Ba(n+1)Zr(n)S(3n+1): RP Chalcogenide Polymorphism

• Ba(n+1)Zr(n)S(3n+1) is a homologous series of Ruddlesden–Popper chalcogenides featuring perovskite slabs interleaved with rocksalt layers.
• The series exhibits layer-dependent octahedral tilt patterns, interface rumpling, and ascending symmetry breaking, which together influence its thermal and electronic behavior.
• Advanced computational methods using machine-learned interatomic potentials predict near-degenerate polymorphs, suggesting that interface engineering can effectively tune material properties.

$\mathrm{Ba_{n+1}Zr_nS_{3n+1}}$ denotes a homologous series of Ruddlesden–Popper (RP) chalcogenides in which blocks of corner-sharing perovskite octahedra are separated by rocksalt-like layers. In the RP structure of general composition $\mathrm{A_{n+1}B_nX_{3n+1}}$, the perovskite slabs here are built from $\mathrm{ZrS_6}$ octahedra and the rocksalt layers are $\mathrm{BaS}$; the index $n$ counts the number of perovskite layers in each slab, from the single-octahedron-thick $n=1$ member to the $n\to\infty$ limit $\mathrm{BaZrS_3}$. Recent work has established that this series is structurally rich, with new polymorphs predicted for each $n$, negative thermal expansion in $n=1$, ascending symmetry breaking in $\mathrm{A_{n+1}B_nX_{3n+1}}$0 and $\mathrm{A_{n+1}B_nX_{3n+1}}$1, and layer-dependent tilt patterns for $\mathrm{A_{n+1}B_nX_{3n+1}}$2 (Kayastha et al., 15 Jul 2025).

1. Crystal chemistry and homologous-series architecture

The series belongs to the RP family $\mathrm{A_{n+1}B_nX_{3n+1}}$3, where perovskite-like slabs and rocksalt-like interlayers coexist in a single layered framework. In $\mathrm{A_{n+1}B_nX_{3n+1}}$4, increasing $\mathrm{A_{n+1}B_nX_{3n+1}}$5 thickens the perovskite block and reduces the density of rocksalt interfaces, while the $\mathrm{A_{n+1}B_nX_{3n+1}}$6 limit recovers the fully connected three-dimensional perovskite $\mathrm{A_{n+1}B_nX_{3n+1}}$7. The primitive cell contains two unique perovskite slabs with a half-cell offset, and this slab/interfacial architecture is central to the structural behavior because the competition between octahedral rotations inside the slab and distortions at the rocksalt interface controls polymorphism across the series (Kayastha et al., 15 Jul 2025).

The principal structural degrees of freedom are octahedral tilts and interface rumpling. The work distinguishes in-phase tilting $\mathrm{A_{n+1}B_nX_{3n+1}}$8, out-of-phase tilting $\mathrm{A_{n+1}B_nX_{3n+1}}$9, rumpling, and in-plane A-site displacements. Rumpling is defined as the out-of-plane displacement of Ba in the rocksalt layer relative to the S plane, with positive amplitude when Ba moves toward the $\mathrm{ZrS_6}$0 octahedra. The reported ground-state rumpling amplitudes are about $\mathrm{ZrS_6}$1–$\mathrm{ZrS_6}$2 Å in the outermost layers, whereas out-of-plane Ba displacements in the slab interior are strongly suppressed. The average Ba displacement across the slab remains zero, so the paraelectric phases carry no net dipole.

A central organizing feature is the slow convergence toward the bulk-perovskite limit. The series becomes clearly perovskite-like only for $\mathrm{ZrS_6}$3, whereas lower-$\mathrm{ZrS_6}$4 members remain strongly interface-dominated. This suggests that finite thickness is not a minor perturbation but a primary control parameter for symmetry, tilt topology, and thermally accessible polymorphism.

2. Ground-state polymorphism and near-degenerate phases

Starting from the 33 tilted RP structures enumerated by Aleksandrov, the structurally resolved ground states split into two regimes. For $\mathrm{ZrS_6}$5, the ground states are low-tilt structures in which slab I carries a single out-of-phase tilt about $\mathrm{ZrS_6}$6 and slab II carries a single out-of-phase tilt about $\mathrm{ZrS_6}$7. For $\mathrm{ZrS_6}$8, the ground states become more perovskite-like, with out-of-phase tilts along both in-plane axes and an in-phase tilt along $\mathrm{ZrS_6}$9 (Kayastha et al., 15 Jul 2025).

$\mathrm{BaS}$0	Ground state	Tilt pattern
1	$\mathrm{BaS}$1	slab I $\mathrm{BaS}$2, slab II $\mathrm{BaS}$3
2	$\mathrm{BaS}$4	slab I $\mathrm{BaS}$5, slab II $\mathrm{BaS}$6
3	$\mathrm{BaS}$7	slab I $\mathrm{BaS}$8, slab II $\mathrm{BaS}$9
4	$n$0	slab I $n$1, slab II $n$2
5	$n$3	slab I $n$4, slab II $n$5
6	$n$6	slab I $n$7, slab II $n$8

These are computationally predicted ground states rather than an experimentally complete phase diagram. The same study emphasizes that it predicts new polymorphs for each $n$9 and that the structural energy landscape is very shallow: for all $n=1$0, a ferroelectric $n=1$1 phase is metastable and lies only $n=1$2–$n=1$3 meV above the paraelectric ground state. A plausible implication is that comparatively weak perturbations may be sufficient to select among nearly degenerate distortions.

The in-plane lattice parameters mirror the same two-regime classification. Low-$n=1$4 members cluster around $n=1$5 Å, whereas high-$n=1$6 members cluster around $n=1$7 Å, consistent with stronger in-plane tilting and hence stronger lattice contraction at larger $n=1$8.

3. Thermal evolution, transition sequences, and anomalous symmetry changes

All members eventually transform to the high-symmetry parent $n=1$9 phase, with untilted octahedra, by about $n\to\infty$0 K. The route to that parent differs sharply with $n\to\infty$1. For $n\to\infty$2, all compounds share the same qualitative sequence of tilt patterns as the parent perovskite $n\to\infty$3: a low-temperature phase with in-plane and out-of-plane tilts, then a phase retaining only out-of-plane tilting in interior layers, and finally the untilted parent. The higher-temperature transition converges toward the 3D perovskite behavior, with transitions near $n\to\infty$4 K and $n\to\infty$5 K as $n\to\infty$6 increases, and the heat-capacity data show in all high-$n\to\infty$7 materials a lower-temperature transition with first-order character and a higher-temperature transition with second-order character (Kayastha et al., 15 Jul 2025).

For the low-$n\to\infty$8 members, the structural evolution is more diverse because the rocksalt interfaces are more densely packed and suppress rotations. The $n\to\infty$9 compound, $\mathrm{BaZrS_3}$0, has ground state $\mathrm{BaZrS_3}$1, undergoes a first-order transition to $\mathrm{BaZrS_3}$2 on heating, and then a second-order transition to $\mathrm{BaZrS_3}$3. The $\mathrm{BaZrS_3}$4 step is unusual because symmetry is lowered with increasing temperature; this ascending symmetry breaking occurs concurrently with negative in-plane thermal expansion. The second-order $\mathrm{BaZrS_3}$5 transition occurs at $\mathrm{BaZrS_3}$6 K, and the negative thermal expansion persists up to that temperature. The data indicate two phase transitions near room temperature, although the lower transition temperature is not quoted numerically in the main text.

The $\mathrm{BaZrS_3}$7 member, $\mathrm{BaZrS_3}$8, is predicted to adopt $\mathrm{BaZrS_3}$9 at room temperature, in good agreement with experiment, and has high-temperature form $n$0. The main text identifies $n$1 as following a low-$n$2 pathway distinct from both $n$3 and $n$4, but does not enumerate every intermediate phase in words. What is explicit is that $n$5 does not display the ascending symmetry breaking highlighted for $n$6 and $n$7.

The $n$8 compound, $n$9, provides a second instance of ascending symmetry breaking. Its ground state is $n=1$0 with tilt pattern $n=1$1; at $n=1$2 K only orthogonal in-plane out-of-phase tilts are present in neighboring slabs. By $n=1$3 K, the system has transformed continuously to a lower-symmetry $n=1$4 phase with tilt pattern $n=1$5, because an additional $n=1$6-axis tilt appears only in the middle layer of each slab, not at the interface. By $n=1$7 K, the in-plane tilts have vanished while out-of-plane tilts remain only in interior layers, and by $n=1$8 K the structure has reached $n=1$9.

For $\mathrm{A_{n+1}B_nX_{3n+1}}$00, $\mathrm{A_{n+1}B_nX_{3n+1}}$01, the ground state is $\mathrm{A_{n+1}B_nX_{3n+1}}$02 with $\mathrm{A_{n+1}B_nX_{3n+1}}$03. At $\mathrm{A_{n+1}B_nX_{3n+1}}$04 K the structure keeps the same space group, but the $\mathrm{A_{n+1}B_nX_{3n+1}}$05-axis tilting is suppressed specifically at the rocksalt interface. This is termed a surface transition because it changes the layer-resolved tilt pattern without changing the global space group. The $\mathrm{A_{n+1}B_nX_{3n+1}}$06 and $\mathrm{A_{n+1}B_nX_{3n+1}}$07 compounds are grouped with $\mathrm{A_{n+1}B_nX_{3n+1}}$08 in the same high-$\mathrm{A_{n+1}B_nX_{3n+1}}$09 regime, with the same two-transition thermodynamics and gradual convergence toward $\mathrm{A_{n+1}B_nX_{3n+1}}$10-like behavior.

4. Microscopic mechanism: octahedral rotations, rumpling, and layer selectivity

The central mechanistic conclusion is that RP polymorphism in this family is controlled by a competition between octahedral rotations and rumpling at the rocksalt interface. This competition is strongest for out-of-plane octahedral rotations at interfacial layers. For $\mathrm{A_{n+1}B_nX_{3n+1}}$11, as the out-of-phase interfacial tilting decreases, the rumpling amplitude increases; rumpling reaches a maximum exactly at the surface tilt transition where out-of-plane tilts at the rocksalt layer disappear. Analysis of the relevant one-dimensional potential-energy surfaces shows inverse coupling: at large rumpling amplitude, the minimum-energy configuration occurs at zero out-of-plane tilt amplitude, denoted $\mathrm{A_{n+1}B_nX_{3n+1}}$12; reducing rumpling allows the out-of-plane rotation to develop (Kayastha et al., 15 Jul 2025).

The physical explanation is steric and coordination-driven. Both rumpling and out-of-plane tilting shorten Ba–S distances, so doing both simultaneously over-coordinates the A-site cation. In the authors’ phrasing, both distortions adjust A–X distances, and their competition explains the surface transitions. This places interface coordination, rather than slab geometry alone, at the center of the polymorphism.

The same framework explains ascending symmetry breaking in $\mathrm{A_{n+1}B_nX_{3n+1}}$13. In $\mathrm{A_{n+1}B_nX_{3n+1}}$14, the in-plane tilt amplitude $\mathrm{A_{n+1}B_nX_{3n+1}}$15 is large in the ground state and suppresses the $\mathrm{A_{n+1}B_nX_{3n+1}}$16-axis tilt $\mathrm{A_{n+1}B_nX_{3n+1}}$17. As temperature rises, $\mathrm{A_{n+1}B_nX_{3n+1}}$18 decreases, making $\mathrm{A_{n+1}B_nX_{3n+1}}$19 energetically favorable in the slab interior, which lowers the symmetry from $\mathrm{A_{n+1}B_nX_{3n+1}}$20 to $\mathrm{A_{n+1}B_nX_{3n+1}}$21. The outermost perovskite layers still do not tilt out of plane because of rumpling at the rocksalt interface. For $\mathrm{A_{n+1}B_nX_{3n+1}}$22 and $\mathrm{A_{n+1}B_nX_{3n+1}}$23, all perovskite layers are adjacent to rocksalt boundaries, so all out-of-plane tilts and the associated symmetry-lowering route are suppressed. For $\mathrm{A_{n+1}B_nX_{3n+1}}$24, the out-of-plane tilt is already active at low temperature, so no analogous heating-induced onset occurs.

The appearance of layer-dependent tilt patterns for $\mathrm{A_{n+1}B_nX_{3n+1}}$25 is singled out as highly unusual for inorganic RP materials. Similar distance-from-interface-dependent tilt effects had previously been reported in simulations of hybrid halide RP compounds, but this is identified as the first report for inorganic RPs. In $\mathrm{A_{n+1}B_nX_{3n+1}}$26, the interior layer alone gains $\mathrm{A_{n+1}B_nX_{3n+1}}$27-axis rotation at $\mathrm{A_{n+1}B_nX_{3n+1}}$28 K; in $\mathrm{A_{n+1}B_nX_{3n+1}}$29, the rocksalt-adjacent layers lose $\mathrm{A_{n+1}B_nX_{3n+1}}$30-axis tilt while the interior retains it. This is a dimensional crossover phenomenon in which the slab interior increasingly behaves like bulk perovskite while the interface remains structurally special.

5. Computational framework, experimental benchmarks, and unresolved assignments

The structural map for $\mathrm{A_{n+1}B_nX_{3n+1}}$31 to $\mathrm{A_{n+1}B_nX_{3n+1}}$32 was generated using a high-accuracy machine-learned interatomic potential. The workflow was to enumerate candidate tilted structures, train a neuroevolution potential implemented in the GPUMD code to hybrid-DFT reference calculations, run molecular-dynamics heating simulations from the $\mathrm{A_{n+1}B_nX_{3n+1}}$33 K ground states, identify phases by projecting onto octahedral tilt phonon modes, and analyze lattice parameters, heat capacity, XRD, tilt angles, and rumpling (Kayastha et al., 15 Jul 2025).

The method was designed to represent the anharmonic potential-energy surface relevant to finite-temperature structural evolution. The main text does not provide numerical fitting errors, train/test force RMSEs, or detailed hyperparameters, placing those in the Supplemental Material. Even so, the validation against available experiments is explicit. Calculated XRD patterns agree strongly with published data for $\mathrm{A_{n+1}B_nX_{3n+1}}$34, confirming the $\mathrm{A_{n+1}B_nX_{3n+1}}$35 phase at $\mathrm{A_{n+1}B_nX_{3n+1}}$36 K. For $\mathrm{A_{n+1}B_nX_{3n+1}}$37, the comparison is less definitive because the relevant transition is second-order and the XRD signatures are expected to evolve gradually; room-temperature $\mathrm{A_{n+1}B_nX_{3n+1}}$38 is assigned tentatively, and single-crystal XRD together with Raman spectroscopy is recommended for firmer determination.

The experimental base remains limited. Only $\mathrm{A_{n+1}B_nX_{3n+1}}$39 to $\mathrm{A_{n+1}B_nX_{3n+1}}$40 have been synthesized so far, and moderate-temperature $\mathrm{A_{n+1}B_nX_{3n+1}}$41 and $\mathrm{A_{n+1}B_nX_{3n+1}}$42 members had been reported in the high-symmetry $\mathrm{A_{n+1}B_nX_{3n+1}}$43 phase. The newer structural analysis predicts lower-symmetry phases on cooling and offers an explanation for why some may have been overlooked. The resulting picture is therefore not a completed experimental phase diagram but a computationally led one with partial experimental corroboration.

6. Functional context, relation to $\mathrm{A_{n+1}B_nX_{3n+1}}$44, and materials-design implications

RP chalcogenides are described as stable, non-toxic candidates for optoelectronic or thermoelectric applications, and the subtle distortions identified in $\mathrm{A_{n+1}B_nX_{3n+1}}$45—changes in tilt pattern, rumpling, and local coordination—are argued to be exactly the kinds of structural changes that can strongly alter band gap, carrier mobility, defect chemistry, and related optoelectronic and thermoelectric properties (Kayastha et al., 15 Jul 2025). Because the accessible polymorphs differ only by small energy scales, controlling octahedral tilting and interface rumpling is presented as a route to property engineering.

The proposed design strategy is to suppress low-temperature octahedral tilting in order to access a wider set of polymorphs and advanced functionalities. In this family, suppression of tilting is tied to enhanced interfacial rumpling, and the suggested routes to amplify rumpling are A-site doping and epitaxial strain. RP chalcogenides with non-centrosymmetric ground states, such as Ca-containing A-site analogues, are pointed to as promising for ferroelectricity. A plausible implication is that the unusually slow convergence to the bulk-perovskite limit creates a broad regime in which interface effects remain active even at relatively large $\mathrm{A_{n+1}B_nX_{3n+1}}$46, thereby enlarging the polymorphic design space.

The $\mathrm{A_{n+1}B_nX_{3n+1}}$47 end member $\mathrm{A_{n+1}B_nX_{3n+1}}$48 provides a useful reference point but should not be conflated with the finite-$\mathrm{A_{n+1}B_nX_{3n+1}}$49 layered phases. $\mathrm{A_{n+1}B_nX_{3n+1}}$50 is the fully connected three-dimensional perovskite with no layer interruption, identified as an orthorhombically distorted perovskite in the $\mathrm{A_{n+1}B_nX_{3n+1}}$51 space group. It has been reported to possess a direct band gap, exceptionally strong near-band-edge light absorption, and good carrier transport, and Ti alloying in $\mathrm{A_{n+1}B_nX_{3n+1}}$52 lowers the band gap from $\mathrm{A_{n+1}B_nX_{3n+1}}$53 eV to $\mathrm{A_{n+1}B_nX_{3n+1}}$54 eV at $\mathrm{A_{n+1}B_nX_{3n+1}}$55 at% Ti, while higher Ti concentration destabilizes the distorted chalcogenide perovskite phase (Wei et al., 2020). That result is directly relevant to the Ba–Zr–S perovskite backbone, but it does not establish that finite-$\mathrm{A_{n+1}B_nX_{3n+1}}$56 members of $\mathrm{A_{n+1}B_nX_{3n+1}}$57 will exhibit the same alloy solubility, the same magnitude of band-gap reduction, or the same stability threshold. A common misconception is to treat the layered series as a simple truncation of $\mathrm{A_{n+1}B_nX_{3n+1}}$58; the structural evidence instead shows that rocksalt interfaces introduce their own rumpling-driven and layer-selective distortions.

Taken together, the series $\mathrm{A_{n+1}B_nX_{3n+1}}$59 is defined less by a single prototype structure than by a coupled set of near-degenerate tilt and interface modes. The low-$\mathrm{A_{n+1}B_nX_{3n+1}}$60 members are interface-dominated and host negative thermal expansion and ascending symmetry breaking, whereas the high-$\mathrm{A_{n+1}B_nX_{3n+1}}$61 members become increasingly perovskite-like but develop previously unreported inorganic layer-dependent tilt patterns and surface transitions. This combination places the series at the intersection of RP crystal chemistry, finite-temperature polymorphism, and interface-controlled functionality.