:
Received 10 January 2012
Received in revised form
12 June 2012Available online 4 July 2012
Keywords:
Domain wall
Magnetic nanotube
Micromagnetism
53/$ - see front matter & 2012 Elsevier B.V. A
x.doi.org/10.1016/j.jmmm.2012.06.028
esponding author at: Depto. Fisica de Ma
idad del Pais Vasco, UPV/EHU, 20018 San Seb
ail addresses: [email protected], sckgus
a b s t r a c t
We present calculations of the different domain wall structures confined in magnetic nanotubes, such
as transverse wall, asymmetric vortex wall, branch fashion wall, and horse-saddle wall. The wall
structures were calculated by micromagnetic simulations. The tube radii R¼50 nm and 100 nm, and
aspect ratios length/radius L/Rr15 were considered. The magnetic phase diagrams of the stability of
different kinds of the domain walls were plotted as function of the tube aspect ratio L/R and the tube
thickness (difference of the outer and inner tube radii).
We found that the domain wall internal structures confined in nanotubes depend mainly on the
sizes of the tubes almost without varying with the tube material parameters. The obtained results offer
a way to design the magnetization structures for the domain wall based devices by controlling the
nanotube geometric parameters.
& 2012 Elsevier B.V. All rights reserved.
1. Introduction
The magnetic domain walls (DW), their internal structures andmagnetic field (or spin polarized electrical current) induced domainwall propagation in confined nanosystems are of basic interest andcritical to the performance of magnetic memory [1] and logicelements [2,3] of nanoscale devices. In domain wall based device akey requirement is to control the spin arrangements in the wall areaand to manipulate the domain walls motions. Therefore, it isimportant to get a quantitative relation of the magnetic structureof the domain walls with the geometry of the confined material forunderstanding of fundamental physics as well as the perspectivetechnological applications of the DW based devices.
Generally, the magnetic domain wall structures are deter-mined by the energy competition and balance of the differentenergy contributions including the exchange, anisotropy andmagnetostatic energies. Since these magnetic energies are notonly determined by the sample material parameters, such as theanisotropy constant Ku, the saturation magnetization Ms, and theexchange constant A, but also by the samples geometry, it leads tothe dependence of the domain wall structures on sample’s sizeand shape for a given sample material.
Furthermore, with reducing magnetic material sizes, thestructure of the domain wall becomes almost independent ofthe material parameters, and becomes determined mostly by the
ll rights reserved.
teriales, Facultad Quımicas,
astian, Spain.
[email protected] (K.Y. Guslienko).
confinement geometry [4]. Therefore, domain wall types canbe modified by controlling the nanostructure dimensions asdemonstrated quantitatively by the micromagnetic simulationsin thin-films, nanowires, and confirmed experimentally by usingspin-polarized scanning tunneling spectroscopy or spin-polarizedscanning electron microscopy [5–8]. These phenomena were alsoobserved in the case of tubular geometry. For long magneticnanotubes with axial direction of magnetic anisotropy the resultsof simulations showed that the magnetization configurations arecharacterized by the uniformly magnetized middle part along thetube axis and the magnetization curling states with the same oropposite rotating senses at the two ends of the tube [9–11,12].These ending curling states can be considered as a Neel typehead-to-head (or tail-to-tail) domain wall, whose center is shiftedoutside the tube faces, and the corresponding domain wall widthstrongly depends on the anisotropy constant Ku [9]. However, forthe relatively short nanotubes (with the aspect ratios length/radius L/Ro15), due to disappearing of the uniformly magnetizedmiddle part the domain walls with certain pattern nucleate at thecenter of the tube to separate two regions with the circularmagnetization rotating in the opposite directions.
For the domain walls confined in nanotubes micromagneticsimulation results have shown that not only the switching fieldvalues but also the magnetization reversal modes are dependenton the sizes of the nanotubes [11,12]. This is because the sizes ofnanotubes determine the relative importance of the exchangeenergy and magnetostatic energy contributions to the totalmagnetic energy of the domain wall. As reducing tube sizes theshort range exchange energy dominates the energy competitionand balance, so the whole magnetization reversal process.
A.P. Chen et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 3912–3917 3913
Increasing the tube sizes the magnetostatic interaction becomesmore important and the reversal modes can be often consideredas nucleation of DWs at the tube ends and their propagation alongthe tube length. The magnetic properties of soft magnetic nano-tubes with a transverse anisotropy were simulated and thestability of their various bamboo-like domain structures wasstudied [13].
It was recently established that DWs in a tubular geometry aremuch more robust than ones in magnetic nanostrips of similarsizes [14]. Moreover, the Walker velocity breakdown of themoving DW can be completely suppressed due to nanotubetopology, and as a result high DW velocities above 1000 m/scould be achieved by application of small magnetic fields.
In this paper we present the micromagnetic simulations of theequilibrium domain wall patterns formed in the relatively shortnanotubes (L/Ro15) of 50 and 100 nm radii with uniaxialanisotropy by using 2D micromagnetic approach.
2. Results of simulations
In the following calculations the cylindrical coordinate system(r–j–z) with the tube axis z parallel to the easy anisotropy axis isused as depicted in Fig. 1. For the finite length L of nanotubes withthe outer radius R, and inner radius Ri, we define the tubethickness as DR¼R�Ri. There are two circular domains near thetube faces (the planes z¼0, L) separated by a domain wallsituated near the middle plane (z¼L/2) of the nanotube. Themagnetization configuration of the domain walls can be found asa result of the minimization of the total nanotube magneticenergy, which consists of the uniaxial anisotropy energy, magne-tostatic energy and the exchange energy.
The simulations were performed using the typical soft magneticmaterial parameters: the exchange constant, A¼10�6 erg/cm, the
Fig. 1. The coordinate system used and sketch of the magnetization distribution in
the cross section of the circular nanotube of radius R and thickness DR (the r–z
plane) presented by different colors in the right side and by vectors in the left side.
The colors describe different signs of the reduced magnetization components
a¼(af, ar, az). (For interpretation of the references to color in this figure legend,
the reader is referred to the web version of this article.)
saturation magnetization, Ms¼103 G, and the magnetic uniaxialanisotropy constant is chosen to be Ku¼5�104 erg/cm3, andKu¼105 erg/cm3. This choice leads to the intrinsic exchange length
of the magnetic material, lex ¼
ffiffiffiffiffiffiffiffiffiffiffiffiA=M2
s
q¼10 nm.
The magnetization distribution within the nanotube isassumed to be of a cylindrical symmetry (no dependence on theazimuthal angle j) due to the tube circular shape and uniaxialanisotropy directed along the tube symmetry axis Oz. Therefore,the micromagnetic structure can be explored by using 2D numer-ical simulations in the r–z plane [10]. A cross-section, DR� L, ofnanotube was subdivided into 2D array of cubic cells with the cellsize 2.5 nm that is smaller than the exchange length of thematerial. The magnetization structure of domain walls calculatedby the 2D simulating method can be presented in two forms: (a)with the vectors arrangement shown by the tube left cross-section; (b) with different colors indicating the magnetization Mcomponents in the cylindrical coordinates M/Ms¼a¼(af, ar, az)shown by the tube right cross-section. The magnetization unitvector a(r, z) is function on the r and z-coordinates.
Fig. 2 shows the domain wall patterns emerged in thenanotubes with radii of 50 and 100 nm, tube thickness normal-ized by the exchange length, DR=lex, in the ranges of smaller than4 and 9, and the tube aspect ratios of L/Ro15, respectively. Wecan distinguish the following DW patterns separating two curlingdomains with opposite chirality: transverse wall, asymmetricvortex wall, branch fashion wall, and horse-saddle wall, whichare described below in detail. We calculated the DW configura-tions, which correspond to the minimal total magnetic energy ofnanotube Etot(R, L/R, DR) as function of the tube radius R, thereduced thickness DR/lex and aspect ratio L/R. We present belowthe simulation results of nanotubes with R¼50 and 100 nm.
2.1. The nanotube radius R¼50 nm
For nanotubes of 50 nm radius the magnetic phase diagram inthe range of the tube reduced thickness DR=lexo4.75 and aspectratio L/Ro10 is presented in Fig. 3. For the thin nanotubeswith DR/lexr1 as we reported in Ref. [10] the lowest energymagnetization configuration (C-state) is characterized by thesame rotating senses of the two curling domains at the nanotubeends. The left panel of Fig. 4 shows the spatial distributions of
Fig. 2. Schematic presentation of the calculated magnetization distribution in the
domain walls (transverse-, vortex-, branch- and horse-saddle walls) in the r–z cross
section of magnetic nanotubes with the radius R¼50 nm, L/R¼3.6 (a) DR¼20 nm, (b)
DR¼40 nm; and R¼100 nm, L/R¼7 (c) DR¼50 nm, (d) DR¼70 nm.
4 5 6 7 80
1
2
3
4
Nan
otub
e th
ickn
ess,
ΔR
/l ex
Nanotube aspect ratio, L/ R
Fig. 3. Magnetic phase diagram of the stable domain wall configurations in
nanotubes with the tube radius R of 50 nm in the ranges of the tube aspect ratio
L/Ro8 and DR=lex o5. Transverse wall (TW), vortex wall (VW), B- and C-states are
marked. The anisotropy constant is Ku¼105 erg/cm3. Inset: the calculated values
of the exchange (Eex, red line), magnetostatic (Em, black line), and uniaxial
anisotropy (Ean, dotted line) energies. (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
φ
φ
Fig. 4. The distribution of the reduced magnetization components (af, az) along
the tube z axis. The tube thickness DR=lex ¼ 1 (a): (1) L/R¼15 (C-state), (2) L/R¼8,
(3) L/R¼7.4 (single vortex state). DR=lex ¼ 2 (b): (1) L/R¼15 (B-state), (2) L/R¼4
(TW). The tube radius is R¼50 nm. The z coordinate is in units of the tube length L.
A.P. Chen et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 3912–39173914
magnetization components af and az along tube length (the Oz axis)for the thin nanotube case. For the thin nanotubes (DR=lexr1) thesame chiralities of two curling domains at the tube ends exclude theemergence of the domain walls forming in the middle of nanotubes.With reducing tube length the increasing demagnetizing energyforces disappearance of the uniformly magnetized middle part of thetube that results in transformation of the whole nanotube to singlecurling (vortex) state, as shown in Fig. 4 (the curve 3, af¼1 andaz¼0), which is the minimal total magnetic energy state for thistube length.
With increasing tube thickness the magnetization configura-tion becomes B-state [10,11] with the narrow end domainsrotating in the opposite directions or with opposite chirality aswe reported in detail in Ref. [11]. With reducing tube length thetransition region between the end domains decreases drasticallyreflecting the demagnetizing energy increase, as shown by thecurve 2 in the right panel of Fig. 4, where the af componentincreases rapidly from �1 to þ1 and az reveals a sharp maximumin the middle of tube.
2.1.1. Transverse domain wall (TW)
For short nanotubes the transverse domain walls are stable inthe middle part of the tube separating two domains with theopposite circular magnetizations as shown in Fig. 2a. The magne-tization distribution in the cross-section DR� L of nanotubes withR¼50 nm, DR=lex ¼ 2, and L/R¼3.2 is shown by color, the positionof the transverse domain wall (TW) is depicted by the interfacebetween the blue and the rose colors presenting respectively twodomains with þej and �ej opposite circular directions. Thetransverse wall structure is shown by the magnetization vectorsa. In the center of the wall region the magnetization vectors arealigned in the easy axis z direction. With extending DW area thenon-uniformity of the magnetization in z direction leads toappearance of the magnetic charges on the wall surface.
In order to reveal the energy competition and balance of DWstructures the minimal total energy Etot for each set (R, L/R, DR) ofthe nanotube geometrical parameters can be represented as asum Etot¼EexþEmþEan of the exchange Eex, magnetosatic Em, andanisotropy energy Ean contributions. This decomposition is usedto determine the main energy responsible for formation of thedifferent DW patterns. For the thick nanotubes with DR=lex ¼ 4 thecalculated values of the exchange, magnetostatic, and uniaxialanisotropy energies are plotted as a function of the tube aspectratio L/R (see inset in Fig. 3). For L/R44.5 there is the TW stabilityregion, where the magnetostatic energy is larger than the exchangeenergy.
For short nanotubes with R¼50 nm the anisotropy energy is sosmall that it can be neglected and, therefore, the energy competi-tion occurs between the magnetostatic energy contributed mainlyby the wall surface magnetic charges and the exchange energyappearing due to non-uniformity of the wall region. With increas-ing tube thickness or reducing tube length both the energiesincrease, but for the transverse DW pattern the magnetostaticenergy is always larger than the exchange energy.
The sensitive tube size dependence of the exchange andmagnetostatic energy contributions to the total nanotube mag-netic energy means that the confined TW width as result of theenergy balance also should be strongly tube size dependent. It isvery different with the classical Bloch or Neel domain walls inbulk magnets for which the wall width is determined only by thematerial parameters. Detailed calculations of the width of thetransverse wall confined in nanotubes will be reported elsewhere.
2.1.2. Vortex domain wall (VW)
If the tube thickness increases to values DR=lexZ4, the tube crosssection is large enough to accommodate a vortex, and also to makethe transverse wall energetically less favorable than in the thinnertube. Reducing the tube length a vortex wall (VW) forms betweentwo domains with opposite circular magnetizations. Fig. 2(b) showsthe magnetization distribution at the tube cross section DR� L ofthe nanotubes with DR¼40 nm and L/R¼3.2, where one core vortexwall pattern appears asymmetrically with respect to the tube axis z.The wall structure shown by the magnetization vectors exhibits acounter-clockwise flux closure vortex wall with the core magnetiza-tion, referring as the polarity [15], pointing to þef, the positivecircumferential direction.
For the relative thick nanotubes with reducing tube length tothe values smaller than L/Ro4.5 the emergence of the vortex walloccurs due to the drastic increase of the magnetostatic energycontributed by the volume magnetic charges. The magnetizationforms a magnetic flux closure configuration inside the tubedomain wall to reduce the magnetostatic energy and the totalenergy of the tube, giving rise to a single vortex structure. Thistransition from the transverse to vortex wall is analogous tosimilar transition in the thin nanostripes increasing stripe width
A.P. Chen et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 3912–3917 3915
[16], where DR plays role of the stripe width. A vortex wallformation in nanotubes can be indicated by the essential increaseof the exchange energy, which becomes equal and then above themagnetostatic energy as shown in inset of Fig. 3 for the particularcase of DR=lex ¼ 4.
The magnetization structure with mirror symmetry withrespect to the tube middle plane becomes energetically unfavor-able due to the essential increase of the exchange energy. Thesystem minimizes its energy by shifting the core of the vortexwall along the tube axis z creating the asymmetry of themagnetization configuration along the tube Oz axis (see Fig. 2b).
2.2. The nanotube radius R¼100 nm
For long nanotubes with R¼100 nm the anisotropy energy hasthe largest value among the three energy terms Eex, Em, and Ean.The domain wall patterns are determined by the competition andbalance among the three energies. Therefore, more different wallpatterns emerge as a result of this competition for the differentnanotube geometrical parameters. Fig. 5 presents the areas ofstability of the DW patterns for the reduced tube thickness
2 4 6 8 10 12 140
1
2
3
4
5
6
Nanotube aspect ratio, L/R
TW
HSWBW
BW
A-TW
A-TW
VW
B-S
C-state S-Vortex
VW
R=100 nm
Nan
otub
e th
ickn
ess,
ΔR
/l ex
VW
Fig. 5. Magnetic phase diagrams of the magnetization states and stable domain wall
vortex (VW)-, branch (BW)-, horse-saddle (HSW) walls, and B-, C-states are marked. Th
Fig. 6. The distributions of the reduced magnetization components af, az along the
Ku¼5�104 erg/cm3. The left panel (the tube thickness is DR¼30 nm): (1) L/R¼5, (2)
R¼100 nm.
DR=lexr6, and the tube aspect ratios of L/Ro14, as well as forDR=lexo8, and L/Ro10, respectively.
2.2.1. Asymmetric transverse wall (A-TW)
The asymmetric transverse walls (A-TW) are observed in thephase diagram within the tube thickness DR=lex intervals of 2–3and 3–4, as tube aspect ratios L/R are smaller than 8 (8) and 13.6(9.6) for Ku¼5�104 erg/cm3 (Ku¼105 erg/cm3). The confinedA-TW pattern keeps the transverse wall structure, but the wallcenter is shifted from the central plane of the tube. The A-TWpatterns formed for different tube thicknesses have a significantdifference in the magnetization configurations for long nanotubesas shown in Fig. 6. For DR¼30 nm the A-TW domain pattern inshort nanotubes with L/R¼5 is compared with the TW pattern forL/R¼8 (Fig. 6, left panel). There is no significant difference.However, the difference is essential for the long nanotubes withthe thickness DR¼40 nm (Fig. 6, right panel).
Let us examine the magnetic energy values (Table 1) corre-sponding to two lengths of nanotubes near the transition from TWto A-TW. Reducing tube length from L/R¼8 to L/R¼5 the exchangeenergy increases considerably, DEex=DEtot ¼ 0:343, versus almost
2 4 6 8 10
0
1
2
3
4
5
6
7
8N
anot
ube
thic
knes
s, Δ
R/l e
x
BW
VW
VW
TWA-TW
B-S
S-Vortex
BW
C-state
BW
R=100 nm
Nanotube aspect ratio, L/R
TW
HSW
configurations in nanotubes with the tube radius R of 100 nm. Transverse (TW)-,
e uniaxial anisotropy constant is Ku¼5�104 erg/cm3 (a) and Ku¼105 erg/cm3 (b).
tube length (the tube axial coordinate z) plotted using the anisotropy constant
L/R¼8. The right panel (DR¼40 nm): (1) L/R¼12, (2) L/R¼14. The tube radius is
Table 1The energy values including the total nanotube magnetic energy, Etot, the exchange
energy Eex, the magnetostatic energy Em and the anisotropy energy Ean in units of
104 erg/cm3 as a function of the tube aspect ratio, L/R. The tube radius is R¼100 nm.
DR¼30 nm and DR¼40 nm for the columns 1–2, and 3–4, respectively.
Nanotube aspect ratio, L/R 5 8 12 15
Etot (104 erg/cm3) 15.093 10.826 9.346 8.440
Eex (104 erg/cm3) 3.584 2.122 1.771 1.409
Em (104 erg/cm3) 7.200 4.588 3.635 3.456
Ean (104 erg/cm3) 4.309 4.116 3.940 3.575
4 6 8 10 120
2
4
6
8
10
Nanotube aspect ratio, L/R
Ene
rgy
dens
ity (1
04 erg/
cm3 )
Fig. 7. Energy contributions to the total magnetic energy of nanotubes with
R¼100 nm, DR=lex ¼ 5 and L/R from 4 to 12 plotted for the magnetic anisotropy
constant Ku¼5�104 erg/cm3 (black) and Ku¼105 erg/cm3 (red), respectively. (For
interpretation of the references to color in this figure legend, the reader is referred
to the web version of this article.)
A.P. Chen et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 3912–39173916
constant anisotropy energy, DEan=DEtot ¼ 0:045. It implies that dueto reducing tube length the increased wall surface charges aremainly eliminated by bending the magnetic moments near the wallsurface that result in increase of the short-range interaction energy.This energy balance causes a slight shift of the transverse wallcenter. As the tube thickness increases to 40 nm A-TW is formed inthe relative long nanotubes. With reducing tube aspect ratio L/Rfrom 14 to 12 an increase of the magnetostatic energy is balancednot only by the exchange but also by the anisotropy energy,and the latter contribution is slightly larger than the former,DEan=DEtot ¼ 0:4034DEex= DEtot ¼ 0:399. The increase of the aniso-tropy energy is responsible for the relatively large shift of the A-TWcenter.
2.2.2. Branch and horse-saddle domain walls
For nanotubes with R¼100 nm, as the tube thickness, DR=lex,increases and becomes more than 5, (see Fig. 5) the domain wallstructure exhibits the branch like DW pattern (BW) shown inFig. 2(c). In the wall region the magnetization vectors along thediagonal line approximating 451 bifurcate into two branches. Atfurther increasing of the tube thickness the wall profile presentshorse-saddle (HSW) like pattern, or a cross-tie wall, as shown inFig. 2(d). The magnetization vectors form the two conjugatehyperbolas, the East–West opening and North–South openingbranches, sharing the same asymptotes to separate two domainswith opposite chiralities þef and �ef [15]. The branch- andhorse-saddle DW positions are located asymmetrically withrespect to the middle tube plane z¼L/2 due to complicatedinterplay between the DW energies and energies of asymmetricalmagnetization domains (the positive DW shift along the z-axiscorresponds to positive chirlaity þef of the upper domain).
For comparison of the influence of the strength of the aniso-tropy constant on wall patterns Fig. 7 depicts the calculatedenergy contributions to the minimal total energy Etot as a functionof the tube aspect ratio in the range from 4 to 12 for thenanotubes with R¼100 nm, DR/lex¼5 (cross-sections of the phasediagrams presented in Fig. 5). For long nanotubes the influence ofthe anisotropy strength on the domain wall patterns is significant.It can be observed in Fig. 7 that for Ku¼105 erg/cm3 the magneto-static and the exchange energy curves cross near the pointL/R¼11, and below this value a VW forms. Then, decreasing L/Rfor the given DR/lex, the VW transforms to BW and HSW.However, for Ku¼5�104 erg/cm3, the crossing of the two energycurves are not observed in the range of L/Ro12 (they cross nearL/R¼15) and TW (A-TW) are not stable.
3. Discussion
For short nanotubes the anisotropy energy is determinedmerely by its strength, the anisotropy constant Ku, and is almostnot varying with the tube length. Therefore, the competitionbetween the magnetostatic energy and the exchange energy still
dominate the energy balance to determine the domain wallpatterns in nanotubes with 100 nm radius. For nanotubes withKu¼105 erg/cm3 in the L/R range of 8–10 the wall pattern is thevortex wall, where the exchange energy is larger than themagnetostatic energy. With reducing L/R value from 8 to 6 thebranch-wall appears. The bifurcation of the magnetizations alongthe diagonal line of the wall region for avoiding surface magneticpoles in the radial boundary results in an essential increase of theexchange energy, meanwhile the magnetostatic energy maintainsa constant value. For the branch domain wall structure the energyterms competition and balance satisfy the relationship of Ean4Eex4Em. At further reducing of the L/R value to 5 (or increasingthe tube thickness) the emergence of HSW is manifested byconsiderable increase of the exchange energy, Eex4Ean4Em,whereas the magnetostatic energy is reduced to the lowest value.The same evolution of the energy terms is observed for the case ofKu¼5�104 erg/cm3.
For the short nanotubes the exchange energy associated withshort-range interaction contributes the most to the total energy ofthe nanotube and depends only on the tube size. It illustrates thefact that the domain wall structures in short and thick nanotubesdepend only on the tube sizes, length and thickness, and areindependent of the material parameters such as the anisotropyconstant, Ku, the saturation magnetization, Ms, or the exchangestiffness, A. It means that the magnetization states of nanotubescan be modified by changing their geometrical parameters asshown by the magnetic phase diagrams in Figs. 3–6. The propertythat the domain wall profiles in nanotubes can be tailored via thetube geometry only is very different from those of Bloch and Neelwalls in bulk materials or magnetic films.
The obtained results can be used to determine the magnetiza-tion reversal modes in a circular Oersted magnetic field of currentflowing through a nanowire parallel to the tube axis inside thetube. According to Figs. 3 and 5 increase of the tube radius leadsto increase of the critical value of the thickness DR to stabilize thevortex or more complicated (branch-, horse-saddle-) domain wall.The similar dependence that was simulated is in Ref. [12] for thetransverse and vortex wall reversal modes as function of R and DR.But the case of the transverse and vortex wall separating axiallymagnetized domains in the long nanotubes was only considered inRef. [12], whereas we considered above the case of the domain wallsseparating circularly magnetized domains stable in the relativelyshort nanotubes. The reversal via nucleation and motion of the
A.P. Chen et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 3912–3917 3917
branch walls and horse-saddle walls in the circular magnetic fieldwould be also possible for thick nanotubes having relatively smallaspect ratio L/R.
We emphasize that the transverse wall and vortex wall hasbeen observed in the magnetic nanostrips, NiFe nanowires, andFe3O4 nanotube arrays [6–8,17,18]. However, no experimentalobservations of the domain walls with more complicated branchand horse-saddle pattern magnetization structures have beenreported so far.
4. Conclusions
Our simulation results reveal that the magnetization struc-tures of the domain walls constrained in the magnetic nanotubesare determined almost by the tube geometry parameters, such asthe tube radius R, tube thickness, DR¼R�Ri, and tube aspect ratioof the length versus radius, L/R. The transverse-, vortex-, branchfashion-, and horse-saddle walls are formed in nanotubesdecreasing the ratio L/R and increasing the tube thickness. Theabove calculated energies and magnetization configurations ofthe different domain walls in circular nanotubes with the uniaxialanisotropy can serve as a benchmark for description of themagnetization reversal in the nanotubes applying a circular oraxial magnetic field.
Acknowledgments
K.G. acknowledges support by IKERBASQUE (the Basque Foun-dation for Science). The work was partially supported by the
Spanish MICINN Grant nos. PIB2010US-00153 and FIS2010-20979-C02-01.
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