dependent effective potential Raman scattering First—principles lattice dynamics Pre-requisite This lecture This summer school Lecture Aims: To better understand “every-day” phonon calculations To better connect theory and practice (Quantum) thermodynamics Note: Interrupt, question and tell me how things can be improved – feedback welcome Thermal transport Wigner thermal transport
Oppenheimer approximation – solve electronic Schrödinger equation for fixed ions • However atoms in a material vibrate around their (T-dependent) equilibrium position • This impacts on the behaviour of a material e.g. band gap, e- mobility, thermodynamics, IR absorption… • In a crystal the vibrations are periodic and can be described as a collective excitation (quasi-particle): phonon
within which the atoms move Potential Energy Surfaces Harmonic Anharmonic (single well) Anharmonic (double well) Potential energy (collective) displacement Anharmonicity * * * * = equilibrium position * Low T High T
i + 1 2 ↵ ij u ↵ i uj + 1 6 ↵ ijk u ↵ i uj uk + . . . Attempted classification of ab-initio methods for phonons method of computing the force constant tensor how the potential energy surface is sampled Finite Displacement Density Functional Perturbation Theory Lattice dynamics – perturbative approaches Ab-initio Molecular Dynamics <latexit sha1_base64="ShRTR+RRK5j4yX7kZf8MwqSfJ+w=">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</latexit> ↵ ij = @2 H @u↵ i @uj = @F↵ i @uj ⇡ @F↵ i (R) F↵ i (0) |R| <latexit sha1_base64="upk25wxl9HMM/29NdK8qe/ZdQvc=">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</latexit> MI ¨ RI = rI["0(R) + VNN(R)] method for extracting vibrational properties <latexit sha1_base64="H84wbuhhMWBtWN4/Qmdhy8etQQQ=">AAACGHicbVDLSgMxFM3UV62vUZdugkVoEeqMiroRim66rGAf0I5DJk3b0MyD5I5YhvkMN/6KGxeKuO3OvzF9IFo9kHA451ySe7xIcAWW9WlkFhaXlleyq7m19Y3NLXN7p67CWFJWo6EIZdMjigkesBpwEKwZSUZ8T7CGN7ge+417JhUPg1sYRszxSS/gXU4JaMk1jyqFttDxDiniS1y5SwpWMcWHuO62gT2A9BN9p9+ZnGvmrZI1Af5L7BnJoxmqrjlqd0Ia+ywAKohSLduKwEmIBE4FS3PtWLGI0AHpsZamAfGZcpLJYik+0EoHd0OpTwB4ov6cSIiv1ND3dNIn0Ffz3lj8z2vF0L1wEh5EMbCATh/qxgJDiMct4Q6XjIIYakKo5PqvmPaJJBR0l+MS7PmV/5L6cck+K53cnObLV7M6smgP7aMCstE5KqMKqqIaougRPaNX9GY8GS/Gu/ExjWaM2cwu+gVj9AUZcZ38</latexit> H( ) = H (0) + Vext( ) Velocity autocorrelation function ?? Effective force constants This lecture
computationally intensive Computationally intensive: phase space sampling & frequency resolution Treating anharmonic effects as a perturbation– validity? Anharmonicity at all orders Limited to smaller amplitude displacements (not suitable at high-T or near a phase transition) Suitable for large amplitude displacements Equilibrium position fixed Equilibrium position can change with temperature
the mode occupation ! 𝑛! 𝑇 from Bose-Einstein statistics Harmonic approximation valid Harmonic approximation invalid Low temperature, low energy eigenstates dominate High temperature, higher energy eigenstates contribute PES as potential in 1D Schrodinger equation à solve to give vibrational eigenstates See e.g. Whalley et al (2016) Phys. Rev. B 94, 220301(R)
Non-restorative force proportional to distance Harmonic approximation invalid Low temperature High temperature Quantifying anharmonicity Anharmonicity measure for materials Knoop et al 2020 Phys. Rev. Materials 4, 083809 Model validity depends on the shape of the PES and the mode occupation ! 𝑛! 𝑇 from Bose-Einstein statistics See e.g. Whalley et al (2016) Phys. Rev. B 94, 220301(R)
, Cartesian directions Taylor expansion of energy to second order <latexit sha1_base64="bvtaQylMevj081R/cKcYhUaCOLA=">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</latexit> H ⇡ H0 + 1 2 ↵ ij u ↵ i uj Unit cells <latexit sha1_base64="9szpkq8anmDamnb8qJ/UL2aCrM4=">AAAB8HicbVDLSgNBEOyNrxhfqx69DAbBi2FXfF2EoBePEcxDkiXMTmaTITOzy8ysEJZ8hRcPinj1c7z5N06SPWhiQUNR1U13V5hwpo3nfTuFpeWV1bXiemljc2t7x93da+g4VYTWScxj1QqxppxJWjfMcNpKFMUi5LQZDm8nfvOJKs1i+WBGCQ0E7ksWMYKNlR5TdI3Uiep6XbfsVbwp0CLxc1KGHLWu+9XpxSQVVBrCsdZt30tMkGFlGOF0XOqkmiaYDHGfti2VWFAdZNODx+jIKj0UxcqWNGiq/p7IsNB6JELbKbAZ6HlvIv7ntVMTXQUZk0lqqCSzRVHKkYnR5HvUY4oSw0eWYKKYvRWRAVaYGJtRyYbgz7+8SBqnFf+icn5/Vq7e5HEU4QAO4Rh8uIQq3EEN6kBAwDO8wpujnBfn3fmYtRacfGYf/sD5/AFKrI9w</latexit> u = r r0 <latexit sha1_base64="aOexMsvG3ngDDR+JQwlZz1oRvrM=">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</latexit> ↵ ij = @2 H @u↵ i @uj = @F↵ i @uj ⇡ F↵ i uj <latexit sha1_base64="jjXYAui7wFj647nYNNJxcunggDU=">AAAB+XicbVDJSgNBEO2JW4zbqEcvjUHwFGbE7RjUg8cIZoHMGHo6NUlrz0J3TSAM+RMvHhTx6p9482/sLAdNfFDweK+KqnpBKoVGx/m2CkvLK6trxfXSxubW9o69u9fQSaY41HkiE9UKmAYpYqijQAmtVAGLAgnN4Ol67DcHoLRI4nscpuBHrBeLUHCGRurYtncDEhnNOo8PXgDIOnbZqTgT0EXizkiZzFDr2F9eN+FZBDFyybRuu06Kfs4UCi5hVPIyDSnjT6wHbUNjFoH288nlI3pklC4NE2UqRjpRf0/kLNJ6GAWmM2LY1/PeWPzPa2cYXvq5iNMMIebTRWEmKSZ0HAPtCgUc5dAQxpUwt1LeZ4pxNGGVTAju/MuLpHFScc8rZ3en5erVLI4iOSCH5Ji45IJUyS2pkTrhZECeySt5s3LrxXq3PqatBWs2s0/+wPr8ARDRk08=</latexit> uj <latexit sha1_base64="pJsh7yb9vjXB5zWeexwrqZ4487g=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2NQEI8RzAOTNfROJsmQ2dllZlYIS/7CiwdFvPo33vwbJ8keNLGgoajqprsriAXXxnW/ndzS8srqWn69sLG5tb1T3N2r6yhRlNVoJCLVDFAzwSWrGW4Ea8aKYRgI1giG1xO/8cSU5pG8N6OY+SH2Je9xisZKDzcd/thGEQ+wUyy5ZXcKski8jJQgQ7VT/Gp3I5qETBoqUOuW58bGT1EZTgUbF9qJZjHSIfZZy1KJIdN+Or14TI6s0iW9SNmShkzV3xMphlqPwsB2hmgGet6biP95rcT0Lv2UyzgxTNLZol4iiInI5H3S5YpRI0aWIFXc3kroABVSY0Mq2BC8+ZcXSf2k7J2Xz+5OS5WrLI48HMAhHIMHF1CBW6hCDShIeIZXeHO08+K8Ox+z1pyTzezDHzifP1TjkLY=</latexit> F↵ i
↵ i0jl exp{iq · [rjl rj0]} Dynamical matrix from fourier transform of Φ <latexit sha1_base64="N5NjdA6SwJOtzMJBp++n2RDJnao=">AAACJ3icdVDLSsNAFJ3UV62vqEs3g0Wom5JUUTdKURcuK9gHNLFMppN26CQTZyZCCf0bN/6KG0FFdOmfOGmz0FYPDJw5517uvceLGJXKsj6N3Nz8wuJSfrmwsrq2vmFubjUkjwUmdcwZFy0PScJoSOqKKkZakSAo8BhpeoOL1G/eEyEpD2/UMCJugHoh9SlGSksd8+yy5ARI9T0/uRvtN39+4Cl0eEB66Lbyb03HLFplaww4S+yMFEGGWsd8cbocxwEJFWZIyrZtRcpNkFAUMzIqOLEkEcID1CNtTUMUEOkm4ztHcE8rXehzoV+o4Fj92ZGgQMph4OnKdEU57aXiX147Vv6Jm9AwihUJ8WSQHzOoOExDg10qCFZsqAnCgupdIe4jgbDS0RZ0CPb0ybOkUSnbR+WD68Ni9TyLIw92wC4oARscgyq4AjVQBxg8gCfwCt6MR+PZeDc+JqU5I+vZBr9gfH0DUoClow==</latexit> D(q)W(q) = !2(q)W(q) Diagonalise to get squared phonon frequencies 𝜔! and eigenvectors W Harmonic energy is expressed as a function of displacement amplitude Q <latexit sha1_base64="qeAYPBdv2p/bWwv1Kmyj14uRXNQ=">AAACEnicbVDLSsNAFJ34rPUVdelmsAgtQkmqqMuimy5bsA9o0jCZTtqhk0yYmYgl5Bvc+CtuXCji1pU7/8bpY6GtBy4czrmXe+/xY0alsqxvY2V1bX1jM7eV397Z3ds3Dw5bkicCkybmjIuOjyRhNCJNRRUjnVgQFPqMtP3R7cRv3xMhKY/u1DgmbogGEQ0oRkpLnlmqFRsl6KA4FvwB1jwLnkEnEAindpZWMoeHZIB6lUav4pkFq2xNAZeJPScFMEfdM7+cPsdJSCKFGZKya1uxclMkFMWMZHknkSRGeIQGpKtphEIi3XT6UgZPtdKHARe6IgWn6u+JFIVSjkNfd4ZIDeWiNxH/87qJCq7dlEZxokiEZ4uChEHF4SQf2KeCYMXGmiAsqL4V4iHSgSidYl6HYC++vExalbJ9WT5vXBSqN/M4cuAYnIAisMEVqIIaqIMmwOARPINX8GY8GS/Gu/Exa10x5jNH4A+Mzx/2jpvQ</latexit> H(Q) ⇡ H0 + 1 2! 2 Q 2 l <latexit sha1_base64="tTaZxfIHJKYcMNXkpbgp1bW9CNQ=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgQcKuinoMevEY0TwgWcLspDcZMzu7zMwKYcknePGgiFe/yJt/4+Rx0MSChqKqm+6uIBFcG9f9dnJLyyura/n1wsbm1vZOcXevruNUMayxWMSqGVCNgkusGW4ENhOFNAoENoLBzdhvPKHSPJYPZpigH9Ge5CFn1Fjpnp88doolt+xOQBaJNyMlmKHaKX61uzFLI5SGCap1y3MT42dUGc4EjgrtVGNC2YD2sGWppBFqP5ucOiJHVumSMFa2pCET9fdERiOth1FgOyNq+nreG4v/ea3UhFd+xmWSGpRsuihMBTExGf9NulwhM2JoCWWK21sJ61NFmbHpFGwI3vzLi6R+WvYuymd356XK9SyOPBzAIRyDB5dQgVuoQg0Y9OAZXuHNEc6L8+58TFtzzmxmH/7A+fwBAFeNnw==</latexit> i, j Atom labels <latexit sha1_base64="YAfU1/KebHI4durrhkUB7zssW8M=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEF1ISFXVZdOOygn1AE8rNdNIOnUnCzEQooeCvuHGhiFu/w51/46TNQlsPXO7hnHuZOydIOFPacb6t0tLyyupaeb2ysbm1vWPv7rVUnEpCmyTmsewEoChnEW1qpjntJJKCCDhtB6Pb3G8/UqlYHD3ocUJ9AYOIhYyANlLPPvCAJ0M4xV5Add4GIAT07KpTc6bAi8QtSBUVaPTsL68fk1TQSBMOSnVdJ9F+BlIzwumk4qWKJkBGMKBdQyMQVPnZ9PwJPjZKH4exNBVpPFV/b2QglBqLwEwK0EM17+Xif1431eG1n7EoSTWNyOyhMOVYxzjPAveZpETzsSFAJDO3YjIECUSbxComBHf+y4ukdVZzL2vn9xfV+k0RRxkdoiN0glx0heroDjVQExGUoWf0it6sJ+vFerc+ZqMlq9jZR39gff4A70KU1g==</latexit> ↵, , Cartesian directions Taylor expansion of energy to second order <latexit sha1_base64="bvtaQylMevj081R/cKcYhUaCOLA=">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</latexit> H ⇡ H0 + 1 2 ↵ ij u ↵ i uj Unit cells
* * Non-restorative force – imaginary phonon frequency Harmonic approximation invalid Low temperature High temperature Harmonic energy is expressed as a function of displacement amplitude Q <latexit sha1_base64="qeAYPBdv2p/bWwv1Kmyj14uRXNQ=">AAACEnicbVDLSsNAFJ34rPUVdelmsAgtQkmqqMuimy5bsA9o0jCZTtqhk0yYmYgl5Bvc+CtuXCji1pU7/8bpY6GtBy4czrmXe+/xY0alsqxvY2V1bX1jM7eV397Z3ds3Dw5bkicCkybmjIuOjyRhNCJNRRUjnVgQFPqMtP3R7cRv3xMhKY/u1DgmbogGEQ0oRkpLnlmqFRsl6KA4FvwB1jwLnkEnEAindpZWMoeHZIB6lUav4pkFq2xNAZeJPScFMEfdM7+cPsdJSCKFGZKya1uxclMkFMWMZHknkSRGeIQGpKtphEIi3XT6UgZPtdKHARe6IgWn6u+JFIVSjkNfd4ZIDeWiNxH/87qJCq7dlEZxokiEZ4uChEHF4SQf2KeCYMXGmiAsqL4V4iHSgSidYl6HYC++vExalbJ9WT5vXBSqN/M4cuAYnIAisMEVqIIaqIMmwOARPINX8GY8GS/Gu/Exa10x5jNH4A+Mzx/2jpvQ</latexit> H(Q) ⇡ H0 + 1 2! 2 Q 2 Phonon mode frequency Pallikara et al 2022 Electron. Struct. 4 033002
2 Frequency Pallikara et al 2022 Electron. Struct. 4 033002 `Mode-mapping’: distort the crystal structure along a particular phonon eigenvector(s) to map out the Potential Energy Surface Low-T Pnma phase geometry
thermal conductivity Harmonic Phonons Non-interacting phonons “Infinite lifetimes” Ionic Forces = 0 at equilibrium Taylor expansion of the potential energy surface <latexit sha1_base64="225n3EBXrpPp2rvYZI4j5EnXitw=">AAACmXicdVHdStxAFJ6kWu1q27WCN94MXQShsCRW1AsLWmlZvFqpq8JmDSezk91xZ5Iwc1JYQt6pz9K7vk0n2SCrtgcGPr6fMzPnRJkUBj3vj+O+Wll9vbb+prWx+fbd+/bWhxuT5prxAUtlqu8iMFyKhA9QoOR3meagIslvo9lFpd/+5NqINLnGecZHCiaJiAUDtFTY/tWjX2gv9FqfaNCfilDcByCzKdC8AaGopFgDK/yyOChrVyEeyvtiYQgijlAu+StYceHDcvLoMTl7Gg0moNR/GlSolsNZ1WucognbHa/r1UVfAr8BHdJUP2z/tjmWK54gk2DM0PcyHBWgUTDJy1aQG54Bm8GEDy1MQHEzKurJlnTPMmMap9qeBGnNLicKUMbMVWSdCnBqnmsV+S9tmGN8MipEkuXIE7a4KM4lxZRWa6JjoTlDObcAmBb2rZRNwc4S7TJbdgj+8y+/BDcHXf+o+/nqsHP2tRnHOtklH8k+8ckxOSM90icDwpwd59T55nx3d91zt+deLqyu02S2yZNyf/wFen/KlA==</latexit> H = H0 + ↵ i u ↵ i + 1 2 ↵ ij u ↵ i uj + 1 6 ↵ ijk u ↵ i uj uk + . . .
Example 3-phonon process high energy phonon low energy phonons <latexit sha1_base64="225n3EBXrpPp2rvYZI4j5EnXitw=">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</latexit> H = H0 + ↵ i u ↵ i + 1 2 ↵ ij u ↵ i uj + 1 6 ↵ ijk u ↵ i uj uk + . . . “But it was also known that [….] the so-called anharmonic terms, were important. These terms cause a coupling between the otherwise independent waves of different length and direction. They are responsible for the absorption of sound waves, which, in the linear approximation, could travel indefinite distances without damping, and for the heat conductivity." -- Rudolf Peierls, Bird of Passage (or see “Recollections of early solid state physics”)
thermal conductivity Harmonic Phonons Non-interacting phonons “Infinite lifetimes” Ionic Forces = 0 at equilibrium Force constant tensors describe the PES 2nd order 3rd order
• Calculating force constants is often the most computationally intensive part of a lattice dynamics calculation • Computational cost is determined by: • Rank of tensor (2nd/3rd/4th order) • Number of atoms in system (supercell expansion) • Crystal symmetry From TDEP Code documentation
Theory Intuitive approach to understand Less intuitive theoretical approach Flexible: can be combined with a variety of functionals, and levels of theory Requires implementation for a particular level of theory Can be split into many smaller jobs: can “game” the computer queues Consists of one larger job which requires significant memory Can only calculate perturbations at the gamma point: supercells commensurate with q are required Perturbation of of any wave vector q possible (note: not available in VASP) Scales poorly with system size Improved scaling with system size; for larger jobs can be computationally cheaper
& dielectric tensor Post-process e.g. dispersion, heat capacity Relax unit cell Extract forces, build and diagonalise dynamical matrix Calculate forces + codes that can stitch it all together, e.g. fhivibes TASKS CODES Knoop et al., (2020) Journal of Open Source Software, 5(56), 2671
the dynamical matrix requires supercells that adequately capture all pairwise interactions • For this reason you may want a cell that is fairly cubic • For this reason you need to do convergence testing <latexit sha1_base64="3mNPpIZLEp1bsup7rXZGzT4MJRc=">AAACh3icbVFNbxMxEPUuhbbhK4UjF4sIqSARdmlVeqlUPg4cg0TaSvGymnW8jVN7d2vPokaW/wo/ihv/Bm8aBG0YydKb9954xuOiUdJikvyK4jsbd+9tbm337j94+Ohxf+fJia1bw8WY16o2ZwVYoWQlxihRibPGCNCFEqfFxcdOP/0ujJV19RUXjcg0nFeylBwwUHn/x6ddpgFnReku/ctvjoFqZsAKgeBzJ+eeHlFWGuAu9Y7ZS4NO51Lnbu69Z7YNSHk2msm10mQehO5qo524ajxz8m8nxqc1Tv7kJviD+/WNPPEZ83l/kAyTZdB1kK7AgKxilPd/smnNWy0q5AqsnaRJg5kDg5Ir4XustaIBfgHnYhJgBVrYzC336OmLwExpWZtwKqRL9t8KB9rahS6Cs5vU3tY68n/apMXyMHOyaloUFb9uVLaKYk27T6FTaQRHtQgAuJFhVspnELaO4et6YQnp7Sevg5O3w/RguPdlf3D8YbWOLfKMPCe7JCXvyDH5TEZkTHi0Eb2K9qL9eDt+Ex/Eh9fWOFrVPCU3In7/Gwapyf8=</latexit> D(q)↵ ij = 1 p mimj X l ↵ i0jl exp{iq · [rjl rj0]} Atom-atom interaction
phonons Maurya et al Physical Review B 96, 134114 (2017) If you want to access a off-Γ wave vector q, you need to map that wave vector q to the Γ-point by using a commensurate supercell Bi4 Ti3 O12 (½, 0, 0)
used to reduce computational cost Lloyd-Williams and Monserrat, Phys. Rev. B 92, 184301 Diagonal elements. Supercell is S11 x S22 x S33 primitive cells. e.g. “2x3x1 supercell” is 6 primitive cells diagonal supercell 4 primitive cells non-diagonal supercell 2 primitive cells (½ , ½)
SlideShare) To fix this either increase size of the FFT grid (plane wave basis) or integration grid (atom-centred basis), or apply a post-hoc correction Numerical approaches can lead to broken translational symmetry and small imaginary frequencies around the gamma point. FC_SYMMETRY = .TRUE. FC_SYMMETRY = .FALSE.
a separation of ions (Coloumb interaction) For TO mode: - E ⊥ q ⇒ E.q = 0 Transverse Optic Longitudinal Optic For LO mode: - E.q is non-zero - E-field adds restoring force. - Frequency is upshifted.
Post-hoc correction using Born effective charge and high-frequency (optical, ion- clamped) dielectric tensors <latexit sha1_base64="NuQAH71QC90N78qoGamxiTSmGf8=">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</latexit> D↵ (jj0, q ! 0) = D↵ (jj0, q = 0) + 1 p mjmj0 4⇡ ⌦0 hP q Z⇤ j, ↵ i hP 0 q 0 Z⇤ j0, 0 i P ↵ q↵✏1 ↵ q . D↵ (jj0, q = 0) + 1 p mjmj0 4⇡ ⌦0 hP q Z⇤ j, ↵ i hP 0 q 0 Z⇤ j0, 0 i P ↵ q↵✏1 ↵ q . LO/TO splitting results from the macroscopic electric field associated with a separation of ions (Coloumb interaction)
UK See Pallikara et al 2022 Electron. Struct. 4 033002 • A very well relaxed structure is a pre-requisite • tighten force convergence criteria for structure relaxation (e.g. to < 0.01 eV Å−1) • Accurate forces are essential • converge forces with respect to the basis set, k-point sampling density and SCF criteria • plane-wave basis: increase cut-off energy by at least 25% above default, up to 2× may be required. • numerical atom-centered basis: the default cut-offs tightened by an order of magnitude. • Avoid interpolation artefacts (imaginary frequencies or flat bands) • use supercell (finite displacement) or q-point grid (DFPT) commensurate with wavevector • Make use of the short (or free queue) • Finite difference consists of several small jobs – possible to be done “for free”
lattice parameters are temperature independent, so cannot predict (the effects of) thermal expansion • At finite temperature, the system will minimise its free energy rather than its lattice internal energy Quasi-harmonic approximation can be used to model the effects of thermal lattice expansion • The fit provides equilibrium volume V(T) and the bulk modulus B(T). From this a number of other properties can be derived: <latexit sha1_base64="5ZNUp6FagQwVsCWvXZnp8J1UWGQ=">AAACAXicbVDLSsNAFJ3UV62vqBvBTbAIFaQk4mtZdOOyQl+QhDCZTtqhM5MwMxFKqBt/xY0LRdz6F+78GydtFtp64MLhnHu5954woUQq2/42SkvLK6tr5fXKxubW9o65u9eRcSoQbqOYxqIXQokp4bitiKK4lwgMWUhxNxzd5n73AQtJYt5S4wT7DA44iQiCSkuBeeAxqIaCZYzwieuxNCC11mnzxA/Mql23p7AWiVOQKijQDMwvrx+jlGGuEIVSuo6dKD+DQhFE8aTipRInEI3gALuacsiw9LPpBxPrWCt9K4qFLq6sqfp7IoNMyjELdWd+r5z3cvE/z01VdO1nhCepwhzNFkUptVRs5XFYfSIwUnSsCUSC6FstNIQCIqVDq+gQnPmXF0nnrO5c1i/uz6uNmyKOMjgER6AGHHAFGuAONEEbIPAInsEreDOejBfj3fiYtZaMYmYf/IHx+QMUPpal</latexit> min[µi(T, P)] <latexit sha1_base64="5ZNUp6FagQwVsCWvXZnp8J1UWGQ=">AAACAXicbVDLSsNAFJ3UV62vqBvBTbAIFaQk4mtZdOOyQl+QhDCZTtqhM5MwMxFKqBt/xY0LRdz6F+78GydtFtp64MLhnHu5954woUQq2/42SkvLK6tr5fXKxubW9o65u9eRcSoQbqOYxqIXQokp4bitiKK4lwgMWUhxNxzd5n73AQtJYt5S4wT7DA44iQiCSkuBeeAxqIaCZYzwieuxNCC11mnzxA/Mql23p7AWiVOQKijQDMwvrx+jlGGuEIVSuo6dKD+DQhFE8aTipRInEI3gALuacsiw9LPpBxPrWCt9K4qFLq6sqfp7IoNMyjELdWd+r5z3cvE/z01VdO1nhCepwhzNFkUptVRs5XFYfSIwUnSsCUSC6FstNIQCIqVDq+gQnPmXF0nnrO5c1i/uz6uNmyKOMjgER6AGHHAFGuAONEEbIPAInsEreDOejBfj3fiYtZaMYmYf/IHx+QMUPpal</latexit> min[µi(T, P)]
thermal lattice expansion Expand and contract structures Run phonopy-qha for post- processing Relax unit cell For each structure calculate thermal properties with harmonic phonons For each structure calculate <latexit sha1_base64="pXKRRmiFpcS4vM+9Gp3Bej2RANE=">AAAB+3icbVDLSgMxFM3UV62vsS7dBIvgqsyIr2XxhcsKfUE7Dpk004YmmSHJiGWYX3HjQhG3/og7/8ZM24W2HggczrmXe3KCmFGlHefbKiwtr6yuFddLG5tb2zv2brmlokRi0sQRi2QnQIowKkhTU81IJ5YE8YCRdjC6yv32I5GKRqKhxzHxOBoIGlKMtJF8u3zj04e0x5EeSp5e3zayzLcrTtWZAC4Sd0YqYIa6b3/1+hFOOBEaM6RU13Vi7aVIaooZyUq9RJEY4REakK6hAnGivHSSPYOHRunDMJLmCQ0n6u+NFHGlxjwwk3lINe/l4n9eN9HhhZdSESeaCDw9FCYM6gjmRcA+lQRrNjYEYUlNVoiHSCKsTV0lU4I7/+VF0jquumfV0/uTSu1yVkcR7IMDcARccA5q4A7UQRNg8ASewSt4szLrxXq3PqajBWu2swf+wPr8ASVvlIc=</latexit> EDFT i e-v.dat thermal_properties.yaml Also see fhi-vibes
workflow. 3-phonon interactions give a finite phonon lifetime which can be used to calculate thermal conductivity Single mode relaxation time approximation to the Boltzmann transport equation In Out Example 3-phonon process high energy phonon low energy phonons Heat capacity Group velocity unit cell volume Num. of q in sum Note: This approach ignores higher order terms (e.g. 4-phonon interactions), volume expansion, and scattering from other sources (e.g. impurities or crystal boundaries). Togo et al., Phys. Rev. B 91, 094306
used to calculate thermal conductivity Now calculating: Similar to the harmonic calc BUT • Many more displacements • New parameter “cutpair” • Computationally demanding post-processing Build supercell Build force constant tensor(s) Post-process to extract properties per- mode Relax unit cell Calculate Forces Generate displacements Converge w.r.t. pair cutoff distance Togo et al., Phys. Rev. B 91, 094306
<latexit sha1_base64="ntzB3nN+NJKp1DNfhVEuERxdFjw=">AAACFnicbVDLSsNAFJ3UV62vqEs3g0VwY0nE17LoQpcV7AOaUm6mk3boTBJmJkIJ+Qo3/oobF4q4FXf+jdM2grYeGDiccy93zvFjzpR2nC+rsLC4tLxSXC2trW9sbtnbOw0VJZLQOol4JFs+KMpZSOuaaU5bsaQgfE6b/vBq7DfvqVQsCu/0KKYdAf2QBYyANlLXPvI0JF2Pm40eYC+WUawj7AUSSOpmqXcNQsCPn3XtslNxJsDzxM1JGeWode1PrxeRRNBQEw5KtV0n1p0UpGaE06zkJYrGQIbQp21DQxBUddJJrAwfGKWHg0iaF2o8UX9vpCCUGgnfTArQAzXrjcX/vHaig4tOysI40TQk00NBwrFJPu4I95ikRPORIUAkM3/FZACmEm2aLJkS3NnI86RxXHHPKqe3J+XqZV5HEe2hfXSIXHSOqugG1VAdEfSAntALerUerWfrzXqfjhasfGcX/YH18Q0EOZ/t</latexit> ⌧ / 1 2nd order calculation 3rd order calculation conservation of energy and momentum; are there available states to scatter into? coupling between states Togo et al., Phys. Rev. B 91, 094306
frequencies Vibrational spectra Free Energies Crystal structure prediction Rahim et al. Chem. Sci., 2020,11, 7904-7909 Ruan et al J. Phys. Chem. C 2020, 124, 4, 2265–2272 Jackson et al, Chem. Sci., 2016,7, 1082-1092
S10 (RP phase) Harmonic phonon spectra à Case Study: High temperature equilibrium of 3D and Ruddlesden-Popper (Ban+1 Zrn S3n+1 ) chalcogenide perovskites Kayastha et al, Solar RRL (2023) 7: 2201078 Primary challenge for chalcogenide perovskites: phase control “The synthesis is pretty damn hard” - Jonathan Scragg, MRS Fall 2022 Dr Giulia Longo
along each mode 3) Calculate dielectric constants for each structure Polarisability tensor Mode eigenvector High-frequency dielectric constant ----> database of Raman spectra for all competing phases in Ba-Zr-S system Prakriti Kayastha Poster presentation Kayastha et al, Solar RRL (2023) 7: 2201078
Ba3 Zr2 S7 and Ba4 Zr3 S10 are energetically accessible during synthesis at high T See Jackson and Walsh Phys. Rev. B 2013 88, 165201 Available online: NU-CEM/ThermoPot phonon calcs
Presentation and example files available here: http://github.com/nu-CEM/phonons_tutorial/ Dr Giulia Longo Northumbria Experimental characterization BaZrS3 Prakriti Kayastha Ab-initio calcs BaZrS3 Poster presentation Dr Jonathan Skelton Uni. Manchester Phonopy- spectroscopy Dr Adam Jackson STFC Ab-initio Thermodyna- mics
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