FERROMAGNETIC RESONANCE LINEWIDTH IN BULK NICKEL FOR VARIOUS ANGLES BETWEEN THE STATIC MAGNETIC FIELD AND THE SURFACE OF THE SAMPLE

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FERROMAGNETIC RESONANCE LINEWIDTH IN BULK NICKEL FOR VARIOUS ANGLES BETWEEN

THE STATIC MAGNETIC FIELD AND THE SURFACE OF THE SAMPLE

W. Anders, E. Biller

To cite this version:

W. Anders, E. Biller. FERROMAGNETIC RESONANCE LINEWIDTH IN BULK NICKEL FOR VARIOUS ANGLES BETWEEN THE STATIC MAGNETIC FIELD AND THE SUR- FACE OF THE SAMPLE. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-774-C1-776.

�10.1051/jphyscol:19711270�. �jpa-00214102�

JOURNAL DE PHYSIQUE Colloque C I , supplment au no 2-3, Tome 32, F'vrier-Mars 1971, page C 1 - 774

FERROMAGNETIC RESONANCE LINEWIDTH IN BULK NICKEL FOR VARIOUS ANGLES BETWEEN THE STATIC MAGNETIC FIELD

AND THE SURFACE OF THE SAMPLE

W. ANDERS and E. BILLER

Sektion Physik der Universitat Munchen, Germany

ResumB. - Des disques de nickel poly- et monocristallin ont kt6 expQimentes a 25

et 8.6 GHz. Les monocris- taux preparks avec soin presentent une largeur de raie descendant jusqu'8 102 Oe dans le cas d'un champ magnetique parallele a la surface, et A o / y calcule a partir du mesurage de AH, s'est trouv6 pratiquement independant de l'angle. La deformation plastique provoque un elargissement de raie qui est fortement dkpendant de I'angle. La relation d'elargisse- ment entre le cas d'un champ parallele et d'un champ perpendiculaire est de 5 : 1 pour des monocristaux. La dkpendance angulaire de Awly dans Ies polycristaux recuits est essentiellement le rksultat de trois apports : la largeur de raie du mono- cristal, l'elargissement decrit par la thkorie des grains independants, et l'apport des dislocations.

Abstract. - Disks of poly- and monocrystalline nickel were investigated at 25 OC at 8.6 GHz. After careful prepa- ration single crystals show linewidths down to 102 Oe in the in-plane case, and Aw/y, calculated from the measured AH,

is found to be nearly independent of the angle. Plastic deformation causes a linebroadening, which is strongly angular dependent. For the parallel and the perpendicular case the increase ratio is 5 : 1 for single crystals. The angular dependence of ^Aw/y in annealed polycrystals results mainly from three contributions : single crystal linewidth, broadening described by the independent grain theory, and a contribution caused by dislocations.

I. Introduction. - Much theoretical and experi- mental work has been done on inhomogeneous broadening of ferromagnetic resonance (FMR) lines, however mainly for ferrites. This paper presents experimental results for the linewidth in bulk polycrys- talline nickel metal. We used the method of altering the angle q between sample surface and dc-magnetic field H (cr geometry-variation >> GV), because with this method we were able to separate different broadening mechanisms [I].

11. Experimental method. - The samples were disks of about 18 mm in diameter and 1 mm thick.

Sample preparation and microwave techniques are described in [l, 21. The linewidth AH is defined as the field separation between the peaks of the derivative absorption curve. If H is not parallel or perpendicular to the sample surface the line shape is distorted by the demagnetizing field, which varies as a function of H because the angle ^cr between the magnetization and the surface is not constant. One can correct the mea- sured AH for this effect by calculating the frequency linewidth Aw/y by the formula Awly = (doly dH) AH.

dw/dH is the total derivative of the resonance equation, calculated at resonance (w = 2 ?lf, y = gyromagnetic ratio). This calculation was achieved by a computer program, taking into account the finite thickness of the samples. To check our formulas [I], we measured the angular dependence of the resonance field HR and AH of polycrystalline ellipsoids of revolution with various ratios of the axes. HR is in close agreement with the calculations, and Aoly as a function of a is the same for all samples within the measurement accuracy, although AH is strongly dependent on the sample shape. a was calculated numerically from the equilibrium condition for known q and H at reso- nance [I].

III. Results and discussions. - Curve 2 in figure 1 shows the average angular dependence of Awly for an annealed and polished polycrystal. A qualitative ana- lysis of Aw/y with rough quantitative estimates is achieved by the combination of temperature and GV-experiments with samples of different properties.

FIG. 1. - Linewidth Ao/y of bulk nickel as a function of the angle a between sample surface and magnetization. 1 & single crystal width, ² polycrystal width, < ^H; >

mean square root of the resonance field for randomly orientated grains, 3 2 curve 1 + 1.3 < H&

4 slightly deformed

single crystal, 5 = curve 4 + 1.2 < H;

1. SINGLE CRYSTAL LINEWIDTH. - Curve 1 in figure 1 represents the linewidth of bulk single crystals of the same size as the polycrystalline samples. In the in-plane case the crystals show widths AH down t o 102 Oe [2], as measured up to now only in whiskers and platelets [3]. If the disks are rotated around the surface normal, the linewidth varies. In the parallel case it varies from about AH = 105 Oe, if H lies in

< ¹⁰⁰ > direction, t o about 140 Oe if H is in < ¹¹⁰ >

for a sample with (110)-surface. Curve 1 is averaged

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711270

FERROMAGNETIC RESONANCE LINEWIDTH IN BULK NICKEL FOR VARIOUS ANGLES BETWEEN C 1 - 775

from several runs, starting from different field direc- tions in the in-plane case.

2. ANISOTROPY BROADENING. - In the discussion of our results we will restrict ourselves to the two limiting cases in the theories of Schlomann [4] and Vrehen et al. [5] : a) the individual grains go through resonance independently ((( independent grain model P), b) crys- talline anisotropy causes a small coupling between the excited mode and spinwaves, and energy is transferred

( G spinwave model ^D). In these two limiting cases the linebroadening should be proportional to I K , I/M and K ? / M ~ , respectively, neglecting higher order constants (Kl is the first anisotropy constant) [4]. Further, the spinwave model results in a linewidth contribution which should be much less for a = 900 than for a = 0, because the broadening mechanism is a two-magnon scattering process.

From our experiments we can deduce that the ani- sotropy broadening in nickel at room temperature must be mainly due to the independent grain mechanism.

This follows from the temperature dependence of the polycrystal width 161 and the relatively slight angular variation of the difference between curves 1 and 2.

This result is in contradiction with theoretical calcu- lations [4, 51, which lead to the condition

for the validity of the independent grain model. This condition does not hold for nickel at room temperature and therefore broadening by two-magnon scattering should dominate. We believe that the discrepancy is caused by the fact that some assumptions of the theory are not fulfilled in our experiment. In ferroma- gnetic metals the microwave field excites mainly spin- waves with wavenumbers k ^% 116 (6 = skindepth x 1 pm), whereas in ferrites the uniform precession is excited. 6 is much smaller than the mean grain size of 50 pm, and therefore the scattering probability is small.

To estimate the broadening by independent grains we have calculated numerically the mean square root

< ^{H: >%} of the resonance field as a function of a.

We made the simplifying assumption that the equili- brium position of the magnetization does not depend on the crystal energy. At room temperature and at internal fields of about 2 kOe this is justifiable for nickel. Since the skindepth is much less than the grainsize we regarded the grains as thin sheets, the surrounding material was assumed to be magnetized homogeneously. The dashed line in figure 1 shows the resulting holy obtained with the parameters

K l / M = - 120 Oe,

K,/M ⁼ - 50 Oe (K, = second anisotropy constant), and M = 475 G. The calculated mean value of the resonance field in the parallel case results in an ani- sotropy shift of 35 Oe, as it was found experimentally 161. This may be regarded as an indication of the permissibility of the simplifications. The angular dependence of < ^{H: >%} is already very similar to the measured curve 2, but there exist still other sources which influence the linewidth.

3. DISLOCATION BROADENING. - We investigated the influence of dislocations on the FMR of nickel, produced by plastic deformation. Figure 2 shows some experimental results for a single crystal, deformed at room temperature by tension with a constant rate of 0.25 mm/min. A more detailed discussion is given in

PI.

FIG. 2. - Angular dependence of the linewidth Aw/y in a plastically deformed single crystal disk with (110)-surface.

Stress direction was < ¹¹⁰ >. ^{z =} resolved shear stress.

The indicated directions refer to the field in the parallel case.

= 3.05 kplmm2

A 0

(100)

a

Oe : ,

⁽¹¹⁰⁾

Investigation of the temperature dependence of AH shows

that there is a linewidth difference 6 H between polycrystal and single crystal even above 200OC, where K, 2 K2 x 0. From the temperature dependence of AH in deformed single crystals it follows that 6Hcorresponds to a single crystal deformed by z x 0.7 kp/mm2. That this is indeed a physically reasonable interpretation is shown by the following considerations. With an orientation factor m = 2.8 [7]

one gets a = z m = 2 kp/mm2. This value conforms with the measured critical stress of polycrystalline nickel sheets [8]. Further, we have determined the disloca- tion density N of the examined polycrystals by means of etchpits and found it to be about 2 x lo7 ~ m - ~ ( ' ) . In single crystals the same N corresponds to a deformation by

= 0.7 kp/mm2. Because of the consistency of these facts we conclude that dislocations inside of the grains cause the linewidth difference above 200 OC.

Therefore to explain the polycrystal linewidth we start with curve 4. Curve 4 results from linear interpo- lation between curve 1 and the average values displayed in figure 2 with z x 0.7 kp/mm2. To curve 4 we add the anisotropy broadening which we assume to be proportional to < ^{H: >%} with an angular indepen- dent factor. The greatest possible contribution is 1.2 < H: ^>%, which leads to the experimentally observed value of the polycrystal width in the mini- mum, see curve 5.

100-

MO-

4. OTHER BROADENING MECHANISMS. - A possible source for the still remaining broadening is surface

t * O +

(111)

A a

b

O A

0 A

+ o + + A A a % &

+ . P 0

O 0

(1)

To be published.

(2)

We are indebted to R. Penzel for performing this investi-

gation.

C 1 - ^{776 W. ANDERS} AND E. BILLER roughness. We did not succeed in electropolishing the

polycrystals as well as the single crystals, some etching at the grain boundaries always occurred. The broaden- ing caused by surface roughness increases with increas- ing a [I]. Therefore this effect is probably responsible for the difference near a = 900. The remaining devia- tion then shows an angular dependence as qualitatively expected for a two-magnon scattering process. Because the deviation disappears above 200°C when the crystal anisotropy approaches zero (compare section 3) we conclude, that the crystal anisotropy is the cause for the scattering processes. But in the case in question the grains themselves are probably not the scattering centres, as pointed out in section 2. Perhaps the scattering centres are the grain boundaries.

IV. Conclusion. - At room temperature the angu- lar dependence of AwJy of a nickel polycrystal can be explained by four main contributions (numbers are rough values for the parallel case) : single crystal linewidth (180 Oe), anisotropy broadening described by the independent grain model (220 Oe), dislocation broadening (70 Oe), and a contribution, which depends on the crystal anisotropy and seems to be due to a two-magnon scattering process (80 Oe).

We are greatly indebted to Prof. Dr. W. Rollwagen for his interest in this work. We would like to thank the Deutsche Forschungsgemeinschaft for financial support, and the Leibniz-Rechenzentrum d. Bayer.

Akademie der Wissenschaften for granting computer time.

References

[I] ANDERS (W.) and BILLER (E.), Z. Angew. Phys., 1969, [5] VREHEN (Q. H. F.), BROESE van GROENOU (A.) and

26, 14. de LAU (J. G . M.), Phys. Rev. _{B, 1970,} 1, 2332.

[2] ANDERS (W.) and BILLER (E.), phys. stat. sol. (a) 1970, [6] ANDERS (W.), BASTIAN (D.) and BILLER (E.), Z. Phys.,

3, k 71. 1969, 224, 113.

131 RODBELL (D. S.), Physics, 1965, 1, 279. 171 SCHWINK (Ch.) and VORBRUGG (W.), 2. Natlcuforsch., [4] SCHLOMANN (E.), Phys. Rev., _1969, 182, 632. 1967, 22a, 626.

[8] BILLER (E.), 2. Naturforsch., 1962, 17a, 559.