Cesium-mediated electron redistribution and electron-electron interaction in high-pressure metallic CsPbI3

Electron-phonon coupling was believed to govern the carrier transport in halide perovskites and related phases. Here we demonstrate that electron-electron interaction enhanced by Cs-involved electron redistribution plays a direct and prominent role in the low-temperature electrical transport of compressed CsPbI3 and renders Fermi liquid (FL)-like behavior. By compressing δ-CsPbI3 to 80 GPa, an insulator-semimetal-metal transition occurs, concomitant with the completion of a slow structural transition from the one-dimensional Pnma (δ) phase to a three-dimensional Pmn21 (ε) phase. Deviation from FL behavior is observed upon CsPbI3 entering the metallic ε phase, which progressively evolves into a FL-like state at 186 GPa. First-principles density functional theory calculations reveal that the enhanced electron-electron coupling results from the sudden increase of the 5d state occupation in Cs and I atoms. Our study presents a promising strategy of cationic manipulation for tuning the electronic structure and carrier scattering of halide perovskites at high pressure.

Three-dimensional (3D) halide perovskites of the form ABX 3 (A = organic or inorganic cation, B = metal cation, and X = halide anion) display remarkable optical and electronic properties that find applications for a wide range of technologies such as photovoltaics 1,2 and light-emitting diodes 3 . In these materials, BX 6 octahedra corner-share to form the 3D perovskite structure. Beyond the perovskite phase, BX 6 octahedra can also connect through face-and edge-sharing and extend into 1D and 2D structures. At ambient conditions, the electronic states of the metal-halide sublattice are typically responsible for the valence band maximum (VBM) and conduction band minimum (CBM) of these ABX 3 phases, and govern their optical absorption [4][5][6] . It is believed that the A-site cation in ABX 3 phases only makes a minimal contribution to the band edge states and plays an indirect role in the electronic structures of these materials 7,8 .
Modulating the electronic structures and searching for intriguing electronic states in ABX 3 phases have attracted considerable experimental and theoretical attention, partly motivated by their structural similarity to the perovskite oxides where exciting electronic states such as ferroelectricity 9 , unconventional superconductivity 10,11 , and topological insulating behavior 12 have been discovered. Recent experimental studies observed ferroelectric-like behavior in (MA)PbI 3 (MA = CH 3 NH 3 + transport properties 16 . Lattice compression has also been found to be a useful means for accessing different electronic states [17][18][19][20][21][22][23][24][25] . Metallic-like behavior was experimentally achieved in compressed (MA)PbI 3 , (FA) PbI 3 (FA = CH(NH 2 ) 2 + ), and δ-CsPbI 3 [26][27][28] . Bandgap closure and reopening were observed in the double perovskite Cs 2 Au 2 I 6 at high pressure 29 . However, almost all these high-pressure experimental efforts were conducted near room temperature. Theoretically, it is predicted that under compression halide perovskites could be topological insulators-e.g., cubic CsPbI 3 [30][31][32] , or superconductors-e.g., Cs 2 Tl 2 X 6 33 . At low temperatures, correlated materials can display fascinating phenomena such as various types of metal-insulator and superconducting transitions due to strong electron-electron (e-e), electron-phonon (e-p), and electron-magnon interaction. The carrier transport and scattering mechanisms have well-known temperature dependences. For instance, in simple metals the e-p interaction plays an important role in the carrier transport, leading to a T 5 dependence of resistivity at low temperatures with a linear temperature dependence above the Debye temperature 34 . For interacting fermion systems, the carrier scattering has been well described by the Fermi liquid (FL) model 35 , in which carriers are considered as quasiparticles and the e-e interaction dominates the low-temperature electrical transport, causing a T n (n = 2) dependence of resistivity for 3D metals. In certain correlated materials, FL behavior has been observed to break down, resulting in non-Fermi liquid (NFL) behavior with n < 2, e.g., in the normal state of high-temperature iron-and cuprate-based superconductors [36][37][38][39] . ABX 3 phases were reported to have a Debye temperature of 100-250 K 40,41 . Hence carrier transport behavior at low temperature will be different from that at above the Debye temperature. At ambient pressure, e-p coupling was believed to be mainly responsible for the scattering of the free charge carriers in ABX 3 -based materials [42][43][44] , although a recent study indicated that e-e interaction had contributions to the low-energy excitons in semiconducting (FA) PbI 3 at ambient pressure 45 . Studying the electronic structures and carrier scattering mechanisms in these materials at low temperature combined with high-pressure conditions represents unexplored territory.
In this work, we study the electronic states in CsPbI 3 over a vast pressure-temperature space of 0.1-186 GPa and 2-300 K. With application of pressure to 80 GPa, insulating δ-CsPbI 3 transforms to a semimetallic and then to a metallic state, accompanied by the completion of a sluggish Pnma-to-Pmn2 1 (δ-to-ε) phase transition from a 1D chain structure to a 3D structure. Deviation from a FL-like state is observed as CsPbI 3 enters the metallic ε phase. With further compression to 186 GPa, the deviation gradually disappears, and the metallic phase behaves as a Fermi liquid. Band structure and orbital occupation calculations reveal sudden increase of the Cs-and I-5d state occupation in the ε phase that strengthen the e-e interaction and induce the FL-like behavior. In contrast to ambient conditions where the A-site cation only has an indirect effect on the electronic properties of the ABX 3 phases, the Cs cation plays a direct and prominent role in the electronic structure of CsPbI 3 at high pressure.

Results
We measured the resistivity of CsPbI 3 over a temperature range of 2-300 K and pressures up to 186 GPa ( Figs. 1 and 2). Sample characterization and pressure calibration details can be found in the supporting information ( Supplementary Figs. 1-4). At ambient conditions, the room-temperature resistivity (ρ RT ) is beyond the measurable range of the instrument since δ-CsPbI 3 is a large bandgap (2.5-2.8 eV) insulator 46,47 . With increasing pressure, the value of ρ RT falls into the measurable range and is 4.5 × 10 3 Ω·cm at 21.5 GPa, accompanied by the color of the sample changing from yellow to red and then to black. With further compression to 80 GPa, ρ RT drops dramatically by more than six orders of magnitude, beyond which a linear reduction of log ρ RT at a rate of (5.3 ± 0.4) × 10 −3 Ω·cm/GPa continues up to 186 GPa, suggesting an electronic transition at 80 GPa. During decompression, ρ RT at~20 GPa is comparable with the values during the compression process ( Supplementary Fig. 5). Upon further decompression, ρ RT is beyond the measurable range again, followed by the color of CsPbI 3 changing from black to red and then to yellow. During recompression, ρ RT decreases with a similar trend to the first compression cycle.
The resistivity-temperature (ρ-T) curves of CsPbI 3 at representative pressures are shown in Fig. 2. At pressures below 50 GPa, the temperature dependence of resistivity (dρ/dT) was negative throughout the temperature range measured (2-300 K), indicating an insulating character. By fitting the linear range of the ln ρ-1/T curves at high temperature to the Arrhenius equation, an activation energy of 134.8 ± 5.3, 59.1 ± 2.1, and 29.5 ± 0.7 meV is obtained at 30, 41, and 50 GPa, respectively. Upon further compression to 61 GPa, the resistivity starts to decrease with cooling (dρ/dT > 0, metallic behavior) near room temperature but reverses its temperature dependence back to dρ/dT < 0 below~255 K, consistent with a semimetallic state. The transition temperature shifts to~135 K at 70 GPa. A linear extrapolation suggests that at a critical pressure of 80.5 GPa, the sign reversal of dρ/ dT at low temperature could be fully suppressed. The experimental result at 80 GPa indeed shows a positive dρ/dT throughout the temperature range of 2-300 K, indicating the electronic transition to a metallic state in compressed CsPbI 3 . A previous study observed metallic-like behavior in CsPbI 3 near room temperature at~40 GPa 28 . However, the limited temperature range, missing experimental details, and their inconsistent structural data of the metallic-like phase make direct comparison difficult.
Detailed analysis of the ρ-T curves reveals remarkable behavior in the metallic phase. For T > 110 K, a linear temperature dependence is obtained in all the ρ-T curves above 80 GPa (Fig. 2a), which is consistent with the well-known simple metal model where the e-p interaction dominates the carrier transport. The ρ-T relationship obeys the following equation, where ρ 0 is the residual resistivity, and B and C are the coefficients of T 5 and T-terms, respectively. Surprisingly, the ρ-T curves below 110 K significantly deviate from this T-dependence (Fig. 2b, and Supplementary Fig 6). Instead, ρ(T) is found to be proportional to T n , where the fitted exponent n is 1.75 ± 0.02 at 80 GPa and increases gradually with pressure to 1.98 ± 0.03 at 186 GPa (Fig. 2c). The observation of ρ(T)~T n where n ≤ 2 in compressed CsPbI 3 at low temperature is anomalous. Such behavior has typically been observed in materials such as high-temperature superconductors and heavy fermion compounds [36][37][38][39] , in which the d or f electron shell is partially filled and e-e interaction is important. However, at ambient conditions δ-CsPbI 3 does not have any d-electrons that can readily serve as charge carriers.
To understand the abnormal low-temperature behavior of the high-pressure metallic phase, we studied the structural evolution of δ-CsPbI 3 as a function of pressure using powder X-ray diffraction (XRD) measurements ( Fig. 3a and Supplementary Figs. 7-14). Below 5.4 GPa, δ-CsPbI 3 crystallizes in a 1D double chain structure with a Pnma symmetry where PbI 6 octahedra edge-share along the b-axis (Supplementary Fig. 10), consistent with the reported structure at ambient conditions 28,48 . Upon compressing to 6.7 GPa, new diffraction peaks (marked as red arrows in Fig. 3a and Supplementary Fig. 7) start to appear and more peaks develop with further compression, indicating the emergence of a high-pressure phase (ε-CsPbI 3 ). This δ-to-ε structural transition is found to be very sluggish where the high-pressure phase grows at the expense of the low-pressure phase. For instance, the peak that appears at~9.8°(red triangle in Fig. 3a) at 6.7 GPa rapidly increases its intensity and becomes the strongest peak above 20.3 GPa, accompanied by a dramatic intensity decrease of the (211) and (212) diffraction peaks of the low-pressure phase. The (211) and (212) reflections are the most intense peaks of the starting structure and eventually disappear by 82 GPa, indicating the completion of the phase transition that spans a pressure window of~75 GPa. With further compression up to 184 GPa ( Supplementary Fig. 9), the XRD patterns are similar except for peak shifts to higher diffraction angles and subtle changes in the peak widths and intensities. Under decompression, CsPbI 3 goes back to the initial δ phase after releasing pressure. Between 6.7 and 82 GPa, new reflections that can be indexed into the high-pressure ε phase gradually appear upon compression, and the relative peak intensities of the δ phase change with pressure. These observations indicate that the structures of the δ and ε phases evolve within this pressure range. Rietveld refinement indicates that an orthorhombic structure with a Pmn2 1 space group fits the diffraction patterns above 82 GPa well ( Supplementary Fig. 11). In contrast to the initial 1D double chain structure, the high-pressure ε phase is a 3D structure with adjacent double chains bonding together where the Pb atoms are eight-and nine-fold coordinated by the I atoms ( Fig. 3c and Supplementary Fig. 10).
Comparing the electrical transport and XRD results, we find that the insulator-semimetal-metal transition and the Pnma-to-Pmn2 1 phase transition are closely related. The pressure at which the structural transition completes coincides with the end pressure of the metallic transition and with that of the change in the ρ RT -P curve, i.e., 80 GPa. Further analysis of the lattice parameters of the Pmn2 1 phase show pronounced deviations between 62 and 82 GPa ( Supplementary  Fig. 12), especially for the a lattice constant. This pressure window is consistent with that of the metallic transition which starts at~61 GPa where CsPbI 3 shows a metallic character above~255 K and completes at~80 GPa where a metallic state is present throughout 2-300 K temperature range. Above 85 GPa, the lattice parameters (Supplementary Fig. 12) and the unit cell volume (Supplementary Fig. 13) of the Pmn2 1 phase change smoothly as a function of pressure and no obvious anomaly is observed.
First-principles density functional theory (DFT) calculations corroborate the experimental results (Fig. 3b). We performed extensive structural searches on CsPbI 3 using the CALYPSO software package, which can predict energetically stable structures of materials at extreme conditions, e.g., high pressure 49 . At 90 GPa, the structural searches find that an orthorhombic Pmn2 1 phase and two monoclinic structures with Cm and C2/m symmetries are energetically stable and have lower enthalpies than the starting Pnma δ-CsPbI 3 . Among these structures, only the Pmn2 1 structure produces a simulated XRD pattern that is consistent with the experimental pattern ( Supplementary  Fig. 11). After structure optimization, the detailed structural parameters of the Pmn2 1 phase agree well with those obtained from XRD refinement (Supplementary Table 1). The difference in the transition pressure between experiments (6.7 GPa, Fig. 3a Fig. 3b) may be due to the nature of the very sluggish phase transition that experimentally begins at 6.7 GPa and completes by~80 GPa. In addition, temperature differences between the XRD experiments (~298 K) and structural searches (0 K) would in part explain the different transition pressures. Our theoretical results indicate that the δ-to-ε structural evolution involves a sequence of Pb-I bond rearrangements within the PbI 6 octahedral chains and between adjacent chains under compression (Fig. 3d, and Supplementary Figs. 15 and 16), consistent with the XRD results that suggest the system is not a simple mixing of the δ and ε phases (Fig. 3a, and Supplementary Figs. 7, 9, 12, and 13). It would rather be visualized as a set of mixed states composed of imperfect, distorted structures of δ-CsPbI 3 and ε-CsPbI 3 that keep evolving with pressure. Our calculations are unable to factor in the complex nature of the mixed phases, which further explains the underestimation of the simulated pressure for the structural and electronic transitions.
Band structure and density of state calculations were performed to study the electronic structure of the high-pressure phase. Because the low-temperature electrical transport results suggest the possibility of d-electrons serving as the charge carriers, the 5dstates of the Cs and I atoms were also included in our calculations which are close to the Cs-6s and I-5p states. At 0 GPa of δ-CsPbI 3 , the VBM is mainly composed of the I-5p and Pb-6s states, while the Pb-6p and I-5p states contribute to the CBM (Fig. 4a and Supplementary   Fig. 17). The A-site Cs atom makes very little contribution to the VBM or CBM, consistent with previous results 6, 28 . The involvement of d-states is negligible except for a very small contribution from the I-5d state to the upper conduction band at~3-4 eV above the CBM. An obvious transition occurs at high pressure as the material undergoes the δ-to-ε phase transition. In the Pmn2 1 ε phase, the electronic states delocalize dramatically, and ε-CsPbI 3 has a zero bandgap at 30 GPa ( Fig. 4b and Supplementary Fig. 18). If not considering the phase transition, our calculations indicate that δ-CsPbI 3 would close the bandgap at 40 GPa. These results further confirm the correlated nature of the structural and electronic transitions in compressed CsPbI 3 . After examining the band structures of ε-CsPbI 3 , we found that the A-site Cs cation significantly affects the electronic structure. The Cs-5d and I-5d states contribute appreciably to the valence and conduction bands, which become more pronounced with pressure up to 180 GPa (dashed lines, Fig. 4b), indicating pressure-induced electron redistribution. Quantitative orbital occupation calculations also support the pressure-induced electron redistribution (Fig. 4c). With the application of pressure, the 5d state occupation of the Cs and I atoms gradually increases, and then a sudden rise is clearly observed when the insulating δ phase transforms into the metallic ε phase (Fig. 4c). Similar electron redistribution has also been observed in the elemental form of Cs 50,51 and I 51,52 at high pressure.

Discussion
The pressure-induced enhancement of the 5d state occupation explains the ρ(T)~T n (n ≈ 2) behavior in compressed CsPbI 3 . Similar to material systems such as high-temperature superconductors and heavy fermion compounds where d or f electrons cause strong e-e interaction [36][37][38][39] , the 5d-electrons in compressed CsPbI 3 enhance the e-e interaction and direct the electrical transport at low temperature. The observed ρ-T curves of the metallic ε phase deviate from the predictions of a simple metal at low temperature (ρ(T)~T 5 ), and show FL-like behavior (ρ(T)~AT n , n ≈ 2), suggesting the presence of strong correlations. The coefficient A, which typically characterizes the strength of the e-e interaction and is related to the effective mass 53,54 , also suggests strong e-e interaction (Fig. 2c). The fitted A value is 1.078 × 10 −8 Ω·cm/ K n at 80 GPa, comparable with those obtained in high-temperature superconductors [54][55][56] . With further compression, the A value gradually decreases to 1.073 × 10 −9 Ω·cm/K n at 186 GPa. Deviation from a FL-like state (n < 2) is also observed at the onset of CsPbI 3 entering the high-pressure metallic phase (Fig. 2c). Many origins for the deviation from FL behavior may be at play in correlated electron systems. Factors such as the Kondo effect 57,58 and lattice deformation and disorder [58][59][60] have been proposed. NFL behavior has also been observed in some unconventional superconductors and heavy fermion metals that was attributed to quantum critical fluctuations 37,39,53 . The Kondo effect usually occurs in correlated metals doped with magnetic impurities. Magnetic impurities are not known to exist in CsPbI 3 ( Supplementary Fig. 1), thus ruling out the Kondo effect as being the likely cause. It is possible that the high-pressure metallic CsPbI 3 phase has defects/disorder in the lattice. However, extrinsic disorder like grain boundaries usually increases the residual resistivity (ρ 0 ) but does not change the temperature dependence of resistivity; intrinsic disorder like lattice defects usually leads to a negative value for the coefficient A 58,59,61 . Hence, we infer that defects and disorder should not be the root cause of the deviation found in the highpressure metallic phase. The electrical transport anomaly in compressed CsPbI 3 seems similar in character to some unconventional superconductors and heavy fermion metals, in which NFL behavior has been observed to exist for a wide range of doping levels between the under-doped and over-doped compositions that may be attributed to proximity to a nearby quantum critical point [36][37][38][39] . Pressure induces the insulator-to-metal transition and increases the carrier concentration in CsPbI 3 , which resemble the evolution from an under-to an over-doped composition in unconventional superconductors and heavy fermion metals. This raises the possibility that the deviation from a FL-like state in the high-pressure metallic ε-CsPbI 3 is related to quantum criticality. In iron-and cuprate-based superconductors, such behavior usually coexists with a superconducting state in the phase diagram 36,37 . Hence, a superconducting transition might occur below 2 K in the highpressure metallic CsPbI 3 . Further experiments are required to understand the origin of this anomaly and search for a superconducting phase in CsPbI 3 and related materials. In summary, by compressing δ-CsPbI 3 to 80 GPa, the initially insulating phase transforms to a semimetallic and then to a metallic state, accompanied by the completion of a sluggish δ-to-ε phase transition from a 1D chain structure to a 3D structure that begins at 6.7 GPa and finishes at~80 GPa. Deviation from a FL-like state is observed in the high-pressure metallic phase, which progressively evolves into a FL-like state at~186 GPa. First-principles DFT calculations indicate that pressure induces electron redistribution which significantly enhances the 5d state occupation of Cs and I in the ε phase that strengthens the e-e interaction and eventually leads to the formation of a FL-like state. In particular, the Cs atom makes a direct and pronounced contribution to the low-temperature electrical properties of CsPbI 3 at high pressure. Our study opens a promising approach of manipulating the A-site cation for tuning the electronic structures and realizing intriguing electronic states in halide perovskites and analog materials at high pressures. Likewise, e-e interaction may be widely attainable in the broad family of halide perovskites, particularly at high pressure, opening routes to access a diversity of electronic states in these materials comparable to that available in the perovskite oxides.

Synthesis of δ-CsPbI 3 samples
Yellow δ-CsPbI 3 samples were synthesized using a solution-based method 62 . Solid PbI 2 (0.46 g, 1.0 mmol) and CsI (0.26 g, 1.0 mmol) powder precursors were mixed in aqueous HI (10.0 mL, 7.58 M, stabilized) in a closed 20 mL glass vial. The solution was then heated on a hotplate to 130°C while stirring until the solid precursors were completely dissolved. The solution was then cooled to room temperature at a rate of 25-35°C/h by turning off the hotplate while stirring the solution, yielding yellow δ-CsPbI 3 powders. The obtained yellow powders were collected using vacuum filtration, rinsed thoroughly with diethyl ether, and dried under reduced pressure for 12 h. The powders were then ground for half an hour before loading for experiments.

Resistivity measurements
Resistivity measurements on yellow δ-CsPbI 3 at simultaneous low temperature (2-300 K) and high pressure up to 186 GPa were conducted using a nonmagnetic diamond anvil cell (made from Be-Cu alloys) with beveled diamonds with an inner and outer culet diameter of 100/300 μm in a physical property measurement system (PPMS DynaCool). PPMS DynaCool offers a high-sensitive resistivity measurement that decreases the data uncertainty, resulting in the error bars being smaller than the symbol size of data points. An insulating gasket was prepared by: (a) pre-indenting a rhenium sheet (~250 μm in thickness) to 30 μm, (b) drilling a 80-μm-diameter hole in the center of the pre-indented area using a 1064-nm laser milling machine, (c) pressing a prepared mixture of epoxy and cubic boron nitride powders (1:10 in weight) into the pre-indented area, and (d) drilling a new 40μm-diameter hole in the center to serve as the sample chamber. Platinum foil electrodes (~4 μm in thickness) in a van der Pauw configuration were then prepared on the insulating gasket by a hand-wiring method 63 . Pressure was calibrated at room temperature using the ruby fluorescence method (0.1-30 GPa) 64 and diamond Raman peaks (30-186 GPa) 65 . No pressure transmitting medium was used to ensure good contact between the sample and the electrodes. Pressure uncertainty of 0.1 GPa and <5 GPa goes with the ruby fluorescence and diamond Raman peak methods, respectively. The pressures above 30 GPa were determined by taking the average values of the pressures at the center and the edge of the samples (Supplementary Fig. 3).

XRD measurements
High-pressure XRD experiments were performed at beamline 12.2.2 of the Advanced Light Source, Lawrence Berkeley National Laboratory, and sector 13 of the Advanced Photon Source (APS), Argonne National Laboratory (ANL) with X-ray wavelengths of 0.4959 Å and 0.3344 Å, respectively. Powdered δ-CsPbI 3 samples, along with a small ruby ball (~5 μm in diameter) and a small piece of Au foil as pressure calibrants were loaded into an 80-μm-diameter (or 40-μm-diameter) sample chamber drilled in the center of a pre-indented tungsten gasket and compressed between a pair of 200-μm (or 100-μm) culet diamond anvils. Neon was loaded as the pressure transmitting medium in the 200-μm culet diamond anvil cell and no pressure medium was used in the 100-μm culet diamond anvil cell.

DFT calculations
Structure prediction of CsPbI 3 at high pressure was performed via a global minimum search of the free energy surface by the swam intelligence-based CALYPSO method 66,67 . First-principles calculations included structural optimization, band structures, density of states, and orbital occupation calculations, and were carried out using DFT with the Perdew-Burke-Ernzerhof exchange-correlation functional and generalized gradient approximation 68 that are implemented in the Vienna ab initio simulation package (VASP) 69 . The projectoraugmented wave pseudopotentials were set with a plane wave energy of 680 eV 70 . A fine Monkhorst-Pack Brillouin zone sampling grid with resolution of 0.04 × 2π Å −1 was applied. Atomic positions and lattice parameters were relaxed until all the forces on the ions were less than 10 −3 eV/Å. The transition path search was performed by using the evolutionary method implemented in the CALYPSO code 71 . In our calculations, relativistic effects, e.g., spin-orbit coupling (SOC), are not included. A proof-of-principle calculation for the ε phase at 30 GPa shows that except for slight broadening of the bandwidth, density of states that includes the SOC effect is comparable with that without considering SOC and the 5d state occupation of the Cs and I atoms remains invariant (Supplementary Fig. 19).

Data availability
All data generated or analyzed during this study are included in this article and its supplementary information files. Additional data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability
The codes for the CALYPSO software are available at http://www. calypso.cn.