Mechanistic insights into the superexchange-interaction driven

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Mechanistic insights into the superexchangeinteraction driven negative thermal expansion in CuO Yuanpeng Zhang, Marshall T. McDonnell, Stuart Calder, and Matthew Tucker J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.9b00569 • Publication Date (Web): 01 Apr 2019 Downloaded from http://pubs.acs.org on April 1, 2019

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Mechanistic insights into the superexchange-interaction driven negative thermal expansion in CuO Author: Yuanpeng Zhang1,2*, Marshall McDonnell1, Stuart Calder1 and Matt Tucker1* 1Neutron

Scattering Division, Oak Ridge National Laboratory (ORNL), Oak Ridge, Tennessee 37831, United States. 2Materials

Measurement Science Division, National Institute of Standards and Technology (NIST), 100 Bureau Drive Gaithersburg, MD 20899, United States. Corresponding Email: [email protected], [email protected]

Abstract: The negative thermal expansion (NTE) in CuO is explained via electron-transfer-driven superexchange interaction. The elusive connection between the spin-lattice coupling and NTE of CuO is investigated by neutron scattering and principal strain axes analysis. The density functional theory calculations show as the temperature decreases, the continuously increasing electron transfer accounts for enhancing the superexchange interaction along [101] – the principal NTE direction. It is further rationalized that only when the interaction along [101] is preferably enhanced to a certain level compared to the other competing antiferromagnetically exchange pathways can the corresponding NTE occur. Outcomes from this work have implications for controlling the thermal expansion through superexchange interaction, via, for example, optical manipulation, electrons or holes doping, etc.

Introduction Thermal expansion is an important consideration in many applied fields ranging from large-scale infrastructure building to high-precision instrumentation1. Most materials expand when heated, as a result of the anharmonic atomic potential where the equilibrium inter-atomic distance will be shifted to a larger value with the increasing of the system’s energy2. However, starting from the discovery of isotropic negative thermal expansion (NTE) in ZrW2O8 by Sleight, et al. in 19963, NTE has been an intensely studied material phenomenon. For NTE, one, two, or all three dimensions of the material shrink with increasing temperature, which is counter to both our intuition and experience. NTE is believed to be connected with various origins, such as geometrical flexibility, ferroelectricity and magnetism, as recently reviewed by M. Dove4, K. Takenaka5 and J. Chen, et al1. Such connections 1


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then provide potential routes for controlling the thermal expansion utilizing NTE materials, which can find various applications such as precision engineered parts, microdevices and thermomechanical actuators, etc.6-9. In recent years, there has been wide research into controllable NTE materials related to magnetic transitions, such as those related to the broadening magnetovolume effect (MVE, in the antiperovskite manganese nitrides10-12), spin frustration13-17, change in the d-orbital involved interactions18-19 and superexchange interactions20-21, etc. For the magnetic effects related NTE, one of the research focuses is the changing of magnetic properties (moments, susceptibility, ordering, etc.) and its interplay with lattice distortion as temperature (T) changes6, 22-26. Specifically concerning the superexchange driven NTE, increasing of atomic distances is beneficial for reducing the on-site Coulomb repulsion (through reducing, e.g. the p-d orbital overlapping) and therefore is potentially accounting for the expanding of lattice as T decreases (i.e. NTE). However, the lattice expansion in turn tends to reduce the magnetic coupling. Therefore, it brings in an outstanding question – how the interplay between superexchange interaction and lattice expansion proceeds as T decreases. Concerning the implication for tuning NTE of superexchange systems in general sense, it is important to resolve such a question. To this aim, we revisit the superexchange involved anisotropic NTE of CuO in this report. The unusual magnetic properties of CuO (more generally, the CuO4 planes contained systems, such as La2CuO427, La2-xBaxCuO428 and La2-xSrxCuO429) has invoked extensive research interest owing to their connection with high temperature superconducting30-33, room-temperature multiferroicity34-37, etc. Considering the NTE study, previous reports38-40 have concluded spin-lattice coupling instead of geometrical flexibility drives the anisotropic NTE behavior for bulk CuO. In 2008, X. G. Zheng, et al41. reported that the colossal NTE (reaching 10-5/K in magnitude) in nano CuO is also attributed to the strong spin-lattice coupling41. However, limited efforts were dedicated upon the interplay between superexchange interaction and lattice distortion as T changes, especially when NTE is involved. Such research could have been impeded partly due to the anomalous magnetism of CuO30, 42-44 and the elusive mechanism concerning the changing of magnetic coupling versus T43, 45-47. Here through comprehensive experimentaltheoretical analyses, we provide a mechanistic insight into the interplay between superexchange interaction and lattice distortion as T changes in and out of the NTE regime. First, we present the extraction of the principal NTE direction and its connection to the magnetic coupling. Then details of exchange interactions are

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examined using density functional theory (DFT) calculations with varying T. It is shown that the onset of anisotropic NTE in CuO is systematically consistent with the Anderson superexchange picture. Finally, based on dihedral angle analysis, it is proposed that flattening of the two types of CuO4 planes accounts for enhancing the magnetic coupling as T decreases in the NTE regime, compensating the effect of atomic distance increasing.

Neutron Diffraction: Rietveld Refinement & Principal Strain Axes Analysis The T dependence of neutron powder diffraction (NPD) data for CuO powder (purchased from SIGMA-ALDRICH, with the particle size of 10 μm) were measured on the HB2A beamline at the High Flux Isotope Reactor at Oak Ridge National Laboratory. Details of the NPD experiment can be found in the supplementary information (SI). The lattice parameters extracted from Rietveld refinements are shown in Fig. 1 (a-c). From inspection of the lattice parameters, several key aspects are addressed here: 1) the lattice constant b shows a linear decre-

Figure. 1. (a-c) Lattice parameters a, b and c obtained from the Rietveld refinement. (d) The Cu-Cu atomic distance along [101]. (e) The direction of the

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principal NTE axis, and (f) the magnitude of the thermal expansion coefficient (TEC) along this axis. -ase as T decreases. 2) In comparison, in the a and c directions, above ~150 K, the lattice constant decreases linearly with decreasing T, and after that, a starts to increase as the temperature decreases and c stays constant. In summary, although there is no isotropic NTE in the bulk CuO (refer to Fig. S1 for the V-T plot), the anisotropic NTE can be observed when T is lowered to below a certain level (~150 K) for a. To determine the directional link between the magnetic ordering and NTE, we first calculate the atomic distances along certain exchange interaction paths. Among the directions analyzed, only the distance along [101] (Fig. 1d, also refer to Fig. 2a for paths illustration) shows a clear increase with decreasing T below ~150 K (refer to Fig. S2 to Fig. S5 for other paths inspected). It is known that the [101] direction has the strongest magnetic coupling among all the distinctive exchange interaction paths (refer to Fig. 2a)48-49. Therefore, the results here show the strong connection between the magnetic coupling and the NTE direction, consistent with previous report38. Since the crystal structure of CuO is monoclinic (with space group C2/c), changes of the lattice parameters a, b and c with T do not exactly reflect the strain (here, the thermal effect) of the system. Extending our analysis of the magnetic ordering coupling to NTE, we carried out principal axes analysis for the strain using PASCal 50. The main idea is that if the thermal strain effect is assumed to be a linear transformation, it can be decomposed into three steps – rotation + scaling + rotation, following the standard procedure of singular value decomposition (SVD) for a linear transformation matrix. The purpose of such a decomposition is to find the orthogonal axes (i.e. the principal axes) along which the strain effect only changes the scale but not direction of the axes51. For the PASCal analysis, we select five adjacent temperature points (used by the program to determine the average strain) and continuously move the selection from low to high temperature. Then, for each selection of the five T points, the corresponding principal NTE axis is obtained for the central T point of the selection. Here, the direction and thermal expansion coefficient (TEC) of the principal negative axis are presented in Fig. 1 (e) and (f), respectively. First, TEC along this axis for nearly all T points below 150 K are negative (i.e. NTE). Secondly, the axis is densely distributed around [10 1] for all T points. Therefore, the PASCal results provide further direct evidence to

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support the strong connection between the magnetic ordering and the anisotropic NTE of CuO, along the principal NTE direction. In fact, the strong spin-lattice coupling in CuO has already been demonstrated by the anisotropic response of both the magnetic and dielectric properties with respect to the applied external field35, 52. For example, the electric polarization observed in CuO within the T range 213 K to 230 K at ambient conditions was attributed to the incommensurate spiral spin ordering, which introduced the study of the high-T multiferroicity of CuO35. However, from the results we have shown so far, the transition T from positive thermal expansion to NTE is around 150 K, which does not agree with either of the two magnetic transition points (230 K – from paramagnetic to incommensurate and 213 K – from incommensurate to commensurate). Experimentally, the magnetic order parameter extracted from our NPD measurements does not show a clear transition around 150 K (see Fig. S7). Nevertheless, the early result by A. Junod et al.53 shows different behaviors in both the effective Debye-temperature and the magnetic specific heat crossing 150 K. Therefore, to further investigate the variation of spin-lattice coupling and its connection to NTE in CuO, we next inspect the changing of the magnetic interaction with T via complimentary density functional theory (DFT) calculations.

Theoretical Work: DFT Calculations The density-functional theory (DFT) based electronic structure calculations were carried out using the Quantum Espresso package with ultra-soft pseudopotentials and plane wave basis sets54. The exchange-correlation energy was calculated using Perdew-Burke-Ernzerhof general gradient approximation (PWE-GGA)55. Additionally, Hubbard U was included to account for the strong correlation effect in this transition metal oxide. The value of U was chosen to be 5 eV, according to previous work by B. G. Ganga, et al48. Considering the ground state magnetic structure of CuO, the self-consistent calculations were carried out on a 2 × 1 × 2 supercell. A 2 × 4 × 1 k-mesh was used for the Brillouin-Zone integration following the Monkhorst-Pack approach. The plane wave basis set cut-off energy was set to 450 eV. For the calculation at each temperature point, the experimentally determined lattice structure obtained from the Rietveld refinements were directly used without further geometrical optimization. For the robustness of such an approach, refer to the article by X. Rocquefelte, et al56. According to the single-band Hubbard model, the Hamiltonian concerning the magnetic coupling is given as: 5


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† 𝑐𝑖𝜎) + 𝑈𝑒𝑓𝑓∑𝑗𝑛𝑗↑𝑛𝑗↓ 𝐻 = ∑(𝑖,𝑗)𝜎(𝑡𝑖𝑗𝑐𝑖𝜎† 𝑐𝑗𝜎 + 𝑡𝑖𝑗𝑐𝑗𝜎

(1)

where the summation is taken over the atoms pair (i, j), 𝑐𝑖𝜎† and 𝑐𝑖𝜎 are the creation and anniliation operators of electron with spin 𝜎 on atom i, 𝑡𝑖𝑗 the electron transfer integral between sites i and j, 𝑈𝑒𝑓𝑓 is the effective onsite Coulombic repulsion and 𝑛𝑘𝜎 the spin density operator for site k with spin σ. Assuming the strongly correlated limit 𝑡𝑖𝑗 ≪ 𝑈𝑒𝑓𝑓, the Hamiltonian given in Eqn. (1) can be reduced to the effective Heisenberg model: 𝐻 = ∑𝑖,𝑗𝐽𝑒𝑓𝑓 𝑖𝑗 𝑆i ⋅ 𝑆j

(2)

th where 𝐽𝑒𝑓𝑓 𝑖𝑗 is the effective Heisenberg coupling constant and 𝑆k (k=i or j) is the k spin vector. The sign and magnitude of 𝐽𝑒𝑓𝑓 𝑖𝑗 determines the property (negative for ferromagnetic – FM or positive for antiferromagnetic – AFM) and the strength of the exchange interaction, respectively.

Figure. 2. (a) Illustration of the five distinctive Js for the CuO system. Here the 2 × 1 × 2 supercell used in the DFT calculation is presented. Atoms shown with squares represent those generated by the periodic boundary condition from the atoms included explicitly (shown with spheres) in the calculation supercell. The O atoms are represented with red and Cu atoms with black (spin up) and grey (spin down). (b) The relative ratios for the difference in AFM coupling coefficients as a function of T. Taking J1 and J2 as an example, we have the following definition  J1 J 2 /  J1 J 2  [( J1  J 2 )  (J1  J 2 )] / (J1  J 2 ) , for the simplicity of T 230

230

T

T

230

230

230

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230



notation. The black dashed lines in (b) are to guide the eyes. Dashed arrows correspond to the ~150K threshold level where anisotropic NTE is observed with decreasing T. The level of error bar (refer to SI for information about the error bar extraction) is given in the bottom panel of (b). Since the size of error bar is smaller than the data point symbols, it is not plotted explicitly along with the data points. The methodology for theoretical estimation of the Js, together with the extracted Js from our DFT calculations, is presented in the SI. Several key findings will be addressed here: 1) The exchange interaction along the [101] direction (J1, negative thus AFM) is the strongest among all the five distinctive Js. Such a result, combined with the atomic distances calculation and the PASCal analysis shown above, shows quantitative evidence of the connection between strong spin-lattice coupling and anisotropic NTE behavior existing in the low T regime. 2) Among the three AFM couplings, the magnitude of J1 is large relative to J3 and J5 across the entire magnetically-ordered T range. Therefore, the question remains why anisotropic NTE (principally along [101], i.e. the J1 direction) only exists for Ts below ~150 K. 3) Not all Js shows the same trend as T lowers – for the three AFM couplings, J1 increases, J5 keeps at a constant level and J3 decreases, respectively as T decreases. Both of the two FM couplings J2 and J4 increases as T decreases. In Fig. 2 (b), the relative ratios between J1 and all the other Js as a function of T are presented. The quantitative result here clearly shows that the exchange interaction along [101] is enhanced to a larger extent with decreasing T. Combined with the onset of anisotropic NTE, one can potentially conclude that for CuO, as T decreases, NTE along the single J1 path could only be observed when the relative ratios between J1 and all the other Js increase to a certain level (i.e. the AFM coupling J1 reaches a threshold value relative to all the other Js), as indicated by the dashed arrows in Fig. 2 (b).

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Figure. 3. The (a) band structure and (b) charge density (for the crystal structure corresponding to 10 K, with the isovalue of 0.007 V-1, in Hartree unit) of the magnetic orbitals. Here the blue color is for Cu atoms and red for O atoms. The translucent ellipsoid indicates the Cu-O-Cu interaction chain, with the two Cu atoms denoted as Cu1 (left) and Cu2 (right), respectively. Here it should be pointed out the calculated Js we have presented based on the reduced Hamiltonian given in Eqn. (2) are effective values, where all interactions are treated implicitly and contained in the effective Js. To further discuss the spinlattice coupling and its fundamental link to NTE in CuO, it is necessary to go beyond the reduced Hamiltonian. Therefore, we base the following discussion back on the single-band Hubbard model (Eqn. 1) within the Anderson superexchange scheme.

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Figure. 4. Bader charge analysis, focusing on the atoms involved in the Cu-O-Cu chain along [101]. Different Bader volumes are shown in both (b) 3D and (c) 2D space. Here the Bader volume belonging to O and considered to be bonded to Cu1 is labeled as OCu1, and the one belonging to Cu1 and considered to be bonded to O is labeled as Cu1O. The Cu1O/OCu1 ratio is then presented in (a). The error bar in (a) was obtained based on the statistics over Bader volumes belonging to all Cu or O atoms with the same surroundings (i.e. the Cu-O-Cu chain along [101]) in the calculation supercell. First, the charge density corresponding to the eight lowest unoccupied bands are shown in Fig. 3. Here, eight Cu atoms with 3d9 configuration in each unit cell contributes eight spin minority orbitals and therefore they are the magnetic orbitals. The charge cloud shapes indicate the magnetic orbitals of Cu are d𝑥2 ― 𝑦2, consistent with previous findings concerning the crystal field splitting48, 57. Moreover, along the [101] chain, the lobes located on the Cu1 atom are much smaller than those on 9


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Cu2 (refer to the translucent pink ellipsoid in Fig. 3b). Also, the charge density on the intermediate O atom is asymmetric, with a relatively smaller lobe on the side closer to the Cu1 atom. This is consistent with the Anderson superexchange interaction model (refer to Fig. S12), where the overlapping between the Cu-d and O-p orbitals on each side of the Cu-O-Cu chain accounts for the electron transfer and establishment of the AFM order, respectively. Furthermore, within the Anderson picture, evolution of the charge density with temperature is examined to reveal the fundamental reason accounting for the onset of NTE. Overall, since the volume keeps decreasing as the temperature lowers (see Fig. S1) and the total number of electrons does not vary in DFT calculation, one would naturally expect an increase of the overall charge density as the temperature decreases. However, the question remains whether the charge density increases uniformly in the whole region or the increase is biased towards certain regions (which infers the electron transfer variation with temperature). To address this question, the whole volume in DFT calculation needs to be divided into separate parts, and here the Bader charge analysis is used58. With Bader analysis, the criterion for separating atomic charges is that the gradient of charge density is zero along the normal of separation surface (i.e. zero-flux surface)58. Since in the current study, we focus on the electron redistribution (as T changes) among atoms, the zero-flux surface is then a natural place to separate atoms from each other. It is found that all the Bader volumes involved in the electron transfer (Cu1, O, Cu1O, and OCu1) along the [101] direction increases with decreasing temperature (see Fig. S11). Here the Bader volume belonging to O and considered to be bonded to Cu1 is labeled as OCu1, and the one belonging to Cu1 and considered to be bonded to O is labeled as Cu1O. When we further calculate the ratio between lobes involved in the electron transfer along the [101] direction, the Cu1O/OCu1 ratio is found to increase as the temperature decreases (Fig. 4a). Thus, the result is an enhancement of the electron transfer from O to Cu1 atom along the [101] direction as the temperature decreases.

Discussion The collective results presented in this paper now resolves the underlying mechanism of NTE in CuO. Initially, the lattice shrinks uniformly as T decreases. When the temperature is lowered to the magnetically ordered region (either below 230 K for the 3D ordering, or just above 230 K for 1D ordering to exist), the strong AFM coupling along the [101] direction is introduced. Accordingly, our DFT 10



calculations shows a continuous increasing of electron transfer along [101]. Within the Anderson superexchange scheme, in such a situation, the system tends to increase the atomic distance, trying to reduce orbital overlapping thus to reduce the on-site Coulomb repulsion. However, as observed experimentally, the anisotropic

Figure. 5. (a & b) The illustration and (c) numerical result of the dihedral angle between the two CuO4 planes changing as the function of temperature. NTE behavior does not immediately show up – NTE only appears when the temperature is below ~150 K. Here, the electron transfer along the three distinctive AFM exchange interaction paths could potentially compete with each other, and therefore the anisotropic NTE could not be observed right below the magnetic transition point. Our DFT calculations then show that the exchange interaction along [101] is preferably enhanced as T further decreases. Then only when the relative ratios between J1 and the other Js (both FM and AFM ones) reaches a certain level does the anisotropic NTE along the single [101] direction occur. Here, concerning the spin-lattice coupling in CuO, it should be mentioned that magnetic frustration along [101] is responsible for establishing the spiral spin ordering and thus the multiferroicity34-35, 59 (through introducing the inverse-symmetry breaking

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Dzyaloshinskii-Moriya interaction60-61). However, in terms of NTE in CuO, the contribution from magnetic frustration should be excluded since the T range where NTE exists (

Mechanistic insights into the superexchange-interaction driven

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