Antiferromagnetism in kagome $\alpha$-Cu$_{3}$Mg(OH)$_6$Br$_2$

TL;DR

This study investigates the antiferromagnetic properties of $ ext{Cu}_3 ext{Mg}( ext{OH})_6 ext{Br}_2$, revealing its magnetic ordering at 5.4 K and suggesting it as a realization of the 2D spin-1/2 Heisenberg kagome model near the ferromagnetic phase.

Contribution

The paper provides experimental evidence of antiferromagnetic order in $ ext{Cu}_3 ext{Mg}( ext{OH})_6 ext{Br}_2$ and links it to the theoretical kagome Heisenberg model, highlighting its position near the ferromagnetic phase boundary.

Findings

Antiferromagnetic order at 5.4 K with moments aligned ferromagnetically within kagome planes.

Ordered magnetic moment of 0.94 μ_B indicates minimal quantum and geometrical fluctuations.

Comparison with haydeeite suggests the material is near the ferromagnetic side of the kagome phase diagram.

Abstract

The antiferromagnetism in $\alpha$-Cu$_3$Mg(OH)$_6$Br$_2$ was studied by magnetic-susceptibility, specific-heat and neutron-diffraction measurements. The crystal structure consists of Cu$^{2+}$ kagome layers with Mg$^{2+}$ ions occupying the centers of the hexagons, separated by Br$^{1-}$ ions. The magnetic system orders antiferromagnetically at 5.4 K with the magnetic moments aligned ferromagnetically within the kagome planes. The ordered moment is 0.94 $\mu_B$, suggesting little quantum and geometrical fluctuations. By comparing the magnetic and specific-heat properties with those of the haydeeite, we suggest that $\alpha$-Cu$_3$Mg(OH)$_6$Br$_2$ may be described by the two-dimensional spin-$1/2$ Heisenberg kagome model and is in the region of the ferromagnetic-order side of the phase diagram.

Figures

Tables

Table 1. Table 1: Nuclear structure parameters of α 𝛼 \alpha -Cu 3 Mg(OD) 6 Br 2 at 10 K. P ​ 6 3 / m ​ m ​ c 𝑃 subscript 6 3 𝑚 𝑚 𝑐 P6_{3}/mmc (No. 194): a = b = 6.28014 ​ ( 20 ) 𝑎 𝑏 6.28014 20 a=b=6.28014(20) Å, c = 6.06283 ​ ( 25 ) 𝑐 6.06283 25 c=6.06283(25) Å, α = β = 90 ∘ 𝛼 𝛽 superscript 90 \alpha=\beta=90^{\circ} , γ = 120 ∘ 𝛾 superscript 120 \gamma=120^{\circ} .

	Site	x	y	z	B (Å2)
Mg	0.00000	0.00000	0.00000	0.281(406)
Cu	0.50000	0.00000	0.00000	0.131(131)
Br	0.33333	0.66667	0.63819(90)	0.117(155)
O	0.82592(32)	0.17418(32)	-0.14100(64)	0.511(120)
D	0.19300(41)	0.80700(41)	0.28666(98)	0.398(140)

Full text

I introduction

The two-dimensional (2D) spin-$1/2$ kagome model has been intensely studied in theories because its rich ground states Sachdev (1992); Jiang et al. (2008); Janson et al. (2008); Yan et al. (2011); Jiang et al. (2012); Messio et al. (2012); Punk et al. (2014); Bieri et al. (2015); Iqbal et al. (2015); Kumar et al. (2015); Gong et al. (2016); Liao et al. (2017); Mei et al. (2017); Wang et al. (2017, 2018). For example, the $S$ = 1/2 Heisenberg antiferromagnetic kagome model (AFKM) can give rise to ferromagnetic (FM) order, different types of antiferromagnetic (AFM) orders and quantum spin liquids (QSLs) Messio et al. (2012). The AFKM may be realized in many minerals of the atacamite group with the molecular formula as Cu3$T$(OH)6$X$2, where $T$ and $X$ are the 3$d$ nonmagnetic transition-metal (Zn, Mg) and the halogen elements (F,Cl,Br), respectively. The most well-known material is the herbertsmithite Cu3Zn(OH)6Cl2, which shows no magnetic order down to at least 30 mK and is suggested to be a QSL Norman (2016); Shores et al. (2005); Bert et al. (2007); Mendels et al. (2007); Helton et al. (2007); Han et al. (2012); Fu et al. (2015); Han et al. (2016a). The structure of $\gamma$-Cu3Mg(OH)6Cl2 is very similar to that of the herbertsmithite and may also host a QSL state Colman et al. (2011). Recently, new materials Cu3Zn(OH)6FBr and Cu3Zn(OH)6FCl with similar structure have been successfully synthesized and shown to possibly host the gapped $Z_{2}$ QSL ground states Feng et al. (2017); Wen (2017); Wei et al. (2017); Feng et al. (2019).

In the above materials, the kagome layers formed by Cu2+ ions are separated with each other by non-magnetic Zn or Mg ions, which may be treated as diamagnetic dilution of the three-dimensional (3D) pyrochlore-like lattice. These non-magnetic ions can also occupy the center of the hexagons in the kagome layers, as found in kapellasite, $\alpha$-Cu3Zn(OH)6Cl2 Colman et al. (2008) and haydeeite, $\alpha$-Cu3Mg(OH)6Cl2 Colman et al. (2010). In this structure, the coupling between the kagome layers is through the weak interlayer O-H-Cl bonding, which should result in highly 2D magnetic properties. It had been suggested that the kapellasite may be a gapless spin liquid or noncoplanar coboc2-type AF order Janson et al. (2008); Fåk et al. (2012), but later measurements found strong Cu/Zn site mixing that makes the AFKM inappropriate Kermarrec et al. (2014). The haydeeite is a rare example of the FM order in the AFKM with $T_{c}$ at 4.2 K Boldrin et al. (2015). Measurements on the single-crystal haydeeite further revealed strong anisotropic behaviors between the in-plane and out-of-plane magnetic properties Puphal et al. (2018). Although the magnetic structure has not been unambiguously solved, the ordered moment is less than 0.2 $\mu_{B}$, suggesting strong quantum fluctuations. Therefore, the haydeeite may be in the proximity to the quantum phase transition between the Heisenberg kagome FM order and the Heisenberg cuboc2 AF order Boldrin et al. (2015).

The idea for the compound of Cu3Mg(OH)6Br2 comes from the substitution of interlayer Cu2+ in the barlowite Cu4(OH)6FBr, which has perfect Cu2+ kagome layers with Cu2+ ions between them and is antiferromagnetically ordered at about 15 K Han et al. (2014); Jeschke et al. (2015); Han et al. (2016b); Feng et al. (2018); Tustain et al. (2018). It has been theoretically suggested that Zn2+ or Mg2+ ions can replace the interlayer Cu2+ in barlowite and thus dilute the AF order to give rise to a QSL state as in herbertsmithite Liu et al. (2015); Guterding et al. (2016). While this proposal has been shown to succeed in Cu3Zn(OH)6FBr Feng et al. (2017); Wen (2017); Wei et al. (2017); Feng et al. (2018); Smaha et al. (2018), no Mg-substituted barlowite has been reported. In this paper, we follow the same route in synthesizing the Zn-substituted barlowite to grow Mg-substituted barlowite. However, the final product is Cu3Mg(OH)6Br2 because MgF2 cannot be dissolved in water. Since it has the same crystal structure as kapellasite and haydeeite, we label it as $\alpha$-Cu3Mg(OH)6Br2. The system orders antiferromagnetically at about 5.4 K but the configuration of the moments within the kagome plane is FM with the ordered moment of 0.94 $\mu_{B}$. Our results suggest that $\alpha$-Cu3Mg(OH)6Br2 is in the FM region of the phase diagram in the AFKM.

II experiments

$\alpha$-Cu3Mg(OH)6Br2 was synthesized by the hydrothermal method as described previously Feng et al. (2017, 2018). The mixture of 1.5-mmol Cu2(OH)2CO3 and 6-mmol MgBr2$\cdot$6H2O was sealed in a 50-ml reaction vessel with 25-ml water, which was slowly heated to 200 ∘C and kept for 12 hours. The polycrystalline samples were obtained by washing the production with the deionized water. To produce deuterated samples, Cu2(OD)2CO3 and heavy water were used in the above process. The Mg content is determined by the inductively coupled plasma mass spectrometer. The magnetic susceptibility and heat capacity were measured by the MPMS and PPMS (Quantum Design), respectively. The magnetic and nuclear structures were determined by neutron diffraction experiments performed on the HB-2A diffractometer at HFIR, USA, with the wavelengths of both 2.4103 and 1.5395 Å.

III results and discussions

Figure 1(a) shows the neutron powder diffraction results at 10 K. There is no structural transition observed since the pattern at 300 K is similar to that at 10 K. Accordingly, the material has a hexagonal structure with the space group of $P6_{3}/mmc$. Detailed refinement results are shown in Table I. The large $R_{wp}$ is mainly due to the use of aluminum can and the presence of impurities. These chemical impurities only exist in the deuterated samples as shown by the room-temperature x-ray measurements. Figure 1(b) gives the nuclear structure. The Cu2+ ions form kagome planes, which are separated by Br1- ions. The Mg2+ ions sit at the centers of the hexagons within the Cu2+ kagome planes.

At 1.6 K, the system becomes magnetically ordered as shown by the new peak in Fig. 2(a). The subtraction between 1.6 and 10 K data provides more magnetic peaks as shown in Fig. 2(b). The k-search method is based on the first 5 possible magnetic peaks. The best result gives out a propagation vector $k$ = (0, 0, 0.5). Sarah was used to check the possible structures from the results of representational analysis based on the nuclear space group and magnetic k vector (0, 0, 0.5). There were 3 irreducible representations (IR or $\Gamma$ 1,3 and 5 ) as a result of the analysis. All the basis vectors were tried. The $\Gamma$ 1 and 3 can not fit the data at all. The best result comes from $\Gamma$5 BV 2. The other basis vectors for $\Gamma$5 other than Basis Vector 2 can not give good fits to the data. The inset of Fig. 2(b) shows the magnetic structure. The moments at Cu2+ positions are confined within the kagome planes and ferromagnetically aligned along b-axis. Along the c-axis, the moments are aligned antiferromagnetically.

Figure 2(c) shows the temperature dependence of the magnetic peak at (0, 0, 0.5) in the nuclear structural notation. Figure 2(d) shows the temperature dependence of the refined moment, which can be fitted by $M_{0}(1-T/T_{N})^{\beta}$. The values of $M_{0}$ and $\beta$ are 0.94 $\pm$ 0.03 $\mu_{B}$ and 0.29 $\pm$ 0.03, respectively. The ordered moment $M_{0}$ at 0 K is consistent with the ordered moment $gS$ for a $S$ = 1/2 system with $g$ = 2. The value of $\beta$ is close to that in a classical 3D Ising (0.326) or Heisenberg (0.367) system.

Figure 3(a) gives the temperature dependence of the magnetic susceptibility $\chi$, which clearly shows an AF phase transition at $T_{N}$ = 5.54 K. No significant difference is found between the field-cooled and zero-field-cooled processes. Above $T_{N}$, $\chi$ still strongly depends on the magnetic field, which suggests the presence of two-dimensional spin fluctuations. The inset shows the temperature dependence of $\chi^{-1}$ at 50 kOe. The fitting at high temperatures gives a Curie temperature $\theta$ of about 34 K and an effective moment of 1.77 $\mu_{B}$, which is close to the value for $S$ = 1/2 with g = 2. Figure 3(b) shows the field dependence of the magnetization $M$. At 2 K, $M$ becomes saturate above 20 kOe with the saturated moment of 0.95 $\mu_{B}$, which also suggests that the system is $S$ = 1/2. By taking the first derivative, two features can be seen at $H_{1}\approx$ 1.4 kOe and $H_{2}\approx$ 8 kOe as shown in Fig. 3(c). The value of $H_{1}$ is similar to the saturated field for field parallel to the kagome planes in haydeeite Puphal et al. (2018), which suggests that the coupling between the kagome planes is weak. These two characteristic fields can also be seen by the hysteresis behavior as shown in Fig. 3(d).

Figure 4(a) shows the specific heat of $\alpha$-Cu3Mg(OH)6Br2 at several magnetic fields. The AF transition results in a large peak, which becomes a broad hump above 3 T due to the suppression of the transition. The background of the specific heat can be estimated by fitting the data from 15 to 30 K with $C=\alpha T^{2}+\beta T^{3}$ as shown by the dashed line in Fig. 4(a). The fitted values for $\alpha$ and $\beta$ are 0.0265 J/mol K3 and 2.86 $\times$ 10${}^{-}4$ J/mol K4, respectively, similar to those in barlowite Han et al. (2014). Figure 4(b) shows the temperature dependence of the entropy released associated with the magnetic transition, which is obtained by the integration of $C/T$ after subtracting the background as described above. The value of $\Delta S_{AF}$ at high temperature is about 3 J/mol Cu K, which is just about half of Rln2. This suggests that spin correlations are formed well above $T_{N}$, consistent with the results in the magnetic susceptibility measurements.

Interestingly, there is an upturn of $C$ below 0.2 K as shown in the inset of Fig. 4(a), which can be fitted as $C\propto T^{-3/2}$. This kind of temperature dependence suggests that the upturn is not from the nuclear schottky anomaly or magnetic impurities. Moreover, the specific heat should move to higher temperatures under fields if it comes from the nuclear schottky anomaly or magnetic impurities. Instead, it is suppressed at 30 kOe and thus should be related to the intrinsic kagome system before the spins are fully polarized by the field. The entropy between 0.07 K to 0.2 K is about 0.12 J/mol K. It is not clear what is the origin of this upturn, but it is rather surprising since the system shows no exotic properties from other measurements.

The above results suggest that $\alpha$-Cu3Mg(OH)6Br2 exhibits 2D kagome ferromagnetism similar to haydeeite $\alpha$-Cu3Mg(OH)6Cl2 Boldrin et al. (2015), although the system orders antiferromagnetically along the c-axis. The ordered moment in haydeeite is just 0.2 $\mu_{B}$ due to strong quantum fluctuations, and it is suggested to sit near the boundary between the FM and cuboc2 type noncoplanar AFM parts in the phase diagram of the kagome lattice for a FM nearest-neighbour interaction Boldrin et al. (2015). The $\alpha$-Cu3Mg(OH)6Br2 is mostly likely in deep region of the FM part as its ordered moment is 0.94 $\mu_{B}$. While this makes $\alpha$-Cu3Mg(OH)6Br2 less interesting due to the lack of frustration effects, it is worth noting that it may be another platform to study the topological bands in the kagome ferromagnet as observed in Cu[1,3-benzenedicarboxylate(bdc)] Chisnell et al. (2015).

IV conclusions

Our systematically studies on the magnetism in $\alpha$-Cu3Mg(OH)6Br2 demonstrate that it orders antiferromagnetically below 5.54 K but the spins ordered ferromagnetically within the kagome planes. The FM state of the kagome planes can be achieved easily by applying a weak magnetic field and its fluctuations survive above $T_{N}$. Our results suggest that $\alpha$-Cu3Mg(OH)6Br2 is in the deep region of the FM part in the phase diagram of 2D AFKM.

Acknowledgements.

This work is supported by the Ministry of Science and Technology of China (Grants No. 2017YFA0302900, No. 2016YFA0300502, No. 2016YFA0300604), the National Natural Science Foundation of China (Grants No. 11874401, No. 11674406, No. 11574359, No. 11674370, No. 11774399, No. 11474330), the Strategic Priority Research Program(B) of the Chinese Academy of Sciences (Grants No. XDB25000000 and No. XDB07020000, No. XDB28000000), and the National Thousand-Young Talents Program of China. Research conducted at ORNL’s High Flux Isotope Reactor was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. Y.W. and Z.F. contributed equally to this work.