- Email: [email protected]
Magnetic superexchange interactions and metamagnetic phase transitions in the single crystals of the spinels ZnCr2−xInxSe4 (0 ≤ x ≤ 0.14)
N ELSEVIER
Journal of Magnetism and Magnetic Materials 166 (1997) 172- 178
~ ~H ~H
Journalof magnetism and magnetic materials
Magnetic superexchange interactions and metamagnetic phase transitions in the single crystals of the spinels ZnCr2_xlnxSe4 (0 < x < 0.14) 1 J. Krok-Kowalski
a,
j. Warczewski a,*, T. Mydlarz b, A. Pacyna c, I. Okofiska-Koztowska d
a University of Silesia, Institute of Physics, ul. Uniwersytecka 4, PL-40007 Katowice, Poland b International Laboratory of High Magnetic Fields and Low Temperature, ul. Gajowicka 95, PL-53529 Wroctaw, Poland c Institute of Nuclear Physics, ul. Radzikowskiego 152, PL-31342 Krakdw, Poland d University of Silesia, Institute of Chemistry, ul. Szkolna 9, PL-40006 Katowice, Poland
Received 14 March 1996; revised 16 July 1996
Abstract The experimental data for the magnetization and the magnetic susceptibility, as well as their dependence on temperature, on the induction of the external magnetic field and on the concentration x have been analyzed. The metamagnetic phase transitions induced by the external magnetic field have been observed as plateau areas and jump-like increments on the magnetization curves at 4.2 K. High values of the effective magnetic moment /.Leff and of the Curie constant C M point to the existence of Cr 2+ ions as well as Cr 3+ ions. The exchange constants, transfer integrals and the spiral angles for the solid solutions under study have been evaluated. Keywords: Spinel; Metamagnetism; Antiferromagnetism; Superexchange
1. Introduction Several authors have studied the magnetic superexchange interactions in magnetic insulators, e.g. Nrel [1], Kanamori [2], Anderson [3], De Gennes [41, Blase [5], Dionne [6] and Srivastava [7]. It turns out that, for simple compounds, the superexchange interactions between localized magnetic moments can be
* Corresponding author. Email: [email protected]; fax: +4832-588431. i Dedicated to Professor Hans-Joachim Seifert of the University of Kassel, Germany, on his 65th birthday.
described with the aid of either one or two exchange constants. However, for the compounds with the spinel structure, and in particular for those having a ferrimagnetic ordering (e.g. ferrites), a larger number of exchange constants is needed. For example, for the inverse ferrimagnetic ferrites possessing three magnetic subarrays, Srivastava analyzed the experimental magnetization and magnetic susceptibility data based on the molecular field approximation, using as many as six exchange constants [7]. He used these exchange constants to evaluate the transfer integrals according to Anderson's superexchange theory [8].
0304-8853/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0304-8 853(96)00448-9
173
J. Krok-Kowalski et al. / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178 6
(a) i + )E
5.2
II X X
X
X
X
X
X
X
x •
'E
x
E 0 3 E
E 0 E
0
0 ,i,,,,, Ill
+ +
4.2
t'-
+
+
+
+
+ x
+
Z n C rz.xl nxSe 4 •
x
=
0,04
+
-I- x = 0 . 0 5
x
~x
ZnCr2.xlnxSe4
= 0.07
• x = 0.04
• x = 0.09 ×x
0
5
10
+X
=0.14
3.2 4
15
t
'
t
h
,
8
= 0.14
i
12
16
Magnetic induction B(T)
Magnetic induction B(T)
Fig. 1. Magnetization isotherms at 4.2 K for all the indium concentrations x. The statistical errors are within the experimental points.
(b) • +
5.2
In this paper we study the magnetic superexchange interactions in single crystals of the spinels ZnCr2_xlnxSe 4, where 0 < x < 0.14, based on the experimental magnetization and magnetic susceptibility data, as well as their dependence on temperature, on the induction of the external magnetic field and on the concentration x. Note that solid solutions with x > 0.14 do not exist, because the ionic radius of In 3+ (0.790 ~, [9]) is much greater than that of Cr 3+ (0.615 ,~ [9]), which leads to a large increase in the number of anion vacancies at x > 0.09, as determined by Seebeck effect studies [11]. The method of Srivastava [7] was applied here to evaluate the transfer integrals.
I
•
t_
• •
+
+
+
•
~
•
+ ~:
+
4-
:a. E 0 E 0
"~
t-.-
4.,~
ZnCr2.=ln,Se, +x=O.05 ~x=O.~ mx=O.~
Fig. 2. Scaled up fragments of the individual magnetization curves: (a) for x = 0.04 and 0.14; (b) for x = 0.05, 0.07 and 0.09. The statistical errors are within the experimental points.
35 4
8
12
Magnetic induction B(T)
174
J. Krok-Kowalski et al. / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178
The compounds under study are simple spinels with the antiferromagnetic simple spiral structure in which both the magnetic chromium ions and substituted nonmagnetic In 3+ ions [10,1 I] occupy exclusively the octahedral positions. ZnCr2Se 4, with its antiferromagnetic simple spiral structure, provides a matrix for the spinel series under study [13,14]. Below the N6el temperature a tetragonal distortion of the crystal structure of ZnCr2Se 4 [12] causes the magnetic moments of Cr 3+ (3d 3) ions to be ordered ferromagnetically in the planes perpendicular to the [001] axis, the orientation of the magnetization being changed from plane to plane with a spiral angle th = 4 2 _ 1° [13,14]. A similar situation also holds for the spinel series studied here.
2. Experiment and results This paper examines the same samples and partly the same measurements as those used in Ref. [10]. The single-crystal samples were provided by Okofiska-Koztowska, as a material for the measurements, as described in Ref. [10]. To recall the latter, the magnetic moment measurements were carried out using the induction method in the temperature range 4.2-300 K and in magnetic fields up to 14 T. The magnetic susceptibility was measured using the Faraday method and a Cahn balance. Our new approach to the data concerns the effects, which are presented in Figs. 1-3 and in Table I. Fig. 1 presents the magnetization isotherms at liquid helium temperature for all indium concentrations x in the samples. These isotherms indicate that the samples are magnetically hard. On each curve plateau areas and jump-like increments in the magne-
tization appear. They testify to the existence of critical magnetic fields at which the magnetization vector component parallel to the external magnetic field shows a spontaneous jump-like increase that is typical of the metamagnetic phase transitions. Scaled up fragments of the individual magnetization curves are presented in Fig. 2(a,b). As an example, Fig. 3 presents the temperature dependence of the molar susceptibility XM and its reciprocal 1//XM for the sample with x = 0.09; for the other indium concentrations the behaviors of XM(T) and 1/XM(T) are very similar. Table 1 presents the values of the molar Curie constant C M, the effective magnetic moment /Zeff, the critical magnetic fields H~l, Hc2 and H¢3 (as interpreted from the magnetization curves), as well as the values of the magnetic moment /x4.2K at B = 14 T, the N6el temperature TN, the paramagnetic Curie-Weiss temperature Ocw (taken from Ref. [10]) and the theoretical saturated magnetic moment /~calc (calculated for Cr 3÷ ions only).
3. Analysis of the exchange constants In the case of the ZnCr2Se 4 matrix the diamagnetic Zn 2+ ions occupy the tetrahedral positions, whereas the magnetic Cr 3+ ions occupy the octahedral positions. The magnetic moments of Cr 3+ ions order antiferromagnetically (simple spiral) in ZnCr2Se 4 below the N6el temperature (T N = 20 K). The high and positive value of the paramagnetic Curie-Weiss temperature (~gcw = 118 K) results from the ferromagnetic ordering of the magnetic moments of Cr 3+ ions in the (001) planes. In the molecular field approximation each of these planes is
Table 1 Magnetic parameters for the samples under study x
/a. at4.2 K, 14 T ( p . B)
TN (K)
tgcw (K)
/Xca,c(P'B)
CM
P'eff (/~a)
Hcl (T)
He2 (T)
He3 (T)
0.0 0.04 0.05 0.07 0.09 0.14
5.98 5.39 5.56 5.46 5.55 4.36
22 [18] 22 [10] 20 [10] 20 [10] 19 [10] 16 [10]
118 [18] 48 [10] 51 [10] 47 [10] 52 [10] 49 [10]
6.00 5.88 5.85 5.79 5.73 5.58
3.74 [181 5.81 5.88 5.92 5.76 6.07
5.47 6.82 6.68 6.88 6.79 6.95
1.2 [221 5 8 10 7 9
5.3 [221 8 12 12 12 12
7 [22] 12 -
[18] [10] [10] [10] [10] [10]
J. Krok-Kowalski et aL / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178
175
~,~ 1.0 @
o.8
~
, : l : i . ;'
06
;" ]
H = 1156Oe Z° = - 0'00459 cm3/m°l 0=52K ~
C = 5.77 K*cm3/mol
" "..
~ df = 6.79 lab T• = 18.6K
/f
~ I-<30 ~
/
/ .
~
~
¢ 20
~04
r~
@
illll
~"-,,,,..,~ 0.0
+
ii
0
i
i
i
it
15~i
4
1
1
1
1
1
ii
i
ii
i i i
lO0
i
5 i
+
i
i
i
t5 +
i
i
i
+
ii
150
TEMPERATURE
i
i
25 l
200
i i i
ii
i
i
i
L i
ii
L
"0
250
(K)
Fig. 3. Temperature dependence of the molar susceptibility XM and its reciprocal I/X M for x = 0.09. The statistical errors are within the experimental points.
treated as a separate subarray, and it is assumed that the spontaneous magnetization in such a plane is induced by the effective molecular field representing the exchange interactions. The magnetic interactions in ZnCr2Se 4 are described using three exchange constants: J0 within each plane mentioned above, Jt between nearest planes, and J2 between next nearest planes. Herpin [15,16] found the following relations: cos ~b
Hs = -
J1 4J2
16J2 sin4 ( ~b/2 ] , m c ]r
(1)
(2)
where H s is the saturating magnetic field, mcr is the magnetic moment of a Cr ion, and ~b is the spiral angle. Lotgering [17] proved that
Ocw = Jo + Jt + J2,
(3)
if one takes into account first three coordination spheres. From the experimental data presented in Table 1 it can be seen that for all the solid solutions studied, the paramagnetic Curie-Weiss temperature
(gcw is almost constant and is equal to about 50 K. Moreover, (gcw is here much lower (of about 68 K) than that for the ZnCr2Se 4 matrix 09cw = 118 K [18]). There is also a considerable lowering of the magnetic moment in the saturation state (/z4.2K at 14 T) with respect to the theoretical values /Zcalc calculated for Cr 3+ ions only. One can then assume that an additional antiferromagnetic interaction should exist which both lowers (gcw and compensates the magnetic moment. Moreover, the values of the molar Curie constant Cm and of the effective magnetic moment /x+ff obtained from the temperature dependence of the magnetic susceptibility (see Table 1) indicate that in these solid solutions Cr 2+ ions appear in addition to Cr 3+ ions. It seems that a more accurate chemical formula for the solid solutions under study is as follows: 3+ 2+ 3+ 2Zn 2+ Cr2_x_2rCr2y In x Se4_y. As one can see, there are twice as many Cr 2+ ions as anion vacancies y. y can be evaluated from the considerations concerning the effective magnetic moment ~['~effand the calculated magnetic moment /Zcalc (see Table 1). For example at x = 0.04, y = 0.14. The magnetic moment of a Cr 2+ ion, which in the high-spin configuration is equal to 4 /zB/ion,
J. Krok-Kowalski et aL / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178
176
always couples (e.g. [19,20]) antiparallel to the magnetic moment of a Cr 3+ ion (equal to 3 /zB/ion), causing reductions in both the magnetic moment in the saturation state and the paramagnetic CurieWeiss temperature ~gcw. It follows from the experimental data (/[Leff ]'/'calc • const) in Table 1 that the concentration of Cr 2÷ ions is approximately the same in all the samples except for x = 0.14. Following this interpretation of the experimental results, one can apply Eqs. (1)-(3) twice to any solid solution under study by assuming that its magnetic structure is composed of both the magnetic structure of the ZnCr2Se 4 matrix and the magnetic ordering following from the additional antiferromagnetic interaction of Cr 3+ and Cr 2+ ions mentioned above. In other words, one can first use Eqs. (1)-(3) separately for the latter interaction as a set of three equations with three unknowns J0, J1 and J2, assuming 4, = 180 °, ~gcw = 118-50 K = 68 K, mcr = 3.5 /z B, and H~ = 27 T. This value of the saturation field H s was observed in the case of the pure antiferromagnetic interaction in Znl_xGa2x/3Cr2Se4 at x = 0.5 [22]. Note that for the ZnCr 2 Se n matrix, H S = 7 T [22]. For x = 0.04, for example, the solution of such a set of equations gives three exchange constants J~ = 100 K, J ~ = - 2 4 K and J ~ = - 6 K, which are responsible for this additional antiferromagnetic interaction of Cr 3+ and Cr 2+ ions. As one can see from Table 1, ~gcw practically does not depend on x. Therefore, we can assume that approximately the same values of J~, J~ and J~ - corresponding to this additional antiferromagnetic interaction - hold for all the solid solutions studied here. -
-
x = 0.04 z L
zL
zL L
y-~
B=OT
Now, the values of the exchange constants for the ZnCr2Se 4 matrix are J0 = 13 K, Jl = 158 K and •/2 = - 5 3 K [21], so that we can evaluate the total exchange constants Jd, J~ and J~ for the solid solutions, as follows:
B--5T
B=8T
--l> Y
J~= Jo + J~=113 K, g] = gl + g~ = 134 K,
g~ = g 2 + g2 = - 5 9 K. Taking Eq. (1), modified with the use of the total exchange constants, we can evaluate the modified spiral angle ~bt: cos c~t = J~/4 J~. So th t = 55 °, which holds approximately for all the indium concentrations x, according to the arguments above. Comparing this value of the spiral angle with that for the ZnCr2Se 4 matrix ( 4 2 _ 1°; see above), one can see that the introduction of even a slight amount of indium into the matrix, i.e. 4% ( x = 0.04), causes a change in the spiral angle of about 13 °. As an example, Fig. 4 presents schematically the magnetic phase transitions forced by the external magnetic field in the sample with x = 0.04, at 4.2 K. These phase transitions are interpreted from Fig. 2(a), where the magnetization curve presented at x = 0.04 reveals the plateau areas and the jump-like increments in the magnetization. For the field B = 0 the magnetic moments lie in the plane, the normal of which makes an angle ~9 with the z-axis. With B increasing from zero the angle ~9 becomes equal to zero and the individual spins create a cone around the B direction. The creation of this cone ends at B = He1 = 5 T. The spin cone angle shows a jumplike change at B = He2 = 8 T, while at B = He3 = 12 T the spins align parallel to the direction of the external magnetic field. This picture is based on the assumption that the value of Hc3 is equal to a saturation field.
4. Evaluation o f the transfer integrals
B=12T
Fig. 4. Scheme of the magnetic phase transitions induced by the external magnetic field in the sample with x = 0.04, at T = 4.2 K.
As mentioned above, the transfer integrals have been evaluated using the method of Srivastava [7]. In
J. Krok-Kowalski et al. / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178 a)
177
following interactions through one p orbital contribute with negative sign to the exchange integral:
z /
o"
"iT
/
I
dx2_y2--px--dxy,
i
/ "2
'Tr O" dxy--py--dxz_y2, "IT
"iT
dxz--pz--dy z. This means that the antiferromagnetic contributions come from the 7r couplings for t2g, and from the tr couplings for eg (the states eg are not occupied in this case). The transfer constant for these interactions is denoted by b ~ . On the other hand, the following interactions with the participation of two p orbitals contribute with positive sign:
yz
o"
o"
dx2_y2--px, py--dx2_y2, x
-
y
Fig. 5. 90 ° exchange interaction: (a) with one p orbital:
dxz~-pz~-dyz; (b) with two p orbitals: dx2_y2Zpx, pyZdx2_y2. The overlapping orbitals are denoted by dotted lines.
(y ,ff dx2_y2--px, p z - - d x y , o" dz~--p z, py--dx2_yz, dxy--Py, Px--dxy,
the case of 90 ° interactions between the magnetic moments localized on chromium ions occupying the octahedral positions, there exist three types of coupling: C r 3 + - C r 3+ ( d 3 - d 3 ) , Cr3+-Cr2+ ( d 3 - d 4) and C r Z + - C r 2+ ( d a - d 4 ) . The electrons interacting magnetically (carrying a magnetic moment) occupy the dxy, dxz and dr z orbitals, whereas the dz2 and dx2_y2 orbitals remain empty. If one assumes, according to Fig. 5, that the ligand atoms are arranged along the z-axis, then t h e dy z and dxz orbitals overlap the Px and py orbitals of the ligands. Two types of coupling related to the electron transfer between the t2g states of two chromium ions through a ligand atom can appear here: these are called 7r and tr. Because of the high value of the ionic radius of Cr 2÷ (73 pm) there can also appear a direct overlap of the dxz a n d dr z orbitals of these cations. According to the Anderson-Kanamori approach the exchange constant JBB between two ions in the octahedral positions is composed of two terms: one positive and one negative. As shown in Fig. 5, the
NIT dxy--Py, p z - - d y z , 'ff 'IT dzx--pz, px--dxy. The transfer constant for the ferromagnetic interactions is denoted boo. Note that the 180 ° interactions break to a high degree in these compounds because of the presence of the nonmagnetic In 3+ ions [10]. In the case of the solid solutions studied here, we can assume that (i) direct weak ferromagnetic exchange interaction occurs only for Cr 2+ ions; (ii) the superexchange interactions d 3 - d 3, d g - d 4 and d 3 - d 4 are identical, which means that the corresponding transfer integrals are also equal: b~,,~,, = b~,~, = b~,,~, and b~,~, = b.~,~, where b~,,~,, is a transfer integral between two Cr 3+ ions, b,~,,~, is a transfer integral between Cr 3+ and Cr 2+ ions, and b~,, = b ~ , is another transfer integral between Cr 3÷ and Cr 2÷ ions (via "rr overlap); and (iii) if the direct interaction between t2g orbitals
178
J. Krok-Kowalski et aL / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178
happens to be significant, then the b ~ interaction is reduced and appears only in the antiferromagnetic coupling constant. The validity of assumptions (ii) is also connected with the cubic symmetry of these compounds, in contrast with the Goodenough model [23], which dealt with ionic compounds containing Mn 3+ ions with orthorhombic symmetry. Taking into account these assumptions, we can write the Srivastava equations for the solid solutions as follows: 1116 J ; = 4 s~3 + s 2
I 2~
6
12(C
1
--~-A+~B+ 10~ +-~2~
[16
)] ,
6
-~-( D - A) + ~( E - B)
4s3+s2+
]
+g(c+zJ
) ,
where
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11]
b 2~ctCttt A ~
similar to the results obtained by Juszczyk [21] for the spinel series Znl_xGaz/3xCrzSe4. This result is probably connected with the striction effects which cause the qr overlap (bond) to be stronger.
u
2t (TF n~
[12]
- - 7
[13] C~
b2tt t - -
b~,, O ~
~ u
[14]
b2,, E=--;
[15]
= 6 eV for Cr 2+ ions, u = 10 eV for Cr 3+ ions, and fi = 8 eV denotes the mean value for Cr 2÷ and Cr 3+ ions. The exchange constant resulting from the direct overlap of the d orbitals of Cr 2+ ions is J~ = 0.081 eV. The transfer integrals b,,,~ and b~,~ evaluated from the above formulas are equal:
[16] [17] [18]
b ~ = 0.26 eV,
[21] [22] [23]
b ~ = 0.58 eV.
cr bonds are usually stronger than "rr bonds, but in this case the situation turns out to be the opposite,
[19] [20]
L. N6el, Ann. Phys. 3 (1948) 137. J. Kanamori, J. Phys. Chem. Solids 10 (1959) 87. P.W. Anderson, Phys. Rev. 115 (1959) 2. P.G. de Gennes, Phys. Rev. 118 (1960) 141. C. Blasse, Philips Res. Rep. 19 (1964) Suppl. 3. G.F. Dionne, J. Appl. Phys. 47 (1976) 4220. G.M. Srivastava, G. Srinivasan and N.G. Nanadikar, Phys. Rev. B 19 (1979) 499. P.W. Anderson, in: Magnetism, vol. I, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1963) p. 25. R.D. Shannon, Acta Crystallogr. A 32 (1976) 751. J. Krok-Kowalski, J. Warczewski, T. Mydlarz, A. Pacyna, A. Bombik, J. Kopyczok and I. Okofiska-Koztowska, J. Magn. Magn. Mater. 111 (1992) 50. T. Grofi, J. Kopyczok, I. Okofiska-Kozlowska, J. Warczewski, J. Magn. Magn. Mater. 111 (1992) 53. R. Kleinberger and R. Kouchkovsky, C.R. Acad. Sci. Paris: Physique des Solides 262 (1966) 628. R. Plumier, J. Physique 27 (1966) 213; R. Plumier, M. Lecomte, A. Miedan-Gros and M. Sougi, Phys. Lett. A 55 (1975) 239. J. Akimitsu, K. Siratori, G. Shirane, M. Izumi and T. Watanabe, J. Phys. Soc. Jpn. 44 (1978) 172. A. Herpin and P. Meriel and J. Villain, J. Phys. Radium 21 (1960) 67. A. Herpin and P. Meriel, J. Phys. Radium 22 (1961) 337. F.K. Lotgering, Solid State Commun. 3 (1965) 347. S. Juszczyk, J. Krok, I. Okofiska-Koztowska, T. Mydlarz and A. Gilewski, J. Magn. Magn. Mater. 46 (1984) 105. H.U Pinch and S.B. Berger, J. Phys. Chem. Solids, 29 (1976) 2091. J. Krok-Kowalski, H. Rej, T. Grofi, J. Warczewski, T. Mydlarz and I. Okofiska-Koztowska, J. Magn. Magn. Mater. 137 (1994) 329. S. Juszczyk, J. Magn. Magn. Mater. 73 (1988) 27. S. Juszczyk, J. Magn. Magn. Mater. 75 (1988) 285. J.B. Goodenough, A. Wold, R.J. Amott and N. Menyuk, Phys. Rev. 124 (1961) 373.
Journal of Magnetism and Magnetic Materials 166 (1997) 172- 178
~ ~H ~H
Journalof magnetism and magnetic materials
Magnetic superexchange interactions and metamagnetic phase transitions in the single crystals of the spinels ZnCr2_xlnxSe4 (0 < x < 0.14) 1 J. Krok-Kowalski
a,
j. Warczewski a,*, T. Mydlarz b, A. Pacyna c, I. Okofiska-Koztowska d
a University of Silesia, Institute of Physics, ul. Uniwersytecka 4, PL-40007 Katowice, Poland b International Laboratory of High Magnetic Fields and Low Temperature, ul. Gajowicka 95, PL-53529 Wroctaw, Poland c Institute of Nuclear Physics, ul. Radzikowskiego 152, PL-31342 Krakdw, Poland d University of Silesia, Institute of Chemistry, ul. Szkolna 9, PL-40006 Katowice, Poland
Received 14 March 1996; revised 16 July 1996
Abstract The experimental data for the magnetization and the magnetic susceptibility, as well as their dependence on temperature, on the induction of the external magnetic field and on the concentration x have been analyzed. The metamagnetic phase transitions induced by the external magnetic field have been observed as plateau areas and jump-like increments on the magnetization curves at 4.2 K. High values of the effective magnetic moment /.Leff and of the Curie constant C M point to the existence of Cr 2+ ions as well as Cr 3+ ions. The exchange constants, transfer integrals and the spiral angles for the solid solutions under study have been evaluated. Keywords: Spinel; Metamagnetism; Antiferromagnetism; Superexchange
1. Introduction Several authors have studied the magnetic superexchange interactions in magnetic insulators, e.g. Nrel [1], Kanamori [2], Anderson [3], De Gennes [41, Blase [5], Dionne [6] and Srivastava [7]. It turns out that, for simple compounds, the superexchange interactions between localized magnetic moments can be
* Corresponding author. Email: [email protected]; fax: +4832-588431. i Dedicated to Professor Hans-Joachim Seifert of the University of Kassel, Germany, on his 65th birthday.
described with the aid of either one or two exchange constants. However, for the compounds with the spinel structure, and in particular for those having a ferrimagnetic ordering (e.g. ferrites), a larger number of exchange constants is needed. For example, for the inverse ferrimagnetic ferrites possessing three magnetic subarrays, Srivastava analyzed the experimental magnetization and magnetic susceptibility data based on the molecular field approximation, using as many as six exchange constants [7]. He used these exchange constants to evaluate the transfer integrals according to Anderson's superexchange theory [8].
0304-8853/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0304-8 853(96)00448-9
173
J. Krok-Kowalski et al. / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178 6
(a) i + )E
5.2
II X X
X
X
X
X
X
X
x •
'E
x
E 0 3 E
E 0 E
0
0 ,i,,,,, Ill
+ +
4.2
t'-
+
+
+
+
+ x
+
Z n C rz.xl nxSe 4 •
x
=
0,04
+
-I- x = 0 . 0 5
x
~x
ZnCr2.xlnxSe4
= 0.07
• x = 0.04
• x = 0.09 ×x
0
5
10
+X
=0.14
3.2 4
15
t
'
t
h
,
8
= 0.14
i
12
16
Magnetic induction B(T)
Magnetic induction B(T)
Fig. 1. Magnetization isotherms at 4.2 K for all the indium concentrations x. The statistical errors are within the experimental points.
(b) • +
5.2
In this paper we study the magnetic superexchange interactions in single crystals of the spinels ZnCr2_xlnxSe 4, where 0 < x < 0.14, based on the experimental magnetization and magnetic susceptibility data, as well as their dependence on temperature, on the induction of the external magnetic field and on the concentration x. Note that solid solutions with x > 0.14 do not exist, because the ionic radius of In 3+ (0.790 ~, [9]) is much greater than that of Cr 3+ (0.615 ,~ [9]), which leads to a large increase in the number of anion vacancies at x > 0.09, as determined by Seebeck effect studies [11]. The method of Srivastava [7] was applied here to evaluate the transfer integrals.
I
•
t_
• •
+
+
+
•
~
•
+ ~:
+
4-
:a. E 0 E 0
"~
t-.-
4.,~
ZnCr2.=ln,Se, +x=O.05 ~x=O.~ mx=O.~
Fig. 2. Scaled up fragments of the individual magnetization curves: (a) for x = 0.04 and 0.14; (b) for x = 0.05, 0.07 and 0.09. The statistical errors are within the experimental points.
35 4
8
12
Magnetic induction B(T)
174
J. Krok-Kowalski et al. / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178
The compounds under study are simple spinels with the antiferromagnetic simple spiral structure in which both the magnetic chromium ions and substituted nonmagnetic In 3+ ions [10,1 I] occupy exclusively the octahedral positions. ZnCr2Se 4, with its antiferromagnetic simple spiral structure, provides a matrix for the spinel series under study [13,14]. Below the N6el temperature a tetragonal distortion of the crystal structure of ZnCr2Se 4 [12] causes the magnetic moments of Cr 3+ (3d 3) ions to be ordered ferromagnetically in the planes perpendicular to the [001] axis, the orientation of the magnetization being changed from plane to plane with a spiral angle th = 4 2 _ 1° [13,14]. A similar situation also holds for the spinel series studied here.
2. Experiment and results This paper examines the same samples and partly the same measurements as those used in Ref. [10]. The single-crystal samples were provided by Okofiska-Koztowska, as a material for the measurements, as described in Ref. [10]. To recall the latter, the magnetic moment measurements were carried out using the induction method in the temperature range 4.2-300 K and in magnetic fields up to 14 T. The magnetic susceptibility was measured using the Faraday method and a Cahn balance. Our new approach to the data concerns the effects, which are presented in Figs. 1-3 and in Table I. Fig. 1 presents the magnetization isotherms at liquid helium temperature for all indium concentrations x in the samples. These isotherms indicate that the samples are magnetically hard. On each curve plateau areas and jump-like increments in the magne-
tization appear. They testify to the existence of critical magnetic fields at which the magnetization vector component parallel to the external magnetic field shows a spontaneous jump-like increase that is typical of the metamagnetic phase transitions. Scaled up fragments of the individual magnetization curves are presented in Fig. 2(a,b). As an example, Fig. 3 presents the temperature dependence of the molar susceptibility XM and its reciprocal 1//XM for the sample with x = 0.09; for the other indium concentrations the behaviors of XM(T) and 1/XM(T) are very similar. Table 1 presents the values of the molar Curie constant C M, the effective magnetic moment /Zeff, the critical magnetic fields H~l, Hc2 and H¢3 (as interpreted from the magnetization curves), as well as the values of the magnetic moment /x4.2K at B = 14 T, the N6el temperature TN, the paramagnetic Curie-Weiss temperature Ocw (taken from Ref. [10]) and the theoretical saturated magnetic moment /~calc (calculated for Cr 3÷ ions only).
3. Analysis of the exchange constants In the case of the ZnCr2Se 4 matrix the diamagnetic Zn 2+ ions occupy the tetrahedral positions, whereas the magnetic Cr 3+ ions occupy the octahedral positions. The magnetic moments of Cr 3+ ions order antiferromagnetically (simple spiral) in ZnCr2Se 4 below the N6el temperature (T N = 20 K). The high and positive value of the paramagnetic Curie-Weiss temperature (~gcw = 118 K) results from the ferromagnetic ordering of the magnetic moments of Cr 3+ ions in the (001) planes. In the molecular field approximation each of these planes is
Table 1 Magnetic parameters for the samples under study x
/a. at4.2 K, 14 T ( p . B)
TN (K)
tgcw (K)
/Xca,c(P'B)
CM
P'eff (/~a)
Hcl (T)
He2 (T)
He3 (T)
0.0 0.04 0.05 0.07 0.09 0.14
5.98 5.39 5.56 5.46 5.55 4.36
22 [18] 22 [10] 20 [10] 20 [10] 19 [10] 16 [10]
118 [18] 48 [10] 51 [10] 47 [10] 52 [10] 49 [10]
6.00 5.88 5.85 5.79 5.73 5.58
3.74 [181 5.81 5.88 5.92 5.76 6.07
5.47 6.82 6.68 6.88 6.79 6.95
1.2 [221 5 8 10 7 9
5.3 [221 8 12 12 12 12
7 [22] 12 -
[18] [10] [10] [10] [10] [10]
J. Krok-Kowalski et aL / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178
175
~,~ 1.0 @
o.8
~
, : l : i . ;'
06
;" ]
H = 1156Oe Z° = - 0'00459 cm3/m°l 0=52K ~
C = 5.77 K*cm3/mol
" "..
~ df = 6.79 lab T• = 18.6K
/f
~ I-<30 ~
/
/ .
~
~
¢ 20
~04
r~
@
illll
~"-,,,,..,~ 0.0
+
ii
0
i
i
i
it
15~i
4
1
1
1
1
1
ii
i
ii
i i i
lO0
i
5 i
+
i
i
i
t5 +
i
i
i
+
ii
150
TEMPERATURE
i
i
25 l
200
i i i
ii
i
i
i
L i
ii
L
"0
250
(K)
Fig. 3. Temperature dependence of the molar susceptibility XM and its reciprocal I/X M for x = 0.09. The statistical errors are within the experimental points.
treated as a separate subarray, and it is assumed that the spontaneous magnetization in such a plane is induced by the effective molecular field representing the exchange interactions. The magnetic interactions in ZnCr2Se 4 are described using three exchange constants: J0 within each plane mentioned above, Jt between nearest planes, and J2 between next nearest planes. Herpin [15,16] found the following relations: cos ~b
Hs = -
J1 4J2
16J2 sin4 ( ~b/2 ] , m c ]r
(1)
(2)
where H s is the saturating magnetic field, mcr is the magnetic moment of a Cr ion, and ~b is the spiral angle. Lotgering [17] proved that
Ocw = Jo + Jt + J2,
(3)
if one takes into account first three coordination spheres. From the experimental data presented in Table 1 it can be seen that for all the solid solutions studied, the paramagnetic Curie-Weiss temperature
(gcw is almost constant and is equal to about 50 K. Moreover, (gcw is here much lower (of about 68 K) than that for the ZnCr2Se 4 matrix 09cw = 118 K [18]). There is also a considerable lowering of the magnetic moment in the saturation state (/z4.2K at 14 T) with respect to the theoretical values /Zcalc calculated for Cr 3+ ions only. One can then assume that an additional antiferromagnetic interaction should exist which both lowers (gcw and compensates the magnetic moment. Moreover, the values of the molar Curie constant Cm and of the effective magnetic moment /x+ff obtained from the temperature dependence of the magnetic susceptibility (see Table 1) indicate that in these solid solutions Cr 2+ ions appear in addition to Cr 3+ ions. It seems that a more accurate chemical formula for the solid solutions under study is as follows: 3+ 2+ 3+ 2Zn 2+ Cr2_x_2rCr2y In x Se4_y. As one can see, there are twice as many Cr 2+ ions as anion vacancies y. y can be evaluated from the considerations concerning the effective magnetic moment ~['~effand the calculated magnetic moment /Zcalc (see Table 1). For example at x = 0.04, y = 0.14. The magnetic moment of a Cr 2+ ion, which in the high-spin configuration is equal to 4 /zB/ion,
J. Krok-Kowalski et aL / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178
176
always couples (e.g. [19,20]) antiparallel to the magnetic moment of a Cr 3+ ion (equal to 3 /zB/ion), causing reductions in both the magnetic moment in the saturation state and the paramagnetic CurieWeiss temperature ~gcw. It follows from the experimental data (/[Leff ]'/'calc • const) in Table 1 that the concentration of Cr 2÷ ions is approximately the same in all the samples except for x = 0.14. Following this interpretation of the experimental results, one can apply Eqs. (1)-(3) twice to any solid solution under study by assuming that its magnetic structure is composed of both the magnetic structure of the ZnCr2Se 4 matrix and the magnetic ordering following from the additional antiferromagnetic interaction of Cr 3+ and Cr 2+ ions mentioned above. In other words, one can first use Eqs. (1)-(3) separately for the latter interaction as a set of three equations with three unknowns J0, J1 and J2, assuming 4, = 180 °, ~gcw = 118-50 K = 68 K, mcr = 3.5 /z B, and H~ = 27 T. This value of the saturation field H s was observed in the case of the pure antiferromagnetic interaction in Znl_xGa2x/3Cr2Se4 at x = 0.5 [22]. Note that for the ZnCr 2 Se n matrix, H S = 7 T [22]. For x = 0.04, for example, the solution of such a set of equations gives three exchange constants J~ = 100 K, J ~ = - 2 4 K and J ~ = - 6 K, which are responsible for this additional antiferromagnetic interaction of Cr 3+ and Cr 2+ ions. As one can see from Table 1, ~gcw practically does not depend on x. Therefore, we can assume that approximately the same values of J~, J~ and J~ - corresponding to this additional antiferromagnetic interaction - hold for all the solid solutions studied here. -
-
x = 0.04 z L
zL
zL L
y-~
B=OT
Now, the values of the exchange constants for the ZnCr2Se 4 matrix are J0 = 13 K, Jl = 158 K and •/2 = - 5 3 K [21], so that we can evaluate the total exchange constants Jd, J~ and J~ for the solid solutions, as follows:
B--5T
B=8T
--l> Y
J~= Jo + J~=113 K, g] = gl + g~ = 134 K,
g~ = g 2 + g2 = - 5 9 K. Taking Eq. (1), modified with the use of the total exchange constants, we can evaluate the modified spiral angle ~bt: cos c~t = J~/4 J~. So th t = 55 °, which holds approximately for all the indium concentrations x, according to the arguments above. Comparing this value of the spiral angle with that for the ZnCr2Se 4 matrix ( 4 2 _ 1°; see above), one can see that the introduction of even a slight amount of indium into the matrix, i.e. 4% ( x = 0.04), causes a change in the spiral angle of about 13 °. As an example, Fig. 4 presents schematically the magnetic phase transitions forced by the external magnetic field in the sample with x = 0.04, at 4.2 K. These phase transitions are interpreted from Fig. 2(a), where the magnetization curve presented at x = 0.04 reveals the plateau areas and the jump-like increments in the magnetization. For the field B = 0 the magnetic moments lie in the plane, the normal of which makes an angle ~9 with the z-axis. With B increasing from zero the angle ~9 becomes equal to zero and the individual spins create a cone around the B direction. The creation of this cone ends at B = He1 = 5 T. The spin cone angle shows a jumplike change at B = He2 = 8 T, while at B = He3 = 12 T the spins align parallel to the direction of the external magnetic field. This picture is based on the assumption that the value of Hc3 is equal to a saturation field.
4. Evaluation o f the transfer integrals
B=12T
Fig. 4. Scheme of the magnetic phase transitions induced by the external magnetic field in the sample with x = 0.04, at T = 4.2 K.
As mentioned above, the transfer integrals have been evaluated using the method of Srivastava [7]. In
J. Krok-Kowalski et al. / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178 a)
177
following interactions through one p orbital contribute with negative sign to the exchange integral:
z /
o"
"iT
/
I
dx2_y2--px--dxy,
i
/ "2
'Tr O" dxy--py--dxz_y2, "IT
"iT
dxz--pz--dy z. This means that the antiferromagnetic contributions come from the 7r couplings for t2g, and from the tr couplings for eg (the states eg are not occupied in this case). The transfer constant for these interactions is denoted by b ~ . On the other hand, the following interactions with the participation of two p orbitals contribute with positive sign:
yz
o"
o"
dx2_y2--px, py--dx2_y2, x
-
y
Fig. 5. 90 ° exchange interaction: (a) with one p orbital:
dxz~-pz~-dyz; (b) with two p orbitals: dx2_y2Zpx, pyZdx2_y2. The overlapping orbitals are denoted by dotted lines.
(y ,ff dx2_y2--px, p z - - d x y , o" dz~--p z, py--dx2_yz, dxy--Py, Px--dxy,
the case of 90 ° interactions between the magnetic moments localized on chromium ions occupying the octahedral positions, there exist three types of coupling: C r 3 + - C r 3+ ( d 3 - d 3 ) , Cr3+-Cr2+ ( d 3 - d 4) and C r Z + - C r 2+ ( d a - d 4 ) . The electrons interacting magnetically (carrying a magnetic moment) occupy the dxy, dxz and dr z orbitals, whereas the dz2 and dx2_y2 orbitals remain empty. If one assumes, according to Fig. 5, that the ligand atoms are arranged along the z-axis, then t h e dy z and dxz orbitals overlap the Px and py orbitals of the ligands. Two types of coupling related to the electron transfer between the t2g states of two chromium ions through a ligand atom can appear here: these are called 7r and tr. Because of the high value of the ionic radius of Cr 2÷ (73 pm) there can also appear a direct overlap of the dxz a n d dr z orbitals of these cations. According to the Anderson-Kanamori approach the exchange constant JBB between two ions in the octahedral positions is composed of two terms: one positive and one negative. As shown in Fig. 5, the
NIT dxy--Py, p z - - d y z , 'ff 'IT dzx--pz, px--dxy. The transfer constant for the ferromagnetic interactions is denoted boo. Note that the 180 ° interactions break to a high degree in these compounds because of the presence of the nonmagnetic In 3+ ions [10]. In the case of the solid solutions studied here, we can assume that (i) direct weak ferromagnetic exchange interaction occurs only for Cr 2+ ions; (ii) the superexchange interactions d 3 - d 3, d g - d 4 and d 3 - d 4 are identical, which means that the corresponding transfer integrals are also equal: b~,,~,, = b~,~, = b~,,~, and b~,~, = b.~,~, where b~,,~,, is a transfer integral between two Cr 3+ ions, b,~,,~, is a transfer integral between Cr 3+ and Cr 2+ ions, and b~,, = b ~ , is another transfer integral between Cr 3÷ and Cr 2÷ ions (via "rr overlap); and (iii) if the direct interaction between t2g orbitals
178
J. Krok-Kowalski et aL / Journal of Magnetism and Magnetic Materials 166 (1997) 172-178
happens to be significant, then the b ~ interaction is reduced and appears only in the antiferromagnetic coupling constant. The validity of assumptions (ii) is also connected with the cubic symmetry of these compounds, in contrast with the Goodenough model [23], which dealt with ionic compounds containing Mn 3+ ions with orthorhombic symmetry. Taking into account these assumptions, we can write the Srivastava equations for the solid solutions as follows: 1116 J ; = 4 s~3 + s 2
I 2~
6
12(C
1
--~-A+~B+ 10~ +-~2~
[16
)] ,
6
-~-( D - A) + ~( E - B)
4s3+s2+
]
+g(c+zJ
) ,
where
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11]
b 2~ctCttt A ~
similar to the results obtained by Juszczyk [21] for the spinel series Znl_xGaz/3xCrzSe4. This result is probably connected with the striction effects which cause the qr overlap (bond) to be stronger.
u
2t (TF n~
[12]
- - 7
[13] C~
b2tt t - -
b~,, O ~
~ u
[14]
b2,, E=--;
[15]
= 6 eV for Cr 2+ ions, u = 10 eV for Cr 3+ ions, and fi = 8 eV denotes the mean value for Cr 2÷ and Cr 3+ ions. The exchange constant resulting from the direct overlap of the d orbitals of Cr 2+ ions is J~ = 0.081 eV. The transfer integrals b,,,~ and b~,~ evaluated from the above formulas are equal:
[16] [17] [18]
b ~ = 0.26 eV,
[21] [22] [23]
b ~ = 0.58 eV.
cr bonds are usually stronger than "rr bonds, but in this case the situation turns out to be the opposite,
[19] [20]
L. N6el, Ann. Phys. 3 (1948) 137. J. Kanamori, J. Phys. Chem. Solids 10 (1959) 87. P.W. Anderson, Phys. Rev. 115 (1959) 2. P.G. de Gennes, Phys. Rev. 118 (1960) 141. C. Blasse, Philips Res. Rep. 19 (1964) Suppl. 3. G.F. Dionne, J. Appl. Phys. 47 (1976) 4220. G.M. Srivastava, G. Srinivasan and N.G. Nanadikar, Phys. Rev. B 19 (1979) 499. P.W. Anderson, in: Magnetism, vol. I, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1963) p. 25. R.D. Shannon, Acta Crystallogr. A 32 (1976) 751. J. Krok-Kowalski, J. Warczewski, T. Mydlarz, A. Pacyna, A. Bombik, J. Kopyczok and I. Okofiska-Koztowska, J. Magn. Magn. Mater. 111 (1992) 50. T. Grofi, J. Kopyczok, I. Okofiska-Kozlowska, J. Warczewski, J. Magn. Magn. Mater. 111 (1992) 53. R. Kleinberger and R. Kouchkovsky, C.R. Acad. Sci. Paris: Physique des Solides 262 (1966) 628. R. Plumier, J. Physique 27 (1966) 213; R. Plumier, M. Lecomte, A. Miedan-Gros and M. Sougi, Phys. Lett. A 55 (1975) 239. J. Akimitsu, K. Siratori, G. Shirane, M. Izumi and T. Watanabe, J. Phys. Soc. Jpn. 44 (1978) 172. A. Herpin and P. Meriel and J. Villain, J. Phys. Radium 21 (1960) 67. A. Herpin and P. Meriel, J. Phys. Radium 22 (1961) 337. F.K. Lotgering, Solid State Commun. 3 (1965) 347. S. Juszczyk, J. Krok, I. Okofiska-Koztowska, T. Mydlarz and A. Gilewski, J. Magn. Magn. Mater. 46 (1984) 105. H.U Pinch and S.B. Berger, J. Phys. Chem. Solids, 29 (1976) 2091. J. Krok-Kowalski, H. Rej, T. Grofi, J. Warczewski, T. Mydlarz and I. Okofiska-Koztowska, J. Magn. Magn. Mater. 137 (1994) 329. S. Juszczyk, J. Magn. Magn. Mater. 73 (1988) 27. S. Juszczyk, J. Magn. Magn. Mater. 75 (1988) 285. J.B. Goodenough, A. Wold, R.J. Amott and N. Menyuk, Phys. Rev. 124 (1961) 373.