PHYSICAL REVIEW B 66, 104419 共2002兲
Critical angles effect:
Vanishing of cubic magnetocrystalline anisotropy in ferromagnetic resonance spectra A. A. Jalali,* S. Kahl, V. Denysenkov, and A. M. Grishin
Condensed Matter Physics, Royal Institute of Technology, S-164 40 Stockholm-Kista, Sweden 共Received 20 November 2001; revised manuscript received 3 June 2002; published 19 September 2002兲 The shapes of ferromagnetic resonance 共FMR兲 spectra of laser deposited epitaxial Y3 Fe5 O12 共YIG兲 films strongly depend on the orientation of the magnetic field with respect to the crystalline axes of the film. Strain and compositional inhomogeneties in the films define local magnetic anisotropies and could be responsible for the complexity of FMR spectra. We show that, in accordance with the Smit-Suhl formula, the contribution from a spatial distribution of the cubic magnetocrystalline anisotropy to the FMR spectra vanishes at certain orientations 共‘‘critical angles’’兲 of the magnetic field. We prove experimentally that it is a necessary condition to orient the magnetic field at the critical angles in order to measure single-line FMR spectra in an inhomogenous laser deposited YIG film. DOI: 10.1103/PhysRevB.66.104419
PACS number共s兲: 76.50.⫹g, 75.30.Gw, 75.50.Gg, 81.15.Fg
I. INTRODUCTION
Single crystal yttrium iron garnet Y3 Fe5 O12 共YIG兲 films grown by liquid phase epitaxy 共LPE兲 were brought to a high level of development as materials for bubble memory devices in the 1970’s1 and remain interesting for microwave applications until now.2 In the last decade, epitaxial YIG films were also grown by pulsed laser deposition 共PLD兲.3 It was shown that laser deposited YIG films exhibit characteristics, which were never observed in films that were grown by LPE. As an example, the field of uniaxial anisotropy of pure YIG films could be varied in a wide range from ⫺850 Oe to 650 Oe by changing the oxygen partial pressure used during PLD.4 For microwave applications, the ferromagnetic resonance 共FMR兲 linewidth determines the microwave loss and should thus be as small as possible. There seem to be material specific angles of the applied magnetic field with regard to the crystalline axes, at which the FMR spectra contain fewer lines or just one. The width of the strongest line in the spectrum can also be reduced at certain angles. It is an attractive thought to partially compensate for the imperfections of laser deposited films by operating them at special angles. The angular dependence of the FMR linewidth in laser deposited YIG films has already been studied in detail and compared to the model of an isotropic film.5 In this paper, we consider the contribution of spatial variations of the cubic magnetocrystalline anisotropy to FMR spectra. Through simple and straightforward calculations we find critical angles 共in analogy to Ref. 6兲, at which the contribution of cubic magnetocrystalline anisotropy to the uniform FMR mode vanishes; this means that spatial inhomogeneities in cubic magnetocrystalline anisotropy will not increase the complexity of the FMR spectra at these angles. We confirm our theoretical results through measurements of FMR spectra of a laser deposited YIG共111兲 film. II. THEORY
For the magnetic film under study, the 关111兴 direction of the cubic lattice is normal to the film plane. It is advanta0163-1829/2002/66共10兲/104419共5兲/$20.00
geous to express all vectors in spherical coordinates 共Fig. 1兲, with polar angle and azimuthal angle from the 关 111 兴 ¯ 0 兴 directions, respectively. and 关 11 The energy of cubic magnetocrystalline anisotropy is7 F1 共 M , M 兲 ⬅K 1 ⫺
冉
1 1 cos4 M ⫹ sin4 M 3 4
冑2 3
冊
sin 3 M sin3 M cos M ,
共1兲
where K 1 is the first constant of cubic anisotropy and the angles M and M denote the polar and the azimuthal angles of the magnetization M(H). The energy of effective uniaxial anisotropy is 2 F2 共 M 兲 ⬅K * u sin M ,
共2兲
where the constant of effective uniaxial anisotropy K * u ⬅K u ⫺2 M 2S includes the energy of uniaxial magnetic anisotropy through K u and the demagnetization energy through 2 M 2S ; M S is the saturation magnetization. The total energy F including Zeeman energy is then given by F⫽⫺HM S cos M cos H ⫺HM S cos共 H ⫺ M 兲 ⫻sin M sin H ⫹F1 共 M , M 兲 ⫹F2 共 M 兲 ,
共3兲
where the angles H and H denote the polar and the azimuthal angles of the external magnetic field H. The frequency of the uniform ferromagnetic resonance for an arbitrary orientation of the applied static magnetic field is given by the Smit-Suhl formula,8,9
冉冊 ␥
2
⫽
1
冋
2F 2F
M 2S sin2 M 2M 2M
⫺
冉
2F MM
冊册 2
,
共4兲
with ␥ as gyromagnetic ratio. The equilibrium orientation of the magnetization vector M(H) can be found from the equations
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©2002 The American Physical Society
PHYSICAL REVIEW B 66, 104419 共2002兲
JALALI, KAHL, DENYSENKOV, AND GRISHIN
冉 冊 ␥H
2
⫽1⫹
⫹
冉 冉
2 F1
1
K 1 2M sin2 H
⫹
F1 2 F1 cot H ⫹ 2 M M
冊
F2 2 F2 cot H ⫹ 2 . K u* M M
冊 共9兲
In Eqs. 共7兲, 共8兲, and 共9兲, all derivatives are taken at the equilibrium position with M ⫽ H and M ⫽ H . Substituting the explicit expressions for the derivatives, we obtain
⫽1⫹ f ⫹ f , ␥H
共10兲
1 f ⫽ 共 1⫹3 cos 2 H 兲 2
共11兲
where FIG. 1. The coordinate system chosen for the 共111兲 oriented film.
F ⬅⫺K 1 冑2 cos 3 M sin3 M cos M M
and
⫺HM S sin H sin M sin共 H ⫺ M 兲 ⫽0
f ⫽⫺
共5兲
and
冉
冑2 F sin 3 M sin4 M ⬅K 1 sin3 M cos M ⫹ M 3 冑2
4 sin2 2 M sin 3 M ⫺ cos3 M sin M ⫺ 4 3
⫹sin2 2 H ⫹
冊
K 1 sin M 2
F1 M
共7兲
and
H⫺ M ⫽
F1 F2 ⫹ . K1 M K* M u
The resonance condition then becomes
11 12冑2
sin 3 H
共8兲
23
sin 2 H sin 3 H . 共12兲 8 冑2 These equations will be used to interpret the experimental results. III. EXPERIMENTAL PROCEDURE
共6兲
In order to approximate the analytical relationship between the resonance frequency and the direction and magnitude of the applied field, we examine the case of a strong magnetic field when the cubic magnetocrystalline and effective uniaxial magnetic anisotropy energies are small compared to the Zeeman energy and can be treated as and disturbances.10 That means ⬅K * u /HM S Ⰶ1 ⬅K 1 /HM S Ⰶ1, which is equivalent to 兩 H ⫺ M 兩 Ⰶ1 and 兩 H ⫺ M 兩 Ⰶ1. In this case, the direction of M that minimizes F is obtained by expanding Eqs. 共5兲 and 共6兲 in terms of and to the first order,
H⫺ M ⫽
冑2
sin H sin 3 H cos3 H ⫺
4 1 ⫻sin 3 H cos H ⫺ cos4 H ⫺ sin4 H 3 2
⫹K * u sin 2 M ⫺HM S 关 cos M sin H cos共 H ⫺ M 兲 ⫺cos H sin M 兴 ⫽0.
3
The theory was compared to measurements on a YIG film prepared by PLD on a single crystal gadolinium gallium garnet Gd3 Ga5 O12 共GGG兲 共111兲 substrate (5⫻5 cm2 ). The KrF excimer laser worked at the wavelength of 248 nm and repetition rate of 40 Hz. The target to substrate distance was 4.5 cm, the laser energy density on the stoichiometric Y3 Fe5 O12 ceramic target 2.5 J/ cm2 , the substrate temperature 650 °C, the ambient oxygen pressure 0.034 mbar, and the deposition time 90 min. Film thickness measured by profilometer was found to be 1.2 m. Magnetic hysteresis loops were traced with a vibrating sample magnetometer; the strong paramagnetic contribution from the GGG substrate was subtracted. The crystalline structure of the YIG film was investigated by x-ray diffraction 共XRD兲 with Cu K ␣ radiation using a three circle diffractometer. scans of 兵 664其 planes were ¯ 兴 and 关 11 ¯ 0 兴 directions performed to determine the 关 112 共parallel to the plane of the film兲 in order to orient the sample for FMR measurements. After the XRD scan, the sample was mounted in the FMR spectrometer with magnetic field ¯ 兴 direction corresponding to azimuthal oriented in the 关 112 angle H ⫽90°. FMR measurements were performed at room temperature
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PHYSICAL REVIEW B 66, 104419 共2002兲
VANISHING OF CUBIC . . .
FIG. 2. -2 XRD scans showing 共444兲 K ␣ 1 and K ␣ 2 Bragg reflections from 1.2 m thick 共solid line兲 and 400 nm thick 共dotted line兲 laser deposited YIG/GGG共111兲 films. The inset shows a scan of the 兵664其 planes of the 1.2 m thick film where the sample and 2 detector angles correspond to peak 1 of the -2 scan.
FIG. 3. Magnetic hysteresis loops of a 1.2 m thick YIG/ GGG共111兲 film. The magnetization is normalized to the saturation magnetization 4 M S ⫽(1450⫾200) G. The coercive fields H C are given in the figure. The dashed lines have been added to guide the eye.
with a broadband FMR spectrometer using a network analyzer as both source and receiver of the microwave signal. The accuracy of the FMR goniometer with respect to the polar angle H is 1°. The polar angle can change by about 1° during a full 360° variation of the azimuthal angle.
film plane and perpendicular to the film plane are shown in Fig. 3; the film is more easily magnetized in the plane. The hysteresis loop with the magnetic field perpendicular to the film plane indicates that different parts of the film switch in different ways. Figure 4 shows FMR spectra at the polar angle H ⫽145° and different azimuthal angles H of the magnetic field. Different numbers of FMR lines were observed at different azimuthal angles. We use the concept of ‘‘full spectral width’’ 共FSW, in analogy to Ref. 12兲 as a measure of simplicity. We define FSW as the distance in units of applied magnetic field between the first and last extremum of the FMR spectrum 共derivative of absorption as function of applied magnetic field兲.
IV. RESULTS AND DISCUSSION
YIG/GGG共111兲 films are exclusively 共111兲 oriented; wide angle -2 scans reveal only 共444兲 and 共888兲 Bragg reflections. The 1.2 m thick film shows 共444兲 reflections 共labeled 1 and 2兲 to the left of the substrate 共444兲 reflections as depicted in Fig. 2. Peaks 1 and 2 do not occur in a film that was made at the same deposition conditions except that the deposition time was only 30 min and film thickness 400 nm 共compare the dotted spectrum in Fig. 2兲. The thin film shows only very little intensity to the left of the GGG peaks, and it is assumed that the main intensity of the 共444兲 reflection from that film is shadowed by the dominating GGG reflections. Probably, peaks 1 and 2 in the spectrum of the thicker film stem from upper layers 400 nm or more above the film-substrate interface. To assume that part of the intensity of the 共444兲 reflection of the 1.2 m thick film is shadowed by the substrate peaks means that the distance between lattice planes (hhh) parallel to the film plane corresponds to a cubic unit cell of lattice parameter 1.238 nm, which is the lattice constant of GGG. In the upper layer, the distance between (hhh) lattice planes is larger by almost 0.5%. There is thus a distribution of the distances between (hhh) lattice planes with film thickness. A distribution of lattice parameters has already been observed in 1 m to 2 m thick laser deposited YIG films grown on Mg-doped GGG substrates 共lattice constants of 1.248 nm兲.11 In our case, this direct observation is not possible due to the good lattice matching between YIG film and GGG substrate. Hysteresis loops with the magnetic field parallel to the
FIG. 4. FMR spectra for a laser deposited 1.2 m thick YIG/ GGG共111兲 film measured at 9.3 GHz for different azimuthal angles H and fixed polar angle H ⫽145°. Close to the points f ( H ⫽145°, H ⫽126° and 174°)⫽0 关compare Eq. 共12兲兴, three FMR lines almost collapse to a single FMR line and full spectral widths 共FSW兲 are small.
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PHYSICAL REVIEW B 66, 104419 共2002兲
JALALI, KAHL, DENYSENKOV, AND GRISHIN
FIG. 5. 共Color兲 Contour plot of f ( H , H ), see Eq. 共12兲. Symbols show experimental data where we have FMR spectra with the smallest full spectral widths. These FMR spectra generally occur at points close to f ( H , H )⫽0.
A spatial distribution of K 1 gives rise to resonance lines at different magnetic fields and thus leads to complex resonance spectra. However, if there is no contribution from K 1 to the position of the resonance line, the spectra do not become more complex, at least not through a spatial distribution of K 1 . In order to support this point of view, we show a contour plot of f as a function of H and H in Fig. 5. The contribution from cubic magnetocrystalline anisotropy to the position of the uniform resonance line vanishes at the zero contour line 共dashed line in Fig. 5兲. At angles close to the zero contour line, we expect the spectra to be simpler, i.e., the FSW’s should be smaller with respect to adjacent spectra. At the critical angles when f ( H , H )⫽0, the resonance condition in the linear approximation 关Eq. 共10兲兴 reduces to the form K u* ⫽H⫹ f . ␥ MS
共13兲
We confirmed the validity of the approximations made in Sec. II for the film investigated here by estimating the average values of K 1 /M S ⬇⫺40 Oe and K * u /M S ⬇⫺800 Oe 共corresponding to ⬇⫺0.01 and ⬇⫺0.2) with the method described in Ref. 13. Our theoretical considerations regarding the critical angles could be confirmed experimentally. Looking back at Fig. 4, FSW’s are smallest for the spectra located in the range ⫺0.5⭐ f ⭐0.5, i.e., close to the zero contour line. The spectrum at H ⫽207°, which is close to the center of one of the cucumber shaped structures in Fig. 5 where f ⬎1, is more complex than the other spectra. The squares and error bars in Fig. 5 correspond to measurements of FSW’s at fixed H ⫽145° when H was varied in steps of 10°. The procedure to present positions of simplest spectra is as follows. As seen in Fig. 4, single-line
FIG. 6. FMR spectra for a laser deposited YIG共111兲 film measured at 9.3 GHz at different polar angles and at the fixed azimuthal ¯ 兴 . The smallest full specangle H ⫽90°, corresponding to H储关 112 tral width 共FSW兲 occurs at H ⫽152°, which is marked by a circle in Fig. 5.
spectra alternate with multiline spectra. Among every group of neighboring single-line spectra, the spectrum with smallest FSW is marked by a square in Fig. 5. The error bars cover azimuthal ranges where spectra have FSW’s that exceed the smallest FSW in the respective group by not more than 10 Oe. It is our general observation that FSW’s become large when H approaches 90°. Therefore, it is best to test our theory far away from H ⫽90°. We would like to emphasize here that a spatial distribution of K 1 is only one among other mechanisms that can cause the splitting and/or broadening of FMR lines. We also varied the polar angle H at fixed azimuthal angle H . Figure 6 shows the angular dependence of FMR spectra ¯ 0 兴 plane with magnetic field applied parallel to the 关 11 ( H ⫽90°). At the angle of H ⫽152°, we have a single uniform resonance with linewidth of 42 Oe. By varying H at fixed H ⫽90°, we go along a line where f changes rapidly; consequently, the change in FSW is very pronounced as can be seen in Fig. 6. If the spatial distribution of cubic magnetocrystalline anisotropy was the only cause of increased FSW’s, spectra with small FSW’s should also appear for polar angles H close to 35° and between 90° and 100°. While there is some decrease in FSW around H ⫽35°, the FSW at H ⫽95° is rather large. This indicates again that more causes of spectral broadening exist. Angles of the spectra with smallest FSW’s measured at fixed azimuthal angles of, respectively, H ⫽30°, 90°, 150°, and 210° are shown as circles in Fig. 5. The general result, which we demonstrated through the examples above, is that no simple spectra occur at angles where 兩 f 兩 ⬎0.5, while not all spectra at 兩 f 兩 ⬍0.5 are simple. V. CONCLUSIONS
We have investigated the influence of spatial variations of the constant of cubic magnetocrystalline anisotropy K 1 on
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CRITICAL ANGLES EFFECT:
PHYSICAL REVIEW B 66, 104419 共2002兲
VANISHING OF CUBIC . . .
the FMR spectra of inhomogeneous laser deposited YIG films. The Smit-Suhl formula allowed us to calculate critical angles ( H , H ) where the cubic magnetocrystalline anisotropy does not contribute to the uniform FMR mode spectrum. Around these orientations of the magnetic
field, we recorded simple FMR spectra with smaller full spectral widths. Our results prove that it is a necessary condition to apply the magnetic field at critical angles in order to measure simple FMR spectra in inhomogenous YIG films.
*Electronic address:
[email protected]; URL: http://www.kth.se/
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