Microscopic Origins of Optical Second Harmonic Generation in

Sep 19, 2014 - Department of Control Engineering, Czech Technical University, Prague, 121 35 Prague 2, Czech Republic. ‡ Department of ... LiSc(SeO3...
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Microscopic Origins of Optical Second Harmonic Generation in Noncentrosymmetric−Nonpolar Materials Antonio Cammarata,*,† Weiguo Zhang,‡ P. Shiv Halasyamani,*,‡ and James M. Rondinelli*,¶,§ †

Department of Control Engineering, Czech Technical University, Prague, 121 35 Prague 2, Czech Republic Department of Chemistry, University of Houston, Houston, Texas 77204, United States ¶ Department of Materials Science and Engineering, Drexel University, Philadelphia, Pennsylvania 19104, United States ‡

S Supporting Information *

ABSTRACT: We use a symmetry-based structural analysis combined with an electronic descriptor for bond covalency to explain the origin of the second-order nonlinear optical response (second harmonic generation, SHG) in noncentrosymmetric nonpolar ATeMoO6 compounds (where A = Mg, Zn, or Cd). We show that the SHG response has a complex dependence on the asymmetric geometry of the AO6 and AO4 functional units and the orbital character at the valence band edge, which we are able to distinguish using an A−O bond covalency descriptor. The degree of covalency between the divalent A-site cation and the oxygen ligands dominates over the geometric contributions to the SHG arising from the acentric polyhedra, and this can be understood from considerations of the local static charge density distribution. The use of a local dipole model for the polyhedral moieties (AO4/AO6, MoO4, and TeO4) can account for a nonzero SHG response, even though the materials exhibit nonpolar structures; however, it is insufficient to explain the change in the magnitude of the SHG response upon A-cation substitution. The atomic scale and electronic structure understanding of the macroscopic SHG behavior is then used to identify hypothetical HgTeMoO6 as a candidate telluromolybdate with an enhanced nonlinear optical response.

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INTRODUCTION The nonlinear optical (NLO) light-matter interaction in second harmonic generating (SHG) materials relies on a two-photon process, whereby frequency doubling of an incident photon occurs (SHG), via a virtual transition facilitated by the asymmetric charge density defined by molecular units within the noncentrosymmetric (NCS) crystal.1 Telluromolybdates with the stoichiometry ATeMoO6 (abbreviated as ATM), where A = Mg, Zn, and Cd, have NCS nonpolar structures, i.e., the materials do not have an inversion center and also do not exhibit macroscopic polarizations (a net dipole moment per unit volume). However, the materials do exhibit SHG with 1064nm incident radiation, and they are type-I phase-matchable. Intriguingly, the materials are either chiral and nonpolar, space group P212121 (MgTM and ZnTM), or achiral and nonpolar, space group P4̅21m (CdTM), despite possessing NLO-active functional building units (FBUs): AO6 or AO4 (depending on the A2+-cation), TeO4, and MoO4 polyhedra (see Figure 1). These facts challenge the established understanding and intuition that one should increase the density and cooperative (additive) alignment of polar FBUs with local electric dipoles:2,3 Polar displacements should be maximized to enhance the nonlinearity, because the SHG response is proportional to the square modulus of the macroscopic electric polarization.2 How does one explain the NLO effects, which are substantially larger than potassium dihydrogen phosphate (KDP),4−7 in these telluromolybdates when the spatial symmetry operations prohibit a net polarization in the crystal? © 2014 American Chemical Society

The nonlinear response has been attributed to the large intrinsic dipole of the TeO4 units and laser-induced atomic distortions to the acentric TeO4 and MoO4 units,7 nonzero contributions from the AO4 tetrahedra,6 or the AO6 octahedra.5 However, spatial symmetry operations require the cancellation of these dipoles as the materials are nonpolar. More recently, the SHG behavior in such quaternary molybdates was explained as arising from induced electronic dipoles from the electromagnetic field of the incident photons during optical excitation, rather than originating in the magnitude of the local dipoles present in the static atomic structure.8 Specifically, Jiang et al.8 identified the second-order dielectric susceptibility of the MoO4 FBU as providing the dominant (more than 90%) contribution to the SHG d ij -tensor coefficientsthe other polyhedra and the divalent A2+ cations were found to contribute much less (if at all). This observation was gleaned from mapping the ATeMoO6 crystal structure onto a flexible-dipole (FD) model, which provides an empirical descriptor (flexibility index) of the ability to (de)form an induced dipole between two ionsan effective estimate of the change in the dipole moment due to valence charge density fluctuations upon optical excitation of the structure. Because the FD model is based on bond-valence charges9 and an average Mo−O bond length normalized by the force required to spatially bind a point charge between the Mo and O ions forming the dipole, it neglects Received: August 6, 2014 Revised: September 18, 2014 Published: September 19, 2014 5773

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Figure 1. Layered structure of ATeMoO6 compounds with distorted TeO4 square pyramids and MoO4 tetrahedra. Panels (a) and (b) depict the chiral and nonpolar structure adopted by A = Mg and Zn (space group P212121), whereas panels (c) and (d) illustrate the achiral and nonpolar structure for A = Cd (space group P4̅21m). Rumpling of the two-dimensional A-cation layers along the c-axis produces the octahedral AO6 units (panel b); the absence of these displacements results in (d) AO4 tetrahedra, because of the ideal two-dimensional (2D) square lattice formed by the divalent A-cations.

the ATM compounds. Rather, the inductive changes to the electronic structure induced by the A-cations are critical to understanding the NLO response. For the latter, we directly investigate the behavior of the charge density along the A−O bond. Our main finding is compatible with previously proposed microscopic models of the SHG response, but is crucially different. We find that the role of the A−O interaction and its bond covalency is the microscopic feature differentiating the size (intensity) of the SHG coefficients (response) in the telluromolybdates. Finally, we use this understanding to design the hypothetical HgTeMoO6 compound, selecting A = Hg to optimize the A−O bond covalency descriptor. Our DFT calculations reveal HgTM should form in the P2 1 2 1 2 1 structure-type with distorted HgO6 FBU and would have the greatest NLO response of those surveyed.

the important role played by the symmetry operations of the crystal in transforming the dipoles at one site in the lattice to another. It also excludes contributions from other FBUs, octapolar moments, the number density of Mo−O bonds or MoO4 tetrahedra, and realistic chemical bonding interactions in ionic−covalent solids. Moreover, because the MoO4 FBU is common to the Mg, Zn, and Cd molybdenum tellurites, the flexibility indexes are almost identical;8 albeit, the structures exhibit distinct NLO responses. The powder second harmonic generation (PSHG) intensities are CdTM > MgTM > ZnTM, which are in disagreement with calculated SHG coefficients and estimated PSHG responses,8 i.e., MgTM > CdTM > ZnTM. Clearly, the A-cation chemistry influences the SHG performance, either through direct changes in the electronic and atomic structure (e.g., via contributions of various parity states to the valence and conduction band edges and local dipoles, respectively) or inductively through more subtle changes in the electronic charge-density distribution. Previous reports have also noted such a change in acentricity, with respect to the size of the A-cation.10−14 In this work, we identify the origin of the SHG response in the ATM materialsMgTeMoO6, ZnTeMoO6, and CdTeMoO6 by disentangling the various contributions (atomic geometry, electronic structure, and the complex interplay thereof) to the SHG response. We use a combination of mode-crystallography analyses and electronic structure calculations based on density functional perturbation theory (DFPT), taking into account our new experimental SHG measurements on the telluromolybdates family. Our approach relies on the fact that the contracted SHG dij-matrix can be decomposed into unique acentric structural contributions15 using crystallographic distortion modes.16 It allows us to deduce the role of various atomic displacements to the SHG response.17−19 We observed a computed SHG response of CdTM > ZnTM > MgTM, in good agreement with our PSHG experiments. It is anticorrelated with the amplitude of the inversion symmetry lifting distortions in the telluromolybdates, and an additional bond covalency descriptor is required to capture the variation in SHG intensity with A substitution. The MoO4 interactions are important, but not fundamental to ranking the SHG response of

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METHODS

Electronic Structure Calculations. Density functional (perturbation) theory [DF(P)T] calculations within the local-density approximation20 (LDA) and the independent electron approximation, neglecting quasi-particle effects, are performed using the ABINIT package.21 To achieve high precision in the relaxed-ion nonlinear dielectric susceptibility, we use a plane-wave cutoff of 800 eV and 5 × 5 × 5 k-point mesh. Norm-conserving pseudo-potentials, generated with the Troullier−Martins scheme, are used for all atoms with the following valence electron configuration: 5s24d10 (Cd), 4s23d10 (Zn), 3s2 (Mg), 6s25d10 (Hg), 5s25p4 (Te), 5s14d5 (Mo), and 2s22p4 (O). Unless specified, we use the experimental atomic positions and crystal structures for all calculations of the ATeMoO6 compounds. For MgTM and CdTM, we use the structures reported in refs 7 and 22, and for ZnTM, we use the structure obtained from our X-ray diffraction (XRD) refinements (see the Supporting Information), which is similar to that found in ref 5. To aid in formulating a predictive model of the SHG response in known telluromolybdates, we also consider hypothetical structures with A-site cation substitutions. For example, by replacing Cd with Zn in the CdTM phase, designated as tet-ZnTM, i.e. ZnTM in space group P4̅21m (the CdTM structure), and analogously, replacing Zn with Cd in the ZnTM geometry. This structure is referred to as orth-CdTM, i.e., CdTM in space group P212121 or that adopted by ZnTM. These computational experiments allow us to disentangle the contributions to the SHG response arising from the acentric atomic geometry and the electronic system. Note that for these hypothetical phases, no structural relaxations 5774

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are performedthe experimental atomic positions of the unsubstituted phases are maintained and the electronic properties computed from those geometries. For the HgTeMoO6 (HgTM) compound, we explored the phase stability of both the P212121 and P4̅21m structures. In each case, the lattice parameters and atomic positions are relaxed until the Hellmann− Feynman forces are minimized to tolerance of a 0.5 meV Å−1. Experimental Synthesis and Characterization. Polycrystalline ATeMoO6 (A = Mg, Zn, and Cd) compounds were synthesized by solidstate reaction techniques. For MgTeMoO6 and CdTeMO6, stoichiometric amounts of MgCO3 and CdCO3, respectively, with TeO2 and MoO3 as reagents were thoroughly ground, pressed into pellets, and heated in air to 400 °C for 20 h, 450 °C for 20 h, and 500 °C for 20 h, with intermittent regrindings. For ZnTeMoO6, the same heating regime and duration were used, but a stoichiometric amount of ZnO was used as reagent. The resultant white powders were found to be pure phases of ATeMoO6 (A = Mg, Zn, and Cd) by powder X-ray diffraction (XRD). Experimental and calculated powder XRD data for all three materials are given in the Supporting Information (SI). Powder-SHG (PSHG) measurements were performed on a modified Kurtz-nonlinear optical (NLO) system using a pulsed Nd:YAG laser with a wavelength of 1064 nm. A detailed description of the equipment and methodology is available in ref 23. As the powder-SHG efficiency has been shown to strongly depend on particle size,24 the materials were ground and sieved into distinct particle size ranges (90 μm). Relevant comparisons with known SHG materials were made by grinding and sieving crystalline α-SiO2 and LiNbO3 into the same particle size ranges. No index matching fluid was used in any of the experiments. Materials Structure and Noncentrosymmetry. The ZnTM and MgTM structures are orthorhombic, crystallizing in the chiral nonpolar NCS space group P212121. Figures 1a and 1b show that the layered structure is composed of alternating TeO4 square pyramids (distorted seesaw geometry, because of the stereoactive Te4+ lone pair) cornerconnected with MoO4 tetrahedra, creating an infinite zigzag chain structure in the ab-plane. (Mo6+ is second-order Jahn−Teller distortion inactive, because of the tetrahedral coordination.) Interleaved between these layers along the c-axis are corner-shared octahedral ZnO6 or MgO6 FBUs, with the A2+ cations occupying the 2b Wyckoff site (..2) displacing in an antipolar manner along the c-axis toward an octahedral edge. The cooperative displacements lead to an oblique 2D A2+-cation net in the ab-plane. The CdTM structure is tetragonal crystallizing in the achiral nonpolar NCS space group P4̅21m. The TeO4···MoO4 chain structure is identical to that in ZnTM and MgTM (see Figure 1c). However, the Cd2+ preference for a four-coordinate geometry results in the telluromolybdate chains interconnected via a layer of corner-connected CdO4 tetrahedra (see Figure 1d). Because of the 2a site symmetry, the zposition of the Cd atoms is constrained to c/2, resulting in an ideal tetrahedral coordination and the absence of rumpling along the c-axis. Therefore, the Cd2+ net yields a square 2D lattice in the ab-plane.

Figure 2. Particle size versus powder-SHG efficiency for MgTM, ZnTM, and CdTM. The curves are drawn to guide the eye and are not fit to the data.

Table 1. SHG Responses for the ATM Structures by Crystal Classa 4̅2m

222 NLO response

ref

MgTM

ZnTM

CdTM

d14 d14 deff exp d14b PSHG (× KTP) PSHG (× α-SiO2)

this work ref 8 this work

6.57 14.04 9.61 11.16c 1.5f 250

8.64 10.61 12.65 4.10d 0.6c 250

14.19 11.75 16.96 14.88e 2d 300

this work

a

First three rows of data in the table reports calculated dij SHG coefficients (pm/V) in the static (ℏω = 0) limit. Last three rows of data compares experimental (exp) powder-SHG (PSHG) data obtained in this work to published data. There is clear disagreement between previously reported experimental measurements and calculated values. While experiments show d14 rankings of CdTM > MgTM > ZnTM, simulations reported in the literature find d14 rankings of MgTM > CdTM > ZnTM, both in disagreement with the trends of our experimental and calculated d14 values. bConverted from PSHG measurements, relative to KTP, for comparison. cData taken from ref 4. dData taken from ref 5. eData taken from ref 6. fData taken from ref 7.

previously reported PSHG measurements are erroneously high. Our PSHG measurements seem more reasonable, given the FBUs in the ATM structures. Beyond these experimental discrepancies, inconsistencies among the calculated static SHG coefficients also exist. Because MgTM and ZnTM are orthorhombic biaxial crystals, they exhibit three independent, nonzero, dij-tensor coefficients: d14, d25, and d36, which are equal if Kleinman’s symmetry condition is valid.25 Similarly, for uniaxial CdTM, the two symmetry-independent coefficients are also equal: d14, d25 = d14, d36. Previous DFT calculations have reported d14 values of MgTM > CdTM > ZnTM, with CdTM being similar to that of ZnTM (see Table 1), whereas we find qualitatively different behavior: Our calculated SHG d14 and deff coefficients indicate that the NLO response of MgTM is less than (and closer to) that of ZnTM, and together both are weaker that than of CdTM with the achiral nonpolar structure (see Table 1). Although our calculations make use of the LDA exchange-correlation functional and those reported previously are based on the generalized-gradient approximation (GGA) of Perdew−Burke−Ernzerhof26 with the Hubbard +U correction,27 it is unusual that the LDA and GGA would produce

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RESULTS AND DISCUSSION SHG Response. Because of the experimental discrepancies in the literature, we remeasured the PSHG efficiencies of our polycrystalline ATM samples. The particle size versus SHG intensity data are shown in Figure 2. As seen, the ATM materials are Type-I phase-matchable. We find that the SHG efficiency of CdTM is larger than MgTM and ZnTM, which are comparable to each other, relative to particle size. The PSHG efficiency of CdTM is 300 × α-SiO2 (7.5 × KDP), whereas MgTM and ZnTM exhibit PSHG efficiencies of 250 × α-SiO2 (6.25 × KDP). It should be noted that the previously reported PSHG measurements revealed efficiencies comparable to or greater than those of KTiOPO4 (KTP); these values are reproduced in Table 1. (Note that the SHG response of KTP is approximately 103 times greater than that of α-SiO2.) Based on our experience and given the structural architecture of the materials, the 5775

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present in the NCS structures. Two of the modes are noncentrosymmetric, transforming as irreps Γ−1 and Γ−2 , while the third mode, Γ+2 , preserves inversion symmetry (see Table 2).

qualitatively different trends in the SHG response: quantitative differences in the amplitudes of the coefficients are more common.28,29 For the hypothetical A-site substituted compounds, we find, for Cd in the ZnTM structure (orth-CdTM), that the d14 and deff values are 12.11 and 17.72 pm/V, respectively. For the tet-ZnTM compound, we obtain SHG responses of d14 = 12.22 pm/V and deff = 14.60 pm/V. We discuss the implications of the cation substitution on the NLO response in these artificial materials in more detail below. Role of Acentric Displacements on SHG. Because of the different crystal structures adopted by the ATM compounds, it is challenging to unambiguously identify the atomic scale displacements that contribute to the SHG response following standard crystal−chemistry approaches. We apply a mode-polarization vector analysis to facilitate this comparison,17 whereby we quantify the distortions present in the static NCS structures using symmetry-adapted modes transforming as irreducible representations (irreps) of the pseudo-symmetric P4/nmm (space group No. 129) structure depicted in Figure 3. The hypothetical

Table 2. Symmetry-Adapted Mode Amplitudes Specified by irrep Label and Isotropy Subgroup (Shown in Parentheses)a MgTM ZnTM CdTM

class

Γ−1 (P4212)

Γ−2 (P4̅21m)

Γ+2 (Pmmn)

222 222 42̅ m

0.870 Å 0.822 Å ···

2.249 Å 2.244 Å 2.034 Å

0.4343 Å 0.3451 Å ···

a

Data obtained from crystallographic analyses of the reported experimental structures relative to a centrosymmetric P4/nmm phase (space group No. 129, setting 2). The atom positions of the P4/nmm are fully relaxed at the DFT-LDA level, keeping fixed the experimental lattice vectors of the noncentrosymmetric phases. Symmetry-forbidden modes are designated by “···”.

Since each irrep acts differently on the atomic positions of the high-symmetry structure, a meaningful way to understand the acentric distortions that produce the two NCS structure relies on examining the physical action of each symmetry mode on the CS structure. The main distortion modes depicted in Figures 3c−e are characterized by (Γ−1 ) O displacements in the ab-plane producing in-phase rotations about the long axis; (Γ2−) antisymmetric stretching of O atoms about the Mo center and antisymmetric scissor (seesaw) O displacements about Te, which shorten the metal−oxygen bonds; and (Γ+2 ) antiparallel A2+ displacement along the c-axis, with the O atoms coordinating Te displacing within the ab-plane to form a rectangular base. Interestingly, our decomposition shows that the Te and Mo atoms do not undergo any acentric displacements, rather the coordinating ligands and A2+-cation are solely responsible for the symmetry reduction from the CS to NCS geometries. Based on the amplitude of each symmetry mode’s contribution to the NCS structure, we can understand the atomic displacements responsible for removing inversion symmetry. First, it is clear that the two antisymmetric modes lift inversion in the 4/nmm point group in different ways: Γ−1 is chiral, leading to point group 422, whereas Γ−2 is achiral and nonpolar (i.e., 42̅ m) and contains both improper rotations and a mirror plane. As shown in Table 2, the chiral and nonpolar structures adopted by MgTM and ZnTM require both NCS modes to condense in the structure. The direct product P4212 ⊕ P42̅ 1m → P212121. Furthermore, the mode described by Γ−2 in both compounds accounts for more than 90% of the total distortions (total mode distortion amplitude is 2.451 and 2.415 Å for MgTM and ZnTM, respectively), which makes it a good structural descriptor for the symmetry-breaking transition. In contrast in CdTM, the mode transforming as Γ−2 is sufficient to obtain the achiral and nonpolar independent of any other distortion, accounting for the complete symmetry reduction. We now calculate the specific-acentric-mode displacement (SAMD) quantity,17 which quantifies the amount of the acentric distortions per unit volume using only contributions from the inversion symmetry breaking mode amplitudes. We obtain SAMD values of 9.52 × 1018, 9.47 × 1018, and 8.04 × 1018 Å/cm3 for MgTM, ZnTM, and CdTM, respectively, which are plotted against our first-principles-computed d14 values in Figure 4. One ̈ would naively anticipate that an enhancement of the structural acentricity, parametrized by the SAMD descriptor, would increase the SHG response, as recently reported15 for the NLO material BaB2O4. However, Figure 4 shows that both the

Figure 3. Illustration of the (a) reference centrosymmetric (CS) ATeMoO6 phase with (b) ideal FBU geometries from which the structural mode decompositions are performed. Note that, in the hypothetical CS, Mo is effectively eight-coordinated, because of the higher symmetry bonding environment. The displacement patterns of the main modes contain (c) cooperative in-phase oxygen displacements in the ab-plane, (d) antisymmetric displacements, and (e) antipolar A2+ displacements along c-axis, accompanied by small oxygen displacements in the ab-plane.

centrosymmetric (CS) reference phase is an immediate nonisomorphic supergroup for both ATM structure types. For each CS phase, we fully relax the free internal degrees of freedom, keeping the cell volume fixed to that of the noncentrosymmetric structure.30 Our decomposition of the ATM compounds, aided by the Isodistort software,31 reveals that up to three modes may be 5776

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Figure 4. Computed SHG d14 coefficient and SHG intensity (inset) as a function of the specific-acentric-mode distortions (SAMD) for the ATM compounds. The linear (broken) line is a guide for the eye to show the anticorrelated relationship between the acentric structural distortions and the SHG response; it is obtained from fitting the experimental structures (filled symbols). Hypothetic structures obtained from A2+cation substitution are noted by open symbols.

calculated d14 value and measured PSHG, relative to α-SiO2 (inset) decrease as the amplitude of the inversion-symmetry lifting structural distortions increases. Jiang et al.8 reported that the NLO response in the ATM family is independent of the local dipole moments imposed by the structural distortions, whereas our symmetry-mode analysis clearly indicates the acentric distortions leading to the canceling dipoles are anti-correlated with the SHG response. Furthermore, we find that, at a fixed atomic geometry, the SHG response also depends on the chemistry of the A-cation in the structure. For example, the d14 coefficient for the hypothetical orth-CdTM compound is 12.11 pm/V, which is approximately 3 pm/V greater than that of ZnTM. The enhancement is obtained without any modification to the atomic positions of the ZnTM structure: only cation-site substitution occurs. Consistent with the anticorrelated SAMD on the SHG response, we find that d14 of tet-ZnTM is reduced relative to CdTM (Figure 4). This observation is in contrast with previous studies that concluded that the role of the A-cation is small, if not negligible, relative to the SHG response, because of the strong ionicity of the divalent cations. Our analyses indicate that an increase in the acentric atomic structure decreases the SHG response, while changes in the A-cation chemistry, i.e., the electronic structure and charge density, can be used to either enhance or suppress the NLO function. Role of Electronic Structure on SHG. We now examine the change in the electronic structure and valence charge density for the nonpolar ATM compounds to obtain a predictive descriptor able to differentiate the SHG response of the chiral and achiral structures. We first calculate the atom- and orbital-resolved electronic density of states (DOS) for all of the above considered structures (Figure 5). This is motivated by the fact that the virtual excitations occur between the top of the valence band maximum (VBM) and the bottom of the conduction band minimum (CBM). All compounds are insulating with band gaps of 2.87, 2.84, and 1.98 eV for MgTM, ZnTM, and CdTM, respectively, which, because of the LDA functional, are underestimated when compared to the experimental values of 3.12 eV,7 3.10 eV,5 and 3.59 eV,6 respectively. The valence band region extends from approximately −6 eV to 0 eV (not entirely shown), and consists

Figure 5. Atom- and orbital-resolved electronic density of states (DOS) for (a) MgTM, (b) ZnTM, (c) CdTM, (d) tet-ZnTM, (e) orth-CdTM, and (f) HgTM structures. In all of the structures, the top of the valence band is largely confined within the energy range from −0.7 eV to 0.0 eV. The Fermi level is set to 0.0 eV.

of orbital contributions from oxygen, bonding Mo 4d states, and Te 5p states. The stereochemically active lone pair of Te4+ is also present in all ATM compounds and produces the locally asymmetric TeO4 coordination environment (Figure 6). These states do not provide any significant orbital contributions near the band edges and should minimally influence SHG response, albeit they may alter the birefringence of the crystal. Closer to the Fermi level (EF), the VBM is dominated by O 2p states with a sharp band edge and a pseudo-gap below the EF, which results in a well-defined energy separation of these 2p-orbitals over the energy range from −0.7 eV to 0 eV. The contribution of orbitals from the A-cations to the valence band region is influenced by the spatial extent of the active valence orbitals, for example, Zn 3d and Cd 4d shown in Figures 5b and 5c, respectively. Because of the relatively short Zn−O (2.147 Å) and Cd−O (2.160 Å) bond lengths in the FBUs, the hybridization with the O 2p states is nonzero and significant over this energy range. The orbital contribution of these cations to the VBM is also higher than that of Mg, albeit the MgTM structure has the largest number of O 2p states near EF. The effect of the coordination geometry and orbital spatial extent on this hybridization is especially evident in the hypothetical compounds; cation substitution of Zn in the CdTM structure (tetZnTM, Figure 5d), results in a reduction of orbital hybridization while an enhancement is observed in orth-CdTM. Remarkably, this change in covalency is correlated with the evolution in the 5777

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the Mo−O−Te bond angle (Figure 7c). For tet-ZnTM, however, we find this excess charge is redistributed on a plane perpendicular to the Zn−O bond axis. Analogous results are obtained by plotting the density difference between the ZnTM and orth-CdTM (not shown). Therefore, by comparing this charge analysis with the DOS to the corresponding SHG coefficients, we find that the SHG response within these nonpolar telluromolybdates is indeed influenced by the orbital character and local environment of A-cation. Interplay of Structure and Bond Covalency on SHG. Recognizing that the SHG response depends on both the tiling of the FBUs and the electronic features of the A-cations, we introduce an approach to isolate the complex interdependence of the coordination geometry, A-cation chemistry, A−O chemical bonding on the SHG. Doing so requires an electronic descriptor that is sensitive to the atomic features of the crystal. Here, we use a recently devised bond covalency parameter to study the peculiarities of the A−O bond. It was shown to be a useful metric for capturing the changes in chemical bonding and electronic structure in octahedrally coordinated transition metals in perovskite oxides.32 The covalency of the A−O bond is defined as *A,O = −|CMA − CM O| and provides a measure of the overlap of the atomic orbitals participating in the bond, obtained through a k-space integration over a defined energy window and projection of the electronic wave functions on the selected atomic species. CMA and CMO are the band center of mass for the A and O orbitals respectively (units are eV). We evaluate *A,O considering only the A and O contribution to the well-defined VBM region described above. Note that the closer this quantity is to zero, the more covalent the bond. Figure 8a shows that, for the experimentally structures, the SHG response is enhanced with increasing A−O bond covalency; it is maximum for the CdTM structure. This trend is nonmonotonic, however, as hypothetical A-site-substituted compounds exhibit a decreases the magnitude of d14, despite a further increase of the covalency. Both trends are consistent with our discussion of the electronic DOS and hybridization, and these are now quantified in a single electronic parameter. According to this trend, we formulate the following general guideline: For large optical nonlinearity, it is desirable to optimize the covalency of the bond formed by the atomic pair contributing states to the valence band, because it leads to increased polarizability of the electronic structure. This outcome and Figure 8 are remarkably similar to what was reported previously for the nonlinear optical susceptibility tensor of semiconducting nonpolar achiral (4̅3m) tetrahedral AB compounds33 more than 40

Figure 6. Electron localization function (ELF) for CdTM, projected on the (a) (100) and (b) (200) planes (cyan color) containing TeO4 and MoO4 polyhedra and nearest-neighbor Cd cations reveal the lobes of nonbonded electron density from the stereochemically active Te4+ lone pair. Note that all Te cations are lone-pair active, only the localized nonbonded electrons appear in the projection plane if it contains the Te atom, i.e., the Te atoms that do not show the lone pair in panel (a) or (b) are not in the plane. The electron lobes are localized far from the Cd−Te interatomic space and are always anti-aligned from one plane to the next independent of the A-cation chemistry.

SHG coefficients of the cation-substituted compounds (Figure 4). Is this correlation coincidental or fundamental to the origin of the SHG response in the telluromolybdates? The discernible change in the electronic structure with substitution of divalent Acations is important, because the optical matrix elements for SHG will be onsite from dipole-allowed transitions of the hybridized O 2p−A 3d (or 4d) states to the O 2p−Mo 4d states. Note that the antibonding Mo 4d states (shaded regions) are also strongly mixed for all ATM compounds throughout the conduction band. All of these effects are difficult to quantify in terms of the FD model, because the A2+ frontier orbitals are effectively integrated out by the oscillating dipole model. To obtain further insight into the role played by the A-cation independent of any structural effects, we plot the difference between the electron charge density of the CdTM and the tetZnTM structures (see Figure 7a), having the same atomic geometry and therefore the same amplitude of acentric distortions (SAMD). We observe that, for cadmium in the CdTM structure, the electronic density about the nearestneighbor oxygen atom is asymmetric and directed along the Cd− O bond axis (Figure 7b), while there is a charge depletion along

Figure 7. (a) Surface plot of the electronic density difference of the CdTM subtracted from the tet-ZnTM structure: yellow and red surfaces represent an excess and a depletion of electrons, respectively. (b) Section of the surface plot in panel (a) containing the A−O bond; RGB gradient shows decreasing charge density. By substituting Cd with Zn, the electron density around the oxygen atom participating in the bond flows toward the A−O bond from the plane orthogonal to it, pointing toward the closest A-cations. (c) Section of the surface plot in panel (a) containing the Te and Mo atoms on a plane orthogonal to the A−O bond; RGB gradient shows decreasing charge, as in panel (b). Cd substitution produces a depletion of charge in favor of the A− O bond. 5778

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difference, and a reduction in both bond covalency and the asymmetrically distributed electron density along the bond axis. Instead, a more spherical distribution about the A ionic core results. Therefore, the structural distortions are key to mediating the hybridization of the electronic states of the A and O ions, while simultaneously fulling the requirement that inversion symmetry is lifted to permit the NLO response. They should not be enhanced, but rather tuned to mediate the optimal covalent bond. In the nonpolar telluromolybdates, the ionic size of the Acation influences the magnitude of the distortion and the strength of the orbital overlap in the chemical bond.

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DISCUSSION Recently, Jiang et al.8 used the quasi-free charge concept to explain the SHG response in ATM compounds. This charge concept employs a f lexibility index that measures the flexibility of a bond. Such a description is based on the bond valence charge that measures the charge on an atom by using the cation−oxygen distance, and on core charges, defined a priori according to the atomistic model used (i.e., pseudo-potentials). This description misses subtle features of the electronic charge distributed along the bond, because of the surrounding environment, since such features are described only by quantities involving the atomic couple forming the bond; moreover, the flexibility index relies on the local geometry that alone is unable to account for purely electronic effects that are crucial in correlated systems. In this sense, the covalency metric used in the present work can give a more-detailed and accurate description, since it makes use of the electronic density distribution, which is an observable quantity determined by all the characteristics of the system (i.e., atom types, geometry, electronic correlation). Thus, our covalency metric accounts for all the electronic and geometric contributions that determine the f inal SHG response. As a final consideration, in the Jiang et al. work,8 they show that the SHG trend of the examined compounds is MgTM > CdTM > ZnTM, in contrast with the experimental observation of the present work. According to that trend, their model predicts no contribution from the Cd−O bonds and a dependence from the Mo−O bond. Indeed, we clearly see from the density difference plots that the A-cation substitution does influence the Mo−O bond charge distribution and it determines the magnitude of the dij coefficients. Design of Enhanced SHG in ATMO Structures. Having established that the *A,O bond covalency is an effective knob to engineer the SHG response in ATeMoO6 compounds, we now seek to identify a descriptor correlated with *A,O that does not require an equilibrium (or known) structure from which it is calculated. An atomic descriptor, which would allow the A-cation to be selected a priori, would then facilitate the prediction of a new telluromolybdate for targeted synthesis. Figure 9 shows the relationship between the experimental structures (CdTM, ZnTM, and MgTM), and the *A,O. We find that the bond covalency increases monotonically with the Acation polarizability, parametrized from ref 34. We use this knowledge, which has been understood to be important in organic molecules35 to design a new ATM compound with enhanced NLO properties. According to the trend shown in Figure 8, we should select an A2+-cation that can form an A−O bond with covalency between −0.025 eV and −0.014 eV. Figure 9 shows that it should also have a polarizability greater than that of Cd2+. Since the polarizability increases as the principal quantum number increases, a promising candidate is Hg, which is

Figure 8. (a) SHG d14 coefficient as a function of the A−O bond covalency (*A,O). The maximum NLO response of the known experimental compounds (filled symbols) occurs for CdTM with strongly hybridized Cd 4d and O 2p states in the VB region. Symbols are as defined in Figure 4; lines are present as a guide for the eye. (b) A−O bond covalency as a function of the SAMD. Structural distortion can be tuned to obtain a target bond covalency value.

years ago: A large SHG response may be obtained when the covalent bond for an atom pair in an acentric geometry and NCS structure is neither overly covalent nor ionic; it should be of an intermediate value, such that the charge density localized along the bond can be polarized under optical excitation. As discussed early on by Levine,33 there is a strong relation between the dij coefficients and the ionicity of the bonds. For AB semiconductors with zincblende structures, he derived an expression for the SHG response and showed that an optimal value of the ionicity of the A−B bond is required to achieve the maximal NLO response. The bond asymmetry depends on both the electronegativity of the ions participating in the bond pair, the distance between atoms (longer bonds are preferred), and the ionic sizes. Note that a highly ionic bond can lead to a large electric polarization in polar compounds because of large mode effective charges and displacements; however, it decreases the bond polarizability and increases the band gap of the material, both of which reduce the SHG response.33 When multiple atomic types are present in the structure, different types of bonds are realized and the effective charge on a specific moiety is determined by local environment. This leads to complexity in multication compounds and motivates our extension of Levine’s approach using quantum-mechanical-derived descriptors. Because the bond covalency and acentric distortions (SAMD) are found to correlate and anticorrelate, respectively, with the SHG response, we examine the interplay between the electronic and atomic descriptors in Figure 8b. For the experimental structures, we find that the ionicity of the compound increases (*A,O deviates from 0) in a nonlinear fashion as the amplitude of the inversion symmetry breaking distortions increases. At the unit-cell level, when SAMD increases the A−O bond is shortened. This leads to an undesirable charge transfer from the A-cation to the ligand, because of the electronegativity 5779

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should be almost twice as large as that of CdTeMoO6, making the HgTM structure a promising NLO material.

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CONCLUSIONS We have identified the microscopic origin of the second harmonic generation (SHG) response in the noncentrosymmetric-nonpolar ATM compounds (MgTeMoO6, ZnTeMoO6, CdTeMoO6), using a symmetry-based structural approach combined with electronic structure calculations, which is able to disentangle the atomic contributions from the electronic interactions responsible for the nonlinear optical (NLO) behavior. Although we found that the SHG response arises in part from the polar MoO4 functional building units (FBUs),36 the electronic contribution to the SHG parametrized by an A−O bond covalency metric is uncovered to be key in differentiating the NLO response in this family of nonpolar materials. In the absence of unique polar axis, we found that the SHG response is anticorrelated with the amplitude of the acentric structural distortions, and one must go beyond the atomic scale contribution to understand the SHG behavior. Indeed, we established a relationship between the A−O bond covalency, which accounts for the orbital hybridization among states derived from these atoms, and the diagonal dij tensor elements to identify that the electronic polarizability of the divalent A-cation largely governs the SHG response through an inductive effect. This theory reconciles inconsistencies among previously proposed mechanisms within a static framework as it considers only the ground-state atomic and electronic structures. Because the bond covalency can be tailored by control over the amplitude of the acentric structural distortions and/or the atomic polarizability of the A-cations, we exploited that understanding to design the telluromolybdate HgTeMoO6, which has not yet been synthesized. Our density functional calculations find the phase to be dynamically stable, and predicts the NLO response to be approximately two times greater than that of the CdTeMoO6 compoundthe telluromolybdate with the largest SHG response in the family. Lastly, we note that, because the methods applied to understand the origin of the SHG response do not rely on structural features or chemistry specific to the telluromolybdates, we believe the SAFOR method15 in combination with this bondcovalency analysis presented here is a particularly powerful approach for explaining the electro-optical properties of NCS materials with mixed ionic−covalent character. This suite of tools may also lead to the rational design of new SHG compounds in diverse material families and chemistries.

Figure 9. *A,O bond covalency as a function of atomic polarizability. Polarizability of the A-site can be experimentally engineered to fine-tune the A−O bond covalency. Symbols and fitting line are as defined in Figure 4.

isoelectronic with Zn and Cd, suggesting that HgTM would also be stable. We thus choose A = Hg and build two model geometries using the ZnTM and CdTM experimental structures as possible polymorphs. We then relaxed the lattice parameters and the atomic positions of each system. We find that the P212121 structure is more than 2 eV lower in energy than the P4̅21m polymorph (see Table 3). The electronic structure of the HgTM Table 3. Crystallographic Data for the Proposed NLO Telluromolybdate HgTeMoO6 (HgTM)a atom

Wyckoff site

x

Hg Mo Te O(1) O(2) O(3)

2a 2b 2b 4c 4c 4c

0 0 0 0.7368 0.2975 0.1991

y

z

1/2 1/2 0.2364 0.2023 0.3012

1/2 0.1752 0.7659 0.7183 0.9487 0.6402

0

a DFT-LDA relaxed lattice parameters: a = 5.2143 Å, b = 5.2156 Å, and c = 9.0605 Å for space group P212121.

system shows features similar to those already discussed above for the isoelectronic ATM compounds (recall Figure 5): The top of the valence band is well-defined in the energy range from −0.7 eV to 0.0 eV, consisting of hybridized O 2p and Hg 5d states, and the conduction band is formed by O 2p and Mo 4d states. The band gap is found to be 1.55 eV at the LDA level. The distortion mode analysis reveals that the HgTM structure has the smallest acentric structural distortion of the surveyed ATM compounds (SAMD = 6.80 × 1018 Å/cm3, compared to that of the other ATM geometries. Since we previously found high *A,O covalency values for less-distorted structures (Figure 4), this result would indicate a Hg−O covalency value near that of the Cd−O bond (Figure 9). Indeed, the calculated covalency value is −0.020 eV, falling in the range that we targeted for enhancing the SHG response. This preliminary analysis supports the hypothesis that the Hg atom is a promising A-cation to enhance the SHG response in telluromolybdates. (Note that this first result is important from a computational point of view, because the evaluation of the dij tensor is computationally demanding.) Next, we used density functional theory (DFT) to compute the dij tensor for the relaxed HgTM compound. We obtained three equivalent nonzero coefficients: d14 = d25 = d36 = 26.64 pm/ V and deff = 39.00 pm/V. According to this prediction, the SHG response of the yet-to-be-synthesized HgTeMoO6 compound

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ASSOCIATED CONTENT

S Supporting Information *

Powder X-ray diffraction analyses and additional crystallographic structure data are available as Supporting Information. This material is available free of charge via the Internet at http://pubs. acs.org/.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A. Cammarata). *E-mail: [email protected] (P. Halasyamani). *E-mail: [email protected] (J. M. Rondinelli). Present Address §

Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA. 5780

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Notes

(26) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. B 1996, 77, 3865−3868. (27) Anisimov, V. I.; Zaanen, J.; Andersen, O. K. Phys. Rev. B 1991, 44, 943−954. (28) Kang, L.; Luo, S.; Huang, H.; Ye, N.; Lin, Z.; Qin, J.; Chen, C. J. Phys. Chem. C 2013, 117, 25684−25692. (29) Lin, Z.; Jiang, X.; Kang, L.; Gong, P.; Luo, S.; Lee, M.-H. J. Phys. D: Appl. Phys. 2014, 47, 253001. (30) The occupied Wyckoff positions in MgTM are: Mg 2a(3/4,1/4,0), Mo 2c(1/4,1/4,z), z = 0.0.351 Te 2c(1/4,1/4,z), z = 0.737, O(1) 4e(0,0,1/2), and O(2) 8j(x,x,z), x = 0.497, z = 0.824. The occupied Wyckoff positions in ZnTM are: Zn 2b(3/4,1/4,0), Mo 2c(1/4,1/4,z), z = 0.082 Te 2c(1/4,1/4,z), z = 0.715, O(1) 8j(x,x,z), x = 0.504, z = 0.343 and O(2) 4d(0,0,0). The occupied Wyckoff positions in CdTM are: Cd 2b(3/4,1/4,1/2), Mo 2c(1/4,1/4,z), z = −0.051, Te 2c(1/4,1/4,z), z = 0.272, O(1) 4d(0,0,0), and O(2) 8j(x,x,z), x = 0.515, z = 0.670. The occupied Wyckoff positions in HgTM are: Hg 2b(3/4,1/4,1/2), Mo 2c(1/4,1/4,z), z = −0.666, Te 2c(1/4,1/4,z), z = 0.259, O(1) 8j(x,x,z), x = 0.010, z = 0.315, and O(2) 4d(0,0,0). (31) Campbell, B. J.; Stokes, H. T.; Tanner, D. E.; Hatch, D. M. J. Appl. Crystallogr. 2006, 39, 607−614. (32) Cammarata, A.; Rondinelli, J. M. J. Chem. Phys. 2014, 141, DOI: 10.1063/1.4895967 (33) Levine, B. F. Phys. Rev. B 1973, 7, 2600−2626. (34) Shannon, R. D.; Fischer, R. X. Phys. Rev. B 2006, 73, 235111. (35) Bailey, R. T.; Cruickshank, F. R.; Pugh, D.; Sherwood, J. N. Acta Crystallogr., Sect. A: Found. Crystallogr. 1991, 47, 145−155. (36) Jeggo, C. R. J. Phys. C 1972, 5, L133. (37) Momma, K.; Izumi, F. J. Appl. Crystallogr. 2008, 41, 653−658.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.C. was supported by ONR, under Grant No. N00014-11-10664. J.M.R. acknowledges the donors of The American Chemical Society Petroleum Research Fund for support (Grant No. 52138-DNI10), and useful discussions with K. Poeppelmeier. P.S.H. and W.Z. thank the Welch Foundation (Grant No. E-1457) for support. This work used the Carbon Cluster at the Center for Nanoscale Materials at Argonne National Laboratory, supported by the U.S. DOE, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. The technical scientific computing support of Dr. Michael Sternberg is thankfully acknowledged. Figures were rendered using VESTA software.37

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