- Lille Wed 4th May 2016 Jarvist Moore Frost, Federico Brivio, Jonathan Skelton, Aron Walsh (University of Bath, UK) Pooya Azarhoosh, Scott McKechnie, Mark van Schilfgaarde (King’s College London, UK) Walsh Materials Design Group, University of Bath, UK [email protected] Dynamic disorder and electron- hole recombination in hybrid halide perovskites
- Lille Wed 4th May 2016 A - Molecular Cation B - {Pb, Sn} X 3 - Halide {I, Br, Cl*} Hybrid Halide Perovskites (ABX 3 ) Weber, Dieter. "CH3NH3PbX3, ein Pb (II)-System mit kubischer Perowskitstruktur/CH3NH3PbX3, a Pb (II)-System with Cubic Perovskite Structure." Zeitschrift für Naturforschung B 33.12 (1978): 1443-1445.
- Lille Wed 4th May 2016 Methylammonium (CH 3 NH 3 +) - MA A closed shell (18 e-) molecular cation with a large electric dipole (2.2 D) J. M. Frost et al, Nano Letters 14, 2584 (2014) Deprotonation (pK a ~ 10): CH 3 NH 3 + → CH 3 NH 2 + H+
- Lille Wed 4th May 2016 ( Videos on YouTube - search for 'MAPI molecular dynamics' ) https://youtu.be/K_-rsop0n5A Incredibly Soft crystal; large distortions of octahedra ➔ MA ion yaw ➔ ...and roll… ➔ ...CH3 clicks ➔ so does NH3 [2x2x2 Pseudo cubic relaxed supercell, lattice parameters held constant during MD (NVT simulation). PBESol Functional at the Gamma point (forces + energies should converge well). dt = 0.5 fs, T = 300 K ]
- Lille Wed 4th May 2016 Cubic? Tetragonal? Orthorhombic? Powder Neutron diffraction allows for a full solution (inc. hydrogens) ➔ 150K 1st order phase transition (Ortho-Tetra) ➔ 2nd order transition to cubic phase Weller et al. Chem. Commun., 2015, DOI: 10.1039/C4CC09944C Received 12 Dec 2014, Accepted 22 Jan 2015
- Lille Wed 4th May 2016 Exciton binding from effective mass theory: Carrier mass & dielectric screening favour free carrier generation (t→infinity) J. M. Frost et al, Nano Letters 14, 2584 (2014) Onsager theory; See Wilsen 1939
- Lille Wed 4th May 2016 Free Charges or large Polarons or small Polarons? (Fröhlich polarons) (Polaron Binding) (Arguments for these follow Landau (1933); from Jones & March (1985), "Theoretical Solid State Physics Vol 2" ) MAPI: (Feynman, 1955) α GaAs: 0.068 CdTe: 0.29 AgCl: 1.84 SrTiO3: 3.77 (Devreese 2005)
- Lille Wed 4th May 2016 CH 3 NH 3 PbI 3 (MAPI for short) Configuration: PbII [5d106s26p0]; I-I [5p6] F. Brivio et al, Physical Review B 89, 155204 (2014) Relativistic QSGW theory with Mark van Schilfgaarde (KCL) Conduction Band Valence Band Dresselhaus Splitting (SOC) [Molecule breaks centrosymmetry]
- Lille Wed 4th May 2016 F. Brivio et al, Physical Review B 89, 155204 (2014) Bands are not parabolic, but… m h */m ~ 0.12 (light holes) m e */m ~ 0.15 (light electrons) [sampled within k B T of band edges] Optical Absorption Hole Effective Mass [110] [112] [111] (Nb: requires sophisticated treatment of k-space grid for sufficient points for fit!)
- Lille Wed 4th May 2016 Absorption: Spin-orbit-coupling flattens the valence band - leading to a large density of states available for direct excitation. A sudden “turn-on”, like 2D band structures. Emission: Holes and electrons quickly thermalise to bottom of band (densities at 1 sun solar flux are very low); indirect radiative recombination is slow. → Have your cake and eat it ← Why MAPI is so great: The Dresselhaus crystal field effect splits the CBM (more than VBM); a spin split indirect gap forms. 75 meV P. Azarhoosh et al., ArXiv 1604.04500 (2016)
- Lille Wed 4th May 2016 Spin-split indirect-gap: 75 meV Biggest contribution where Xi(r) is large, near the Pb (Z=82) nucleus. Driven by the crystal (electric) field. Weaker effect at I (Z=53) on 5p- orbital, flattens bands. → Electric field at nucleus
- Lille Wed 4th May 2016 Calculate radiative recombination rate: QSGW band structure (120x120x120 K-point mesh). Direct transitions only. Fermi-Dirac distribution for the electrons / holes within their band (full thermalisation).
- Lille Wed 4th May 2016 Predictions: Spin-split indirect-gap leads to 300 X decrease in bi-molecular recombination. Weak indirect gap ~75 meV below direct; should not be present in Orthorhombic phase (<150K). B coeff. varies strongly as a function of intensity (you can't do a 'global fit' to TRPL data over many decades) Faster recombination expected in Sn analogue due to reduced Spin Orbit Coupling - it should be more direct gap like. Lasing threshold can be directly explained by intensity dependence of B. Epitaxial / ferroelectric manipulation should affect optical properties. Spin split indirect gap → may be a new design feature for novel solar cell materials. Present where {Sb,Bi,Pb} + ferroelectric distortion. P. Azarhoosh et al., ArXiv 1604.04500 (2016)
- Lille Wed 4th May 2016 Back to real space: STARRYNIGHT Classical Metropolis algorithm simulation of cage:cage dipole interactions. Analytic Hamiltonian, interaction strength parameterised by DFT. ( APL Materials 2 (8), 081506, 2014. Open source on GitHub )
- Lille Wed 4th May 2016 Display direction of Dipole by point on HSV sphere p (Nb: Simulation linear scaling + very fast; here I present 2D slices of ~20x20, as any larger and you can't see what's going on!)
- Lille Wed 4th May 2016 https://github.com/WMD-group/StarryNight Metropolis (local spin move) Monte Carlo code written in C99. Efficient & on lattice → millions of moves per second. Analysis code built in, and additional Julia post processing tools. Open source!
- Lille Wed 4th May 2016 Cagestrain=25 meV → Semi-ordered Ferroelectric ground state; Intermediate long range order (dynamic) at finite T Increasingly long range partial ferroelectric ordering as T drops. Reaches the orthorhombic phase transition before full ordering. 0K 128K 64K 256K 384K Radial order parameter ( Ziman 1979 ) [ R→ infinity value is equal to the Landau order ]
- Lille Wed 4th May 2016 Recombination vs. Mobility ? This is structureless disorder (just density of states), using models more suitable from low mobility materials (from amorphous silicon). At 300 K the reduction in recombination (x 100) due to charge segregation is balanced by an estimate of the reduction in mobility (x 100) caused by trapping / detrapping. Further work will look at structure and see whether 'ferroelectric highways' allow for greater mobility than expected of Gaussian Disorder Model.
- Lille Wed 4th May 2016 Starrynight Conclusions ➔ Ground state dependent on details of Hamiltonian terms. Our errors here could be +-200%. ➔ We observe exponentially decaying long range partial ferroelectric ordering. ➔ Continuous inter-converting domains at finite temperature. ➔ Considerable (+-150 meV) electrostatic potential fluctuations. Statistical mechanics models indicate what behaviour is possible. Experiment will show that which is present. https://github.com/WMD-Group/StarryNight
- Lille Wed 4th May 2016 Collaborators:- Piers Barnes, Aurel Leguy, Andrew McMahon - Imperial College London Mark van Schilfgaarde, Scott McKechnie, Pooya Azarhoosh - King's College London Piers Barnes Aurelien Leguy Mark van Schilfgaarde Pooya Azarhoosh WMD Group, Bath Acknowledgments:- EPSRC - EP/K016288/1 EPSRC Archer - EP/L000202 University of Bath HPC http://go.bath.ac.uk/wmd
- Lille Wed 4th May 2016 R→ infinity value is the same as the Landau order (but SNR has increased by R^2 !) Define a radially-dependent autocorrelation function of the dipoles Ziman - Models of Disorder, 1979
- Lille Wed 4th May 2016 Reproduce Adriaenssens 1997 result (numerically) Sum of rates (proport. to densities) Direct evaluation via partition functions
- Lille Wed 4th May 2016 Zero field (open circuit) Dipole potential (2D FFT) Short Circuit Field Dipole potential (2D FFT) Same figures, presented side by side...
- Lille Wed 4th May 2016 “Natural” Valence Band Alignments K. T. Butler et al, Materials Horizons, Advance Article (2015) Similar to other thin-film PV materials Band gap engineering through A, B or X site modification
Solid state physics of hybrid Perovskites 13th May 2015 New Spiro? Calculating Ionisation Potential of SPIRO-OMeTAD and twelve methoxy isomers and polymethoxy derivatives, simple vacuum hybrid DFT (Delta SCF) calculations. Modular design of SPIRO-OMeTAD analogues as hole transport materials in solar cells Alexander T. Murray, Jarvist M. Frost, Christopher H. Hendon, Christopher D. Molloy, David R. Carbery and Aron Walsh Chem. Commun., 2015, Advance Article DOI: 10.1039/C5CC02129D Received 12 Mar 2015, Accepted 23 Apr 2015
- Lille Wed 4th May 2016 a Baikie T., et al., Synthesis and crystal chemistry of the hybrid perovskite (CH 3 NH 3 ) PbI 3 for solid- state sensitised solar cell applications, J. Mater. Chem. A, 1, 5628-5641 (2013). b Stoumpos, C. C., Malliakas, C. D. & Kanatzidis, M. G. Semiconducting tin and lead iodide perovskites with organic cations: phase transitions, high mobilities, and near-infrared photoluminescent properties. Inorg. Chem. 52, 9019–9038 (2013). c Weller M. T., et al., Complete structure and cation orientation in the perovskite photovoltaic methylammonium lead iodide between 100 and 352K Chem. Comm., DOI:10.1039/c4cc09944c (2015) d Kawamura Y., Mashiyama H., Hasebe K., Structural study on cubic-tetragonal transition of CH 3 NH 3 PbI 3 , J. Phys. Soc. Japan. 71, 1694-1697 (2002). † Note: due to the manner in which orientational disorder is fitted to neutron diffraction data, this bond length represents an underestimate. To refine the orthorhombic structure, Weller et al use fixed bond lengths of 1.46Å (C-N), 1.13Å (C-H) and 1.00Å (N-H).
- Lille Wed 4th May 2016 Ortho. DFT, with 150 K Expt data. Cage Cation Experimental data: Oliver J. Weber, Mark T. Weller, (Bath) Alejandro R. Goni (ICMAB, Barcelona), Aurelien M. A. Leguy, Piers R. F. Barnes (Imperial, London) Aurelien Leguy ICMAB, Barcelona Imperial College London ?
- Lille Wed 4th May 2016 3 mid-energy range MA hydrogen modes Most molecular modes are the same in vacuum (by DFT calculation), as in the solid state. Low-frequency molecular modes (methyl clicker) seem highly affected by environment (900 → 300 cm-1 ). Good be a useful probe of local packing / ordering.
- Lille Wed 4th May 2016 Nudged elastic band activation energies, of vacancy mediated diffusion; from DFT / PBESol in MD equilibriated Supercells Iodine Vacancy mediated diffusion: Ea = 0.58 eV
- Lille Wed 4th May 2016 T= 0 K (Ground State - but a bit out of eqm, due to MC) CageStrain = 0 ---> Anti-Ferroelectric (The potential at a site from the dipole on the nearest neighbour (= 1 in the internet units of Starrynight) is simply 0.165 V.)
- Lille Wed 4th May 2016 Band Gap variation (Gamma, PBESol) during MD Fluctuation in the eigenvalues → motion coupling into energy levels. This is a thermodynamically sampled renormalisation of the electron energies (electron-phonon coupling)
- Lille Wed 4th May 2016 A total of 58 ps (2319 frames) of data was used for analysis, after an equilibration run of 5 ps. This generated 18547 unique MA alignment vectors.
- Lille Wed 4th May 2016 Incredibly Soft crystal; large distortions of octahedra ➔ MA ion yaw ➔ ...and roll… ➔ ...CH3 clicks ➔ so does NH3 Do electronic structure calculations on 'perfect' 'equilibrated' crystals have any real meaning for MAPI? [2x2x2 Pseudo cubic relaxed supercell, lattice parameters held constant during MD (NVT simulation). PBESol Functional at the Gamma point (forces + energies should converge well). dt = 0.5 fs, T = 300 K ] Molecular Dynamics (MAPI is as soft as Jelly.)
- Lille Wed 4th May 2016 Electrostatic potential projected onto electron density isosurface (iso value = 0.02) 0.6 0.375 0.3 Electrostatic Potential Vacuum calculation CCSD/cc-pVTZ on b3lyp/6-31g* geom. Gaussian09 2.2 Debye Φ E 18 electrons,closed shell = happy quantum chemical days (Frost, Nano Letters 2014) Hartrees (x 27.211 V)
- Lille Wed 4th May 2016 Interaction Energy Vacuum dielectric (effect through empty cage gap) On-lattice dipoles, spacing of 6.29Å. Point dipole approximation * = 25 meV * MA Dipole moment massively dominates in polarisation tensor, point approximation possibly not valid r=6.29Å 2.2 Debye (Frost, 2014, Nano Letters)
- Lille Wed 4th May 2016 Calculation of cage strain term (elastic response)... Rotating single MA in 4x4x4 supercell (+ relaxing intermediates) gives you a ~ near-neighbour dot product local strain / elastic response term. = 25-75 meV / nearest neighbour [[ with significant error bars ]]
- Lille Wed 4th May 2016 ➢ Optimised on-lattice C99 code ; ◦ up to ~10 million MC moves / second ➢ 2D version released May 2014; (Frost, APL Mater 2014) ◦ 3D version + many extra analyses, ~June 2015 ◦ OPENMP ~July 2015 (doesn't help much…) STARRYNIGHT
- Lille Wed 4th May 2016 Landau Order Param Naive definition poor for dipoles: <<<<<>>>>> = 0 Replace with radial distribution function based order param: ( Ziman 1979 ) [ R→ infinity value is equal to the Landau order ]
- Lille Wed 4th May 2016 Why is radiative recombination so slow? POLARON POLARON NORALOP Slightly indirect band gap. Hypothesise slow recombination due to:- 1) Electrostatic potential fluct (proposed Nanoletters, Starrynight model APLMater, now studying rates) 2) Indirect gap recombination due to SOC (Pooya Azarhoosh, KCL; calculating rates)
- Lille Wed 4th May 2016 Electrostatic Pot. Fluct. Recombination Model • Thomas-Fermi model of electron density… ◦ Not ready for presentation... • Monte-Carlo (classical) model of electro hopping; conceptual issues with defining rates and motion, but coupled dipoles: electron transport would be easy. • First step: Classical Stat. Mech. population of charge carriers → effective slow down of bimolecular recombination
- Lille Wed 4th May 2016 Basic Stat. Mech. V h+ h+ h+ e- e- Recombination... Recombination proportional to electron and hole densities at a site (Langevin).
- Lille Wed 4th May 2016 Why is MAPI an efficient solar cell? ◆ Almost absent non-radiative recombination ◦ Few mid gap defects (fortitude?) ◆ Slow radiative recombination • Unusual for a direct gap material • ? Slightly-indirect gap due to Rashba splitting • ? Electrostatic potential fluctuations reduce recombination ◆ Sufficient mobility to get charges out • But not that high considering effective mass (~50 cm2/Vs vs. 1000 cm2/Vs for CdTe) ◦ ? Reduced by electrostatic fluct?
- Lille Wed 4th May 2016 Mobility (Boltzmann distribution of electrons, at 300 K) Simple thermal de- trapping model, with assumed percolation threshold.
- Lille Wed 4th May 2016 Polarisation Applied field Elastic Cage Strain = 25 meV No cage strain - columnar Anti- ferroelectric ~1GHz Scan 10 nm sample Nb: Time in Monte Carlo is ill defined Assume 3 ps / MC move Nb: Fields enormous! ~75 V Lossy Dielectric? Ferroelectric?
- Lille Wed 4th May 2016 ~1MHz Scan 10 nm 2D sample Nb: Time in Monte Carlo is ill defined Assume 3 ps / MC move Polarisation Caveat: 2D simulation! - (simulation time limitations) ~2.5 V
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