Journal of Magnetism and Magnetic Materials 97 (1991) 297-304 North-Holland
297
Effective zero field splitting in the ground doublet in ErBa 2Cu30x M.X. H u a n g , E.M. Jackson, S.M. B h a g a t * Department of Physics and Astronomy, Centerfor Superconductivity Research, University of Maryland, College Park, MD 20742-4111, USA
L.C. G u p t a , A.K. R a j a r a j a n a n d R. V i j a y a r a g h a v a n Tata Institute of Fundamental Research, Bombay, India Received 4 October 1990
We report the frequency and magnetic-field dependence of the microwave absorption in ErBa2Cu 3Ox for 1.3 K < T < 100 K. At temperatures below about 10 K, there is a huge increase in the absorption at zero field. Surprisingly, this giant effect comes from the electron spin resonance involving the lowest-lying Kramers doublet of the Er 3+ ion. The ESR is broad and highly distorted, presumably due to a wide distribution of slowly varying dipolar fields which, in turn, leads to an effective zero field splitting of about 1 K.
Perovskite cuprates of the type RiBa2Cu3Ox, where R is Y or a rare earth, have attracted a tremendous amount of attention because of the occurrence of superconductivity at high temperatures for 6.5 < x < 7. It is also agreed that for 6 < x < 6.5 the Cu ions align antiferromagnetically so that these compounds provide interesting possibilities for studying the interplay between magnetism and superconductivity. Apart from Pr, the R ion appears to play the role of a spectator as far as superconductivity is concerned. However, their magnetic behavior is interesting in its own right and many studies have been so directed. In the R = Er case, which is of primary interest here, it is well established [1-4] that for x = 7, there is a transition to an antiferromagnetic state at 0.6 K and that the 3D order is induced by essentially 2D ordering in the CuO planes. Neutron scattering studies [1] have also been used to establish at least some of the positions of the expected 8 Kramers doublets of the E r 3+ ion (4115/2). The overall crystal-field splitting is found to be exceptionally * Author to w h o m all correspondence should be addressed.
large. In our laboratory we have been systematically studying the low lying states in the RIBa2Cu 3Ox compounds using microwave absorption. Microwave absorption is a very sensitive probe not only of the onset of superconductivity, but also of the magnetic ground state so that one can simultaneously study both phenomena. During these measurements, it was recently [5] discovered that there was a huge increase in the microwave absorption in both superconducting and non-superconducting ErBa2Cu30 x at temperatures below --- 10 K. The anomalous absorption ( P a) rose steadily as the temperature was lowered to 1.3 K. No such anomaly was found in Y1Ba2Cu3Ox. In order to explore this phenomenon further we have measured Pa as a function of temperature and applied field. The measurements indicate that one is observing an electron spin resonance which is so broadened and distorted by dipolar interactions that there is an effective splitting of ~ 1 K in the ground doublet of the Er 3+ ions in zero field. The samples were made using standard heatand-grind techniques and were fully oxygenated
0304-8853/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
298
M.X. Huang et al. / Zero fieM splitting
Table 1 Characteristics of Er]Ba2Cu30 x samples
I
Vacuum treatment x a idometric (£) analysis
b (,~)
c (~.)
Fully oxygenated 360°C/6h 400°C/6h 500°C/6h 600°C/6h 650°C/6h
3.8845 3.8795 3.8765 3.8558 3.8557 3.8556
1i.6632 11.6751 11.6835 11.7939 11.8076 11.8066
6.95 6.78 6.65 6.20 6.04 6.01
3.8158 3.8201 3.8297 3.8552 3.8544 3.8551
I
250
Tc (K) 200
91 90 64 -
initially to o b t a i n E r B a 2 C u 3 0 7. Subsequently, oxygen d e p l e t i o n was a c c o m p l i s h e d b y a n n e a l i n g in v a c u u m for 6 h at ever increasing t e m p e r a t u r e s between 330 a n d 650 o C. T h e oxygen c o n t e n t was m e a s u r e d b y an i o d o m e t r i c m e t h o d . A l l the samples were single p h a s e to = 5% as d e t e r m i n e d b y X - r a y diffraction. T a b l e 1 shows the m e a s u r e d lattice constants, oxygen c o n t e n t a n d Tc o f the samples. It is i m p o r t a n t to note that all the s a m p l e s used in the present e x p e r i m e n t s were micron-size powders. T h e m i c r o w a v e m e a s u r e m e n t t e c h n i q u e is described in detail elsewhere [6,7]. H e r e we n o t e only that the m e a s u r e d q u a n t i t y is Pc, the p o w e r reflected f r o m the cavity at its r e s o n a n t frequency. M e a s u r e m e n t s were m a d e at 10, 25.5 a n d 36.5 G H z in a p p l i e d fields f r o m 0 O e to 12 k O e with 1.3 K < T < 100 K. Pa is so large that no m o d u l a tion was necessary to observe it so n o n e was applied. In any case, when there is a sizable abs o r p t i o n at " z e r o " field, it is i n a d v i s a b l e to use the m o d u l a t i o n technique. The a n o m a l o u s increase occurs in all s a m p l e s of E r B a 2 C u 3 0 x with x < 6.8 (figs. 1 a n d 2). T h e effect is m u c h w e a k e r in t h e fully o x y g e n a t e d ( x = 6.95) m a t e r i a l where only a very small increase in Pa is o b s e r v e d b e l o w 4 K. C o m p a r i s o n of the l o w - t e m p e r a t u r e d a t a to the r o o m - t e m p e r a ture m e a s u r e m e n t s shows that t h e c h a n g e in Pc is very large, larger t h a n the d r o p in a b s o r p t i o n seen at T~ in the: s u p e r c o n d u c t i n g samples (fig. 2). It is believed that the change from t e t r a g o n a l s y m m e try is not responsible for the weakness of the a b s o r p t i o n in the x = 6.95 samples since s a m p l e s with x = 6.65 a n d 6.78 which exhibit a large effect
Pc (A. U.)
100 a ErlBa 2 CuaO 6
150
50
10 G H z
0
50
100 T (K)
Fig. 1. Reflected power (PC) at the resonance frequency (cf. fig. 1 in ref. [6]), of a cavity containing ErBa2Cu306 at 10 GHz in zero field. Note the low temperature increase in absorption, Pa- P~ is in arbitrary units. The line is a guide to the eye.
are a l r e a d y o r t h o r h o m b i c . It is also n o t e w o r t h y that s u P e r c o n d u c t i v i t y a n d the a n o m a l o u s a b s o r p tion (which we will show is caused b y m a g n e t i c splitting of the Er 3÷ levels) coexist in s a m p l e s with 6.65 _< x ~ 6.8. T h e a b s o r p t i o n is r e l a t e d to the presence of m a g n e t i c Er ions. S a m p l e s of R B a 2 C u 306, where R = Ho, Y, Eu a n d Pr, have b e e n tested [5] a n d o n l y H o B a 2 C u 3 0 6 (in which H o is m a g n e t i c ) shows a n y hint of a similar effect. T h e c h a n g e in 160
i
l
140 ]
l
I
o&~8Oo~%o
120:
0
I
I
I
I
ErlBa 2 Cu 3 06.65 36.5 GHz
o
Pc (A U) 100 8O
6o[~
o
00000
0 0 0 0
0
o
f
I I I I I i 610 810 100 120 140 160 180 200 40 T (K) Fig. 2. Same as fig. 1 except for ErBa2Cu306.65.
40 ~ © 0 20
M.X. Huang et al. / Zero fieM splitting 140 [
I
I
I
120
I
J
I
I
I
I
I
I
I
I
Er 1Ba2Cu306 36.5 GHz
100 i~~ A 80 ~
0
0
10
I
I
I
I
20
30
40
50
1
I
60 70 T (K)
t
I
80
90
100 110 120
Fig. 3. Same as fig. 1 except at 36.5 GHz.
PrBa2Cu306 is in the wrong direction [5]. That is, absorption decreases at low temperatures. EuBa2Cu306 and YBa2Cu306, where Eu 3+ and y3+ have no moment, show no effect. Since all the samples with x < 6.8 behave similarly (both with regard to temperature and field dependence of Pa), the rest of this discussion will be limited to samples with x = 6.01 except as noted. Figs. 1 and 3 show Pc as a function of temperature at 10 and 36.5 GHz, respectively. The units of absorption are arbitrary so no direct comparison of the size of Pa, defined by the relation Pa = Pc(T) - Pc (minimum), can be made between different frequencies, but it is large compared to the room-temperature absorption at all frequencies above 10 GHz. The temperature dependence of Pa is proportional to 1 / T (fig. 4). Such a temperature dependence is consistent with Pa being proportional to tanh(c/2kT), the expected result for a Boltzman distribution for two states separated by e, when << kT. Considering that Pa becomes nonzero for T < (10-25) K and is proportional to 1 / T down to 1.3 K, this suggests [8] that c - - 1 K. This is totally unexpected since all previous measurements [1,3,4] suggested that the lowest lying levels form a Kramers doublet with the next doublet being roughly 100 K higher. The magnetic origin of the splitting responsible for Pa was further confirmed by observation of the magnetic field dependence of Pa- Figs. 5 a - c show the effect Of applying a dc field (normal to
299
the microwave h-field) at 1.3 K at frequencies of 10, 25 and 36 GHz, respectively. At high fields, Pa vanishes, that is Pc reduces to roughly its high temperature ( -- 10-20 K) value. The peak (marked by arrows in figs. 5a-c) observed at = 3, 8 and 12 kOe at 10, 25 and 36.5 GHz, respectively, is an electron-spin resonance, possibly due to Cu 2 ÷ ions. It disappears when the microwave field is aligned with the static field. Note that there is a peak in the Pa vs. field curves at 25.5 and 36.5 G H z but none at 10 GHz. As the temperature is increased the maximum absorption decreases but the peak (4.8 kOe at 36.5 GHz, for instance) occurs at the same field, i.e. the field dependence of Pa is independent of T. Again, these observations are totally different from those anticipated for a Kramers doublet. At 77 K, the absorption is independent of applied field up to 12 kOe. Qualitatively, the results can be understood as follows. At low frequencies a sufficiently broad ESR line may cause absorption even at zero field. In zero field, both the clockwise and counterclockwise rotating parts of the rf field will contribute to the signal. As the Zeeman field is increased the contribution from the counterclockwise term will be reduced, but if the linewidth is very large compared to the resonance field one can still get a significant absorption near zero field. This will shift the apparent peak toward lower values. Thus,
I
A
I
I
I
I
I
I
I
Erl Ba 2Cu306.65 36.6 GHz
×,o
ErlBa2Cu306
f
A
10 GHz (A U )
01 0
I
I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1/T (K)-1 Fig. 4. Pa for ErBa2Cu306 vs. 1 / T at 10 GHz. The line is a guide to the eye.
300
M.X. Huang et al. / Z e r o field splitting
a
Erl Ba2Cu306 10 GHz 1.3 K
(A U)
0
5
10
Ha(kOe)
15
Er 3+ ions. In the sequel we present two ways of analyzing the data. First, a cursory examination of the lineshapes (figs. 5a-c) reveals their remarkable similarity to N M R lineshapes which arise from interacting spins and are amply discussed in the literature [9,10]. Whereas powder spectra are extremely difficult to interpret in quantitative detail one can make some headway by following the method developed in ref. [9]. Instead of the usual schemes for calculating the moments of the line (a very difficult task for a very broad line, especially for the higher order moments) they suggest that the lineshape be cast in the empirical form Pa(0)l, Ha) = P0(O)l) e x p ( - h H ~ - .Ha4),
b ErlBa2Cu306 I (A U)
I
N
25.5 GHz
\ I ~ H l a ±
0
5
hrf 10
H a (kOe) ErlBa2Cu306 36,5 GHz ~ 1.3K ~ a ± h:f C
Pa ~ (AU) 0
5
(1)
w h e r e . > 0 but h may be positive or negative. If < 0 there will be a peak at H a > 0 while if X > 0 the absorption will decrease monotonically with increasing H a . The parameters obtained from the data of figs. 5 a - c are given in table 2 and the quality of the fit can be seen from figs. 6a-c. The primary advantage of using eq. (1) is that the moments of the line are immediately obtainable in terms of h and . . For example, the first two moments are given by the relations m 2 = ( 8 . ) - 1 / 2 0 _ 3/2 ( ~ ' / [ 2 "
10
X 0 _ 1/2 ( X / [ 2 .
11/2)
]1/2),
(2)
H a (kOe) Fig. 5. (a) The field dependence of Pa at 10 GHz and 1.3 K. The arrow indicates the position of a spin resonance at (presumably due to Cr 2+ ) 3 kOe; (b) the field dependence of Pa at 25.5 GHz and 1.3 K. Note the peak in absorption near 3 kOe. The arrow indicates the position of a spin resonance at 8 kOe; (c) the field dependence of Pa at 36.5 GHz and 1.3 K. Note the peak in absorption near 5 kOe. The arrow indicates the position of a spin resonance at 12 kOe.
at 36.5 G H z the ESR peak is at 4.8 kOe, just below ~0/'y = 4.9 kOe (assuming a ground state moment of 5.3,B). At 25.5 GHz, ~o/~ = 3.4 kOe and Pa peaks at 2.8 kOe, while at 10 G H z the maximum absorption is, at best, close to zero field, well below ~0/'~ = 1.4 kOe. In the present case the highly distorted ESR signals are produced by a wide distribution of dipolar fields among the
M4 = (3.)
°-
5/2(?t/[2"
11/2)D_ 1/2( ? t / t 2 , ] 1/2),
(3) where D,(x) are parabolic cylinder functions [11]. It is clear that the lineshape is controlled by the ratio ( ~ / , ) and that eq. (1) encompasses a very wide variety of line profiles. For the present discussion, we note that Pa is maximum when H a
Table 2 Parameters from the fits of Pa(H) to eq. (1) f (GHz)
?, (kOe - 2)
/t (kOe - '*)
36.5 25.5 10
- 0.06 - 0.02 0.07
0.0013 0.0015 0.003
M.X. Huang et aL / Zero field splitting 160
i
140 120 100
\
<~ 80 m~ 60 40
I
8
Er123 t 10GHz Fit to Eq. 1 20 k = 0.07 kO62 ~t= 0.003 kOe-4 0 I I 0 0.5 1 1.5 2 2.5 3
I
3.5
4
4.5
H a (kOe)
301J
v ~
I b
- - ~
25
I ~-0.02
] i Er123 at 25 GHz Fit to Eq. 1 kOe"2 ~t= 0.0015 kOe"4
20 15 10
",.
5 0
2
0
4
6
1
8
10
12
H a (kOe)
70
60
/
50
,._6.
40
j~
0Y 3o 2010
o 0.0
\
/ I C
Er123 at 36.5 GHz __ Fit to Eq. 1 L=-0.057kO~2 ~t= 0.0013 kOda
I
I
I
I
I
I
1.0
2.0
3.0
4.0
5.0
6.0
\ 7.0
8.0
9.0 10.0
H a (kOe) Fig. 6. (a) A c o m p a r i s o n o f P a ( H a ) at 10 G H z a n d 1.3 K w i t h eq. (1) (full line) for the i n d i c a t e d p a r a m e t e r values; (b) s a m e as (a) b u t f o r 25.5 G H z ; (c) s a m e as (a) b u t f o r 36.5 G H z .
equals (--~//2bt) 1/2, SO that when ?~ = 0 the peak will occur at H a = 0 . One defines c o as the frequency (energy) where A is zero. A linear leastsquares fit reveals that this "zero" field splitting c o = (23 5: 2) G H z or equivalently, (1.1 + 0.1) K. The crystal field in ErBa 2Cu 3 O x has axial symmetry and has been partially worked out by Walters, et al. [2,12]. Er 3+ has J = ~ , and the crystal field will split its 16 levels into eight doublets. Each doublet will consist of two states differing only in being composed of states having Jz
301
values of opposite sign. Each state in the doublet is a mixture of four Jz states whose Jz labels differ by four, e.g. one doublet would be written as a ]+~-)+bl+7)+clT-½)+dl-T-~) where one state has the upper sign and one has the lower sign. There will be four linearly independent combinations of these states with different coefficients ( a i, bi, c i, d,). The other four doublets will be c o m p o s e d of states w i [ - ~ ) + xi[ +-I) + Y~I -T- 3) + z~ I -T- ~ ) . The lowest-lying doublet must consist of a mixture of states such as to allow A jz = 1 transitions. The magnitude of Pa makes higher-order transitions implausible as its cause. According to ref. [2], the next lowest doublet is at = 10 meV (110 K) and will not contribute to the absorption or mix with the lowest two states at the temperatures of interest here. In fact, neutron scattering measurements [2] below the Nrel temperature (0.6 K) strongly suggest that in ErlBa 2Cu 3 0 7 the Er 3 + moments behave like a 2D s p i n - ½ Ising system. The slight variations in crystal symmetry in samples with differing oxygen content has no apparent effect on the ESR. Measurements of the field dependence of Pa at 36.5 G H z show that the peak absorption is at the same field for all samples with 6.01 < x < 6.78. Simply interpreted, this implies that the present data do not support the recent report by Maletta et al. [13] that the Er moment reduces with reducing x. An effective magnetic field must be postulated to split the lowest Kramers doublet produced by the crystal field. This field is unlikely to be due to the Cu 2÷ dipole moments. The dipolar fields of the Cu ions nearest to an Er site cancel due to symmetry in the antiferromagnetic state which exists in Er1Ba2Cu30 x for x < 6.5. For x > 6.5 the Cu m o m e n t is do not order so they could give rise to dipole fields but the Cu moment is rather small. Instead, the large Er 3÷ dipolar field which is presumed responsible for the 0.6 K Nrel temperature [2-4] (TN) at x = 7 is most likely also responsible for the = 1 K energy splitting observed here. Calculations of the dipolar field at an Er site given various configurations of the Er 3÷ moments confirm that fields of the necessary magnitude ( = few kOe) m a y occur. Since the Er 3÷ moments are not ordered above 1 K, one expects the fields
302
M.X. Huang et al. / Zero fieM splitting
to fluctuate with time and average to zero. However, under appropriate conditions they modify the ESR spectrum in a profound way. The effects of fluctuating fields have been discussed thoroughly in the literature. The calculations of interest here are due to K u b o and Toyabe (KT) [14] wherein they considered the effects of fluctuations when the magnitude of the r a n d o m fields is comparable to, or larger than, the Zeeman fields. Since the present experiments reveal no temperature dependence in the ESR line profiles, the K T analysis for the static case (very slow modulation) appears to be the most appropriate. For the transverse and longitudinal resonance spectra at nonzero frequency ~0 they find, respectively,
Ix~ =
2~
-
2~--~°0
I-
o~o~---o+ w2o~-----~
(1 + -A2- + - -A4 ) exp [ - i ~"°"1-¢'d0) 2]}, tot90
6020~2
~
(4)
Table 3 Values of parameters for the KT [14] fits oJ (GHz)
Transverse resolution (Pa ± = Ixx)
10 25.5 36.5
0.6±0.2 1.8±0.1 2.2±0.2
(kOe) 2.0±0.3 2.1±0.1 2.2±0.1
~°*°o exp - ~
~
J
(5)
,
where A 2 / 7 2 = ( H x 2 ) = ( H ~ ) = ( H ~ ) are the mean-square fluctuating fields and ~Oo= "I,H~. • All the observed lineshapes (figs. 5a-c) are reproduced qualitatively by eq. (4). Fig. 7 shows a comparison between I ~ and the 25 G H z line. The 1.0 ~ "3
~
\
0.8
\
0.6
< ¢~
k
25.5GHz ~ 1.3 K
k
0.4 0.2 0
ErBa2Cu3O6.01
~
X N
2
4 6 Ha ( kOe )
8
10
Fig. 7. Comparison of the calculated KT [14] transverse spectral function, Ix~, eq. (4), with the experimental results at 25.5 GHz and 1.3 K. The parameters values are o~/A =1.8, A / ' r = 2.1 kOe.
(kOe) 0.6±0.2 1.8±0.1 2.1±0.2
0.9±0.1 1.9±0.1 1.7±0.2
values of A and 7 required to fit the data are listed in table 3. One notes that the theory reproduces the 25 G H z data most successfully (least spread in parameter values). The best overall choice appears to be A = (16 + 2) G H z and ~, = (8 + 1) G H z / k O e . A 7 of (8 + 1) G H z / k O e corresponds to a magnetic moment of/~ = (5.7 + 0.7)# B, which is somewhat higher than the value estimated from the slight upturn in the low-temperature specific heat (4.5/%) in ref. [3] but in better accord with that obtained from neutron scattering measurements (5.3gB) in ref. [1]. At zero field
Ixx=Izz=(2v)" - 1 / 2 Je x p ( - ~ +
Longitudinal signal (Pall = Izz)
~°~2),
(6)
which has a peak at ~ / A = 1.4 and therefore implies c o = 1.4A = (22 -+ 2) GHz, in good agreement with that deduced above from the lineshape analysis using eq. (1). Applying the external field ( H a ) along the microwave magnetic field (hrf) changes the field dependence of Pa and provides another method for estimating A. A qualitative explanation follows. Since Ajz = _ + 1 transitions are forbidden when h rr is parallel to the static field and, from the standard crystal field splitting analysis, the two states involved in the absorption have no Jz in common, only the component of the effective internal field ( H i) perpendicular to the rf field will be important. Then without loss of generality, h rf may be chosen along the x axis and H i along the z axis. N o w application of an external field along the x axis (parallel orientation) will rotate the total effective static field, H i + H a, towards the x axis. This will reduce the efficacy of the microwaves in causing Aft = _+1 transitions since only
M.X. Huang et aL / Zero field splitting ErlBa2Cu306 Pa
~
25 GHz
(A U )
~
,
1.3K
~
hrf 5
10
H a (kOe) Fig. 8. P.(H) vs. Ha with //~llhre. Note that Pa(H.) has no peak and decreases more rapidly with field here than in fig. 5b where H a 2. h rf"
the component of the rf field perpendicular to the total field will be effective. In contrast, application of H a perpendicular (perpendicular orientation) to h rf, in the y - z plane, will change the magnitude of the total field, but H i + H a will remain in the y - z plane so the effectiveness of the microwaves will not be changed. The net effect is to cause the absorption intensity to decrease rapidly with increasing external field when nallhrf. A typical set of data is shown in fig. 8 and bears comparison with the data of fig. 5b. A simple estimate of the magnitude of H i was obtained as follows. In the perpendicular case the angle (0) between the total equivalent static field and the rf field is always ~r/2 while in the parallel case we have tan 0 = H i / H a. As argued above, Pali/Pa&
-
sinZ0 - 1 / [ 1 + (H2/HiZ)] ,
whence Hi = n a / ~ / e a ±//Pall- 1 .
(7)
Unfortunately, the form of this function yields very unreliable results at large and small H a be1.0 ~ - . I ,-
0.8
_X
Er Ba 2 Cu 3 O6.01 36.5 GHz
0.6
,r\
1.3 K
o-~,\
0.4
303
cause Pa "/Pall = 1. However, at intermediate fields and especially in the vicinity of the peak in Pa ± one gets H i = (3.0 +_ 0.3) kOe for x = 6.01 and 6.2 at 25 and 36 G H z [15]. At higher values of x, the field variation of Pa± and Pall also contains a component due to the field dependence of absorption in the superconducting background and therefore eq. (7) cannot be used. As anticipated, this value is consistent with the contention that two components of .the random field contribute to H i, that is, H i = v~-A. One can also interpret the parallel signal in terms of the longitudinal resonance function Izz Of KT, eq. (5). As in the transverse case, it is found that Iz~ provides an excellent qualitative description of the Pall data at all frequencies. Fig. 9 shows Izz with ~0/A = 2.1, A / y = 1.7, compared with the 36.5 G H z observations. Again, the quantitative fit is most precise (least spread in values displayed in table 3) at 25.5 GHz. The abnormally large value of ~, required to explain the 10 G H z results is not understood. In conclusion, we have shown that the large zero-field microwave absorption observed at low temperatures in ErlBa2Cu30 x arises from a highly broadened and distorted electron-spin-resonance line profile caused by slowly varying dipolar fields. Concomitantly, there is an effective zero field splitting of about 1 K in the ground Kramers doublet of the Er 3+ ion. Further elucidation of this phenomenon is being sought in a study of the microwave absorption in Yl_xErxBa2Cu306 compounds.
Acknowledgements We have greatly benefited from discussions with J. Barak, M.A. Manheimer, S. Tyagi, A. Gould and G. Shaw.
\,.
~= 0.2
X
0
I
0
2
I
References i "~'- ~
4 6 H a ( kOe )
~-
8
__J
10
Fig. 9. Comparison of the calculated K T [14] longitudinal spectrum Izz, eq. (5), with the observed signal at 36.5 G H z and 1.3 K. The parameter values are to/A = 2.1, A/y = 2.1 kOe.
[1] U. Walter, S. Fahy, A. Zettl, S. Louie, M. Cohen, P. Tejedor and A.M. Stacy, Phys. Rev. B 36 (1987) 8899. [2] J.W. Lynn, T.W. Clinton, W - H Li, R.W. Erwin, J.Z. Liu, K. Vandervoort and R.N. Shelton, Phys. Rev. Lett. 63 (1989) 2606.
304
M.X. Huang et al. / Zero field splitting
[3] B.D. Dunlap, M. Slaski, D.G. Hinks, L. Soderholm, M. Beno, K. Zhang, C. Segre, G.W. Crabtree, W.K. Kwok, S.K. Malik, I.K. SchuUer, J.D. Jorgensen and Z. Sungaila, J. Magn. Magn. Mat. 68 (1987) L139. [4] H.P. van der Meulen, J.J.M. Franse, Z. Tarnawski, K. Kadowaki, J.C.P. Klaasse and A.A. Menovsky, Physica C 152 (1988) 65. [5] E.M. Jackson, S.M. Bhagat, L.C. Gupta, A.K. Rajarajan and R. Vijayaraghavan, Bull. A.P.S. 35 (1990) 715. [6] E.M. Jackson, S.B. Liao, J. Silvis, A.H. Swihart, S.M. Bhagat, R. Crittenden, R.E. Glover III and M.A. Manheimer, Physica C 152 (1988) 125. [7] M.L. Spano and S.M. Bhagat, J. Magn. Magn. Mat. 24 (1981) 143. [8] One must not take this determination too seriously since, as we shall see, the "levels" are severely broadened and the "splitting" is purely a consequence of the broad distribution of random fields.
[9] J.G. Powles and B. Carazza, Magnetic Resonance, eds. C.K. Coogan, N.S. Ham, S.N. Stuart, J.R. Pilbrow and G.V.H. Wilson (Plenum Press, New York, 1970). [10] See for instance, A. Abragam, The Principles of Nuclear Magnetism (Clarendon Press, Oxford, 1961). [11] M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Tables (National Bureau of Standards, Galthersburg, 1964) p. 685. [12] See for instance, M.T. Hutchings, Solid State Physics, vol. 16 (Academic press, New York, 1965). [13] H. Maletta, E. P~Srschke and T. Chattopadhyay, Physica C 166 (1990) 9. [14] R. Kubo and T. Toyabe, in: Magnetic Resonance and Relaxation, ed. R. Blinc (North-Holland, Amsterdam, 1967) p. 810. [15] The presence of the Cu spin resonance around 3 kOe in the 10 GHz data makes analysis of that data less dependeble.