The role of lattice mismatch on the emergence of surface states
in 2D hybrid perovskite quantum wells
M. Kepenekian, B. Traore, J.-C. Blancon, L. Pedesseau, H. Tsai, W. Nie, C. C. Stoumpos,
M. G. Kanatzidis, J. Even, A. D. Mohite, S. Tretiak, and C. Katan
Univ Rennes, ENSCR, INSA Rennes, CNRS, ISCR – UMR 6226, F-35000 Rennes, France
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Univ Rennes, INSA Rennes, CNRS, FOTON – UMR 6082, F-35000 Rennes, France
Department of Materials Science and Nanoengineering, Rice University, Houston, TX 77005, USA
Department of Chemistry, Northwestern University, Evanston, IL 60208, USA
Surface states are ubiquitous to semiconductors and significantly impact the physical properties and consequently the performance of optoelectronic devices. Moreover, surface effects are strongly amplified in lower dimensional systems such as quantum wells and nanostructures. Layered halide perovskites (LHPs) are 2D solution-processed natural quantum wells, [1, 2, 3] where optoelectronic properties can be tuned by varying the perovskite layer thickness. They are efficient semiconductors with technologically relevant stability. [4, 5, 6, 7] Here, a generic elastic model and electronic structure modelling are applied to LHPs heterostructures with various layer thickness. We show that the relaxation of the interface strain is triggered by perovskite layers above a critical thickness. This leads to the release of the mechanical energy arising from the lattice mismatch, which nucleates the surface reorganization and consequently the formation of lower energy edge states. These states, which are absent in 3D perovskites, dominate the optoelectronic properties of LHPs and are anticipated to play a crucial role in the design of LHPs for optoelectronics devices.
Surfaces and interfaces are known to play a central part in the performances of classical semiconductor based devices. [8, 9, 10] This holds true for the recently emerged halide perovskites. [11, 12] The 2D members of the family, layered halide perovskites (LHPs) have superior photo- and chemo- stability compared to their 3D counterparts. They show strong promise in high performance optoelectronic devices such as photovoltaics, field effect transistors, electrically injected light emission and polarized optical spin injection. [4, 13, 14, 5, 6, 7] Their properties depend on the number n of MX octahedra that span the perovskite layer (M is a metal, X a halogen). As in classical semiconductors,  surface and interface structures can have a strong influence on the properties of LHPs.  While experimental results exists, especially in Ruddlesden-Popper perovskites (RPPs) of general formula A’AMX (A and A’ being cations), there is no simple model to predict and control LHP surface properties. Here, we show that optical properties of RPPs are decisively impacted by surface relaxations occurring for structures with n2. We rationalize these features by considering LHPs as heterostructures built from the n=1 monolayered perovskite A’MX and the n= 3D AMX. This picture leads to understanding of physical phenomena underpinning surface reconstruction and concomitant modifications of electronic structure, and allows to formulate the design principles of LHP materials optimized for optoelectronics, solid-state lighting or photovoltaics.
We start with evaluating the elastic energy density accumulated in LHPs by developing an elastic model for the bulk structure (see Method and Supplementary Information for details). Our model was constructed based on the theory of elasticity in classical semiconductor heterostructures  by identifying the LHP structure with a multi-quantum well system (Fig. 1a) with alternating stacking of 3D perovskite layers L1 (AMX, of thickness n-1) and of 2D perovskite monolayers L2 (single octahedron, n=1). This combination forms an interface between two structurally-different layers, equivalent to a so-called L1/L2 heterostructures with a coherent interface (lattices are continuous across the interface in two directions).  To illustrate this general concept, we consider the family of RPP of general formula (BA)(MA)PbI that can be synthesized in phase-pure form (only one n-value). [18, 19, 20]
Fig. 1b represents experimentally observed variations of the in-plane average lattice parameter as a function of n for the native (BA)(MA)PbI heterostructure, as well as the out-of plane lattice parameters for the end members of the homologous series L1 ((n-1)MAPbI) and L2 ((BA)PbI). As qualitatively predicted based on elasticity, the in-plane lattice expansion from n=1 to n=, gives rise to an out-of-plane lattice contraction in both L1 and L2 layers. However, the experimental variation of the in-plane parameter is noticeably steep, the in-plane parameter of MAPbI (n=) being almost already recovered for n=2. A similar steep variation is observed for the out-plane lattice parameter of the L2 layer. Interestingly, quantitative agreement between experimental results and elastic model predictions can only be obtained when a very low effective stiffness of the L2 layer is considered (Fig. 1c). The origin of such low stiffness can be traced back to the octahedra tilt angles extracted from the (BA)(MA)PbI RPP experimental structures (Table S1). The octahedra tilt angles in L2 structure are indeed more important than in the L1 layers. In other words, the mechanical energy is more efficiently relaxed in the RPPs structures by rotation of those octahedra that are directly in contact with the flexible organic cations, than by Pb-I bond elongation.
Classic theory of elasticity predicts that, for a heterostructure L1/L2 with a large lattice mismatch between L1 and L2, the structure may undergo a reorganization for a critical layer thickness, to form nanostructures at the surface in order to relax the accumulated bulk mechanical energy.  From the above results, the elastic energy density in RPPs with varying perovskite thickness n was computed (Fig. 1d). We observed a maximum elastic energy density of 0.16 MPa for the RPP n=2, and a monotonic reduction of this energy with increasing n, which ultimately vanishes for bulk 3D perovskite (n). Therefore, elastic energy density arising from the interface is expected to have direct consequences over surface properties for RPPs with low n-values.
We gained further microscopic insight by modelling the structural relaxation at the relevant surfaces of the (BA)(MA)PbI RPP (n=1-4) using density functional theory (DFT), which allows direct simulation of all structural distortions (see Method for computational details). Applications using RPPs as active materials mainly employ two different orientations; either in-plane or out-of-plane with respect to the substrate or an interface layer (Fig. 2a). The most relevant surface of the RPP is then the (101) surface (Fig. 2b), [4, 15] which we model here on specifically designed slabs (see Supplementary Text I and Fig. S4), labelled as bulk-like and surface, for varying thickness n=1 to 4. The calculated changes in the surface structure were represented by (i) the contraction/expansion of the octahedron slabs close to the surface in the (101) direction (Fig. 2c), and (ii) the in-plane and out-of-plane tilting of the octahedra close to the surface (Fig. 2d).  This representation highlights our early conclusion drawn from the elastic model that rotational degree of freedom of the octahedra play an essential role in relaxing the internal elastic energy, in contrast with classical semiconductor descriptions where local strain tensor suffices. 
Fig. 2c shows the variation along the (101) direction of the distance h between octahedral slabs close to the perovskite surface. The reference value of h is obtained from the bulk-like region fixed in our DFT calculations (45 distance in Fig. 2c). The evolution of the inter-slab distance yields two opposite behaviours for n=1,2 and n2. For n2, the surface slab expansion is accompanied by a contraction of the sub-surface slabs, which leads to a decoupling of the top surface octahedron slab from the sub-surface ones. On the other hand, for n=1,2, expansion of octahedra slabs was observed in the entire surface region. A similar distinct behaviour between n=1,2 and n2 was noted by analysing the surface relaxation in RPPs occurring through in-plane and out-of-plane tilting of octahedra (Fig. 2d). In fact, surface octahedra in n=1,2 yield almost no rotational degree of freedom, whereas n2 systems exhibit significantly larger tilting of surface octahedra. The drastic change of surface behaviour, when increasing the perovskite layer thickness from n=2 to n=3 attests to a significant change of surface flexibility, which helps structural relaxation of the internal elastic energy.
We evaluate the impact of these surface relaxation processes on the electronic and optical properties of RRPs by comparing band structures and wavefunctions at the surface and in the bulk (Fig. 3). The electronic band structure still presents a direct bandgap at the surface as compared to the bulk but with variation of the bandgap energy (Fig. 3a,b and Fig. S5). We observe that the bandgap blueshifts by 70 and 150 meV for n=1 and 2 respectively and redshifts by 120 and 70 meV for n=3 and 4, respectively. The accuracy of our approach is supported by (i) the excellent agreement between the calculated exciton properties in the bulk-like region with previously reported experimental results for the same materials (see Supplementary Text II),  and (ii) the similar pattern in the optical bandgap shift between the RPP layer surface with respect to the bulk (Fig. S6).  According to surface relaxation results, lattice expansion at the (101) surface with relatively small octahedral tilting leads to a bandgap blueshift, whereas sub-surface lattice compression with significant octahedral distortions results in a redshift of the bandgap due to appearance of in-gap electronic states.
In order to understand the microscopic impact of the structural changes at the surface on each type of charges, localized density of states (LDOS) of the valence band maximum (hole) and conduction band minimum (electron) were computed (Fig. 3c and Fig. S7a,b). For all n-values, surface relaxation leads to hole wavefunctions repelled away from the surface to the bulk. A similar behaviour is observed for electrons for n=1,2. In sharp contrast, for n2, the electron gets localized mainly at the top (101) surface slab. Concomitantly, the preferential direction of electronic coupling switches from (010) to (101). From the barycenters of electron (z) and hole (z) LDOS profiles (Fig. S7c,d), we inspect separation of carriers (z=zz) and demonstrate that upon appearance of in-gap states, the electron and hole get separated (Fig. 3d). The effect is maximum for n=3, z=13.2 Å (5.5 Å for n=4). Its impact on optical activity is estimated by computing Kane energies  for bulk-like and relaxed slabs (see Method and Table S3). They reflect oscillator strengths of the optical-transitions and show a systematic reduction by 50%, 85%, 30% and 95% for the 4 lowest excitations of n=3 RPP (Table S3). Such electron-hole separation at the surface is consistent with the longer photoluminescence lifetime of low-energy states reported recently.  Fig. 3e summarizes our understanding of the formation of these low-energy states (LES) in RPPs with n2, which primarily stems from surface relaxation that strongly localizes the electron at the surface and facilitates dissociation of the strongly bound bulk exciton.
LES result from the release of the strain-induced elastic energy at the L1/L2 interface (Fig. 1). From our elastic model, the amount of energy accumulated in the materials is directly dependent on the amplitude of lattice mismatch between layers in the heterostructure L1/L2 and as a result, tuning the RPP structure and composition can lead to drastic changes of surface properties. Using this general approach, the internal elastic energy density accumulated in the bulk of LHPs can be estimated for any composition and perovskite layer thickness. From a practical perspective, understanding the relaxation of the stored elastic energy at the surface of the LHP materials is of paramount importance and presents a perfect platform for the systematic and comprehensive evaluation and screening of LHP compounds with defined functionalities for novel devices. This concept is illustrated by changing organic cation A’ in RPPs (Fig. 4a). For example, replacing BA with CHNH (NoA), which has a significantly smaller lattice mismatch,  results in the reduction of the elastic energy density of the RPP composite by more than an order of magnitude (Fig. 4b). This would prevent formation of LES and preserve the bulk Wannier exciton. By contrast, RPPs based on an organic cation inducing a larger mismatch, namely (4Cl-CHNH)PbI (4Cl-PhA),  undergoes increased strain (Fig. 4a), thus larger elastic energy density that should favour significant (101) surface relaxation suitable for e-h carrier separation.
In summary, we simulated edge (surface) relaxation effects in layered hybrid perovskite materials and discovered a critical layer thickness above which the surface reorganization becomes significant. This consequently leads to the formation of lower energy electronic states rationalizing and confirming experimental observations.  Our modelling is based on the first generic elastic model for LHPs accounting for the internal elastic energy accumulated in the material bulk and is further demonstrated using electronic structure calculations of the surface relaxation of perovskite layers. Our observation of electronic bandgap shifts and exciton dissociation at the surface, depending on the layered perovskite structure distinguishes these materials from their 3D APbI (A=cation; n=) counterparts and pave the way to unique tailored properties and functionalities for optoelectronic applications.
-  Saparov, B. & Mitzi, D. B. Organic-inorganic perovskites: Structural versatility for functional materials design. Chem. Rev. 116, 4558–4596 (2016).
-  Pedesseau, L. et al. Advances and promises of layered halide hybrid perovskite semiconductors. ACS Nano 10, 9776–9786 (2016).
-  Li, W. et al. Chemically diverse and multifunctional hybrid organic-inorganic perovskites. Nat. Rev. Mater. 2, 16099 (2017).
-  Tsai, H. et al. High-efficiency two-dimensional ruddlesden-popper perovskite solar cells. Nature 536, 312–317 (2016).
-  Yuan, M. et al. Perovskite energy funnels for efficient light-emitting diodes. Nature Nanotech. 11, 872–877 (2016).
-  Wang, N. et al. Perovskite light-emitting diodes based on solution-processed self-organized multiple quantum wells. Nature Photon. 10, 699–704 (2016).
-  Odenthal, P. et al. Spin-polarized exciton quantum beating in hybrid organic-inorganic perovskites. Nature Phys. 13, 894–899 (2017).
-  Zwanenburg, F. A. et al. Silicon quantum electronics. Rev. Mod. Phys. 85, 961 (2013).
-  del Alamo, J. A. Nanometre-scale electronics with III-V compound semiconductors. Nature 479, 317–323 (2011).
-  Hwang, H. Y. et al. Emergent phenomena at oxide interfaces. Nature Mater. 11, 103–113 (2012).
-  Haruyama, J., Sodeyama, K., Han, L. & Tateyama, Y. Surface properties of CHNHPbI for perovskite solar cells. Acc. Chem. Res. 49, 554–561 (2016).
-  Murali, B. et al. The surface of hybrid perovskite crystals: A boon or bane. ACS Energy Lett. 2, 846–856 (2017).
-  Mitzi, D. B., Feild, C. A., Harrison, W. T. A. & Guloy, A. M. Conducting tin halides with a layered organic-based perovskite structure. Nature 369, 467 (1994).
-  Kagan, C. R., Mitzi, D. B. & Dimitrakopoulos, C. D. Organic-inorganic hybrid materials as semiconducting channels in thin-film field-effect transistors. Science 286, 945 (1999).
-  Blancon, J.-C. et al. Extremely efficient internal exciton dissociation through edge states in layered 2d perovskites. Science 355, 1288–1292 (2017).
-  Vurgaftman, I. & Meyer, J. R. Band parameters for III–V compound semiconductors and their alloys. J. Appl. Phys. 89, 5815 (2001).
-  Bimberg, D., Grundmann, M. & Ledentsov, N. N. Quantum Dot Heterostructures (John-Wiley and Sons, 1999).
-  Ruddlesden, S. N. & Popper, P. New compounds of the KNiF type. Acta Cryst. 10, 538–539 (1957).
-  Ruddlesden, S. N. & Popper, P. The compound SrTiO and its structure. Acta Cryst. 11, 54–55 (1958).
-  Stoumpos, C. C. et al. Ruddlesden-popper hybrid lead iodide perovskite 2d homologous semiconductors. Chem. Mater. 28, 2852–2867 (2016).
-  Davies, J. H., Bruls, D. M., Vugs, J. W. A. M. & Koenraad, P. M. Relaxation of a strained quantum well at a cleaved surface. J. Appl. Phys. 91, 4171–4176 (2002).
-  Blancon, J.-C. et al. Strongly bound excitons in ruddlesden-popper 2d perovskites, arXiv:1710.07653.
-  Kane, E. O. Semiconductors and Semimetals (Academic Press, 1966).
-  Lemmerer, A. & Billing, D. G. Synthesis, characterization and phase transitions of the inorganic-organic layered perovskite-type hybrids [(CHNH)PbI], n=7, 8, 9 and 10. Dalton Trans. 41, 1146–1157 (2012).
-  Liu, Z. et al. Crystal structure of bis(4-chloroanilinium) tetraiodoplumbate(ii), (ClCHNH)PbLi. Z. Kristallogr. NCS 219, 457–458 (2004).
The work in France was supported by Agence Nationale pour la Recherche (TRANSHYPERO project) and was granted access to the HPC resources of [TGCC/CINES/IDRIS] under the allocation 2017-A0010907682 made by GENCI. The work at Los Alamos National Laboratory (LANL) was supported by LANL LDRD program (J-C.B., W.N., S.T., A.D.M.) and was partially performed at the Center for Nonlinear Studies. The work was conducted, in part, at the Center for Integrated Nanotechnologies (CINT), a U.S. Department of Energy, Office of Science user facility. Work at Northwestern University was supported by grant SC0012541 from the U.S. Department of Energy, Office of Science. C.C.S. and M.G.K. acknowledge the support under ONR Grant N00014-17-1-2231.
M. K., S.T. and C.K. conceived the idea, designed the work, and wrote the manuscript. J.E developed the semi-empirical BSE approach and the elastic model. M.K. performed the DFT calculations with support from B.T. and L.P. C.K. and M.K. analysed the data and provided insight into the mechanisms. M.G.K. and C.S.S. lend their expertise in chemistry. A.D.M., J.C., H. T. W. N. supplied knowledge from an application perspective. All authors contributed to this work, read the manuscript and agree to its contents, and all data are reported in the main text and supplemental materials.