Supporting Information:
Play the heavy: an effective mass study
for α-Fe2O3 and corundum oxides
Ofer Neufeld1 and Maytal Caspary Toroker2,*
1
The Nancy and Stephen Grand Technion Energy Program and 2Department of
Materials Science and Engineering, Technion - Israel Institute of Technology, Haifa
32000, Israel
*Corresponding author: E-mail: [email protected] , Tel.: +972 4 8294298.
In this supporting information file we provide our codes for calculating the density of
states (DOS) effective carrier masses in and the band effective carrier masses. Additionally
we provide the relaxed geometries for all the corundum oxides and plots of the band structure
along high symmetry lines for each oxide, and an example comparison of the ab-initio
calculated DOS with the fitting of eq. (1) and (2) for Fe2O3.
1. DOS effective mass script:
The script is written in Wolfram Mathematica v9.0. It can be implemented on any
semiconductor, and is currently set to work automatically with VASP outputs, with the
exception of user inputted numerical parameters, such as cutoffs etc. The VASP output files
must come from a static calculation, that is set NSW = 0, and with site projected DOS, that is
LORBIT = 10. Additionally, the code only runs for ISPIN = 2 at the moment, and only in a
spin symmetric case. It can be extended easily enough to include other cases. The required
input files are a geometry file, POSCAR, a DOS file, DOSCAR, and the OUTCAR file.
Required user inputs are the width of states to be included in the least squares fit around the
CBM and VBM, named "tailC" and "tailV", respectively in the script. Default values are
0.25eV. Script is attached here as images, but also the notebook itself is available for
download.
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2. Band effective mass script:
The script is written in Wolfram Mathematica v9.0. It can be implemented on any
semiconductor, and is currently set to work automatically with VASP outputs, with the
exception of user inputted numerical parameters. The VASP output files must come from a
static calculation, that is set NSW = 0. Additionally, the code only runs for ISPIN = 2 at the
moment, and only in a spin symmetric case. It can be extended easily enough to include other
cases. The required input files are a geometry file, POSCAR, an eigenvalue file, EIGENVAL,
and the OUTCAR file. Required user inputs are the radius of the cutoff spheres in reciprocal
space around the band edges, named "RsphereCB" and "RsphereVB", respectively in the
script. Default values are 0.2-0.25 2π/Å. Script is attached here as images, but also the
notebook itself is available for download.
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3. Band structure along high symmetry lines in the corundum oxides:
We provide the spin up band structure in al oxides calculated with PBE+U for Fe 2O3 and
Cr2O3, and with PBE for Al2O3, Ga2O3, and In2O3. Spin down band structure is symmetric
and not displayed. High symmetry lines are chosen in the Brillouin zone according to the
rhombohedral crystal symmetry group, specified in Figure S.1.
Figure S.1 – PBE and PBE+U spin up band structures along high symmetry lines for corundum oxides: (a)
Fe2O3, (b) Cr2O3, (c) Al2O3, (d) Ga2O3, and (e) In2O3. Fermi level is set to zero energy, and the conduction
and valence band edges are marked with dashed lines. Only the bottom and top ten bands are shown
around the fermi level. The rhombohedral crystal high symmetry line paths in the Brillouin zone are: Γ-LB1|B-Z-Γ-X|Q-F-P1-Z|L-P, with: Γ=(0,0,0), L=(1/2,0,0), B 1=(1/2,1-η,η-1), B=(η,1/2,1-η), Z=(1/2,1/2,1/2),
X=(ν,0,-ν), Q=(1-ν,ν,0), F=(1/2,1/2,0), P1=(1-ν,1-ν,1-η), P=(η,ν,ν), and with η=(1+4cos(α))/(2+4cos(α)), and
ν=(3/4- η/2), such that α is the rhombohedral lattice angle.
4. DOS comparison with fitting ideal DOS of eq. (1) and (2):
We provide an example fitting of the ab-initio calculated DOS with PBE+U for Fe2O3
with eqs. (1) and (2). Deviation from this ideal behavior is observed, but not strongly, and
hence the fitting method is very much applicable.
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Figure S.2 – PBE+U calculated DOS for Fe2O3 against eq. (1) and (2) fit, from which the DOS effective mass
is extracted. Zero energy is set to the valence band edge, and the band edges are indicated with vertical
lines.
5. Relaxed Geometries for the corundum oxides:
The PBE+U relaxed geometries for the primitive rhombohedral unit cells for the
corundum oxides are given below in the following order: title, lattice vectors in Cartesian
coordinates, atom types and numbers, respectively, and atom locations in fractional
coordinates with respect to the lattice vectors. The atom coordinates are ordered according to
the ordering given in the atom type row (format of VASP simulation package).
5.1. Relaxed geometry of α-Fe2O3, total energy: -68.5159 eV:
4.8444900000000004 -2.5354199999999998 0.0000000000000000
4.8444900000000004 2.5354199999999998 0.0000000000000000
3.5175500000000000 0.0000000000000000 4.1862000000000004
O Fe
6 4
Selective dynamics
Direct
0.7498726041710242 0.4440874186904651 0.0561194694885145
0.0560393085618500 0.7499875849464246 0.4439813920253073
0.4438747971226960 0.0559050290947383 0.7501422321542748
0.2501202154875486 0.5559133298089876 0.9438734449210813
0.9439717187361367 0.2500091186349067 0.5559885106347267
0.5561312539358880 0.9441194729224236 0.2498709482010568
0.8539550998867682 0.8539854044845754 0.8540525980530376
0.1459991536261711 0.1459746566419966 0.1459294668509372
0.6459591664499271 0.6459916068977307 0.6460604875893736
0.3540766630220062 0.3540263588777606 0.3539814310816922
5.2. Relaxed geometry of corundum α-Al2O3, total energy: -74.8610 eV:
5.1724425793591839 -0.0074523469388156 -0.0050289210832583
2.9420307501810070 4.2542705512656953 -0.0051218826117188
2.9420811141061347 1.5396939079804972 3.9658120591309740
O Al
6 4
Selective dynamics
Direct
0.7499750373575509
0.0558734862619588
0.4440895189815421
0.2500232702517167
0.9441277901159992
0.5559151661777406
0.8523611745710085
0.1476338666895245
0.4441761510689801
0.7499929529282809
0.0558222254294165
0.5558271163015718
0.2500030603380665
0.9441837265107083
0.8523519503648203
0.1476511279037425
0.0559037956279056
0.4441377083181450
0.7500290730825157
0.9440901643884132
0.5558567269436523
0.2499744433407827
0.8523525609477787
0.1476536101502290
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0.6476494808818600 0.6476489520197006 0.6476382610764091
0.3523511897111220 0.3523427181347216 0.3523636371241707
5.3. Relaxed geometry of corundum α-Cr2O3, total energy: -79.3681 eV:
5.4176959295383771 -0.0395351612109077 -0.0270712562431112
3.0554617384974194 4.4742599277885766 -0.0270977683233927
3.0554234698453038 1.5991389048953486 4.1786494932114433
O Cr
6 4
Selective dynamics
Direct
0.7499820140065195 0.4477824270069022 0.0522689886948388
0.0522468292356406 0.7499964924077519 0.4477604568917462
0.4477281901456467 0.0522148510871504 0.7500201721142190
0.2500170553477977 0.5522172260899505 0.9477302963745038
0.9477524715024792 0.2500023409187754 0.5522381737753207
0.5522739612054584 0.9477875764886814 0.2499807971503865
0.8485495259092993 0.8485478200554059 0.8485448728297864
0.1514489494739880 0.1514525079254446 0.1514557965061351
0.6514556530881900 0.6514530268159529 0.6514502071966776
0.3485453310850037 0.3485457122039941 0.3485502194663880
5.4. Relaxed geometry of corundum α-Ga2O3, total energy: -60.1883 eV:
5.4146904981496489 -0.0299520974114722 -0.0204591125255243
3.0616232243826871 4.4661436376859154 -0.0204505268728511
3.0616310808111074 1.6023531182387412 4.1688653494181196
O Ga
6 4
Selective dynamics
Direct
0.7499978170910779 0.4486766475613067 0.0513340684831647
0.0513316239418771 0.7499990863417949 0.4486703077358811
0.4486641078695612 0.0513233755183933 0.7500029767312313
0.2500022361611798 0.5513235974134147 0.9486656014016006
0.9486685416550813 0.2500009590611683 0.5513294799085173
0.5513358991431190 0.9486766879989261 0.2499970395400126
0.8545644638307479 0.8545647756409807 0.8545630261322685
0.1454352880144327 0.1454350613906641 0.1454371372928165
0.6454372364503698 0.6454355436948163 0.6454351424165949
0.3545627668425837 0.3545642463785441 0.3545652013579144
5.5. Relaxed geometry of corundum In2O3, total energy: -55.8186 eV:
5.9106331783087889 -0.0718486246904649 -0.0491559887879992
3.3098705642021251 4.8975633104642853 -0.0491116142643254
3.3098370642489381 1.7323042178293289 4.5811824553948997
O In
6 4
Selective dynamics
Direct
0.7499987946033997 0.4549727421670511 0.0450354915116051
0.0450303323169408 0.7500004508877254 0.4549710152473736
0.4549641999711227 0.0450262933758978 0.7500006957331422
0.2500012337264863 0.5450272121674615 0.9549645498944841
0.9549696515565316 0.2499996027553024 0.5450290098037911
0.5450357494828353 0.9549736695111335 0.2499992288567654
0.8562791529869500 0.8562771361691190 0.8562772643286110
0.1437208284859324 0.1437228988415100 0.1437227292829988
0.6437212305316962 0.6437239293562200 0.6437213524778116
0.3562788073381284 0.3562760457685812 0.3562786438634049
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