Magnetic ground-state of strongly frustrated pyrochlore anti-ferromagnet Er2Sn2O7

Journal of Magnetism and Magnetic Materials 361 (2014) 175–181

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Magnetic ground-state of strongly frustrated pyrochlore anti-ferromagnet Er2Sn2O7 J. Alam, Y.M. Jana n, A. Ali Biswas Department of Physics, University of Kalyani, Kalyani, Nadia 741235, W.B., India

art ic l e i nf o

a b s t r a c t

Article history: Received 8 May 2013 Received in revised form 17 February 2014 Available online 3 March 2014

The experimental results of bulk magnetic susceptibility, isothermal magnetization and magnetic specific heat of the frustrated anti-ferromagnetic pyrochlore Er2Sn2O7 are simulated and analyzed using appropriate crystal-field (CF) and molecular field tensors at Er-sites in the mean-field approach, which are found to be anisotropic along and normal to the local 〈111〉 axis of the frustrated tetrahedral spinstructure. Er-spins are constrained to lie in the local [111] easy-plane in the anti-ferromagnetic phase. Isothermal magnetization can be described by angle-averaged magnetization of the polycrystalline sample, expressed in terms of anisotropic g-tensors and exchange tensors. Temperature-dependent characteristic exchange splitting of the single-ion ground doublet describes the magnetic specific heat satisfactorily. The smaller exchange-splitting may cause rapid fluctuations of Er-spin moments in Er2Sn2O7 and hence long-range ordering of Er-spins is not exhibited by Er2Sn2O7 down to 100 mK, unlike in Er2Ti2O7. & 2014 Elsevier B.V. All rights reserved.

Keywords: Pyrochlore anti-ferromagnet Crystal-field Single-ion susceptibility Random exchange interaction

1. Introduction Frustration of spin–spin interactions on the pyrochlore lattice structure has captured imagination of the condensed matter community for the last two decades due to novel and exotic magnetic ground-states, e.g., spin-ice, spin-liquid and spin-glass phases [1,2] found in these spin-structures. In the pyrochlore compounds, R2M2O7 (R3 þ ¼rare-earth, M4 þ ¼transition or nontransition metals), R- and M-spins individually reside on an infinite network of corner-sharing tetrahedral. Strong spin frustration is, therefore, inherent in these systems due to combination of such topological structure and nearest-neighbor (n.n.) anti-ferromagnetic (AFM) Heisenberg exchange interactions, and induces a large degree of spin-degeneracy into the ground-state. As a consequence, the magnetic system remains in the paramagnetic phase with no long-range order (LRO) observed down to T ¼0 K. However it was found that mere existence of local AFM spin frustration is neither a necessity, nor a sufficient condition for magnetic frustration, and real materials often enter into a longrange ordered state promoted by few other weaker intrinsic perturbations in the spin-Hamiltonian, mainly coming from the directional anisotropies of single-ion crystal-field (CF) interactions at the R-site and exchange interactions, dipolar and further n.n.

n

Corresponding author. Mobile: þ 91 983 099 7571. E-mail addresses: [email protected], [email protected] (Y.M. Jana).

http://dx.doi.org/10.1016/j.jmmm.2014.02.086 0304-8853 & 2014 Elsevier B.V. All rights reserved.

exchange interactions, as well as thermal and/or quantum fluctuations (‘order-by-disorder’ mechanism). The role of local CF anisotropy is notably significant: strong axial (along the local 〈111〉 axis of tetrahedral) CF anisotropy favors a disordered ‘spin-ice’ state in the presence of the effective ferromagnetic coupling in R2Ti2O7 and R2Sn2O7 (R¼Ho, Dy) [1], while a strong easy-plane ([111] plane) anisotropy leads to a Néel state in Er2Ti2O7 [3], and conventional long-range order in Gd2M2O7 where M¼ Ti, Sn, Zr, Hf. In this context, the Gd-based pyrochlores demand special attention on their own merits, as they exhibit very intriguing and contrasting behaviors. Gd2Ti2O7 displays two successive phase transitions at Tc1 ¼ 1.02 K and Tc2 ¼0.74 K to multi-k structures. Specific heat Cv(T) of Gd2Ti2O7 shows an unconventional T2 dependence below Tc2, where the dynamic Gd-spins align collinearly within the [111] plane but very weakly [4,5]. On the other hand, Gd2Sn2O7 corresponds to the Palmer and Chalker (PC) ground-state with k ¼(0, 0, 0) spin structure which exhibits a long-range ordered phase transition at Tc  1 K, below which gapped magnetic excitation spectrum in Cv  exp(  Δ/T) behavior is observed [6,7]. It was, therefore, surmised that the differences between the spin structures and spin dynamics in the low-temperature ordered phases of Gdpyrochlores may be characterized by anisotropic exchange tensors along and normal to the n.n. bond separation and appreciable single-ion planar anisotropies [8]. Experimental observations of several unconventional properties of many pyrochlore magnetic materials have been published world-wide, which have given birth to urge for theoretical analysis

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J. Alam et al. / Journal of Magnetism and Magnetic Materials 361 (2014) 175–181

and justifications of these properties vis-à-vis frustration of spins. Er-based pyrochlore Er2Sn2O7 is one such material of immense interest which was studied by means of magnetic susceptibility and isothermal magnetization [9], neutron scattering, and specific heat measurements [10]. Er3 þ -spins in Er2Sn2O7 show no signs of longrange order down to T¼ 100 mK, although it has appreciable AFM spin–spin interactions reflected through the negative Curie Weiss temperature θCW ¼  14 K [9,10]. To compare Er2Sn2O7 with its isomorphous compound Er2Ti2O7, we noted that the latter compound is an 〈XY〉 anti-ferromagnet having θCW ¼  22 K which orders in the [111] plane by order-through-disorder mechanism at TN ¼ 1.173 K [3]. The frustration index which measures the degree of spin-frustration (or degree of disability of order) is defined by f¼|θCW |/TN [1] and is found to be 18.7 for Er2Ti2O7. To the first approximation, if we assume that both these Er-compounds are fashioned by similar crystal-field (disregarding their slightly different lattice constants, a0 ¼ 10.352 Å and 10.087 Å for Er2Sn2O7 and Er2Ti2O7 [11], respectively) and isotropic exchange interactions (despite their different strength of θCW), f would be the same for Er2Sn2O7 which implies that the Er-stannate should also exhibit long-ranged ordering at any finite temperature Tr1 K. However, in contrast to this prediction, the lack of LRO in the Er-stannate definitely points that order-by-disorder mechanism, which splits the degenerate ground-state manifold and stabilizes a particular ordering wave-vector in Er2Ti2O7 [12], is either absent or quiescent due to different local crystalline environments (hence different local CF anisotropies along 〈111〉 axis and its normal direction) and slight change in the internal molecular pffiffiffi field tensors (due to different n.n. bond separations r n:n: ¼ a0 = 8) along and perpendicular to the n.n. bond separation. In the present work, we, therefore, address ourselves to explore the effect of anisotropies in the part of CF and internal molecular fields at the site of magnetic Er-ions lying onto the tetrahedral lattice network in Er2Sn2O7 on the observed physical properties, e. g., magnetic susceptibility, magnetization [9], and magnetic specific heat [10] and to find out the possible origin of lack of LRO in Er2Sn2O7. To this end, we proceed as follows. In Section 2, we construct a Hamiltonian within the frame work of a CF theory appropriate to the 4f11 electronic configuration of Er3 þ -ion and a mean-field approximation by adopting dipolar and exchange interactions among Er3 þ -ions situated on the tetrahedral sublattice. An exact relation between the bulk and single-ion susceptibility was used to simulate the experimental susceptibility of Er2Sn2O7. In Section 3, we analyze our calculated results, and finally in Section 4, further discussion of the above analyses followed by conclusions is given.

2. Theoretical outline The R-ions in the pyrochlore lattice are located within a trigonally (D3d) distorted scalenohedra formed by eight O2  anions, of which six equally spaced O2  (48f) lie on an equatorial plane and two O2  (8b) are normal (i.e., along 〈111〉 direction) to this plane [11]. The magnetic properties of the central R-ion are, therefore, influenced by a D3d crystal-field which causes the local magnetic susceptibility to be different along the local 〈111〉 axis and in the perpendicular plane, thereby producing a single-ion CF magnetic anisotropy at the R-site. Further, the directions of the local symmetry axes for four R-ions occupying the vertices of a particular tetrahedron are nonequivalent and are different from the crystallographic axis as well as from the n.n. bond directions. As a result, the measured (bulk) susceptibility differs from the above local (site) susceptibility at the R-site. To derive a relation between the bulk and local single-ion susceptibilities, taking into account the long-range dipolar and anisotropic exchange interactions

among R-ions in appropriate manner, we construct the total Hamiltonian of a single R-ion in crystalline lattice within the frame work of the CF theory in combination with a mean-field approximation shown as follows [13]: ! ! H t ðiÞ ¼ H FI ðiÞ þ H CF ðiÞ þ H Z þ μ ðiÞ  H int ðiÞ ð1Þ where HFI and HCF are, respectively, free-ion and D3d CF contribu! ! tions. H Z ¼  g J μB H ef f  J represents the Zeeman coupling between ! the 4f R-ion and the effective magnetic field H ef f (external field ! corrected for demagnetization effects). H int is the local internal magnetic field (in the absence of the external field) at the R-site appearing due to exchange and magnetic dipole–dipole interactions. The spinspin exchange tensor is defined by three principal independent components which couple the n.n. spins along (λ||) and normal (λ ? ,x, λ ? , y) to the bond direction [13]. The free-ion and CF components are written as [14,15] " # 3 ! ! i H FI þ H CF ¼ ∑ E ei þ ζ so L  S þ αLðL þ 1Þ þ β GðG2 Þ þ γ GðR7 Þ i¼1

þ ∑U kq Bkq ðk ¼ 2; 4; 6; q ¼ 0; 7 3; 7 6Þ

ð2Þ

The symbols bear their usual meanings. Bkq is the even-parity CF parameter (CFP) [15]. Because of the electrostatic, spin–orbit and configuration interactions, the free-ion terms of Er3 þ are mixed and form intermediately-coupled (IC) states. The energy matrix of Ht is formed with 330 |LSJ〉 basis states with the principal LS percentages derived from the 16 different low-lying Russell– Saunders terms, e.g., 4S, 4D, 4F, 4G, 4I, 2P, 2D(1), 2D(2),2F(1), 2F(2), 2 (1) 2 (2) 2 (1) 2 (2) 2 2 G , G , H , H , I, K, corresponding to the 4f11 electronic configuration of Er3 þ -ion [16]. We diagonalized the Hamiltonian (2) by treating all the terms on equal footing following the standard procedure and found that 16-fold degenerate groundmultiplet 4 I 15=2 of Er3 þ -ion splits into 8 Kramers doublets in the D3d symmetry. The energy values and wave-functions are then used to evaluate single-ion susceptibilities χ sj ( j ¼||, ? to the local 〈111〉 quantization axis of the crystal-field) using Van Vleck's expression. χ sj is next used to renormalize the effective site

χj and simulate bulk susceptibility χ of Er2Sn2O7 λ|| and λ ? (for simplicity, we assume λ ? ,x ¼ λ ? , y ¼ λ ? ): χs χ jj ¼ jj ½1 þ ð3λjj  λ ? þ2q  pÞχ s? ; ð3Þ Δ

susceptibilities

as functions of CFP's Bkq, and exchange interactions

χ? ¼

χ s? ½1 þ ð2λ ? þ p qÞχ sjj ; Δ

ð4Þ

where the symbols are defined in Refs. [8,13]. The bulk susceptibility equals χ ¼ ðN a =3Þðχ jj þ 2χ ? Þ, where Na is Avogadro number. The demagnetizing factor was taken to be N ¼0.99 70.03 for polycrystalline Er2Sn2O7 [17]. 3. Results and analysis 3.1. DC magnetic susceptibility The temperature dependence of the dc magnetic susceptibility χ of polycrystalline samples of Er2Sn2O7 was measured by Matsuhira et al. [9] and Sarte et al. [10] in the temperature range 1.8–300 K, as shown in Fig. 1. All these measurements agree qualitatively and obeys a CW law above  20 K with the values of θCW   14 K and μeff ¼ 9.55 μB/Er. The estimated μeff obtained from the hightemperature zone agrees with the free-ion value of 9.59 μB/Er for the ground-multiplet 4 I 15=2 of Er3 þ -ion. No sharp features were observed in χ down to 0.2 K [9] which can be identified as a signature of long-range ordering of Er-spins. Only below 20 K,

J. Alam et al. / Journal of Magnetism and Magnetic Materials 361 (2014) 175–181

1.5

8 1/χ (mole/emu)

χ (emu/mole)

Table 1 Crystal-field parameters (Bkq) (in cm  1) and exchange interaction constants (λ|| and λ ? ), their isotropic (λiso ¼ (λ|| þ 2λ ? )/3) and anisotropic (λanis ¼ λ||–λ ? ) combination (in units of T/μB) in Er2Sn2O7 (present work) and Er2Ti2O7 (from Refs. [13,19]) are given. The CFP's given in spherical tensor notations Bqp in [13] are converted to the

10

1.2

0.9

0.6

0.3

present form using Table 4 of [15].

6

Parameters

4 2 0

0

20

40 60 T (K)

80

100

0.0 0

50

177

100

150

200

250

300

B20 B40 B43 B60 B63 B66 λ|| λ? λiso λaniso

Er2Sn2O7

Er2Ti2O7

(Present work)

[13]

[19]

 855 78 810 78 10607 10  910 7 9 285 7 3  10357 10  0.018  0.035  0.029 0.017

 867.6 2707.6 772.8  973.4 587.6  919.7  0.0218  0.0741  0.0567 0.0523

 10007 7 950 7 10  9007 7  8007 14 2707 3  650 7 10

T (K) Fig. 1. dc magnetic susceptibility χ of Er2Sn2O7 – calculated values (solid curve), taking account of anisotropies of CF and molecular field, match with the experimental results (O) [9]. Calculated susceptibilities (dashed curve) in the crystal-field (χCF) match with the measured values down to T  80 K. Inset shows 1=χ in the range of 2 K o T o 100 K.

downward bend of 1=χ of Er2Sn2O7 showing a different decreasing rate of 1=χ on cooling was found with the value of θCW   3.2770.01 K. The variation of θCW and 1=χ over the temperature range marks the thermal population of higher CF states in addition to the effect of anti-ferromagnetic spin–spin interactions to be significant at lower temperatures [18]. It must be mentioned here that for isomorphous compound Er2Ti2O7, whose CF scheme may differ from Er2Sn2O7 due to slightly different local crystalline environment, inelastic neutron scattering experiments showed an ordered moment μord ¼3.01 μB/Er at 50 mK [3]. Such a reduced value of μord compared to the value of μeff was attributed to the quasi-classical ordering of local 〈XY〉-type anti-ferromagnet having spins fixed by the CF in the [111] easy-plane. Such quasi-classical order gives rise to the quantum fluctuations of the single-ion ground CF doublet, splits the degenerate ground-state manifold and lowers the free energy of a particular ordering wave vector in Er2Ti2O7 [10,12]. In this scenario, a crystal-field calculation is, therefore, required to determine the nature of the ground-state for Er2Sn2O7. Since firstly, the lattice parameters and geometry of the oxygen-surroundings of Er-ion in Er2Sn2O7 and Er2Ti2O7 are very close [11]; secondly, the thermal variation of χ of these two Er-compounds does not differ very much; and thirdly, CF is the most effective perturbation on the free-ion Hamiltonian (HFI) particularly in the high-temperature range; we have substituted the CFP's of Er2Ti2O7 reported by Dasgupta et al. [19] and Malkin et al. [13] (Table 1) in Eq. (2) and calculate single-ion susceptibilities χ sj and bulk susceptibility χ . Since neither set of Bkq's can simulate the observed thermal variation of χ and isothermal magnetization well, we vary these starting parameters in a selfconsistent way with the introduction of constraint of exchange interactions to fit the magnetic results. The best-fitted CF parameters and anisotropic spin–spin exchange tensor (λ||, λ ? ) are given in Table 1, and corresponding CF energy levels, wavefunctions of the ground multiplet 4 I 15=2 of Er2Sn2O7 are given in Table 2. Calculated single-ion CF susceptibility χCF agrees appreciably with the observed results in the temperature range of T ¼80– 300 K, but differs remarkably below 50 K (Fig. 1). Since the ground multiplet 4 I 15=2 is not a pure LS state and also because of the anisotropic singe-ion CF susceptibility, we assume the Heisenberg exchange among Er-ions in Er2Sn2O7 to be anisotropic along and

normal to the n.n. bond, as was done for Er2Ti2O7 in Ref. [13]. Inclusion of anti-ferromagnetic exchange tensor, λ|| ¼ 0.018 T/mB and λ ? ¼ 0.035 T/mB, in the spin-Hamiltonian (1) results in the renormalized bulk susceptibility which matches very well with the experimental data down to 2 K, shown in the inset of Fig.1. The renormalized components of the site susceptibility tensors, χ ? 4 χ||, and the ratio χ ? /χ|| E16 at 10 K (compared to the corresponding value of χ ? /χ||  3.6 for Er2Ti2O7 [19]) imply that the magnetic anisotropy of Er2Sn2O7 is easy-planar type such that Er-spins are confined effectively in the local [111] plane due to the coordinated effect of crystal-field interactions, and spin–spin exchange and dipolar interactions among Er-spins. Using the value of isotropic exchange constants

λiso ¼(λ|| þ2λ┴)/

3 ¼  0.029 T/mB, dipolar contribution θdip ¼ ðμ2ef f =3kB Þð16π ð1 

NÞ=3vÞ and exchange contribution θex ¼ ð2μ2ef f =kB Þλiso to the Curie –Weiss temperature θCW [13] are estimated and found to be, respectively, 0.0115 K and 3.54 K for Er2Sn2O7. Hence the Curie– Weiss temperature θCW ¼ θdip þ θex   3:53 K which agrees reasonably with the value obtained from the CW fitting of the susceptibility result below 20 K. The components of the magnetic moments, along and perpendicular to the local 〈111〉 direction, of the ground Kramers doublet Ψ g ¼ 0:39j 7 11=2i 8 0:24j 7 5=2i  0:755j 8 1=2i 7 0:334j 8 7=2i þ 0:

μ|| ¼ 7 μ ? ¼ 7 4.22 μB. The ground-state is well-isolated from the next upper state at Δg E58 K, and hence can be approximated as

283j 8 13=2i are found to be anisotropic, respectively, 0.264 μB,

an effectively spin-1/2 system. The effective magnetic moment of the ground-state of Er3 þ -spin in Er2Sn2O7 is given by μeff;g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμ2jj þ 2μ2? Þ=3  7 3:45 μB . Thus the value of μeff,g of Er-spins in Er2Sn2O7 is 15% larger than the observed ordered moment μord ¼ 3.01 μB/Er at 50 mK in Er2Ti2O7 which however is reduced from the single-ion value due to zero-point quantum fluctuations [3,12]. The anisotropy of the Landé g-factor of the ground doublet is calculated from the above CF wave-functions and found to be g|| ¼ 0.53 and g ? ¼8.45. We calculate the normalized expectation value of 〈J jj 〉=J  0:03 and 〈J ? 〉=J  0:94 for Er2Sn2O7. For comparison, these values in Er2Ti2O7 are also estimated to be 〈J jj 〉=J ¼ 0:016 and 〈J ? 〉=J  0:86 from the given CF wave-functions [19]. It is noted that the average value of 〈J j 〉=J, where j¼ ||,? to the 〈111〉 quantization axis, signifies fluctuations of the spin-moments away from the quantization axis or plane [20]. Significant low-energy spin fluctuations were observed in the inelastic spectrum of Er2Sn2O7 at 100 mK [10,21]. The difference of 〈J ? 〉=J in the two compounds, Er2Ti2O7 and

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J. Alam et al. / Journal of Magnetism and Magnetic Materials 361 (2014) 175–181

Table 2 Calculated CF energies (Ei) compared with the measured energies of few low-lying doublets [10], eigen-functions (ψi) of the eight Kramer's doublets of the ground multiplet j4 I 15=2 〉 of Er-ion in Er2Sn2O7 obtained using the CFP's given in Table 1. Calculated [13,19] and measured [3] CF energies of Er2Ti2O7 are also given for comparison. Er2Sn2O7 (present work)

Er2Ti2O7 ψi

Ei(K) Expt. [10]

Theory

0

0

59.16 7 0.1

58.6 70.6

87.177 0.23

89.5 7 1.0

198.44 71.2

194 72.0 495.87 5.0

560.3 75.5 636.2 76.0 705 7 7.0

Ei(K) Theory [13]

0.39 |7 11/2S 8 0.24 |7 5/2S  0.755 |8 1/2S7 0.33 | 8 7/2S þ 0.28 | 8 13/2S  0.06 |7 15/2S7 0.37 | 7 9/2S 8 0.76 |7 3/2S  0.51 |8 9/2S 7 0.27 |7 13/2S þ 0.47 | 7 7/2S  0.62 |8 5/2S7 0.54 | 8 11/2S 8 0.85 | 713/2S  0.165 |7 7/2S  0.09 |8 5/2S7 0.47 | 8 11/2S 8 0.29 | 713/2S þ0.30 |7 7/2S 8 0.63 | 71/2S  0.34 | 8 5/2S 8 0.54 | 8 11/2S 8 0.25 | 715/2S 8 0.42 |7 11/2S 7 0.6 | 73/2S7 0.6 | 8 9/2S  0.12 | 713/2S7 0.72 |7 7/2S 7 0.64 | 8 5/2S þ 0.16 | 8 11/2S 8 0.95 | 715/2S 8 0.20 |7 3/2S 8 0.12 | 8 9/2S

Er2Sn2O7, may indicate more directionality of the Er-spin magnetic moments, being normal to the 〈111〉 axis, in Er2Sn2O7 than in Er2Ti2O7 and may be attributable to the large proportions of the |MJ ¼ 71/2〉 component,  75% in Er2Sn2O7 compared to 56% in Er2Ti2O7 [3,19], in the CF ground-state

Ei(K) Expt. [3]

0

[19]

0

0

74.04

74.4

74

85.8

83.74

90

177.3

179.86

138

591.4

325

617.3

337

670.5

470

1001.44

684

increases to 0.82 cm  1 [3] or 0.93 cm  1 [19] for Er2Ti2O7, |MJ ¼ 7 1/2〉 contribution in the ground-state decreases to 56% [3] or 51% [19], and hence directionality of Er-spins with respect to the local [111] plane decreases.

Ψg. This directional property is also mani-

fested through the ratio χ ? /χ|| and the value of the ordered moments found for these two isomorphous compounds. The planar anisotropy of Er2Sn2O7 demonstrated in the susceptibility and ground-state magnetic moments can easily be understood from the following simplified interpretation of the value and sign of the second-order axial parameter B20 which plays a determining role in the CF Hamiltonian [2],1 if one uses the Stevens Operator Equivalent (SOE) approach instead of the irreducible tensor operator formalism used in the present work. In the SOE technique of describing CF, the axial term is B02 O02 where O02  f3J 2z  JðJ þ 1Þg and B02 is a numerical factor equal to the product of the Stevens operator equivalent factor, αJ, and CF potential having the form A02 〈r 2 〉 [14,15]. When the sign of B02 is negative, the CF ground-state is comprised mainly of highest | 7MJ〉 component and the spins are fixed along the easy-axis 〈111〉; on the other hand, when B02 is positive, j7 M min 〉 component J is the most contributing component in the ground CF state and favors an easy-planar anisotropy [2]. B20 is found to be  855 cm  1 for Er2Sn2O7 and αJ ¼4/1575 for Er3 þ -ion [14] and using the relation B02 ¼ ðB20 αJ =  2:73252Þ[15], we obtain the positive value of B02 E 0.8 cm  1. This implies that Er-spins in Er2Sn2O7 are confined in the [111] easy-plane with major contribution (  75%) coming from |MJ ¼ 71/2〉 in the CF ground-state Ψg (Table 2). As B02 1 One must be careful that the crystal-field Hamiltonian is parameterized by a set of parameters Bkq (k ¼ 2, 4, 6; q ¼0, 7 3, 76) appropriate to the CF symmetry at the site of magnetic ion. These parameters contribute differently and remarkably to the thermo-magnetic properties particularly at low temperatures, and hence determination of the reliable CF parameters and CF level pattern is a challenging problem from the available experimental results of magnetic susceptibility, magnetization, specific heat, optical spectra and inelastic neutron scattering etc. (see for example, Refs. [3, 8, 13, 18–20]). However, the value of the second-order axial parameter B20 is particularly significant, which varies linearly with the ionic radius of M-site ions in pyrochlores and also with the diminishing radius of the unfilled 4f shell [8, 13] thereby changing the lattice dimensions and CF properties, and therefore, B20 is used to study the systematic variation of CF over any pyrochlore series.

3.2. Isothermal magnetization Isothermal magnetization M(H) of polycrystalline Er2Sn2O7 with varying magnetic field at T ¼2, 6, 30 K was measured by Matsuhira et al. [9]. At 2 K, the value of magnetization increases gradually up to 2.5 T above which the increment becomes slow and finally saturated at the value 4.2 μB/Er around 5 T. However for higher temperatures, M(H) increases and no saturation arrives even at 5 T. Such behavior did not match with a Brillouin curve and may be influenced by the low-lying CF doublets. The CF ground-state is well separated by Δg  60 K from the next doublet which is larger than the splitting of the ground and first excited state in the saturation field of Hsat ¼5 T, being μeff,g Hsat E 11.6 K and μeff,1st Hsat E 1.4 K, respectively. We can, therefore, assume that the excited states do not merge and mix with the ground CF doublet state in presence of the external field H r5 T, and hence the field variation of the magnetization of the polycrystalline sample can be obtained powder-averaging of the magnetization ! over the angle θ between the effective magnetic field H ef f at the site of a magnetic ion and the local 〈111〉 direction using the following expression [8,22] appropriate for a well-isolated anisotropic ground doublet (spin-1/2 system) Z 1 π MðHÞ ¼ MðH; θÞ sin θdθ ð5Þ 2 0 where

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2jj cos 2 θ þ g 2? sin 2 θ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 , μB ðHeff þ zλiso μ Þ g2jj cos 2 θ þ g2? sin 2 θ A tanh@ 2kB T

1 MðH; θÞ ¼ μB 2

! þ χ VV ðH eff þ zλiso μ Þ

ð6Þ

The above expressions are found to describe the observed isothermal magnetization at T(K)¼ 2, 6, 30 quite well, as shown in Fig. 2, with the values of g|| ¼0.56 70.02 and g ? ¼8.4 70.2,

J. Alam et al. / Journal of Magnetism and Magnetic Materials 361 (2014) 175–181

4

temperature paramagnetic phase and an anti-ferromagnetism at lower temperatures, followed by a very sharp peak observed at Tp  0.35 K. In this context it is worthy to mention that this specific heat peak is not associated with the nuclear hyperfine interactions, since hyperfine contribution to the specific heat shows a peak below 100 mK, as was found in Er2Ti2O7 [19].We demonstrate below that a split in the ground CF doublet due to spin–spin exchange interactions may be the origin of such unconventional peak in specific heat characteristics of Er2Sn2O7 which, though, inhibits any long-range-ordering down to 100 mK. We present a qualitative description of Cmag/T behavior, on the assumption that the molecular field at Er-site breaks the 2-fold Kramers degeneracy of the anti-ferromagnetic ground CF state of Er-ions, and below 2 K, only the ground-state is thermally populated by Er-spins. If δ(T) is the splitting of the doublet state, then the magnetic specific heat can be written as

2K 6K

M ( μΒ/Er)

3

30K

2

1

0

0

1

2

3

4

5

H (T)

C mag ðδ; TÞ ¼  T

Fig. 2. Isothermal magnetization of Er2Sn2O7 measured at 2 K, 6 K and 30 K [9] are simulated by powder-averaging the magnetization over the field-direction (see text).

5 0.70

δ 0( Τ )

3

-1

0.00

1

2

3

4

5

T(K)

1

0 0

5

10

15

¼ Rðδ=TÞ2

eδ=T

ð7Þ

ð1 þ eδ=T Þ2

where F(δ, T) is the Helmholtz free energy of the spin-system arising due to the distribution of spins in the 2-level Schottky manifold. We assume that the 〈XY〉-type anti-ferromagnetic spins are arranged in small clusters of tetrahedral in a correlated manner with the restriction that the sum over the spins on a ! particular tetrahedron ∑ s j ¼ 0, which are randomly oriented

represents probability distribution for the splitting δ(T) and has a Gaussian form, above and below Tp, corresponding to the observed peak in specific heat

-0.70 0

∂T

2

relative to each other. Any individual spin ‘feels’ an effective molecular field which, instead of having a well-defined value, has a random character and hence follows a probability distribution [23]. We, therefore, multiply Eq. (7) by a factor P(T), which

-0.35

2

∂2 F

j

Tc

0.35

-2

-1

C pT (JK mol )

4

179

20

25

T (K) Fig. 3. Observed specific heat Cp/T (O) shows a Schottky anomaly down to  3.5 K below which an unconventional rise occurs with a peak at  0.35 K [10]. Schottky anomaly is fitted (dashed line) using the CF level pattern of the ground-multiplet 4 I 15=2 of Er3 þ -ion, given in Table 2. A split in the ground doublet due to exchange interactions which is temperature-dependent and follows a Gaussian probability distribution may be the origin of such unconventional specific heat rise (solid curve). Inset shows the temperature-dependent exchange splitting, which vanishes at the temperature Tc, above which the Er-spin system enters into the paramagnetic phase.

λiso ¼  0.031 70.001 T/μB, and Van Vleck paramagnetism χVV E0.1 emu/mole ¼ 0.18 μB/T. These values agree with the values obtained above from the CF simulation of the magnetic susceptibility. 3.3. Specific heat Magnetic specific heat (Cmag) of Er2Sn2O7 as a function of temperature in the range 0.3–25 K was reported by Sarte et al. [10]. The specific heat has two distinct anomalies: (i) a broad shoulder down to  3.5 K corresponding to a multilevel Schottky anomaly which can be fitted using the CF level pattern of the ground-multiplet 4 I 15=2 of Er3 þ -ion (Fig. 3), and (ii) an anomalous rise in Cmag/T below 2 K, showing a transition between the high-

PðδÞ ¼ Cexp½  ðδ  δ0 Þ2 =2s2G 

for δ Z0

PðδÞ ¼ Cexp½  ðδ þ δ0 Þ2 =2s2G 

for δ r0

ð8aÞ ð8bÞ R1

where C is the normalization factor defined by  1 PðδÞdδ ¼ 1. Further we assume that the molecular field vanishes in the paramagnetic regime where all the spins act independently (in the crystal-field) [23], so that the specific heat is given by the Schottky distribution of spins over the low-lying CF doublet. It is, therefore, necessary to introduce a temperature-dependence of the random exchange field according to a power law [24]

δ0 ðTÞ ¼ δ0 ð1  T=T c Þa

ð9aÞ

sG ðTÞ ¼ sG ð1 T=T c Þa

ð9bÞ

δ0 represents the absolute splitting of the ground doublet at T ¼0 K in absence of randomness of spins. Temperature dependence of the exchange splitting is plotted in the inset of Fig.3, which vanishes at the temperature Tc, above which the Er-spin system enters into the paramagnetic regime. sG defines the standard deviation of the splitting from its average value given by δ0. The splitting parameters δ0, sG, and Tc can be determined from the numerical simulation of the observed magnetic specific heat below 3 K using the following formula taking account of Eq. (9): Z þ1 PðTÞ C mag ðδÞdδðTÞ ð10Þ C mag ðTÞ ¼ 1

Fig. 3 displays a theoretical fitting of Cmag/T using above model which reproduces the specific heat peak at 0.34 ( 70.01) K, adopting the mean-field critical exponent α ¼0.5 and using the values of δ0 ¼1.4 K, sG E1.42 K and Tc ¼ 3.4 K. The ‘characteristic energy separation’ for the Gaussian probability distribution of the splitting between the two magnetic levels of the ground doublet is

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J. Alam et al. / Journal of Magnetism and Magnetic Materials 361 (2014) 175–181

defined by sG þ δ0 [24] and is found to be 2.82 K. Characteristic exchange splitting can also be evaluated using the molecular-field Hamiltonian [25,26] at Er-site in the ground CF state as follows: H mol ðiÞ ¼ ∑ λij ð  μef f ;g /μg S þ 1=2/μg S2 Þ

ð11Þ

j

Summation is done over n.n. spins (z¼ 6). The value of the splitting is found to be 2.78 K which thus agrees well with the above specific heat parameter. The temperature-dependent splitting due to the molecular field was also observed for other anti-ferromagnetic materials, Tb2Ti2O7 [27], Tb2Nb2O7 [18] having pyrochlore structure and FeBr2 [25], LaCoO3 [28], ErNi5 [29] in different crystal-field environments. Finally, though the statistical model of Klein and Brout is ideally suited for the spin-glass systems in which randomness of spins and frustration are caused either due to impurities (as in dilute Cu–Mn) [23] or due to topological arrangement of spins (e.g. in R2Mo2O7, RQGd, Sm) [24], this model was successfully applied to simulate the magnetic specific heat behavior of R2V2O7, R ¼Y, Lu, [26,30] below and above the ferromagnetic transition temperature Tc assuming a probability distribution and temperature dependence of the ground-state splitting due to the molecular field at V-site.

4. Discussion and conclusions We have simulated and analyzed the experimental results of bulk magnetic susceptibility, isothermal magnetization and magnetic specific heat, published earlier, of the frustrated anti-ferromagnetic pyrochlore compound Er2Sn2O7. The dc susceptibility is simulated adopting an exact relation between single-ion susceptibility tensors and site susceptibility tensors in the mean-field approach, assuming anisotropic Heisenberg exchange interactions (λ|| and λ ? ) among nearest-neighbor (n.n.) Er-spins and using a set of CF parameters defining D3d symmetry. The magnetization is well described taking account of angle-averaged magnetization of the polycrystalline sample, expressed in terms of anisotropic g-tensors and exchange tensors. The magnetic specific heat of Er2Sn2O7 is fitted assuming a probability distribution of the exchange splitting of the ground doublet following the Klein–Brout statistical model. Since the specific heat displays a Schottky anomaly down to Tc  3.5 K and an anomalous peak at Tp 0.35 K, these two temperatures served as additional constraints in the fitting procedure. The CF ground-state of Er-spins in Er2Sn2O7 can be considered to be an isolated doublet to the first approximation, with major contribution coming from |J, Mmin 〉¼ |15/2, 71/2〉 component. This J fact qualitatively implies that the Er-magnetic moments in Er2Sn2O7 lie in a plane normal to the local 〈111〉 (D3d) symmetry axis due to the anisotropies arising from the single-ion CF and internal molecular field at the magnetic Er-site forming the tetrahedral network. The total CF splitting (ΔCF) of the ground multiplet 4 I 15=2 of Er-ion in Er2M2O7 increases as the lattice constant decreases, if Sn-site is replaced by Ti-ions (Table 1). Exchange interactions are found to be comparatively weaker and less anisotropic in Er2Sn2O7 due to larger n.n. bond separation of Er-spins arranged in the tetrahedral network than its titanate counterpart, causing smaller value of the Curie–Weiss temperature in Er2Sn2O7. Though the local CF environments are identical for the titanate and stannate compounds, slight changes in the distortion of the MO6-octahedra, local crystalline environment and nearestneighbor bond separation due to change in the Shannon radius [31] of Sn4 þ (0.69 Å) and Ti4 þ -ions (0.605 Å) in Er2Sn2O7 and Er2Ti2O7 give rise to the remarkable variations in the magnetic and specific heat properties of these two compounds, manifested

through the crystal-field parameters (more specifically, through B20), CF splitting (ΔCF and Δg), ordered or saturation magnetization of Er-spins, and anisotropic values of the exchange interaction tensors. Smaller values of the separation energy (Δg) of the ground doublet from the higher ones and the smaller values of the effective exchange interactions (λiso) in Er2Sn2O7 result in smaller value of the temperature-dependent characteristic splitting of the CF ground doublet of Er-ions. Smaller energy splitting of the single-ion ground doublet may lead to the rapidly fluctuating magnetic moments in Er2Sn2O7 inhibiting any long-range ordering in polycrystalline Er2Sn2O7 down to 100 mK, in sharp contrast to Er2Ti2O7 where the order-by-disorder mechanism favors Er-spins to stabilize along a particular ordering wave vector denoted ψ2 due to the appreciable splitting of the ground-state manifold [3,10,12,32]. Recently in an independent study, Guitteny et al. [33] demonstrated the existence of short-range correlated domains of magnetic moments in Er2Sn2O7 which exhibits a freezing below 200 mK in the Palmer–Chalker (PC) configurations, hence quite different from the ψ2 ordering selected in Er2Ti2O7 [32]. The PC ground-state is realized for an XY-type pyrochlore antiferromagnet having isotropic exchange and long-range dipolar interactions among magnetic moments. In conclusion, we describe the dc magnetic susceptibility, isothermal magnetization and specific heat of pyrochlore Er2Sn2O7 employing anisotropic crystal-field and molecular field in the mean-field approach. Er-spins are effectively constrained to lie in the local [111] easy-plane of the frustrated tetrahedral spinstructure in the anti-ferromagnetic phase. The CF ground-state of Er-ion is a doublet which is split by the spin–spin exchange coupling to form new magnetic states which gives rise to unconventional rise in the magnetic specific heat in Er2Sn2O7.

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