Origins of bond and spin order in rare-earth nickelate bulk and heterostructures

TL;DR

This paper investigates the mechanisms behind bond and spin order in rare-earth nickelates, revealing charge fluctuations drive bond order in bulk and spin fluctuations dominate in heterostructures, with results matching experimental data.

Contribution

It provides a detailed theoretical analysis of charge and spin responses in RNiO3, clarifying the origins of bond and magnetic order in bulk and heterostructures.

Findings

Charge fluctuations drive bond order in bulk RNiO3.

Magnetic order in bulk is due to localized moments, not spin fluctuations.

In heterostructures, spin fluctuations dominate, leading to spin-density-wave states.

Abstract

We analyze the charge- and spin response functions of rare-earth nickelates RNiO3 and their heterostructures using random-phase approximation in a two-band Hubbard model. The inter-orbital charge fluctuation is found to be the driving mechanism for the rock-salt type bond order in bulk RNiO3, and good agreement of the ordering temperature with experimental values is achieved for all RNiO3 using realistic crystal structures and interaction parameters. We further show that magnetic ordering in bulk is not driven by the spin fluctuation and should be instead explained as ordering of localized moments. This picture changes for low-dimensional heterostructures, where the charge fluctuation is suppressed and overtaken by the enhanced spin instability, which results in a spin-density-wave ground state observed in recent experiments. Predictions for spectroscopy allow for further experimental testing of our claims.

Equations

$$\begin{array}{rl}H=& \sum _{\mathbf{k}\sigma ab}{ϵ}_{\mathbf{k}ab}{c}_{\mathbf{k}a\sigma }^{†}{c}_{\mathbf{k}b\sigma }+U\sum _{ia}{n}_{ia\uparrow }{n}_{ia\downarrow }+U^{\prime }\sum _i{n}_{ia}{n}_{ib}\\ & +J\sum _{i\sigma {\sigma }^{\prime }}{c}_{ia\sigma }^{†}{c}_{ib{\sigma }^{\prime }}^{†}{c}_{ia{\sigma }^{\prime }}{c}_{ib\sigma }+J^{\prime }\sum _i{c}_{ia\uparrow }^{†}{c}_{ia\downarrow }^{†}{c}_{ib\downarrow }{c}_{ib\uparrow },\end{array}$$

$${\hat{\chi }}^{c/s}={\hat{\chi }}^0(\mathbb{I}\pm {\hat{\chi }}^0{U}^{c/s}{)}^{-1}$$

$${\chi }_{a{a}^{\prime }b{b}^{\prime }}^0(\mathbf{q},\mathrm{i}{\Omega }_n)=-\frac{1}{\beta }\sum _{\mathbf{k}m}{G}_{a{b}^{\prime }}^0(\mathbf{k},\mathrm{i}{\omega }_m)G_{b{a}^{\prime }}^0(\mathbf{k}+\mathbf{q},\mathrm{i}{\omega }_m^{\prime })$$

$$c_{\mathbf{r}}^{†}=\sqrt{\frac{2}{N_{xy}(N_L+1)}}\sum _{\mathbf{k}}{c}_{{\mathbf{k}}_{xy}{k}_z}^{†}{e}^{-\mathrm{i}{\mathbf{k}}_{xy}{\mathbf{r}}_{xy}}\sin k_z{r}_z$$

$${\chi }^0(\mathbf{q},\mathrm{i}{\Omega }_n)=-\frac{1}{\beta N_L}\sum _{{\mathbf{k}}_{xy},k_z,k_z^{\prime },m}{\alpha }_{k_z,k_z^{\prime },q_z}{G}^0({\mathbf{k}}_{xy},k_z,\mathrm{i}{\omega }_m)G^0({\mathbf{k}}_{xy}+{\mathbf{q}}_{xy},k_z^{\prime },\mathrm{i}({\omega }_m+{\Omega }_n))$$

Full text

Introduction.— Understanding the mechanisms behind collective orders and excitations in solids is a pivotal topic in current condensed-matter research. The interplay between various electronic degrees of freedom at different time and energy scales gives rise to virtually unlimited variety of properties such as metal-insulator transitions (MIT), multiferroicity and superconductivity. One example of long-standing interest are the rare-earth nickelates $R$NiO3, which exhibit complex ordering phenomena depending on the NiO6 octahedra tilts and distortions controlled by the radius of rare-earth ion $R$ Torrance et al. (1992); Medarde (1997); Catalan (2008). For the smallest $R=$ Lu, $R$NiO3 goes through a MIT at $T_{c}\simeq$ 600 K, accompanied by a rock-salt type bond order of NiO6 octahedra at wave vector $\bm{q}_{c}=(1/2,1/2,1/2)$ (in units of $2\pi/a$ with $a$ the pseudocubic lattice constant) with alternating Ni-O bond lengths. An antiferromagnetically ordered phase follows at much lower temperature $T_{s}\simeq$ 130 K with an unusual $\bm{q}_{s}=(1/4,1/4,1/4)$. The temperature difference between the two transitions decreases with increasing $R$ size and disappears at $R$ = Nd with $T_{c}=T_{s}\simeq$ 200 K. LaNiO3, with the largest $R$, remains metallic at all temperatures. This complex phase diagram can be further enriched by newly developed controlled growth of oxides with atomic precision Hwang et al. (2012). Recent experiments have shown that via strain, dimensionality, and symmetry control in epitaxial films and heterostructures, the phase boundaries can be shifted and different order parameters can be selectively altered Scherwitzl et al. (2011); Liu et al. (2011); Boris et al. (2011); Frano et al. (2013); Lu et al. (2016); Hepting et al. (2014); Catalano et al. (2015); Hoffman et al. (2013); Meyers et al. (2016); Kim et al. (2016). The quasi-two-dimensional heterostructures, for instance, show a pure spin-density-wave (SDW) ground state without bond order Frano et al. (2013); Lu et al. (2016); Hepting et al. (2014)—remarkably different from the bulk.

The complex phase behavior of the nickelates and the apparent dichotomy between the bulk and heterostructures pose several theoretical challenges archetypical for transition metal oxides. The outstanding challenge is to understand the relation between the structural and electronic transitions. Recent discussions in the context of “negative charge transfer” insulators Mizokawa et al. (1991) have shown that the bond order is indispensable for understanding the MIT of the $R$NiO3. Constraining the system to the experimentally observed bond-ordered state, an insulating ground state was found in small-cluster Johnston et al. (2014); Green et al. (2016), mean-field Lau and Millis (2013); Johnston et al. (2014), and dynamical mean-field Park et al. (2012); Subedi et al. (2015) calculations. However, the origin of the essential bond order, or its absence in low-dimensional heterostructures, has remained obscure.

In this Letter, we address this crucial issue by examining—on equal footing—the charge- and spin response functions in the unordered metallic phase for the $R$NiO3 series with multiorbital random phase approximation (RPA) Takimoto et al. (2004); Graser et al. (2009) in a two-band Hubbard model. We identify a dominating charge response at $\bm{q}_{c}$ originated from inter-orbital fluctuations in the Ni-$e_{g}$ states, which can drive the system into the bond order via strong electron-phonon coupling Medarde et al. (1998). The instability increases with increasing $Pbnm$ (or $R\bar{3}c$ for $R$ = La) distortion and naturally explains the $R$ dependence of the ordering temperature $T_{c}$ in bulk $R$NiO3. The previously assumed primary spin instability Lee et al. (2011a, b), on the other hand, remains marginal in all bulk $R$NiO3. We further show that charge fluctuations are suppressed in spatially confined heterostructures below certain thickness, and a concomitant increase in the spin response can give rise to the experimentally observed SDW ground state without bond order Frano et al. (2013); Lu et al. (2016); Hepting et al. (2014).

The Hamiltonian and multiorbital RPA.— We consider an effective two-band model Kubo (2007) for the Ni-$e_{g}$ orbitals

[TABLE]

where $c^{\dagger}_{ia\sigma}$ ($c^{\dagger}_{\bm{k}a\sigma}$) creates an electron at site $i$ (momentum $\bm{k}$) in orbital $a$ with spin $\sigma=\uparrow,\downarrow$. The orbital indices $a,b\in\{d_{3z^{2}-1},d_{x^{2}-y^{2}}\}$ label the $e_{g}$ Wannier functions. $\epsilon_{\bm{k}ab}$ is the hopping matrix including the chemical potential. The number operators $n_{ia\sigma}=c^{\dagger}_{ia\sigma}c_{ia\sigma}$ and $n_{ia}=n_{ia\uparrow}+n_{ia\downarrow}$. The coupling constants $U$, $U^{\prime}$ denote the strength of intra- and inter-orbital Coulomb repulsion, and $J$, $J^{\prime}$ the intraorbital exchange and pair hopping. The RPA charge and spin susceptibilities are then given as

[TABLE]

where the matrix elements of the bare susceptibility $\hat{\chi}^{0}$ reads

[TABLE]

with $\beta=1/T$ the inverse temperature and $G^{0}_{ab^{\prime}}$ the bare Green’s function. $\omega_{m}$ and $\omega_{m}^{\prime}=\omega_{m}+\Omega_{n}$ are the fermionic Matsubara frequencies. $U^{c}$ and $U^{s}$ are the the bare vertices coupling to charge- and spin-type of fluctuations, respectively, with matrix elements $U^{c}_{aa^{\prime}bb^{\prime}}=(U,-U^{\prime}+2J,2U^{\prime}-J,J^{\prime},0)$ and $U^{s}_{aa^{\prime}bb^{\prime}}=(U,U^{\prime},J,J^{\prime},0)$ when ($a=a^{\prime}=b=b^{\prime}$, $a=b^{\prime}\neq a^{\prime}=b$, $a=a^{\prime}\neq b=b^{\prime}$, $a=b\neq a^{\prime}=b^{\prime}$ and otherwise). The total charge/spin susceptibility is then $\chi^{c/s}=\frac{1}{2}\sum_{ab}[\chi^{c/s}_{abba}]$.

While the interaction constants are often adopted as tuning parameters Takimoto et al. (2004); Graser et al. (2009), it would be favorable to take parameters most relevant to the specific materials at hand. Such effective parameters can be calculated from first principles using the constrained RPA Aryasetiawan et al. (2006). For LuNiO3 the values in the $e_{g}$ subspace are calculated by Seth et al. Seth and Georges (2016) as $U=1.65$ eV, $J=0.33$ eV, $U^{\prime}=U-2J$ and $J^{\prime}=J$. These values are considerably smaller than the typical $R$NiO3 bandwidth of $\sim$3 eV Sup —a parameter regime that RPA is well suited for. It is, however, important to note that RPA ignores crucial vertex corrections and overestimates the instabilities when using bare interaction parameters. Therefore we use the renormalized values given by the particle-particle vertex equation $\hat{\bar{U}}=\hat{U}(\mathbb{I}+\hat{U}\hat{\Gamma}^{p})^{-1}$ with $\Gamma^{p}_{aa^{\prime}bb^{\prime}}(\bm{q},\mathrm{i}\Omega_{n})=\frac{1}{\beta}\sum_{\bm{k}m}G^{0}_{ab^{\prime}}(\bm{k},\mathrm{i}\omega_{m})G^{0}_{ba^{\prime}}(-\bm{k}+\bm{q},\mathrm{i}\omega_{m}^{\prime})$. Such an approach has shown to reproduce correctly the exact susceptibilities obtained by quantum Monte Carlo methods in Hubbard models Chen et al. (1991); Bulut et al. (1993). We arrive at static renormalized values at $T=300$ K with $\bar{U}$ = 1.02 eV, $\bar{U}^{\prime}$ = 0.70 eV, $\bar{J}$ = 0.17 eV and $\bar{J}^{\prime}$ = 0.13 eV by averaging over the $R$NiO3 series. While the exact values of these parameters have a certain material and temperature dependence, we have checked that the variation does not change the results substantially. For simplicity we keep the interaction parameters fixed throughout this Letter unless otherwise noted.