Electronic Supplemental Materials π π΄ π π - RSC
15
Electronic Supplemental Materials for Thermodynamic stability of stoichiometric LaFeO 3 and BiFeO 3 : hybrid DFT study Eugene Heifets 1 , Eugene Kotomin 1,2 , A. A. Bagaturyants 3,4 , and Joachim Maier 1 1 Max Planck Institute for Solid State Research, Heisenbergstr. 1, Stuttgart 70569, Germany 2 Institute for Solid State Physics, The University of Latvia, Riga, 8 Kengaraga str., Riga 1063, Latvia 3 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye shosse 31, Moscow 115409 Russia 4 Photochemistry Center RAS Federal State Institution, Federal research center Crystallography and Photonics Russian Academy of Science, 7a Novatorov St., Moscow, 119421 Russia Basis Sets Calculations of the electronic structure for crystals are performed using expansion of one-electron crystalline orbitals into linear combination of Bloch functions built from local atom- (;) (;) centered functions : ( - - ) , (;) = β , () (;) , (;) = β ( - - ) β
where A ΞΌ is vector describing position of atom ΞΌ in unit cell, g denotes summation over all lattice vectors R g . Sets of such local functions ( or atomic orbitals: AOs ) for each atom constitute basis sets Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics. This journal is Β© the Owner Societies 2016
Transcript of Electronic Supplemental Materials π π΄ π π - RSC
Electronic Supplemental Materials
for
Thermodynamic stability of stoichiometric LaFeO3 and BiFeO3 :
hybrid DFT study
Eugene Heifets1, Eugene Kotomin1,2, A. A. Bagaturyants3,4, and Joachim Maier1
1Max Planck Institute for Solid State Research, Heisenbergstr. 1, Stuttgart 70569, Germany
2Institute for Solid State Physics, The University of Latvia, Riga, 8 Kengaraga str., Riga 1063, Latvia
3National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye shosse 31, Moscow 115409 Russia
4Photochemistry Center RAS Federal State Institution, Federal research center Crystallography and Photonics Russian Academy of Science, 7a Novatorov St., Moscow, 119421 Russia
Basis Sets
Calculations of the electronic structure for crystals are performed using expansion of one-electron crystalline orbitals into linear combination of Bloch functions built from local atom-ππ(π;π) ππ(π;π)
centered functions :ππ(π - π΄π - π π)
,
ππ(π;π) = βπ
ππ, π(π)ππ(π;π)
,
ππ(π;π) = βπ
ππ(π - π΄π - π π)πππ β π π
where AΒ΅ is vector describing position of atom Β΅ in unit cell, g denotes summation over all lattice vectors Rg . Sets of such local functions ( or atomic orbitals: AOs ) for each atom constitute basis sets
Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics.This journal is Β© the Owner Societies 2016
employed in the periodical electronic structure calculations. These local functions are expressed, in their turn, as linear combination of individually normalized Gaussian type functions centered on the same atom:
,ππ(π) =
ππΊ
βπ
πππΊ(πΌπ; π)
where exponents Ξ±j and coefficients dj are optimized and tabulated for future use in calculations; nG is the number of Gaussian type functions in such an expansion. These values for O, Sr, Al, Fe and Bi atoms were optimized and employed in this work. Tables with them are provided below. Groups (or βshellsβ) of AOs with the same principal n and orbital l quantum numbers are described together using the same sets of exponents Ξ±j and coefficients dj . The AOs in such groups differ by magnetic quantum number m. In order to save computational time shells with l=0 (s-shells) and l=1 (p-shells) can be described jointly (βsp-shellsβ) using the same set of exponents, but the expansion coefficients are determined separately for each orbital quantum number. Shell types are defined with respect to shellβs orbital quantum number:
s-type: l=0; sp-type: l=0, 1; p-type: l=1; d-type: l=2.
The tables with basis sets contain also values of occupation numbers (or charges) q for respective shells used as initial guess for calculations of initial electron density of atoms.
Table S1. All-electron basis set for O atoms. Nuclear charge 8; 6 shells in the basis set.
Shell # Shell type Initial occupation q nG Ξ±j dj(s) or dj
(d) dj(p)
1 s 2.0 826591.0 0.0003385434069.71 0.002548773943.177 0.012944513270.377 0.05183780788.7216 0.16437203232.0427 0.385512.4302 0.5623075394.97625 0.350155411
2 sp 6.0 462.9777 -0.006945266 0.0063389314.9059 -0.076543237 0.04316704.61171 -0.132826481 0.1572791.60313 0.379 0.347
3 sp 0.0 10.568730 1.0 1.0
4 sp 0.0 10.184511 1.0 1.0
5 d 0.0 11.38730 1.0
6 d 0.0 10.450917 1.0
Table S2. All-electron basis set for Al atoms. Nuclear charge 13; 9 shells in the basis set.
Shell # Shell type Initial occupation q nG Ξ±j dj
1 s 2.0 870510.0 0.000226
10080.0 0.00192131.0 0.0110547.5 0.0509163.1 0.169754.48 0.368819.05 0.3546 5.402 0.0443
2 s 2.0 599.4276 -0.022806331.9927 -0.10753916.7322 -0.09935394.44895 0.3308392.52919 0.7675
3 s 2.0 11.13055 1.0
4 s 0.0 10.319814 1.0
5 p 6.0 5318.906 0.0020740274.9861 0.016259023.9910 0.06916008.77431 0.1880583.37141 0.2995
6 p 1.0 11.30896 1.0
7 p 0.0 10.449274 1.0
8 d 0.0 12.32903 1.0
9 d 0.0 10.480048 1.0
Table S3. All-electron basis set for Fe atoms. Nuclear charge 26; 12 shells in the basis set.
Shell # Shell type Initial occupation q nG Ξ±j dj
1 s 2.0 8247426.1 0.00022346237388.81 0.001707318612.280 0.008675842490.788 0.0337587834.2824 0.102984307.9567 0.2328971120.8824 0.30883248.77426 0.1434
2 s 2.0 6661.0824 -0.00705923207.1095 -0.051348282.81511 -0.11646419.49410 0.36794010.02012 0.5673475.141981 0.2833
3 s 2.0 418.69281 -0.04289049.991180 -0.2402025.279362 -0.1184352.189743 0.749492
4 s 2.0 10.9679509 1.0
5 s 0.0 10.3750287 1.0
6 p 6.0 62098.776 0.000982813497.1310 0.00814291160.7620 0.0396483060.14883 0.12684024.41218 0.24617610.29450 0.223
7 p 6.0 430.50802 -0.047286416.31452 -0.006806634.300890 0.2442034.105736 0.620533
Continued on the next page.
Table S3. Continuation.
Shell # Shell type Initial occupation q nG Ξ±j dj
8 p 0.0 1
1.71010 1.0
9 p 0.0 1
0.654329 1.0
10 d 6.0 4
65.3814 0.0141740
18.8060 0.0909531
6.57480 0.288138
2.48433 0.496742
11 d 0.0 1
0.902707 1.0
12 d 0.0 1
0.272140 1.0
Table S4. Basis set for Fe atoms for use with ECP10MDF (http://www.tc.uni-koeln.de/cgi-bin/pp.pl?language=en,format=crystal09,element=Fe,job=getecp,ecp=ECP10MDF ) , accessed July, 2015). Core charge 16; 9 shells in the basis set.
Shell # Shell type Initial occupation q nG Ξ±j dj
1 s 2.0 421.2607 0.1209579.62721 -0.4940954.87997 -0.1945392.21708 0.749492
2 s 2.0 11.00711 1.0
3 s 0.0 10.404291 1.0
4 p 6.0 452.0945 0.0054005612.9358 -0.1103223.46391 0.3935691.65959 0.620533
5 p 0.0 10.780300 1.0
6 p 0.0 10.344346 1.0
7 d 6.0 450.8759 0.01719716.43746 0.1077496.062923 0.3098282.349843 0.496742
8 d 0.0 10.875471 1.0
9 d 0.0 10.270016 1.0
Table S5. Basis set for Sr atoms for use with ECP28MDF ( http://www.tc.uni-koeln.de/cgi-bin/pp.pl?language=en,format=crystal09,element=Sr,job=getecp,ecp=ECP28MDF , accessed July, 2015). Core charge 10; 8 shells in the basis set.
Shell # Shell type Initial occupation q nG Ξ±j dj
1 s 2.0 429.5132 -0.002093905.68229 0.2228973.19253 -0.5937960.660957 0.6
2 s 2.0 10.328483 1.0
3 s 0.0 10.0835463 1.0
4 p 6.0 410.8491 0.009811983.03450 -0.3905442.57683 0.2671020.747431 0.650
5 p 0.0 10.307114 1.0
6 p 0.0 10.0911895 1.0
7 d 0.0 10.821952 1.0
8 d 0.0 10.251302 1.0
Table S6. Basis sets (without and with f-orbitals included) for La atoms for use with ECP46MWB (http://www.tc.uni-koeln.de/cgi-bin/pp.pl?language=en,format=crystal09,element=La,job=getecp, ecp=ECP46MWB , accessed July, 2015). Core charge 11; 9 (without f-orbitals) or 11 (with f-orbitals) shells in the basis sets.
Shell # Shell type Initial occupation q nG Ξ±j dj Ξ±j dj
1 s 2.0 45.697401 -0.110665 5.709411 -0.1202303.624281 0.658589 3.687094 0.6555952.274880 -1.012191 2.253480 -1.0223810.483542 0.973100 0.488543 0.973100
2 s 1.0 10.447390 1.0 0.462438 1.0
3 s 0.0 10.220245 1.0 0.223669 1.0
4 p 6.0 47.705635 -0.002963 7.752729 -0.0030273.170832 0.309799 3.146075 0.3174612.536505 -0.504328 2.557021 -0.4978140.548362 0.532700 0.548053 0.532700
5 p 0.0 10.588306 1.0 0.629794 1.0
6 p 0.0 10.220429 1.0 0.221977 1.0
7 d 2.0 42.907882 0.028653 3.103559 0.0282462.052341 -0.085212 2.096736 -0.0859540.598340 0.142846 0.595537 0.1357750.382845 0.393600 0.398908 0.393600
8 d 0.0 10.207869 1.0 0.228168 1.0
9 d 0.0 10.122188 1.0 0.125418 1.0
10 f 0.0 1- - 3.621134 1.0
11 f 0.0 1- - 0.637270 1.0
Table S7. Basis set for Bi atoms for use with ECP60MDF ( http://www.tc.uni-koeln.de/cgi-bin/pp.pl?language=en,format=crystal09,element=Bi,job=getecp,ecp=ECP60MDF , accessed July, 2015). Core charge 23; 10 shells in the basis set.
Shell # Shell type Initial occupation q nG Ξ±j dj
1 s 2.0 439.5919 0.024348923.8503 -0.19954815.2582 0.5460906.99865 -0.874930
2 s 2.0 49.44357 0.03100065.88051 -0.2259631.83688 0.9089240.965753 0.501350
3 s 0.0 10.300010 1.0
4 s 0.0 10.108980 1.0
5 p 6.0 410.7359 -0.05865947.47103 0.1390832.09302 -0.1967440.953500 -0.153806
6 p 3.0 10.258083 1.0
7 p 0.0 10.0780702 1.0
8 d 10.0 416.6468 0.007825027.01713 -0.06832162.34957 0.3398641.14364 0.485831
9 d 0.0 10.550606 1.0
10 d 0.0 10.255790 1.0
Table S8. Lattice parameters and relative energy differences (ΞE) of LaFeO3 phases.
Space group Basis sets a Lattice parameters bPhase
Obtained from
Magnetic order label # At Fe At La a, Γ b, Γ c, Γ
Z cΞE,
meV/f.u.
Ortho-rhombic Expt.63 AFM:G Pnma 62 5.560 7.850 5.551 4
FM ECP no f 5.637 7.929 5.603 4 212AFM:A 5.637 7.906 5.600 4 138AFM:C 5.604 7.928 5.596 4 64AFM:G 5.606 7.909 5.593 4 0
FM all-e no f 5.643 7.937 5.609 4 235AFM:A 5.646 7.909 5.605 4 155AFM:C 5.609 7.938 5.600 4 71AFM:G 5.613 7.916 5.596 4 0
FM ECP with f 5.631 7.900 5.578 4 205AFM:A 5.633 7.876 5.575 4 134AFM:C 5.602 7.902 5.569 4 62AFM:G 5.606 7.882 5.566 4 0
FM all-e with f 5.638 7.909 5.583 4 231AFM:A 5.641 7.880 5.580 4 150AFM:C 5.606 7.911 5.574 4 70AFM:G 5.610 7.888 5.570 4 0
cubic Expt.61 AFM:G Pm3m 221 7.852 7.852 7.852 8FM ECP no f 7.897 7.897 7.897 8 1003
AFM:A 7.899 7.899 7.870 8 434AFM:C 7.875 7.875 7.895 8 341AFM:G 7.875 7.875 7.875 8 256
FM all-e no f 7.910 7.910 7.910 8 730AFM:A 7.911 7.911 7.878 8 608AFM:C 7.883 7.883 7.910 8 396AFM:G 7.885 7.885 7.885 8 272
FM ECP with f 7.874 7.874 7.874 8 1174AFM:A 7.875 7.875 7.846 8 538AFM:C 7.849 7.849 7.873 8 443AFM:G 7.851 7.851 7.851 8 356
FM all-e with f 7.886 7.886 7.886 8 844AFM:A 7.888 7.888 7.854 8 581AFM:C 7.859 7.859 7.887 8 474AFM:G 7.861 7.861 7.861 8 377
a The computations with the ECP and all-electron (all-e) basis sets on Fe ion and with ECP at La ion with f-orbitals included or absent in the basis set for La were performed. b Angles Ξ±=Ξ²=Ξ³=90ΒΊ in the unit cells of all crystal structures. c Z is a number of formula units in unit cell.
Phase Diagrams
(a) ECP on Fe:
500 1000 1500 2000
500 1000 1500 2000
-5
-4
-3
-2
-1
0
-7 -6 -5 -4 -3 -2 -1 0
-7 -6 -5 -4 -3 -2 -1 0
-30 -20 -10 0 10
-30 -20 -10 0 10
-5
-4
-3
-2
-1
0
0.2 p0
0.1 p0
0.01 p0
100 p0
10 p0
10-30 p0
10-20 p0
10-10 p0
1010 p0
1 p0
O , eV
T, K
T, K
Fe, eV
Fe, eV
5
41
32100 K
1500
K
1000
K
2000
K
500 K
log10(p/p0)
O
, eV
log10(p/p0)
(b) All-electron Fe:
500 1000 1500 2000
500 1000 1500 2000
-5
-4
-3
-2
-1
0
-7 -6 -5 -4 -3 -2 -1 0
-7 -6 -5 -4 -3 -2 -1 0
-30 -20 -10 0 10
-30 -20 -10 0 10
-5
-4
-3
-2
-1
0
0.2 p0
0.1 p0
0.01 p0
100 p0
10 p0
10-30 p0
10-20 p0
10-10 p0
1010 p0
1 p0
O , eV
T, K
T, K
Fe, eV
Fe, eV
5
41
32100 K
1500
K
1000
K
2000
K
500 K
log10(p/p0)
O
, eV
log10(p/p0)
FIG. S1. (Color online) Phase diagrams for LaFeO3: (a) based on ECP treatment and (b) based on all electron treatment of Fe atoms. The energies of used to build these diagrams are provided in Table VII in the paper. The diagrams based on calculations without f-functions in basis set at La. The lines, pointed by numbers in circles describe conditions of forming of: (1) FeO, (2) Fe2O3, (3) Fe3O4, (4) La2O3, (5) La metal. Green area marks the LaFeO3 stability region. Side panels at (a) and (b) serve to convert values of oxygen chemical potential to easy observable values of T and oxygen partial pressure similarly to Figs. 1 and 2..
2Op
Stability Regions
Figure S2. Region of BiFeO3 stability (green) on the map of O atoms chemical potential βππ
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0
-1
-2
-9
-3
-8
-7
-4
-6
-5
0 500 1000 1500 2000-30
-25
-20
-15
-10
-5
0
5
100 500 1000 1500 2000
-30
-25
-20
-15
-10
-5
0
5
10
Pressure, Log
10 (p/p0 )
Temperature, K
Pre
ssur
e, L
og10
(p/p
0)
Temperature, K
ECP, B3PW
defined by Eq. (3) in the main text. Contour lines start from eV and go in the negative βππ = 0direction with increment of 0.5 eV. The stability region based on calculations with ECP on Fe. The dashed lines replace contour lines for eV. The value of corresponding to βππ = β 2 βππ
reduction of Bi2O3 to metallic Bi (-1.99eV) practically coincides with this value.
(a)
0 500 1000 1500 2000
0 500 1000 1500 2000
-30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
-15
-10
-5
0
5
10
Pressure, Log10 (p/p
0 )
Temperature, K
Pres
sure
, Log
10(p
/p0)
Temperature, K
ECP at Fe, without f-orb. at La, B3PW
(b)
0 500 1000 1500 2000
0 500 1000 1500 2000
-30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
-15
-10
-5
0
5
10
Pressure, Log10 (p/p
0 )Temperature, K
Pres
sure
, Log
10(p
/p0)
Temperature, K
All-electron at Fe, without f-orb. at La, B3PW
(c)
0
-1
-2
-3
-3
0 500 1000 1500 2000
0 500 1000 1500 2000
-30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
-15
-10
-5
0
5
10
Pressure, Log
10 (p/p0 )
Temperature, K
Pre
ssur
e, L
og10
(p/p
0)
Temperature, K
ECP at Fe, with f-orb. at La, B3PW
FIG. S3. (Color online) Regions of LaFeO3 stability (green) vs temperature and oxygen pressure on the maps of oxygen chemical potential defined by Eq.(3). (a) The stability O
region based on calculations with ECP at Fe and without f-orbitals at La. (b) The same for calculations performed using Fe all-electron basis set and without f-orbitals at La. (c) The stability region based on calculations with ECP at Fe and with f-orbitals at La. In this figure the dashed lines correspond to the conditions, where FeO coexists with metallic iron. The dotted lines correspond to coexisting Fe2O3 and Fe3O4. The dot-dash line represents conditions, where Fe3O4 coexists with FeO. Corresponding reduction or oxidation occurs, when conditions cross these lines.
![Global Optimisation of Chiller - AIRAHΒ Β· Plant simulation πΗ π=πΗcwππ€(ππβππ) β’where, πΗcwis the mass flow rate of the chilled water [kg s-1]](https://static.fdocuments.us/doc/165x107/5e77b1d764dcef24141fa6da/global-optimisation-of-chiller-airah-plant-simulation-cwaoe.jpg)
