Ferromagnetic resonance, magnetic susceptibility ...

JOURNAL OF APPLIED PHYSICS 109, 083926 (2011)

Ferromagnetic resonance, magnetic susceptibility, and transformation of domain structure in CoFeB film with growth induced anisotropy S. A. Manuilov,1 A. M. Grishin,1,a) and M. Munakata2 1

Department of Condensed Matter Physics, Royal Institute of Technology, SE-164 40 Stockholm-Kista, Sweden Energy Electronics Laboratory, Sojo University, Kumamoto 860-0082, Japan

2

(Received 28 September 2010; accepted 26 January 2011; published online 22 April 2011) Field dependence of magnetic susceptibility in nanocrystalline CoFeB film was studied in a wide frequency range from 500 kHz to 15 GHz. Anomalies of the susceptibility were detected exciting CoFeB film with a solenoidal coil, microwave strip line, in the tunable microwave cavity as well as employing magneto-optical domains imaging. Critical spin fluctuations in the form of “soft” modes were observed in a whole range of orientations of magnetic field perpendicular to the “easy” magnetic axis. A sequence of domain structure transformations was extensively examined in a “hard” direction in in-plane magnetic field reduced below the field of uniaxial anisotropy Hp ¼ 535 Oe. At first, uniformly magnetized state in CoFeB film transforms to stripe domains separated by low angle Neél domain walls (DWs) parallel to the “hard”-axis. Then, at critical field Hcr ¼ 232 Oe, Neél DWs gradually convert to the“easy”-axis oriented Bloch DWs loaded with vertical Bloch lines (VBLs). After field reversal at H ¼ Hcr, backward conversion of VBL-loaded Bloch DWs to Neél DWs results in instantaneous energy release and sharp anomaly of magnetic susceptibility. Appearance of critical spin fluctuations accomplishes domains transformation to the uniformly C 2011 American Institute of Physics. [doi:10.1063/1.3559732] magnetized state at H ¼ 535 Oe. V I. INTRODUCTION

Among a large number of soft magnetic amorphous and nanocrystalline compositions synthesized so far, Fe-Co-Nibased amorphous alloys possess superior soft magnetic properties: high saturation magnetization 4pMs, high magnetic susceptibility v, and low coercive field Hc. These materials in the form of rapidly quenched transition-metal-metalloid amorphous alloys (metallic glasses) have been extensively studied due to various potential applications.1,2 The role of transition-metal constituents has been infallibly established. E.g., Co substitution of Fe increases 4pMs whereas duties of metalloids like B, P, or Si still need profound elucidation. Having smaller atomic radii they promote achievement of high packing density and low free volume. Also, in combination with transition metals they have negative free energy of formation, impeding crystal nucleation. Short range exchange interaction between magnetic atoms provides spontaneous magnetization meanwhile a long range structural disorder renders magnetosoft properties making amorphous material to behave as an isotropic ferromagnet. Recent advent of magnetic tunnel junctions revived the interest to amorphous magnets. Record-high room temperature tunneling magnetoresistance observed in CoFeB magnetic tunnel junctions with AlOx3 and MgO4 barriers as well as enhanced tunneling spin polarization in CoFeB compared to pure CoFe5 have a great potential for various spintronic devices. Besides high saturation magnetization magnetic anisotropy is often required. E.g., to operate as a magnetoresistive sensor or spin-torque oscillator, spinvalve devices manipulate magnetic anisotropy in pinned and free CoFeB layers. Since 1955,6 the application of magnetic field was a)

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employed during films synthesis/annealing to introduce inplane uniaxial magnetic anisotropy. Typically it ranges 20 to 50 Oe. Very strong in-plane anisotropy, up to 700 Oe, was achieved in CoFeB films sputtered onto the rotating glass substrate with the magnetic “hard”-axis appeared to be along the direction of substrate motion.7 In the “easy”-axis direction, these films show relatively high coercive field Hc 10 – 100 Oe caused by domain walls pinning, as was revealed by magneto-optical Kerr microscopy.8 We used amorphous CoFeB-SiO2 film with the field of uniaxial anisotropy Hp ¼ 535 Oe as a sample material to study spin dynamics and magnetic susceptibility in a wide frequency range from 500 kHz to 15 GHz. The paper is organized in the following way. In Sec. II, we present the experimental details on the films processing, nanocrystalline film structure by x-ray diffraction, and static magnetic film properties. General formulas for ferromagnetic resonance (FMR) frequencies in various orientations of external magnetic field are derived in Sec. III. In Sec. IV, FMR spectroscopy reveals “soft” magnetic modes and earns parameters of magnetic anisotropy. Measurements of magnetic susceptibility at various orientations of external magnetic field are carried out in solenoidal coil, resonant cavity, and with a microwave strip line in Sec. V. In Sec. VI we present theory of Bloch and Neél domain walls transformation. Magnetooptical images of domains are collected in Sec. VII. In Sec. VIII, we compare experimental observations with theoretical predictions and summarize results in Sec. IX. II. SAMPLES PREPARATION

The details of hetero-amorphous films processing have been published elsewhere.7 In brief, composite (Co0.27Fe0.62 B0.11)1-x–(SiO2)x films were deposited by synchronous

109, 083926-1

C 2011 American Institute of Physics V

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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Manuilov, Grishin, and Munakata

triple-RF magnetron sputtering from SiO2, Fe, and Co0.666 Fe0.074B0.26 targets onto the Matsunami #7050 glass substrate mounted on the rotating holder. The direction of the substrate motion hereafter will be referred to as the “hard” direction since it appeared to be parallel to the magnetic “hard”-axis. Comparative experiments were performed for two samples: 4 lm thick CoFeB-reach and 0.48 lm thick CoFeB-poor films with x ¼ 0.235 and 0.34, respectively. Amorphous film structure was verified by x-ray diffraction (XRD). Figure 1 shows for comparison XRD patterns of blank glass substrate and two hetero-amorphous CoFeB– SiO2 films. CoFe (110) Bragg reflection is fitted with the Lorentzian contour that has a full-width at half of the maximum as wide as 6.6 degrees. Using Scherrer formula we estimate the perpendicular to the film surface grain size to be around 1.3 nm. Hereinafter, all the figures present results for 4 lm thick (CoFeB)1x–(SiO2)x film with x ¼ 0.235. The difference in the properties of CoFeB-poor film with x ¼ 0.34 will be specially noted in Sec. VIII. Figure 2 shows magnetic hysteresis loops measured with a vibrating sample magnetometer (VSM) at room temperature in two orientations of in-plane magnetic field. Loops clearly demonstrate growth-induced in-plane magnetic anisotropy. The saturation magnetization is 4pMs ¼ 10.5 kGs and coercive field in the “easy”-axis direction is Hc ¼ 50 Oe. Temperature dependence of coercive field conforms to a superferromagnetic state.9 In the “hard” direction magnetization saturates at Hp ¼ 535 Oe. Later in Sec. IV, using FMR spectroscopy we identify parameter Hp as in-plane uniaxial anisotropy field. Besides growth-induced magnetic anisotropy these films demonstrate also anisotropy of electric transport properties. CoFeB-reach film showed strong in-plane anisotropic magnetoresistance superimposed upon isotropic giant magnetoresistance (GMR), whereas CoFeB-poor sample possesses GMR effect only.10

J. Appl. Phys. 109, 083926 (2011)

FIG. 2. (Color online) VSM recorded hysteresis loops in (CoFeB)0.765– (SiO2)0.235 film. In-plane magnetic field is parallel to the “hard”- (square symbols h) and “easy”- (circular symbols ) magnetic axes. Coercive field Hc ¼ 50 Oe and saturation field Hp ¼ 535 Oe are shown for hysteresis loops traced, respectively, in “easy” and “hard” directions. Letters in square boxes notify the positions where magneto-optical images in Fig. 13 gallery were taken. Inset shows the orientation of spherical coordinate system in relation to the “hard”- and “easy”-axes.

III. GENERAL EQUATIONS

To characterize magnetic anisotropy we examined properties of CoFeB-SiO2 films in a wide range of frequencies. Films have high saturation magnetization 4pMs ¼ 8 12 kGs. Therefore, to obtain reliable characteristics of magnetic anisotropy from spectroscopic measurements it is vital to account misalignment of magnetization M and magnetic field H vectors that occurs in relatively weak magnetic fields. Following Stoner-Wohlfarth theory for the coherent M rotation, we start with the expression for the total free energy density for uniformly magnetized film: F ¼ HMs ½sin hH sin hM cosðuH uM Þ þ cos hH cos hM þ 2pMs2 cos2 hM Kp sin2 hM sin2 uM :

FIG. 1. (Color online) X-ray diffraction H-2H scans of blank Matsunami #7050 glass substrate and two hetero-amorphous (Co0.27Fe0.62B0.11)1-x– (SiO2)x films in Cu Ka radiation. Glass and (CoFeB)0.66–(SiO2)0.34 film patterns are offset, respectively, by þ 5 and þ 3 cps for clarity. Lorentzian contour with a full-width at half of the maximum of 6.6 deg fits a broad CoFeB (110) Bragg reflection.

(1)

It includes Zeeman energy as the first term, the second term is the stray (demagnetizing) field energy, and the last term is the energy of in-plane uniaxial anisotropy Kp. The orientation of M and H vectors is defined by spherical coordinates with the corresponding subscripts M and H: polar angle h and azimuth u angle that is reckoned from the “hard”-axis (see inset to Fig. 2). The equilibrium orientation of the magnetization vector M(H) must be determined from the condition of the minimum of free energy: oF HMs sin hH sin hM sinðuH uM Þ ouM 1 Hp Ms sin2 hM sin 2uM ¼ 0; (2.1) 2 oF HMs ½cos hH sin hM sin hH cos hM cosðuH uM Þ ohM 1 2pMs2 sin 2hM Hp Ms sin 2hM sin2 uM ¼ 0: (2.2) 2 Here we introduced the field of in-plane uniaxial magnetic anisotropy Hp ¼ 2Kp =Ms .


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In saturated ferromagnet the magnetization vector M experiences precession with the characteristic FMR frequency xres derived by Suhl and Smit:11

xres c

2

" 2 2 # 1 o2 F o2 F o F : ¼ 2 2 ohM ouM Ms sin hM oh2M o2M

(3)

h is the gyromagnetic ratio expressed through Here c ¼ glB = the g-factor, Bohr magneton lB and Planck constant h. Twice differentiating free energy F from Eq. (1) and using the equilibrium conditions from Eqs. (2.1) and (2.2) we present the expression for FMR frequency as follows: xres 2 cos hH 4pMs Hp cos2 uM ¼ cH cos hM H H Hp cos hH 4pMs cos2 hM cos2 hM sin2 uM cos hM H H 2 Hp cos hM sin 2uM : (4) 2H To obtain numerical solution of three coupled equations Eqs. (4), (2.1), and (2.2) we used MATLAB program. In four special geometries, it turns out to express FMR spectra with the reduced formulas: •

FIG. 3. (Color online) Magnetic “phase diagram” for the film with in-plane uniaxial anisotropy (4pMs ¼ 11.5 kGs, Hp ¼ 535 Oe). External magnetic field H is tilted at angle hH from the normal to the film plane toward the “hard”-axis. The area above the boundary curve Hb ðhH Þ corresponds to uniformly magnetized state whereas the shaded area below Hb ðhH Þ is the domain state. Triangular symbols show the positions of “soft” modes detected at frequency of 465 MHz. They marked with 14 small triangular symbols grouped horizontally in Fig. 4.

•

In perpendicular magnetic field (hH ¼ 0) it is always uM ¼ p=2 and there are two ranges of magnetic field where the polar angle hM is defined as follows: H • if H 4pMs þ Hp , then cos hM ¼ 4pMs þHp and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u 2 # u H t xres ¼ c Hp 4pMs þ Hp 1 ; 4pMs þ Hp

( Hb ðhH Þ ¼ Hp

•

if H 4pMs þ Hp , then hM 0 and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi xres ¼ c ðH 4pMs Þ H 4pMs Hp :

Hp 1 4pMs þ Hp

2 #)1=2 ;

magnetization vector M becomes orthogonal to the “easy”-axis ðuM ¼ 0Þ and

•

(5.2)

H Hb ðhH Þ ¼ Hp f 1 cos2 hH ½1 ð4pMs pþHp Þ2 g1=2 H

xres

1=2 sin hH ¼c H Hp ½H cosðhH hM Þ sin hM 4pMs cos 2hM 1=2 :

•

1=2

•

(5.5)

Magnetic field H rotates being orthogonal to the “hard”axis (uH ¼ p=2), then always uM ¼ p=2, vector M approaches vector H toward the normal to the film plane ðH=4pMs þ Hp Þ sinðhM hH Þ ¼ sin hM cos hM , and xres

4pMs þHp vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 2 " 2 # u Hp H H ; t1 þcos2 hH 1 4pMs þHp Hp Hp (5.3)

if

(above the shaded area in Fig. 3), then vector M, being orthogonal to the “easy”-axis ðuM ¼ 0Þ, continues to approach vector H toward the normal to the film plane: ðH=4pMs Þ sinðhM hH Þ ¼ sin hM cos hM and

and

1 cos hH

xres ! 0;

Magnetic field H rotates being always perpendicular to the “easy”-axis (uH ¼ 0): • if H Hp f1 cos2 hH ½1 ðHp =ð4pMs þ Hp ÞÞ2 g1=2 (shaded area in Fig. 3), then vector M deviates out-offilm plane toward “hard”-axis: H cos hH ; cos hM ¼ 4pMs þ Hp H sin hH sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos uM ¼ 2ffi Hp H 1 cos2 hH 4pMs þ Hp

xres ¼ cHp1=2

" 2

(5.4)

(5.1) •

at the boundary curve defined by equation

1=2 sin hH H cosðhH hM Þ ¼c H þ Hp sin hM

1=2 : ð4pMs þ Hp Þ cos 2hM

(5.6)

Magnetic field rotates in film plane (hH ¼ p=2), then always hM ¼ p=2, vector M approaches vector


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H: ð2H=Hp Þ sinðuM uH Þ ¼ sin 2uM and

1=2 xres ¼ c H cosðuH uM Þ þ 4pMs þ Hp sin2 uM

1=2 H cosðuH uM Þ Hp cos 2uM : (5.7) These equations we used to plot resonance condition xres versus H or vice versa Hres versus x for arbitrarily oriented magnetic field H (for various hH and uH ). IV. FMR SPECTROSCOPY

The dependence of FMR frequency xres on the external magnetic field H is presented in Fig. 4 for five different geometries: vector H is perpendicular to the film plane hH ¼ 0, H is in plane along the “easy”- (hH ¼ 90 ; uH ¼ 90 ) and “hard”-axis (hH ¼ 90 ; uH ¼ 0) directions as well as magnetic field was tilted toward “easy”- (hH ¼ 27 ; uH ¼ 90 ) and “hard”-axis (hH ¼ 27 ; uH ¼ 0). Solid lines depict theoretical curves calculated with formulas (5.1)–(5.7). When magnetic field is perpendicular to the “easy”-axis, FMR spectra always demonstrate occurrence of the “soft” modes. Corresponding xres versus H spectral curves in Fig. 4 create specific “beaks” where FMR frequency xres goes to zero. As shown in Figs. 3 and 4, “soft” mode position Hb(hH ) depends on the orientation of vector H. Field Hb(hH ) is a characteristic field of reorientation magnetic phase transition. Here uniformly magnetized state becomes unstable regarding the nucleation of magnetic domains. These are two equivalent domains with in-plane projection of magnetization M parallel and antiparallel to the “hard”-axis. Experimental xres versus H data are superimposed in Fig. 4 over theoretical spectra. They were obtained by several methods in Secs. IV and V; thus, we show them with different symbols. The experimental points rest themselves along the theoretical curves in a reasonably plain manner.

FIG. 4. (Color online) Five FMR spectra in CoFeB film for different orientations of external magnetic field H: perpendicular to the film (hH ¼ 0 ), tilted from the film normal at angle hH ¼ 27 toward the “easy”(uH ¼ 90 ) and “hard”- (uH ¼ 0 ) axes, as well as two parallel to the film plane geometries: H is directed along the “hard”- (hH ¼ 90 ; uH ¼ 0 ) and “easy”- (hH ¼ 90 ; uH ¼ 90 ) axes. Solid lines are computed using formulas in Eqs. (5.1)–(5.7). Experimental data collected with a tunable resonant cavity and solenoidal coil are presented with square h and circular symbols , respectively. The horizontal group of 14 small triangular symbols depicts positions of the “soft” modes excited in CoFeB film in solenoidal coil at 465 MHz.

J. Appl. Phys. 109, 083926 (2011)

Fourteen small triangular symbols grouped horizontally in Fig. 4 at frequency of 465 MHz depict positions of “soft” modes detected at 14 different orientations of magnetic field tilted from the normal n of the film plane toward the “hard”axis (hH var; uH ¼ 0). These experimental points are shown also in Fig. 3. They nicely fit the boundary curve H ¼ Hb(hH ) up to the field of 4 kOe. At stronger magnetic field (higher resonance frequencies) spin excitation with a solenoidal coil (see experiment details in Sec. V) becomes less effective. Signal of differential susceptibility weakens hence the measurements precision degrades. Magnetic anisotropy in CoFeB films was examined by angular resolved FMR spectroscopy carried out in the rectangular microwave cavity in TE1012 mode at frequency x/2p ¼ 14.8 GHz (Ku band, k/4 ¼ 5.1 mm). CoFeB samples were shaped to 5 mm diameter disk and mounted 3.5 mm away from the cavity end. Use of 2-circle goniometer holder enables in-plane sample rotation (azimuthal uH - scan) and tilting of magnetic field out of film plane (polar hH - scan). Figure 5 exemplifies typical differential FMR absorption lines. There DH defines peak-to-peak FMR linewidths in out-of-plane and two in-plane geometries of magnetic field orientation. As always, the narrowest FMR linewidth is observed when magnetic field is oriented in the “hard” directions: along in-plane “hard”-axis and perpendicular to the film plane that is a global “hard” magnetic direction. This undeniably indicates the broadening of FMR lines is caused by various magnetic nonhomogeneities: shape, size, and distribution of CoFe nanocrystals. FMR recorded in the resonant cavity using TE1012 mode at 14.8 GHz is marked with three square bold symbols in Fig. 4. Figure 6 shows hH dependencies of FMR field Hres when magnetic field was rotated (hH ¼ var) within two planes: being perpendicular to the “easy”- (uH ¼ 0) and “hard”-axis (uH ¼ p=2) direction. Experimental data shown with symbols are fitted to theoretical curves from

FIG. 5. (Color online) Differential FMR absorption @v00 /@H in CoFeB-SiO2 film recorded at frequency x/2p ¼ 14.8 GHz in three geometries: perpendicular magnetic field (right frame) and in-plane magnetic field (left frame) parallel to the “easy”- and “hard”-axes. DH shows peak-to-peak FMR linewidths.


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FIG. 6. (Color online) Polar angle dependence of the resonance field Hres(hH) recorded at x/2p ¼ 14.8 GHz rotating magnetic field within two planes: being perpendicular to the “easy”- (uH ¼ 0 , square symbols h) and “hard”- (uH ¼ 90 , circular symbols *) axes. Solid lines are theoretical curves computed with Eqs. (5.5), (5.6), and fitting parameters that are shown as the shorthand. Inset shows in-plane angular dependence of the resonance field Hres (hH ¼ 90 ; uH ) at x/2p ¼ 14.8 GHz. Symbols * stand for experimental data and solid line depicts theoretical curve calculated with Eq. (5.7) using the same fitting parameters that those revealed from the polar angle hH-scans.

Eqs. (5.5)–(5.6). Within a 5% error we obtained g ¼ 2.09, 4pMs ¼ 11.5 kGs, Hp ¼ 535 Oe, and Kp ¼ 2.24 105 erg/cm3 for the energy of in-plane uniaxial anisotropy. About a 9% difference of 4pMs obtained from FMR and VSM measurements can be relied upon the standard 10% inaccuracy of VSM data (determined mainly by uncertainty of the film area estimation) or negative out-of-plane magnetic anisotropy Heff 1 kOe. The latter might be induced during the growth of 4 lm thick CoFeB film. As an additional check, we recorded angular dependence of the resonance field Hres when magnetic field was rotated in film plane (hH ¼ p=2; uH ¼ var). It is shown with symbols in inset to Fig. 6. A small “wobbling” of the sample holder distorts the 180 periodicity of Hres (uH ) function. High saturation magnetization of CoFeB sample enhances this effect. Solid line depicts theoretical curve calculated with Eq. (5.7) using the same fitting parameters as we obtained from the polar hH-scan in the mainframe of Fig. 6. V. MAGNETIC SUSCEPTIBILITY

Transformation of domain structure in CoFeB films was studied recording magnetic susceptibility v by several methods in a wide range from hundreds kHz to microwave frequencies. Hereafter, in all Figs. 7–10 ascending and/or descending direction of the sweep of magnetic field H are indicated by arrows near the experimentally recorded curve or within the title of magnetic field abscise axis. A. Solenoidal coil

We found excitation of CoFeB films with a rectangular solenoidal coil to be an effective method to characterize field dependence of magnetic susceptibility up to 6 GHz. The coil

FIG. 7. (Color online) In-plane RF field hRF is parallel to the “hard”-axis. (a) Differential magnetic susceptibility recorded in solenoidal coil sweeping magnetic field H parallel to the “easy”-axis. Arrows mark ascending and descending branches of magnetization loop. (b) Field dependence of S11 parameter measured for TE104 and TE109 longitudinal modes in rectangular microwave cavity. Descending branch of magnetization loop at H || “easy”axis is displayed and marked by the arrow at the title of abscissa axis. Snowflake symbols * notify positions of peaks of ferromagnetic resonance. Encircled irreversible jumps display the instantaneous reversal of the magnetization at Hc ¼ 50 Oe.

with a size of 8 6 2 mm3 has an empty inductance L ¼ 1 mH and resistance R ¼ 800 mX. Circular shaped 5 mm diameter CoFeB disk sample was placed inside the coil. HP8656B signal generator feds the coil while broad band diode detector terminates the RF output. External magnetic field was modulated by 160 Hz signal of 1-2 Oe amplitude and rectified RF signal from the detector was lock-in amplified. Orientation of RF-field hRF, that is always parallel to the film surface, can be arbitrarily varied in relation to the “easy”-axis and external magnetic field H directions. Figures 7(a) and 8 present traces of the differential susceptibility @v/@H recorded at ascending and descending branches of magnetization loop. When external field H is parallel to the “easy”-axis, differential susceptibility in Fig. 7(a) looks very similar to the second derivative of the magnetization loop in Fig. 2, i.e., @v/@H ¼ @ 2M/@H2, except the coercive field strength. M-H hysteresis loop in Fig. 2 was recorded in VSM in quasistatic regime whereas the dependence @v/@H versus H was recorded sweeping magnetic field with the rate of 3–6 Oe/s. Relaxation of magnetic domain structure causes this difference in coercive fields. Previously, measuring the temperature dependence of the coercive field Hc in CoFeB-SiO2


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FIG. 8. (Color online) Differential magnetic susceptibility recorded in solenoidal coil sweeping magnetic field H parallel to the “hard”-axis. (a) RF field is parallel to the external field and the “hard”-axis hRF || H. (b) RF field is perpendicular to the external field and the “hard”-axis hRF \ H. Arrows show sweep direction of magnetic field. Ovals mark the irreversible peaks of susceptibility occurred at the critical field Hcr ¼ 6 232 Oe.

films, it was found that below the blocking temperature Hc is strongly dependent on a measurement time. Kinetic approach was developed to model the relaxation of the magnetization of the ensemble of interacting Stoner-Wohlfarth nanoparticles. Numerical solution of the kinetic equation shows that

FIG. 9. (Color online) Field dependence of S11 parameter in microwave cavity with RF field perpendicular to the film plane. Descending branch of M-H loop is shown. Magnetic field sweep parallel to the “hard”-axis reveals three different anomalies: FMR marked with snow-flake symbols *, reversible absorption peaks at Hp ¼ 6 535 Oe, and irreversible instanteneous peak of microwaves reflection at Hcr ¼ 232 Oe.

J. Appl. Phys. 109, 083926 (2011)

FIG. 10. (Color online) 2D plots of frequency and field dependence of the intensity of S11 parameter. Descending and ascending branches of M-H loop marked, respectively, with / and ! arrows in the abscissa titles. Dense color bent bands marked with snow-flake symbols * display FMR spectra in magnetic field parallel to the “easy”- (a) and “hard”-axes (b). Bright vertical lines at Hc ¼ 6 50 Oe in a-plot exhibit very low magnetic susceptibility at the flattop of square magnetization loop at H || “easy”- axis (see Fig. 2). Dense color vertical lines at Hp ¼ 6 535 Oe in b-plot display enhanced microwave absorption by critical fluctuations at the lability boundary of uniformly magnetized state. Very narrow dense and following bright vertical lines occurred after field reversal at Hcr ¼ 6 232 Oe correspond to Bloch-toNeél DWs transformations.

field oriented ferromagnetic nanoparticles spontaneously order themselves into the superferromagnetic state manifested by increased coercivity. When temperature decreases below the blocking temperature, the interparticle interaction grows. It leads to the crossover from the fast collective reversal to the slow highly coercive relaxation of the magnetization.9 In collinear geometry of RF and external field hRF || H || “easy”-axis, the anomaly of the susceptibility has the same shape as in Fig. 7(a) at hRF \ H though it is hardly visible on the noise background. Field dependence of susceptibility becomes much more complicated if in-plane magnetic field is oriented parallel to the “hard”-axis. As seen in Figs. 8(a) and 8(b), RF response measured by solenoidal coil is strong in both cases: hRF \ H and hRF || H. Differential susceptibility @v/@H as a function of magnetic field has four anomalies at H ¼ 6 535 Oe and 6 232 Oe. Their positions are frequency independent within the whole explored frequency range 500 kHz to 12 GHz (see below in Fig. 10(b) the measurements performed with a microwave strip line). Ascending and descending @v/@H versus H curves consist of two very different contributions. The first is completely reversible (nonhysteretic) component which is the odd function of magnetic field, i.e., @v(H)/@H ¼ @v(H)/@H. This component consists of two peaks that occur at H ¼ 6 535 Oe. Complete reversibility signifies exact reproducibility of these peaks if a minor loop


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is swept with a small field amplitude around H ¼ 6 535 Oe. Also, these peaks possess odd H-function symmetry both within the same ascending/descending branch as well as a cross-branch symmetry: @v(H)/@H |ascending ¼ @v(H)/ @H |descending. Position of these peaks precisely coincides with a magnitude of in-plane uniaxial field Hp ¼ 535 Oe obtained using FMR spectroscopy. The second component, that we marked with ovals in Figs. 8(a) and 8(b), exhibits itself by two peaks at critical field Hcr ¼ 6 232 Oe. The shape of these peaks is different at Hcr ¼ þ 232 Oe and – 232 Oe within the same ascending/ descending branch of hysteresis loop though they have a cross-branch symmetry: @v(H)/@H |ascending ¼ @v(H)/ @H |descending. Small amplitude sweep of magnetic field around Hcr ¼ 6 232 Oe results in opening of minor hysteresis loop. Such behavior is specific for irreversible transformations of magnetic domain structure that we discuss in details in Sec. VIII. B. Resonant cavity

To explore field dependence of magnetic susceptibility at higher frequencies we examined characteristics of TE102TE109 longitudinal modes in tunable rectangular cavity. Absorption and resonance frequency in cavity were measured recording S11-parameter with the vector network analyzer HP8722D in frequency range 6.7 to 10 GHz. This method enables characterization of the sample in all possible mutual orientations of external magnetic field H, RF field hRF and the axis of magnetic anisotropy. To enhance signalto-noise ratio statistical sampling was acquired that significantly increases measurement time. Therefore, cavity-related scans in Figs. 7(b) and 9 were performed sweeping magnetic field with the average rate of 1 Oe/s in descending direction only. The latter is indicated by an arrow in the title of abscissa axis. Grid lines opposite to zero ordinate serve as a reference for S11-parameter. Resonant cavity tests reveal two types of anomalies of v(H) dependence: frequency dependent and independent. Frequency dependent singularities of S11-parameter are positioned in magnetic field strengths Hres(x) that correspond to FMR. They are marked with snow-flake symbols * in Figs. 7(b) and 9. Also, eight ferromagnetic resonances of TE102-TE109 modes are shown with a square symbols in Fig. 4. Positions of frequency independent anomalies coincide with coercive field Hc ¼ 50 Oe and uniaxial field Hp ¼ 535 Oe when magnetic field is parallel to the “easy”- [Fig. 7(b)] and “hard”-axis (Fig. 9), respectively. In both cases the orthogonal orientation of RF field was chosen hRF \ H. C. Broadband microwave spectroscopy

It is interesting to infallibly discriminate domain contribution to magnetic susceptibility from the uniform ferromagnetic resonance. For this purpose we employed broad frequency band excitation of magnetic film with a microstrip. CoFeB sample was put in close contact to the short ended open air 6 mm long copper strip line. Up to 12.5 GHz it induces a relatively uniform RF field inside the film. Strip line was fed with HP8722D vector network analyzer and

J. Appl. Phys. 109, 083926 (2011)

S11-parameter was recorded in 2–12 GHz frequency range as magnetic field H was swept. To distinguish variations of film susceptibility, the calibration of the sample loaded strip line has been performed in in-plane magnetic field H ¼ 3.4 kOe. This field guarantees the uniform film magnetization. The intensity of S11-parameter was plotted as a two dimensional (2D) frequency and magnetic field strength function S11(x, H). Figure 10(a) presents 2D S11(x, H) plot when magnetic field is oriented parallel to the “easy”-axis. Dark rising bands marked with a white dash line and snow-flake symbol * show strong FMR microwave absorption. They visualize the xres versus H spectrum for hH ¼ 90 ; uH ¼ 90 geometry from Fig. 4. FMR in the uniformly magnetized film is interrupted by instantaneous reversal of magnetization at coercive field |H| ¼ Hc indicated by a bright vertical line. 2D S11(x, H) plots recorded in descending and ascending magnetic field parallel to the “hard”-axis are collected in Fig. 10(b). Dark absorption bands bent upward and marked with a snow-flake symbol * start from the uniaxial field Hp ¼ 6535 Oe. They accord with the FMR spectrum xres versus H for hH ¼ 90 ; uH ¼ 0 geometry from Fig. 4. Bright vertical bands in a weak magnetic field are present in a whole frequency range. They have a sharp boundary when magnetic field after reversal reaches a critical value Hcr ¼ 232 Oe. Opposite boundary is blurred. It indicates gradual conversion of domain walls topology. The detail discussion follows in Sec. VIII. VI. DOMAIN STRUCTURE

Observed anomalies of magnetic susceptibility we rely upon the transformation of domain walls (DWs) structure. To model DWs properties we follow quantitative approach suggested by Neél12 and generalized for external magnetic field case by Middelhoek.13 Neél simplified model suggests that DW, that connects two regions of nearly uniform magnetization (domains), pierces a whole film in the form of infinite elliptic cylinder with the main axes: D (DW width) and t (film thickness). Magnetization vector M gradually turns in the wall, while its direction inside adjacent domains is defined by “bulk” equilibrium conditions from Eqs. (2.1)(2.2). In magnetic field parallel to the “hard”-axis H k Ox (hH ¼ p=2; uH ¼ 0) there are two equilibrium orientations of vector M: uM ¼ 6 uo where uo ¼ cos1 h, h ¼ H/Hp, and hM ¼ p=2. Between these two directions vector M could rotate in the film plane (Neél DW) or flip out of the film plane in such a way that perpendicular to the DW plane component of the magnetization Mn is kept constant (Bloch DW). Here axis On is perpendicular to the DW plane. Besides Zeeman and in-plane uniaxial anisotropy terms from Eq. (1), DW density energy contains also exchange energy ðA=M2 Þð@M=@nÞ2 with exchange constant A and magnetostatic energy 12MHs. Rigorous calculation of spatially inhomogeneous stray field Hs is a fairly complicated problem. Its solution was found in recent decades using micromagnetic computations. To overcome this obstacle Neél invented barely sufficient model approximating interior DW magnetization by its average value hence using a well


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^ known formula Hs¼ 4p NM for uniformly magnetized ^ ellipsoid with the tensor of demagnetizing coefficients N. Middelhoek13 generalized the model and successfully caught the main features of thickness and field dependent transformation of DW from the Neél-type in very thin to the Blochtype in thicker films. Following Middelhoek, Torok et al.14 corrected him showing the Neél-to-Bloch DWs transformation occurs gradually through the superposition of Neél and Bloch components in the wall. However, the main Middelhoek’s model assumption of the uniform magnetization inside the DW leads to the field independent DW width. This conclusion contradicts observations made by Lorentz electron microscopy though they confirmed a general sequence of DWs transformations.14 We modernize Neél-Middelhoek model suggesting that ^ ðnÞ follows a spatial a stray field in DW H s ðnÞ ¼ 4pNM distribution of the magnetization. This assumption, known as a Winter approximation,15 wholly corresponds to the finite element analysis of magnetostatic problem when arbitrarily shaped inhomogeneously magnetized sample is considered as a layered stack of uniformly magnetized thin sheets. As a result, the magnetostatic energy 12 M ðnÞH s ðnÞ ¼ 2p2 ^ ðnÞ contains spatially varrying M ðnÞ instead of avM ðnÞNM erage magnetization. Symmetry dictates an existence of three types of simple domain walls in magnetic field parallel to the “hard”- axis H k Ox: Neél and Bloch DWs parallel to the “easy”-axis and Neél DW along the “hard”- axis. A. Neél domain walls

In Neél-type DW the magnetization vector M lies always in film plane hM ¼ p=2 whereas azimuth angle uM experiences rotation from uo to þ uo: uM ¼ 2uo n=D; D=2 n D=2:

(6)

Neél DW free energy density can be expressed as follows: dFN ¼ HMs ðcos uM cos uo Þ Kp sin2 uM sin2 uo cos2 uM 2uo 2 t 2 : (7) þ2pMs þA D t þ D sin2 uM Here Nnn ¼ t=ðt þ DÞ is the demagnetizing factor for infinite elliptic cylinder in n-axis direction.16 Upper cos2 uM and lower sin2 uM functions in the braces stand, respectively, for DW parallel to the “easy”- and “hard”-axis direction. DW width is much smaller than film thickness D t. Therefore, calculating total DW energy per unit area FN[erg/cm2] we integrate dFN from Eq. (7) considering the elliptic cylinder as a flat parallel plate: ð 2 1 D þuo dFN dn¼ dFN duM ¼4A cos1 h 2uo uo D D=2 ( pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi!) Hp h 1h2 3h 1h2 þpMs2 16 þ D: 1þ2h2 1 cos h 4pMs cos1 h

FN ¼

ð D=2

(8)

Upper “plus” and lower “minus” signs in the braces stand for DW parallel to the “easy”- and “hard”-axis direction, correspondingly. Minimizing DW energy FN with respect to D, we obtain the width and energy of Neél DW: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DN ðhÞ ¼ 2 A=pMs2 cos1 h ( pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi!)1=2 2 h 1h2 Hp 2 3h 1h 16 þ 1þ2h ; cos1 h 4pMs cos1 h qffiffiffiffiffiffiffiffiffiffiffiffi FN ðhÞ ¼ 4 pAMs2 cos1 h ( pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi!)þ1=2 h 1h2 Hp 3h 1h2 2 þ 1þ2h 16 cos1 h 4pMs cos1 h (9) In Fig. 11, we plot the energy FN and the width DN of Neél DW as functions of in-plane magnetic field h ¼ H/Hp using previously obtained (CoFeB)0.765–(SiO2)0.235 film parameters: 4pMs ¼ 11.5 kGs, Hp ¼ 535 Oe and settling A ¼ 2.8 10-6 erg/cm.17 FN and DN plots shown with solid lines correspond to descending branch of magnetization curve when magnetic field parallel to the “hard”-axis H k Ox (positive h) decreases below Hp ¼ 535 Oe. At H ¼ Hp (h ¼ 1) in-plane magnetization vector M starts to deviate from the “hard”-axis Ox in opposite (clockwise and anticlockwise) directions in adjacent domains. As a result, the pffiffiffiffiffiffiffiffiffiffiffi nucleation of a low-angle 2uo ¼ 23=2 1 h ! 0 boundary between two equivalent magnetic domains occurs. Nearby the saturation field, the energy of the DW that is parallel to the “hard”-axis [lower sign “minus” in Eqs. (9)] decreases pffiffiffiffiffiffiffiffiffiffiffi proportionally to ð1 hÞ. This is faster compared to 1 h in DWk“easy”-axis (upper sign “plus”). DWk“hard”-axis has lower energy since the same 2uo angle rotation occurs at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi finite width D ¼ 6A=pMs2 in contrast to very narrow width pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi D ¼ 2 A=pMs2 1 h ! 0 of “easy”-axis oriented DW. Anisotropy term Hp =4pMs in the braces in Eq. (9) gives a small, less than 10%, contribution to a stray field slightly reducing the energy FN and increasing DN that become noticeable only after field reversal at h < 0.

B. Bloch domain wall

Stripe domains can be separated also by Bloch DW parallel to the “easy”-axis Oy. In magnetic field parallel to the “hard”-axis H k Ox, magnetization in domains interior deviates from DW direction Oy toward H at angle 6 uo ¼ 6 cos1 h. In “true” Bloch DW, there is no change of magnetization component Mx that is perpendicular to the wall plane and is continuous in adjacent domains and DW: Mx ¼ Ms sin hM cos uM ¼ Ms cos uo :

(10)

This means the rotation of magnetization vector M from one side of DW at x ¼ D/2 to another one at x ¼ þD/2 can be defined by an angle variable on the cone-shaped shell 0 wM ¼ pðx=D þ 1=2Þ p. Spherical angles hM ; and uM relate to uo and wM through Eq. (10) and


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Nzz ¼ D=ðt þ DÞ in perpendicular to film plane direction. Integrating dFB over the DW width we obtain: FB ¼

ð D=2

dFB dx ¼

D p

ðp

dFB dwM Hp 2 2 1 2 2 2 þ 2pMs h þ ¼p A 1h 1 h D: (12) 8pMs D D=2

0

The contribution from stray field energy 2pMz2 was neglected since D=t Hp =4pMs :

(13)

After minimization of FB with respect to the D, we obtain the width and energy of Bloch DW: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 Hp 2 2 2 2 1h ; DB ðhÞ ¼ pA=2Ms 1 h h þ 8pMs qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi þ1=2 Hp FB ðhÞ ¼ 2p 2pAMs2 1 h2 h2 þ 1 h2 : 8pMs (14) Bloch DW energy FB(h) and width DB(h) plots we added to Fig. 11 for comparison with FN and DN. For our 4 lm thick (CoFeB)0.765–(SiO2)0.235 film the criteria in Eq. (13) is fulfilled for DW width DB below 186 nm. As seen from Fig. 11(a), this approximation remains valid in the whole range of magnetic fields 0 < h < 1.

C. Domain wall transformations

FIG. 11. (Color online) Field dependence of DWs width (a) and energy (b) computed for (CoFeB)0.765–(SiO2)0.235 film using Eqs. (9) and (14). Pure Bloch DW parallel to the “easy”-axis could exist in the entire range of magnetic fields Hp < H < Hp . Here Hp ¼ 535 Oe is the field of in-plane uniaxial anisotropy. For descending M-H branch, energy FN and width DN are shown for two Neél DWs: parallel to the “hard”- and “easy”-axes. Within the range þHcr < H < þHp (Hcr ¼ 232 Oe is a critical field) Neél DW parallel to the “hard”- axis has the lowest energy. Magnetization vector M in the center of this Neél DW is parallel to the magnetizing field. Bloch DW || “easy”- axis becomes energetically favorable in the range Hcr < H < þ Hcr . After field reversal in the range Hp < H < Hcr , another Neél DW that is antiparallel to the “hard”- axis has the energy lower than Bloch DW. This Neél DW has in the center M vector antiparallel to the direction of initial magnetization. Field dependence of this Neél DW energy FN(H) is shown with a dashed line.

cos hM ¼ sin uo sin wM . Representing hM and uM through the uo and wM angles we can write Bloch wall energy density as follows: dFB ¼ Kp sin2 uo sin2 wM þ A sin2 uo ð@wM =@xÞ2 þ2pMs2 t D cos2 uo þ 2pMs2 sin2 uo sin2 wM : (11) tþD tþD Here stray field energy is presented with two terms 2pMx2 and 2pMz2 . The latter one contains the demagnetizing factor

Comparison of the energies of DWs in Fig. 11(b) shows domain walls experience structural transformations when magnetic field H k “hard”-axis is swept within the range Hp < H < þHp . If we start from the “virgin” completely demagnetized state (H ¼ 0, M ¼ 0) the Blochp DW is realized. It has the lowffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi est energy FB ð0ÞFN ð0Þ ¼ Hp =4pMs 0:2 and exists until increasing magnetic field reaches the value Hcr ¼ 232 Oe. Then, Bloch DW transforms to Neél DWk“hard” axis that exists up to a film saturation at H ¼ Hp ¼ 535 Oe. Here Neél DW disappears since the pmisalignment ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of M vectors in adjacent domains 2uo ¼ 23=2 1 H=Hp ! 0 goes to zero. At descending branch of magnetization curve when magnetic field crosses Hp ¼ þ535 Oe, uniformly magnetized state becomes unstable regarding the appearance of equivalent domains separated by a low angle Neél DWk“hard”axis. With a field decrease, energy of Neél DW grows and at H ¼ Hcr becomes equal to Bloch DW energy. At this field Neél DW transforms to a Bloch-type wall. This transformation is accompanied with sudden changes of domain wall structure. Theory predicts an abrupt increase of DW width from about 29 to 46 nm [see Fig. 11(a)]. Also, magnetization vector M in DW center flips out of film plane at the angle 90 hcr ¼ cos1 Hcr =Hp 64 . With a further field decrease this angle gradually grows reaching 90 at H ¼ 0. Crossing zero field, Bloch DW remains energetically favorable until a nucleation of Neél DW with the magnetization


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FIG. 12. (Color online) Schematic presentation of Neél-to-Bloch DW transformation occurred in magnetic field parallel to the “hard”-axis at H ¼ þ Hcr . Jagged DW separates two adjacent magnetic domains. Low angle Neél DW (NDW) in the center has the M vector parallel to magnetic field H. Three kinks of Bloch DW (BDW) appeared. They have magnetization vector M flipped out from the film plane: upward and downward in the central BDW and two marginal BDWs, respectively. BDWs grow while NDWs shrink converting to the vertical Bloch lines when magnetic field reaches zero strength.

antiparallel to the “hard”-axis happens at H ¼ Hcr ¼ 232 Oe, Therefore at the descending branch of a magnetizing cycle domain walls experience four structural transformations: at saturation field H ¼ 6 Hp and at critical field H ¼ 6 Hcr . In ideal thermodynamically equilibrium conditions these transformations should have reversible character. However, as we will see now, the nucleation of Bloch DWs with different chirality at H ¼ þ Hcr makes the transformation at critical field H ¼ Hcr to be irreversible. The point is a nontrivial change of DW topology from “hard”-to-“easy” axis orientation occurring at H ¼ Hcr . In Fig. 12 we show the schematic of intuitive model of such DW reorientation at the descending branch of M-H loop. Arrows show the orientation of M vectors in two adjacent domains. Magnetic field H is parallel to the “hard”-axis Ox and its strength just falls below the critical field H < Hcr. Besides two segments of Neél DW (NDW), three kinks of Bloch DW (BDW) parallel to the “easy”-axis already appeared. Magnetization vector M flips out from the film plane, respectively, upward in the central BDW kink and downward in two marginal BDW kinks. Although the spatial distribution of magnetization vector M in Neél and Bloch DWs is very different, the transformation occurs as a continuous process via a nucleation of a zero-length kink of opposite-type DW. At the initial branch of magnetization process from the virgin state, small kinks of NDW segments k H appear at H ¼ þHcr intermitting Bloch DW. And vice versa, small kinks of BDW k“easy”axis intermit continuous Neél DWk“hard”-axis when the field crosses Hcr at descending branch of M-H curve. A key issue is a fact that all Neél DWs that appear have magnetization vector M to be parallel to the magnetizing

J. Appl. Phys. 109, 083926 (2011)

field HkOx and preserve this direction up to the transformation to the Bloch DWs at H ¼ þ Hcr . Correspondingly, all emerging segments of Bloch DW inherit positive Mx-component of magnetization vector M. However, their Mz component could have different sign positive or negative for M rotating clockwise or anticlockwise with respect to the “hard”-axis Ox. Since the nucleation of BDW kinks happens stochastically, at zero field domain walls appear to consist of BDW segments with the magnetization vector M directed upward and downward to the film plane. Opposite BDW segments meet each other and develop Vertical Bloch Line (VBL) inside a domain wall. These VBLs are short pieces of Neél-type DWs as remains of corresponding NDW segments. All of them have vector M to be parallel to the inherited direction of magnetizing field HkOx. As a result, below the critical field H ¼ þ Hcr domain structure appears to be in a metastable state since Bloch DWs loaded with VBLs have higher energy than pure Bloch DW. So, the magnetization process performed from the virgin state in magnetic field in excess of Hcr has irreversible character. At H ¼ 0 stripe domains never come back to the virgin completely demagnetized state separated by Bloch DWs. However, if being in a metastable state we increase magnetic field from zero again, then each VBL works as a nucleus and at H ¼ þ Hcr gives a birth of Neél DW segment. It means Neél DW to VBL loaded Bloch DW and vice versa transformation at H ¼ þ Hcr occurs reversibly. Now let us consider what will be happening at field reversal. It is obvious that VBLs with M antiparallel to H become energetically unfavorable. Therefore at H ¼ Hcr ¼ 232 Oe, they should disappear via a transformation of VBL loaded Bloch DW to Neél DW which is antiparallel to the “hard”-axis. In Fig. 11(b) the energy of Neél DW FN(H) with M k - Ox is shown with a dash line. After field reversal at H ¼ Hcr , FN(H) equals to and then becomes smaller than FB(H) when a field strength increases. As it was shown earlier, domain walls transformation at H ¼ þ Hcr from the virgin state with stripe domains k “easy”-axis to the stripes k “hard”-axis occurs continuously and sequentially. At first, the nucleation of kinks of Neél DWk“hard”-axis occurs. Then, Neél kinks increase their length while BDW segments shrink (Fig. 12). Opposing to this case, there is no continuous scenario of the transformation of Neél DWs magnetized parallel to the “hard”-axis direction (solid line in Fig. 11) into the oppositely magnetized Neél DWs (dash lines in Fig. 11) that become energetically favorable after field reversal. Let us pay close attention to the following details that occur at descending branch of magnetization curve. Below þ Hcr, kinks of Bloch DWs magnetized upwardly and downwardly to the film plane appear to be separated by contracting NDW segments with 2uo < 180o and M k þ Ox in the middle of the wall. At zero field, NDW segments degenerate into 180 VBLs inside the Bloch DW k “easy” axis. After field reversal, kinks of Neél DWs antiparallel to the “hard”-axis could appear spontaneously. They have M k - Ox in the middle of the wall. Their energy FN(H) decreases with field strength increase following a dash line in Fig. 11 and meets FB(H) at H ¼ Hcr. As for VBL, the


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magnetization vector inside it experiences more and more rotation 180o < 2uo < 360o . VBL energy arises rapidly since the rotation angle 2uo continues to grow approaching 360 at H ¼ Hp. All VBLs have M k þ Ox inside. They cannot convert themselves into the lengthening Neél DWs with M k Ox having different chirality, respectively, clock- and anticlockwise rotation regarding the Oz axis. It means due to rapidly increasing energy VBL loaded Bloch DWs disappear irreversibly at fields that precede H ¼ Hcr. We arrive to the conclusion that DWs transformation at the same branch of magnetization curve has different character at H ¼ þ Hcr and H ¼ Hcr . At descending M-H branch, gradual nucleation of VBLs at H ¼ þ Hcr requires an additional energy whereas their sudden destruction around H ¼ Hcr is accompanied with energy release. As we show in Sec. VIII, these theoretical predictions are firmly consistent with experimentally observed anomalies of magnetic susceptibility and visualized domain structure. VII. DOMAINS VIZUALIZATION

To visualize domains in CoFeB sample we employed magneto-optical (MO) imaging technique based on the Faraday effect in MO active film, hereinafter indicator. We used 6 lm thick bismuth-doped iron garnet (Y,Bi)3(Fe,Ga,Sc)5O12 (BIG) film with the perpendicular anisotropy grown onto the Gd3Ga5O12(111) substrate by liquid phase epitaxy. Tightly contacted indicator and CoFeB film were placed in electromagnet that supplies magnetic field parallel to the film plane. MO images in indicator were observed in polarized light reflected from the surface of CoFeB film. Three MO images in Figs. 13(a)–(c) reveal domain structure in (CoFeB)0.765–(SiO2)0.235 sample at different strength of magnetic field parallel to the “easy”-axis. Native labyrinth domains in BIG indicator become readily seen when Figs. 13 are zoomed in. Stripe domains in BIG have a period about 6 lm, possess crystallographic[111] C3-axis 120 rotational symmetry orienting themselves along one of the equivalent [112], [121], and [211] directions in the film plane. External magnetic field and stray field from domains in CoFeB film cause distortions of labyrinth domains in BIG indicator. Image in Fig. 13(a) reproduces virgin stripe domains parallel to the “easy”-axis in CoFeB film that was completely demagnetized by applying an alternating slowly decreasing magnetic field. In Fig. 2 for magnetization M-H loop, the corresponding point H ¼ 0, M ¼ 0 is marked with a letter a. Stripes in CoFeB film have period about 50 lm. In Fig. 13(a), BIG indicator show primarily a uniform contrast along the vertical axis that is “easy” direction in CoFeB film except short cross-ties at some of the boundaries. This uniform contrast evidences a full saturation of BIG indicator caused by a uniform perpendicular component of stray magnetic field from underlying CoFeB sample. A uniform Bloch DW parallel to the “easy” axis produces such field at the surface of CoFeB film. Short cross-ties have a period of 6 lm. These are the remains of labyrinth domains in BIG indicator oriented by in-plane component of stray field that create a closure of magnetic flux between the neighboring oppositely directed Bloch DWs.

FIG. 13. (Color online) Gallery of magneto-optical images of domains in (CoFeB)0.765–(SiO2)0.235 film visualized with 6 lm thick bismuth-doped iron garnet (Y,Bi)3(Fe,Ga,Sc)5O12 film indicator. Zooming in each image makes native labyrinth domains in the indicator to be readily seen. The letters (a) to (l) notify the positions at the magnetization M-H loop in Fig. 2 where the images were recorded. Images (a) to (c) and (d) to (l) were taken in magnetic field parallel to the “easy”- and “hard”-axis, correspondingly. (a) Virgin state of stripe domains in CoFeB film completely demagnetized in slowly decreasing ac-field. (b) Saturated CoFeB film does not produce any magnetic relief noticeable in the indicator. (c) Close to H ¼ Hc after field reversal stripe domains with VBL loaded Bloch DWs of alternating polarity (chirality) become visible. (d)–(l) The sequence of domains transformation at descending M-H branch from uniformly magnetized state [image (d)] through sudden appearance of stripe domains when VBL loaded Bloch DWs convert to Neél DWs [image (h)] and finally to magnetically saturated state [image (l)]. In Ref. 18 one could find MO movie which shows a whole continuous cycle of film magnetization at H || “easy”- axis from the virgin to the saturated state.

Magnetic field of 200–300 Oe applied in “easy” direction uniformly magnetizes CoFeB film up to saturation. Saturation magnetization 4pMs remains also as a remnant at H ¼ 0. At these states, marked with letters b in Fig. 2, BIG film does not show any traces of magnetic relief coming from underlying CoFeB film. In Fig. 13(b) one can see only native labyrinth domains in BIG indicator. Stripe domains in CoFeB film become visible again after field reversal close to coercive field Hc ¼ 6 50 Oe. Magnifying Fig. 13(c) we could see stripes in CoFeB have the same period, on the average, though DWs look differently compared to Fig. 13(a). In a virgin state a only few stripes have cross-tielike structure with jagged DWs. In the c-state, all CoFeB stripe domains connect each other in the “hard”-axis direction with multiple crossroads. Optical contrast intermitting along the “easy” axis indicates that Bloch DWs in CoFeB film contain Bloch segments with alternating upward


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and downward direction of the magnetization separated with VBLs. Stray field from in-plane M component inside the VBL makes entangled crossroad labyrinths in BIG indicator. MO movie in18 shows the dynamics of domain wall transformation that occurs in magnetic field parallel to “easy” direction. Photo gallery in Figs. 13(d)–(I) presents MO images of domain structure in CoFeB film collected at the descending branch of M-H loop in the “hard”-axis direction. If magnetic field strength exceeds Hp ¼ 535 Oe, there is only a native labyrinth pattern in BIG indicator [Figs. 13(d) and 13(l)] denoting saturated in-plane magnetization in CoFeB film. Figures 13(d)–(g) do not show any magnetic relief coming from CoFeB when magnetic field decreases to zero. There is only a slight change of labyrinth domains in BIG: external field stretches them a little whereas they own virgin 120

rotational symmetry at H ¼ 0. MO images in Figs. 13(a) and 13(g) depict the most striking difference in CoFeB domain structure. According to M-H loops in Fig. 2, both of them correspond to demagnetized state: M ¼ 0, H ¼ 0. Figure 13(a) shows zero net virgin magnetization at H ¼ 0 in the form of regular stripe domains in CoFeB. They were achieved demagnetizing film in the “easy”-axis ac-field. In contrast, BIG indicator in Fig. 13(g) does not reveal any magnetic relief in CoFeB film demagnetized from the saturation by decreasing strength of magnetic field parallel to the “hard”-axis. This signifies completely different scenario of magnetization reversal that takes place at H k“hard”-axis. Domain structure might has a feature size beyond the spatial resolution of BIG indicator. Stripe domains in CoFeB suddenly appear after magnetic field reversal. Zooming in Fig. 13(h), one concludes perpendicular component of stray field from CoFeB film changes the widths of labyrinth domains in BIG indicator. Strong average optical contrast appears and visualizes stripes in CoFeB. Further increase of magnetic field strength gradually attenuates the contrast [Fig. 13(j)–(k)]. Finally, stripes disappear again when magnetic field reaches Hp ¼ 535 Oe [Fig. 13(I)]. VIII. DISCUSSION

Experimental microwave FMR spectra, field anomalies of magnetic susceptibility, and domains visualization in CoFeB film presented, respectively, in details in Secs. IV, V, and VII confirm the model of DWs transformation developed in Sec. VI. Magnetization process in “easy” direction is featured by sudden disappearance of magnetic domains oriented oppositely to magnetizing field. This transformation occurs at coercive field Hc ¼ 6 50 Oe. It is distinctly revealed by MO indicator and accompanied by instantaneous peaks of magnetic susceptibility [Fig. 7(a)]. DW pinning at various magnetic defects causes coercivity. This mechanism does not depend on spin precession hence the anomaly is observed in a whole range of frequencies from 500 kHz to 12 GHz as seen in Figs. 7 and 10(a). When CoFeB film is magnetized in the “hard” direction magnetic susceptibility experiences sharp frequency inde-

J. Appl. Phys. 109, 083926 (2011)

pendent changes four times. The anomaly at Hp ¼ 6 535 Oe has the same shape in positive and negative fields at descending and ascending branches of M-H loop. This is the manifestation of the “soft” mode from Eqs. (5.3)–(5.5) when Neél-type low angle DWs k H transform to the uniformly magnetized state. The process cannot be visualized by MO image technique since low angle Neél DWs do not produce perpendicular stray fields that can be detected by BIG indicator. The anomaly of v at saturation field H ¼ Hp has the character of critical spin fluctuations that present in a whole frequency range and cause enhanced absorption displayed by dark vertical bands in Fig. 10(b). The anomaly of susceptibility at H ¼ 6 Hcr has different shape and interchange its position at descending and ascending branches of magnetization loop. Small peak occurs when magnetic field strength decreases approaching zero. This is a transformation of Neél DWs k “hard” axis to VBL loaded Bloch DWs k “easy” axis. It goes on continuously thus has a reversible character. Frequent alternation of magnetic poles (Mz component) in upward and downward directed Bloch DWs nullifies the average stray field perpendicular to the film plane. As a result, such VBL loaded DW produces obscure magnetic relief that is sightless in labyrinth domains of the indicator [Figs. 13(d)–(g)]. According to our model, sudden disappearance of VBLs occurs after field reversal at H ¼ Hcr for descending and at H ¼þ Hcr for ascending branch. It accomplishes the conversion of Bloch DWs k “easy” axis to Neél DWs k “hard” axis. Magnetization vector M falls down to the film plane, instantaneously releases large energy and induces voltage signal in pick-up coil depicted as a strong anomaly of susceptibility in Figs. 8. Broadband microwave strip spectroscopy also detects this event. In Fig. 10(b), it is exhibited at first by increased microwave loss (vertical narrow dense color line) and then by zero absorption (bright vertical line) at H ¼ Hcr. Beyond H ¼ Hcr, when the Blochto-Neél DW transformation is completed, absorption gradually increases reaching its maximum value for critical fluctuations at H ¼ Hp. Very similar effects we observed also in 0.48 lm thick (CoFeB)0.66–(SiO2)0.34 film. The only difference was lower saturation magnetization 4pMs ¼ 8.6 kGs and field of uniaxial anisotropy Hp ¼ 165 Oe. Therefore, Bloch-to-Neél DW transformations were observed in the field H ¼ Hcr ¼ 73 Oe that with a 5% accuracy coincides with Hcr calculated from the Eqs. (9) and (14) for FN(Hcr) ¼ FB(Hcr). IX. CONCLUSIONS

Spin dynamics in nanocrystalline CoFeB films with growth induced anisotropy was studied in a wide frequency range from 500 kHz to 15 GHz. Broad band FMR spectroscopy revealed “soft” magnetic modes in a whole range of orientations of magnetic field perpendicular to the “easy” magnetic axis. The most interesting transformations of domain structure were observed in a “hard” direction in a weak magnetic field below the field of uniaxial anisotropy Hp. When field decreases, uniform magnetization in CoFeB film splits at H ¼ þHp to domains separated by low- angle


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Neél DWs parallel to the “hard”-axis. Then, at critical field H ¼ þ Hcr walls gradually transform to the Bloch DWs parallel to the “easy”-axis loaded with VBLs. After field reversal at H ¼Hcr, Bloch DWs disappear. Their conversion to Neél DWs is accompanied with instanteneous release of VBLs energy and strong anomalies of susceptibility in a wide range of frequencies. Growing critical spin fluctuations accomplish domains transformation to the uniformly magnetized state at H ¼ Hp. ACKNOWLEDGMENTS

This work was partially supported by the Vetenskapsra˚det (Swedish Research Council), Swedish Institute, and Science and Innovation Agency of Russian Federation CK @ 02.740.11.5179. 1

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