Massive band gap variation in layered oxides through cation ordering

The electronic band gap is a fundamental material parameter requiring control for light harvesting, conversion and transport technologies, including photovoltaics, lasers and sensors. Although traditional methods to tune band gaps rely on chemical alloying, quantum size effects, lattice mismatch or superlattice formation, the spectral variation is often limited to <1 eV, unless marked changes to composition or structure occur. Here we report large band gap changes of up to 200% or ~2 eV without modifying chemical composition or use of epitaxial strain in the LaSrAlO4 Ruddlesden-Popper oxide. First-principles calculations show that ordering electrically charged [LaO]1+ and neutral [SrO]0 monoxide planes imposes internal electric fields in the layered oxides. These fields drive local atomic displacements and bond distortions that control the energy levels at the valence and conduction band edges, providing a path towards electronic structure engineering in complex oxides. Understanding and controlling the electronic band gap of a material is vital for many electronic and optoelectronic applications. Towards this aim, this study shows how huge band gap variations can arise by manipulating the electrostatic interactions via cation ordering in correlated oxide materials.

I n superlattices and heterostructures, the Coulombic interaction between adjacent atomic layers of (nominally) charged and/or neutral planes can create internal electric fields 1-3 that induce or alter many functional electronic, ferroic and optical properties [4][5][6][7][8][9] . In the majority of atomically layered oxides explored to date, however, the heterostructures are constructed by interleaving two or more bulk materials, often perovskite 'blocks' with well-defined three-dimensional connectivity [10][11][12][13][14][15] . Rarely are the starting oxides natural heterostructures themselves, for example, Aurivillius, Dion-Jacobson or the Ruddlesden-Popper (RP) phases. The latter (ABO 3 ) n /(AO) (also denoted as A n þ 1 B n O 3n þ 1 ) RP structure has n ABO 3 perovskite blocks stacked along the [001] direction with an extra sheet of AO rocksalt layers interleaved every n perovskite layers. This geometry disconnects the BO 6 octahedra along one direction and imposes severe constraints on the nearest-neighbour interactions 16 , promoting anisotropy in the structure-derived electronic, transport and magnetic properties, which contrasts sharply with the 3D perovskite analogues [17][18][19][20] . Indeed, the RP structure dimensionality 21,22 , that is, systematic stacking of the n layers along [001] direction, has recently been exploited 23 in combination with epitaxial strain to achieve highly tunable low-loss dielectrics. Yet, the range over which band gap control may be achieved in oxide heterostructures derived from two-dimensional building blocks remains less explored and largely unknown.
In n ¼ 1 RP oxides, there are three types of A-cation arrangements that produce [001]-cation ordering without changing the number of formula units per cell relative to the A 2 BO 4 aristotype 24 . We denote the stoichiometry of the ordered variants as AA 0 BO 4 , where A and A 0 are two chemically distinct cations with 100% site occupancy. In the context of this work, A, A 0 and B are La 3 þ , Sr 2 þ and Al 3 þ cations, respectively, and the corresponding n ¼ 1 RP oxide stoichiometry is denoted as LaSrAlO 4 . The A-cation ordering scheme produces the longrange periodic ordering of [LaO] 1 þ and [SrO] 0 planes in the crystal structure as shown in Fig. 1. Normally, such long-range chemical orderings are experimentally detected using diffraction measurements through the appearance of superlattice reflections, which are forbidden in a fully disordered or solid solution phase. As there are three different ways to order [LaO] 1 þ and [SrO] 0 planes without cell multiplication, we introduce two structural parameters, l and x, that provide a measure for classifying the orderings, which have both the same composition and number of formula units. This scheme can also describe longer period sequences not directly examined herein, and is also useful to describe the interlayer interactions across disconnected BO 6 units 25 . l and x capture the relative proximities of the positively AI (2)   DFT-PBEsol ground states for the three cation ordered configurations (a) Z ¼ 1/3, (b) Z ¼ 1/2 and (c) Z ¼ 2, where Z ¼ x/l. x (distance between two nearest chemically inequivalent [LaO] 1 þ and [SrO] 0 layers) and l (distance between two chemically equivalent [LaO] 1 þ (or [SrO] 0 ) layers) are defined in the schematic of charged atomic AO and BO 2 layers that corresponds to the atomic arrangement in (a). Note that for the Z ¼ 1/3 arrangement in (a), there are two ways to define x as described in ref. 25. Here we take x to be the minimum value, because the interactions from the nearest-neighbour monoxide layers are unscreened by an interleaved perovskite block. The Z ¼ 1/2 configuration shown in (b) has two crystallographically inequivalent Al sites. In (c) the coordination of Al is reduced to fivefold square pyramidal. Colour coding and naming scheme based on the local atomic structure in parentheses: (a) red box (polar), (b) blue box (non-polar) and (c) purple box (anti-polar). Bond lengths and angles are given in units of angstroms and degrees, respectively. It is the delicate balance of electrostatic interactions among these layers that provides a handle to direct the Al-O bond distortions and ionicity in LaSrAlO 4 through A-site ordering. The ratio of these parameters, Z ¼ x/l, is used to distinguish each phase by layer sequence; a local structural description of how Z distinguishes each variant is described below and emphasized by the colour-coding scheme in Fig. 1. Here, we seek to understand the effects of l and x on the electronic band gap in LaSrAlO 4 and predict its change on ordering. Experiments of the solid solution phase (Z ¼ 0) report an indirect band gap (E g ) near B4-5 eV (refs [26][27][28], whereas the proposed A-site ordered structures examined herein remain to be synthesized and are unknown. More recently, we computationally demonstrated the importance of A-site cation ordering and Z parameter on the rock-salt layers of n ¼ 1 RPs in metallic LaANiO 4 with correlated electrons, where A ¼ Ca 2 þ , Sr 2 þ and Ba 2 þ (ref. 25). The ordering sequence of A-site cations on (001) AO monoxide planes in LaANiO 4 RP oxides alters the electrostatic interactions, through a so-called 'electrostatic chemical strain' (ECS) effect 25 , and was used to manipulate the 3d-orbital structure at the Fermi level. It is termed an ECS effect, because of the manner in which these artificial oxides are constructed. The unit cell is obtained by ordering nominally positively charged [LaO] 1 þ and neutral [AO] 0 þ metal oxides, thereby creating a continuously varying 'electrostatic' potential due to dipolar interactions derived from [LaO] 1 þ and [AO] 0 þ ordering not appearing in the solid solution material. Our density functional theory (DFT) calculations revealed that the bond stresses 29 from the chemically imposed local electric fields are accommodated by changes to the bond lengths and bond angles. It is the change in bond length between an ordered and disordered (solid solution) system relative to the disordered system that leads to the local microscopic 'strain' within the ECS framework. Furthermore, the charge state of the ions and their relative sizes, which we refer to as chemistry of the element, were also identified to be important. Thus, the interplay of electrostatics, bond strain and chemistry manifests as an ECS in layered oxides with ordered cation sublattice.
In this work, we focus on a prototypical insulating RP oxide, LaSrAlO 4 , and show that A-cation ordering, without any change to stoichiometry, enables deterministic control over the electronic band gap of the material (E g ). We report that the arrangement of the AO monoxide [LaO] 1 þ and [SrO] 0 layers within a single-unit cell directs the magnitude of E g through internal electric fields, which drive bond distortions producing large reductions in the conduction band minimum: it varies by B2 eV in magnitude over the monoxide layering sequences explored. Our work establishes heterovalent A-cation ordering as a novel 'control knob' in addition to epitaxial strain and layer dimensionality (n) to design the properties of artificial complex oxides from modular building blocks that are naturally layered.

Results
Energetics. We obtain equilibrium LaSrAlO 4 cation ordered structures by systematically exploring the energetics of various lattice modes obtained from computed phonon dispersion curves for each sequence. Our frozen phonon calculations indicate the high symmetry structures of all three LaSrAlO 4 sequences are dynamically stable (see Supplementary Figs 1 -3). Figure 1 depicts the ground state structures for the three A-site ordered variants. No AlO 6 octahedral tilting or rotations occur, only bond elongations and rumpling distortions, that is, the Al and O atoms are displaced away from the Al-O equatorial plane. On comparing the total energies for the three sequences, the Z ¼ 2 sequence is found to be the most stable ordered variant (Fig. 1c), with the Z ¼ 1/3 and 1/2 sequences 4 and 242 meV per formula unit (f.u.), respectively, higher in energy (see Supplementary Fig. 4).
Ground-state structures and dynamical charges. The groundstate structure for the Z ¼ 1/3 sequence is polar (space group I4mm, Fig. 1a) owing to the A-cation compositional order removing inversion symmetry 24 . The lattice tetragonality (c/a) is 3.324, and the AlO 6 octahedra have four crystallographically equivalent equatorial (eq) Al-O eq and two unique (one short and one long) apical (ap) Al-O ap bonds, which give rise to an electric polarization of 7.2 mC cm À 2 along the [001] direction as obtained from the Berry phase method. The asymmetry in the Al-O ap bond lengths arises from the negatively charged [AlO 2 ] 1 À layer being interleaved between a positively charged [LaO] 1 þ and a neutral [SrO] 0 layer. The O eq atoms in the [AlO 2 ] 1 À layer are attracted towards the positively charged [LaO] 1 þ layer, which simultaneously buckles the O eq -Al-O eq bonds so they deviate from 180°. Interestingly, we find no anomalously large or small Born effective-charges Z* (relative to the nominal charge) in the ground state (Table 1); however, anisotropy between the in-plane (Z Ã xx and Z Ã yy ) and out-of-plane Z Ã zz is evident. In contrast, the Z ¼ 1/2 LaSrAlO 4 structure is centrosymmetric and non-polar (space group P4/mmm). Figure 1b shows that, although the AO monoxide sequence splits the Al sites into two distinct AlO 2 sublattices, we find that the c/a value (3.318) is close to that of Z ¼ 1/3. Unlike the Z ¼ 1/3 sequence, where the O eq -Al-O eq bond angles are o180°and there are two Al-O ap bonds, here there is no rumpling (+O eq -Al-O eq ¼ 180°). At each site, the two Al-O ap bonds are also of equal length; however, the relative lengths of the Al-O ap bonds differ between the two AlO 6 octahedra. In the [LaO|Al(1)O 2 |LaO] and [SrO|Al(2)O 2 |SrO] perovskite-like blocks, the Al-O ap bonds are 1.96 and 2.12 Å, respectively, indicating a markedly different coordination and chemical environment owing to the cation order (Fig. 1b). Indeed, while the in-plane Z Ã xx (and Z Ã yy ) for both Al sites is close to the nominal ionic charge of þ 3e, the out-of-plane Z Ã zz components show significant deviations from the nominal value that depend on the adjacent monoxide layer chemistry. The Z Ã zz values are 2.57e and 3.70e for Al(1) and Al (2) It is also worth noting that the magnitude of the anisotropy DZ for La and Sr is greater in the Z ¼ 2 sequence relative to the Z ¼ 1/3 sequence (Table 1). However, we find DZ(La)o 0, similar to Z ¼ 1/3.
Internal electric fields. In Fig. 2, we show the macroscopically averaged 32,33 local nuclear ionic potential, excluding contributions of the exchange-correlation and Hartree terms, along the [001] direction obtained from our first-principles calculations for each sequence. The built-in internal electric fields, which are proportional to the gradient in the potential, are the source for the cation-ordering-driven bond-length modulations. The dipolar interactions provide the driving force for the cooperative atomic displacements (as shown in Fig. 1), which occur to minimize the interatomic forces for a given Z in LaSrAlO 4 . Figure 2a,c clearly depicts the different electric fields that occur across the AlO 2 layers in the Z ¼ 1/3 and 2 sequences (arrowed). The steeper slope produces a larger asymmetry in the Al-O ap bonds, that is, the magnitude of local electric field across the AlO 2 layers is greater for Z ¼ 2 than in the Z ¼ 1/3 sequence. In addition, the field does not cancel in Z ¼ 1/3 owing to the polar crystal structure. Interestingly, in the Z ¼ 1/2 sequence the macroscopic average profile of the two perovskite-like blocks, [SrO|AlO 2 |SrO] and [LaO|AlO 2 |LaO] are markedly different (Fig. 2b). The local minimum in the macroscopic averages do not coincide in Z ¼ 1/2 (for example, see filled symbols indicating Al sites), unlike the other two sequences. We anticipate these variations in the internal electric field from cation ordering to alter the positions of the valence and conduction band edges. Atom  Electronic structure and band gaps. Figure 3 shows the electronic band structures obtained from our DFT-PBEsol simulations for the three cation-ordered sequences. All three configurations are found to be insulating, with the top of the valence band (VB) and bottom of the conduction band (CB) formed mainly from O 2p and La 5d states, respectively. Note that the bandwidths of the O 2p and La 5d states are also altered as a function of Z, but more remarkably, the magnitude of the indirect band gap (E g , indicated by the blue arrows) is significantly affected by the AO layering sequence.
The polar Z ¼ 1/3 LaSrAlO 4 structure has the largest E g (3.0 eV), followed by the Z ¼ 2 sequence, which is slightly reduced (2.8 eV). Moreover, the Z ¼ 1/2 structure with two distinct A-and B-site sublattices has a significantly reduced E g of 1 eV. Note that the inclusion of the La 4f-states in the pseudopotentials increased the E g of Z ¼ 1/2 by 0.5 eV, but has a negligible effect on the electronic gaps of Z ¼ 1/3 and 2 sequences. For Z ¼ 1/2 LaSrAlO 4 , we find two important changes to the band edges that are notably absent in both Z ¼ 1/3 and 2 sequences. Figure 3b shows a shift in the band edges and an enhanced dispersion of both the O 2p states from G-M-X, originating from the O(4) eq atoms in the [SrO|AlO 2 |SrO] block in the VB, and the La 5d states at the bottom of the CB. The combination of these two effects, which are less pronounced in Z ¼ 1/3 and 2, leads to the massive 2 eV band gap reduction. For comparison, we also report the E g values obtained using an all-electron DFT scheme in Table 2 (see the row labelled Wien2k and the Methods section for details). Although the band gap values of the Z ¼ 2 and Z ¼ 1/3 configurations using the full potential method remain essentially unaltered relative to the ultrasoft pseudopotential values (see Table 2), the inclusion of the La 4f states increases the band gap by 0.6 eV in Z ¼ 1/2. This finding is consistent with results using pseudopotentials that explicitly treat the 4f states described above.
Our calculations with the HSE06 functional, which includes a fractional contribution of exact-Fock exchange, and with the Green's function G 0 W 0 method are known to more accurately reproduce band gaps 34,35 also find a similar 2.3 eV reduction for the Z ¼ 1/2; albeit, as expected, the absolute value of the gap differs between methods ( Table 2). The direct G-G band gaps also show similar behaviour, with that the Z ¼ 1/2 sequence considerably smaller than that of the Z ¼ 1/3 and 2 sequences.
Origin of gap variation. To identify the atomic features responsible for the band gap reduction, we probe the sensitivity of the electronic structure to various lattice distortions appearing in the equilibrium Z ¼ 1/3, 1/2 and 2 structures. We performed a set of computations starting with the experimental structure of the Z ¼ 0 LaSrAlO 4 solid solution, where the La-and Sr cations are randomly distributed. The Z ¼ 0 LaSrAlO 4 belongs to I4/mmm space group and its lattice constants a and c are 3.754 and 12.64 Å, respectively. With the Z ¼ 0 structure, we ordered La and Sr on the A-sites corresponding to the Z ¼ 1/3, 1/2 and 2 sequences.
We next performed self-consistent DFT calculations using ultrasoft pseudopotentials at the PBEsol level of theory, where we neither relaxed the internal coordinates nor the unit cell geometry and calculated the electronic band gap. In this way, we are able to isolate the change in the electronic structure independent of the induced lattice and bond distortions to identify the [LaO] 1 þ and [SrO] 0 ordering effect on the band gap. We found that the E g of the Z ¼ 1/3 and 2 sequences reduced from 3.0 and 2.8 eV, respectively in the ground state (see Table 2) to values of 2.77 and 0.16 eV, respectively. In contrast, the Z ¼ 1/2 sequence became metallic. These calculations when compared with the results obtained from allowing differential ionic relaxations due to the chemical ordering reveal two key effects: (i) the cation ordering drives large band gap variations to such an extent that the Z ¼ 1/2 sequence becomes metallic and (ii) the critical role of lattice relaxations, that is, variation in the bond lengths (strains) and angles, in response to the cation ordering sequences restores the gap. Our findings highlight the important interplay of electrostatics and local bond strain by [LaO] 1 þ and [SrO] 0 ordering to tune the electronic properties of LaSrAlO 4 . The cation order drives a reduction in the E g from the large electric fields (as shown in Fig. 2), whereas the screening provided by differential atomic relaxations 36 reduces these field strengths and maintains the insulating behaviour of LaSrAlO 4 . As the variation in the fields for each sequence is different (due to different Z values), we arrive at the variation in the E g ( Table 2).   ARTICLE Furthermore, we also explored the sensitivity of the conduction band edge, (that is, La 5d states) for the Z ¼ 1/2 sequence with the smallest E g . We cooperatively displaced the two O ap atoms in the [LaO|AlO 2 |LaO] block, such that P4/mmm symmetry is preserved. Supplementary Fig. 5 shows that as the two Al-O ap bond lengths decrease (increase) relative to the ground state configuration, the energy difference between the minimum in the macroscopically averaged local ionic potential between the two perovskite-like blocks increases (decreases). This bond contraction (elongation) results in an increase (decrease) of the indirect band gap (Supplementary Fig. 5b-d.) As the induced Al-O displacements are produced by the buckling of the LaO monoxide layers in response to the electrostatic effect, our computational exercises conclusively show that the E g changes arises directly from the ordering of [LaO] 1 þ and [SrO] 0 monoxide layers in the [001] direction-the cation-ordering-driven bond length modulations provide a direct route to tune band structure using internal electric fields without elemental substitution.

Discussion
Using this principal finding of massive variations in the E g trend without any changes to the stoichiometry through AO monoxide layering induced internal fields, we establish cation ordering as a novel means for electronic structure engineering in materials with well-defined two-dimensional planes. As transition metal oxides exhibit many functional properties, including anisotropic transport, strong electron-lattice coupling (superconductivity and magnetoresistance), relative ease of ion-exchange and intercalation, low thermal conductivity and ferroelectricity (to name a few), this understanding could lead to layered oxides as a platform for multilfunctional and adaptive electronic materials.
To realize our theoretical predictions, we suggest thin film growth methods like oxide molecular beam epitaxy (MBE) to be well-suited for experimentally achieving the proposed cation ordering variants [37][38][39] . Recently, A-site cation ordered LaSrNiO 4 and LaSrMnO 4 RP oxides were experimentally grown with precise atomic control using oxide-MBE techniques 40,41 . Leveraging, this approach with misfit strain from epitaxial growth or by tuning the magnitude of the cation-orderingdriven bond length modulations via large A-site valence differences offers a potentially transformative paradigm to rationally design band gaps, absorption and emission edges, or transport properties for photo-and electrocatalysis, where such layered Ruddlesden-Popper and Dion-Jacobson phases are already finding use, yet remains far from ideal.

Methods
Electronic structure calculations. Density functional theory calculations were performed within the generalized gradient approximation (GGA) PBEsol exchange-correlation functional 42 as implemented in the plane-wave pseudopotential code, Quantum ESPRESSO (QE) 43 . In calculations involving QE, the core and valence electrons were treated with ultrasoft pseudopotentials 44 with the inclusion of nonlinear core corrections with a planewave basis (60 Ry cutoff). See Supplementary Note 1 and Supplementary Table 1 for additional calculation details. The following valence electronic configurations were used for our Troullier-Martin 45 pseudopotentials, La: 5s 2 5p 6 5d 1 6s 2 ; Sr: 4s 2 4p 6 5s 2 ; Al: 3s 2 3p 1 ; and O: 2s 2 2p 4 . The Brillouin zone integration was performed over a 8 Â 8 Â 5 Monkhorst-Pack k-point mesh 46 centred at G. The atomic positions and the cell volume were relaxed until the Hellmann-Feynman forces and stress tensor o2 meV Å À 1 and 0.1 kbar, respectively, were achieved.
Lattice dynamical calculations were performed using PHONOPY 47 on a 2 Â 2 Â 2 supercell with forces obtained from self-consistent total energy calculations using the finite displacement method (see Supplementary Note 2). The electric polarization (DP) for the polar LaSrAlO 4 variant is calculated using the Berry phase method 48 . The Born effective-charge tensors (Z*), which characterize the influence of long-range Coulombic interactions, were calculated using the density functional perturbation theory (DFPT) for the equilibrium structures 49 .
We also used the all-electron code Wien2k 50 to validate the suitability of our pseudopotentials for accurately describing the electronic band structure. Wien2k calculations were also performed within the PBEsol exchange-correlation functional. The muffin-tin sphere radii were chosen as follows; La: 2.2, Sr: 2.0, Al: 1.7, O: 1.6 a.u. and we used a plane-wave cutoff of RK MAX ¼ 7. The Brillouin zone integration was performed over a 14 Â 14 Â 4 Monkhorst-Pack k-point mesh 46 centred at G. We relaxed the lattice using MSR1a for optimization of positions 51 . Note that the relaxed lattice and internal coordinates from Wiek2k agreed well with the Quantum ESPRESSO simulations.
Given the inherent limitation in the DFT exchange-correlation functionals in accurately predicting band gaps, we also used the hybrid exchange-correlation functional HSEsol 52 , and a beyond-DFT (G 0 W 0 ) method as implemented in the BerkeleyGW 53-55 code to substantiate the band gap variations observed with changes in Z. Normconserving pseudopotentials constructed using the OPIUM pseudopotential package (http://opium.sourceforge.net/) were used for the HSEsol and G 0 W 0 calculations.
DFT calculations were performed on a 8 Â 8 Â 5 mesh to obtain well-converged single-electron wave functions using the GGA-PBEsol functional as a starting guess for calculating the quasiparticle energies and wavefunctions within the G 0 W 0 method. The process of computing the quasiparticle energies requires two steps: first, calculation of the static or frequency-dependent polarizabilities (w) and dielectric functions within the random phase approximation (RPA). The generalized plasmon-pole model is used to extend the dielectric function to finite frequency. We used a dielectric cutoff energy of 50 Ry to solve for w, from which the RPA dielectric matrix is calculated. Second, calculation of the screened Coulomb interaction, W, and the one-particle Green's function, G, from the inverse dielectric matrix (obtained from Step 1) and DFT eigenfunctions, respectively. These quantities are then used to evaluate the self-energy operator (S) and obtain the quasiparticle energies. We used screened coulomb and bare coulomb cutoff values of 10 and 25 Ry, respectively. Additional details on the methodology can be found in the ref. 55.
For Z ¼ 1/3 and 2 sequences, the complex flavour of the BerkeleyGW code was used. In contrast, we used the real flavour for the Z ¼ 1/2 sequence. In Supplementary Table 3, the convergence of the indirect band gap (E g ) as a function of number of conduction or empty bands is given.