Predicting physical properties of crystalline solids

CSC Spring School in Computational Chemistry 2015

2015-03-13

Antti Karttunen Department of Chemistry

Aalto University

Ab initio materials modelling • Methods based on quantum mechanics • Two major “branches” of methods

– Ab initio molecular orbital theory (MO) – Density functional theory (DFT)

• No system-dependent parametrization required – Only the universal physical constants and the unit cell coordinates of the

system are required to predict the properties of the system – Choosing the right level of theory for a chemical problem is a demanding task

• With the help of powerful computational resources, modern materials modelling techniques can be used to – Assist in the interpretation and explanation of experimental results – Predict the existence and properties of new materials and molecules

• The predictive power of the modern materials modelling techniques enables computational materials design – Most effective in close collaboration with experimental work

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Level of theory • The level of theory determines the reliability of the results:

1. How the electron-electron interactions are described (=”method”) 2. How a single electron is described (=”one electron basis set”)

• The computational resource requirements depend on – The level of theory – The size of the model system (number of atoms) – The type of the model system (molecular or periodic in 1D/2D/3D)

• Here the focus is on calculations based on Density Functional Theory (DFT) – Currently the most practical computational approach for solids – Typically 101 – 103 atoms in the unit cell

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Nucleus e- e-

1. 2.

Models: Three states of matter

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Gas Solid Liquid

• Going from left to right:

– Accurate, non-parametrized molecular and materials modelling research generally becomes more difficult

– Reliable interpretation of the experiments also often becomes somewhat more difficult

– Direct comparison between quantum chemical calculations and experiment becomes more difficult

• The main challenge for liquid systems: finite temperature (T > 0)

Solid state models • Bulk (3D), surfaces (2D), polymers (1D) • Periodic models are defined using a unit cell

(lattice vectors + atomic positions) • Amorphous solids are much more challenging

and comparable to liquids in difficulty!

5 Carbon fullerene (0D) Carbon nanotube (1D)

Diamond (3D)

Graphite (3D) Graphene (2D)

How to find structural data? • ICSD (Inorganic Crystal Structure Database)

– Crystal structures of inorganic compounds – Currently over 173 000 structures (2015-03-11), over 6 000 added every year – Available at http://icsd.fiz-karlsruhe.de/ (if your university has a license)

• CSD (Cambridge Structural Database) – Crystal structure data for organic and organometallic compounds – Over 710 000 structures in the 2015 release (> 40 000 new structures / year) – Mostly molecular crystals – Available at http://webcsd.ccdc.cam.ac.uk/ (if your university has a license)

• Search the literature, the papers often include – Unit cell parameters – Atomic positions

• Or build it from scratch!

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Builders / Visualization • Accelrys Materials Studio (available via http://research.csc.fi/software) • VESTA (http://jp-minerals.org/vesta/en/) • Mainly for visualization: VMD / Jmol

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Introduction to CRYSTAL • A general-purpose program for studying crystalline solids • Development began already in the late 1970s, first public release in 1988

– Symmetry handling was implemented extremely well very early to speed up the calculations

• Both ab initio wavefunction and DFT methods implemented – First program to enable hybrid DFT calculations on solids (1998)

• Based on local Gaussian-type basis sets • Electron-correlated ab initio calculations on solids are also possible: CRYSCOR

– Local-MP2, feasible for systems with 100+ atoms in the unit cell

Models in CRYSTAL • Periodic 3D/2D/1D/0D systems with full symmetry treatment

– 3D crystalline solids (230 space groups) – 2D films and surfaces (80 layer groups) – 1D polymers and nanotubes (75 rod groups + helical symmetries) – 0D molecules (32 crystallographic point groups)

• Geometry manipulation and transformation with automatic symmetry handling – 3D to n3D - supercell creation – 3D to 2D - slab parallel to a selected crystalline face (hkl) – 3D to 0D - cluster from a perfect crystal (H-saturated) – 3D to 0D - extraction of molecules from a molecular crystal – 2D to 1D - building nanotubes from a single-layer slab model – 2D to 0D - building fullerene-like structures from a single-layer slab model – 3D to 1D, 0D - building nanorods and nanoparticles from a perfect crystal – 2D to 0D - construction of Wulff's polyhedron from surface energies – Insertion, displacement, substitution, deletion of atoms

GTOs and Plane waves

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Local Gaussian-type orbitals (GTO)

• Traditional choice in quantum chemistry

• Particularly suitable for molecular calculations

• Relative performance: 0D > 1D > 2D > 3D

• Good for wavefunction methods and hybrid DFT

• Typical programs: CRYSTAL, Gaussian

Periodic (augmented) plane-waves

• Traditional choice in material physics

• Particularly suitable for 3D periodic calculations

• Relative performance: 3D > 2D > 1D > 0D

• Less suitable for wavefunction methods

• Typical programs: VASP, Quantum Espresso, ABINIT

Comparison of CRYSTAL and VASP

• CRYSTAL: PBE0/SVP; VASP: HSE/PAW (≈PBE0/PAW)1

CRYSTAL (GPa) VASP (GPa) Exp. (GPa)

C 470 (+6.0%) 467 (+5.4%) 442

Si 102 (+2.9 .. +7.1%) 99 (+4.2 .. +0.0%) 94.8 .. 99

Ge 71.8 (–2.5 .. –4.3%) 71 (–3.7 .. –5.5%) 73.6 .. 74.9

Sn 45 43 N/A

CRYSTAL (Å) VASP (Å) Exp. (Å)

C 3.563 (-0.1%) 3.549 (-0.5%) 3.567

Si 5.455 (+0.5%) 5.435 (+0.1%) 5.431

Ge 5.716 (+1.0%) 5.682 (+0.4%) 5.657

Sn 6.564 (+1.1%) 6.561 (+1.1%) 6.491

Lattice constant a

Bulk modulus B0

1 Hummer K.; Harl, J.; Kresse, G. Phys. Rev B 2009, 80, 115205.

P

P

P

P

P

P

Physical properties in CRYSTAL14

12 1 R. Dovesi, et al. Int. J. Quant. Chem. 2014, 114, 1287

Orbital/band energies

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Molecular orbitals in a benzene molecule (HOMOs) For solids, we take periodicity into account by plotting E(k), where k is a wave vector Energy bands in bulk silicon (band-projected electron density, isovalue 0.02 a.u.)

Band structure and band gap

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Copper: metal, partially filled bands

Silicon: semiconductor, energy gap between occupied and non-occupied bands

NaCl: insulator, large energy gap between occupied and non-occupied bands

Occupied bands

Empty bands

IR and Raman spectra

• Both IR and Raman intensities are available

• Together with an analysis of the normal modes (e.g. Jmol), allows for very detailed interpretation

Experimental and predicted Raman spectrum of Ba(BrF4)2

Ivlev, S.; Sobolev, V.; Hoelzel, M.; Karttunen, A. J., Müller, T.; Gerin, I.; Ostvald, R.; Kraus, F. Eur. J. Inorg. Chem., 2014, 6261.

Elastic properties (1)

• Bulk modulus: The resistance to uniform compression

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0

100

200

300

400

500

α (dia) I II III IV V VI VII VIII H II-4H I-100 II-100 IV-100

Bulk

mod

ulus

(GPa

)

Carbon Silicon Germanium Tin

Karttunen, A. J.; Härkönen V. J.; Linnolahti, M.; Pakkanen, T. A. J. Phys. Chem. C 2011, 115, 19925–19930.

P

P

P

P

P

P

Elastic properties (2)

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• Young’s modulus: A measure of the stiffness of a material

AF

L ∆L

Thermal expansion

• Quasiharmonic approximation (utilize frequency / volume dependence) • Calculations carried out with Quantum Espresso

18 Härkönen, V. J.; Karttunen, A. J. Phys. Rev. B 2014, 89, 024305.

Thermal conductivity

• Second and third-order force constants determined with Quantum Espresso • Boltzmann Transport Equation solved iteratively with ShengBTE

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1 P. Giannozzi et al. J. Phys. 2009, 21, 395502. 2 W. Li, J. Carrete, N. A. Katcho, N. Mingo, Comp. Phys. Commun. 2014, 185, 1747–1758.

ZnO lattice thermal conductivity (κl)

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1 T. Tynell et al., J. Mater. Chem. A 2014, 2, 12150; 2 G. A. Slack, Phys. Rev. B 1972, 6, 3791. 3 A. J. Karttunen, T. Tynell, M. Karppinen, manuscript

0

50

100

150

200

100 200 300 400 500 600 700 800 900 1000

κ l (W

/(m

K)

T (K)

κ (exp. film - Tynell)

κ (exp. Bulk - Slack)

κ_xx (DFT-PBE)

κ_zz (DFT-PBE)

Thermoelectricity

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Thermoelectricity (2) • Band structure from CRYSTAL • Band structure dependent thermoelectric quantities from Boltzmann transport

equation (BoltzTraP, Comput. Phys. Commun. 2006, 175, 67)

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I12[As12Ge56]

Karttunen, A. J.; Fässler, T. F. ChemPhysChem 2013, 14, 1807

How to start? • A CRYSTAL14 evaluation copy is available free of charge for Linux & Windows

operating systems. All functionalities of the full version are active, the number of atoms in the primitive cell is just restricted to 20. – http://www.crystal.unito.it/how-to-get-a-copy.php – http://www.crystal.unito.it/tutorials/

• CASTEP + Materials Studio (via CSC) • Quantum Espresso (large user base, active development)

– http://www.quantum-espresso.org/ – http://www.quantum-espresso.org/tutorials/

• GPAW (available at CSC) • CP2K (available at CSC, great for ab initio molecular dynamics) • Gaussian09

– Robust, but very slow if you try to run it with the defaults. Please read at least http://scuseria.rice.edu/gau/PBC-Guide.pdf

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