First-Principles calculations of the structural and electronic properties of the high-K dielectric HfO 2 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 Corporation 2 Fritz-Haber-Institut der Max-Planck-Gesellschaft 3 Physics Department, Yarmouk University

Introduction HfO 2 Direct tunneling current I g = I 0 exp ( - t ) Fundamental properties about HfO 2 by first-principles calculations ( structural and electronic properties) I DSAT   * C ox   * K / t IgIg Gate electrode VgVg Band offset E Si IgIg t e- high-K material with large physical thickness I DSAT (1) high dielectric constant ( HfO 2 ~25, SiO 2 ~ 4 ) (2) wide bandgap ( HfO 2 ~6eV, SiO 2 ~ 9eV) (3) good thermal stability (amorphous phase) Scaling of MOS-FET Transistor Speed Low power consumption Manufacturing costs Demand

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 a

Outline 1. Pseudo potential and calculation method 2. Structural property (cubic-HfO 2, tetragonal-HfO 2 ) 3. Electronic property (cubic-HfO 2, tetragonal-HfO 2 ) 4. Comparison between cubic and tetragonal phase 5. Summary

Pseudo-potentials (Oxygen, Hafnium) Troullier-Martins scheme Oxygen (1s 2 2s 2 2p 4 ) valance electrons: 2s 2 2p 4 Hafnium ( [Xe]4f 14 5d 2 6s 2 ) valance electrons: 5s 2 5p 6 5d x 6s y eigenvalue transferabilityx=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 Radial Densities for Hf Rc

DFT-LDA calculation The all-state-preconditioned conjugate gradient scheme (CCG) for structural calculation The 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-HfO2Bulk modulus of c-HfO2 SFHIngX (Plane wave basis set)

Structural Properties 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 state Bottom of conduction band Hf 5d state Band gap ~ 4eV Partial density of states (LDA)

Kohn-Sham equation: (ground state properties) Quasiparticle equation (GW calculation): First-order correction: GW approximation  self-energy=  =iGW G = one-particle Greens function, W = screened Coulomb interaction GW correction for band structure calculation

Electronic Properties (cubic phase) kzkz kyky kxkx b1b1 b3b3 b2b2 L W  X K Band energy (eV) * Y 2 O 3 (0.15) HfO 2 (0.85) J.Appl.Phys vol, (2002) GW correction ~1.8 eV Eg (direct) LDA+GW LDA

Electronic Properties (tetragonal phase) kzkz kyky kxkx b1b1 R Z  A M b2b2 b3b3 Band energy (eV) Band transition indirect (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 a a c (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 ) a a c a=5.15 Å (fixed ), c/a = (fixed) dz/c factor effect Band energy (eV) dz/c reflects the difference between 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)

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Eigen value transferability test for Hf pseudo-potential Error = Pseudo - all electron Hf Hf + 5d occupancy 6s occupancy Hf Hf +

Other theoretical calculations

Band gap vs lattice constant (cubic phase)

Comparison between cubic and tetragonal phase kzkz kyky kxkx b1b1 R Z  A M b2b2 b3b3 Band energy (eV) (tetragonal M = cubic X) (1) Transition is different (2) Band gap is increasing Tetragonal Cubic (a=5.15Å  Eg (direct)

Band transition change (tetragonal phase)

GW calculation 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

Hedin`s GW approximationSpace-time method* convolutions multiplications FFT *Rieger et al. CPC 117, (1999) real space, energy domain real 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 Energy level -2.9eV -5.3eV -35.7eV -67.2eV -17.0eV 5d5d 6s6s 4f4f 5p5p 5s5s fhi98pp- program

Anisotropy in tetragonal phase The head of dielectric matrix