Magnetization process of atacamite: a case of weakly coupled $S = 1/2$ sawtooth chains

TL;DR

This study combines experimental and theoretical methods to analyze atacamite's magnetic behavior, revealing a unique magnetization plateau and complex field-dependent magnetic order due to weakly coupled sawtooth chains.

Contribution

It provides a detailed model of atacamite's anisotropic sawtooth chains and uncovers a novel magnetization process unrelated to known plateaus.

Findings

Observation of a magnetization plateau at about half saturation

Identification of a field-driven canting mechanism in weakly coupled chains

Full characterization of the antiferromagnetic order in atacamite

Abstract

We present a combined experimental and theoretical study of the mineral atacamite Cu$_2$Cl(OH)$_3$. Density functional theory yields a Hamiltonian describing anisotropic sawtooth chains with weak 3D connections. Experimentally, we fully characterize the antiferromagnetically ordered state. Magnetic order shows a complex evolution with the magnetic field, while, starting at 31.5 T, we observe a plateau-like magnetization at about $M_{\rm sat}/2$. Based on complementary theoretical approaches, we show that the latter is unrelated to the known magnetization plateau of a sawtooth chain. Instead, we provide evidence that the magnetization process in atacamite is a field-driven canting of a 3D network of weakly coupled sawtooth chains that form giant moments.

Equations

$$\mathcal{H}=\sum_{i}J\mathbf{S}_{i}\cdot\mathbf{S}_{i+2}+J^{\prime}\big{(}\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}+\mathbf{S}_{i+1}\cdot\mathbf{S}_{i+2}\big{)}-\mathbf{h}\cdot\mathbf{S}_{i}.$$

Full text

Frustrated low-dimensional quantum spin systems offer a unique opportunity to study complex quantum phases lacroix2011 ; starykh2015 ; wosnitza2016 ; savary2017 . In the search for novel and exotic ground and field-induced states, such as spin liquids, magnetization plateaus or nematic phases, a multitude of models have been studied, including the kagome lattice, the diamond chain or the frustrated $J_{1}$-$J_{2}$ chain harris1992 ; takano1996 ; tonegawa1987 . Experimental efforts to identify materials to test these theoretical concepts are exemplified by work on natural minerals such as herbertsmithite, azurite or linarite olariu2008 ; kikuchi2005 ; rule2011 ; jeschke2011 ; willenberg2016 ; inosov2018 . Through this combined effort a new level of insight into complex topics of quantum magnetism is achieved.

The $\Delta$-, or sawtooth chain represents one of the fundamental models of frustrated quantum magnetism. It consists of a chain of spin triangles, with the Hamiltonian

[TABLE]

$\mathbf{S}_{i}$ represents a spin $S=1/2$ at site $i$; the sites $i$ and $i+2$ are neighbors in the chain “spine”, while $J^{\prime}$ is the interaction between spine sites and the sawteeth tips. h is the external magnetic field. This model has been studied theoretically for decades hamada1988 ; kubo1993 ; otsuka1995 ; nakamura1995 ; nakamura1996 ; sen1996 ; maisinger1998 ; blundell2003 ; zhitomirsky2004 ; tonegawa2004 ; derzhko2004 ; chandra2004 ; richter2004 ; inagaki2005 ; richter2008 ; hida2008 ; hao2011 ; krivnov2014 ; dmitriev2015 ; dmitriev2016 ; dmitriev2018 . Real materials, however, are inevitably more complex than this simplified model. Delafossite and euchroite have more than two relevant couplings bacq2005 ; kikuchi2011 , certain metalorganic systems have a ferromagnetic intra-spine $J$ inagaki2005 ; schnack2018 , and in Rb2NaTi3F12 the $\Delta$-chain is coupled to an antiferromagnetic (AFM) chain Jeschke2019 . In this Letter, based on a combined experimental and theoretical study of atacamite, Cu2Cl(OH)3, we show that its magnetic behavior originate from an intricate and rather unusual 3D connectivity of $\Delta$-chains, not previously addressed.

We have measured magnetization, magnetic susceptibility and specific heat of atacamite in fields up to 13 T. Neutron scattering was carried out at the HZB BER II reactor using the instruments E2, E5 and HFM/EXED for fields up to 25 T E2paper ; heinze2018 ; HFMEXEDpaper . We have determined both the magnetic and crystallographic structures of our mineral single crystals SM ; xtal ; sears1995 ; brown1995 . In addition, we have performed a high-field magnetostriction and magnetization study in fields up to 65 T at the Pulsed Field Facility of the NHMFL, Los Alamos. In the present work, we focus on data taken in magnetic fields applied along the crystallographic $b$ axis.

Atacamite magnetically orders at low temperatures and we found a complex field-induced spin reorientation behavior. Magnetic fields of $\sim 30$ T suppress the ordered state, taking the system to nearly half its saturation magnetization, where it persists up to the highest field reached in this study. To rationalize these results, we have investigated the electronic structure and magnetic interactions using Density Functional Theory (DFT) with full potential local orbital (FPLO) basis koepernik1999 and generalized gradient approximation (GGA) functional perdew1996 ; electronic correlations on Cu2+ were accounted for by the GGA+$U$ method Liechtenstein1995 . The Hamiltonian thus obtained consists of strongly coupled Cu $\Delta$-chains, forming a weakly coupled network. We consider the uncoupled chains in a magnetic field using infinite system time-evolving block decimation (iTEBD) vidal2007 as well as exact diagonalization (ED). The results justify our subsequent evaluation of the magnetization process within a 3D mean-field approximation (MFA), and accounting for the inter-chain coupling.

Atacamite Cu2Cl(OH)3 crystallizes in a $Pnma$ orthorhombic structure (lattice constants $a=6.02797$ Å, $b=6.86383$ Å, $c=9.11562$ Å; Fig. 1 (a)) SM ; parise1986 ; zheng2005 . There are two inequivalent Cu sites (dark (Cu(1)) and light (Cu(2)) blue spheres). Previously, this crystal structure was derived from a network of pyrochlore tetrahedra built up by Cu2+ ions zheng2004 ; zheng2005 ; zenmyo2013 . Our DFT calculations, however, indicate that the symmetry of the magnetic Hamiltonian is dramatically lower than the one anticipated from the bond lengths only. Indeed, the bonds derived from the 1st, 2nd and 3rd pyrochlore coordination shells vary in length by $\pm 10\%$ within each set, but the calculated exchange parameters (corresponding to 4, 6 and 7 distinct Cu-Cu distances, respectively), vary by two orders of magnitude. As we show below, our calculated Hamiltonian provides an excellent explanation of the experimental observations.

First evidence of the existence of ordered magnetism in atacamite was reported previously heinze2018 ; zheng2004 ; zheng2005 ; zenmyo2013 . Furthermore, we present zero-field specific heat measurements, with an anomaly indicating a magnetic transition at $T_{\mathrm{N}}=8.4$ K (Fig. 2 (a)). An antiferromagnetic anomaly is also observed at $T_{\mathrm{N}}=8.4$ K in the low-field (0.1 T) susceptibility as maximum in $d(\chi T)/dT$ (Fig. 2 (b)) impurity . In neutron diffraction, we find magnetic intensity below a slightly higher $T_{\mathrm{N}}=8.9$ K with a magnetic propagation vector $\mathbf{q}=(1/2,0,1/2)$ (Fig. 2 (c)).

We also detect an additional hump in the specific heat at $T\sim 5$ K (Fig. 2 (a)) hinting at a more complex temperature evolution of the magnetic state, involving, for instance, spin reorientations. A calculation of the magnetic entropy from our data (ignoring a phonon contribution) gives a value $\sim 0.65\,R\,\ln(2)$ at $T_{\mathrm{N}}$ SM . Such a small value is typical for magnetically ordered states in frustrated magnets with the magnetic entropy being distributed over the temperature scale set by the dominant coupling strengths, here $J_{4}$ and $J_{3}$ (Fig. 1 (c)).

In magnetic fields $\mathbf{H}\parallel b$, the features in the specific heat and the susceptibility are shifted to lower temperatures and the AFM anomaly is sharpened (Fig. 2 (a)–(b)). This shift is supported by neutron scattering in 6.5 T (Fig. 2 (c)). Field-dependent neutron scattering at the HFM/EXED instrument yields a suppression of AFM order at 24 T ($T=3.5$ K) (Fig. 2 (d)). Altogether, an external magnetic field leads to a suppression of the AFM phase, with $T_{\mathrm{N}}$ fully suppressed in $\sim 30$ T.

The low-$T$ susceptibility in high fields is larger than in low fields (Fig. 2 (b)). It reflects a metamagnetic transition occurring at a few Tesla (see below). Since for the other crystallographic directions we find no such transition, it suggests that the $b$ axis is the easy magnetic axis and this is a spin-flop transition SM . This is consistent with our refined magnetic structure.

From the magnetic Bragg peak intensities, we derive the magnetic structure in Fig. 1 (b) SM ; bertaut1986 . On the Cu(1) site, the ordered magnetic moments of $0.34(4)\,\mu_{\mathrm{B}}$ are arranged in a nearly perfect AFM pattern with the Cartesian components $\mu_{\mathrm{ord,Cu(1)}}(x,y,z)=\left[0.09(9),0.04(2),0.32(7)\right]\,\mu_{\mathrm{B}}$. The ordering vector corresponds to alternating signs for the $x$ and $z$ moment components, while the $y$ component stays the same within the same chain. The angle between two Cu(1) neighbors is thus $\theta=166.3^{\circ}$, close to $180^{\circ}$. All Cu(2) sites carry a moment $\mu_{\mathrm{ord,Cu(2)}}(x,y,z)=\left[0,0.59(2),0\right]\,\mu_{\mathrm{B}}$ where moments are parallel to $y$ within one set of sawtooth sites of a single chain (details in Ref. SM ).

To assess the magnetic phase diagram, we used magnetometry in pulsed magnetic fields for $\mathbf{H}\parallel b$ jaime2017 ; detwiler2000 . In Fig. 3 (a)–(b) we summarize the magnetostriction and magnetization, respectively. Below $T_{\mathrm{N}}$ and fields of $\mu_{0}H_{1}\lesssim 4$ T, a kink in the magnetization indicates a spin-flop transition. When increasing temperature the kink becomes weaker and shifts to higher fields in the AFM phase SM . For temperatures below $\sim 5$ K, a weak shoulder appears (inset Fig. 3 (b)), which corresponds to shallow minima in the magnetostriction SM . This might indicate a splitting of the spin-flop transition due to a weak three-axes exchange anisotropy.

Immediately after the spin-flop transition, $M(H)$ grows linearly with $dM/dH\approx 0.013\,\mu_{\mathrm{B}}/$T, but starts bending upwards up to a field of $\mu_{0}H_{2}=31.5$ T, where the slope reaches $0.042\,\mu_{\mathrm{B}}/$T (Fig. 3 (b)). After that, a wide magnetization plateau-like behavior at about $0.45\,\mu_{\mathrm{B}}$/Cu sets in. The plateau, also detected in the magnetostriction (Fig. 3 (a)), reaches up to highest measured fields and is not perfectly flat, but rising at a rate of $0.001\,\mu_{\mathrm{B}}/$T.

From our data, we construct the magnetic phase diagram of atacamite for $\mathbf{H}\parallel b$ (Fig. 4). The AFM phase exists below $T_{\mathrm{N}}$ and up to $\sim 30$ T. It is separated into a low-field regime with the magnetic structure described before and a high-field regime for fields above the spin-flop transition. In the limit $T\to 0$ K, the suppression of AFM order possibly coincides with the appearance of a magnetization plateau-like behavior. To fully establish the magnetic phase diagram in this field region, it requires a determination of the magnetocaloric effect in pulsed magnetic fields nomura2020 ; MCE . At highest fields of 65 T the system is still far from saturation.

It is now instructive to establish the Hamiltonian of atacamite and connect it to the observations. To this end, we used an energy mapping technique Glasbrenner2015 ; Iqbal2018 ; Ghosh2019 to calculate 17 exchange interactions, derived from the first three coordination shells of the parent pyrochlore structure SM . Only six of them exceed 1 K, or 0.3% of the dominant coupling $J_{4}$ (Fig. 1 (c)). A Curie-Weiss temperature $\Theta_{\mathrm{CW}}$ calculated from our $J$s for $U=8.24$ eV, typical for Cu2+, matches the experiment heinze2018 ; zheng2005 . Two antiferromagnetic interactions stand out: $J_{4}=336$ K and $J_{3}=102$ K, which bind Cu(1) and Cu(2) atoms into anisotropic $\Delta$-chains (compare Eq. (1) with $J\equiv J_{4}$ and $J^{\prime}\equiv J_{3}$).

Based on these findings, we consider a single $\Delta$-chain (Eq. (1)). Fig. 3 (c) shows iTEBD results, complementary ED results are described in SM . For $0.5\leq J^{\prime}/J\leq 2$ we observe the famous quantum half-magnetization plateau richter2004 ; richter2008 ; Metavitsiadis2020 . However, it is practically invisible for $J^{\prime}/J\lesssim 0.5$, relevant for atacamite. Moreover, its field scale is of $O(J)\sim O(250$ T$)$. Therefore, while tempting, the observed flattening of $M(H)$ around 31.5 T in atacamite is not related to the half-magnetization plateau physics. On the other hand, the 3D exchange among the chains proves to be relevant. The second finding in Fig. 3 (c) is far more striking and has not been appreciated before: in the small-$J^{\prime}$ gapless phase, e.g., at $J^{\prime}/J=0.5$, the low-$h$ susceptibility appears singular and the magnetization approaches a non-quantized finite value as $h\rightarrow 0$. For the relevant $J^{\prime}/J\sim 0.3$, the inset of Fig. 3 (c) shows iTEBD versus increasing imaginary simulation times $\tau=n_{\tau}d\tau$ in terms of $d\tau=0.01/J$. Since iTEBD inherits the limit of system size $N\rightarrow\infty$ by construction, and by identifying $\tau^{-1}\sim T$ with a quasi-temperature, we extract the following order of limits $\lim_{h\rightarrow 0}\lim_{N\rightarrow\infty}\langle S^{z}\rangle_{T}:=M_{0}(T)$ from the inset. For $T\neq 0$, we find $M_{0}(T)=0$, however as $T\rightarrow 0$, very likely, $M_{0}(T=0)/\mathrm{cell}\approx 0.435\neq 0$, all of which is consistent with Mermin-Wagner’s theorem. Rephrasing, we seem to observe ferromagnetic order at $T=0$ for the $\Delta$-chain at $J^{\prime}/J\sim 0.3$. This likely holds for the entire small-$J^{\prime}$ gapless phase. This is consistent with ED SM and with a classical treatment of the $\Delta$-chain.

In the MFA, the ground state of a single $\Delta$-chain has the same pattern as observed experimentally, with $\theta=360^{\circ}-2\arccos\left(-\frac{J^{\prime}}{2J}\right)=162.5^{\circ}$, in excellent agreement with our neutron data, with the Cu2+ net moment, $M_{\mathrm{eff}}=(M_{2}-M_{1}J^{\prime}/2J)/2$ SM . If the moment of each copper is taken to be $M_{2}=M_{1}=1\,\mu_{\mathrm{B}}$, then $M_{\mathrm{eff}}^{\mathrm{MFA}}=0.42\,\mu_{\mathrm{B}}$. We know from experiment though that these moments are suppressed by fluctuations to $M_{1}=0.34\,\mu_{\mathrm{B}}$, $M_{2}=0.59\,\mu_{\mathrm{B}}$. This reduces the net moment to $M_{\mathrm{eff}}(H=0)=0.27\,\mu_{\mathrm{B}}$/Cu SM . On the other hand, $M_{\mathrm{eff}}^{\mathrm{MFA}}=0.42\,\mu_{\mathrm{B}}$ agrees very well with the magnetization at $\sim 30$ T, indicating that in such fields the fluctuations are mostly quenched. In the following we used $M_{\mathrm{eff}}(H)$ and $M_{1,2}(H)$ linearly interpolating between the two limits.

We are now in a position to describe an effective 3D magnetic model that can be addressed by classical mean-field calculations. These treat the $\Delta$-chains as emergent, rigid macroscopic objects, carrying a large magnetic moment. Classically, the latter arises primarily from Cu(2) moments being aligned ferromagnetically along $b$ (Fig. 1 (b)). These large moments are AFM stacked into a 2D crystal and coupled via the small subleading exchange interactions. Their projection onto the $ac$ plane forms an anisotropic triangular lattice with three effective AFM couplings, $J_{\rm B}\lesssim J_{\rm A}\ll J_{\rm C}$ SM . For our selected value of $U=8.24$ eV, $J_{\rm C}\approx 8$ K, $J_{\rm A}\approx 0.5$ K and $J_{\rm B}\approx 0$ K. The classical ground state of this model is collinear with Néel order along C and B, and FM order along A, as observed experimentally (Fig. 1 (b)). We note, however, that $J_{\rm A,B}$ rapidly rise with decreasing $U$, and at $U=7$ eV (still an admissible value for Cu${}^{2+})$ $J_{\rm A}\approx J_{\rm B}\approx J_{\rm C}\approx 10$ K. In that case, the MFA ground state would have been the 120∘ order.

We focus on the 2D collinear Néel order along the C direction. Since $b$ is the easy axis, the MFA predicts a spin-flop at low fields $\mathbf{H}\parallel b$, whereupon all magnetic moments rigidly rotate so that the Cu(2) moments are $\perp\mathbf{H}$. The classical spin-flop field is $\mu_{0}H_{1}=\frac{M_{2}(0)}{M_{\mathrm{eff}}(0)}\sqrt{2KJ_{\Delta-\Delta}}$, where $K$ measures the uniaxial anisotropy (given here for simplicity as an effective single-site term), and the effective coupling $J_{\Delta-\Delta}\approx J_{\rm C}+J_{\rm B}\approx 8.5$ K for $U=8.24$ eV. To reproduce the experimentally observed $\mu_{0}H_{1}\sim 3.5$ T one needs $K\sim 0.04$ K (a typical energy scale for Cu2+ Tranquada1989 ). The low symmetry of atacamite allows also for some in-plane magnetic anisotropy. A possible splitting of $H_{1}$ into two close transitions likely reflects such anisotropies.

As the field increases, the spin-flopped state gradually cants, generating a net magnetic moment of $HM_{\mathrm{eff}}^{2}(H)/M_{2}^{2}(H)J_{\Delta-\Delta}$. In an uncorrected MFA, $M(H)$ is linear. However, accounting for quantum fluctuations and their gradual quenching with field leads to deviation from linearity. These deviations are visible in experiment SM . At a field $H_{2}$ the moments cant into the “plateau” configuration, where all $\Delta$-chains are ordered ferromagnetically, and the total moment is $M=M_{\mathrm{eff}}^{\mathrm{MFA}}$. For our calculated parameters, $\mu_{0}H_{2}=30.1$ T, to be compared to the experimental value of 31.5 T. In this state, the total moment does not remain constant but keeps rising as $M_{\mathrm{eff}}(H)=(1/2-J^{\prime}/4J+H/4J)\mu_{\mathrm{B}}$. The differential susceptibility $dM/dH$, calculated this way, is much smaller than in the experiment, yet is qualitatively consistent with the latter SM .

The overall dependence of $M(H)$ as calculated in MFA, adjusted for the quenching of the fluctuations, and using the DFT exchange couplings, exhibits an excellent agreement with the experiment (Fig. 3 (d)), giving credence to the calculation and to the described scenario.

Finally, let us discuss the finite-$T$ phase diagram. Since temperature effects might slightly change the ratios between $J_{\rm A}$, $J_{\rm B}$, $J_{\rm C}$, we note that a tuning towards the region $J_{\rm A}\approx J_{\rm B}\approx J_{\rm C}$ opens the possibility of multiple phases with various degrees of non-collinearity, some of them only emerging at finite temperatures starykh2015 . While our current observations do not yield hints as to the specific nature of this phase, we note that the phase diagram for the simple isotropic triangular lattice is similar to our Fig. 4 (see Fig. 3 in Ref. starykh2015 ). This is a subject for future investigations.

We have studied the natural mineral Cu2Cl(OH)3 and found that it is well described as a weakly coupled asymmetric triangular lattice of $S=1/2$ $\Delta$-chains. We find an unusual magnetic behavior, with a magnetization deceivingly reminiscent of the quantum half-magnetization plateau, which however turns out to be a classical effect, well described by MFA. A magnetic Hamiltonian derived from first-principles calculations predicts a spin flop, a magnetization plateau, and weak deviation from the plateau behavior in high fields. This compound therefore represents a unique example of strongly-coupled 1D ferromagnetic objects coordinated by weak and anisotropic 2D interactions. We hope that this discovery will encourage more studies of this class of magnetic models.