Recent progress of two‐dimensional lead halide perovskite single crystals: Crystal growth, physical properties, and device applications

State Key Discipline Laboratory of Wide Band Gap Semiconductor Technology, Shaanxi Joint Key Laboratory of Graphene, Advanced Interdisciplinary Research Center for Flexible Electronics, School of Microelectronics, Xidian University, Xi'an, China Key Laboratory of Applied Surface and Colloid Chemistry, Ministry of Education, Shaanxi Key Laboratory for Advanced Energy Devices, Shaanxi Engineering Lab for Advanced Energy Technology, Institute for Advanced Energy Materials, School of Materials Science and Engineering, Shaanxi Normal University, Xi'an, China Dalian National Laboratory for Clean Energy, iChEM, Dalian National Laboratory for Clean Energy, Dalian Institute of Chemical Physics, University of the Chinese Academy of Sciences, Chinese Academy of Sciences, Dalian, China

Perovskite material was discovered by Gustav Rose in 1839 and named after the Russian mineralogist Lev Perovski. [16][17][18] "Perovskites" does not refer specifically to the complex oxides of calcium and titanium (CaTiO 3 ), but rather to a series of compounds with the chemical formula ABX 3 . For the organic-inorganic halide perovskite, A in ABX 3 is usually an organic group, such as CH 7 N 2 + (FA) or CH 3 NH 3 + (MA), B is a metal cation, like Pb 2+ or Sn 2+ , and X is a halide anion, such as Cl − , Br − , or I − . [19][20][21][22] Compared to the structure of three-dimensional (3D) perovskite, the chemical formula of two-dimensional (2D) homologous perovskite is (A 0 ) m A n − 1 B n X 3n + 1 , where A 0 is a monovalent (m = 2) or divalent (m = 1) long-chain organic cation, and the elements represented by A, B, X are the same as those of 3D perovskite ( Figure 1A). 24,25 The relationship between 2D and 3D perovskites contains two aspects: 1 In 3D perovskite, octahedral BX 6 4− is connected in a corner-sharing way and forms the cage in which the small organic group A is locked. 19 The tolerance factor (TF) can be used to reflect the stability and distortion of the 3D perovskite crystal structure, and the octahedral factor (OF) is used to determine whether the B site has the right size to fit the X 6 octahedron, and it is also an important factor in evaluating the structural stability. According to 26 : where r A , r B , and r X are the radius of the organic cation, metal cation, and halogen anion, respectively. The TF value of a typical 3D perovskite is 0.8 to 1. 20 The OF value of the stable 3D perovskite structure should be between 0.442 and 0.895. 27 When the larger organic group A 0 is substituted for the smaller group A, the cage is broken (TF deviates from the range of 3D perovskite) and a 2D perovskite of formula A 0 m BX 4 is formed in which the layers of the organic group are surrounded by two layers of corner-sharing metal halide octahedra, like a sandwich. 28 2 From the point of view of the formula, when n is equal to 1, the formula becomes A 0 m BX 4 . When n equals 2, the bread on each side changes from a corner-sharing octahedron to a single layer of perovskite. When n equals 3 or 4, the number of perovskite layers on each side is respectively 2, 3, and so forth (when n is not equal to 1 or infinity, it is called quasi-2D perovskite [29][30][31][32] ); as n approaches infinity, by ignoring the layer of the large organic group on the edge of the structure, the formula is (ABX 3 ) n , with a common 3D perovskite single crystal lattice structure. Note that organic and inorganic layers are stacked in turn to form a graphene-like layered structure, and the interlayer interaction is via van der Waals forces. 33 The 2D perovskite is a very large material system. According to octahedral connection mode, it can be divided into three types: (100)-oriented, (010)-oriented and (001)-oriented 2D perovskites (Table 1). 25 The (110)-oriented material requires small, highly symmetric A 0 cation to maintain structural stability. 38,39 (111)-oriented material is a class of defective 2D perovskite lacking metal cations (B sites). 40 The (100)-oriented 2D perovskite 25,41 is widely studied at present and can be divided into three types (determined by the cation at the F I G U R E 1 (PEA) 2 (MA) n − 1 Pb n I 3n + 1 perovskite as an example. A, Schematic diagram of the unit cell structures for different n values, which shows the evolution of dimensionality from 2D (n = 1) to 3D (n = ∞). B, DFT simulation of the formation energy of perovskites with different n values in different atmospheres. Source: Reproduced with permission: Copyright 2016, American Chemical Society 23 A 0 site): the Ruddlesden-Popper (RP) phases, 28, 42 the Dion-Jacobson (DJ) phases, 43,44 and the alternating cation in the interlayer space (ACI) type. 45,46 (a) RP phases. The octahedral element between layers is offset by (1/2, 1/2), like (BA) 2 (MA) 2 Pb 3 I 10 . 34,47 (b) DJ phases. In DJ phase (3AMP)(MA) 2 Pb 3 I 10 , the octahedrons do not show any shift (0, 0), but are perfectly stacked. 35 (c) ACI type. The octahedron shows the displacement of (1/2, 0). 35 In addition to conventional 2D perovskite materials, the lead-free double perovskite, like typical symmetrical face-centered cubic Cs 2 AgBiBr 6 which has a formula form of A 2 B I B 0III X 6 , can be considered as another type of 3D perovskite ABX 3 . [48][49][50][51][52] However, it is worth noting that the 2D double perovskite structure, such as (BA) 4 (AgBi)Br 8 ((100)-orientation), the octahedron formed by two metal cations is alternately arranged to form an inorganic layer, and then form 2D structure with large organic cation layer. 53 In recent years, great progress has been made in the preparation and application of 2D perovskite materials, such as the simple spin-coating method to prepare various perovskite films for photovoltaics, photodetectors, and other applications. [54][55][56][57][58][59][60] However, the resulting polycrystalline film has a stability problem and is easily eroded by oxygen and humidity ( Figure 1B). 58,[61][62][63] Therefore, it is very important to find an efficient method to prepare 2D single crystals with good air and humidity stability. Poglitsch and Weber first reported that 3D single-crystalline perovskites can be produced by cooling an HX-based solution containing perovskites in 1987. 64 The preparation method of 3D perovskite single crystals has gradually matured: Huang's group applied a top-seeded solution growth (TSSG) method; Cheng's and Soek's groups used antisolvent-assisted crystalline methods; Bakr's group developed the antisolvent vapor-assisted crystallization (AVC) method 17, 65, 66 ; Liu's group discovered the inverse temperature crystallization (ITC) method 13 ; and so forth. For 2D perovskite, the growth of microcrystalline thin films (MCTFs) and nanocrystalline materials has been reported. However, although researchers put forth considerable effort, they failed to prepare large-area 2D single crystals.
In this review, the methods for preparing 2D perovskite single crystals have been summarized. Moreover, the device applications of 2D perovskite single crystals are discussed, and future research directions and developments are proposed.

| METHODS OF 2D PEROVSKITE CRYSTALLIZATION
The strategies and basic principles of 2D perovskite single-crystal growth are introduced. Some crystal formation mechanisms (steady nucleation rate and crystal growth rate) are given. In this review, the 2D perovskite refers to organic and inorganic lead halide 2D perovskite unless stated otherwise.

| Methods of preparation of 2D perovskite single crystals
The most widely used method to prepare organicinorganic lead halide 2D perovskite single crystals is the solution process. Compared with the perovskite polycrystalline films, the growth of single crystal is more difficult and takes longer. The growth of single crystals is based on the solubility variation vs temperature or solvent content, but the conditions need to be well controlled. Several methods are introduced for the growth of 2D perovskite single crystals.

| Cooling HX-based precursor solutions/acid precipitation method
This method requires that the solubility of the perovskite in HX-based (X = Cl, Br, I) solutions is positively correlated with temperature. [67][68][69] Poglitsch and Weber used this cooling method to grow 3D perovskite single crystals in 1987. 64 They gradually decreased the temperature of an HX-based perovskite precursor solution from a high temperature (>90 C) to room temperature to obtain single crystals of 2D or 3D perovskite. The cooling rate affects the growth of single crystals greatly. By cooling the boiling HX-based precursor to room temperature, a series of homologous 2D perovskite crystals, such as (CH 3 (CH 2 ) 3 NH 3 ) 2 (CH 3 NH 3 ) n − 1 Pb n I 3n + 1 ((BA) 2 (MA) n − 1 Pb n I 3n + 1 ) (n = 1, 2, 3, 4, ∞), were prepared by the Kanatzidis group. 34 Rapid cooling leads to multiple nucleation sites with the formation of a large number of small crystals. Therefore, strict control of the cooling rate is important for obtaining high-quality, large, single crystals. Bakr's group prepared (PEA) 2 PbI 4 ·(MAPbI 3 ) n -1 (n = 1, 2, 3) single crystals by cooling 90 C precursor solutions to room temperature at a rate of 1 C/h. 70 The size of the single crystals reached the millimeter level ( Figure 2A). However, when n is greater than 3, the 2D perovskite longer forms a single crystal, but grows in the form of a mixture, due to the large solubility difference between PEA and MA in the same solvent. Kanatzidis et al prepared millimeter-sized plate-like crystals of (3AMP) (MA) n − 1 Pb n I3 n + 1 and (4AMP)(MA) n − 1 Pb n I3 n + 1 by this cooling method ( Figure 2B). 35  The induced peripheral crystallization (IPC) method proposed by Liu's group utilizes prior edge evaporation to create seed crystals. The solubility of perovskite in γ-butyrolactone (GBL) solvent increases with temperature ( Figure 3A,B). 73 Perovskite single crystals prepared by conventional methods are usually in bulk form, but IPC method provides a good way to prepare single crystal membranes with controllable thicknesses and growth in the parallel orientation ( Figure 3E). The apparatus for IPC is simple: a uniform substrate, a top glass, and a thermostat. The prepared saturated solution is dropped onto the preheated substrate, covered with the top glass, and placed at a constant temperature. When GBL volatilizes at the edge of the two glass plates, the inner solution is sealed by the edge crystals. At this time, when the temperature decreases, the solubility of perovskite decreases and crystal growth begins at the edge (the specific steps are shown in Figure 3D). Using this method, Liu's group grew (PEA) 2 PbI 4 single-crystal membranes (SCMs) on glass and polyethylene terephthalate (PET) substrates, and the size of the single crystals reached centimeters (>75 × 35 mm 2 ) ( Figure 3C). This IPC method limits the growth height of the single crystal in a confined space to obtain 2D perovskite single crystals on the flexible substrate (PET) ( Figure 3F,G). Therefore, it is feasible to apply it to flexible devices. As the temperature decreases, the solubility of perovskite decreases to supply the growth of precipitated crystals. However, cooling too quickly will lead to multiple crystal nucleation sites, so large perovskite single crystals cannot be obtained. Huang's group synthesized large-area quasi-2D perovskite thin single crystals BA 2 MA 2 Pb 3 I 10 by the space-confined method. 74 In perovskite precursor, perovskite ions form complexes with solvents. Nonwetting substrates differ from wetting substrates in that they are less attractive to the perovskite ions in the attraction complex, and thus the horizontal transport of ions along the confined space can be improved, which is beneficial for the large-scale growth of 2D perovskite in the horizontal direction (Figure 3H-K). Fu's group prepared (BA) 2 (MA) n − 1 Pb n X 3n + 1 (n = 1, 2, 3) single-crystal films with millimeter lateral size and nanometer thickness by modifying the growth method of the space-constrained aqueous solution. 76 The precursor is an HI-based solution. When the H 2 O and HI in the precursor on two substrates are completely evaporated at 80 C, 2D perovskite single crystal membranes can be obtained.
Song's group adopted an inkjet printing method in which they formed perovskite seeds on the substrate, put the seed stamp on the solution added to the substrate, F I G U R E 3 Legend on next page. and grew perovskite single crystal films at a fixed site ( Figure 3L). 75 This method is applicable to the growth of various perovskite monocrystalline films and can change the distribution of perovskite precursor ions by affecting mass transport, thus effectively inhibiting random nucleation. The growth location of the single crystal film is related to the seed, and random single-crystal growth is inhibited.

| AVC method
The AVC method proposed by Cheng's and Soek's groups, also known as the solvent engineering approach and fast crystallization deposition, is used to prepare perovskite films. 65,66 By taking advantage of the solubility difference of perovskite in different solvents, the perovskite was dissolved in solvents with high solubility (such as N,N-dimethylformamide [DMF], DMSO, GBL), and the antisolvent with low solubility (such as chlorobenzene, benzene, diethylether) was transported to the solution to crystallize perovskite. 77 When perovskite films nucleate, the antisolvent can be added directly. When a 2D or 3D single crystal nucleates, the antisolvent vapor is continuously transported to the solution to promote crystallization ( Figure 4A). Ren's group reported the preparation of (PEA) 2 PbBr 4 perovskite single crystals by the AVC method. DMF was used as the solvent of (PEA) 2 PbBr 4 precursor solution which was stored in an atmosphere of chlorobenzene vapor. 78 After several days, millimeter-sized (PEA) 2 PbBr 4 2D perovskite single crystals could be synthesized by controlling the rate of the antisolvent vaporization ( Figure 4B-E). The key point of this approach is to find the optimal rate of the antisolvent vaporization. Evaporation that is too fast will cause multiple crystal nucleation sites and even affect the crystal quality, while evaporation that is too slow will prolong the growth cycle. In Deleporte's group, (PEA) 2 PbI 4 single crystal was prepared by combining the AVC method with a "space-confined" method as the "Antisolvent Vaporassisted Capping Crystallization" (AVCC) method (Figure 4F,H). The crystallization of (PEA) 2 PbI 4 perovskite with GBL as the solvent and dichloromethane (DCM) as the antisolvent is better than that with DMF as the solvent ( Figure 4G,I). 79

| Slowly evaporating at constant temperature/controlled-evaporation method
With the solvent evaporating, the concentration of the solute increases gradually. When the concentration reaches the critical saturation concentration, the solute begins to precipitate, which is the principle used in the controlled-evaporation method. As the solvent continues to evaporate, the solute continues to crystallize. The solute concentration remains at the saturated state for subsequent continuous crystallization.
Based on this simple physics principle, Raghavan et al grew RP phase (BA) 2 (MA) n − 1 Pb n I 3n + 1 (n = 1,2, and 3) by solution-growth technique of slowly evaporating at constant temperature (SECT). 80 The scale of single crystal is up to millimeter and has good crystallinity and spectral uniformity. Liu's group utilized the controlled this evaporation method to fabricate (PEA) 2 PbBr 4 2D perovskite single crystals at room temperature. 33 By controlling the rate of solvent evaporation at a fixed temperature, a nucleation seed forms in a gradually oversaturated solution. The next step is to control the growth conditions to prevent the formation of other nucleation seeds and ensure a large crystal is growing from the existing nucleation seed. With the temperature fixed at 23 ± 0.5 C, centimeter-sized (≈27 × 11 mm 2 ) (PEA) 2 PbBr 4 2D perovskite single crystals ( Figure 4J-M) can be obtained in the DMF-based precursor after 20 days. By studying the growth process of single crystals, they found the process is divided into two stages: there is no crystal formation in the first stage, where only the solvent evaporates; in the second stage, the solution becomes oversaturated and crystallization is observed, with the rate of the solvent (DMF) evaporation lower than its counterpart was in the first stage. At the same time, the fixed growth temperature of the single crystals needs to be optimized. If the F I G U R E 3 A, Solubility of (PEA) 2 PbI 4 SCM in GBL as a function of temperature. B, Solubility derivative with respect to temperature for (PEA) 2 PbI 4 SCM in GBL. C, Photo of the (PEA) 2 PbI 4 SCM sample (73 × 35 mm 2 ). D, Schematic illustration of the IPC procedure to grow 2D layered (PEA) 2 PbI 4 SCM. E, Photos of a (PEA) 2 PbI 4 SCM taken at different stages of the growth process. F, Photo of a piece of (PEA) 2 PbI 4 SCM, 3.5 μm in thickness, wrapped around a small tube (1.6 cm in diameter) to show its flexibility. G, A photo showing the flexing angle measurement for the (PEA) 2 PbI 4 SCM. H, Schematic of the experimental procedure: 20 μL of (BA) 2 (MA) n − 1 Pb n I 3n + 1 hydriodic acid solution drop on a 2 × 2 μm glass substrate covered with another. The device was placed in a drying oven at 80 C until all H 2 O and HI volatilized. I-K, Bright field microscopy images and fluorescence pictures (inset) of (BA) 2 (MA) n − 1 Pb n I 3n + 1 (n = 1, 2, 3) singlecrystal films, respectively. L, Schematic illustration of the scalable growth of single-crystal perovskite thin film arrays. Scale bar, 1 mm Source: A-G, Reproduced with permission: Copyright 2018, Springer Nature Publishing AG. 73  temperature is raised to 30 C, a large number of stacked crystallites will be produced.

| Surface-tension-controlled crystallization
Via the difference between the growth rate on the surface of the solution and the growth rate in the solution, Liu's group obtained some inch-sized single-crystal blocks ( Figure 5F). 81 By using the same solute and solvent as those in the IPC method and using different methods to control the surface tension and a hot stage, (PEA) 2 PbI 4 block crystal with a regular shape was grown ( Figure 5A-C). The solubility of (PEA) 2 PbI 4 materials in γ-butyrolactone (GBL) solvent increases with increasing temperature. With decreasing temperature (cooling rate of 0.5 C/h from 95 C), the solubility of (PEA) 2 PbI 4 in the prepared solution decreases, and the crystal grows on the surface of the solution due to the fast transfer at the surface ( Figure 5D). The growth rate on the solution surface is higher than that in the solution. As the temperature drops further, the crystals grow so that the weight of the crystals exceeds the buoyancy, and the crystals sink completely into the solution, falling to the bottom of the bottle ( Figure 5E). The cooling process continues down to 30 C, and large crystal blocks of regular shape can be obtained.
Priya's group prepared a series of perovskites (C 4 H 9 NH 3 ) n (CH 3 NH 3 ) n − 1 Pb n I 3n + 1 (n = 1,2,3,4,∞) from 2D to quasi-2D and then 3D, and studied the nucleation mechanism at the air-water interface in 2016. 82 The nucleation rate at the surface of the solution is greater than that in the solution. This conclusion is also confirmed in the 3D perovskite. 83

| Models of the crystal growth
Crystal growth in supersaturated solution can be divided into two stages: (a) the formation of the crystal nucleus; and (b) the formation of a large single crystal by the continuous growth of the crystal nucleus. The nucleation stage in perovskite crystals is largely different from the classical nucleation in solidification. In solidification, F I G U R E 5 Crystallization of (PEA) 2 PbI 4 perovskite single crystals. A, The Gibbs free energy change ΔG total as a function of particle radius. B, Graph illustrating the lower nucleation barrier for the solution surface compared with that in the solution volume. C, Schematic of the single crystal staying afloat on the solution surface. D, Schematic of the surface tension-controlled crystallization process. E, Photographs of the (PEA) 2 PbI 4 perovskite single crystals grown at different temperatures. F, Corresponding photographs of (PEA) 2 PbI 4 perovskite single crystals completed at different temperatures Source: Reproduced with permission: Copyright 2019, Elsevier Inc 81 molecules come together to directly form a crystal nucleus, which can be totally reversed to molecules if the experimental condition reverses. However, molecules assemble to form irreversible micelles, and then the micelles join together to form a reversible crystal nucleus in perovskite crystallization. The second stage, which affects the quality of single crystals, is controlled by two aspects: (a) the solute diffuses to the crystal surface in the F I G U R E 6 Legend on next page. solution, and (b) the solute deposits on the crystal surface.
By using density functional theory (DFT) and classical nucleation theory, the crystallization phenomenon can be further understood. 82 The crystallization process is divided into two stages, nucleation and crystal growth, corresponding to the model of single crystal suitable for solution growth described below.

| The steady-state nucleation rate
When the solution is oversaturated, the solute will precipitate out to form small micelles, and the micelles assemble to form a nucleus. Therefore, by vaporizing the solute or changing the temperature, we can control the solution to reach the oversaturated state and thus obtain the seed crystals and make the crystal grow. Equation (3) is used to describe the stable nucleation rate on the substrate. 84 where ν is the steady nucleation rate, ω is the frequency at which the solute molecules diffuse from the solution to the critical nucleus, Γ is the nonequilibrium Zeldovich factor, N 0 is the density of adsorption sites on the crystal surface, ΔG is the Gibbs free energy change for the nucleus formation, also known as the nucleation barrier, k B is the Boltzmann constant, and T is the solution temperature. From Equation (2), the nucleation rate is affected by many factors and can be divided into two parts: the exponential part exp − ΔG k B T and the nonexponential part ωΓN 0 . The value of ωΓN 0 depends on the supersaturation of the solution. When temperature is the main regulating factor of single crystal growth, the influence of the exponential part on the rate is dominant, and the influence of the nonexponential part can be ignored as a constant. 86,87 As the crystal grows, the exponential part changes with temperature.
The driving force in the crystallization process of a supersaturated solution is related not only to the thermodynamic temperature, but also to the natural logarithmic ratio of the concentration of the supersaturated solution. Regardless of the concentration of the solution, there are small grains precipitated and dissolved in the solution. When the solution is in the supersaturation region, the precipitation is larger than the dissolution, which will immediately cause the formation of small grains with more defects. As it enters the unsaturated zone, precipitation is less than dissolution, and the resulting crystal begins to slowly dissolve in solution. 77 Therefore, it is crucial to control the proper solution concentration on the solubility curve: when the concentration is higher than saturation at a temperature, the nucleation rate increases sharply, and when the concentration is below saturation, the nucleation rate is zero ( Figure 6A). When the supersaturation is higher, the nucleation rate is faster. The mass change of the precursors with evaporation time is recorded while forming single crystal (PEA) 2 PbBr 4 with the controllable evaporation method. 33 The mass loss curve is clearly divided into two segments, and no crystallization is found in the first part ( Figure 6B,C). In the second part, the solution is supersaturated and crystallization is observed. The intersection of these two lines is the critical concentration of nucleation. By controlling the temperature to a certain fixed value, the crystal growth rate remains stable, thus obtaining high-quality single crystals.
In the surface-tension-controlled crystallization method, an interesting phenomenon has been observed experimentally-the formation of a single crystal film on the surface of the solution in the initial state. This is easily explained by a simplified model of a single solute A and a single solvent S. A nanocrystallite contains many solute S molecules, which are the essence of the forming nucleus. It is assumed that a molecule A and j molecules F I G U R E 6 A, Schematic diagram of solubility and supersolubility; B, Mass and concentration of (PEA) 2 PbBr 4 precursor solution as a function of evaporation time; C, Mass and growth rate of (PEA) 2 PbBr 4 as a function of time; D, Schematic illustration of the alignment of butylammonium cation surfactant at the water-air interface for templating the nucleation. E, Experimental setup during nucleation. F, Molecular interaction between precursor molecules (red) and water molecules (yellow). Lower interaction energy is expected for the surface layer molecules due to the surface tension effect compared to those in bulk solution. G, Experimental setup during crystal growth. Graphic illustrations of H, the lower nucleation barrier, I, the larger free energy changes during crystal growth, and J, the higher growth rate of the precursor molecules at the water-air interface compared to those in bulk solution. K, Solubility of (PEA) 2 PbI 4 in GBL as a function of temperature; L, Solubility derivative with respect to temperature for (PEA) 2  S can form an A-S j complex in the solution through the complex binding energy (E c ). 82,83 The concentration of A-S j complex is relative to the energies of A (ε A >0) and S (ε s >0, ε is the cohesive energy), which is determined by E c . 81 The molecules are subjected to surface tension on the surface of the solution, and E cs , ε As and ε ss are lower than E c , ε A and ε s , respectively. 82 The nucleation barrier in solution can be expressed as 81,82 : The nucleation barrier at the solution surface can be expressed as 81,82 : where γ is the surface energy of crystallite, ε is the cohesive energy, ε A is the energy of the A-molecule, N A is the mole fraction of isolated A, and ε Surf is the surface tension-related elastic energy per A-molecule in the surface layer. On the surface of the solution, the decrease of the A-molecules is due to the surface tension, making ε AS = ε A − ε Surf , and ε Surf > 0. 83 Equations (4) and (5) show that the nucleation barrier in solution, ΔG Solution , is larger than that at the surface ΔG Surface ( Figure 5A, Figure 6H-J). By combining Equations (4) and (5) with Equation (3), it can be concluded that the nucleation rate on the surface of the solution, ν Surface , is higher than that in the solution, ν Solution , so the solvent molecules will form the film on the solution surface first. The existence of surface tension leads to the reduction of surface molecules, which reduces the nucleation barrier and molecular binding energy of the crystal, and the nucleus is formed on the solution-air surface and can serve as a template to guide subsequent crystal growth.

| The crystal growth rate
The steady-state nucleation rate is used to predict the deposition of micelles on the crystal surface on the microscopic scale, and the crystal growth rate is used to control the crystal growth from the macroscopic change of crystal quality. The growth rate of the crystal determines the quality of the crystal. There is a general understanding that the lower the growth rate, the better the quality of the crystal. 77,82 That is why perovskite films fabricated by spin coating are polycrystalline-the solution in spin coating has a high nucleation rate (multipoint nucleation) and high crystal growth rate (limited grain size).
The continuous growth of crystal includes two processes: solute diffusion from the solution to the crystal surface and solute deposition on the crystal surface. 77 The diffusion rate in the former is an important factor (the diffusion rate varies exponentially with the temperature). For a solution system, control of the diffusion rate can be achieved by controlling the temperature. 89 The solute deposition process is more difficult to control than the diffusion rate. Crystallization kinetics shows that when solute concentration is controlled above the solubility curve (when slightly oversaturated), high-quality crystals can be grown on already-formed crystals or seeds without excess nucleation (nucleation point). 90 According to the oversaturation model in Equation (6), the growth rate of the crystal mass is directly proportional to the first-order derivative of solubility and the temperature ramp rate. 91 where m is the mass of the crystal, C is the solution concentration, and V is the volume of the solution, which changes very little throughout the crystallization process, so it can be regarded as a constant. Take the example of (PEA) 2 PbI 4 single crystals growing by the induced-peripheral-crystallization (IPC) method of cooling GBL-based precursor solution. The solubility of (PEA) 2 PbI 4 in GBL is positively proportional to temperature ( Figure 6K), 73 and its first derivative with respect to temperature is fitted and shown in Figure 6L. Figure 6L shows that the second derivative of solubility changes greatly from 3.45 × 10 −4 to 8.47 × 10 −4 with respect to temperature near 75 C. By choosing 25 C to 80 C as the temperature range, the first derivative of solubility with temperature can be expressed as 73 : where k is approximately a fixed value of 3.45 × 10 −4 , and b is the intercept of the fitting line with a fixed value.

T A B L E 2
Shifts of the emission peak with the wavelength of excitation light from small to large The temperature drop rate is fixed at a constant value of 1 C h −1 . Since the volume, V, and the rate of temperature change with time, dT/dt, during crystal growth can be regarded as constants, the change of crystal mass with time depends on the value of the first derivative of solubility with respect to temperature in solution. Therefore, the crystal growth rate can be simplified as 73 : where A is a simplified constant. As the temperature drops from 80 C to 25 C at a rate of 1 C h −1 , the rate of crystal growth is slowing. The decreasing growth rate can not only produce high-quality single crystal, but also restrict the growth of single crystal thickness, control the growth rate in different directions, and make the single crystal grow along the side selectively. 77 High-quality single crystals can be obtained by choosing the proper temperature ramp rate. Because the temperature of the solution can be well controlled, the growth of high-quality single crystals can be obtained. 73 Most of the preparation of single crystal is by the solution method. By adjusting the temperature, the solution concentration can be maintained at a slightly supersaturated condition, so as to obtain high-quality single crystal. However, the controlled-evaporation method keeps the solution concentration in the supersaturated zone while the crystal grows (the solvent decreases) and the solvent evaporates continuously (at a certain temperature). At the beginning, the solution concentration is lower than the solubility and in the unsaturated zone. From Figure 6B, it can be seen that at 23 C, the solvent evaporation rate changes from 3.77 to 2.78 mg/h, and the solution concentration changes from 0.873 to 1.073 g/mL. 29 The intersection of the two lines in Figure 6B is the critical saturation of the solution at 23 C, which indicates that the solution is in the slightly supersaturated region when the crystal is growing.
Regardless of the growth method, a stable growth rate is important for obtaining high-quality single crystals. Whether through temperature regulation or solvent evaporation, it is based on the regulation of the solubility curve region corresponding to the solution concentration. Maintaining a slightly supersaturated solution concentration inhibits excessive nucleation while preventing the dissolution of the already-grown nuclei and achieves a slower growth rate.

| Applications of 2D perovskite single crystals
2D perovskite single-crystal not only has the excellent photoelectric properties of three-dimensional perovskite, but also has better moisture resistance because of its larger hydrophobic organic cation group, making it a better choice for various devices, such as memory and photodetectors. Several applications of 2D perovskite singlecrystals are summarized in the following section.

| Optoelectronic properties
In this section, we introduce some fundamental photoelectric properties of semiconductor materials that are of great benefit to the rational application of 2D perovskite single crystal materials. Because they have fewer grain boundaries and defects, single crystals have better photoelectric properties than MCTFs, such as lower trap density, longer carrier lifetime, and so forth.

Electrical properties
The trap-state density (n trap ) is the key parameter of semiconductor material quality. The lower the value, the better quality of the perovskite single crystal. To evaluate the quality and electrical properties of perovskite single crystals, the space charge limited current (SCLC) method is used by measuring the current-voltage (I-V) characteristics in the dark. This curve is divided into three regions: (a) The electrode is in ohmic contact with the perovskite, and the current changes linearly with the voltage; (b) the current increases nonlinearly with the change of voltage. At this stage, the trap states inside the perovskite are filled with injected carriers; and (c) the current is proportional to the square of the voltage, when the carrier drift velocity is saturated. There is a kink point at the intersection of the ohmic region and the trap-filled limit area. At this time, the trap states inside the material are just filled by the injected carriers, and there are no excess injected carriers. Therefore, n trap can be calculated from the trapfilled limit voltage (V TFL ) 92,93 : where e is the electron charge, L is the device thickness, ε is the relative dielectric constant of the perovskite, ε 0 is the vacuum permittivity, and n trap is the electron-trap density or hole-trap density, depending on whether it is an electron-only device or a hole-only device. The carrier lifetime (τ) is the lifetime of the nonequilibrium carriers produced by photoinjection and electroinjection, reflecting the crystal defects, and is defined as the average time of carrier existence. The equation is 88 : where τ r is the radiative lifetime, and τ nr is the nonradiative lifetime. The main factors affecting the lifetime of nonequilibrium carriers are the recombination mechanisms of carriers (direct recombination, indirect recombination, surface recombination, Auger recombination, and so forth). The recombination centers caused by harmful impurities and defects in materials greatly influence the nonequilibrium carrier lifetime. In order to increase the nonequilibrium carrier lifetime, harmful impurities and defects should be removed. Single-crystalline materials have fewer defects and thus have longer lifetimes than MCTFs. 73 Transient absorption spectra and time-resolved fluorescence spectra are applied to measure the fluorescence lifetime. The longer the fluorescence lifetime, the fewer defects and the better crystallinity of the material.
Carrier mobility, μ, is an important parameter of semiconductors, and it describes the motion of carriers in an electric field. Two pathways can take to measure μ: one utilizes the I-V curve of the SCLC, from which it can be calculated according to Mott's SCLC theory in the third stage 3,94 ; the second is to determine the type of semiconductor conduction, as well as the concentration and mobility of the majority carriers by the Hall effect.

Optical properties
By analyzing the absorption spectrum, we can determine the absorption onset and bandgap of the (PEA) 2 PbX 4 single crystals. For example, when X changes from Br to I, the absorption onset of the absorption spectrum redshifts from 428 nm to 550 nm, which means that (PEA) 2 PbI 4 has a wider range of visible light absorption than (PEA) 2 PbBr 4 . 33 The position of the valence band edge relative to the vacuum energy level can be estimated by using the results of ultraviolet photoelectron spectroscopy (UPS) measurements. Combining this with the bandgap, the value of the conduction band edge can be further obtained.
The photoluminescence (PL) quantum yield is defined as the ratio of the number of photons emitted by a luminous substance after absorption to the number of photons absorbed from the excitation source. The smaller the full width at half maximum (FWHM) of the PL spectra, the better consistency of the substance and the higher color purity and narrower emission bandwidth of the sample. Dou et al showed that the stable PL peak of 2D perovskite single crystal varies with the thickness of the single crystal. 95 When the thickness of the single crystal increases, the blue shift trend appears in PL peak.
It is worth noting that the emission peaks of both (PEA) 2 PbBr 4 and (PEA) 2 PbI 4 shift with the change of the excitation wavelength. This characteristic feature can also be observed in MCTFs (Table 2). 33,73 (Table 3) summarizes basic optoelectronic properties of single crystal (SC), microcrystalline thin film (MCTF), and nanoplates of (PEA) 2 PbI 4 and (PEA) 2 PbBr 4 . Raghavan et al explored homologous (BA) 2 (MA) n − 1 Pb n I 3n + 1 (n = 1, 2, and 3) materials, and they believed that the low-threshold lasing behaviors with different emission wavelengths at room temperature might be the cause of this phenomenon. 80

| Solar cells
At present, 3D perovskite polycrystalline or nanocrystalline thin films are mostly used as the light absorption layers in PSCs, and the device power conversion efficiency (PCE) has already exceeded 25%. 1 However, a number of key problems, such as the lack of resistance to water, light and heat, and many grain boundaries and defects of polycrystalline and nanocrystalline thin films, have hindered large-scale industrial production of PSCs. PSCs with single-crystal thin film as the light absorption layer have mainly been constructed with MAPbI 3 5, 96 and MAPbBr 3 . 6,97,98 Although the PCE of 3D single-crystal PSCs has increased rapidly to 21.09% in the past 23 years, it is still far lower than that of polycrystalline silicon cells. [99][100][101] For pure 2D perovskite single-crystals, however, the wide band gap (>2.3 eV) makes them less suitable for photovoltaic applications. 102 Karunadasa's group first proposed the method of mixing 2D perovskite with conventional 3D perovskite in a single absorption layer to overcome the high bandgap value, reduce the exciton binding energy, and improve the efficiency of charge collection. 61 They applied (PEA)2PbI 4 ·(MAPbI 3 ) 2 as the light absorption layer. And the PCE was 4.73% (AM1.5G) with V oc of 1.18 V. Meanwhile, the device exhibited good stability for over 46 days in air at a relative humidity level (RH) of 52%. In the study of homologous (CH 3 (CH 2 ) 3 NH 3 ) 2 (CH 3 NH 3 ) n − 1 Pb n I 3n + 1 family, Stoumpos et al showed that when n < 5, the crystal is more likely to grow freely along the plane of the inorganic layer and grows slowly in the direction of vertical accumulation; when n = 5, the crystals appear to accumulate vertically ( Figure 7C). 103 The DFT calculation of the noncentrosymmetric crystal showed that BA 2 MA 4 Pb 5 I 16 (n = 5) is a direct band gap with or without spin-orbit coupling (SOC) (Figure 7A,B). By using the hot-casting device fabrication process, the prefabricated BA 2 MA 4 Pb 5 I 16 crystals were redissolved in DMF to achieve a planar photovoltaic device, and the PCE reached 8.71% (Figure 7D-F). Overall, the quasi-2D perovskites are used to improve the charge collection efficiency of PSCs, and enhance the device stability under environmental conditions. 47, 53, 104, 105

T A B L E 3
Basic optoelectronic properties of single crystal (SC), microcrystalline thin film (MCTF), and nanoplates of (PEA) 2  In addition, since the orientation of spin-coated polycrystalline films is random, many efforts have been made to optimize the crystal orientation and increase the grain size to reduce the effect of defects on solar cell performance. 62,106 For 2D perovskite crystal growth, it is better to make the grain orientation more favorable for carrier transport. However, the growth direction of 2D RP phase perovskite single crystal (n < 5) tends to grow along (00 hour) plane preferentially. 33,92,103 Therefore, it is worth exploring the method of directly growing quasi-2D single crystal with favorable electron and hole transport direction to reduce defects and improve device performance. Moreover, the synthesis of n > 5 (the growth direction of the trend in the horizontal and vertical direction is not large) quasi-2D perovskite single crystal should also be tried.

| Memories
Ren's group used 2D Ruddlesden Popper phase hybrid lead bromide perovskite single crystals ((PEA) 2 PbBr 4 with bandgap of $2.9 eV) as materials for low-working-current nanodevice applications. 78 The 2D perovskite single crystal is fabricated by the AVC method, stripped into thin layers and combined into a graphene/2D crystal/Au vertical structure to study the electrical properties in resistance memory elements ( Figure 8A). Monte Carlo simulations 110 based on Br − ions and vacancy movement to form filaments supported the TEM observations ( Figure 8B-D). The resistive memory has the lowest operation current (10 pA, Figure 8E), which can overcome the disadvantage of the large dark current of the current resistive memory materials, such as HfO x , 111,112 AlO x , 113 TaO x , 114 TiO x , 115 MoS 2 , 116 and so forth ( Figure 8F). In addition, the memory also demonstrated 400 fJ/spike synaptic operation, very close to the energy consumption of biological synapses. Therefore, the 2D perovskite single crystal can be used to realize very low energy consumption in neuromorphic computation for massively parallel information processing similar to that of the human brain. In the future, the back-propagation neural network based on a 2D perovskite resistance memory array will have extremely high energy efficiency, which is the basis of high-performance and low energy applications.

| Photodetectors
Due to the excellent optical properties, including fluorescence and exciton effects, of 2D perovskite single crystals, they are more widely used to make high-performance photodetectors with high responsivity and low dark current.
Peng's group reported the first photodetector based on 2D (C 4 H 9 NH 3 ) 2 PbBr 4 perovskite crystals. 109 The perovskite crystals with the domain size of several to tens of micrometers and the thickness of tens of nanometers ( Figure 8I) were synthesized using a solution-processed method and grown on arbitrary flat substrates. Then, the photodetectors were formed with single-crystalline graphene films as the source-drain top electrodes (Figure 8G-8 hours). When the incident wavelength is below 510 nm (photon energy 2.43 eV), the photocurrent of the device was very large and generated obvious response. At the same time, the optical response of the device depended on the variation of incident optical power and bias voltage. The devices possessed extremely low dark current ($10 −10 A), a high ON/OFF current ratio (up to 10 3 ), and high responsivity ($2100 A/W) ( Figure 8J-L).
Liu's group reported an "induced peripheral crystallization" (IPC) method to fabricate a large-area (73 × 35 mm 2 , exceeding 2500 mm 2 , 0.6 μm thickness) 2D (PEA) 2 PbI 4 perovskite single-crystalline flexible membrane for photosensors in 2018. 73 According to the oversaturation model and the plot of the first-order derivative with respect to temperature of (PEA) 2 PbI 4 , the temperature ramp rate is regulated to 1 C h −1 for growing single crystals along lateral dimensions and limiting the growth thickness. The structure of the flexible photosensors is Au/crystal/Au, and they exhibited very high external quantum efficiency (EQE) of 26 530%, responsivity (R) of 98.17A W −1 and detectivity (D*) as high as 1.62 × 10 15 cm·HZ 1/2 W −1 (Jones) (Figure 9A-F).
In the same year, Liu's group reported another method called the "controlled-evaporation method" utilizing a controlled-evaporation process to grow welldefined large-size (exceeding 200 mm 2 ) 2D (PEA) 2 PbBr 4 single crystals. 33 According to the model of the steadystate nucleation rate, the optimized single-crystal growth temperature is 23 ± 0.5 C. Planar-type UV photodetectors based on (PEA) 2 PbBr 4 single crystals with Au interdigitated electrodes showed excellent detection performance, such as extremely low dark current, high ON/OFF ratio and detectivity, and fast response rate (shown in Table 4) ( Figure 9G-I). Additionally, the environmental and irradiation stability are excellent as shown through humidity testing and UV-source switch testing. The performance of this UV detector is the best among the detectors using zinc oxide (ZnO), 117 titanium dioxide (TiO 2 ) 118 , and other state-of-the-art materials, 94 which paves the way for ultrafast optical computing and optical communications using the large-size, high-quality singlecrystal materials.

| Field-effect transistors
The layered structure of 2D and quasi-2D perovskite results in carrier transport anisotropy. The mobility of carriers in the layered plane composed of inorganic octahedrons is much higher than that perpendicular to the plane (the tunneling probability of carriers through interlaminar organic macromolecules is lower). 119 By taking advantage of this material property, field-effect transistors (FETs) can be made with a surface channel as the transport path. The majority of FETs using perovskite single crystals are 2D FETs due to restriction of the 3D material thickness. 120-122 Chen's group fabricated a series of 2D FETs using (BA) 2 (MA) n − 1 Pb n I 3n + 1 (n = 1, 2, 3) in   Figure 10A), and showed unipolar n-type carrier transport characteristics. 123 As the material changes from 2D to quasi-2D (n value from 1 to 3), the carrier mobility increases ( Figure 10E). This can also be seen in the output characteristic curves ( Figure 10B-D).
Li's group also used the same material (BA) 2 (MA) n − 1 Pb n I 3n + 1 (n = 3) to construct 2D FETs with controlled material phase purity through mechanical exfoliation. 126 Perovskite FETs are combined with the good lightabsorbing properties of the material itself to produce photoresponsive perovskite FETs. Peng's group made the first 2D perovskite single crystal photodetector based on an FET structure ( Figure 10F). 109 Rand's group used graphene to cover 2D I-containing perovskite single-crystal (PEA) 2 PbI 4 to inhibit the spontaneous loss of iodine, reduce crystal degradation and improve the overall stability of the device ( Figure 10G). 124 In addition to I-based materials, there are also many Sn-based perovskite FETs, such as phenylethyl ammonium tin iodide (PEA) 2 SnI 4 , and alkyl ammonium tin iodide (C n H 2n + 1 NH 3 ) 2 SnI 4 (n from 4 to 12). [127][128][129][130][131][132] However, as Sn 2+ is easily oxidized to Sn 4+ in 2D perovskite, the environmental stability is worse than that of Ibased perovskite, so the single-crystal devices are less reported. 133 Li's group prepared (C 6 H 5 C 2 H 4 NH 3 ) 2 CsSn 2 I 7 single-crystal FETs with millimeter size and high fieldeffect hole mobility and proved that the material lacked Sn 4+ with high purity and high quality ( Figure 10H). 125 The introduction of large organic cations enhances the humidity resistance of perovskite and makes the device more stable. The structure of 2D materials is bound to be better than 3D counterparts in FET applications. Table 5 summarizes parameters of single-crystal perovskite-based FETs.

| CONCLUSIONS AND OUTLOOK
2D perovskite single-crystal materials are receiving increasing attention because of their remarkable properties and superior potential in optoelectronic applications. Their most notable features are natural quantum wells and excellent moisture resistance compared with their 3D counterparts.
2D perovskite single-crystal materials have many excellent properties, including long fluorescence lifetime, fewer defects and high photoluminescence quantum yield. It is worth noting that 3D perovskite single-crystal offer the ability to change the elements at three sites (different organic groups, metals, halogens) and to adjust the components to obtain materials with different properties. On the contrary, 2D perovskite materials have two positions of organic molecules, which greatly increase the diversity of the materials. By changing the type of large organic cations, the band gap of 2D materials can be adjusted to obtain desired optoelectronic properties. Therefore, the diversity of 2D perovskite single-crystal elements can be used to obtain different material properties, which further extends the application range of 2D perovskite single-crystal materials.
For 2D perovskite single crystals, the future research directions can be considered from among the following: 1. For some 2D materials, existing fabrication methods require finding a specific solvent to dissolve perovskite. For example, in the IPC method, (PEA) 2 PbBr 4 is dissolved in GBL, and the solubility increases with temperature. To use this method, one needs to find a solvent in which the solubility changes with temperature. Therefore, general solvents will be sought that can be used to make various perovskite single crystals. 2. The growth method needs to be further optimized to facilitate commercialization. Many growth methods of 2D perovskite single crystals have been discussed here, but all of them require a long time to grow large-area single crystals, so the growth time needs to be further optimize and shortened to facilitate future large-scale production. 3. The properties of 2D perovskite materials can be utilized to seek more device applications. The method of fabricating large-area flexible 2D perovskite single crystals paves the way for applications in high-performance, flexible, single-crystalline electronics and wearable devices including displays, touch sensing devices, transistors, and so forth. For 2D/3D mixed PSCs, it is worth exploring to find a method to directly grow quasi-2D perovskite single crystal to reduce defects and improve the power conversion efficiency. Regarding FETs, it has been reported that they are all made on 2D/quasi-2D perovskite single crystals below the millimeter level, so it is necessary to develop simple and large-area single-crystal membranes with controllable thickness for further FET applications.