first-principles calculations of the structural and electronic properties of the high-k dielectric...
DESCRIPTION
HfO 2 crystallized structure cubic phase tetragonal phase monoclinic phase a a a c a b c a 1 = (0, a/2, a/2) a 2 = (a/2, 0, a/2) a 3 = (a/2, a/2, 0) a 1 = (-a/2, a/2, 0) a 2 = (a/2, a/2, 0) a 3 = (0, 0, c) Oxygen Hafnium Hafnium : seven-fold coordinated Oxygen : three-fold coordinated four-fold coordinated a aTRANSCRIPT
First-Principles calculations of the structural and electronic properties of the high-K dielectric HfO2
Kazuhito Nishitani 1,2, Patrick Rinke 2, Abdallah Qteish 3,Philipp Eggert 2, Javad Hashemifar 2,Peter Kratzer 2, and Matthias Scheffler 2
1 Corporate Manufacturing Engineering Center, Toshiba Corporation2 Fritz-Haber-Institut der Max-Planck-Gesellschaft3 Physics Department, Yarmouk University
Introduction
HfO2
Direct tunneling current Ig = I0 exp (- t )
Fundamental properties about HfO2 by first-principles calculations( structural and electronic properties)
IDSAT * Cox * K / t
IgGate electrode
VgBand offset
ESiIg
t
e-
high-K material with large physical thickness
IDSAT
(1) high dielectric constant ( HfO2 ~25 , SiO2 ~ 4 )(2) wide bandgap ( HfO2 ~6eV, SiO2 ~ 9eV)(3) good thermal stability (amorphous phase)
Scaling of MOS-FET
Transistor SpeedLow power consumptionManufacturing costs
Demand
HfO2 crystallized structure
cubic phase tetragonal phase monoclinic phase
a aa
c
ab
c
a1= (0, a/2, a/2)a2= (a/2, 0, a/2)a3= (a/2, a/2, 0)
a1= (-a/2, a/2, 0)a2= (a/2, a/2, 0)a3= (0, 0, c)
<Unit cell> <Unit cell>
Oxygen
Hafnium
Hafnium : seven-fold coordinated
Oxygen : three-fold coordinated four-fold coordinated
a
a
Outline
1. Pseudo potential and calculation method
2. Structural property (cubic-HfO2, tetragonal-HfO2)
3. Electronic property (cubic-HfO2, tetragonal-HfO2)
4. Comparison between cubic and tetragonal phase
5. Summary
Pseudo-potentials (Oxygen, Hafnium)
Troullier-Martins scheme
Oxygen (1s2 2s2 2p4)
valance electrons: 2s2 2p4
Hafnium ( [Xe]4f 14 5d26s2 )
valance electrons: 5s2 5p6 5dx 6sy
eigenvalue transferability x=3, y=0
non-linear core correction (Rc =0.7 a.u)
ghost states local component = s wave
local component = p wave
Atomic wave function of 5shell for Hf
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 2 4 6
r (a.u)
U(r)
5d5p5s
0
50
100
150
200
0 1 2 3 4
r (a.u)
4p r2 n
(r)
pseudomodel coretrue core
Radial Densities for Hf
Rc
DFT-LDA calculation
The all-state-preconditioned conjugate gradient scheme (CCG) for structural calculationThe state by state conjugate gradient scheme (DIIS_CCG) for band calculation
Ecut = 70Ry
k-points = 4 x 4 x 4 Monkhorst-Pack grid(irreducible k-points=10 and 6 for cubic and tetragonal phase)
Lattice constant of c-HfO2 Bulk modulus of c-HfO2
4.95.05.15.25.3
0 50 100
Ecut (Ry)
aLat
(Å)
0.0100.0200.0300.0400.0
0 50 100
Ecut (Ry)
B0
(GPa
)
SFHIngX (Plane wave basis set)
Structural Properties
DFT-LDA LAPW EXP * Error
a (Å) 4.95 5.00 5.08 2.6% (1.1%)
B0 (GPa) 280 277 no-data (1.1%)
DFT-LDA Exp ** Error
a (Å) 4.94 5.15 4.1%
c (Å) 5.054 5.289 4.4%
c/a 1.023 1.027 0.4%
dz (Å) 0.202 no-data no-data
dz /c 0.040 no-data no-data
B0 (GPa) 274.6 no-data no-data
a
dz
a
c
Structural parameters are in good agreement with experimental values (within 5%)
Cubic phase
Tetragonal phase
*J.Amer.Ceram.Soc.53,264 (1970)
**J.Amer.Ceram.Soc.55,482 (1972)
Electronic Properties (cubic phase)
Top of valance band O 2p stateBottom of conduction band Hf 5d stateBand gap ~ 4eV
Partial density of states (LDA)
0
2
4
6
8
10
-80 -60 -40 -20 0 20Energy (eV)
DO
S (A
rb. u
nits
)
O(s)O(p)Hf (s)Hf (p)Hf (d)
)()()()()]()([ LDALDALDALDAXC
LDAH rrrrrr kkkk nnnn VVh
)(')'(),',()()]()([ QPQP3QPQPQPH rrrrrrr kkkkk nnnnn rdVh
)()(,',)( LDALDAXC
QPLDALDAQP rrrrr kkkkk nnnnn V
QPLDAkk nn
Kohn-Sham equation: (ground state properties)
Quasiparticle equation (GW calculation):
First-order correction:
GW approximation self-energy==iGWG = one-particle Greens function, W = screened Coulomb interaction
GW correction for band structure calculation
Electronic Properties (cubic phase)
kz
ky
kx
b1
b3
b2 L
W
X
K
Band energy (eV)
Eg
DFT-LDA 3.71 eV
LAPW 3.55 eV
LDA +GW 5.51 eV
EXP * 5.8 eV
* Y2O3 (0.15) HfO2 (0.85)
J.Appl.Phys vol, 91 4500 (2002)
GW correction ~1.8 eV
Eg (direct)
LDA+GW
LDA
Electronic Properties (tetragonal phase)kz
ky
kx
b1
RZ
A
Mb2
b3
Eg
DFT-LDA 4.11 eV
LDA +GW 5.82 eV
Band energy (eV)
Band transitionindirect (A to )
GW correction ~1.7 eV
Eg (indirect)
LDA+GW
LDA
c/a factor effect
a=5.15 Å (fixed ), dz/c = 0.0 (fixed)
Band energy (eV)Cubic (c/a=1.00)
c/a=1.027
aa
c
c/a transition Eg
1.00 M to M 3.58
1.01 M to M 3.51
1.02 M to M 3.43
1.027 M to M 3.37
(1) Transition is same
(2) Band gap is decreased
(tetragonal M = cubic X)
Tetra (dz/c=0.04)
c/a=1.027 (dz/c=0.00 )
aa
c
a=5.15 Å (fixed ) , c/a = 1.027 (fixed)
dz/c factor effect
dz/c transition Eg
0.00 M to M 3.37
0.01 M to 3.48
0.02 M to 3.69
0.03 A to 3.89
0.04 A to 4.11
Band energy (eV)
dz/c reflects the differencebetween cubic and tetragonal
(1) DFT-LDA reliability
(2) GW correction
(3) The comparison between cubic and tetragonal phase
Summary
Change from direct to indirect gap is due to internal oxygen relaxation
Cubic phase : LDA+GW (5.5 eV), LDA (3.7eV), experiment (5.8eV)
Structural properties are in good agreement with LAPW and experiment
Band gap is underestimated compared with experiment
Tetragonal phase : LDA+GW (5.8 eV), LDA (4.1eV)
Thank you for your attention
Eigen value transferability test for Hf pseudo-potential
Error = Pseudo - all electron
-100.0
-50.0
0.0
50.0
100.0
0 1 2 3
5d occupancy
Erro
r (m
eV) 5s
5p5d6s
HfHf+
-100.0
-50.0
0.0
50.0
100.0
0 1 2 3
6s occupancy
Erro
r (m
eV) 5s
5p5d6s
5d occupancy
6s occupancy
HfHf+
Other theoretical calculations
Author Paper Hf pp pp Exc content
Xinyuan Zhao PRB vol65, 233106 (2002) 5s 5p 5d 6s Ultrasoft LDA/GGA Structure, epsilonDemkov Phys.Stat.sol 226, 57(2001) no-data CASTEP LDA SrTiO3, HfO2, ZrO2, structure,bandDabrowski MR vol41, 1093 (2001) no-data norm-conserving LDA/GW GW band structure, Pr psuedopotentialFiorentini PRL vol 89, 266101 (2002) semi-core VASP GGA Band offset, formatin energy, epsilonNieminen group PRB vol65, 174117 (2002) 5d(3) 6s(1) VASP GGA Vacancy, interstial defectJoongoo Kang PRB vol68, 054106 (2003) no-data norm-conserving LDA/GGA Structure, phase transitionRignanese PRB vol69, 184301 (2004) 5s 5p 5d 6s norm-conserving LDA Silicates, structure,epsilonLowther PRB vol60, 14485 (1999) no-data norm-conserving LDA StructureRobertson group PRB vol92, 057601 (2004) no-data CASTEP LDA ZrO2 Interface, Band offsetRobertson group APL vol84, 106 (2003) no-data CASTEP LDA HfSiON, Band offsetRobertson group JAP vol92, 4712 (2002) no-data CASTEP LDA many High-K, Band offsetRobertson group JVST.B18, 1785 (2000) no-data CASTEP LDA many High-K, Band offsetJomard PRB vol59, 4044 (1999) semi-core Ultrasoft LDA ZrO2, structureFinnis PRL vol81, 5149 (1998) no-data no-data no-data ZrO2, structureKralik PRB vol57, 7027 (1998) semi-core norm-conserving LDA/GW ZrO2, GW band structureFrench PRB vol49, 5133 (1994) no-data no-data no-data ZrO2, band structureParlinski PRL vol78, 4063 (1997) no-data CASTEP no-data ZrO2, structure, phonon dispersion
Band gap vs lattice constant (cubic phase)
a (Å) Eg (eV)4.80 4.214.90 4.024.95 3.955.00 3.855.08 3.715.10 3.675.20 3.51 3.00
3.203.403.603.804.004.204.40
4.70 4.80 4.90 5.00 5.10 5.20 5.30aLat (Å)
Eg (e
V)
Comparison between cubic and tetragonal phasekz
ky
kx
b1
RZ
A
Mb2
b3
Band energy (eV)
(tetragonal M = cubic X)
transition Eg
cubic M to M 3.58
tetra A to 4.11
(1) Transition is different
(2) Band gap is increasing
Tetragonal
Cubic (a=5.15Å
Eg (direct)
Band transition change (tetragonal phase)
c/a=1.00 VB CBA 1.9603 6.7306M 2.1542 5.7345G 2.1540 5.7342
c/a=1.01 VB CBA 1.9460 6.5721M 2.1287 5.6377G 2.0219 5.6450
c/a=1.02 VB CBA 1.9279 6.4090M 2.0992 5.5300G 1.8925 5.5496
c/a=1.027 VB CBA 1.9147 6.2951M 2.0787 5.4508G 1.8056 5.4648
dz/c=0.00 VB CBA 1.9147 6.2951M 2.0787 5.4508G 1.8056 5.4648
dz/c=0.01 VB CBA 1.8847 6.3326M 2.0102 5.5154G 1.7444 5.4911
dz/c=0.02 VB CBA 1.7986 6.4441M 1.8211 5.6965G 1.5768 5.5156
dz/c=0.03 VB CBA 1.6688 6.6253M 1.5496 5.9643G 1.4537 5.5592
dz/c=0.04 VB CBA 1.4967 6.8532M 1.2179 6.281G 1.3847 5.611
GW calculation
<Convergence Parameter>
Number of empty states = 800 states for correlation part
k points = 4 x 4 x 4
ecut off = 36Ha / 20Ha for cubic (exchange / correlation)
36 Ha / 24Ha for tetragonal (exchange /correlation)
GWST
Space-time method
Space - time method
Hedin`s GW approximation Space-time method*
k k
kk rrrrn n
nn
iG
LDALDA
;,
;,;,;, rrrrrr GGdP
;,;, 3 rrrrrrrr Prd
;,;, 13 rrrrrr rdW
ieGWd ;,;,;, rrrrrr
LDA LDA*n
unoccLDA exp kkk rr n
nni
LDAn
LDA*n
occLDAn exp;, kkkk rrrr
n
iiG
iGiGiP ;,;,;,0 rrrrrr RRR
convolutions
GGGGGG kG
kGGk
,;,;, iPi
iiW ;,,;, 1 GGGGGGG
kkk
iWiGi ;,;,;, rrrrrr RRR
multiplications
FFT
FFT
*Rieger et al. CPC 117, 211-228 (1999)
real space, energy domainreal space reciprocal space, imaginary time
LAPW method
(1) inside atomic sphere
l max =10 Hf (l =0, 1, 2 : APW+lo, l>2 : LAPW)O (l = 0, 1 : APW+lo, l>1 : LAPW)
Muffin tin radius ( Hf = 2.0 a.u, O =1.7 a.u )
(2) interstitial region
Plane wave cut off = 21.1Ry
Wien 2K
Hf atom energy level
n l occupation Enl(eV)1 0 2 -65232.72 0 2 -11157.12 1 6 -9800.43 0 2 -2541.63 1 6 -2136.43 2 10 -1655.94 0 2 -511.44 1 6 -378.44 2 10 -206.54 3 14 -17.05 0 2 -67.25 1 6 -35.75 2 2 -2.96 0 2 -5.3
Energy level
-2.9eV-5.3eV
-35.7eV
-67.2eV
-17.0eV
5d6s
4f
5p
5s
fhi98pp- program
Anisotropy in tetragonal phase
The head of dielectric matrix
iw dir1 dir2 dir3 anisotropy(%)0.0318 6.29 6.29 6.04 4.190.1663 5.07 5.07 4.94 2.670.4031 3.14 3.14 3.11 1.080.7338 2.07 2.07 2.06 0.441.1464 1.58 1.58 1.57 0.181.6259 1.34 1.34 1.34 0.072.1552 1.21 1.21 1.21 0.032.7150 1.14 1.14 1.14 0.023.2850 1.10 1.10 1.10 0.013.8448 1.08 1.08 1.08 0.004.3741 1.06 1.06 1.06 0.004.8536 1.05 1.05 1.05 0.005.2662 1.04 1.04 1.04 0.005.5969 1.04 1.04 1.04 0.005.8337 1.04 1.04 1.04 0.005.9682 1.03 1.03 1.03 0.007.7587 1.02 1.02 1.02 0.00
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10
0 5 10
iwan
isot
ropy
(%)