The Spin Gap in Malachite Cu2 (OH) 2CO3 and Its Evolution Under Pressure
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The spin gap in malachite Cu2(OH)2CO3 and its evolution under pressure Stefan Lebernegg,1, ∗ Alexander A. Tsirlin,1, 2 Oleg Janson,1, 2 and Helge Rosner1, y 1Max Planck Institute for Chemical Physics of Solids, N¨othnitzerStr. 40, 01187 Dresden, Germany 2National Institute of Chemical Physics and Biophysics, 12618 Tallinn, Estonia (Dated: March 22, 2018) We report on the microscopic magnetic modeling of the spin-1/2 copper mineral malachite at ambient and elevated pressures. Despite the layered crystal structure of this mineral, the ambient- pressure susceptibility and magnetization data can be well described by an unfrustrated quasi- one-dimensional magnetic model. Weakly interacting antiferromagnetic alternating spin chains are responsible for a large spin gap of 120 K. Although the intradimer Cu{O{Cu bridging angles are considerably smaller than the interdimer angles, density functional theory (DFT) calculations revealed that the largest exchange coupling of 190 K operates within the structural dimers. The lack of the inversion symmetry in the exchange pathways gives rise to sizable Dzyaloshinskii-Moriya interactions which were estimated by full-relativistic DFT+U calculations. Based on available high-pressure crystal structures, we investigate the exchange couplings under pressure and make predictions for the evolution of the spin gap. The calculations evidence that intradimer couplings are strongly pressure-dependent and their evolution underlies the decrease of the spin gap under pressure. Finally, we assess the accuracy of hydrogen positions determined by structural relaxation within DFT and put forward this computational method as a viable alternative to elaborate experiments. PACS numbers: 75.50.Ee,75.10.Jm,75.30.Gw,61.50.Ks I. INTRODUCTION Regarding this long and prominent history of mala- chite, surprisingly little is known about its magnetism. Janod et al.22 reported the sizable spin gap of about Cu-based minerals enjoy close attention of researchers 130 K and proposed a one-dimensional model of bond- working in the fields of geology and solid-state physics alternating spin chains. This model emerges naturally alike. Intricate crystal structures underlie complex ar- from the crystal structure of malachite,23 where CuO rangements of the spin- 1 Cu2+ ions that, in turn, trig- 4 2 plaquettes form Cu O dimers by edge-sharing and fur- ger interesting low-temperature quantum effects and 2 6 1 ther link into chains along the [201] direction by corner- exotic ground states. For example, herbertsmithite 22 1 sharing. According to Janod et al., the stronger cou- Cu3Zn(OH)6Cl2, the best available spin- 2 kagom´esys- 1{5 pling should run between the structural dimers because of tem, shows putative spin-liquid ground state. Dioptase the larger Cu{O{Cu angle that promotes antiferromag- Cu6Si6O18 · 6H2O demonstrates unusually strong quan- netic (AFM) superexchange. This suggestion, inferred tum fluctuations on a non-frustrated three-dimensional from the well-known Goodenough-Kanamori-Anderson spin lattice.6 Azurite Cu (CO ) (OH) reveals a 1 - 3 3 2 2 3 (GKA) rules,24{26 is in line with many recent studies of plateau in the magnetization and presumably hosts a rare 2+ 7{10 Cu -based compounds, where structural Cu2O6 dimers magnetic topology of the diamond spin chain. Linar- do not match the spin dimers and show weak magnetic ite PbCuSO4(OH)2 is an excellent material prototype of couplings, only.27{31 the strongly frustrated spin chain.11,12 Finally, volbor- thite Cu3V2O7(OH)2·2H2O that was originally consid- Here, we present a detailed density functional theory ered as a kagom´ematerial13{15 reveals a more complex (DFT)-based microscopic study of malachite and support and still enigmatic frustrated spin lattice.16,17 the ensuing model by magnetization measurements com- Malachite is arguably the best known Cu secondary bined with quantum Monte-Carlo (QMC) simulations of thermodynamic properties. Contrary to Janod et al.,22 mineral typically formed in the oxidation zone of Cu de- 2+ posits as weathering product of Cu sulphides. The ear- we argue that malachite is a rare Cu system where arXiv:1308.4866v1 [cond-mat.str-el] 22 Aug 2013 liest source of copper (minerals quarried together with structural dimers match the spin dimers. We discuss the malachite were a convenient flux that facilitated the origin of this effect, and provide a comprehensive pic- smelting),18 it was extensively used as ornamental stone ture of magnetic exchange parameters, including both and as a green pigment19 since antiquity. The related fa- isotropic and anisotropic exchange couplings. Eventu- mous blue Cu-carbonate azurite transforms to malachite ally, we take advantage of available high-pressure crystal structures of malachite32 and investigate the pressure de- by absorption of water and loss of CO2. This transfor- mation known as "greening" is responsible for greenish pendence of the exchange couplings and the ensuing spin instead of blue skies on some historical frescos.20 More gap. recently, malachite and its Zn-substituted versions were The paper is organized as follows. In Sec. II, the recognized as a convenient precursor of mixed CuO{ZnO applied experimental and theoretical methods are pre- catalysts.21 sented. The crystal structure of malachite is described in 2 AFM Sec. III. The experimental and computational results for evaluated in second order perturbation theory as Jij = the ambient-pressure malachite are provided in Sec.IV. 2 4tij=Ueff. Our predictions for the evolution of the spin gap under Alternatively, the full exchange couplings Jij, compris- pressure are presented in Sec.V. Finally, discussion and ing ferromagnetic (FM) and AFM contributions, can be summary are given in Secs.VI and VII, respectively. derived from total energy differences of various collinear spin arrangements evaluated in spin-polarized supercell calculations within the mean-field DFT+U formalism. II. METHODS For the double counting correction, a fully localized limit approximation was used and the on-site Coulomb repul- For our experimental studies we used a natural sam- sion and onsite Hund's exchange of the Cu(3d) orbitals ple of needle-shaped malachite from Tsumeb, Namibia. are chosen as Ud = 8.0±1.0 eV and Jd = 1.0 eV, respec- The sample was investigated by laboratory powder x-ray tively, similar to parameter sets we have used previously 6,10 diffraction (XRD) (Huber G670 Guinier camera, CuKα 1 for other cuprates. radiation, ImagePlate detector, 2θ = 3 − 100◦ angle The anisotropic exchange was calculated with the range). High-resolution low-temperature XRD data were full relativistic version of GGA+U provided by VASP collected at the ID31 beamline of the European Syn- with Ud = 9.5 eV, Jd = 1.0 eV and the default projector- chrotron Radiation Facility (ESRF, Grenoble) at a wave- augmented wave (\PAW-PBE") pseudopotentials41 on a length of about 0.35 A.˚ The chemical composition was de- 4×4×4 k-mesh. For each exchange (J and J 0) 36 mag- termined by the ICP-OES method.33 The magnetization netic configurations (four configurations for each matrix was measured with a Quantum Design MPMS SQUID element of the exchange matrix) were calculated.42 The magnetometer in a temperature range of 2-400 K in fields Ud parameter of GGA+U was chosen so that the isotropic up to 5 T. For measurements up to 14 T, a vibrating sam- exchanges Jij obtained from VASP agree with those from ple magnetometer setup of a Quantum Design PPMS was the FPLO calculations. The 1.5 eV offset in the Ud values used. arises from the different exchange-correlation functionals Electronic and magnetic structure calculations were and different basis sets used by the two codes. performed within DFT by using the full-potential local- Quantum Monte Carlo (QMC) simulations were per- orbital code fplo9.07-41.34 Local density (LDA)35 and formed using the codes loop43 and dirloop sse44 from generalized gradient approximations (GGA)36 were used the software package alps-1.3.45 Magnetic susceptibility for the exchange-correlation potential together with a and magnetization of the two-dimensional model were well converged k-mesh of 5×5×5 points for the crys- simulated on finite lattices comprising up to N = 1024 tallographic unit cell and about 100 points for super- spins, using periodic boundary conditions. For simula- cells. For the optimization of hydrogen positions, in ad- tions in zero field, we used 200 000 sweeps for thermal- dition to fplo, the Vienna Ab initio Simulation Package ization and 2 000 000 sweeps after thermalization. For (VASP5.2)37 was used in combination with LDA, GGA, finite-field simulations, 40 000 and 400 000 sweeps were revPBE,38 DFT-D39 and HSE0640 exchange-correlation used, respectively. functionals. For a full relaxation of all atomic positions in the high-pressure structures, we employed the GGA+U method implemented in VASP. III. CRYSTAL STRUCTURE Strong electronic correlations were included in two dif- ferent ways: First, by mapping the LDA bands onto an Malachite crystallizes in the monoclinic space effective one-orbital tight-binding (TB) model. Thereby, group P 21=a with the lattice constants a = 9.5020 A,˚ the transfer integrals tij of the TB-model are evaluated b = 11.9740 A,˚ c = 3.240 A˚ and the monoclinic angle ◦ 23 as nondiagonal matrix elements between Wannier func- β = 98.75 . Nearly planar CuO4 plaquettes form tions (WFs). These transfer integrals tij are further in- doubly bridged Cu2O6 dimers by edge-sharing (Fig.1). troduced into the half-filled single-band Hubbard model The dimers themselves share common corners and form ^ ^ P H = HTB + Ueff i n^i"n^i#, with Ueff being the effec- slightly twisted chains running along the [201] direction. tive onsite Coulomb repulsion. In case of half filling and The Cu{O{Cu bridging angles within the dimers are ◦ ◦ for the strongly correlated limit tij Ueff, as realized rather different with 94.7 and 106.4 , respectively, in malachite (see Table IV), the Hubbard model can be resulting in an average bridging angle of 100.5◦.