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Static and dynamic X-ray resonant magnetic scattering studies on magnetic
domains
Soriano, J.M.
Publication date
2005
Link to publication
Citation for published version (APA):
Soriano, J. M. (2005). Static and dynamic X-ray resonant magnetic scattering studies on
magnetic domains.
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3 3
A NN
XRMS
STUDY OF
DISORDEREDD MAGNETIC STRIPE
DOMAINSS IN
0 - G D F E
THIN
FILMS S
X-rayX-ray resonant magnetic scattering (XRMS) has been used to investigate the structure ofof magnetic stripe domain patterns in thin amorphous GdFe films. Under the influence ofof a perpendicular magnetic field, the scattered intensity displays a smooth transition fromfrom a structure factor of correlated stripes to the form factor of isolated domains. We
derivederive an expression that relates the total scattered intensity of XRMS to the absolute valuevalue of the magnetization. Furthermore, we show how the strong circular dichroism inin the scattered intensity can be used to probe the domain wall structure. Finally, we findfind that domain theory is applicable to pre-aligned stripes, but loses relevance with
increasingincreasing orientational disorder of the stripe system.
3.1.. Introduction
Magnetizationn reversal in thin films is a subject that is rich in physics [2] andd highly relevant for the data recording industry. For switching timescales downn to nanoseconds, magnetic reversal involves the field-driven, thermally-assistedd nucleation of incipient domains which then grow via domain wall prop-agationn [71]. The dynamics of this process depends strongly on the film thick-nesss and the properties of the magnetic material in question. Furthermore, the
filmm homogeneity determines the density of nucleation centers and the strength off domain wall pinning. The interplay between these factors leads to a profu-sionn of possible domain structures [2].
Amongg these, one of the most studied are stripe domains found in thin filmss displaying perpendicular anisotropy [5, 13, 72, 73, 74, 75, 76, 77, 78]. In thesee cases, the local exchange and anisotropy interactions favor a single do-mainn state with the magnetization saturated perpendicular to the film plane. Thee resulting long-range demagnetizing field leads to a break up of this sin-glee domain, at the cost of the creation of domain walls. Given the right com-binationn of thickness and magnetic properties, highly correlated alternating bandss of up and down magnetization form. Such stripe systems can have quite complexx structures, the alternating up-down strips being separated by Bloch walls,, possibly capped with closure domains that further minimize the dipo-larr energy [79]. Additionally, the two-dimensional domain pattern depends on thee magnetic history of the system. After demagnetization in a perpendicular field,, the stripe pattern is disordered, looking much like a human fingerprint, whereass after in-plane saturation highly aligned stripes appear, oriented in the saturatingg field direction.
Sincee the seminal work of Kooy and Enz [5] the theoretical description off magnetic stripe domains in applied fields has steadily improved, both for in-planee [72], and out-of-plane [73, 75, 76, 77, 80] magnetization loops. Such theoreticall approaches, dubbed domain theory models, use a Fourier expansion off the magnetic structure to calculate the dipolar energy. Furthermore, they as-sumee perfect translational order. However, these models rely on an estimation off the domain wall energy and do not describe wall structures precisely. Al-ternatively,, the true domain structure can be calculated accurately using micro-magneticc finite element methods [74, 81, 82], which however require the stripe periodd as an input parameter. The smallest domain period is roughly twice the domainn wall width, which scales with the square root of the film thickness.
Inn order to test domain models in thin films, experimental methods are requiredd that give access to the three-dimensional magnetic structure with nanome-terr resolution. In this context, magnetic stripe morphology has been studied ex-tensivelyy using Magneto-Optical Kerr Effect microscopy (MOKE) [5,83], which iss the most suitable technique for the study of the evolution of stripe patterns
Ann XRMS study of disordered magnetic stripe domains in a-GdFe thin films 41
Figuree 3.1: Schematic view of aligned stripe domains (a) and reversed domain (b). The upp and down domains (vertical arrows), Bloch walls (0, ©) and closure domains (hor-izontall arrows) are depicted.
inn external fields. More recently, Magnetic Force Microscopy (MFM) has en-abledd much higher resolution studies in zero or moderate fields [84, 85, 86, 87]. Anotherr new technique, transmission X-ray microscopy (TXM) [88, 89, 90], ex-ploitss the strong magneto-optical contrast at certain X-ray absorption transi-tions.. Since the resolution of the zone plate lenses is still limited, resulting in a laterall resolution of 25 nm, X-ray resonant magnetic scattering (XRMS) [13, 22, 38,60,61,63,65,66,, 67,68,91,92,93,94,95,96,97] is an interesting and simpler alternativee that could potentially give higher resolution.
Inn scattering experiments one loses the phase information of the wave fieldd coming from the sample, thus obtaining - by definition - ensemble aver-agedd information. In this chapter we show that despite this phase problem, XRMSS is a powerful tool for the study of domain structures, especially when thesee are periodic.
Dataa from two 42 nm thick GdFe films are compared here, whereby the filmss themselves differ mainly in their saturation magnetization. We show that thee resulting change in dipolar interactions leads to a quite different domain
widthh and magnetization loop. Furthermore, for the sample with the smallest stripee period, we compare the differences in behavior starting from either the alignedd or the disordered initial stripe structure. We find that the behavior of the alignedd case is described quite well by a domain theory model [76], in contrast too a recent similar study [63]. This is ascribed to the absence of strong pinning centerss in the flat and structureless amorphous layers that are considered here. Furthermore,, in the case of aligned stripes, we show that the scattered intensity displayss strong circular dichroism, which can be used to estimate the size of the Blochh wall magnetization surrounding isolated stripes. In addition, we derive a generall relation between the total scattered intensity and the absolute value of thee magnetization.
Thee layout of the chapter is as follows: after introducing the magnetism inn rare-earth transition-metal thin films, Section 3.2 introduces the magnetic sys-temm under investigation and describes the experimental details. In Section 3.3, wee relate the experimental results and their discussion, split into sub-sections dealingg with scattering results (A) for the zero-magnetization state and field-dependentt scattering curves (B), the theoretical description of stripe diffraction patterss in the small-angle limit (C) and the field dependence of the total scatter-ingg intensity (D), interpretation of the scattering curves (E), the effect of disorder (F)) and domain period and magnetization (G). Finally, we close with conclu-sionss in Section 3.4.
3.1.1.. Magnetism of amorphous GdFe thin films
Amorphouss GdFe films have been described [98, 99, 100] both as ferri-magnetss (e.g. Gd) and as sperimagnets (magnetic structure of a two-subnetwork amorphouss magnet where the moments of one or both subnetworks are dis-persedd over a range of angles around the magnetization direction).
Forr the case of RE-Fe amorphous alloys, ferromagnetic (F) and antifer-romagneticc (AF) interactions were deduced from susceptibility, magnetization andd specific heat data within the Fe subnetwork due to different interatomic Fe-Fee distances [4, 101]. These competing interactions provoke the creation of twoo sets of Fe spins within the Fe subnetwork, the F and AF Fe subnetworks. Thee number of nearest neighbours in bcc and fee Fe is respectively 8 and 12, and theyy are ferro- and antiferromagnetic. Since the number of adjacent atoms
in-Ann XRMS study of disordered magnetic stripe domains in o-GdFe thin films 43 creasess with the disorder in the atomic positions, so will do the AF subnetwork.
AA non-collinear structure in the Fe-subnetwork spins was observed for A-GdFee thin films [102], with typical apex half angles of 40°. This spin structure cann be understood as a visualization of the F/AF Fe subnetworks, as a result of thee short-range exchange interactions of different signs. A plethora of different valuess for the exchange interactions is found in the literature [4, 98, 102, 103]. Despitee this, all of them coincide in giving | J f ep e | one order of magnitude larger thann | JcdFel' a nd an almost negligible Gd-Gd exchange coupling [102]. A-GdFee was firstly assumed to be a pure ferrimagnet due to the S = 0 charac-terr of the Gd3 + ion [104, 105]. However, it was later pointed out that, close to thee ferrimagnetic compensation composition xc (the composition where the Fe
andd Gd subnetwork magnetization cancel each other), GdFe may be sperimag-neticc [98], with the Gd spins no longer collinear but distributed in a cone-like shape. .
Amorphouss GdFe films are structurally highly disordered on a length scalee of several interatomic separations, while being extremely flat and defect-freee on a length scale larger than a few nanometers [4]. In our samples, these propertiess are reflected in a particularly high degree of perfection of the aligned stripee systems.
3.2.. Experimental
3.2.1.. Samples: fl-GdFe thin films
Gdi-jFe** magnetic thin films with x — 0.83 (sample A) and 0.81 (sam-plee B) were grown by electron beam evaporation on a rotating sample holder att l x l 0 ~9 mbar. The two selected compositions lie on the Fe-rich side of the ferrimagneticc compensation composition xc ~ 0.76 [106]. A thickness of 42 nm
wass chosen to give approximately l/e absorption at the Gd M5 resonance using thee calculated cross sections due to Thole et al. [27]. As supports we used 100 nmm thick commercially available Si3N4 windows, which have a transmission of ~95%% at the Gd M5 resonance energy. The magnetic films were capped with a 22 nm Al protection layer in order to prevent oxidation. X-ray diffraction scans showedd no trace of structural order in the GdFe films. Rutherford back scatter-ingg was used to determine the film thicknesses, as well as to check composition
Tablee 3.1: Magnetic properties of the Gdi-^Fe^ films. Sample Sample A A B B X X (%Fe) (%Fe) 0.83 3 0.81 1 Ms s (kA/m) (kA/m) 221 1 150 0 nil l DDnuc nuc {mT) {mT) 160.3 3 228.6 6 °nuc °nuc (mT) (mT) 89 9 7.4 4 KKu u (105//m3) ) 0.18 8 0.17 7
a n dd homogeneity. A t o m i c force microscopy (AFM) m e a s u r e m e n t s s h o w e d that t h ee surfaces w e r e free of pinholes, flat and structureless on the 1 n m scale.
-0.100 -0.O5 -0.00 0.05 0.10
-0.033 -0.02 -0.01 0.00 0.01 0.02 0.03
HH00H(T) H(T)
Figuree 3.2: Perpendicular-field magnetization loops of Gdi_xFey with x = 0.81 (sample A,, top) and x = 0.83 (sample B, bottom). Insets: schematic cross section of magnetic fluxx patterns at remanence and close to saturation showing principal stripe magne-tizationn (vertical arrows) with their stray field (ellipses), Bloch walls (+) and closure magnetizationn (horizontal arrows).
Ann XRMS study of disordered magnetic stripe domains in a-GdFe thin films 45
Tablee 3.2: Magnetic properties used and obtained from the model explained in Sect.. 3.3.7. Q Q 0.58 8 1.21 1 V* V* 2.73 3 1.82 2 7& 7& 2.5 5 12.3 3 k k {nm) {nm) 13.9 9 33.2 2 1w 1w (lO-tj/m(lO-tj/m22) ) 8.5 5 9.3 3 ö ö (nm) (nm) 37.2 2 42.9 9 A A {W-{W-l2l2J/m) J/m) 2.6 6 3.2 2 Ac c 0.33 3 0.79 9 Thee magnetization loops of both samples as measured with vibrating samplee magnetometry (VSM) are shown in Fig. 3.2. Their form is typical for stripee domain samples: a low-field hysteresis-free region separating two trian-gularr hysteretic regions. The insets represent typical low and high field cross sectionss of the magnetic flux pattern. The values of the saturation magnetiza-tionn Ms and the anisotropy constant Ku estimated from the in-plane nucleation
fieldd B„uc are listed in Table 3.1. The main distinction between the samples are
thee much smaller perpendicular nucleation field B^uc and the 33% smaller
sat-urationn magnetization of sample B due to it being closer to the compensation compositionn xc.
Thee remanent domain states were imaged with Magnetic Force Microscopy (MFM).. The MFM images were affected to some extent by tip-domain interac-tions,, visible as horizontal discontinuities. Stable images appeared typically onlyy after a few sweeps over the field of view even when using low moment tips,, indicating low domain wall pinning, and illustrating a limitation of MFM forr domain characterization. The measured periods are listed in Table 3.3.
3.2.2.. Small-angle X-ray scattering setup
Thee X-ray experiments reported here were carried out at the soft X-ray beamlinee ID08 at the European Synchrotron Radiation Facility [107]. The Apple III undulator source offers complete control over the polarization. The exper-imentss were performed with modest energy resolution AE/E < 10~3 and a beamm size of 100 ^m.
Fig.. 3.3 shows the layout of the experimental setup. The incident inten-sityy Jo was monitored by reading the drain current from the refocusing mirror. Eitherr a horizontal 1 mm wire (Fig. 3.4-d, e) or a knife edge (panel f) were used ass beamstops. The scattered intensity was recorded 50 cm behind the sample.
Tablee 3.3: Stripe periods as obtained from MFM and XRMS, and correlation length to periodd ratio from XRMS.
Sample Sample AA dis Aord Aord Bdis Bdis TT (MFM) (MFM) (nm) (nm) 232 2 160 0 835 5 T (XKMS) ) (nm) (nm) 253 3 162 2 934 4 Ê/T T 2 2 8 8 1.2 2
Ass a detector we used a P20 phosphor-coated (5 /mi thick, 1 }im grain size) vac-u vac-u mm window with a 12 bit CCD camera. For the present experiments, a TV lens combinedd with a 5 mm macro-ring was used, giving a field of view of ~15 mm andd a 10 }im resolution.
Thee co-ordinate system employed here is such that (x, y) define the sam-plee plane and z is parallel to the light propagation direction k. In the longitu-dinall geometry, the field B = fioH is applied parallel to k, while in transverse geometryy B is parallel to x. The sample was mounted in a room-temperature rotatablee holder, allowing us to preset the initial alignment of the stripe lattice too an ordered or disordered lattice by saturating the sample in-plane or out-of-planee respectively.
Beamstop p
CCD D
Phosphorr Sample Pinholes screenn or
diode e
Slits s
Figuree 3.3: Layout of the scattering experiments. The incoming synchrotron beam is de-finedd with the help of slits and pinholes. The sample could be magnetized transversely too the beam with a yoked horizontal magnet (not shown). The beamstop eliminates thee straight-through beam allowing the scattered intensity to be measured either with a photodiodee or with a fluorescent layer on a vacuum window, imaged by a visible-light CCDD camera.
A nn XRMS s t u d y of disordered magnetic stripe d o m a i n s in a-GdFe thin films 47
155 -20 -10 0 10 qxx (urn"1)
Figuree 3.4: Left: MFM images of the remanent magnetic domain patterns of sample A (a)) and B (c) after out-of-plane saturation, (b) Id. of the aligned pattern obtained after in-planee magnetization for sample B. Right: corresponding measured 2D diffraction patterns. .
3.3.. Results and discussion
3.3.1.. Scattering curves at remanence
Inn order to make a connection between the real space domain patterns ass measured with MFM and the reciprocal space scattering data, we compare threee important domain patterns in Fig. 3.4. After out-of-plane magnetic satura-tion,, both samples have a remanent domain pattern (panels a, c) that consists of completelyy disordered meandering bands with appreciable domain branching andd truncation. On the other hand, after saturation in an in-plane field, sample AA shows a remarkably perfect aligned stripe system (panel b).
Onn the right of these MFM images we show the corresponding XRMS scatteringg patterns. The disordered stripe patterns produce diffuse rings of scat-teredd intensity [13, 108, 109] (panels d, f), while the aligned stripe system pro-ducess very sharp and intense diffraction peaks (panel e), which at the Gd M5 resonancee contain even 5% of the transmitted primary beam. The used beam stopss gave scattering vector ranges of (0.004-0.05) and (0.01-0.20) ran"1 for sam-plee A and B respectively. The diffuse background was eliminated before further dataa processing by subtraction of exposures taken at magnetic saturation.
Angularr integration of these images gives the dependence of the scat-teredd intensity on the scattering wavevector transfer c\r, reproduced in Fig. 3.5.
Thee disordered pattern of sample A (panel a) shows clear first and third diffrac-tionn orders, whereas sample B in this case only displays a very broad maximum. Thee ordered lattice of sample A, shown in Fig. 3.5-b, produces a series of very pronouncedd diffraction peaks, shown here up to the fifth order and marked withh small arrows. The peak width increases linearly with the diffraction order duee to residual disorder in the stripe lattice [110].
Fromm the peak position of these curves, the average domain period r cann be obtained, while the width gives the correlation length £, i.e. the distance overr which the periodicity exists. As shown in Table 3.3, the period values from XRMSS are up to 10% larger than the MFM values, the difference being mainly duee to calibration errors. Also, in sample A the period of the aligned lattice is 30%% smaller than in the disordered lattice, due to the absence of branchings and otherr lattice faults.
Ann XRMS study of disordered magnetic stripe domains in a-GdFe thin films 49 c c 13 3 - Q Q I— — CO O CO O c c m m CD D i— — CD D
s s
o o 0.01 1 0.022 0.03 qrr (nm') 0.04 4 0.05 5Figuree 3.5: XRMS intensity curves obtained by azimuthal integration of the 2D pat-terns,, (a) and (c): the disordered remanent magnetic domains of samples A and B. (b)
Id.Id. for the aligned case for sample A taken with left- (black) and right-circular (grey)
polarization. .
3.3.2.. Field-dependent scattering curves
Field-dependentt XRMS is a convenient and direct way to monitor the evolutionn of the domain pattern over these magnetization loops. In general, thee scattered intensity consists of a series of concentric rings, the higher or-derss showing up more strongly in systems in which the stripe period is better defined.. This is exemplified in Fig. 3.6-a-b, which shows the evolution of the angularlyy integrated scattered intensity between the nucleation and saturation
0.011 0.03 0.05 0.05 0.10 0.15 0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20
qr( n m ' )) qr (nm') q, (nm'
1
) qr( n m ' )
Figuree 3.6: Field-dependent evolution of the scattered intensity for the disordered do-mainss of samples B (a) and A (b) and for the aligned stripes in sample A (c), taken with left-- (black) and right-(grey) circularly polarized beams. Data are displayed on a loga-rithmicc scale. The asymmetry y^rp for the ordered case of sample A is shown using a linearr scale in (d). The value of the applied field in mT is shown next to the traces.
p o i n t ss for b o t h s a m p l e s . For the sake of clarity not all curves are given. In both casess w e observe a clear evolution of the p e a k positions a n d intensities. Sample AA s h o w s m o r e h i g h e r o r d e r peaks, reflecting the larger correlation length in the stripee lattice. In b o t h cases the peaks d i s a p p e a r at high fields a n d ultimately t h ee curves d e v e l o p into a broad structure with intensity m i n i m a that m o v e to h i g h e rr qr. Later o n w e will identify this structure with the form factor of the
r e m a i n i n gg reversed d o m a i n s as s h o w n in Fig. 3.1-b.
Fig.. 3.6-c s h o w s the field d e p e n d e n c e for s a m p l e A w h e n starting from t h ee aligned stripe lattice. In this case the initial diffraction spots of Fig. 3.4-e r e m a i nn well defined u p to saturation, w i t h a constant a n g u l a r w i d t h of 5°. This i m p l i e ss that the reversed stripes are p u s h e d apart, b u t remain grosso modo paral-lel.. Also, in this case t h e scattered intensity observed for left a n d right circularly b e a m ss 1+ a n d ƒ_ s h o w a p r o n o u n c e d circular dichroism. The circular dichroic
Ann XRMS study of disordered magnetic stripe domains in a-GdFe thin films 51 asymmetryy ratio, defined as j+~ j ~ , is shown in Fig. 3.6-d. Initially the asymme-tryy curve is strong only at the even diffraction orders, but towards high fields a broadd structure develops with a zero crossing that is located at the qr position
off the minima in the corresponding diffraction curves in Fig. 3.6-c.
Inn Fig. 3.7 we compile the field dependent data for both samples: the reducedd magnetization mz = Mz/Ms, the domain period T obtained from the
wavee vector of the first order peak, the correlation length of the stripe lattice obtainedd from the peak width, normalized to the period £ / T and the total scat-teredd intensity. The width of the reversed domain wj given in panel (c) will be discussedd later.
3.3.3.. Stripe diffraction patterns in the small-angle limit
Inn Sect. 2.2 we showed how the polarization dependence of resonant magneticc scattering at the RE M4 5 edges could be expanded as function of the charge,, circular and linear dichroic scattering cross sections (Eq. 2.8).
Wee can rewrite this equation in the form of a simple Jones matrix, first intro-ducedd by Hill and McMorrow [111]. In the small angle scattering limit, which iss applicable since the domain sizes are at least 150 times the wavelength, this givess the simple expression
00 -imz\ p(1) + / m\ mxmy\ f ( :
'm'mzz 0 J \mxmy m2, )
Heree we choose as an orthogonal basis two polarization directions x, y lying in thee plane of the sample and the light propagation direction along the normal direction,, k / / z .
Iff a domain structure m(r) is present in the sample, the resonant terms F ^ will causee part of the incoming plane waves Eo to be scattered out of the incident beam,, where the far field scattered amplitude is the Fourier transform of the wavee field Eout (r) just after the sample. For samples thinner than one absorption
length,, the latter is given to good approximation by taking the integral of Eq. 3.1 overr the sample thickness (i.e. we neglect the distortion of the wavefront as it iss travelling through the sample). Introducing the contrast functions gz(y) — JJQQ mz(y,z)dz and g\j(y) — J0 tni(y,z)mj(y,z)dz (i,j — x,y), where t is the film
thickness,, we obtain
-scat -scat (3.1) )
EoHf(r)) = f(i)) + \&
xx
8xv\ f (2)
1.0 0 a.a. 0.5 ii o.o -1.0 0 33
-J?
500 0 rr (nm ) C OO J * oo o oo o 2000 [ b \ \ff
j
j»j»
i
^ * 4 ^ ^ ^ 80 0 75 5 70 0 65 5 60 0 55 5 50 0c c
--\ --\ V V \ \ \w w
V V
>»» » 1.0 0 0.8 8ee
X * ^
0.611 // 0.4uu / / 0.2 2 nn n \ \ \ \ \\ J :: K ^ -100 0 -500 0 50 Ho»Ho» (mT) 1000 -20 -100 0 10 /A/A00HH (mT) 20 0Figuree 3.7: From top to bottom, a) the VSM hysteresis loops (full line) together with thee magnetization value obtained from the scattering data (symbols), b) the period T, c)) the reversed domain size, w^, estimated from the minima of the form factor, d) the Is'' order peak FWHM to period ratio and e) the total integrated intensity (symbols), togetherr with the 1 — {tnz)2 curves, calculated from the VSM data. Left panels: sample
A,, right panels: sample B. represents data obtained from the disordered state, A from thee aligned one. Black/grey: increasing/decreasing field.
Ann XRMS study of disordered magnetic stripe domains in a-GdFe thin films 53 Noticee we have left out the non-magnetic term of Eq. 3.1 as it only contributes too the average attenuation. Since at any position in the sample, the Bloch wall magnetizationn mx is symmetric with respect to the film midplane in 2, while the
closuree domain magnetization my is antisymmetric, the contrast function gxy
vanishes.. The Fourier transform of Eq. 3.2 is
00 -iGz(q)\ (1) (GXM) 0 )F(2)~
iGz(q)) 0 J \ 0 Gyy(q)J '
wheree Gz(q), GIX(q) and Gyy(q) are the Fourier transforms of the
correspond-ingg contrast functions. The far field scattered intensity J(q) = | E ( q ) |2i s then givenn by the absolute square of this amplitude.
Wee will now narrow the discussion to the case of the aligned stripes in samplee A, which we will approximate by a perfectly periodic set of stripe do-mainss magnetized along the z direction, with translational symmetry in the y direction.. The Bloch walls in this case are magnetized along the x direction, and cann be capped with closure domains with magnetization in the y direction, as indicatedd in the insets of Fig. 3.2. Since the structure is invariant along the x direction,, the Fourier transform in this direction yields a trivial delta function. Iff the incident X rays are linearly polarized along the x direction, the total inten-sityy is given by
IIxx = I0{\FWGZ\2+\FWGXX\2\, (3.3)
withh a similar expression for y polarization. For circularly polarized incident lightt with helicity , the total intensity is:
ii = J O J I F ^ ^ + K I F W G ^ ^ + IFWG^I2)
Re [(FMGZ)*FW{GXX + Gyy)] } (3.4)
Thee last helicity dependent term is an interference between the polarization-rotatingg F^ term and the polarization-conserving F^ term. It can give cir-cularr dichroism if it is non-zero, which happens if Gz(qy) has the same spatial
frequenciess as Gxx(qy) or Gyy(qy). The resulting asymmetry is: I+I+ _ ^ Re [(F(1)GZ)*F(2)(G« + Gj,y)]
h T rr = |F(i)G2|2 + i ( | F ( 2 ) G „ P + |F(2)Gyy|2)- ( 3-5 )
-w/2-w/2 y w/2
qqyyw/ji w/ji qqyyw/jr w/jr
Figuree 3.8: Generic real space scattering contrast functions for a single up domain (A) andd an in-plane (Bloch or closure) magnetization component (B) as shown in Fig. 3.1 andd their Fourier transforms (C) and (D). Corresponding form factor for circular and linearr polarized light for |F( 2>/F( 1 ,|2 = 0.164 (fto;=1184 eV) [18] are shown in (E) and thee asymmetry ratio fT~ j ~ in (F). The domain wall to domain width ratio is 2:9.
Itt s h o u l d b e n o t e d that at the transition metal L2,3 edges F^2' is small with re-spectt to F ^ , m a k i n g the linear dichroic effects h a r d to observe, while at the G d M 4 55 resonance they can h a v e similar a m p l i t u d e s [16, 14] a n d the a s y m m e t r i e s reachh 2 5 % in o u r circularly polarized data, as s h o w n in C h a p t e r 2. We will first d e t e r m i n ee w h a t information can be obtained from this dichroism.
D e n o t i n gg the total period as T = iuu + wA, w h e r e wu a n d wd are the u p
Ann XRMS study of disordered magnetic stripe domains in a-GdFe thin films 55 G2(0): :
mzmz
=w=w
ss
=lL=lL
Tgz{y)dy=Gz{0)Tgz{y)dy=Gz{0)
--
(3,6)
Forr a periodic system, Gz(qy) will have maxima at qy = Inn/j with n any in-teger.. At remanence however (mz> = 0, so that wu = wd = r/i and the only
non-zeroo terms of Gz(qy) are at qy = Inn/r with n odd, explaining the strong
oddd numbered diffraction peaks in Fig. 3.5. Since at remanence the period of mj andd mj is half that of mz, Gxx(qy) and Gyy(qy) are non-zero only at qy = 2nn/r
withh n even, which show up as the second-order peak in Fig. 3.5-a-b. Therefore, noo interference between the in-plane and out-of-plane components exists at re-manence.. This is not true in out-of-plane fields, when Gz(qy) contains terms
withh any integer n, which will interfere with Gxx and Gyy giving rise to the
dichroicc asymmetry observed in sample A in the ordered case. The absence of dichroismm in the disordered cases is not completely understood, although the mostt likely explanation is the large disorder.
Clearly,, the one-dimensional periodic model used so far is inappropriate forr the broad structure observed at high fields, which can be interpreted as the summ of form factors of all the uncorrelated reversed domains. To illustrate this, wee show the contrast functions of an isolated reversed domain of width w and theirr Fourier transforms in panels (A-B) and (C-D) of Fig. 3.8. Again, the inter-ferencee between the scattering from the out-of-plane domain and the in-plane magnetizationn components (Bloch wall and closure) surrounding it leads to a helicity-dependentt scattered intensity (panel E) and asymmetry ratio (panel F), whichh closely resemble the observed line shapes. The position of the minima of thee form factor and the zero-crossings of the asymmetry curves are positioned att multiples of (2n/w) and therefore are a simple means to determine the
aver-agee size of the reversed domains.
3.3.4.. Field dependence of total scattered intensity
Ass shown in the bottom panels of Fig. 3.7, the normalized total scattered intensityy is approximated very well by the function 1 - \mz |2, depicted by a full
linee for the two field directions (grey and black). Although this behaviour was alreadyy present in recent work from other groups [92,109], this agreement has neverr been microscopically justified. It can be explained by applying Parseval's
theoremm to the contrast function g:(y)'
~~ I \gz{y)\2dy= / | G2( ^ ) |2^ = |G2(0)|2+ / \Gz(qy)\2dqy.
11 J0 J - c o Jqy^Q
Thee term on the left side of the equation is the average of the squared reduced magnetizationn (w2), which equals unity if the in-plane magnetization of the domainn walls can be neglected. According to Eq. 3.6, |GZ(0) |2 = (mz)2 while the
integrall over qy ^ 0 is the integrated scattered intensity. Normalizing to the
maximumm scattering at remanence, we find:
Thee result implies that the scattered intensity can be used to measure the abso-lutee value of the magnetization for any sample containing mainly out-of-plane domains. .
Inn the bottom panels of Fig. 3.7 the intensity predicted by this expres-sionn is compared to the measured intensity at the diode. A quite good match is obtainedd and we have found the same agreement in many other samples. Es-pecially,, it is worth noting that the field dependence of the scattered intensity is thee same for the disordered and aligned cases of sample A, while the period for thee two cases is completely different.
3.3.5.. Interpretation of scattering curves
Kooyy and Enz [5] found that the magnetization loop of stripe systems hass a reversible part at low fields, which is characterized by reversible adjust-mentss of the relative width of the up and down domain without changes in the overalll pattern. This process continues until the reversed domains cannot be compressedd more, which happens when they reach a minimum width of about twoo domain wall widths. From that point on, stripes are eliminated, causing an increasee of the domain period and a loss of long range correlation in stripe po-sition.. The stripes then break up into segments [77], which gradually shorten to magneticc bubbles, located at favorable (i.e. low anisotropy) pinning sites, and aree eventually annihilated by progressively higher fields. On returning from saturation,, bubbles nucleate at much lower fields Bnuc and rapidly finger out
too fill the film surface. This difference in annihilation and creation leads to the appearancee of a distinctive hysteresis in the magnetization loop, characterized
Ann XRMS study of disordered magnetic stripe domains in fl-GdFe thin films 57 byy the triangular shape near magnetic saturation. The main difference between thee two samples considered here (Fig. 3.7-a) is the range of the reversible region andd the nucleation field, both larger in sample A.
Thiss description, developed for MOKE data, can also be applied to the presentt data. Starting with the aligned initial state of sample A (Fig. 3.7, left) the diffractionn curves show well-defined higher order peaks with an initial correla-tionn length of eight stripes. In the reversible region, the system mainly adapts too the field by increasing the up domain width wu with respect to w^ while
keepingg the lattice period more or less constant. This causes the appearance of evenn orders in the Fourier transform of mz, which mix with the scattering of the
in-planee components, producing appreciable circular dichroism as discussed above.. The angular width (not shown) of the diffraction spots stays constant withh field over the whole field range, indicating that the stripes remain parallel andd that we do not reach the bubble state, which should scatter isotropically. However,, above ~20 mT, the transverse correlation drops linearly with field (seee Fig. 3.7-d) to one stripe width, indicating the complete loss of correlation in thee position of the isolated stripes.
Thee scattering and asymmetry curves become gradually dominated by thee form-factor shape of the reversed domains with the intensity minima mov-ingg to higher q. The reversed domain width w, as estimated from the zero-crossingg of the dichroic asymmetry (Eq. 3.5), displayss a gradual decrease from thee half-period to 50 nm, with a trend to much smaller sizes. For an ideal stripe system,, the magnetization can be calculated from the reversed domain size and thee period. The result is compared with the reduced magnetization in Fig. 3.7-a, thee agreement being quite satisfactory for lower fields, proving the correctness off the average reversed domain size obtained this way. Clearly, in the bubble regimee this approach is no longer valid.
Byy fitting the measured asymmetry curves with a model as in Fig. 3.8-F,, and neglecting possible closure, we obtain a domain wall width of about 30 nm.. With data extending over a more extended range and by using linearly polarizedd light it should also be possible to get very precise information on the reversedd domain and the spin structure of the adjacent Bloch wall and closure magnetizationn [112].
3.3.6.. The effect of disorder
Althoughh the aligned stripes can be qualitatively described with the one dimensionall lattice model, the finite peak widths indicate the presence of dis-orderr in the stripe lattice. Hellwig et al. [63] applied a model [110] to describe moderatee disorder in the domain period of the aligned structure. An important conclusionn from this work is that disorder causes a peak shift towards lower q-values,, implying that the position of the first order peak tends to overestimate thee real domain period. However, actual fitting of the diffraction curves with thiss model turns out to be possible only in the most ordered case of sample A nearr remanence. From this fit we obtainn a Gaussian distribution in the domain periodd with a standard deviation equal to 5% of the domain period and a do-mainn wall width of 19 nm, which is considerably smaller than the 30 nm width off the walls surrounding the uncorrected reversed domains at high fields.
Fittingg with this one-dimensional model becomes impossible for the dis-orderedd case of sample A (dots, Fig. 3.7-a). The peak width at remanence corre-spondss to a correlation length of only two stripes. In applied fields, the diffrac-tionn rings broaden much faster than in the aligned case, and quickly merge in thee form factor structure of uncorrected stripes. The period, correlation length andd scattered intensity all show quasi-parabolic field dependence with only lit-tlee hysteresis. The period directly after nucleation is twice as big as at rema-nencee and always much larger than that of the aligned case due to imperfec-tionss in the lattice such as branching and end points, which prevent the sys-temm from reaching the equilibrium domain period. Also, due to disorder, the circularr dichroism becomes washed out over the whole scattering pattern and becomess too small to be observable. Over a limited range the form factor shape iss clear enough to extract a reversed domain size, but the magnetization calcu-latedd from this is too large, probably mainly coming from an underestimation off the period [63].
Samplee B is clearly much more disordered, showing only a single broad diffractionn peak over much of the magnetization loop. The average domain dis-tancee decreases rapidly down to a minimum at 9 mT and then increases to val-uess that fall beyond our minimum vector-transfer range. The correlation length presentss a maximum at 8 mT, but it barely deviates from unity. This behav-iorr might indicate that the magnetization arranges in a collection of domains
Ann XRMS study of disordered magnetic stripe domains in fl-GdFe thin films 59 thatt look somewhat in between a pure bubble and a stripe domain, as seen byy others [113]. Another indication of this is the wide field range at which a clearr reversed domain size can be observed. Nevertheless, the ratio of reversed domainn size to period again produces a quite acceptable estimation of the mag-netizationn (Fig. 3.7-a). Also, in contrast with sample A, the period shows large hysteresis,, implying that the domain pattern is far from equilibrium over the wholee loop.
3.3.7.. Domain period and magnetization
Thee Kooy and Enz model [5] predicts the field dependence of stripes inn perpendicular fields by treating the demagnetization energy in terms of a Fourierr expansion and then minimizing the total free energy to obtain the sam-plee magnetization and domain period at equilibrium. The parameters involved aree the saturation magnetization Ms, the uniaxial anisotropy constant KU/ the exchangee stiffness constant A and the film thickness t. They enter the expres-sionss in the form of two dimensionless parameters: the reduced anisotropy ma-teriall constant
QQ = Y- = - H S 2 ' ( 3-8 )
KKdd fi0Mf
thatt gives the ratio between the anisotropy energy and the demagnetization energy,, and the reduced characteristic length
thatt is a measure for the domain wall energy j w = 4yM.iCu. Here lc is the
char-acteristicc length. The model assumes that Q ;§> 1, that the film thickness is att least several times larger than the period at remanence and that the domain walll width is negligible compared to the period. The energy lowering due to the tiltingg of the magnetization close to the film surface is approximated by intro-ducingg an effective rotational permeability JA* — 1 + i [2]. Gehanno et al. [76] extendedd this model with a better approximation for the demagnetizing energy densityy in order to make it applicable to films with thickness smaller than the domainn period and Q ^ 1. They obtained analytical expressions for the depen-dencee of the reversible normalized magnetization mz — Mz/Ms and the domain periodd T on the reduced field h — H/}IQMS:
E E c c
0.055 0.10 0.15 0.20 0.25 0.30 0.00 0.05
hh h
0.10 0
Figuree 3.9: Comparison of the reduced reversible magnetization (top) and calculated domainn period (bottom) versus reduced field with the experimental data from sample AA (left) and B (right), with dots for the disordered and triangles for the ordered stripes. Calculatedd and experimental results are shown in grey and black respectively.
r(h)=^tr(h)=^tXX°°rr sec {^mz(h)y
Thee reduced magnetic susceptibility is:
X°r X°r
dmdmz zh=0 h=0 TT TT
exp(7rAcc +ƒ(ƒ))
(3.11) )
(3.12) ) wheree ƒ (r) is a slowly varying function of r — | ( 1 +
-7^)-Thee reduced susceptibility at remanence ^ can be obtained from the slopee of the out-of-plane magnetization loop (see Table 3.2). By using the ob-tainedd value and \i* as input parameters in Eq. 3.12, we determined the char-acteristicc length lc, the domain wall energy density j w = 4y/AK„ = 2K^lCr the domainn wall width Ö = nyjA/Ku, the exchange constant A = and the re-ducedd characteristic length Ac =lc/t (see Table 3.2). We find that in sample A
Ann XRMS study of disordered magnetic stripe domains in a-GdFe thin films 61 bothh Q and Ac are two times smaller than in sample B, reflecting the difference
inn saturation magnetization.
Thee reversible magnetization and period calculated from Eqs. 3.10-3.11 aree compared to the magnetization measured with VSM and X-ray derived pe-riodss in Fig. 3.9. In the case of sample A, the period, shown in panel (b), at remanencee is now only four times the thickness, and the correlation length is upp to eight periods in this case. Furthermore, it shows very little hysteresis. Indeed,, when magnetizing from the aligned situation, the magnetization and domainn period are predicted accurately by the model up to quite high fields. Onn the one hand, this is proof of the retention of a high degree of alignment up too the field where stripes become unstable with respect to dots. On the other hand,, this success proves the suitability of the model to describe systems with periodd a few times larger than the film thickness even for very thin films.
Thiss is in marked contrast to what was found in XRMS experiments on alignedd stripes in Co/Pt multilayered films, where the magnetization adapted too the field by annihilating stripes, leaving surrounding stripes at the same positionn [63], We ascribe the differences between the two studies to the fact thatt the Co/Pt films have a more poly crystalline structure, whereas the amor-phouss GdFe films studied here were flat and structurally homogeneous on the nanometerr scale. Clearly, in films with such a low level of defects, domain the-oryy is valid.
Ass was mentioned before, the disorder of the stripe lattice in sample A causess an increase of the remanent period with a factor 1.5 as compared to the alignedd case (Fig. 3.9-b). In our view this is purely caused by the branchings andd truncations of the disordered stripe system taking up more space, rather thann by domain wall pinning. Strictly speaking, the Gehanno description is ap-plicablee only to perfectly aligned one dimensional lattices while the disordered structuree is clearly two dimensional. Still, applying a brute force scaling of the calculatedd period for the aligned case by a factor 1.5 produces a remarkably goodd agreement with the data.
Forr sample B, the model gives a magnetization intermediate between the twoo branches of the hysteresis loop, but clearly is inadequate in describing the fieldd dependence of the average domain distance. The low saturation
magneti-zationn implies much lower dipolar interactions between the up and down do-mains.. This leads to a very large domain period compared to the film thickness
(To/t(To/t ~ 22) and a very poor correlation (see p. 309 in Ref. [2]). Furthermore, the
largee hysteresis in domain period shows that the dipolar interactions are weak comparedd to the residual domain pinning. We conclude that in this sample thee domain structure is not in the equilibrium state assumed in the continuum model. .
3.4.. Conclusions
Inn this chapter we explored the possibilities of soft X-ray resonant mag-neticc scattering in the study of nanometer-scale magnetic domain structures, us-ingg magnetically striped GdFe thin films as a testing ground for this promising technique.. We have worked out a description of the resonant X-ray scattering processs for circularly polarized light in the forward geometry for the case where thee linear dichroic term of the scattering cross section is important, as is the case att the rare earth M4 5 edges. Using this description, we then went on to explain thee origin of the different scattering features in terms of the out-of-plane and the in-planee magnetization components. Furthermore, via application of Parseval's theorem,, a general relation between the scattered intensity and the expectation valuee of the modulus of the magnetization could be derived, under the condi-tionn that the volume of domain walls and closure domains is small.
Ourr analysis shows that the amount of information that can ultimately bee extracted from the scattering data is limited by the degree of disorder. In the mostt ordered case -the aligned stripes after in-plane saturation- the scattered intensityy shows marked circular dichroic asymmetry. At low fields this asym-metryy is located at the position of the even diffraction orders, and is the result of interferencee of the scattering from the out-of-plane and in-plane magnetization componentss which involve the F ^ and F^ terms of the resonant scattering length.. At high fields, the scattering becomes dominated by the form factor of thee reversed domains. The magnitude of the dichroic asymmetry provides a directt way to determine the domain wall width. Moreover, the zero-crossings off the asymmetry are a direct measure of the reversed domain width which, in combinationn with the domain period obtained from the peak positions, yields a secondd independent measure of the absolute value of the magnetization. This valuee turns out to be in quite good agreement with the magnetization obtained
Ann XRMS study of disordered magnetic stripe domains in a-GdFe thin films 63 fromm VSM measurements.
Regardingg the field-induced evolution of the domains in the GdFe sys-tem,, we found that upon magnetizing from the aligned initial state, the domains remainn parallel to each other up to the field at which they collapse into bubbles. Inn this respect, the magnetization and domain period evolve as predicted by the domainn theory of Gehanno et al. [76].
Evenn in the disordered case, we have been able to follow the average domain period,, the correlation length and the reversed domain size from nucleation to saturation.. The different behavior of the two samples considered can be as-cribedd to their difference in saturation magnetization.
