Ferromagnetic resonance of gadolinium doped calcium vanadium garnets
A. K. Srivastava and M. J. Patni
Citation: J. Appl. Phys. 81, 1863 (1997); doi: 10.1063/1.364015
View online: http://dx.doi.org/10.1063/1.364015
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Published by the American Institute of Physics.
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Ferromagnetic resonance of gadolinium doped calcium vanadium garnets
A. K. Srivastava and M. J. Patnia)
Department of Material Science and Metallurgical Engineering, Indian Institute of Technology, Powai,
Bombay 400 076, India
~Received 18 January 1996; accepted for publication 13 September 1996!
The ferromagnetic resonance ~FMR! line shapes of Gd-substituted calcium vanadium garnets
Y1.62z Ca1.4Gdz Fe4V0.4Zr0.6O12 ~0.8<z<1.4! have been studied as the function of the Gd content.
The samples had been synthesized by two different routes, with porosities varying from 0.23% to
8.9%. The FMR linewidth, DH, in low porosity samples is small ~15–20 Oe! and becomes seven
to eight times higher in high porosity samples. The linewidth can be explained on the basis of
Schloemann’s theory of anisotropy and porosity broadening in polycrystalline materials. The line
shape is Lorentzian for low porosity and Gaussian for high porosity samples. This variation of the
FMR line shape with porosity is explained on the basis of the stochastic theory. © 1997 American
Institute of Physics. @S0021-8979~96!07924-8#
I. INTRODUCTION
The ferromagnetic relaxation in polycrystalline calcium
vanadium garnets has been extensively studied by several
workers.1–4 This study has shown that the large increase in
linewidth in polycrystalline garnets compared to single crystals arises due to porosity and random orientation of anisotropy axes within crystallites. The ferromagnetic resonance
~FMR! linewidth, DH, in polycrystalline garnets is then expressed as
DH5DH int1DH a 1DH p ,
~1!
where DH int is the intrinsic linewidth, DH a and DH p are line
broadening due to anisotropy, and porosity, respectively. The
line broadening DH a and DH p are obtained from the spin
wave theory. Following this approach, Patton5 evaluated the
different contributions to linewidth in ultralow-loss indium
substituted calcium vanadium ~In–CaV! polycrystalline garnets with saturation magnetization, 4p M s , of 1000 G and
DH of 2 Oe. His analysis indicates that at room temperature
DH of 2 Oe is comprised of DH int;0.4 Oe, DH a ;0.2 Oe,
DH p ;0.9 Oe, and DH imp;0.5 Oe, where DH imp is the contribution from impurity. In principle, DH imp could be reduced by taking higher purity of yttrium oxide.
The spin wave theory was also applied to estimate the
anisotropy and porosity contributions in large linewidth materials ~DH;400 Oe!. It was found that there was good
agreement with theory for the anisotropy broadening but not
so good for the porosity broadening.6
Warin et al.7 have studied the variation of linewidth over
a large frequency range ~0.5–8 GHz! and the change in FMR
line shape with rf power level. They have concluded that the
variation of DH with frequency and rf power cannot be explained on the basis of the existing theories.
For the reasons discussed above, there exists a wide scatter of data on FMR linewidths in polycrystalline samples and
line shapes are rarely reported. Iglesias et al.8 observed
asymmetry of the ferromagnetic resonance peak in LiZnTi
ferrites and have attributed it to the anisotropy field from the
a!
Electronic mail: [email protected]
J. Appl. Phys. 81 (4), 15 February 1997
presence of nonstoichiometric divalent ions. Dorsey et al.9
have shown that though spherical samples of both barium
ferrite and yttrium iron garnet ~YIG! have line shapes which
are symmetrical, the thin films of these materials have an
asymmetrical line shape due to Suhl instability.
In this article, the dependence of FMR line shape on
porosity has been studied. A low loss garnet composition for
which the anisotropy field is small compared to saturation
magnetization (H a !4 p M s ) has been chosen. To vary the
porosity without affecting the homogeneity we have used
two different processing techniques, the conventional ceramic and citrate-gel route. The former has been used to give
samples with relatively larger porosity in the range of 2%–
9%, while the latter is used to produce samples with small
porosity, less than 0.3%. The FMR line for the sample with
small porosity is observed to fit to a Lorentzian expression,
while the line for the sample with large porosity fits better to
a Gaussian expression. This observation has been explained
on the basis of the stochastic theory of line shape in magnetic
resonance developed by Kubo.10
II. LINE SHAPE ANALYSIS
In the presence of large anisotropy fields (H a @4 p M s )
each grain resonates independently, consequently the resonance curve manifests as a broad envelope to individual
grain resonances. The linewidth and line shape are then not
related to the basic relaxation mechanism and depend largely
on the grain size distribution.11 Contrary to this, the relaxation phenomena for H a !4 p M s is well described by the
spin wave theory. Classically the line broadening due to anisotropy should be of the order of 2H a . However, Schloemann found that the linewidth is narrowed by dipolar interactions and for spherical samples is given by12
DH a 52.07
S D
H 2a
4pM s
.
~2!
The contribution to linewidth from porosity obtained by
Schloemann is13
DH p 51.5~ 4 p M s ! p,
0021-8979/97/81(4)/1863/5/$10.00
© 1997 American Institute of Physics
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~3!
1863
where p is the porosity.
Values of DH a have been obtained from Eq. ~1! using
experimental values of DH, p, and 4 p M s and taking
DH int;0.3 Oe. The DH is measured using an electron spin
resonance ~ESR! spectrometer with field modulation at rf
close to 10 GHz. Spherical samples are placed at the center
of a TE102 rectangular cavity. The external static magnetic
field is perpendicular to the rf field and is swept from 0 to 5
kOe, keeping the frequency and the input rf power to cavity
constant. The derivative of rf power absorbed by the sample
with respect to the static field (d P/dH) is plotted as a function of the steady field. The ratio of the power absorbed by
the sample under investigation to the power dissipated by the
cavity is given by14
P
54 p h x 9 Q 0 ,
P0
~4!
9 ~ H ! 54 p 2 M s g ~ H ! ,
x6
~5!
where
DH 1/2
.
p @~ DH 1/2! 2 1 ~ H7H 0 ! 2 #
~6!
9 is the susceptibility for the circular polarization.
Here x 6
The positive sign denotes the rotation of the magnetic rf field
in the same sense as the precessional motion of the dipole
and so leads to a singularity at H5H 0 . For FMR at a constant excitation frequency, v, as in the present case, the resonance field for a spherical specimen is given by H 05v/g.
Here g is the gyromagnetic ratio, H is the applied dc field,
DH 1/2 is the half width at half power maxima, and g(H) is a
normalized Lorentzian curve. The term 2DH 1/2 gives the
value of DH in Eq. ~1!.
As stated earlier, the observed FMR line shape, g(H),
for small porosity samples fits to a Lorentzian @Eq. ~6!#. For
high porosity samples, however, it fits better to a Gaussian
expression
g~ H !5
1
A2 ps
F
exp
G
2 ~ H2H 0 ! 2
,
2s2
~7!
where
s5
DH 1/2
& ~ ln 2!
H ~ t ! 5H 0 1H 1 ~ t ! ,
where the average of the fluctuating field is zero
H 1 ~ t ! 50.
~10!
.
~8!
v ~ t ! 5 v 01 v 1~ t !
The stochastic theory has been successfully used to account for the narrowing of nuclear magnetic resonance
~NMR! lines due to the thermal motion of nuclei. This theory
has also been used to explain the exchange narrowing of
dipole broadened lines in electron paramagnetic resonance.
J. Appl. Phys., Vol. 81, No. 4, 15 February 1997
~11!
with
v̄ 1 50.
~12!
We wish to evaluate the shape of the resonance line intensity
I~v2v0! in the presence of the modulation v1 . This is described in terms of two characteristic parameters of the stochastic process, amplitude of modulation D, and correlation
time of modulation, tc . These are defined below.
~a! If P~v1! denotes the probability distribution of finding the spins with frequency v1 in a large number of independent magnetic spins in the ensemble, the amplitude of
modulation, D, is given by
D 25
Ev
2
2
1 P ~ v 1 !d v 15 ^ v 1& .
~13!
Thus D is a measure of the magnitude of the modulation.
~b! To define the correlation time of modulation, tc , a
correlation function is defined by
c~ t !5
1
^ v 1 ~ t ! v 1 ~ t1 t ! & .
D2
~14!
The correlation time tc is now given by
t c5
Ec
`
0
~15!
~ t ! dt.
It is shown in Ref. 10 that there exists two limiting cases
for which there is a definite form of the line shape. These are
fast and slow modulation processes.
~i! Fast modulation ~D•tc !1!. In this case the product of
D and tc is small compared to 1 and the shape of the line is
given by a Lorentzian function
I~ v2v0!5
III. THE STOCHASTIC THEORY
1864
~9!
The spin in the field H(t) precesses with frequency
where x9 is the imaginary part of the complex susceptibility
x5x82j x 9 of the sample, Q 0 is the unloaded Q of the cavity, and h is the fill factor, v s /V. Here v s is the sample
volume, V is the cavity volume, and v s !V. The magnitude
of x9 contains the shape function g(H) and can be obtained
from the equation of motion for the magnetization vector.
For the Landau–Lifshitz equation of motion
g~ H !5
In Eq. ~2!, the dipolar fields in FMR cause narrowing of the
anisotropy linewidth by (H a /4p M s ). The analysis of line
shape may therefore give information on the relaxation phenomena in ferromagnetic systems which even today are not
fully understood. We briefly outline below the shape of the
line in magnetic resonance expected on the basis of the stochastic theory. In this, we follow closely the treatment given
by Kubo.10
Consider the precession of a spin in a steady magnetic
field, H 0 , on which a random fluctuating field H 1 (t) is superimposed
D v 1/2
,
p @~ v 2 v 0 ! 2 1D v 21/2#
~16!
where
D v 1/25D 2 t c 5 g DH 1/2 .
~16a!
But as D•tc !1, we obtain
D v 1/2!D.
~16b!
A. Srivastava and M. Patni
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This shows that due to fast modulation, the Lorentzian line,
which should have a half width of D, gets narrowed by a
factor D•tc .
The expression for DH a in Eq. ~2! can be understood on
the basis of Eq. ~16a!. Without dipole narrowing the spins
within randomly oriented crystallites will experience an anisotropy field which will vary from 1H a to 2H a around the
applied field H 0 . The term DH a is then equal to 2H a . When
dipolar narrowing occurs the linewidth is reduced by a factor
H a /4p M s . The physical reasoning is as follows. Suppose
that the local field has a value for a short time tc and then
I,
changes randomly to 6H a . As the magnetization vector, M
precesses around H 0 , the spin experiences fluctuation in anisotropy energy. This change occurs against the dipolar interaction which aligns the spins along the easy direction. The
correlation time tc can then be taken as 1/( g 4 p M s ). The
linewidth is thus reduced by
D• t c 5
Ha
.
4pM s
~17!
~ii! Slow modulation ~D•tc @1!. In this case the line shape is
Gaussian
I~ v2v0!5
1
A2 p D
H
exp 2
J
~ v2v0!2
.
2D 2
~18a!
Here
D5D v 1/2~ A 2 ln 2! 51.17D v 1/2 .
~18b!
The linewidth Dv1/2 then is nearly equal to the amplitude of
modulation, D.
Occasionally it is found that near the resonance field
region the line shape is Lorentzian, while in the wings, it is
Gaussian. This is explained10 on the assumption that the line
shape within regions varies as follows:
u v 2 v 0 u ,D
Lorentzian,
u v 2 v 0 u .D
Gaussian.
~19!
The present observation that the shape of FMR line for low
porosity samples is Lorentzian and for high porosity is
Gaussian, is similar to the NMR result on dipolar linewidth
in magnetically diluted rigid lattices. If the magnetic moments are distributed at random on lattice points it can be
shown15 that the NMR line shape is Gaussian for concentration f .0.1 and Lorentzian for f ,0.01. In the present case,
for porosity p,0.5% the line is Lorentzian and for p.5% it
is Gaussian. A possible physical explanation is that in a low
porosity sample most spins see a weak local fluctuating field
which causes narrowing of the line. In this case a major
contribution to the line shape comes from spins near the pore
which see an effective field H 0 1H loc , where H loc5H a 1H p .
Here H p is the field due to porosity. For low porosity materials H a dominates over H p and narrowing of the line is
given by Eq. ~17!.
For high porosity samples H p dominates over H a , as the
correlation time is determined by the time dependence of
H loc . The field due to porosity can be taken as time independent. H a , however, depends on the angle between the magnetization vector and the easy axis within the crystallite. As
J. Appl. Phys., Vol. 81, No. 4, 15 February 1997
the magnetization vector precesses around H 0 1H loc , the anisotropy field varies with time. The correlation time is then
given by t c 51/( g DH a ), where DH a is linewidth due to anisotropy. Hence
D• t c 5 g DH p •
1
DH p
5
.
g DH a DH a
~20!
From Eq. ~20! it follows that when DH p is large compared to
DH a , the line shape is Gaussian. The Lorentzian line shape
g(H) in Eq. ~6! follows from the equation of motion of
magnetization and agrees with the intensity function
I~v2v0! in Eq. ~16!. However, the Gaussian function @Eq.
~18a!# cannot be obtained from the equation of motion. This
could arise due to the dominance of the dipolar field over the
anisotropy field.
IV. EXPERIMENTAL RESULTS
We have synthesized Y1.62z Ca1.4Gdz Fe4V0.4Zr0.6O12
~0.8<z<1.4! garnets. For these compositions the value of
4 p M s at room temperature is in the range of 750–1100 G.
Samples were synthesized using conventional ceramic and
citrate-gel methods and were coded Cn-(z) or CIT-(z).
Here C and CIT denote ceramic and citrate-gel routes, respectively, z denotes the concentration of gadolinium substitution, and n denotes the batch number of the ceramic
sample. FMR measurements were performed on a Varian E
line century E-112 spectrometer. Samples were spherical in
shape ~diam;1.0–1.8 mm!. The field calibrated resonance
absorption spectra were obtained using an XY recorder.
Tetra-cyano-ethylene ~TCNE! with g52.002 77 was used to
provide the reference g factor. All measurements reported
here were made either at 9.07 or 9.50 GHz. The saturation
magnetization was measured using an EG&G, PARC, VIBRATING SAMPLE MAGNETOMETER ~VSM! model
no. 4500. The values of porosities and 4 p M s at room temperature are given in Table I for samples prepared by the
citrate-gel route. These samples have FMR linewidths in the
range of 15–20 Oe. Samples of the sample composition prepared by ceramic route have DH seven to eight times higher.
The FMR line shape of low linewidth sample is Lorentzian
and that of high linewidth is Gaussian. The procedure to fit
an observed line shape to a Lorentzian is as follows. The
experimental points are fitted to a Lorentzian derivative
dP
A ~ H2H 0 !
5
,
2
dH @ DH 1/21 ~ H2H 0 ! 2 # 2
~21!
where from Eqs. ~4!, ~5!, and ~6!
A522 ~ 4 p h !~ 4 p M s ! Q 0 P 0 DH 1/2 .
~22!
Taking into consideration the gain, G, of the spectrometer
A5
kQ 0 P 0 h M s DH 1/2
,
G
~23!
where k is an arbitrary constant. Note that the Lorentzian
curve in Eq. ~6! is normalized to unity. We take a trial value
of A, H 0 , and DH 1/2 and plot values of d P/dH using Eq.
~21! with H varying at an interval of 2 Oe. This is then
A. Srivastava and M. Patni
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1865
TABLE I. Resonance field ~H 0!, half linewidth ~DH 1/2!, A, and g eff for the system
Y1.62z Ca1.4Gdz Fe4V0.4Zr0.6O12 obtained by line shape analysis. The term d represents the deviation of the
observed data points from Lorentzian. FMR spectra for samples with z50.8 and 1.0 were taken at 9.07 GHz and
with z51.2 and 1.4 were taken at 9.50 GHz.
z
4pM S
~G!
p
~%!
H 0 ~Oe!
60.1
DH 1/2 ~Oe!
60.02
A
~arbitrary units!
g eff
d
0.8
1.0
1.2
1.4
1080
890
840
750
0.23
0.27
0.26
0.28
3235.5
3255.5
3406.5
3398.0
8.60
9.89
8.92
14.10
22 430
97 300
116 772
440 700
2.006 51
1.9879
1.9863
1.9910
0.288
0.607
0.912
1.14
compared with the experimental values of d P/dH. The standard root mean square deviation is calculated using
d 25
1
N
N
(
n51
~ x en 2x tn ! 2 ,
~24!
where x en and x tn are the nth experimental and theoretical
points, respectively. Using a computer program A, H 0 , and
DH 1/2 are so chosen that d2 in Eq. ~24! has a minimum value.
Figure
1
gives
the
FMR
spectrum
of
Y0.8Ca1.4Gd0.8Fe4V0.4Zr0.6O12 synthesized by the citrate-gel
route. The values of H 0 , DH 1/2, A, and the deviation d obtained using Eqs. ~21! and ~24! are given in Table I. This
table also includes values of these parameters for z51.0, 1.2,
and 1.4 prepared by the citrate-gel route whose fitting for the
FMR spectra are similar to Fig. 1. The g eff by this technique
could be measured with an accuracy of 0.05% and DH with
an accuracy of 0.5%.
In low loss ferromagnetic specimen, in addition to the
main resonance, there also occurs an additional resonance
which originates from the coupling of the electromagnetic
field to the magnetostatic ~MS! wave mode. In the present
case ~Fig. 1!, this occurs for MS-~531! mode at a field of
3132 Oe. The index ~NMR! for the magnetostatic mode is
based on Walker’s notation.16 This line is also Lorentzian
and has DH 1/2 of 6.495 Oe. Its amplitude is one order of
magnitude smaller ~A51.6933103! compared to the main
resonance ~A52.2433104!.
The FMR spectrum of C32~0.8! prepared by the ceramic technique is compared with the best fitting to a Lorent-
FIG. 1. ESR spectra of CIT-~0.8! with a Lorentzian best fit.
1866
J. Appl. Phys., Vol. 81, No. 4, 15 February 1997
zian and a Gaussian curve in Figs. 2~a! and 2~b!, respectively. This sample has a porosity of 8.7% while the sample
prepared by citrate-gel route has only 0.23%. The parameters
used to fit these two curves are given in Table II.
The d value for Gaussian fit is almost 1/3 of the Lorentzian and the experimental points in the wings as well as in
the center fit well. For the Lorentzian curve there is a large
discrepancy between calculated and experimental values in
the wings. The reason is that the C32~0.8! sample satisfies
the slow modulation criterion while CIT-~0.8! satisfies the
fast modulation criterion. For example, for CIT-~0.8!, using
relation ~3! and the data from Table I, we obtain DH p 53.73
Oe. On substituting this value in Eq. ~1! and using
DH int;0.3 Oe, we get DH a 513.17 Oe. This value when
FIG. 2. ~a! ESR spectra of C32~0.8! with a Lorentzian best fit. ~b! ESR
spectra of C32~0.8! with a Gaussian best fit.
A. Srivastava and M. Patni
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TABLE II. Values of H 0 , DH 1/2, A, g eff , and d obtained from fitting of the
observed spectrum to a Lorentzian and a Gaussian for z50.8 prepared by
ceramic route. The spectrum was recorded at 9.07 GHz. The porosity of the
sample is 8.7% and 4 p M S is 807 G at room temperature.
A
H 0 DH 1/2
~Oe! ~Oe! ~arbitrary units!
Lorentzian 3215 63.7
Eq. ~3!
Gaussian 3215 49.49
Eq. ~4!
7
3.43310
2000
case, from Eq. ~20!, D• c 5 d H p /DH a 5105.3/21.854.8@1
and the line shape is Gaussian. This is in agreement with the
result in Fig. 2~b!.
V. CONCLUSION
d
g eff
Remarks
2.006 11.36 Wings not fitting
2.006
3.76
Wings fit well
The FMR line shapes of Gd-substituted calcium vanadium garnets have been examined for samples with small
and large porosities. The low porosity sample gives a Lorentzian curve while the high porosity sample gives a Gaussian
curve. These observations have been explained on the basis
of the stochastic theory of line shape in magnetic resonance.
1
substituted in Eq. ~2! gives H a 582.9 Oe. C32~0.8! has
p58.7% and 4 p M s 5807 G. This gives DH p 5105.3 Oe.
The observed values of DH(52DH 1/2) for this sample using
the Lorentzian fitting is 127.4 Oe and hence DH a 521.8 Oe.
This value of DH a is of the same order as that obtained for
CIT-0.8. Using this value in Eq. ~2! we obtain H a 591.76
Oe. These values of H a are small compared to the value of
H a obtained by Patton17 who, for a similar garnet
Y2.36Ca0.64Fe2.68V0.32O12 with 4 p M s 51200 G, finds
H a 5114 Oe.
For the CIT-~0.8! sample, the porosity contribution to
the linewidth, DH p , is small compared to the anisotropy
contribution, DH a . In this case, from Eq. ~17!,
D• t c 5H a /4p M s 582.9/108050.077!1. This then satisfies
the condition for fast modulation. The line is Lorentzian in
agreement with the observation in Fig. 1.
For C32~0.8!, DH p is almost five times of DH a . In this
J. Appl. Phys., Vol. 81, No. 4, 15 February 1997
H. H. Van Hook, J. J. Green, F. Euler, and E. R. Czerlinsky, J. Appl. Phys.
39, 730 ~1969!.
2
G. Winkler, P. Hansen, and P. Holst, Philips Res. Rep. 27, 151 ~1972!.
3
G. P. Espinosa and S. Geller, J. Appl. Phys. 35, 2551 ~1964!.
4
Y. Machida, H. Saji, T. Yamadaya, and M. Asanuma, IEEE Trans. Magn.
MAG-10, 613 ~1974!.
5
C. E. Patton, IEEE Trans. Magn. MAG-8, 433 ~1972!.
6
Q. H. F Vrehen, J. Appl. Phys. 40, 1849 ~1969!.
7
D. Warin, J. C. Mage and W. Simonet, J. Appl. Phys. 55, 2452 ~1984!.
8
A. Iglesia, I. V. Guerasimenko, S. Diaz, and A. Gonzaler, Phys. Status
Solidi A 140, 221 ~1993!.
9
P. Dorsey, J. B. Sokoloff, and C. Vittoria, J. Appl. Phys. 74, 1938 ~1993!.
10
R. Kubo, Fluctuation, Relaxation and Resonance in Magnetic Systems,
edited by D. Ter Haar ~Oliver and Boyd, London, 1962! pp. 23–68.
11
E. Schloemann, J. Phys. Chem. Solids 6, 257 ~1958!.
12
E. Schloemann, J. Phys. Chem. Solids 6, 242 ~1958!.
13
E. Schloemann, AIEE Spec. Pub. T-91, Proceedings of the Conference on
Magnetics and Magnetic Materials, 1956 ~unpublished!, p. 600.
14
J. P. Pilbrow, Transition Ion Electron Paramagnetic Resonance ~Clarendon, Oxford, 1990!, pp. 187–195.
15
C. Kittel and E. Abrahams, Phys. Rev. 90, 238 ~1958!.
16
L. R. Walker, J. Appl. Phys. 29, 318 ~1958!.
17
C. E. Patton, Phys. Rev. 179, 352 ~1969!.
A. Srivastava and M. Patni
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1867