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A method for determination of higher-order magnetic anisotropy constants — Importance of the cubic K3 and K4 for certain energy levels models
Journal of Magnetism and Magnetic Materials 30 (1983) 285-294 North-Holland Publishing Company
285
A M E T H O D FOR D E T E R M I N A T I O N OF H I G H E R - O R D E R MAGNETIC A N I S O T R O P Y
C O N S T A N T S - I M P O R T A N C E OF T H E CUBIC K 3 AND K 4 FOR CERTAIN ENERGY LEVEL M O D E L S Czeslaw RUDOWICZ * Institut ffir Physikalische und Theoretische Chemie, Universiti~tErlangen-Niirnberg, D- 8520 Erlangen, Fed. Rep. Germany Received 14 June 1982; in revised form 23 August 1982; in 2nd revised form 8 September 1982
A method for determination of magnetocrystalline anisotropy constants of arbitrary order is proposed. The method is based on a least squares fitting of a phenomenological anisotropy energy for a given symmetry truncated at an arbitrary order term to a theoretical anisotropy energy computed exactly for a given energy level model. Several applications of the method to cubic systems are considered. The study reveals that the widely used expressions in the literature for the cubic constants K~ and K 2 in terms of free energy for the three symmetry direction are of rather limited validity only. The higher-order cubic constants g 3 , K 4 and K 5 are determined besides the usual K~ and K 2 in temperature range 0 to 300 K. The importance of the higher-order terms with respect to the first term in the cubic anisotropy energy is discussed. The results show that the cubic constants K 3 and K 4 cannot be neglected for most of the energy level models studied at certain temperatures.
1. I n t r o d u c t i o n
The magnetocrystalline anisotropy energy or shortly the anisotropy F a of a magnetic crystal is by definition the part of the free energy F of the crystal that depends on the direction of magnetization M (for a review see refs. [1-4]). Thus the anisotropy Fa can be expanded phenomenologically in terms of the direction cosines (1, m, n) - a of M with respect to a choosen system of axes in crystal. For a cubic crystal up to the 12th power in a t expressed with respect to the cubic axes we have [3] (see also appendix): F a ( a ) = g l t ~ -F K2Q + g 3 t ~ 2 + g4c~ Q + K s Q 2
(1)
+ K 6 t ~ 3 "4- . . . .
where +
+
2 2
Ot2~t3 ,
Q ~ ~lOt2~t3,
(2)
and the quantities K~ are called the anisotropy * On leave from: Solid State Division, Institute of Physics, A. Mickiewicz University, 60-769 Poznan, Poland. Present address: The Australian National University, Research School of Chemistry, P.O. Box 4, Canberra, A C T 2600, Australia.
constants. The corresponding expansions for hexagonal and tetragonal systems are available in the literature with various accuracy [1,3,5,6]. In the early days of the single-ion theory of magnetic anisotropy (see, e.g. refs. [7-9]) only the K 1 and K 2 in (1) have been given consideration [10]. It has been widely accepted since then that K 2 is much smaller then K 1 and the series in (1) converges rapidly. Therefore the higher-order terms in (1) have been simply neglected by most of authors (for a review see refs. [1-3,10-12]). Nevertheless, in some cases studied, the constant K 3 ( a n d / o r K 2) appeared, at least at certain temperatures, to be greater than or comparable in magnitude with KE(KI) for cubic systems [13-20,2]. Also for tetragonal [21] and hexagonal [22] systems the higher-order anisotropy terms have been found of importance. The fact that the torque and the ferromagnetic resonance measurements often yield divergent results for the anisotropy constants [10-12] may also be attributed to the neglect of the higher-order terms. One expects that the values of K~'s derived from experimental data as well as those derived from theoretical models depend strongly on the approximation made in expansion
0304-8853/83/0000-0000/$03.00 © 1983 North-Holland
C. Rudowicz / On higher-order magnetic anisotropy constant,s
286
series (like (1) for the cubic case). The present paper originated from out investigations on the anisotropy of Fe 2+ ion in Y I G : S i [23] in view of the recently observed spin-reorientation [24]. The well-known approximated expression for K l and K 2 in terms of F[100], F[110] and F I l l l] (cf. refs. [1,2]) turned out to give erroneous results for that case. Hence we have applied a numerical method for determination of the magnetocrystalline anisotropy constants of arbitrary order. The method is based on a least square fitting of the phenomenological and the theoretical anisotropy energy. The aim of this paper is to present this method and to study the higher-order cubic anisotropy for several energy level models. This study provides a theoretical evidence of the importance of the cubic K 3 and K 4 c o n s t a n t s in certain temperature ranges for the models considered. The results reveal that the values of K, derived from various fitting procedures differ significantly. The method proposed here can be applied to any other symmetry system, too. The detailed results for Fe 2+ in Y I G : S i ( G e ) will be published elsewhere [25].
2. Free energy and magnetic anisotropy For a uniform distribution of magnetic ions over the magnetically inequivalent sites in crystal the free energy per magnetic ion at a given temperature T is given by [1-3,5]
F[hkl]
kT n
In Z exp i=1
j
kT
'
(3)
where n is the number of magnetically inequivalent sites, E,j[hkl] denote the energy levels of an ion at the ith site, where the magnetization M is along the direction [hkl]. The energies are to be found by solving a model Hamiltonian and thus depend on the parameters specific to the model. Once they are known explicitly one can derive the explicit expressions for the anisotropy constants by expanding the In and exp functions in (3) with some approximations as valid for the model and the temperature range considered. However, for more complicated models this way turns out to be too cumbersome.
There is no way to derive the K,'s explicitly when the energies can be calculated only numerically. Then for cubic systems one can resort to the relations [ 1,2,9] K, = 4 ( r [ 1 1 0 ] - F [ 100]),
K2=9(3F[lll]+F[lOO]-3F[llO]),
(4)
which, however, hold exactly only if K 3 and higher-order constants are equal to zero. Most authors have used the relations (4) without any consideration of the higher-order constants. The discussion of K 3 and K 4 at 0 K [15] shows the limited validity of the assumption K, = 0 for i >/3. On the other hand, the existing experimental data (see introduction) suggest that the assmption IK, I >> IK, + 11, i.e. more appropriately formulated IK, f,I >> [K~+ i~ + 11, where ~ is the corresponding angledependent function, can no longer be taken for granted. Below we propose a method for theoretical determination of the magnetic anisotropy constants which can be very useful in solving the question of importance of the higher-order terms. The method is presented in application to cubic systems and the idea can be easily extended to other symmetry systems.
3. Least squares method The present model [25] of energy levels of Fe 2 ÷ in Y I G : Si(Ge) is based on a Hamiltonian consisting of the molecular field (h = g#BHex) and the zero-field spin Hamiltonian with eight parameters Bq~k) [26]. The five energies E,j[hkl] for the spin S = 2 have been derived by perturbation theory [25]. The explicit expressions for K~ and K 2 could be derived only at T = 0 K [23]. The magnetization M in Y I G : S i undergoes a spin-reorientation transition from the axis [100] at low T to the axis [111] at high T [24]. Hence it is necessary to consider the free energy F of Fe z + ion versus the angle 6 that M makes with the direction [100] in the plane (011) contaning the [100] and [111] direction. For this case we have in (1) a~ - l = COS ¢~, a 2 ~ m = a 3 --= n = (1/~f2)sin 6 and thus
C. Rudowicz / On higher-order magnetic anisotropy constants
= c°s28 sin28 + ¼ sin48 = ft ( 8 ) ,
Fa (ernl/i o n)
O = ¼ c0s28 sin48 = f 2 ( 6 ) .
(5)
The cubic anisotropy Fa(8 ) can be written up to the p th term as P
Ff(8) = E
Ktft(8),
(6)
1=1
fi
fl
0.30
0.25
0.20
0.1 5 f3
0.10
0.05
f2 f6 .
11001
10
T=IOK
0.9
0.8 0.7 0.6 0.5
where from (1) we have f3 = fl 2, f4 = fl f2, f5 = f ~ and f6 = f 3 . The functions f, are plotted in fig. 1. The function f5 attains too small values to be shown in fig. 1 (f5(54.74 °) = 0.00137). In the ferromagnetic resonance the anisotropy Fa(0 ) in the plane (li0), with 0 being the angle between M and the axis [001], is usually considered. All results for F~(8) apply, however, also to Fa(O ) for the equivalence f , ( 0 ) - f , ( 8 ) holds. For a given set of the values of the Bq~k) 's and h the calculations have followed two ways. The theoretical free energy Ft(8, T) has been computed from (3) with ~ varying in steps 5 ° between 0 ° and 50 ° and with the last 8 = 54.74 ° (for the direction [111]). Independently the free energy F[100], F [ l l 0 ] and F [ l l l ] has been c o m p u t e d from (3)
o
287
.
20
,
30 5 (deg)
f~
/,0
50 I' [111]
Fig. 1. The angle-dependent function f in the expansion of the cubic anisotropy energy versus the angle 8 between magnetization and the direction [100] in the plane (011).
O.Z. ÷ 0.3 0.2 +
+ +
0.1 0 [loo]
1
10
20
3'o
~o
5(deg)
50lI~II
Fig. 2. Theoretical anisotropy, energy for Fe 2÷ in YIG:Si(Ge) versus the angle 8 with h = 265, Bot2) = 14.6, B~2)= -5.0, B2t2~= -4.3, B0t4)=0.190, BI4)=-0.050, B2t4'=0.0035, B3~4'= - 0 . 0 0 0 2 a n d B4(4) = 0 . 0 0 0 4 c m - i.
and then K I and K 2 from (4). With these values of K 1 and K 2 the curve F P ( 8 ) for p = 2 in (6) and f , ( 6 ) from (5) has been plotted and for comparison the points F,t(Sq)= F t ( S q ) - F 0, where F 0 is taken to be Ft(8 = 0 °) as being the m i n i m u m of Ft(Sq) for all 8q considered. A striking feature observed for some sets of the Bqtk) 's and h at certain temperatures is a p r o n o u n c e d discrepancy between the 'exact' points and the ' a p p r o x i m a t e d ' curve. It is illustrated in fig. 2 for one set of the Bq~k) 's and h choosen so that K 1 and K 2 at 10 K are close to the experimental values for Fe E+ in Y I G : Ge at 4.2 K [16]. The above results show that for the present model the approximation p = 2 in (6) is wrong at high T. Hence the assumption K i - 0 for i >~ 3 is invalid and the relations (4) are inapplicable to this case, at least at high T. The question arises if the theoretical anisotropy energy F,t(8) for our model has an overall cubic higher-order character. This has led us to the idea of a least square fitting of the set of points (F~t(Sq)) to the form (6) with various degrees of approximation from p - - 2 to p = 6. F r o m the condition for m i n i m u m of the
C. Rudowic, / On higher-order magnetic' anisotropv constants
288
quantity
4. Numerical results
N
We= Z {Ef(3q)-Ea'(3q)} 2,
(7)
q=l
where N is the n u m b e r of t h e 6q values considered, one obtains a set of p-linear equations which can be solved for the K,'s. We denote the K,'s fitted in this way by K,c, where e = a to e for p = 2 to 6, respectively. If the usual approximation p = 2 in (6) and (1) is sufficient for a given model at certain T then KI~ and K2a fitted in the above way should be equal to, respectively, K 1 and K 2 calculated from (4). The goodness of the p t h approximation can be measured by standard deviation o ( o 2 = Wp/N). We have tested by the above method several energy level models. The results are presented in the next section.
4.1. The present model for a 3d6(3d 4 ) ion in distorted trigonal symmetry sites In order to discuss the importance of the terms
Ki, fi(3q) in (6) we calculate the percentage R,,(3q) = IKi, fi(3q)/Klefl(3q)l x 100%
(8)
with respect to the first term in (1) for each 8q. As it is seen from fig. 1 the functions f,(8) reach the m a x i m u m for 3 = 54.74 °. Hence the greatest value o f R,~(C~q) in (8) is reached also for 6q = 54.74 °. These values, i.e. R for the direction [111], are given in brackets in tables 1-4 (zero means R less than 0.5%). In all tables 1-4 the anisotropy constants are in c m - 1 / i o n and temperature in K. The results for the present model suitable for Fe z+ in Y I G : Ge are presented in table 1.
Table 1 T h e a n i s o t r o p y c o n s t a n t s for the p r e s e n t m o d e l of e n e r g y levels of Fe 2 + in Y I G : Si(Ge). T h e values of the p a r a m e t e r s B~tx ) a n d h are as i n d i c a t e d in fig. 2 T
10
Kt K2 o2
4.192 - 12.55(33) 4.4E-11
K~ K2, , o2
4.192 -12.55(33) 1.0E-13
Klb K2b o2
4.192 -- 12.55(33) 0.0002(0) 5 . 0 E - 16
KI¢ K2c K3c K4¢ 02 Kid K2d K3a K4d Ksd 02
K3b
100
150
200
250
2.534 - 5.392(24) 2.1E-04
1.628 - 2.890(20) 3.6E 04
0.9899 - 1.561(18) 4.8E 04
0.5672 0.9011(18) 5.6E 04
2.329 -3.472(17) 5.3E 06
1.422 0.9569(7) 4.6E-06
0.8043 +0.1700(2) 3.4E 06
0.4063 +0.5984(16) 2.3E 06
3.380 -- 11.22(37) 1.239(12) 1.1E-08
2.246 -- 7.777(38) 1.655(24) 2.0E 08
1.345 --4.977(41 ) 1.545(38) 1.6E-08
0.7390 3.248(49) 1.314(59) 9.7E-09
/).3517 -- 2.256(71 ) 1.097(103) 5.8E-09
4.192 - 12.55(33) 0.0003(0) 0.0004(0) 4.3E-18
3.374 - 9.810(32) 0.9593(9) - 1.558(2) 9.6E-10
2.238 - 5.806(29) 1.265(19) - 2.172(4) 2.8E-10
1.337 3.236(27) 1.201 (30) 1.918(5) 5.0E 11
0.7332 - 1.880(28) 1.043(47 ) - 1.507(8) 8.9E-12
0.3473 I. 197,(38) 0.8878( 85 ) 1.167(12) 1.9E 12
4.192 -- 12.55(33) 0.0004(0) 0.0003(0) 0.0016(0) 1 . 5 E - 17
3.376 -- 8.339(27) 0.5419(5) -- 1.250(1) -- 9.275( 1) 4 . 6 E - 11
2.240 -- 5.000(25) 1.037( 15 ) -- 2.004(3) 5.082( 1) 9 . 6 E - 12
1.338 -- 2.896(24) 1.105(28) 1.847(5) -- 2.145( 1) 1 . 2 E - 12
0.7335 -- 1.735(26) 1.002(46 ) -- 1.477(7) -- 0.9125( 1) 1.1E-13
0.3474 1.130(36) 0.8689( 83 ) 1.153(12) 0.4192(0) 1.1E 14
3.578 - 9.292(29) 6.6E-05 3.441 -8.002(26) 3.0E-06
300
C. Rudowicz/ On higher-order magnetic anisotropy constants The discrepancy between the points E~t(Sq) and the curve F.,(8) with K I and K 2 (4) seen in fig. 2 is reflected in table 1 as a disagreement between K~, K~a and K 2, K2a. For high T is K2~ eve~a, of opposite sign to K 2. Therefore it is necesary to consider higher terms in (6). As is seen from table 1, the approximations p = 3, 4 and 5 provide successively substantial i m p r o v e m e n t in o 2 over the K~ and K 2 calculated from (4) and over the fitting with p = 2. In table 1 a 'saturation' effect can be observed, namely, the approximation p = 5 is only slightly better than that with p = 4. The contribution Ksd f5 is, however, less than 1% of Kid fl for all T and therefore the termination of the series in (6) at p = 4 is well justified for the present model. The results indicate the importance of the K 3 term for T in the range 150 to 300 K, whereas the K 4 term becomes greater than 10% of the Kt term only for T approaching 300 K. For each approximation the respective curves E l ( & T) have been plotted, however, for their features we omit these curves from fig. 2. The curves for p = 2 pass partially above and below the points in fig. 2, for p = 3 nearly through and for p = 4 and 5 exactly through the points for a given T. A more instructive measure of the goodness of a fit is the value of 02 in table 1.
4.2. The anisotropic ground doublet model for Fe 2 + in trigonal symmetry sites Within the framework of this model [27,28] the lowest two energies are given by
Eij = ,j2~(cos20i + f 2 ) 1/2,
(9)
where ~ = - 1, ~2 "~ 1, ~ is the s p i n - o r b i t coupling constant, f is treated as an adjustable p a r a m eter and 0, is the angle between M and the local trigonal axis. This model has been widely applied for Fe 2+ in Y I G : Si(Ge) but some difficulties still remain (cf. refs. [23-26,16]). We have performed the calculations for the three different values of ?~ and f2 [28,29,16]. Diagrams as in fig. 2 for the previous model reveal the discrepancy for the present model, now at low temperatures. The results with ~ and f 2 for Fe z+ =
289
in Y I G : S i [29,28] are listed in table 2a and b, respectively. The results with ~ and f 2 for Fe 2+ in Y I G : G e [16] are included into ref. [25]. These results indicate that for this model the K 3 and K 4 term is of some importance at low temperatures, whereas the K 5 term can be neglected. It is also found (see table 2b and ref. [25]) that for low values of f 2 [28,16] the fitting with p = 5 gives at T = 40 K and below, K 2 and K 3 of opposite sign comparing with the results of the fittings with p = 2, 3 and 4.
4.3. Low-lying magnetic doublet model for Ru s+ (4d s) in trigonal symmetry sites The two lowest energy levels of Ru 3+ in trigohal sites are given by [30]
Eij=,j½gt.tl~Hex [ (1 + 202.)2cos20i + ansin20i] i / 2
(10) where ej and 0i have the same meaning as in (9) and the parameter a is related to the one-electron parameters: v - of the trigonal field and f - of the spin-orbit coupling. The i m p o r t a n c e of the higher-order terms in (1) at low T for this model has been pointed out in ref. [31], although no higher Ki, except K I and K2, has been derived. The value gl~BH~x -- 500 c m - i and v / ~ = - 0 . 6 [31 ] or - 1.0 [30] can be used for Ru 3 + in Y I G . Some results are presented in tables 3 and 4. For this model the fitting with p = 5 yield only slightly beter o2 than that with p = 4. The contribution of the Ksd term drops very sharply with temperature, being negligible for T > 50 K. The terms K3c and K4c contribute even more than the term Kzc at 10 K and remain important up to T = 125 K (cf. table 4).
Remarks on the fitting with p = 6 for the models
(1)-(3) The fittings with p = 6 have also been performed for the above models. The results seem, however, to be unreliable. Unexpectedly high contributions Kie f with respect to Klef~ are obtained for all T for each of these models. That is, even at low T for the first model and at high T for the second and third model, where from the successive
C. Rudowicz / On higher-order magnetic anisotropy constants
290
Table 2 T h e a n i s o t r o p y c o n s t a n t s for the d o u b l e t m o d e l for Fe 2 + in Y I G : S i (a) X = - 5 1 e r a - l a n d T
f2=0.728
10
K~
100 5.351
150
3.353
K2
- 8.079(17)
- 3.364(11)
02
3.1E-05
1.9E-06
K~ K z~ 02
5.264 - 7.237(15) 3.5E-06
3.331 - 3.150(11) 1.1E-07
K~b
5.197
K2b
-- 10.73(23)
- 3.780(13)
K3b
1.340(9) 7.2R-08
0.2419(2) 1.4E-09
o2
3.319
1.804 - 1.172(7) 9.1E-08
200 0.9921 - 0.4311 (5)
250
300
0.5825
0.3647
- 0.1775(3)
- 0.0814(2)
5.2E-09
7 . 7 E - 10
3 . 3 E - 11
1.799 - 1.125(7) 3.5E-09
0.9909 - 0.4198(5) 1.5E-10
0.5821 0.1743(3) 1.0E-11
0.3646 - 0.0804(2) 9.8E-13
1.797
0.9905
0.5820
0.3646
0.4427(5)
- 0.1802(3)
- 0.0822(3)
0.0423( 1) 2 . 4 E - 11
0.0088(0) 5 . 9 E - 13
0.0023(0) 2.2 E - 14
0.0007(0) 1.2E- 15
- 1.236(8)
K i~ K2¢ K3~
5.182 -6.988(15) 0.6021(4)
3.316 -3.257(11) 0.1387(1)
1.797 - 1.167(7) 0.0287(1 )
0.9904 -0.4320(5) 0.0067(0)
0.5820 -0.1781(3) 0.0019(0)
0.3646 -0.0817(2) 0.0006(0)
K4c o2
- 4.117(3) 3 . 5 E - 11
- 0.5755(1 ) 6 . 7 E - 14
- 0.0759(0) 1 . 6 E - 15
- 0.0118(0) 2 . 6 E - 17
- 0.0023(0) 2 . 6 E - 18
- 0.0005(0) 2 . 6 E - 18
Kid K2a
5.182 - 6.751(14)
3.316 - 3.269(11)
1.797 - I. 169(7)
0.9904 - 0.4322(5)
0.5820 - 0.1782(3)
0.3646 - 0.0817(2)
K3a K4d Ksd
02
(b),X= -71cm T K1
0.1420( 1)
0.0293( 1)
0.0067(0)
0.0019(0)
0.0006(0)
-- 0.5780(1) 0.0728(0) 1.3E-14
-- 0.0763(0) 0.0122(0) 4.1E-17
-- 0.0118(0) 0,0012(0) 3.1E-18
-- 0.0023(0) 0.0004(0) 1.8E-18
-- 0.0005(0) 0.0001 (0) 2.6E-18
100
150
200
300
0.5347(3) -- 4.067(3) - - 1.499(0) 1.2E-11
landf2=0.25(c=0.5)
10
50
3.655
1.355
K2 o2
- 50.50(28) 4.3E-03
- 45.62(27) 2.8E-03
- 20.97(19) 2.4E-04
- 7.597(13) 1.3E-05
- 2.886(9) 8.1 E - 0 7
- 0.566(5) 7.6E-09
Kl~ K2~ o2
19.02 - 41.59(24) 9.8E-04
18.24 - 38.13(23) 5.8E-04
11.97 - 18.62(17) 2.9E-05
6.545 - 7.045(12) 8.9E-07
3.640 - 2.745(8) 3.8E-08
1.354 - 0.5529(5) 2 . 3 E - 10
Klh K2b K3b o2
17.92 -- 99.26(62) 22.16(41) 2.7E-05
17,39 -- 82.50(53) 17.05(33) 1.5 E - 0 5
11.78 -- 28.62(27) 3.842(11) 6.1 E - 0 7
6.512 8.798(15) 0.6737(3) 1.3 E - 0 8
3.633 -- 3.107(10) 0.1389(1) 3.4 E - 10
1.353 -- 0.580(5) 0.0109(0) 8.5 E - 13
Kl~
17.62 - 28.23(18) 8.132(15) - 78.25(16) 6.2E-07
17.16 - 28.33(18) 6.349(12) - 59.68(13) 1.5E-07
11.74 - 17.76(17) 1.697(5) - 11.97(4) 2 . 8 E - 10
6.505 - 7.237(12) 0.3654(2) - 1.720( 1) 4 . 2 E - 13
3.632 - 2.850(9) 0,0883( 1) - 0.2827(0) 2 . 3 E - 14
1.353 - 0.5675(5) 0."0084(0) - 0.0141(0) 4 . 5 E - 17
11.74 -- 17.09(16) 1.507(4) - 11.83(4) - 4.209(0) 9.1 E - 11
6.505 - 7.258(12) 0.3716(2) - 1.725(1) 0.1384(0) 2 . 2 E - 13
3.632 -- 2.858(9) 0.0903(1) - 0.2842(0) 0.0463(0) 8 . 9 E - 16
1.353 -- 0.5679(5) 0.0085(0) - 0.0142(0) 0.0021(0) 7 . 8 E - 18
Kzc K3~ K4~ o2 Kid K2d K3o K4d
K~d o2
19.91
17.68 + 7.996(5) - 2.150(4) -- 70.66(15) -- 228.5(5) 6.5E-08
18.99
17.19 - 10.43(7) + 1.271(2) - 55.93(12) - 112.9(3) 1.9E-08
12.22
6.603
C. Rudowicz / On higher-order magnetic anisotropy constants Table 3 T h e a n i s o t r o p y c o n s t a n t s for m a g n e t i c d o u b l e t m o d e l for Ru 3 ~(4d 5) in Y I G T KI K2 O2
10 34.65 -115.8(37) 4.8E-02
Kla Kz~ 02
32.09 - 89.43(31) 1.4E-02
(g#BHex =
500 c m - i
291
v/~
= --0.6)
100
150
200
250
300
16.84 -33.11(22) 7.1E-04
8.134 -10.58(14) 2.8E-05
4.393 -3.771(10) 1.6E-06
2.526 -1.533(7) 1.3E-07
1.563 -0.6988(5) 1.4E-08
16.40 - 28.85(20) 1.2E-04
8.224 - 9.724(13) 2.4E-06
4.372
2.520
- 3.570 8.3E-08
- 1.476(7) 4.8E-09
Kih
27.86
16.02
8.169
K2b
-- 311.1(124)
-- 49.1(34)
-- 12.61(17)
--4.107(10)
1.110( 5 ) 3.8E -08
0.2063(2) 8 . 3 E - 10
0.0497( 1) - 3 . 0 E - 11
0.0148(0) 1 . 7 E - 12
K3b
85.19(102) 4.4E-04
o2 K I~ K2c
K3c K4c o2 Kid K2d g3d Kad Ksd o2
7.750(16) 2.7E-06
4.361
1.561 - 0.6799(5) 4 . 2 E - 10 1.560 --0.7184(5)
26.72 - 39.70(17) 31.58(39) -- 299.0(41) 5.8E-05
15.92 - 26.00(18) 3.204(7) - 25.35(6) 3.8E-09
8.157 - 9.902(13) 0.5746(2) - 2.987( 1) 1.1 E - 12
4.360 - 3.706(9) 0.1270( 1) - 0.4421 (0) 5 . 3 E - 14
2.517 - 1.528(7) 0.0345(0) - 0.0845(0) 1.9E- 15
1.560 - 0.7001 (5) 0.0112(0) - 0.0201 (0) 1 . 0 E - 16
27.35 + 316.8(129) -- 69.59(85) -- 224.4(30) -- 2248. (34) 4.6E-06
15.93 -- 23.32(16) + 2.445(5) -- 24.79(6) -- 16.87(0) 8.2E-10
8.157 -- 9.913(14) 0.5778(2) -- 2.989(1) 0.0728(0) I.IE~I2
4.360 -- 3.717(9) 0.1302(1) -- 0.4444(0) 0.0702(0) 3.4E-15
2.517 -- 1.531(7) 0.0351 (0) -- 0.0850(0) 0.0133(0) 2.9E-17
1.560 -- 0.7007(5) 0.0113(0) -- 0.0202(0) 0.0035(0) 5.3E-18
approximations p = 2, 3, 4 and 5 it follows that the higher-order terms should be negligible (cf. tables 1-3). Moreover the values of Ki~ (p = 6) depend strongly on the accuracy of the Fat(Sq) used. For the model (2) and (3) the computations of Fat(Sq) were incorporated into the fitting programs and hence the maximum available accuracy of 12 significant digits was achieved. For example, for the model (2) with X = 5 1 cm -1 and f 2 = 0 . 7 2 8 we
obtain then Kie at T = 1 0 K equal to 5.182, -20.43(44), 3.955(25), 57.49(41), -93.84(7), -13.68(29) for i = 1 to i = 6, respectively. Corresponding results when Fat(Sq) entered externally the fitting program with the accuracy of 7 significant digits are as follows: 5.182, -28.88(62), 6.068(39), 95.53(68), -150.9(12), -22.13(47). In both cases o z equals 1.1E-11. For both of the above Kie sets the anisotropy energy (6) yields
Table 4 T h e a n i s o t r o p y c o n s t a n t s for m a g n e t i c d o u b l e t m o d e l for R u 3 + ( 4 d 5) in Y I G T
2.518 -- 1.605(7)
10
100
(gl~BHex =
500 c m
i, v/~
= - 1.0)
150
200
250
300
KI K2 o2
66.82 - 285.3(47) 3.7E-01
32.60 - 82.23(28) 7.6E-03
16.91 - 28.72(19) 3.8E-04
9.240 - 10.93(13) 2.5E-05
5.427 - 4.641 (10) 2.3E-06
3.404 - 2.179(7) 2.8E-07
Kic K 2,: g3c K4c O2
39.34 - 61.41 (17) 125.8(107) - 1066.(100) 3.9E-03
29.25 - 50.47(19) 11.60(13) - 109.3(14) 4.8E-07
16.28 - 24.57(17) 2.258(5) - 15.58(4) 3 . 8 E - 10
9.095 - 10.37(13) 0.5413(2) - 2.609(1) 9 . 0 E - 13
5.387 - 4.562(9) 0.1555(1) - 0.5387(0) 7.9 E - 14
3.390 - 2.170(7) 0.0521(1) - 0.1346(0) 5 . 0 E - 15
292
C. Rudowicz / On higher-order magnetic anisotropy constants
Table 5 The anisotropy constants for orbital singlet model for Fe 2+ at trigonal sites in spinels T
10
100
150
200
250
300
K~ Kz
0.1830 0.0864
0.1717 0.0746
0.1408 0.0501
0.1057 0.0291
0.0756 0.0157
0.0523 0.0081
Kh, Kza
0.1830 0.0864
0.1717 0.0744
0.1409 0.0497
0.1058 0.0286
0.0756 0.0152
0.0523 0.0076
identical curves passing exactly through the theoretical points. On the other hand, for the fittings with p = 2 to p = 5 nearly the same values of K , are obtained for the model (2) and (3) at all T in the case of accuracy of 12 as well as 7 significant digits for the values o f Fat(~q). For the model (1) the values of Fat((~q) entered the fitting programs with the accuracy of 8 significant digits for T = 10 to 200 K and 7 sd for T = 250 to 300 K. Corresponding results for p = 2 to p = 5 are collected in table 1. The results based on the values of Eat(~q) rounded up to 5 significant digits are very close to those in table 1. However, completely different values of K,~, i.e. for the fittings with p = 6, are obtained in the two cases. Keeping in mind that the t e r m K 6 f 6 arises only from higher-order expansions in the free energy (3) we consider the values Kie from the fitting with p = 6 only as possible mathematical solution of the relevant set of linear equations, being of no physical meaning rather. 4.4. Orbital singlet model with S = 2 f o r Fe e + at trigonal sites in spinels
The details on the five energy levels of Fe 2+ considered within the framework of this model [32] are to be found in ref. [33]. Using the values D = 4.4, i.e. B0( 2 ) = - 2 . 9 , and B 0 ( 4 ) : 0 . 0 1 5 , B0( 4 ) = 0.009 and gltBHex = 340 cm i [33] we obtain from (4) K 1 and K 2 very close to K ~ and K z a , respectively (cf. table 5). Therefore the approximation with p = 2 in (6) appears to be sufficient for this model.
dependance of the theoretical anisotropy energy presented in this paper has enabled a study of the higher-order anisotropy terms and their temperature dependance. For most of the energy level models considered the first two cubic anisotropy constants K 1 and K 2 are found insufficient to describe the magnetic anisotropy in the whole temperature range. Consequently the applicability of the relations (4) for K~ and K 2 in terms of F[100], F[110] and F[111] based on the neglect of higher-order anisotropy constants is strongly limited. These relations should only be applied when the correctness of the basic assumption has been proved. Our method provides a practical tool to solve the later question. In a number of papers the higher-order magnetic anisotropy terms have, however, been neglected without any consideration. The discrepancy between the theory and experiment encountered by some authors may be due to the omission of the higher-order anisotropy from consideration. It seems to be the case for the interpretation of the magnetic anisotropy of rare earth ions [17,19] and of Fe 2+ in CdCr2S 4 [34].
Acknowledgements The author is very much in debt to Professor E. K6nig for the hospitality during his stay at Erlangen. Thanks are due to the Alexander von Humboldt Foundation for a researcfa fellowship. Critical reading of the manuscript by Dr. V.P. Desai is also gratefully acknowledged.
5. Discussion
Appendix
The numerical method for determination of the magnetic anisotropy constants from the angular
In this appendix we discuss the relations between the expansion of the anisotropy energy (1)
C. Rudowicz / On higher- order magnetic anisotropy constants
and the other expansions used in the literature. The expansion in terms of the spherical (or cubic) harmonics with the anisotropy coefficients ki [35,1] (to distinguish from the anisotropy constants K~) has several advantages in the theory of magnetic anisotropy over the conventional expansion used here. For instance, it has enabled to relate the anisotropy and magnetization through the power laws [35,36]. On the other hand, the expansion in terms of the direction cosines ai is the most convenient for analyzing experimental data [ 1,2,11,12], especially the torque data [37,38]. The expansion (1) has been generalized in terms of the symmetrical polynomials up to an arbitrary order [37]. The symmetrical polynomials S and P [37,38] are equivalent to the functions • and Q in (2), respectively, within the proportionality factor. Explicit form of the expansion up to the 24th order in a~ has been given in ref. [38]. The expansion (1) corresponds exactly to the first six terms of the expansion [38], whereas in a review article [39] it is written with K 5 interchanged with K 6. Some authors, i.e. refs. [18,40], use the expansion truncated at the third term, however, in a slightly different form: ra
1
4
4
+
+
(A.1) where we use superscript 1 to distinguish these anisotropy constants from the corresponding ones in (1) [1]. Using the identity 4 2= 2Q + (aaa42+ O~140~4_~_Ot2a 4 43) w e f i n d t h e r e l a t i o n s
K~ = K1,
K 1 = K 2 -]- 2 K 3 ,
K 1 = K 3.
(A.2)
It is evident that the use of only two basic functions • and Q is more convenient than the form (A.1), particularly if the higher-order terms are to be taken into account in the expansion of the anisotropy energy. It is worthwhile to mention that the method presented here can be straightforwardly used for determination of the anisotropy coefficients for the cubic as well as other symmetry systems.
293
Note added
After this paper has been submitted for publication the author has derived analytical expressions for the constants K~, K 2, K 3 and g 4 in terms of free energy F[hkl] for some five choosen [hkl] directions. The present results for the Kit's with p = 4, i = 1-4, show an excellent agreement with the corresponding analytical ones for the model (1) and (2). For details see ref. [25]. It is also worthwhile to mention the paper by U. Atzmony and M.P. Dariel (Phys. Rev. B13 (1976) 4006) which has only recently come to our attention. The method used by these authors for determination of the constants K), i = 1, 2 and 3, eq. (A.1), is basically equivalent to our approximation p = 3. From the present study it follows that the values of K21 and KJ may considerably change if the other higher-order constants were taken into account.
References [1] M.I. Darby and E.D. Isaac, IEEE Trans. Magn. MAG-10 (1974) 259. [2] P. Hansen, in: Proc. Intern. School of Physics Enrico Fermi, Course LXX, Physics of Magnetic Garnets, Varenna, Italy, 1977 (North-Holland, Amsterdam, 1978) p. 56. [3] L. Kowalewski and C. Rudowicz, Magnetocrystalline Anisotropy of Ionic Oxide Compounds (Scientific Publications of A. Mickiewicz University, Poznan, 1978) in Polish. [4] G.T. Rado, J. Appl. Phys. 50 (1979) 7285. [5] R.R. Birss, Symmetry and Magnetism (North-Holland, Amsterdam, 1966). [6] V.M. Volkov, E.F. Kustov, T.K. Makietov and G. Steczko, Phys. Stat. Sol. (b) 104 (1981) 649. [7] K. Yosida and M. Tachiki, Progr. Theoret. Phys. 17 (1957) 331. [8] W.P. Wolf, Phys. Rev. 108 (1957) 1152. [9] J.C. Slonczewski, Phys. Rev. 110 (1958) 1341. [10] J. Kanamori, in." Magnetism, Vol. I, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1963) p. 127. [11] Landoldt-Bbrnstein Tables, New Series, Vol. III/12, Part a (1978) and Part b (1980); Vol. 11I/4, Part a (1970) and Part b (1970), eds. K.H. Hellwege and A.M. Hellwege (Springer, Berlin). [12] G. Winkler, Magnetic Garnets (Vieweg, Braunschweig, 1981). [13] R.F. Pearson and R.W. Teale, Proc. Phys. Soc. (London) 75 (1960) 314.
294
C. Rudowic'~ / On higher-order magnetic anisotropy constants
[14] G. Elbinger, Phys. Stat. Sot. 5 (1964) 87. [15] P. Hansen, Phys. Rev. B5 (1972) 3737. [16] P. Hansen, W. Tolksdorf and J. Schuldt, J. Appl. Phys. 43 (1972) 4740. [17] U. Antzmony, M.P.Dariel, E.R. Bauminger, D. Lebenbaum, [. Nowik and S. Ofer, Phys. Rev. B7 (1973) 4220. [18] P. Escudier, Ann. de Phys. 9 (1975) 125. [19] M. Huq, Phys. Stat. Sol. (a) 65 (1981) K29. [20] T. Jagielinski and M. Berkowski, Phys. Star. Sol. (a) 27 (1975) KI7. [21] H. Nagata, T. Miyadai and S. Miyahara, IEEE Trans. Magn. MAG-8 (1972) 451. [22] P.H. Bly, W.D. Corner, K.N.R. Taylor and M.I. Darby, J. Appl. Phys. 39 (1968) 1336. [23] C. Rudowicz, J. Appl. Phys. 53 (1982) 593. [24] G. Balestrino, S. Geller, W. Tolksdorf and P. Willich, Phys. Rev. B22 (1980) 2282. [25] C. Rudowicz, Z. Naturforsch, (a) submitted. [26] C. Rudowicz, Phys. Rev. B21 (1980) 4967.
[27] J.C. Slonczewski, J. Appt. Phys. 32 (1961) 253 S. [28] T.S. Hartwick and J. Smit, J. Appl. Phys. 40 (1969) 3995. [29] R.W. Teale, D.W. Temple and D.I. Weatherley, J. Phys. C3 (1970) 1376. [30] P. Hansen, Phys. Rev. B3 (1971) 862. [31] P. Hansen, Phys. Star. Sol. (b) 47 (1971) 565. [32] R. Gerber and G. Elbinger, J. Phys. C3 (1970) 1363. [33] C. Rudowicz and L. Kowalewski, Physica 80B (1975) 517. [34] B. Hoekstra, R.P. van Stapele and A.B. Voermans, Phys. Rev. B6 (1972) 2762. [35] H.B. Callen and E. Callen, J. Phys. Chem. Solids 27 (1966) 1271. [36] E. Callen, in: Proc. 31M 3 Conf. Montreal (1982). [37] G. Aubert, Y. Ayant, E. Belorizky and R. Casalegno, Phys. Rev. B14 (1976) 5314. [38] R. Gersdorf and G. Aubert, Physica 95B (1978) 135. [39] E.P. Wohlfarth, in: Ferromagnetic Materials, vol. 1, ed. E.P. Wohlfarth (North-Holland, Amsterdam, 1980) p. 1. [40] G. Aubert, J. Appl. Phys. 39 (1968) 504.
285
A M E T H O D FOR D E T E R M I N A T I O N OF H I G H E R - O R D E R MAGNETIC A N I S O T R O P Y
C O N S T A N T S - I M P O R T A N C E OF T H E CUBIC K 3 AND K 4 FOR CERTAIN ENERGY LEVEL M O D E L S Czeslaw RUDOWICZ * Institut ffir Physikalische und Theoretische Chemie, Universiti~tErlangen-Niirnberg, D- 8520 Erlangen, Fed. Rep. Germany Received 14 June 1982; in revised form 23 August 1982; in 2nd revised form 8 September 1982
A method for determination of magnetocrystalline anisotropy constants of arbitrary order is proposed. The method is based on a least squares fitting of a phenomenological anisotropy energy for a given symmetry truncated at an arbitrary order term to a theoretical anisotropy energy computed exactly for a given energy level model. Several applications of the method to cubic systems are considered. The study reveals that the widely used expressions in the literature for the cubic constants K~ and K 2 in terms of free energy for the three symmetry direction are of rather limited validity only. The higher-order cubic constants g 3 , K 4 and K 5 are determined besides the usual K~ and K 2 in temperature range 0 to 300 K. The importance of the higher-order terms with respect to the first term in the cubic anisotropy energy is discussed. The results show that the cubic constants K 3 and K 4 cannot be neglected for most of the energy level models studied at certain temperatures.
1. I n t r o d u c t i o n
The magnetocrystalline anisotropy energy or shortly the anisotropy F a of a magnetic crystal is by definition the part of the free energy F of the crystal that depends on the direction of magnetization M (for a review see refs. [1-4]). Thus the anisotropy Fa can be expanded phenomenologically in terms of the direction cosines (1, m, n) - a of M with respect to a choosen system of axes in crystal. For a cubic crystal up to the 12th power in a t expressed with respect to the cubic axes we have [3] (see also appendix): F a ( a ) = g l t ~ -F K2Q + g 3 t ~ 2 + g4c~ Q + K s Q 2
(1)
+ K 6 t ~ 3 "4- . . . .
where +
+
2 2
Ot2~t3 ,
Q ~ ~lOt2~t3,
(2)
and the quantities K~ are called the anisotropy * On leave from: Solid State Division, Institute of Physics, A. Mickiewicz University, 60-769 Poznan, Poland. Present address: The Australian National University, Research School of Chemistry, P.O. Box 4, Canberra, A C T 2600, Australia.
constants. The corresponding expansions for hexagonal and tetragonal systems are available in the literature with various accuracy [1,3,5,6]. In the early days of the single-ion theory of magnetic anisotropy (see, e.g. refs. [7-9]) only the K 1 and K 2 in (1) have been given consideration [10]. It has been widely accepted since then that K 2 is much smaller then K 1 and the series in (1) converges rapidly. Therefore the higher-order terms in (1) have been simply neglected by most of authors (for a review see refs. [1-3,10-12]). Nevertheless, in some cases studied, the constant K 3 ( a n d / o r K 2) appeared, at least at certain temperatures, to be greater than or comparable in magnitude with KE(KI) for cubic systems [13-20,2]. Also for tetragonal [21] and hexagonal [22] systems the higher-order anisotropy terms have been found of importance. The fact that the torque and the ferromagnetic resonance measurements often yield divergent results for the anisotropy constants [10-12] may also be attributed to the neglect of the higher-order terms. One expects that the values of K~'s derived from experimental data as well as those derived from theoretical models depend strongly on the approximation made in expansion
0304-8853/83/0000-0000/$03.00 © 1983 North-Holland
C. Rudowicz / On higher-order magnetic anisotropy constant,s
286
series (like (1) for the cubic case). The present paper originated from out investigations on the anisotropy of Fe 2+ ion in Y I G : S i [23] in view of the recently observed spin-reorientation [24]. The well-known approximated expression for K l and K 2 in terms of F[100], F[110] and F I l l l] (cf. refs. [1,2]) turned out to give erroneous results for that case. Hence we have applied a numerical method for determination of the magnetocrystalline anisotropy constants of arbitrary order. The method is based on a least square fitting of the phenomenological and the theoretical anisotropy energy. The aim of this paper is to present this method and to study the higher-order cubic anisotropy for several energy level models. This study provides a theoretical evidence of the importance of the cubic K 3 and K 4 c o n s t a n t s in certain temperature ranges for the models considered. The results reveal that the values of K, derived from various fitting procedures differ significantly. The method proposed here can be applied to any other symmetry system, too. The detailed results for Fe 2+ in Y I G : S i ( G e ) will be published elsewhere [25].
2. Free energy and magnetic anisotropy For a uniform distribution of magnetic ions over the magnetically inequivalent sites in crystal the free energy per magnetic ion at a given temperature T is given by [1-3,5]
F[hkl]
kT n
In Z exp i=1
j
kT
'
(3)
where n is the number of magnetically inequivalent sites, E,j[hkl] denote the energy levels of an ion at the ith site, where the magnetization M is along the direction [hkl]. The energies are to be found by solving a model Hamiltonian and thus depend on the parameters specific to the model. Once they are known explicitly one can derive the explicit expressions for the anisotropy constants by expanding the In and exp functions in (3) with some approximations as valid for the model and the temperature range considered. However, for more complicated models this way turns out to be too cumbersome.
There is no way to derive the K,'s explicitly when the energies can be calculated only numerically. Then for cubic systems one can resort to the relations [ 1,2,9] K, = 4 ( r [ 1 1 0 ] - F [ 100]),
K2=9(3F[lll]+F[lOO]-3F[llO]),
(4)
which, however, hold exactly only if K 3 and higher-order constants are equal to zero. Most authors have used the relations (4) without any consideration of the higher-order constants. The discussion of K 3 and K 4 at 0 K [15] shows the limited validity of the assumption K, = 0 for i >/3. On the other hand, the existing experimental data (see introduction) suggest that the assmption IK, I >> IK, + 11, i.e. more appropriately formulated IK, f,I >> [K~+ i~ + 11, where ~ is the corresponding angledependent function, can no longer be taken for granted. Below we propose a method for theoretical determination of the magnetic anisotropy constants which can be very useful in solving the question of importance of the higher-order terms. The method is presented in application to cubic systems and the idea can be easily extended to other symmetry systems.
3. Least squares method The present model [25] of energy levels of Fe 2 ÷ in Y I G : Si(Ge) is based on a Hamiltonian consisting of the molecular field (h = g#BHex) and the zero-field spin Hamiltonian with eight parameters Bq~k) [26]. The five energies E,j[hkl] for the spin S = 2 have been derived by perturbation theory [25]. The explicit expressions for K~ and K 2 could be derived only at T = 0 K [23]. The magnetization M in Y I G : S i undergoes a spin-reorientation transition from the axis [100] at low T to the axis [111] at high T [24]. Hence it is necessary to consider the free energy F of Fe z + ion versus the angle 6 that M makes with the direction [100] in the plane (011) contaning the [100] and [111] direction. For this case we have in (1) a~ - l = COS ¢~, a 2 ~ m = a 3 --= n = (1/~f2)sin 6 and thus
C. Rudowicz / On higher-order magnetic anisotropy constants
= c°s28 sin28 + ¼ sin48 = ft ( 8 ) ,
Fa (ernl/i o n)
O = ¼ c0s28 sin48 = f 2 ( 6 ) .
(5)
The cubic anisotropy Fa(8 ) can be written up to the p th term as P
Ff(8) = E
Ktft(8),
(6)
1=1
fi
fl
0.30
0.25
0.20
0.1 5 f3
0.10
0.05
f2 f6 .
11001
10
T=IOK
0.9
0.8 0.7 0.6 0.5
where from (1) we have f3 = fl 2, f4 = fl f2, f5 = f ~ and f6 = f 3 . The functions f, are plotted in fig. 1. The function f5 attains too small values to be shown in fig. 1 (f5(54.74 °) = 0.00137). In the ferromagnetic resonance the anisotropy Fa(0 ) in the plane (li0), with 0 being the angle between M and the axis [001], is usually considered. All results for F~(8) apply, however, also to Fa(O ) for the equivalence f , ( 0 ) - f , ( 8 ) holds. For a given set of the values of the Bq~k) 's and h the calculations have followed two ways. The theoretical free energy Ft(8, T) has been computed from (3) with ~ varying in steps 5 ° between 0 ° and 50 ° and with the last 8 = 54.74 ° (for the direction [111]). Independently the free energy F[100], F [ l l 0 ] and F [ l l l ] has been c o m p u t e d from (3)
o
287
.
20
,
30 5 (deg)
f~
/,0
50 I' [111]
Fig. 1. The angle-dependent function f in the expansion of the cubic anisotropy energy versus the angle 8 between magnetization and the direction [100] in the plane (011).
O.Z. ÷ 0.3 0.2 +
+ +
0.1 0 [loo]
1
10
20
3'o
~o
5(deg)
50lI~II
Fig. 2. Theoretical anisotropy, energy for Fe 2÷ in YIG:Si(Ge) versus the angle 8 with h = 265, Bot2) = 14.6, B~2)= -5.0, B2t2~= -4.3, B0t4)=0.190, BI4)=-0.050, B2t4'=0.0035, B3~4'= - 0 . 0 0 0 2 a n d B4(4) = 0 . 0 0 0 4 c m - i.
and then K I and K 2 from (4). With these values of K 1 and K 2 the curve F P ( 8 ) for p = 2 in (6) and f , ( 6 ) from (5) has been plotted and for comparison the points F,t(Sq)= F t ( S q ) - F 0, where F 0 is taken to be Ft(8 = 0 °) as being the m i n i m u m of Ft(Sq) for all 8q considered. A striking feature observed for some sets of the Bqtk) 's and h at certain temperatures is a p r o n o u n c e d discrepancy between the 'exact' points and the ' a p p r o x i m a t e d ' curve. It is illustrated in fig. 2 for one set of the Bq~k) 's and h choosen so that K 1 and K 2 at 10 K are close to the experimental values for Fe E+ in Y I G : Ge at 4.2 K [16]. The above results show that for the present model the approximation p = 2 in (6) is wrong at high T. Hence the assumption K i - 0 for i >~ 3 is invalid and the relations (4) are inapplicable to this case, at least at high T. The question arises if the theoretical anisotropy energy F,t(8) for our model has an overall cubic higher-order character. This has led us to the idea of a least square fitting of the set of points (F~t(Sq)) to the form (6) with various degrees of approximation from p - - 2 to p = 6. F r o m the condition for m i n i m u m of the
C. Rudowic, / On higher-order magnetic' anisotropv constants
288
quantity
4. Numerical results
N
We= Z {Ef(3q)-Ea'(3q)} 2,
(7)
q=l
where N is the n u m b e r of t h e 6q values considered, one obtains a set of p-linear equations which can be solved for the K,'s. We denote the K,'s fitted in this way by K,c, where e = a to e for p = 2 to 6, respectively. If the usual approximation p = 2 in (6) and (1) is sufficient for a given model at certain T then KI~ and K2a fitted in the above way should be equal to, respectively, K 1 and K 2 calculated from (4). The goodness of the p t h approximation can be measured by standard deviation o ( o 2 = Wp/N). We have tested by the above method several energy level models. The results are presented in the next section.
4.1. The present model for a 3d6(3d 4 ) ion in distorted trigonal symmetry sites In order to discuss the importance of the terms
Ki, fi(3q) in (6) we calculate the percentage R,,(3q) = IKi, fi(3q)/Klefl(3q)l x 100%
(8)
with respect to the first term in (1) for each 8q. As it is seen from fig. 1 the functions f,(8) reach the m a x i m u m for 3 = 54.74 °. Hence the greatest value o f R,~(C~q) in (8) is reached also for 6q = 54.74 °. These values, i.e. R for the direction [111], are given in brackets in tables 1-4 (zero means R less than 0.5%). In all tables 1-4 the anisotropy constants are in c m - 1 / i o n and temperature in K. The results for the present model suitable for Fe z+ in Y I G : Ge are presented in table 1.
Table 1 T h e a n i s o t r o p y c o n s t a n t s for the p r e s e n t m o d e l of e n e r g y levels of Fe 2 + in Y I G : Si(Ge). T h e values of the p a r a m e t e r s B~tx ) a n d h are as i n d i c a t e d in fig. 2 T
10
Kt K2 o2
4.192 - 12.55(33) 4.4E-11
K~ K2, , o2
4.192 -12.55(33) 1.0E-13
Klb K2b o2
4.192 -- 12.55(33) 0.0002(0) 5 . 0 E - 16
KI¢ K2c K3c K4¢ 02 Kid K2d K3a K4d Ksd 02
K3b
100
150
200
250
2.534 - 5.392(24) 2.1E-04
1.628 - 2.890(20) 3.6E 04
0.9899 - 1.561(18) 4.8E 04
0.5672 0.9011(18) 5.6E 04
2.329 -3.472(17) 5.3E 06
1.422 0.9569(7) 4.6E-06
0.8043 +0.1700(2) 3.4E 06
0.4063 +0.5984(16) 2.3E 06
3.380 -- 11.22(37) 1.239(12) 1.1E-08
2.246 -- 7.777(38) 1.655(24) 2.0E 08
1.345 --4.977(41 ) 1.545(38) 1.6E-08
0.7390 3.248(49) 1.314(59) 9.7E-09
/).3517 -- 2.256(71 ) 1.097(103) 5.8E-09
4.192 - 12.55(33) 0.0003(0) 0.0004(0) 4.3E-18
3.374 - 9.810(32) 0.9593(9) - 1.558(2) 9.6E-10
2.238 - 5.806(29) 1.265(19) - 2.172(4) 2.8E-10
1.337 3.236(27) 1.201 (30) 1.918(5) 5.0E 11
0.7332 - 1.880(28) 1.043(47 ) - 1.507(8) 8.9E-12
0.3473 I. 197,(38) 0.8878( 85 ) 1.167(12) 1.9E 12
4.192 -- 12.55(33) 0.0004(0) 0.0003(0) 0.0016(0) 1 . 5 E - 17
3.376 -- 8.339(27) 0.5419(5) -- 1.250(1) -- 9.275( 1) 4 . 6 E - 11
2.240 -- 5.000(25) 1.037( 15 ) -- 2.004(3) 5.082( 1) 9 . 6 E - 12
1.338 -- 2.896(24) 1.105(28) 1.847(5) -- 2.145( 1) 1 . 2 E - 12
0.7335 -- 1.735(26) 1.002(46 ) -- 1.477(7) -- 0.9125( 1) 1.1E-13
0.3474 1.130(36) 0.8689( 83 ) 1.153(12) 0.4192(0) 1.1E 14
3.578 - 9.292(29) 6.6E-05 3.441 -8.002(26) 3.0E-06
300
C. Rudowicz/ On higher-order magnetic anisotropy constants The discrepancy between the points E~t(Sq) and the curve F.,(8) with K I and K 2 (4) seen in fig. 2 is reflected in table 1 as a disagreement between K~, K~a and K 2, K2a. For high T is K2~ eve~a, of opposite sign to K 2. Therefore it is necesary to consider higher terms in (6). As is seen from table 1, the approximations p = 3, 4 and 5 provide successively substantial i m p r o v e m e n t in o 2 over the K~ and K 2 calculated from (4) and over the fitting with p = 2. In table 1 a 'saturation' effect can be observed, namely, the approximation p = 5 is only slightly better than that with p = 4. The contribution Ksd f5 is, however, less than 1% of Kid fl for all T and therefore the termination of the series in (6) at p = 4 is well justified for the present model. The results indicate the importance of the K 3 term for T in the range 150 to 300 K, whereas the K 4 term becomes greater than 10% of the Kt term only for T approaching 300 K. For each approximation the respective curves E l ( & T) have been plotted, however, for their features we omit these curves from fig. 2. The curves for p = 2 pass partially above and below the points in fig. 2, for p = 3 nearly through and for p = 4 and 5 exactly through the points for a given T. A more instructive measure of the goodness of a fit is the value of 02 in table 1.
4.2. The anisotropic ground doublet model for Fe 2 + in trigonal symmetry sites Within the framework of this model [27,28] the lowest two energies are given by
Eij = ,j2~(cos20i + f 2 ) 1/2,
(9)
where ~ = - 1, ~2 "~ 1, ~ is the s p i n - o r b i t coupling constant, f is treated as an adjustable p a r a m eter and 0, is the angle between M and the local trigonal axis. This model has been widely applied for Fe 2+ in Y I G : Si(Ge) but some difficulties still remain (cf. refs. [23-26,16]). We have performed the calculations for the three different values of ?~ and f2 [28,29,16]. Diagrams as in fig. 2 for the previous model reveal the discrepancy for the present model, now at low temperatures. The results with ~ and f 2 for Fe z+ =
289
in Y I G : S i [29,28] are listed in table 2a and b, respectively. The results with ~ and f 2 for Fe 2+ in Y I G : G e [16] are included into ref. [25]. These results indicate that for this model the K 3 and K 4 term is of some importance at low temperatures, whereas the K 5 term can be neglected. It is also found (see table 2b and ref. [25]) that for low values of f 2 [28,16] the fitting with p = 5 gives at T = 40 K and below, K 2 and K 3 of opposite sign comparing with the results of the fittings with p = 2, 3 and 4.
4.3. Low-lying magnetic doublet model for Ru s+ (4d s) in trigonal symmetry sites The two lowest energy levels of Ru 3+ in trigohal sites are given by [30]
Eij=,j½gt.tl~Hex [ (1 + 202.)2cos20i + ansin20i] i / 2
(10) where ej and 0i have the same meaning as in (9) and the parameter a is related to the one-electron parameters: v - of the trigonal field and f - of the spin-orbit coupling. The i m p o r t a n c e of the higher-order terms in (1) at low T for this model has been pointed out in ref. [31], although no higher Ki, except K I and K2, has been derived. The value gl~BH~x -- 500 c m - i and v / ~ = - 0 . 6 [31 ] or - 1.0 [30] can be used for Ru 3 + in Y I G . Some results are presented in tables 3 and 4. For this model the fitting with p = 5 yield only slightly beter o2 than that with p = 4. The contribution of the Ksd term drops very sharply with temperature, being negligible for T > 50 K. The terms K3c and K4c contribute even more than the term Kzc at 10 K and remain important up to T = 125 K (cf. table 4).
Remarks on the fitting with p = 6 for the models
(1)-(3) The fittings with p = 6 have also been performed for the above models. The results seem, however, to be unreliable. Unexpectedly high contributions Kie f with respect to Klef~ are obtained for all T for each of these models. That is, even at low T for the first model and at high T for the second and third model, where from the successive
C. Rudowicz / On higher-order magnetic anisotropy constants
290
Table 2 T h e a n i s o t r o p y c o n s t a n t s for the d o u b l e t m o d e l for Fe 2 + in Y I G : S i (a) X = - 5 1 e r a - l a n d T
f2=0.728
10
K~
100 5.351
150
3.353
K2
- 8.079(17)
- 3.364(11)
02
3.1E-05
1.9E-06
K~ K z~ 02
5.264 - 7.237(15) 3.5E-06
3.331 - 3.150(11) 1.1E-07
K~b
5.197
K2b
-- 10.73(23)
- 3.780(13)
K3b
1.340(9) 7.2R-08
0.2419(2) 1.4E-09
o2
3.319
1.804 - 1.172(7) 9.1E-08
200 0.9921 - 0.4311 (5)
250
300
0.5825
0.3647
- 0.1775(3)
- 0.0814(2)
5.2E-09
7 . 7 E - 10
3 . 3 E - 11
1.799 - 1.125(7) 3.5E-09
0.9909 - 0.4198(5) 1.5E-10
0.5821 0.1743(3) 1.0E-11
0.3646 - 0.0804(2) 9.8E-13
1.797
0.9905
0.5820
0.3646
0.4427(5)
- 0.1802(3)
- 0.0822(3)
0.0423( 1) 2 . 4 E - 11
0.0088(0) 5 . 9 E - 13
0.0023(0) 2.2 E - 14
0.0007(0) 1.2E- 15
- 1.236(8)
K i~ K2¢ K3~
5.182 -6.988(15) 0.6021(4)
3.316 -3.257(11) 0.1387(1)
1.797 - 1.167(7) 0.0287(1 )
0.9904 -0.4320(5) 0.0067(0)
0.5820 -0.1781(3) 0.0019(0)
0.3646 -0.0817(2) 0.0006(0)
K4c o2
- 4.117(3) 3 . 5 E - 11
- 0.5755(1 ) 6 . 7 E - 14
- 0.0759(0) 1 . 6 E - 15
- 0.0118(0) 2 . 6 E - 17
- 0.0023(0) 2 . 6 E - 18
- 0.0005(0) 2 . 6 E - 18
Kid K2a
5.182 - 6.751(14)
3.316 - 3.269(11)
1.797 - I. 169(7)
0.9904 - 0.4322(5)
0.5820 - 0.1782(3)
0.3646 - 0.0817(2)
K3a K4d Ksd
02
(b),X= -71cm T K1
0.1420( 1)
0.0293( 1)
0.0067(0)
0.0019(0)
0.0006(0)
-- 0.5780(1) 0.0728(0) 1.3E-14
-- 0.0763(0) 0.0122(0) 4.1E-17
-- 0.0118(0) 0,0012(0) 3.1E-18
-- 0.0023(0) 0.0004(0) 1.8E-18
-- 0.0005(0) 0.0001 (0) 2.6E-18
100
150
200
300
0.5347(3) -- 4.067(3) - - 1.499(0) 1.2E-11
landf2=0.25(c=0.5)
10
50
3.655
1.355
K2 o2
- 50.50(28) 4.3E-03
- 45.62(27) 2.8E-03
- 20.97(19) 2.4E-04
- 7.597(13) 1.3E-05
- 2.886(9) 8.1 E - 0 7
- 0.566(5) 7.6E-09
Kl~ K2~ o2
19.02 - 41.59(24) 9.8E-04
18.24 - 38.13(23) 5.8E-04
11.97 - 18.62(17) 2.9E-05
6.545 - 7.045(12) 8.9E-07
3.640 - 2.745(8) 3.8E-08
1.354 - 0.5529(5) 2 . 3 E - 10
Klh K2b K3b o2
17.92 -- 99.26(62) 22.16(41) 2.7E-05
17,39 -- 82.50(53) 17.05(33) 1.5 E - 0 5
11.78 -- 28.62(27) 3.842(11) 6.1 E - 0 7
6.512 8.798(15) 0.6737(3) 1.3 E - 0 8
3.633 -- 3.107(10) 0.1389(1) 3.4 E - 10
1.353 -- 0.580(5) 0.0109(0) 8.5 E - 13
Kl~
17.62 - 28.23(18) 8.132(15) - 78.25(16) 6.2E-07
17.16 - 28.33(18) 6.349(12) - 59.68(13) 1.5E-07
11.74 - 17.76(17) 1.697(5) - 11.97(4) 2 . 8 E - 10
6.505 - 7.237(12) 0.3654(2) - 1.720( 1) 4 . 2 E - 13
3.632 - 2.850(9) 0,0883( 1) - 0.2827(0) 2 . 3 E - 14
1.353 - 0.5675(5) 0."0084(0) - 0.0141(0) 4 . 5 E - 17
11.74 -- 17.09(16) 1.507(4) - 11.83(4) - 4.209(0) 9.1 E - 11
6.505 - 7.258(12) 0.3716(2) - 1.725(1) 0.1384(0) 2 . 2 E - 13
3.632 -- 2.858(9) 0.0903(1) - 0.2842(0) 0.0463(0) 8 . 9 E - 16
1.353 -- 0.5679(5) 0.0085(0) - 0.0142(0) 0.0021(0) 7 . 8 E - 18
Kzc K3~ K4~ o2 Kid K2d K3o K4d
K~d o2
19.91
17.68 + 7.996(5) - 2.150(4) -- 70.66(15) -- 228.5(5) 6.5E-08
18.99
17.19 - 10.43(7) + 1.271(2) - 55.93(12) - 112.9(3) 1.9E-08
12.22
6.603
C. Rudowicz / On higher-order magnetic anisotropy constants Table 3 T h e a n i s o t r o p y c o n s t a n t s for m a g n e t i c d o u b l e t m o d e l for Ru 3 ~(4d 5) in Y I G T KI K2 O2
10 34.65 -115.8(37) 4.8E-02
Kla Kz~ 02
32.09 - 89.43(31) 1.4E-02
(g#BHex =
500 c m - i
291
v/~
= --0.6)
100
150
200
250
300
16.84 -33.11(22) 7.1E-04
8.134 -10.58(14) 2.8E-05
4.393 -3.771(10) 1.6E-06
2.526 -1.533(7) 1.3E-07
1.563 -0.6988(5) 1.4E-08
16.40 - 28.85(20) 1.2E-04
8.224 - 9.724(13) 2.4E-06
4.372
2.520
- 3.570 8.3E-08
- 1.476(7) 4.8E-09
Kih
27.86
16.02
8.169
K2b
-- 311.1(124)
-- 49.1(34)
-- 12.61(17)
--4.107(10)
1.110( 5 ) 3.8E -08
0.2063(2) 8 . 3 E - 10
0.0497( 1) - 3 . 0 E - 11
0.0148(0) 1 . 7 E - 12
K3b
85.19(102) 4.4E-04
o2 K I~ K2c
K3c K4c o2 Kid K2d g3d Kad Ksd o2
7.750(16) 2.7E-06
4.361
1.561 - 0.6799(5) 4 . 2 E - 10 1.560 --0.7184(5)
26.72 - 39.70(17) 31.58(39) -- 299.0(41) 5.8E-05
15.92 - 26.00(18) 3.204(7) - 25.35(6) 3.8E-09
8.157 - 9.902(13) 0.5746(2) - 2.987( 1) 1.1 E - 12
4.360 - 3.706(9) 0.1270( 1) - 0.4421 (0) 5 . 3 E - 14
2.517 - 1.528(7) 0.0345(0) - 0.0845(0) 1.9E- 15
1.560 - 0.7001 (5) 0.0112(0) - 0.0201 (0) 1 . 0 E - 16
27.35 + 316.8(129) -- 69.59(85) -- 224.4(30) -- 2248. (34) 4.6E-06
15.93 -- 23.32(16) + 2.445(5) -- 24.79(6) -- 16.87(0) 8.2E-10
8.157 -- 9.913(14) 0.5778(2) -- 2.989(1) 0.0728(0) I.IE~I2
4.360 -- 3.717(9) 0.1302(1) -- 0.4444(0) 0.0702(0) 3.4E-15
2.517 -- 1.531(7) 0.0351 (0) -- 0.0850(0) 0.0133(0) 2.9E-17
1.560 -- 0.7007(5) 0.0113(0) -- 0.0202(0) 0.0035(0) 5.3E-18
approximations p = 2, 3, 4 and 5 it follows that the higher-order terms should be negligible (cf. tables 1-3). Moreover the values of Ki~ (p = 6) depend strongly on the accuracy of the Fat(Sq) used. For the model (2) and (3) the computations of Fat(Sq) were incorporated into the fitting programs and hence the maximum available accuracy of 12 significant digits was achieved. For example, for the model (2) with X = 5 1 cm -1 and f 2 = 0 . 7 2 8 we
obtain then Kie at T = 1 0 K equal to 5.182, -20.43(44), 3.955(25), 57.49(41), -93.84(7), -13.68(29) for i = 1 to i = 6, respectively. Corresponding results when Fat(Sq) entered externally the fitting program with the accuracy of 7 significant digits are as follows: 5.182, -28.88(62), 6.068(39), 95.53(68), -150.9(12), -22.13(47). In both cases o z equals 1.1E-11. For both of the above Kie sets the anisotropy energy (6) yields
Table 4 T h e a n i s o t r o p y c o n s t a n t s for m a g n e t i c d o u b l e t m o d e l for R u 3 + ( 4 d 5) in Y I G T
2.518 -- 1.605(7)
10
100
(gl~BHex =
500 c m
i, v/~
= - 1.0)
150
200
250
300
KI K2 o2
66.82 - 285.3(47) 3.7E-01
32.60 - 82.23(28) 7.6E-03
16.91 - 28.72(19) 3.8E-04
9.240 - 10.93(13) 2.5E-05
5.427 - 4.641 (10) 2.3E-06
3.404 - 2.179(7) 2.8E-07
Kic K 2,: g3c K4c O2
39.34 - 61.41 (17) 125.8(107) - 1066.(100) 3.9E-03
29.25 - 50.47(19) 11.60(13) - 109.3(14) 4.8E-07
16.28 - 24.57(17) 2.258(5) - 15.58(4) 3 . 8 E - 10
9.095 - 10.37(13) 0.5413(2) - 2.609(1) 9 . 0 E - 13
5.387 - 4.562(9) 0.1555(1) - 0.5387(0) 7.9 E - 14
3.390 - 2.170(7) 0.0521(1) - 0.1346(0) 5 . 0 E - 15
292
C. Rudowicz / On higher-order magnetic anisotropy constants
Table 5 The anisotropy constants for orbital singlet model for Fe 2+ at trigonal sites in spinels T
10
100
150
200
250
300
K~ Kz
0.1830 0.0864
0.1717 0.0746
0.1408 0.0501
0.1057 0.0291
0.0756 0.0157
0.0523 0.0081
Kh, Kza
0.1830 0.0864
0.1717 0.0744
0.1409 0.0497
0.1058 0.0286
0.0756 0.0152
0.0523 0.0076
identical curves passing exactly through the theoretical points. On the other hand, for the fittings with p = 2 to p = 5 nearly the same values of K , are obtained for the model (2) and (3) at all T in the case of accuracy of 12 as well as 7 significant digits for the values o f Fat(~q). For the model (1) the values of Fat((~q) entered the fitting programs with the accuracy of 8 significant digits for T = 10 to 200 K and 7 sd for T = 250 to 300 K. Corresponding results for p = 2 to p = 5 are collected in table 1. The results based on the values of Eat(~q) rounded up to 5 significant digits are very close to those in table 1. However, completely different values of K,~, i.e. for the fittings with p = 6, are obtained in the two cases. Keeping in mind that the t e r m K 6 f 6 arises only from higher-order expansions in the free energy (3) we consider the values Kie from the fitting with p = 6 only as possible mathematical solution of the relevant set of linear equations, being of no physical meaning rather. 4.4. Orbital singlet model with S = 2 f o r Fe e + at trigonal sites in spinels
The details on the five energy levels of Fe 2+ considered within the framework of this model [32] are to be found in ref. [33]. Using the values D = 4.4, i.e. B0( 2 ) = - 2 . 9 , and B 0 ( 4 ) : 0 . 0 1 5 , B0( 4 ) = 0.009 and gltBHex = 340 cm i [33] we obtain from (4) K 1 and K 2 very close to K ~ and K z a , respectively (cf. table 5). Therefore the approximation with p = 2 in (6) appears to be sufficient for this model.
dependance of the theoretical anisotropy energy presented in this paper has enabled a study of the higher-order anisotropy terms and their temperature dependance. For most of the energy level models considered the first two cubic anisotropy constants K 1 and K 2 are found insufficient to describe the magnetic anisotropy in the whole temperature range. Consequently the applicability of the relations (4) for K~ and K 2 in terms of F[100], F[110] and F[111] based on the neglect of higher-order anisotropy constants is strongly limited. These relations should only be applied when the correctness of the basic assumption has been proved. Our method provides a practical tool to solve the later question. In a number of papers the higher-order magnetic anisotropy terms have, however, been neglected without any consideration. The discrepancy between the theory and experiment encountered by some authors may be due to the omission of the higher-order anisotropy from consideration. It seems to be the case for the interpretation of the magnetic anisotropy of rare earth ions [17,19] and of Fe 2+ in CdCr2S 4 [34].
Acknowledgements The author is very much in debt to Professor E. K6nig for the hospitality during his stay at Erlangen. Thanks are due to the Alexander von Humboldt Foundation for a researcfa fellowship. Critical reading of the manuscript by Dr. V.P. Desai is also gratefully acknowledged.
5. Discussion
Appendix
The numerical method for determination of the magnetic anisotropy constants from the angular
In this appendix we discuss the relations between the expansion of the anisotropy energy (1)
C. Rudowicz / On higher- order magnetic anisotropy constants
and the other expansions used in the literature. The expansion in terms of the spherical (or cubic) harmonics with the anisotropy coefficients ki [35,1] (to distinguish from the anisotropy constants K~) has several advantages in the theory of magnetic anisotropy over the conventional expansion used here. For instance, it has enabled to relate the anisotropy and magnetization through the power laws [35,36]. On the other hand, the expansion in terms of the direction cosines ai is the most convenient for analyzing experimental data [ 1,2,11,12], especially the torque data [37,38]. The expansion (1) has been generalized in terms of the symmetrical polynomials up to an arbitrary order [37]. The symmetrical polynomials S and P [37,38] are equivalent to the functions • and Q in (2), respectively, within the proportionality factor. Explicit form of the expansion up to the 24th order in a~ has been given in ref. [38]. The expansion (1) corresponds exactly to the first six terms of the expansion [38], whereas in a review article [39] it is written with K 5 interchanged with K 6. Some authors, i.e. refs. [18,40], use the expansion truncated at the third term, however, in a slightly different form: ra
1
4
4
+
+
(A.1) where we use superscript 1 to distinguish these anisotropy constants from the corresponding ones in (1) [1]. Using the identity 4 2= 2Q + (aaa42+ O~140~4_~_Ot2a 4 43) w e f i n d t h e r e l a t i o n s
K~ = K1,
K 1 = K 2 -]- 2 K 3 ,
K 1 = K 3.
(A.2)
It is evident that the use of only two basic functions • and Q is more convenient than the form (A.1), particularly if the higher-order terms are to be taken into account in the expansion of the anisotropy energy. It is worthwhile to mention that the method presented here can be straightforwardly used for determination of the anisotropy coefficients for the cubic as well as other symmetry systems.
293
Note added
After this paper has been submitted for publication the author has derived analytical expressions for the constants K~, K 2, K 3 and g 4 in terms of free energy F[hkl] for some five choosen [hkl] directions. The present results for the Kit's with p = 4, i = 1-4, show an excellent agreement with the corresponding analytical ones for the model (1) and (2). For details see ref. [25]. It is also worthwhile to mention the paper by U. Atzmony and M.P. Dariel (Phys. Rev. B13 (1976) 4006) which has only recently come to our attention. The method used by these authors for determination of the constants K), i = 1, 2 and 3, eq. (A.1), is basically equivalent to our approximation p = 3. From the present study it follows that the values of K21 and KJ may considerably change if the other higher-order constants were taken into account.
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294
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