J. Phys. Chem. Solids
Pergamon Press 1969. Vol. 30, pp. 1745-1750.
Printed in Great Britain.
CONSTANTS IN ORTHOFERRITES YFeO, AND LuFeO,*
G. GORODETSKY Department of Electronics, The Weizmann Institute of Science, Rehovot, Israel (Received
Abstract-The exchange interaction with nearest neighbors J,, next nearest spectroscopic splitting factor g were determined for two orthoferrites YFeO, temperature analysis of the paramagnetic susceptibility yielded the following Jl/k = -27.2? l”K, J,/k = -1.2 ?0.2”K, g = 2.05 kO.1, and for LuFeO,, J,/k -1%0.19(, R = 2.0070.05. Compatible values of J, were found from the low bility measurements and Ntel point. 1. INTRODUCTION
YFeO, and LuFeO, represent a class of weak ferromagnets with a perovskite structure, in which the magnetically ordered ions lie approximately on a simple cubic lattice. Each iron ion is coupled through the O”- ions by a superexchange to six nearest neighbors (n.n.) in an essenrially antiferromagnetic ordering[ 1,2]. Weak ferromagnetism arises from a canting of the predominantly antiferromagnetic spins, due to an antisymmetric exchange term, -Dti( Si x Sj). This interaction tends to tilt the two interacting spins St, Sj from a strict antiferromagnetic alignment in a plane perpendicular to the constant sector Dij. In the orthoferrites studied the vector D = F,Fji coincides THE
with the b-axis of the orthorhombic unit cell, the spontaneous magnetic moment lies along the c-axis and the antiferromagnetic direction is parallel to the u-axis. Evidence indicating the existance of an antiferromagnetic coupling through more than one diamagnetic ion has been given by Blasse for isostructural compounds in which the nearest neighbors to each iron ion on the Fe”+ sublattice are replaced by dia*The research reported in this document has been sponsored in part by the Air Force Materials Laboratory Research and Technology Division AFSC through the European Office of Aerospace Research, United States Air Force Contract F61052-67C-0040.
neighbors J2, and the and LuFeO,. A high results: For YFeO,, = -24.2 I+ l”K, J,/k = temperature suscepti-
magnetic ions; e.g. SrsFeSbOG. Here, the next nearest neighbors (n.n.n.) exchange interaction is found to be -3 per cent of the n.n. interaction in rare earth orthoferrites. We therefore might expect a similar magnitude of n.n.n. coupling in the orthoferrites studied, YFeO, and LuFeO,. In Section 2 of this article the experimental techniques are described. In Section 3 the derivation of the n.n. and the n.n.n. exchange interaction is carried out from the measurements of the paramagnetic susceptibility, using the high temperature expansion.? We also calculate the n.n. exchange interaction from measurements of the low temperature perpendicular susceptibility. In Section 4 the results are compared with those obtained from the NCel point. 2. EXPERIMENTAL
The magnetic susceptibility of some single crystals+ of LuFeO, and Y FeO, was measured at 4*2”K, and above the transition temperature TN, (622” and 643”K, respectively)  to 1000°K. The measurements at 4.2”K were carried out using a motor driven vibrating tpreliminary results of the exchange constants in LuFeO, found by the high temperature analysis, have been used already for the expression of the paramagnetic susceptibility by a Padt approximant . *The single crystals were kindly provided by J. P. Remeika from The Bell Telephone Laboratories.
sample magnetometer. The sampie was vibrated between pick-up coils mounted in a bore of a superconducting magnet wherein the magnetic field is applied parallel to the axis of vibration and the axis of the pick-up coils. The magnetic susceptibility was measured along the two principle crystallographic axes perpendicular to the antiferromagnetic a-axis. The results obtained at 4*2”K for YFeO, are: x6 = (2-8 tO.1) lo-” (e.m.u.lmole), xC = (2.7 30.1) lO+ (e.m.u.1 mole) and for LuFeO,, x~, xc = (Z-S& 0.2) 10-;’ (e.m.u.imole). Where xs and xc donote the perpendicuIar susceptibilities along the orthorhombic axes b and c respectively. The magnetic susceptibility was also measured in powder samples above the NCel point up to 1500°K. These measurements were done by a balance magnetometer. The value of the paramagnetic susceptibility was obtained from the measured susceptibiIity by substracting the calculated diamagnetic contribution. The results obtained are presented in Fig. 1. Susceptibility measurements from above TX up to -lOOO°K were also done by a vibrating sample magnetomete wherein the magnetic field was applied perpendicular to the direction of vibration. The data obtained by the vibrating sample magnetometer agree to within 2 per cent with those obtained using the balance method. The anisotropy in the susceptibility above the NCel temperature was measured in single crystals using a torquemeter. It was found that for TIT,. > 1.08, the anisotropy (xi-xj)i xj < 0.01 (i.j= o.h.c). This amount of anisotropy is negligible for our purposes. 3. ANALYSIS OF THE EXPERIMENTAL
(i) High temperature susceptibility The high temperature series expansion is an exact asymptotic series expansion for the susceptibility when T > T.?. Rushbrooke and Wood(7J have calculated the first six terms for the nearest neighbors Heisenberg ferromagnet and antiferromagnet. Wojtowicz and Joseph [S], DaltonL91
Fig. I. The inverse paramagnetic susceptibility x’““ vs. tempe~ture for LuFeO, and YFeO,. The value of x’~’ were obtained by subtking the calculated diamagnetic contribution from the experimental values measured by a balance magnetometer.
and Pirnie et &IO] have calculations to next nearest actions for which, in the external magnetic field H, Hamiltonian has the form:
extended these neighbors interpresence of an the Heisenberg
The first term represents the n.n. interaction with an exchange integral J1. The summation is carried out over all pairs of n.n. The second term represents the n.n.n. with an exchange integral J2, where the summation extends over all pairs of n.n.n. Zeeman energy for the entire crystal is represented by the third term. Here, pR denotes Bohr magneton and R is a spectroscopic splitting factor. Assuming that the ratio CI= J,jJ, = O-03 deduced previously for a similar perovskite structure, Sr,FeSbO, as noted above, holds also for the orthoferrites we find that the antisymmetric exchange energy and the anisotropy energy are signifi-
cantly smaller than the n.n.n. exchange interaction [ 1 l- 131. We shall therefore confine our discussion to the Hamiltonian (1) and thus the high temperature series for the inverse susceptibility may be written in the form : x-’ = T/C x ’ ( 1-t b,,( J,/kT ),, (J,/kT )“I) U,,n (2) where T denotes temperature, C is the Curie constant and k is the Boltzman constant. The prime in the summation sign exclude m = n = 0. Numerical expressions for the coefficients b,, up to n + m = 6 have been tabulated by Pirnie et al.[ IO]. The determination of the exchange integrals J,, J, and the Curie-Weiss constant C was undertaken by fitting equation (2) to the experimental results, using a least squares method to obtain the best fit [8,14]. The fitting procedure was carried out systematically for points with physical validity in a plane defined by J1 and C. Values of J, were chosen for each set (fl, C) to give least mean square deviation [ 141. The three parameters were then determined from the coordinates J, +J2 and C of the point that had the smallest deviation. The error in determining J,, Jz and C due to the convergence properties can be estimated by calculating these parameters in different temperatures regions where the convergence of the power series equation (2) is fairly rapid. The results obtained for Y FeO,, are presented in Table 1. No well defined minima were found in the region MOOoK< T < lSOO”K,the accuracy in determining J1, Jz and C for each temperature region is better than the spread of the results given in Table 1 (e.g. in the region 2.5 > TIT, > l-5, a deviation of 1 per cent in J, and C increases the root mean square deviation, by a factor -2. At this point (J,, C) a deviation of 10 per cent in Jz causes the same increase in the root mean squares deviation). The same analysis was undertaken for LuFeO,. The results obtained from the temperature region 850°K < T < 1500°K are J,/k = -24*2”K1 Jz,ik = -1.O”K, C = 4.35 (e.m.u. “K;lmolef.
Table Curie fitting to the
1. The exchange constants and the constant in YFeO:, as obtained by the high temperature series expansion experimental data in various temperature regions
Temperature 1500 > 1500 > 1.500> 1500 >
T T T T
> > > >
region 850°K 900°K 950°K 1000°K
27.2 27.4 27.8 27.9
- 1-2 - 1.2 --1.2 - 1.4
4.61 4.63 4.67 4-69
It is of interest to consider the molecular field approximation in the same temperature range. Using a least squares fitting the Curie constant C and the paramagnetic CurieWeiss temperature 0, can be determined from the well known relation xc”) = C/ (T - 0,) where x(p) is the paramagnetic susceptibility. It was found that 0, = - 1400°K for YFeO, and 8, = - 1200°K for LuFeO,. The exchange constants were calculated from 8, and T, using the following relations for a simple cubic lattice [ 151: T N =2s(S+1)(-6J 3k tjU=2S(S+1)(6J 3k
1+12J) 2 1
Here S = 5/2 is the spin value of a S-state ion, Fe3+. The results obtained for J1, J2 and the spectroscopic splitting factor g = ~3kC~~~~~~ (S -t- 1) are given in Table 2. (ii) Low temperature susceptibility According to the molecular field theory for a weak ferromagnet, the susceptibility along the orthorhombic axes, b and c (perpendicular to the antiferromagnetic a-axis) is temperature independent and given by [ 131:
Here, x denotes a molar pe~endi~ul~ susceptibility. Equation (4) is the same as the
(2) Neel temperature (3) Perpendicular susceptib~~ity (4) Sub~a~tice maenetizatian
(1f P~ama~netic High temperature expansion ~ushbrooke & Wood Molecular field & Green-function R.P.A. ~pinwav~~~9] -23”1 -25”2
constants with n.n. and n.n.n., J,lk and 3,/k respectively splitting factor g, us derived by various theories
Table 2, The exchange
and the spectroscopic
well known expression for the perpendicular susceptibility of a pure antiferromagnet. An identical expression for x1 in an antiferromagnet is given by the random phase approximation [ 161. Other approaches such as Callen decoupling and the Oguchi spin wave theories yield within the first order approximation the same results[l6]. The numerical values are summarized in Table 2. 4. DISCUSSION
The analysis of the paramagnetic susceptibility by the high temperature expansion was found to be a useful approach for determining the exchange interactions and the g factor in the orthoferrites Y FeO, and LuFeO,. In both cases the quantities J1, J, and g, given in Table 2, yielded a well defined minima in the fitting procedure. The results obtained for the n.n. interaction are in reasonable agreement with those obtained from the low temperature susceptibility (see Table 2). The ratio (Y= JZ/J1 = 0.05, obtained for the two perovskites is supported by the value found in the isomorphic structure Sr,FeSbO,. The g factor calculated from the Curie constant C by the high temperature analysis (gYFeOo = 2.05 2 0.1, gl,uFe,,a= 2*00+ 0.05) are in the agreement with the expected one for an S-state ion. By comparing these results with the one obtained by the molecular field theory (see Table 2) it is apparent that the latter are rather poor; J, in this case is significantly too high. Additional experimental data from which the exchange interaction can be derived is the NCel temperature. Rushbrooke and Wood [ 171 have determined the antiferromagnetic ordering temperature for a Heisenberg antiferromagnet having only n.n. interaction, TN(~ = 0) ((u = JZ/Jl) from the high temperature staggered susceptibility. The result is:
Tahir-Kheli et al.[ 181 using the random phase approximation. In the region where (Y+ 1, this dependence to the first approximation is linear in (Y.For a simple cubic lattice one finds m=
A similar result is obtained from the molecular field theory, where m = - 2.0. The solution of equation (6) yields. T,(a)
The exchange constant J, was calculated from TV(a) using equations (5) and (7). The results obtained for J1 are presented in Table 2. Eibschiitz et al.[ 191 have evaluated the n.n. exchange interaction in the orthoferrites using Oguchi spin wave theory. The value of J, was extracted by fitting of the experimental low temperature sublattice magnetization to the dependence by spin wave theory. The results obtained for J1 (see Table 2) are in agreement with those derived from the low temperature susceptibility. It should be noted that the values of the exchange constant J, obtained from the low temperature region are on the average lower than those obtained from the high temperature region. The dependence of the exchange integral on the thermal expansion could be roughly estimated from the relation a In TN/d In u = 2 that was found for the orthoferrites. Here Y denotes the volume of a unit cell. The use of an average thermal expansion coefficient for ceramics (- 5 . lo-+, cubic) should yield a decrease of - 5 per cent over the entire temperature range. This effect would lead to a better agreement between the observed values for J,.
Tj,(a! = 0) = 5/192( J,/k) (zI - 1) X [lLS(S+l)-1][1+0*63/z,S(S+l)].
The dependence of the transition temperature TN(~) on the ratio J,/J, was calculated by
author thanks Professor R. Pauthenet for his kind hospitality in CNRS. Grenoble, where part of the reported measurements were carried out. He is indebted to Professor S. Shtrikman for guidance and helpful suggestions.
REFERENCES I. TREVES D., EIBSCHUTZ M. and COPPENS P.. Phys. Left. 18,2 16 (1965). 2. TREVES D., J. uppl. Phys. 36, (part 2) 1033 (1965). 3. BLASSE G., Philips Res. Rep. 20,327 (I 956). 4. GORODETSKY G., Solid State Commun. 6, 159
(1968). 5. FLANDERS P. J. and DOYLE W. D., Rev. scient. Instrum. 33.69 1 (1962). 6. SELWOOD P. W., In Magnetochemistry, 2nd Edn., p. 78. Inter-science, New York (I 956). G. S. and WOOD P. J., Molec. 7. RUSHBROOKE Phys. 1,257(1958). 8. WOJTOWICZ P. J. and JOSEPH R. I., Phys. Rev. 135,A1314(1964). 9. DALTON N. W.. Proc. phys. Sot. 88,659 (1966).
K., WOOD P. J. and EVE J.. Molec. Phy.v. IO. PIRNIE II, 551 (1966). D., Phys. Rev. 125, 1843 ( 1962). 11. TREVES 12. HERMANN G. F.. Phys. Rev. A133, 1334(1964L 13. ~~~R~DE~TKY~..S~~TRI~~ANS..TENNENBAUM J. and TREVES D.. To be published. R. I.. Phys. Rev. Al38, 144 I (I 965). 14. JOSEPH T.. YOSHIDA K. and KUBO R.. 15. NAGAMIYA Advun. Phys. 4, I (1956). B. and CALLEN H. B., Phys. Rev. 16.ANDERSON A136, 1068 (1964). G. S. and WOOD P. J.. Molec. 17. RUSHBROOKE Phys. 6,409 ( 1963). CALLEN H. B. andJARRET H.. 18. TAHIR-KHELI, J. Phys. Chem. Solids 27,23 (1966). M., SHTRIKMAN S. and TREVES 19. EIBSCHUTZ D., Phys. Reu. 156.562 (I 967).