Magnetic Properties of New Triangular Lattice Magnets A${_4}$B'B${_2}$O$_{12}$

TL;DR

This study investigates the magnetic behaviors of A4B'B2O12 triangular lattice magnets, revealing how different transition metal ions and chemical pressure influence their ferromagnetic or antiferromagnetic ground states.

Contribution

It provides new insights into how spin number and chemical pressure affect magnetic ordering in geometrically frustrated triangular lattice magnets.

Findings

Co2+ and Ni2+ samples show ferromagnetic order

Mn2+ sample exhibits antiferromagnetic order

Chemical pressure tunes ferromagnetic transition temperature

Abstract

The geometrically frustrated two dimensional triangular lattice magnets A${_4}$B'B${_2}$O$_{12}$ (A = Ba, Sr, La; B' = Co, Ni, Mn; B = W, Re) have been studied by x-ray diffraction, AC and DC susceptibilities, powder neutron diffraction, and specific heat measurements. The results reveal that (i) the samples containing Co$^{2+}$ (effective spin-1/2) and Ni$^{2+}$ (spin-1) ions with small spin numbers exhibit ferromagnetic (FM) ordering while the sample containing Mn$^{2+}$ (spin-5/2) ions with a large spin number exhibits antiferromagnetic (AFM) ordering. We ascribe these spin number manipulated ground states to the competition between the AFM B'-O-O-B' and FM B'-O-B-O-B' superexchange interactions; (ii) the chemical pressure introduced into the Co containing samples through the replacement of different size ions on the A site finely tunes the FM ordering temperature of the system. We attribute this effect to the modification of the FM interaction strength induced by the change of the O-B-O angle through chemical pressure.

Tables

Table 1. Table 1: Structural parameters for the Co containing compounds at room temperature (space group R-3mH ) determined from refined XRD measurements.

Refinement	Atom	Site	x	y	z	Occupancy
Sr4CoRe2O12      ${\chi }^2$ = 2.29      (a)	Sr1	6c	0	0	0.12982(5)	0.16666
	Sr2	6c	0	0	0.29351(6)	0.16666
	Co	3a	0	0	0	0.08333
	Re	6c	0	0	0.42186(3)	0.16667
	O1	18h	0.50289(50)	0.49711(50)	0.12312(14)	0.50
	O2	18h	0.49249(53)	0.50751(53)	0.29174(19)	0.50
	$a$ = $b$ = 5.54464(26) (Å), $c$ = 26.73815(126) (Å)
	Overall B-factor = 1.332 (Å2)
Sr2La2CoW2O12      ${\chi }^2$ = 2.58      (b)	Sr1	6c	0	0	0.13151(12)	0.08333
	La1	6c	0	0	0.13151(12)	0.08333
	Sr2	6c	0	0	0.29159(15)	0.08333
	La2	6c	0	0	0.29159(15)	0.08333
	Co	3a	0	0	0	0.08333
	W	6c	0	0	0.42266(10)	0.16667
	O1	18h	0.49750(140)	0.50250(140)	0.12427(38)	0.50
	O2	18h	0.50466(156)	0.49533(156)	0.29473(48)	0.50
	$a$ = $b$ = 5.62838(5) (Å), $c$ = 26.69634(44) (Å)
	Overall B-factor = 1.127 (Å2)
Ba2La2CoW2O12      ${\chi }^2$ = 2.91      (c)	Ba1	6c	0	0	0.13426(8)	0.08333
	La1	6c	0	0	0.13425(8)	0.08333
	Ba2	6c	0	0	0.29018(9)	0.08333
	La2	6c	0	0	0.29018(9)	0.08333
	Co	3a	0	0	0	0.08333
	W	6c	0	0	0.41694(8)	0.16667
	O1	18h	0.50022(89)	0.49977(89)	0.11969(26)	0.50
	O2	18h	0.47565(77)	0.52436(77)	0.29193(36)	0.50
	$a$ = $b$ = 5.67644(22) (Å), $c$ = 27.35413(106) (Å)
	Overall B-factor = 1.743 (Å2)
Ba3LaCoWReO12      ${\chi }^2$ = 1.43      (d)	Ba1	6c	0	0	0.12939(6)	0.12500
	La1	6c	0	0	0.12944(6)	0.04167
	Ba2	6c	0	0	0.29533(6)	0.12500
	La2	6c	0	0	0.29533(6)	0.04167
	Co	3a	0	0	0	0.08333
	W	6c	0	0	0.42185(6)	0.08333
	Re	6c	0	0	0.42185(6)	0.08333
	O1	18h	0.51068(82)	0.48932(82)	0.12143(23)	0.50
	O2	18h	0.48365(80)	0.51635(80)	0.29260(32)	0.50
	$a$ = $b$ = 5.70429(9) (Å), $c$ = 27.675722(54) (Å)
	Overall B-factor = 1.757 (Å2)
Ba4CoRe2O12      ${\chi }^2$ = 4.51      (e)	Ba1	6c	0	0	0.12911(8)	0.16666
	Ba2	6c	0	0	0.29435(9)	0.16666
	Co	3a	0	0	0	0.08333
	Re	6c	0	0	0.42089(8)	0.16667
	O1	18h	0.50262(108)	0.49738(108)	0.12325(29)	0.50
	O2	18h	0.47513(97)	0.52487(97)	0.28794(41)	0.50
	$a$ = $b$ = 5.72455(33) (Å), $c$ = 27.76966(161) (Å)
	Overall B-factor = 1.694 (Å2)

Table 2. Table 2: Structural parameters for Ba 2 La 2 NiW 2 O 12 and Ba 2 La 2 MnW 2 O 12 at room temperature (space group R-3mH ) determined from refined NPD measurements.

Refinement	Atom	Site	x	y	z	Occupancy
Ba2La2NiW2O12      (a)	Ba1	6c	0	0	0.13311(48)	0.08333
	La1	6c	0	0	0.13311(48)	0.08333
	Ba2	6c	0	0	0.29408(53)	0.08333
	La2	6c	0	0	0.29408(53)	0.08333
	Ni	3a	0	0	0	0.08333
	W	6c	0	0	0.41928(65)	0.16667
	O1	18h	0.49819(82)	0.50181(82)	0.11674(28)	0.50
	O2	18h	0.47851(76)	0.52149(76)	0.29363(36)	0.50
	$a$ = $b$ = 5.66471(25) (Å), $c$ = 27.37551(179) (Å)
	Overall B-factor = 1.263 (Å2)
Refinement	Atom	Site	x	y	z	Occupancy
Ba2La2MnW2O12      (b)	Ba1	6c	0	0	0.13971(71)	0.08333
	La1	6c	0	0	0.13971(71)	0.08333
	Ba2	6c	0	0	0.29399(60)	0.08333
	La2	6c	0	0	0.29399(60)	0.08333
	Mn	3a	0	0	0	0.08333
	W	6c	0	0	0.41675(92)	0.16667
	O1	18h	0.48087(116)	0.51913 (116)	0.11743(34)	0.50
	O2	18h	0.49090(107)	0.50910(107)	0.29269(40)	0.50
	$a$ = $b$ = 5.73298(37) (Å), $c$ = 27.41336(275) (Å)
	Overall B-factor = 1.928 (Å2)

Table 3. Table 3: Summary of magnetic properties (transition temperatures T C C {}_{\text{C}} , T N N {}_{\text{N}} , Curie-Weiss constant θ CW subscript 𝜃 CW \theta_{\text{CW}} , μ eff subscript 𝜇 eff \mu_{\text{eff}} , saturation magnetization M S S {}_{\text{S}} , and gyromagnetic ratio g 𝑔 g ) of A 4 B’B 2 O 12 compounds

Compound	T${}_{\text{C}}$/T${}_{\text{N}}$ (K)	${\theta }_{\text{CW}}$ (K)	${\mu }_{\text{eff}}$ (${\mu }_{\text{B}}$)	M${}_{\text{S}}$ (${\mu }_{\text{B}}$)	$g$
Sr4CoRe2O12	1.0	1.0	3.91	1.5	3.0
Sr2La2CoW2O12	1.26	1.5	3.95	2.0	4.0
Ba2La2CoW2O12	1.28	1.6	4.01	2.1	4.2
Ba3LaCoReWO12	0.83	0.78	3.89	1.9	3.8
Ba4CoRe2O12	0.87	0.61	3.87	1.6	3.2
Ba2La2NiW2O12	6.2	25.5	3.19	$\approx$ 2	$\approx$ 2
Ba2La2MnW2O12	1.7	-10.7	5.73	–	–

Full text

I INTRODUCTION

Geometrically frustrated magnets have attracted much attention for the novel properties they exhibit at low temperatures such as spin ice, spin liquids, non-collinear ground states, and fractional gauge fields ramirez ; Balents ; Greedan . Of particular interest is the two dimensional (2D) triangular lattice magnets (TLMs) as, despite its simple structure, it can be host to a multitude of ground states collins ; htlaf2 ; htlaf1 . There are several examples: (i) Ba3CoSb2O9ba3coz ; ba3cos ; jeffery ; nmr , with equilateral Co2+ (effective spin-1/2) triangular lattice, exhibits a 120 degree ordered state at zero field and an up up down phase with applied field. Although its spin wave spectrum can be reasonably described by the XXZ model with a small easy plane anisotropy, the quantum spin fluctuations still lead to abnormal magnon decay and line broadening that cannot be accounted for by linear and nonlinear spin wave theories jma ; (ii) Ba3BNb2O9 (B = Co2q ; yokota , Nihwang , Mnlee ), CuCrO2cucro2 ; kimura , and RbFe(MoO4)2RbFe ; RbFe2 exhibit interesting multifeorroic properties in the 120 degree ordered state; (iii) 2H-AgNiO2 AgNiO ; AgNiO2 is a rare TLM that shows ordering with a collinear alternating stripe pattern which could be related to its strong easy axis anisotropy; (iv) AAg2M(VO4)2 (A = Ba, Sr; M = Co, Ni)AgVO ; BaAg are TLMs possessing ferromagnetic (FM) orderings due to superexchange via the bridging vanadates on a triangular lattice; (v) NaVO2 navo1 ; navo is a rare TLM exhibiting ordering of the V3+ orbitals. Consequently, this orbital ordering relieves the geometrical frustration and leads to long range magnetic ordering; (vi) shown recently, the exotic quantum spin liquid (QSL) state is realized in YbMgGaO4 YMG1 ; YMG2 ; YMG3 ; YMG4 with an effective spin-1/2 Yb3+ triangular lattice in which the spin anisotropy and next-nearest neighbor interactions play important roles.

To investigate all of these intriguing ground states, the discovery and exploration of new examples of TLMs is necessary. From a materials engineering perspective, one possible way to realize the 2D triangular lattice is by a stacked layer model, for example in the perovskite structurestack ; prv . The basic perovskite structure (ABO3) can be considered as consisting of 3 layers of AO3 with layers of B ions in between. The A and B site ions form corner sharing octahedra with the surrounding oxygens. Alternately, adjacent layers may consist of edge sharing oxygen octahedra, opening up further stacking mechanisms to be explored in a plethora of compounds stack . Of interest in realizing true 2D behavior is the presence of vacant layers where neither a magnetic nor non-magnetic ion resides. This vacancy helps to ensure that interlayer interactions are small compared to intralayer interactions.

Based on this principle, we examine new 2D TLMs utilizing close-packed stacking of perovskite layers. A promising structure is A4B'B2O12 (A = Ba, Sr, La; B’ = Co, Ni, Mn; B = W, Re). In this structure, using A4CoB2O12 as the example shown in Fig. 1, the magnetic Co2+ ions and nonmagnetic W6+/Re7+ ions occupy the octahedral sites in the perovskite layers in an ordered fashion at the (3$a$) and (6$c$) positions, respectively. Thus, the CoO6 ocatahedral layer forms a Co2+-triangular lattice in the $ab$ plane. Here the Co octahedra are corner sharing with the adjacent W/Re octahedra and this Co layer is sandwiched by two (W/Re)O6 layers. Moreover, the octahedral interstices between the two adjacent W/Re layers are vacant to accommodate the strong electric repulsion of the W6+/Re7+ ions. The separation of the layers of Co2+ by two layers of W6+/Re7+ and one vacant layer significantly reduces the interplane interactions and enhances the 2D nature of this structure. More intriguingly, due to the flexible chemistry in this structure, we can easily (i) introduce chemical pressure into the system by changing the lattice parameters. For example, within the set of samples, Sr4CoRe2O12, Sr2La2CoW2O12, Ba2La2CoW2O12, Ba3LaCoReWO12, and Ba4CoRe2O12, we expect the lattice parameters will change accordingly as we replace Sr2+ ions with larger La2+ and even larger Ba2+ ions; (ii) change the spin numbers of the system by varying magnetic the B’ ions. For example, within the set of samples Ba2La2CoW2O12, Ba2La2NiW2O12 and Ba2La2MnW2O12, the spin numbers change from effective spin-1/2 (Co2+) to spin-1 (Ni2+), and spin-5/2 (Mn2+). Therefore, the A4B'B2O12 structure provides an ideal platform to study the magnetic properties of new TLMs and how the perturbations, such as chemical pressure and spin numbers, affect them. So far, while the structures of several A4B'B2O12 members have been reported stack2 ; bala1 ; bala2 and the susceptibility of Ba2La2MnW2O12balamn has been measured down to 2 K, there are no detailed studies on their magnetic properties.

In this paper, we examine the structural and magnetic properties of these two sets of samples by using x-ray diffraction (XRD), AC and DC susceptibility ($\chi_{\text{AC}}$, $\chi_{\text{DC}}$), powder neutron diffraction, and specific heat measurements. The results reveal that the magnetic ground state is FM for samples containing Co2+ and Ni2+ ions with small spins but antiferromagnetic (AFM) for the Mn2+ containing sample. We propose that this ground state change is due to the competition between the different superexchange interactions in the structure. We also find that the chemical pressure finely tunes the transition temperature and exchange interaction in the Co samples, but does not drastically change the ground state as the spin number does.

II EXPERIMENTAL

In total, seven polycrystalline samples of A4B’B2O12 were prepared by the standard solid state reaction method. They are Sr4CoRe2O12, Sr2La2CoW2O12, Ba2La2CoW2O12, Ba3LaCoReWO12, Ba4CoRe2O12, Ba2La2NiW2O12 and Ba2La2MnW2O12. Stoichiometric amounts of BaCO3/SrCO3, La2O3 (pre-dried at 980 ∘C overnight), CoCO3/ NiO/MnO, and WO3/Re metal were mixed in agate mortars, compressed into pellets, and annealed for 20 hours at temperatures of 1000 ∘C for Ba4CoRe2O12, 1050 ∘C for Ba3LaCoReWO12, 1100 ∘C for Sr4CoRe2O12, and 1250 ∘C for Ba2La2CoW2O12, Sr2La2CoW2O12, Ba2La2MnW2O12, and Ba2La2NiW2O12.

Powder XRD patterns were performed at room temperature using a HUBER imaging-plate Guinier camera 670 with Ge monochromatized Cu K$\alpha$1 radiation (1.54059 Å). High-resolution neutron powder diffraction (NPD) measurements were performed by a neutron powder diffractometer, HB2A, at the High Flux Isotope Reactor (HFIR) of the Oak Ridge National Laboratory (ORNL). Around 3 g of powder was loaded in an Al-cylinder can and mounted in a close-cycled refrigerator. We used a neutron wavelength of $\lambda$ =1.5405  Å  with a collimation of 12′-open-6′. Both the XRD and NPD patterns were analyzed by the Rietveld refinement program FullProf Fprof . DC magnetic susceptibility measurements were performed at temperatures of 2-300K using a Quantum Design superconducting interference device (SQUID) magnetometer with an applied field of 0.5 T. AC susceptibility was measured with the conventional mutual inductance technique with a homemade setupAC . The specific heat data was obtained using a commercial physical property measurement system (PPMS, Quantum Design).

III RESULTS

Cobalt containing compounds– The refined XRD patterns of five Co containing compounds are shown in Fig. 2. All compounds show pure phases with the space group R-3mH. Used as a continuing example, Sr4CoRe2O12 has lattice constants of $a$ = 5.5446(3)  Å  and $c$ = 26.7382(12) Å. The refined structural parameters are listed in Table I. It is obvious that the lattice parameters are governed by the size of A site ions, which increase as strontium is replaced by lanthanum and subsequently replaced by barium. We also tested the refinement by involving the site mixing between Co and W/Re sites. However, the refinements with the extra variable show no improvement from the refinementof the fully ordered structure used here. The results indicate that the site disorder in the compounds is not significant. It could be below 5%, within our XRD resolution.

The inverse DC susceptibility for all five Co containing members is shown in Fig. 3 (a-e). All 1/$\chi_{DC}$ $\sim$ $T$ curves show a slope change around 80 K. Continuing to use Sr4CoRe2O12 as an example, linear fitting from 150 K to 300 K yields a Curie-Weiss constant of $\theta_{\text{CW}}$ = -17 K and $\mu_{\text{eff}}$ = 4.72 $\mu_{\text{B}}$ while the linear fitting from 4.0 K to 15 K yields $\theta_{\text{CW}}$ = 1.0 K and $\mu_{\text{eff}}$ = 3.91 $\mu_{\text{B}}$. This change of effective magnetic moment indicates a spin state transition of the Co2+ ions from high spin (S = 3/2) to low spin (S = 1/2). This transition is due to the octahedral environment of the Co2+ ions, where the combination of the crystal field and spin orbital coupling lead to a Kramers doublet ground state with effective S = 1/2, as described by Low low and further examined by Lloret et al lloret . Thus the positive $\theta_{\text{CW}}$ at low temperatures suggests FM exchange interactions of the effective spin-1/2 Co2+ ions. The spin state transition and positive $\theta_{\text{CW}}$ were also observed for the four other Co compounds by similar linear fittings. The DC magnetization data was taken at 1.8 K, shown in Fig. 3 (a-e) insets. All data consistently shows a tendency of saturation around $\mu_{0}$Hs $\approx$ 3 T. To account for Van Vleck paramagnetism, a linear fit of high field data was used to calculate the saturation magnetization (M${}_{\text{S}}$). For Sr4CoRe2O12, the extrapolation yields a value of M${}_{\text{S}}$ = 1.52 $\mu_{\text{B}}$/Co2+ and a powder-averaged gyromagnetic ratio $g$ = 3.04. For the four other compounds, the obtained $g$ is around 3 $\sim$ 4.14. All of the magnetic parameters are summarized in Table III.

The real part of the AC susceptibility measurements taken under different DC fields for the Co compounds are shown in Fig. 4. Continuing use of Sr4CoRe2O12 as an example, its $\chi_{\text{AC}}$ shows a fast increase with decreasing temperature below 1.5 K followed by a broad peak. Here we define the transition temperatures as the local minima position in the first derivative of $\chi_{\text{AC}}$, which is 1.0 K for Sr4CoRe2O12. With increasing DC field, this broad peak is suppressed and shifts to higher temperatures. This increase of the transition temperature under applied field suggests a FM nature which is consistent with the DC susceptibility results. The four other Co compounds show similar $\chi_{\text{AC}}$ behavior with different transition temperatures, which are listed in Table III. Therefore, all five Co compounds show FM ordering around 1 K. One noteworthy feature is that for Sr2La2CoW2O12 and Ba2La2CoW2O12, the $\chi_{\text{AC}}$ shows a second peak at lower temperatures below 1 K when a 0.05 T DC field was applied. Whether this feature represents a second magnetic ordering to make the ordering process two-stepped or magnetic domain movements in the polycrystalline sample is not clear at this stage.

Ba2La2NiW2O12– The Rietveld refinement performed on the NPD pattern (Fig. 5(a)) measured at room temperature confirms the pure phase with space group R-3mH for Ba2La2NiW2O12. The refined structural parameters are $a$ = 5.6647(3)  Å  and $c$ = 27.3755(18) Å. The detailed structural parameters are listed in Table II. The linear fitting of the inverse DC susceptibility from 150 to 300 K (Fig. 5(b)) yields a positive $\theta_{\text{CW}}$ = 25.5 K and $\mu_{\text{eff}}$ = 3.19 $\mu_{\text{B}}$, indicating FM interactions. As shown in the insert of Fig. 5(b), the $\chi_{AC}$ measured at 50 Oe clearly shows a sharp transition at $T_{\text{C}}$ = 6.2 K (defined as the minimum in $d\chi_{AC}/dT$). With increasing field, this transition becomes broader and shifts to higher temperatures, which again verifies its FM nature. The DC magnetization data taken at 2.0 K (Fig. 5(c)) clearly shows a hysteresis loop and a saturation value at M${}_{\text{S}}\approx$ 2 $\mu_{\text{B}}$. Under zero field, the specific heat data (C${}_{\text{P}}$, Fig. 5(d)) shows a sharp peak at 6.2 K in agreement with $T_{\text{C}}$ from $\chi_{AC}$. Under applied DC fields, this peak broadens out and occurs at higher temperatures. The transition temperature (the peak position) increases linearly with the increasing field as shown in the insert of Fig. 5(d). All of this data, such as the positive $\theta_{\text{CW}}$, fast increase of $\chi_{AC}$ and peak of C${}_{\text{P}}$ at 6.2 K, the hysteresis loop of magnetization, and the increase of $T_{\text{C}}$ under fields, consistently show that Ba2La2NiW2O12 is a spin-1 system with long range FM ordering at 6.2 K.

Ba2La2MnW2O12–The Rietveld refinement performed on the NPD pattern measured at room temperature (Fig. 6(a)) confirms the pure phase with space group R-3mH for Ba2La2MnW2O12. The refined structural parameters are $a$ = 5.7330(4)  Å  and $c$ = 27.4134(28) Å. The detailed structural parameters are listed in Table II. The linear fitting of the inverse DC susceptibility from 150 to 300 K (Fig. 6(c)) yields a negative $\theta_{\text{CW}}$ = -10.7 K and $\mu_{\text{eff}}$ = 5.73 $\mu_{\text{B}}$, indicating AFM interactions. The specific heat data (Fig. 6(c)) shows a sharp peak at $T_{\text{N}}$ = 1.7 K, which should represent AFM long range ordering.

The $\chi_{\text{AC}}$ was measured as a function of temperature under varying applied DC fields (Fig. 7(a)) and as a function of DC field under applied temperatures (Fig. 7(b)). Here, transitions were again found using local minima of d$$\chi_{\text{{AC}}}/$dT$ and d$$\chi_{\text{AC}}/$d$H. As shown in Fig. 7(a), at zero field, the temperature dependence of $\chi_{\text{AC}}$ shows no significant feature but a weak slope change around 1.7 K, which is consistent with the $T_{\text{N}}$ observed from the specific heat. Then, with H = 2 T, there is a peak that appears at 1.5 K. Thereafter, this peak position shifts to higher temperatures first and then shifts to lower temperatures with increasing DC field, as indicated by the curved arrow in Fig. 7(a). Meanwhile, the field scan performed at 0.3 K shows two slope changes around 4 T and 11 T. These two features are more clearly visible as the two sharp valleys from the $d\chi_{\text{AC}}$/$d$H curve (Fig. 7(c)). Here we define the minima positions as H${}_{\text{c1}}$ = 3.75 T and H${}_{\text{c2}}$ = 11.6 T. With increasing temperature, these two features become broader and shift to lower fields. Above $T_{\text{N}}$, they disappear. A magnetic phase diagram of Ba2La2MnW2O12 was constructed by the transition temperatures and critical field values obtained from C${}_{\text{P}}$ and $\chi_{\text{AC}}$, as shown in Fig. 8.

IV DISCUSSION

It is obvious that the spin number of the magnetic B’ ions significantly affects the magnetic ground states of the studied A4B’B2O12 compounds, which manifest FM ordering for the Co2+ (S = 1/2) and Ni2+ (S = 1) compounds but AFM ordering for the Mn2+ (S = 5/2) compound. To understand this drastic ground state change, we examine the superexchange interactions of the B’2+ ions in the structure. Using the theoretical framework laid down by KanamoriKanamori , a qualitative description of the superexchange interaction between magnetic cations on an octahedral site can be determined from the orbital configurations of the magnetic cations and the nonmagnetic, bridging anions. In A4B’B2O12, octahedra of B’O6 are corner sharing with octahedra of BO6 via an oxygen, providing two pathways for intralayer superexchange. These paths are B’2+-O2--O2--B’2+ with the other being B’2+-O2--W6+-O2--B’2+ or B’2+-O2--Re7+-O2--B’2+, as show in Fig. 9(a). The superexchange pathway along B’2+-O2--O2--B’2+ is commonly found in other magnetic oxides, where the interaction is AFM.

Meanwhile, for W6+ and Re7+ ions on octahedral sites, the cubic crystal field splits their degenerate $4f$ orbitals (the filled outermost orbitals) into three groups as shown in Fig. 9(b). The group of $f_{x^{3}}$, $f_{y^{3}}$, and $f_{z^{3}}$ with $t_{1g}$ symmetry mainly participate in the orbital hybridization with the O2--$2p$ orbitals due to geometrical reasons. Thus, one possible situation for the B’2+-O2--W6+-O2--Cr3+ exchange path, which is similar for the Re case, is shown in Fig. 9(c). Here we consider the superexchange interaction between the spins on the $d_{x^{2}-y^{2}}$ orbitals of the B’2+ ions and assume that the B’2+, O2- and W6+ ions are on the same line and that the O2--W6+-O2- bond angle is 90∘. In this situation, spin 1 on the left B’2+ ion is transferred to the molecular orbital composed of the $p_{y}$ orbitals of the O2- 2$p$ orbitals and the $f_{y^{3}}$ orbitals of the W6+ $4f$ orbitals; meanwhile, spin 2 on the right B’2+ ion is transferred to the molecular orbital composed of the $p_{x}$ orbitals of O2- and the $f_{x^{3}}$ orbitals of W6+ ions. Due to Hund’s rules, these two spins on the $f_{y^{3}}$ and $f_{x^{3}}$ orbitals in the W6+ ions have to be parallel. Then, after these two spins are transferred back to the B’2+ ion, a FM superexchange interaction between them is formed. With these two different superexhanges in the system, it is natural to propose that the ground state change is due to the competition between them. This means the FM interaction exceeds the AFM interaction for spin-1/2 and 1 systems, but AFM interactions are stronger for the spin-5/2 system.

Since the $t_{1g}$ symmetry of the $4f$ orbitals here is identical to the symmetry of the $p$ orbitals of 4$p$ or 3$p$ orbitals, similar FM superexchange interactions should also occur for 3$d$-2$p$-4$p$(or 3 $p$)-2$p$-3$d$ paths. Several other TLMS with layered perovskite structures have been observed to possess such FM superexchange interactions. Similar competitive FM and AFM superexchanges are observed in Ba3CoNb2O9yokota , where the Co2+-O2--Nb5+-O2--Co2+ superexchange interaction involving the Nb5+ 4$p$ orbitals is FM, and it opposes the AFM Co2+-O2--O2--Co2+ interaction leading to a weaker AFM interaction. This manifests in a low AFM transition temperature as well as a low saturation field. Alternatively, AAg2M(VO4)2 (A=Ba, Sr; M=Co, Ni)AgVO has a stronger FM superexchange via Co2+-O2--V5+-O2--Co2+ than AFM superexchange via Co2+-O2--O2--Co2+ and possesses an FM transition.

Next, we look into the chemical pressure effect among the Co samples. In general, the increasing lattice parameter should decrease the exchange interactions and therefore lead to a low transition temperature. However, this is not the case here. Instead, with increasing lattice parameter, the $T_{\text{C}}$ first increases from Sr4 to Sr2La2 and then to the Ba2La2 sample, afterwards it decreases to Ba3La and to the Ba4 sample. To explore the more general rule behind this chemical pressure effect, we turn to more detailed structural information. Figure 10 shows the O-W/Re-O angle dependence of $T_{\text{C}}$. The general trend is that with increasing angle (or as its deviation from 90∘ decreases), the $T_{\text{C}}$ and $\theta_{\text{CW}}$ increases. Here, we make two assumptions: (i) $\theta_{\text{CW}}=[-zJs(s+1)]/(3k_{B})$, allowing J to be extracted from $J/k_{B}=(2/3)\theta_{\text{CW}}$; (ii) since all Co samples show FM transitions, the dominant exchange should be FM. We assume that the chemical pressure affects the FM interaction more strongly than the AFM interaction. As we discussed above for the FM interaction, it is built on the assumption that the O2--W6+-O2- bond angle is 90∘. The strength of this FM interaction should decrease as the O-W-O angle deviates from 90∘. The larger this deviation is, the weaker this FM interaction is, and therefore the lower $T_{\text{C}}$ and J should be, since we assume that the AFM interaction does not change. These are the cases for the Ba3La and Ba4 samples. On the other hand, the smaller this deviation is the stronger this FM interaction is and therefore the higher $T_{\text{C}}$ and J should be. These are the cases for the Ba2La2 and Sr2La2 samples. Therefore, this O-W/Re-O angle dependence of $T_{\text{C}}$ and J also confirms our proposal that the system’s magnetic properties are controlled by two different superexchanges.

Finally, we comment on the phase diagram of Ba2La2MnW2O12. One common ground state at zero field observed for studied triangualr lattice antiferromagnets (TLAFs), such as Ba3CoSb2O9 ba3coz and Ba3BNb2O9 (B = Co2q ; yokota , Nihwang , Mnlee ), is the 120 degree ordered state. The phase diagrams of these TLAFs consistently show a canted 120 degree, up up down, oblique, and polarized phases with increasing applied magnetic field. Comparing the phase diagram of Ba2La2MnW2O12 to theirs, the overall trend is similar. Therefore, we tend to ascribe its zero field ground state to be 120 degree ordering. With increasing field it enters the canted 120 degree phase below H${}_{\text{c1}}$, up up down phase above H${}_{\text{c1}}$, and then polarized phase above H${}_{\text{c2}}$. One difference here is that we do not observe the oblique phase boundary between the up up down and polarized phases for Ba2La2MnW2O12. One possibility is that our AC measurements do not have the resolution to detect this phase boundary due to the polycrystalline nature of the sample.

V CONCLUSION

We have examined the structural and magnetic properties of a family of A4B’B2O12 compounds with a 2D magnetic triangular lattice. Due to the competition between two kinds of superexchange interactions (one FM and one AFM) in the structure, the ground states of the triangular lattice can be efficiently tuned by varying the spin numbers of the magnetic B’2+ ions, which is FM ordering for the Co and Ni samples with small spin numbers and AFM for the Mn samples with a large spin number. Moreover, the chemical pressure can finely tune the FM ordering temperature and effective exchange interactions among the Co sample by changing the local structure, mainly by changing the O-W/Re-O angle to tune the FM interaction strength. These findings demonstrate that A4B’B2O12 is a new platform for TLMs waiting for future exploration to study geometrically frustrated magnetism.

Acknowledgements.

R.R. and H.D.Z. thank the support from NSF-DMR through Award DMR-1350002. J.M. thanks the support of the Ministry of Science and Technology of China (2016YFA0300500). The research at HFIR/ORNL was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. The work at NHMFL is supported by NSF-DMR-1157490 and the State of Florida.