Richard Hennig, University of Florida

1. rhennig@ufl.edu
h-p://hennig.mse.ufl.edu
AI for Materials Science
August 7-8, 2018 • Gaithersburg, MD
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Richard G. Hennig, University of Florida
MPInterfaces - High throughput framework for 2D materials
sulfur
16
S32.065
selenium
34
Se78.96
tellurium
52
Te127.60
platinum
78
Pt
195.08
tungsten
74
W
183.84
molybdenum
42
Mo95.96
chromium
24
Cr51.996
tantalum
73
Ta
180.95
niobium
41
Nb92.906
vanadium
23
V50.942
hafnium
72
Hf
178.49
zirconium
40
Zr91.224
titanium
22
Ti47.867
silicon
14
Si28.086
carbon
6
C12.011
oxygen
8
O15.999
zinc
30
Zn65.38
cadmium
48
Cd112.41
mercury
80
Hg
200.59
beryllium
4
Be9.0122
magnesium
12
Mg24.305
calcium
20
Ca40.078
phosphorus
15
P30.974
arsenic
33
As74.922
antinomy
51
Sb121.76
nitrogen
7
N14.007
aluminum
13
Al26.982
gallium
31
Ga69.723
indium
49
In114.82
boron
5
B10.811
germanium
32
Ge72.64
tin
50
Sn118.710
lead
82
Pb
207.2
GASP - Gene?c algorithm
and machine learning
for structure predic?ons
(b) 2D Pb-O System(a) 2D Sn-O System
0.0 0.2 0.4 0.6 0.8 1.0
O fraction
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
(i)
(ii)
(iii)
Pb O
2D GA
Bulk
0.0 0.2 0.4 0.6 0.8 1.0
O fraction
1.5
1.0
0.5
0.0
Enthalpy(eV/atom)
(i)
(ii)
(iii)
Sn O
2D GA
Bulk
Open source available at h-ps://github.com/henniggroup
Data available at h-p://materialsweb.org
VASPSol - Ab ini?o methods
for solid/liquid interfaces
Solvated Water in DMC
Method Dielectric energy Cavitation energy Solvation energy
DFT -19 mHa
DMC -20(1) mHa
Classical DFT* 4.90 mHa
Classical DFT+DMC -15(1) mHa
Expt5 -10 mHa
5. T. Truong and E. Stefanovich, Chem. Phys. Lett. 240. 253 (1995).* Classical DFT from Ravishankar Sundararaman
Pathways Towards a Hierarchical Discovery of Materials

2. rhennig@ufl.edu
h-p://hennig.mse.ufl.edu
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August 7-8, 2018 • Gaithersburg, MD
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Richard G. Hennig, University of Florida
MPInterfaces - High throughput framework for 2D materials
sulfur
16
S32.065
selenium
34
Se78.96
tellurium
52
Te127.60
platinum
78
Pt
195.08
tungsten
74
W
183.84
molybdenum
42
Mo95.96
chromium
24
Cr51.996
tantalum
73
Ta
180.95
niobium
41
Nb92.906
vanadium
23
V50.942
hafnium
72
Hf
178.49
zirconium
40
Zr91.224
titanium
22
Ti47.867
silicon
14
Si28.086
carbon
6
C12.011
oxygen
8
O15.999
zinc
30
Zn65.38
cadmium
48
Cd112.41
mercury
80
Hg
200.59
beryllium
4
Be9.0122
magnesium
12
Mg24.305
calcium
20
Ca40.078
phosphorus
15
P30.974
arsenic
33
As74.922
antinomy
51
Sb121.76
nitrogen
7
N14.007
aluminum
13
Al26.982
gallium
31
Ga69.723
indium
49
In114.82
boron
5
B10.811
germanium
32
Ge72.64
tin
50
Sn118.710
lead
82
Pb
207.2
GASP - Gene?c algorithm
and machine learning
for structure predic?ons
(b) 2D Pb-O System(a) 2D Sn-O System
0.0 0.2 0.4 0.6 0.8 1.0
O fraction
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
(i)
(ii)
(iii)
Pb O
2D GA
Bulk
0.0 0.2 0.4 0.6 0.8 1.0
O fraction
1.5
1.0
0.5
0.0
Enthalpy(eV/atom)
(i)
(ii)
(iii)
Sn O
2D GA
Bulk
Open source available at h-ps://github.com/henniggroup
Data available at h-p://materialsweb.org
Search for materials
• Materials structure and microstructure
• Thermodynamic stability
• Structure searches for 2D materials by
datamining, chemical subs?tu?on,
evolu?onary algorithms, machine learning
Pathways Towards a Hierarchical Discovery of Materials

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Acknowledgment
• MPInterfaces and novel 2D materials: K. Mathew, A. Singh, M. Ashton, J. Paul, D. Gluhovic, H. Zhuang,
J. Gabriel, M. Blonsky, M. Johannes, R. Ramanathan, R. Duan, Z. Ziyu, F. Tavazza, S. SinnoQ, D. Stewart
• GASP gene?c algorithm and machine learning: B. Revard, W. Tipton, A. Yesupenko, S. Honrao, S. Xie,
A. M. Tan, B. Antonio
• Financial support by NSF, DOE, NIST
• Computa?onal resources: HiPerGator@UF, NSF XSEDE, and Google Cloud PlaXorm

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Part I: Hierarchy of Materials Structures

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Materials Hierarchy - Structure ClassificaXon
Acta Cryst. A48, 928 (1992)
Long-range order present? Measured by sharp diffrac?on peaks.
Yes
Crystal
No
Glassk =
nX
i=1
hia⇤
i
<latexit sha1_base64="iNkRgWQbNxgQAb2Yjad7nXc4efo=">AAACEnicbZBNS8MwHMZTX+d8q3r0EhyCXkYrgl4GA0E8TnAvsHYlzdItLElLkgqjFPwGXvwqXjwo4tWTN7+NWbeDbj4Q+PE8/5D8nzBhVGnH+baWlldW19ZLG+XNre2dXXtvv6XiVGLSxDGLZSdEijAqSFNTzUgnkQTxkJF2OLqa5O17IhWNxZ0eJ8TnaCBoRDHSxgrs08wLIzjKYQ16KuVBRmtu3hNwGFBYRKjnIaUDmgd2xak6heAiuDOogJkagf3l9WOcciI0Zkiprusk2s+Q1BQzkpe9VJEE4REakK5BgThRflaslMNj4/RhFEtzhIaF+/tGhrhSYx6aSY70UM1nE/O/rJvq6NLPqEhSTQSePhSlDOoYTvqBfSoJ1mxsAGFJzV8hHiKJsDYtlk0J7vzKi9A6q7pO1b09r9SvH6Z1lMAhOAInwAUXoA5uQAM0AQaP4Bm8gjfryXqx3q2P6eiSNavwAPyR9fkD7zGdbw==</latexit><latexit sha1_base64="iNkRgWQbNxgQAb2Yjad7nXc4efo=">AAACEnicbZBNS8MwHMZTX+d8q3r0EhyCXkYrgl4GA0E8TnAvsHYlzdItLElLkgqjFPwGXvwqXjwo4tWTN7+NWbeDbj4Q+PE8/5D8nzBhVGnH+baWlldW19ZLG+XNre2dXXtvv6XiVGLSxDGLZSdEijAqSFNTzUgnkQTxkJF2OLqa5O17IhWNxZ0eJ8TnaCBoRDHSxgrs08wLIzjKYQ16KuVBRmtu3hNwGFBYRKjnIaUDmgd2xak6heAiuDOogJkagf3l9WOcciI0Zkiprusk2s+Q1BQzkpe9VJEE4REakK5BgThRflaslMNj4/RhFEtzhIaF+/tGhrhSYx6aSY70UM1nE/O/rJvq6NLPqEhSTQSePhSlDOoYTvqBfSoJ1mxsAGFJzV8hHiKJsDYtlk0J7vzKi9A6q7pO1b09r9SvH6Z1lMAhOAInwAUXoA5uQAM0AQaP4Bm8gjfryXqx3q2P6eiSNavwAPyR9fkD7zGdbw==</latexit><latexit sha1_base64="iNkRgWQbNxgQAb2Yjad7nXc4efo=">AAACEnicbZBNS8MwHMZTX+d8q3r0EhyCXkYrgl4GA0E8TnAvsHYlzdItLElLkgqjFPwGXvwqXjwo4tWTN7+NWbeDbj4Q+PE8/5D8nzBhVGnH+baWlldW19ZLG+XNre2dXXtvv6XiVGLSxDGLZSdEijAqSFNTzUgnkQTxkJF2OLqa5O17IhWNxZ0eJ8TnaCBoRDHSxgrs08wLIzjKYQ16KuVBRmtu3hNwGFBYRKjnIaUDmgd2xak6heAiuDOogJkagf3l9WOcciI0Zkiprusk2s+Q1BQzkpe9VJEE4REakK5BgThRflaslMNj4/RhFEtzhIaF+/tGhrhSYx6aSY70UM1nE/O/rJvq6NLPqEhSTQSePhSlDOoYTvqBfSoJ1mxsAGFJzV8hHiKJsDYtlk0J7vzKi9A6q7pO1b09r9SvH6Z1lMAhOAInwAUXoA5uQAM0AQaP4Bm8gjfryXqx3q2P6eiSNavwAPyR9fkD7zGdbw==</latexit><latexit sha1_base64="iNkRgWQbNxgQAb2Yjad7nXc4efo=">AAACEnicbZBNS8MwHMZTX+d8q3r0EhyCXkYrgl4GA0E8TnAvsHYlzdItLElLkgqjFPwGXvwqXjwo4tWTN7+NWbeDbj4Q+PE8/5D8nzBhVGnH+baWlldW19ZLG+XNre2dXXtvv6XiVGLSxDGLZSdEijAqSFNTzUgnkQTxkJF2OLqa5O17IhWNxZ0eJ8TnaCBoRDHSxgrs08wLIzjKYQ16KuVBRmtu3hNwGFBYRKjnIaUDmgd2xak6heAiuDOogJkagf3l9WOcciI0Zkiprusk2s+Q1BQzkpe9VJEE4REakK5BgThRflaslMNj4/RhFEtzhIaF+/tGhrhSYx6aSY70UM1nE/O/rJvq6NLPqEhSTQSePhSlDOoYTvqBfSoJ1mxsAGFJzV8hHiKJsDYtlk0J7vzKi9A6q7pO1b09r9SvH6Z1lMAhOAInwAUXoA5uQAM0AQaP4Bm8gjfryXqx3q2P6eiSNavwAPyR9fkD7zGdbw==</latexit>
n=d
Periodic crystal
n>d
Aperiodic crystal
Forbidden
rota?ons
Yes
Quasicrystal
No
Incommensurate crystal
• Lack of transla?onal symmetry complicates methods
• Concepts of unit cell(s), ?lings, laece sites
• Complica?on due to par?al occupancy of laece sites
Structures lacking periodicity require new computaXonal methods.

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Materials Hierarchy - Defects by Dimensionality
3D - Volume defects
Precipitates, voids, inclusions
0D - Point defects
present in thermodynamic equilibrium
1D - Line defects
present in thermodynamic equilibrium
Disloca?ons
Disclina?ons
2D - Area defects
Stacking faults, twin, grain,
and interface boundaries, surfaces
High-throughput methods for 1D to 3D defects needed

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Materials Hierarchy - Microstructure
Acta Cryst. A48, 928 (1992)
We thus define microstructure as all the existing defects
in a material, which are not in thermodynamic equilibrium
(according to type, number, distribution, size, and shape).
Peter Haasen, Physical Metallurgy

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• Combina?on of crystal/amorphous structure, point defects, and microstructure
• Microstructure is processing dependent
• Modelling of processing requires
dimensionality reducXons
Opportunity for machine learning
• Issue is lack of good data and
objecXve funcXons
• Overcome by merging Physics and AI?
• Incorporate knowledge of possible phases
and defects requires mul?scale approaches
• Opportunity to develop and use
defect databases
Materials Hierarchy - Microstructure
Acta Cryst. A48, 928 (1992)
How to prodict microstructures from phases, defects, and processing?

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Part II: Materials Energy Landscapes

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Mixed representaXon of materials structure space (real space representaXon)
Dimensionality of materials structure space
• Determined by number of atoms: 3Natom + 3 for crystals, 3Natom for molecules and clusters
Complexity of materials structure space
• Exponen?al increases with number of atoms and number of species
• Reduced by space and point group symmetries
Materials Structure Space
Integer variables:
• Number of atoms
• Atomic species
Real variables:
• Atom posi?ons
• Laece parameters (for crystalline structures)
Energy landscape space has conXnues and integer variables

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ComputaXonal structure predicXon based on opXmizaXon
• Stable structure at (p, T) Lowest Gibbs free energy, G(p, T) = E(V, S) – TS + pV
• Gibbs free energy of a structure is an integral over the basin of the local minimum
- Expensive to calculate
- Difficult to explore
due to discon?nui?es
- Can contain minima not present
in the energy landscape
• No similar objec?ve func?on exists
for microstructure, which is
processing path dependent
Minimum Gibbs Free Energy Principle
Structure
Energy
G
E
Free energy landscape space is disconXnuous and difficult to search

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ComputaXonal structure predicXon based on energy minimizaXon
• Combined op?miza?on over real and integer variables
• For mul?-component materials, we need to consider
stability against compe?ng phases
➡ Concept of convex hull
Minimum Energy Principle
00.2 0.4 0.6 0.81
Zr Fraction, XZr
-0.20
-0.15
-0.10
-0.05
0.00
FormationEnergy
E0
ZrCu
• Density-func?onal theory offers balance of speed and accuracy
• Pseudopoten?als and plane-wave basis (VASP)
InterpolaXve
(Semi-)empirical methods
ExtrapolaXve and predicXve
First-principles or ab-ini7o methods
Structure for mulX-component materials requires knowledge of compeXng phases
Energy methods

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Bell–Evans–Polanyi principle
• Highly exothermic chemical reac?ons have low ac?va?on energies
• Low-energy basins are expected to occur
near other low-energy basins
• These regions are referred to as ‘funnels’
General Features of the Energy Landscape

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Probability distribuXon of energies of local minima
• Basins with lower-energy minima have larger hyper-volumes
• Related to similar elas?c constants and
vibra?onal frequencies for different structures
• Power law probability distribu?on of these
hyper-volumes
• Order in the arrangement of basins
of different sizes
• Smaller basins filling gaps
between larger ones
General Features of the Energy Landscape
For each system, we ran two sets of simulations. First,
constant-temperature molecular dynamics was performed
and by regularly minimizing configurations generated along
the trajectory we obtained pmin͑E,T͒ and hence ⍀min͑E͒ ͑and
a Gaussian fit to it͒. We chose temperatures well above the
glass transition where equilibrium sampling of the liquid is
easy to obtain. Second, Metropolis Monte Carlo simulations
were performed on the transformed landscape, from which
pmin
trans
͑E,T͒ and hence A͑E͒ was obtained. As a check of the
sampling, pmin
trans
͑E,T͒ distributions were collected at several
temperatures. The resulting area distributions were in good
agreement, confirming the reliability of the approach.
In Fig. 3͑a͒, A͑E͒ is depicted for these three systems. In
agreement with our expectation that the deeper, more con-
nected minima should have larger basins of attraction, the
basin areas decrease very rapidly with increasing energy in
an approximately exponential manner. The probability distri-
butions for the hyperareas of the basins of attraction are il-
lustrated in Fig. 3͑b͒. It is apparent that the distributions
10
10
10
1
-1 -0.99 -0.98 -0.97 -0.96 -0.95 -0.94 -0.93
A/Amax
Emin−E /
BLJ
Dz−10
−5
−15
(a)
Si
(b)
BRIEF REPORTS PHYSICAL REVIEW E 75, 037101 ͑2007͒
Binary Lennard-Jones
Amorphous silicon
Dzugutov liquids
Massen & Doye, Phys. Rev. E 75, 037101 (2007)Searching for the larger basins reduces effecXve dimensionality.

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CorrelaXon between energy and symmetry
• Low (and high) energy minima tend to correspond to symmetrical structures
• High symmetry of low-energy minima supported by the ubiquity of crystals
Example: 55-atom Lennard-Jones clusters (D. Wales ’98)
Lowest energy Two highest energy
Symmetry and Structural MoXfs
The Rule of Parsimony: “The Number of essen7ally different kinds
of cons7tuents in a crystal tends to be small.” (Linus Pauling 1929)
( )D.J. WalesrChemical Physics Letters 285 1998 330–336 335
Symmetry further reduces problem dimensionality.

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Frequency of Space Groups
Space Group Pnma P21/c Fm-3m Fd-3m
Frequency 7.4% 7.2% 5.6% 5.1%
Examples Fe3C, CaTiO3 ZrO2 Cu C, Cu2Mg
Inorganic systems show different space group frequencies
• 67% of about 100,000 compounds occur in only 24 space groups
For crystals of small organic molecules
• 75% of about 30,000 compounds occur in only five space groups
• 29 space groups only have one entry and 35 space groups none at all
Space Group P21/c P-11 P212121 P21 C2/c
Frequency 36% 14% 12% 7% 7%
Some space groups are much more common than others in crystals
• Note: bcc is not among one of the top 24 space groups

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Specific Features of Energy Landscapes
Chemical consideraXons
• Know a great deal about the chemistry of the systems we study
• Know which atomic types prefer to bond to one another
• Approximate bond lengths
• Likely coordina?on numbers of the atoms
ResulXng empirical rules
• Hume-Rothery rules
• Laves rules for intermetallics
• Pauling rules for ionic materials
• Peefor structure maps
...
How can we uXlize this informaXon to accelerate structure searches?

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Crystal Structure PredicXon Methods
Limited success of convenXonal opXmizaXon methods
• Guess of structures based on symmetry and chemistry (trial-and-error)
• Simulated annealing (molecular dynamics, cooling from high temperature)
Advances in opXmizaXon methods:
• Accelerated molecular dynamics, Voter et al. ’97
• Basin hopping, Wales et al.
• Minima hopping, search for neighboring minima, Goedecker ’04
• Metadynamics, Parinello et al. ’02
• Random search, randomly sampling phase space, Pickard & Needs ’06
• EvoluXonary algorithms
Genera?on of structures, Bush et al. ’95, Woodley et al. ’99, Oganov ’06
• ParXcle Swarm OpXmizaXon, Ma et al. ’10
• Datamining of structure databases, Ceder et al.
Configuration coordinate
3N dimensional
Energy
Molecular dynamics
Configuration coordinate
3N dimensional
Energy
Relaxation of random configurations
Mutation
Heredity
Extension of methods to defects complicated by increased complexity.

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Part III: Materials Informa?cs for 2D Materials

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H. L. Zhuang and RGH, JOM 66, 366 (2014)
Structure
ProperXes
Processing Performance
Materials
Selection
Structure
Properties
Processing
Data Mining
Structural Stability
Dynamic Stability
Optotelectronics
Substrates
Layered bulk materials
Formation Energy
Phonon Spectrum
Bandgap, Offsets,
Absorption, Excitons
Adsorption,
Strain, Doping
Genetic Algorithm
New Compositions
and Structures
Performance
Chemical Stability Pourbaix Diagrams
Electronic Devices
Energy Applications
I-V Characteristic
Carrier Mobility
Light conversion,
Catalytic Activity
Magnetism
Chemical Substitutions
Electrochemistry
Etching, Solutions
Domains, Critical T
Materials InformaXcs of 2D Materials

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Structure and Stability of 2D Materials
• ClassificaXon of 2D materials
• Criteria for stability ΔEf < 200 meV/atom
• Methods for 2D materials discovery
‣ Datamining for exfolia?on
‣ Evolu?onary algorithm searches
‣ Chemical subs?tu?ons and etching
• Characteriza?on of 2D materials
‣ Pourbaix diagrams
‣ Photocatalysis
‣ 2D half-metals
‣ Spin-orbit and 2D magne?sm
‣ Substrate control of phase stability

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ClassificaXon of 2D Materials
C (Graphene) P (Phosphorene) 2H-MoS2
1T-MoS2
Litharge SnO Ti2
CO2
(MXene)
a) Crystalline 2D Materials c) Amorphous 2D SiO2
b) Quasicrystalline 2D
BaTiO3
DefiniXon: A 2D material is a material with finite thickness in one dimension
and an essenXally infinite extent in the other two dimensions.
•Classifica?on into crystalline and amorphous materials (long-range order)
•2D crystals include periodic and aperiodic crystals based on point group
•Aperiodic 2D crystals include incommensurate crystals and quasicrystals
Review article: Paul, Hennig, et al. J. Phys.: Cond. Matter 2017
Cockayne, Mihalkovič, Henley, Phys. Rev. B 93, 020101(R)

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Thermodynamic Stability of 2D Materials
FormaXon energy
• Comparison of VASP PAW using vdW-DF optB88
with QMC and RPA shows accuracy of ΔEf of
30 meV/atom1
Gf = G2D min
phases
G3D
approximated by Ef = E2D min
phases
E3D
2D materials are metastable
• Energy can always be lowered by van
der Waals interac?ons for mul?-layers
StabilizaXon by substrates
• Substrates used to nucleate and grow 2D materials
• Stabiliza?on through physi- and chemisorp?on
ΔEf < 200 meV/atom
for free-standing
2D materials
min over single or mixture of phases
1 T. Björkman et al., Phys. Rev. Le-. 108, 235502 (2012).

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Structure and Stability of 2D Materials
• Classifica?on of 2D materials
• Criteria for stability ΔEf < 200 meV/atom
• Methods for 2D materials discovery
‣ Datamining for exfoliaXon
‣ Chemical subs?tu?ons and etching
‣ Evolu?onary algorithm searches
• Characteriza?on of 2D materials
‣ Pourbaix diagrams
‣ Photocatalysis
‣ 2D half-metals
‣ Spin-orbit and 2D magne?sm
‣ Substrate control of phase stability

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Datamining to Discover 2D Materials
from mpinterfaces.utils import
get_structure_type
layered = []
for s in structures:
if get_structure_type(s) ==
“layered”:
layered.append(s)
Ashton, Paul, Sinno- & Hennig. Phys. Rev. Le-. 118, 106101 (2017)
h-ps://arxiv.org/abs/1610.07673 (2016)

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Datamining to Discover 2D Materials
from mpinterfaces.utils import
get_structure_type
layered = []
for s in structures:
if get_structure_type(s) ==
“layered”:
layered.append(s)
Hennig et al., arXiv:1610.07673 (10/2016)
Marzari et al., arXiv:1611.05234 (11/2017)
Cheon, Reed, et al., Nano Lett. (2/2017)
Ashton, Paul, Sinno- & Hennig. Phys. Rev. Le-. 118, 106101 (2017)
h-ps://arxiv.org/abs/1610.07673 (2016)

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1 Björkman, et al. Phys. Rev. Lett. 2012
2 Lebègue, et al. Phys. Rev. X 2013
Searching for Layered Compounds
The ideal case Search for materials with:
You’ll find about 100 layered materials in the ICSD1,2
•low packing fractions
•gaps along c-axis
Reality
Gaps along other axes High packing fractionsNo planar gaps at all Gaps along multiple axis
a) Co2
NiO6
(mp-765906) b) V2
O5
(mp-25643)
c) Ge2
Te5
As2
(mp-14791) d) Li-WCl6
(mp-570512)
c
b
c
a
a) Co2
NiO6
(mp-765906) b) V2
O5
(mp-25643)
c) Ge2
Te5
As2
(mp-14791) d) Li-WCl6
(mp-570512)
c
b
c
b
c
b
c
a
a) Co2
NiO6
(mp-765906) b) V2
O5
(mp-25643)
c) Ge2
Te5
As2
(mp-14791) d) Li-WCl6
(mp-570512)
c
b
c
a
Co2NiO6
mp-765906
V2O5
mp-25643
a) Co2
NiO6
(mp-765906) b) V2
O5
(mp-25643)
c) Ge2
Te5
As2
(mp-14791) d) Li-WCl6
(mp-570512)
c
b
c
b
c
b
c
a
Ge2Te5As2
mp-14791
Li-WCl6
mp-570512

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The Topology Scaling Algorithm
Cluster of 9 atoms
1×1×1 cellCo2NiO6
mp-765906
2×2×2 cell
Cluster of 36 atoms
N2 scaling Layered Material
Ashton, Paul, Sinnott & Hennig. Phys. Rev. Lett. 118, 106101 (2017)

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Layered Materials in MaterialsProject
Layered materials: 1,560 (2%)All materials: about 65,000
quinary (1.2%)unary (0.6%)
binary
ternary
quaternary
40.8%
10.1%
47.3%
quinary
other (1.2%)unary (0.7%)
binary
ternary
quaternary16.4%
7.8%
23.4%
50.5%
b) Compositional complexity of
layered materials (this work)
c) Compositional complexity of all
materials (Materials Project database)
ABC2
AB3
AB
ABC
AB2
Other
4.9%
7.8%
5.9%
11.8%
19.6%
50.0%
(a) Stoichiometries of 2D materials
)
y quinary
other (1.2%)unary (0.7%)
binary
ternary
quaternary16.4%
7.8%
23.4%
50.5%
c) Compositional complexity of all
materials (Materials Project database)
826/1560 have unique monolayers and are nearly stable with Ehull < 50 meV/atom.
IdenXfied 625 2D materials with energy below 150 meV/atom
Binary and ternary compounds more likely to be layered.
Exfoliation energy
Ashton, Paul, Sinnott & Hennig. Phys. Rev. Lett. 118, 106101 (2017)

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Online Database: hQps://materialsweb.org
Use the topology-scaling algorithm
on your own structures
Browse 2D materials
00765
Ashton, Paul, Sinnott & Hennig. Phys. Rev. Lett. 118, 106101 (2017)

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Online Database: hQps://materialsweb.org
Structural, electronic and thermodynamic
data available for all materials
Unique crystal structures for each 2D material
stoichiometry in the DB
Ashton, Paul, Sinnott & Hennig. Phys. Rev. Lett. 118, 106101 (2017)

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Structure Prototypes
mp-21405)
mp-27440)
p-2798)
mp-938)
p-2231)
Figure 6
b) P (mp-157)
a) C (mp-48)
d) Sb (mp-104)
e) Bi (mp-23152)
c) As (mp-158)
a) GaSe (mp-568263) b) GaS (mp-9889)
d) AuSe (mp-2793) e) BN (mp-604884)
m) CuI (mp-570136)
c) InSe (mp-21405)
g) AuBr (mp-505366)
j) CuBr (mp-22917) l) ZrCl (mp-27440)
o) SiP (mp-2798)
f) GeTe (mp-938)
n) TlF (mp-720)
h) SnSe (mp-8936)
k) CuTe (mp-20826)
i) SnS (mp-2231)
Figure 4 Figure 6
b) P (mp-157)
a) C (mp-48)
d) Sb (mp-104)
e) Bi (mp-23152)
c) As (mp-158)
Binary A-B 2D Materials
Elementary 2D Materials
…
Ashton, Paul, Sinnott & Hennig. Phys. Rev. Lett. 2017

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Using Crystal Structure Templates
Ashton, Paul, Sinnott & Hennig. Phys. Rev. Lett. 2017
Unique AB crystal structures Genetic algorithms
Element substitution

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Structure and Stability of 2D Materials
• Classifica?on of 2D materials
• Criteria for stability ΔEf < 200 meV/atom
• Methods for 2D materials discovery
‣ Datamining for exfolia?on
‣ EvoluXonary algorithm searches
‣ Chemical subs?tu?ons and etching
• Characteriza?on of 2D materials
‣ Pourbaix diagrams
‣ Photocatalysis
‣ 2D half-metals
‣ Spin-orbit and 2D magne?sm
‣ Substrate control of phase stability

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GeneXc Algorithm Search for 2D Materials
ModificaXon for
0D, 1D, 2D structure search
h-ps://github.com/henniggroup/gasp-python
W. W. Tipton, RGH, J. Phys.: Cond. Ma-er 25, 495401 (2013)
B. C. Revard, W. W. Tipton, A. Yesypenko, R. G. Hennig, PRB 93, 054117 (2016)
Surface
Solvent or Vacuum
⇒+
Solvent or Vacuum
Begin
Create Pool from
Initial Population
No Yes
Create Initial
Population
Done!
Create Offspring
Organism
Pre-Evaluation
Development
Structure Relaxation and Energy
Evaluation with External Code
Post-Evaluation
Development
Convergence
Achieved?
Add Offspring
to Pool
Grand canonical geneXc algorithm
Variable number of atoms and composiXon

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GeneXc Algorithm – 2D Tin and Lead Oxides
h-ps://github.com/henniggroup/gasp
W. W. Tipton, RGH, J. Phys.: Cond. Ma-er 25, 495401 (2013)
B. C. Revard, W. W. Tipton, RGH, Topics in Current Chemistry (2014)
B. C. Revard, W. W. Tipton, A. Yesypenko, R. G. Hennig, PRB 93, 054117 (2016)
(i) HB hexagonal
(ii) Litharge
(iii) 1T
ConfirmaXon of our previous predicXon of litharge structure for 2D SnO and PbO
New predicXon of 1T structure for 2D SnO2 and PbO2
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
O fraction
− 1 .5
− 1 .0
− 0 .5
0 .0
Enthalpy(eV/atom)
(i)
(ii)
(iii)
Sn O
2 D GA
Bulk
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
O fraction
− 1 .2
− 1 .0
− 0 .8
− 0 .6
− 0 .4
− 0 .2
0 .0
0 .2
(i)
(ii)
(iii)
Pb O
2 D GA
Bulk
2D Sn-O Phases 2D Pb-O Phases

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GeneXc Algorithm – 2D Tin and Lead Oxides
h-ps://github.com/henniggroup/gasp
W. W. Tipton, RGH, J. Phys.: Cond. Ma-er 25, 495401 (2013)
B. C. Revard, W. W. Tipton, RGH, Topics in Current Chemistry (2014)
B. C. Revard, W. W. Tipton, A. Yesypenko, R. G. Hennig, PRB 93, 054117 (2016)
2D Sn-O Phases 2D Pb-O Phases
The 2D SnO2 and PbO2 1T phases are stable in water for ionic concentraXons of 10–6!
SnO2
(2D)
HSnO2
-
SnO3
2-
SnOH+
Sn2+
SnO(OH)+
+1
-1
0
E(V)
0 2 4 6 8 10 12 14
pH
-2
Pb2+
PbOH+
PbO2
(2D)
HPbO2
-
+2
+1
-1
0
E(V)
0 2 4 6 8 10 12 14
pH
-2

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Part IV: Machine Learning of Energy Landscapes

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Structure representaXon (features) should ideally fulfill three criteria
(i) Invariance with respect to choice of unit cell and crystal symmetry
(ii) Uniqueness, so no two different crystal structures have the same vector representa?on
(iii)ConXnuity, such that the energy difference between two crystal structures with vector
representa?ons x1 and x2 goes to zero in the limit || x1 − x2 ||→ 0
Machine Learning Surrogate Models
Machine Learning Regression
• Takes a vector x ∈ Rn as input and return a scalar y
• Must first construct a vector-based data representa?on of the crystal structure that
encodes relevant physical informa?on, i.e. chemical iden?ty and posi?on of the atoms
S. Honrao, RGH at al., submi-ed (2018)

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ParXal Radial DistribuXon FuncXons
gAB(r) =
1
NA
NAX
i=1
1X
j=1
1
rn
exp
"
r dAB
ij
2
2 2
g
#
⇥(dc dAB
ij )
• Captures primary distance dependence of bonds
• Criteria:
+ Invariance
+ ConXnuity
- Uniqueness
• Cannot dis?nguish between homometric structures,
i.e. structures of iden?cal atoms that exhibit
the same set of interatomic distances
Li4Ge
representation of a given structure as input to quickly evaluates
the structure’s energy.
Motivation
Calculating the energies of crystal structures involves
performing arduous quantum-mechanical calulcations. For
calculating energies of larger sets of crystals, this computational
time can take an hour up to a month.
Machine learning techniques enable us to calculate the energies
of a smaller set of structures, and develop a surrogate model
describing the relationship between structure and energy. This
model can then be used to quickly compute the energy of other
candiate structures. The goal is to reduce the amount of time
required to perform energy calculations.
Support Vector Regression Energy Model
Support Vectors
Non-support Vectors
Fit
Machine Learning
: Distance between and atoms
: Number of atoms
: Gaussian Width
Atoms in 2-D Space Generation of
: Atoms of Type
: Atoms of Type
Crystal Structures Represented As:
: Bin Size
80
100
120
140
Pair Correlation: Li4Ge
(Li,Ge)
(Ge,Ge)
(Li,Li)
Examples:
S. Honrao, RGH at al., submi-ed (2018)

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ParXal Radial DistribuXon FuncXons
Ge-Ge
Li-Ge
Li-Li
Distance (Å)
0 5 10 15 20
gAB
(r)
0.5
0.4
0.3
0.2
0.1
0.0
gAB(r) =
1
NA
NAX
i=1
1X
j=1
1
rn
exp
"
r dAB
ij
2
2 2
g
#
⇥(dc dAB
ij )
• Captures primary distance dependence of bonds
• Criteria:
+ Invariance
+ ConXnuity
- Uniqueness
• Cannot dis?nguish between homometric structures,
i.e. structures of iden?cal atoms that exhibit
the same set of interatomic distances
Li4Ge
S. Honrao, RGH at al., submi-ed (2018)

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candiate structures. The goal is to reduce the amoun
required to perform energy calculations.
Contact Info
Support Vector Regression Energy Model
Optimization
Support Vectors
Non-support Vectors
Fit
Machine Learning
Machine Learning
Machine-learning models
• Use kernel-ridge regression (KRR), ε-support vector regression (SVR), and neural networks
Input data preprocessing
• Feature scaling of components of input vector xi = giAB(r) in training set
to obtain zero mean and unit standard devia?on
• Standardizing each set of components avoids the norm from
being biased towards vector components with higher variance
• 30% of data for fieng, 70% for tes?ng
x0
ij =
xij µi
i
S. Honrao, RGH at al., submi-ed (2018)

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0.4 0.3 0.2 0.1 0.0 0.1 0.2
DFT Energy [eV/atom]
0.4
0.3
0.2
0.1
0.0
0.1
0.2
PredictedEnergy[eV/atom]
(a) Unrelaxed formation energies
Machine Learning Methods
Algorithm MAE RMS R2
KRR 12.7 20.4 0.98
SVR 13.6 20.8 0.98
NN 12.8 20.2 0.98
• Similar predic?on error for
different machine-learning techniques
• NN most demanding
• SVR is computa?onally most efficient
and provides best tradeoff
between complexity and predic?on error
SVR
Error on
tes?ng data
S. Honrao, RGH at al., submi-ed (2018)

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0.4 0.3 0.2 0.1 0.0 0.1 0.2
DFT Energy [eV/atom]
0.4
0.3
0.2
0.1
0.0
0.1
0.2
PredictedEnergy[eV/atom]
(b) Relaxed formation energies
Learning of Basin of AQracXons from Unrelaxed Structures
• Predic?on of the relaxed energies (minima) from unrelaxed structures
SVR
Error on
tes?ng data
Algorithm MAE RMS R2
KRR 12.7 20.4 0.98
SVR 13.6 20.8 0.98
NN 12.8 20.2 0.98
KRR-min 11.8 20.3 0.98
SVR-min 13.4 20.9 0.98
• Similar accuracy for minima predic?on
• Useful for screening of GA structures to avoid
costly DFT relaxa?ons
S. Honrao, RGH at al., submi-ed (2018)ML learning of minima has similar accuracy as learning of energy landscape

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Richard G. Hennig, University of Florida
MPInterfaces - High throughput framework for 2D materials
sulfur
16
S32.065
selenium
34
Se78.96
tellurium
52
Te127.60
platinum
78
Pt
195.08
tungsten
74
W
183.84
molybdenum
42
Mo95.96
chromium
24
Cr51.996
tantalum
73
Ta
180.95
niobium
41
Nb92.906
vanadium
23
V50.942
hafnium
72
Hf
178.49
zirconium
40
Zr91.224
titanium
22
Ti47.867
silicon
14
Si28.086
carbon
6
C12.011
oxygen
8
O15.999
zinc
30
Zn65.38
cadmium
48
Cd112.41
mercury
80
Hg
200.59
beryllium
4
Be9.0122
magnesium
12
Mg24.305
calcium
20
Ca40.078
phosphorus
15
P30.974
arsenic
33
As74.922
antinomy
51
Sb121.76
nitrogen
7
N14.007
aluminum
13
Al26.982
gallium
31
Ga69.723
indium
49
In114.82
boron
5
B10.811
germanium
32
Ge72.64
tin
50
Sn118.710
lead
82
Pb
207.2
GASP - Gene?c algorithm
and machine learning
for structure predic?ons
(b) 2D Pb-O System(a) 2D Sn-O System
0.0 0.2 0.4 0.6 0.8 1.0
O fraction
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
(i)
(ii)
(iii)
Pb O
2D GA
Bulk
0.0 0.2 0.4 0.6 0.8 1.0
O fraction
1.5
1.0
0.5
0.0
Enthalpy(eV/atom)
(i)
(ii)
(iii)
Sn O
2D GA
Bulk
Open source available at h-ps://github.com/henniggroup
Data available at h-p://materialsweb.org
Pathways Towards a Hierarchical Discovery of Materials
Search for materials
• Materials structure and microstructure
• Need for new methods for aperiodic
structures, non-equilibrium defects and
processing
• Possible role of ML for dimensionality
reduc?on, cannot violate physics/thermodynamics