Defects in materials. Manish Jain. July 8, Department of Physics Indian Institute of Science Bangalore 1/46

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1 1/46 Defects in materials Manish Jain Department of Physics Indian Institute of Science Bangalore July 8, 2014

2 Outline 2/46 Motivation. Computational methods. Defects in oxides. Why are defects challenging? Defects in high-κ dielectric HfO 2. Defects in technologically important material TiO 2. Defects for quantum computing.

3 Outline 3/46 Motivation. Computational methods. Defects in oxides. Why are defects challenging? Defects in high-κ dielectrics HfO 2. Defects in technologically important material TiO 2. Defects for quantum computing.

4 Motivation High-κ dielectric material HfO2 Moore's law about size of transistor. HfO2 has replaced SiO2 in todays MOSFET devices. κsio2 3.9 κhfo /46

5 High-κ dielectric materials Motivation 5/46 High defect density. Charge trapping by defects in the interfacial layer or oxide threshold voltage instability 1. Oxygen-related defects vacancy 2 (V O ) or interstitials (I O ) major cause. Study the stability of point defects V O. 1 G. Ribes et. al., IEEE Trans. Device Mater. Reliab. 5, 5 (2005). 2 K. Shiraishi et. al., Jpn. J. Appl. Phys., Part 2 43 L1413 (2004); H. Park et. al., IEEE Electron Device Lett. 29, 54 (2008).

6 6/46 Motivation TiO 2 Photovoltaic material in solar cells 1. Photocatalytic material hydrolysis of water 2. Memresistive switches for non-volatile memory 3. Oxygen vacancies are believed to play a very important role in all these applications. 1 B. O'Regan and M. G ratzel, Nature (London) 353, 737 (1991). 2 A. Fujishima and K. Honda, Nature (London) 238, 37 (1972). 3 J. J. Yang, M. D. Pickett, X. Li, D. A. A. Ohlberg, D. R. Stewart, and R. S. Williams, Nature Nanotechnology 3, 429 (2008).

7 Outline 7/46 Motivation. Computational methods. Defects in oxides. Why are defects challenging? Defects in high-κ dielectrics HfO 2. Defects in technologically important material TiO 2. Defects for quantum computing.

8 Methods Ground-state properties 8/46 Properties that are intrinsic to a system with all its electrons in equilibrium. Density functional theory is the standard model for understanding ground-state properties. Total energy is a functional of the charge density. Kohn-Sham formulation: Map the interacting many-electron problem to non-interacting electrons moving in a self-consistent eld. ( V ionic(r) + V Hartree (r) + V xc (r) ) ψ(r) = ɛ ψ(r) Local density approximation Generalized gradient approximation

9 Methods Excited-state properties 9/46 Spectroscopic properties that involve experiments creating an excited particle above the ground state. Concept and formalism of the interacting particle Green's function (G). Many-body perturbation theory is the standard model for understanding excited-state properties. G 1 = G Σ GW approximation to the self-energy (Σ). Σ = = igw W = ε 1 v

10 Outline 10/46 Motivation. Computational methods. Defects in oxides. Why are defects challenging? Defects in high-κ dielectric HfO 2. Defects in technologically important material TiO 2. Defects for quantum computing.

11 Why are defects challenging? 11/46 Potentially strong electron-electron correlations. Can be open-shell systems. Multiple localized, interacting electrons. Lattice relaxation eects. Screening from the host. Mimicking the system by isolated cluster may be incorrect. Experiments often involve excited-state properties (deep level transient spectroscopy or optical absorption etc.) Computational diculty isolated defect.

12 Outline 12/46 Motivation. Computational methods. Defects in oxides. Why are defects challenging? Defects in high-κ dielectric HfO 2. Defects in technologically important material TiO 2. Defects for quantum computing.

13 High-κ dielectric materials Denitions Formation energy of an oxygen-vacancy: E f [ R] = E[ R] E ref E O 2 From slides by C. G. Van de Walle 13/46

14 High-κ dielectric materials Denitions Formation energy of an oxygen-vacancy: E f q[ R q ](E F ) = E q [ R q ] E ref E O 2 + q(e F + E v ) From slides by C. G. Van de Walle 13/46

15 High-κ dielectric materials Denitions 14/46 Formation energy of an oxygen-related defect in hafnia : E f q[ R q ](E F ) = E q [ R q ] E HfO2 n O µ O + q(e F + E v ) where µ O is the oxygen chemical potential.

16 High-κ dielectric materials Denitions 14/46 Formation energy of an oxygen-related defect in hafnia : E f q[ R q ](E F ) = E q [ R q ] E HfO2 n O µ O + q(e F + E v ) where µ O is the oxygen chemical potential. Charge transition level : ε q/q-1 = Fermi energy where defect q q 1. E f q +1 0 VBM ε +1/0 E F CBM

17 High-κ dielectric materials Denitions 14/46 Formation energy of an oxygen-related defect in hafnia : E f q[ R q ](E F ) = E q [ R q ] E HfO2 n O µ O + q(e F + E v ) where µ O is the oxygen chemical potential. Charge transition level : ε q/q-1 = Fermi energy where defect q q 1. = E f q-1 [ R q-1 ](E F = 0) E f q[ R q ](E F = 0) E f q +1 0 VBM ε +1/0 E F CBM

18 High-κ dielectric materials Methodology DFT + GW methodology. Formation energy E f q 1 E f q 1 ( R q 1) E f q E f q( R q) ε q/q 1 R q 1 R q Generalized coordinate M. Hedström, A. Schindlmayr, G. Schwarz, and M. Scheer, Phys. Rev. Lett. 97, (2006); P. Rinke, A. Janotti, M. Scheer, and C. G. Van de Walle, Phys. Rev. Lett. 102, (2009); M. Jain, J. R. Chelikowsky and S. G. Louie, Phys. Rev. Lett. 107, (2011). 15/46

19 High-κ dielectric materials Methodology DFT + GW methodology. Formation energy E f q 1 E f q 1 ( R q 1) E f q E f q( R q) ε q/q 1 R q 1 R q Generalized coordinate M. Hedström, A. Schindlmayr, G. Schwarz, and M. Scheer, Phys. Rev. Lett. 97, (2006); P. Rinke, A. Janotti, M. Scheer, and C. G. Van de Walle, Phys. Rev. Lett. 102, (2009); M. Jain, J. R. Chelikowsky and S. G. Louie, Phys. Rev. Lett. 107, (2011). 15/46

20 High-κ dielectric materials Computational details 16/46 Three fold coordinated (V O(3) ) and four fold coordinated (V O(4) ) vacancies in charge states {0, +1, +2} 96 atom super cells. Norm conserving pseudopotentials - Hf 5s 2 5p 6 5d 2 6s 2 and O 2s 2 2p Ry energy cuto for wavefunctions. PBE functional for structural energies. `One shot' GW done within generalized plasmon pole model done with BerkeleyGW. Bulk structural parameters and band gap (6.00 ev) in good agreement with experiment. SIESTA for large supercells to get electrostatic corrections.

21 17/46 PBE and quasiparticle level diagram - VO(3) 5 Energy (ev) Energy (ev) V +2 O(3) V +1 O(3) V 0 O(3) 0 V +2 O(3) V +1 O(3) V 0 O(3) PBE GW

22 18/46 Electrostatic corrections - VO(3) GW charge transition levels (ev) ε +2/ ε +2/ ε +1/ ε +1/ Formation energy E f q-1 E f q-1 ( R q-1 ) E f q ( R q) ε E f q R q-1rq Generalized coordinate

23 18/46 Electrostatic corrections - VO(3) GW charge transition levels (ev) ε +2/ ε +2/ ε +1/ ε +1/ Formation energy E f q-1 E f q-1 ( R q-1 ) S. Lany and A. Zunger, Phys. Rev. B (2010) E f q ε E f q ( R q) R q-1rq Generalized coordinate

24 18/46 Electrostatic corrections - VO(3) εdefect εv (ev) 3 2 Number of atoms in supercell V +2 O(3) V +1 O(3) V +1 O(3) V 0 O(3) V 1 3 PBE defect levels. 96 GW charge transition levels (ev) ε +2/ ε +2/ ε +1/ ε +1/ Formation energy E f q-1 ( R q-1 ) E f q ( R q) ε E f q-1 E f q R q-1rq Generalized coordinate

25 18/46 Electrostatic corrections - VO(3) εdefect εv (ev) 3 2 Number of atoms in supercell V +2 O(3) V +1 O(3) V +1 O(3) V 0 O(3) V 1 3 PBE defect levels. 96 GW charge transition levels (ev) w/o with cor. cor. ε +2/ ε +2/ ε +1/ ε +1/ Formation energy E f q-1 ( R q-1 ) E f q ( R q) ε E f q-1 E f q R q-1rq Generalized coordinate

26 High-κ dielectric materials Formation energy of VO Relative formation energy (ev) VBM V O(3) V +2 O V 0 O V +1 O CBM Fermi energy (ev) 1 M. Jain, J. R. Chelikowsky and S. G. Louie, Phys. Rev. Lett. 107, (2011). 19/46

27 High-κ dielectric materials Formation energy of VO Relative formation energy (ev) VBM V O(3) V O(4) V +2 O V 0 O V +1 O CBM Fermi energy (ev) 1 M. Jain, J. R. Chelikowsky and S. G. Louie, Phys. Rev. Lett. 107, (2011). 19/46

28 High-κ dielectric materials Formation energy of VO Relative formation energy (ev) VBM V O(3) V O(4) VBM V +2 O V 0 O V +1 O CBM Si CBM Fermi energy (ev) 1 M. Jain, J. R. Chelikowsky and S. G. Louie, Phys. Rev. Lett. 107, (2011). 19/46

29 High-κ dielectric materials Formation energy of vacancies Formation Energy (ev) Oxygen rich V 2 + O(3) V 2 + O(3) Oxygen poor V 1 + O(3) V 1 + O(3) V 0 O(4) V 0 O(4) Fermi Energy (ev) 1 M. Jain, J. R. Chelikowsky and S. G. Louie, Phys. Rev. Lett. 107, (2011). 20/46

30 Defect charging energy U 21/46 U, defect charging energy, is dened as the energy of the reaction: 2V + V 2+ + V 0 U = E f +2 ( R +2 ) + E f 0 ( R 0 ) 2E f +1 ( R +1 ) = ε +2/+1 + ε +1/0

31 Comparison with previous studies 22/46 All values are in ev. V O(3) V O(4) ε +2/+1 ε +1/0 U ε +2/+1 ε +1/0 U GGA HSE PBE GW 3 (24 atoms) GW 4 (96 atoms) J.L. Lyons, A. Janotti and C.G. Van de Walle, Microelectronic Engineering 88, 1452 (2011). 2 P. Broqvist and A. Pasquarello, Appl. Phys. Lett. 89, (2006). 3 E.-A. Choi, and K. J. Chang, Appl. Phys. Lett., 94, (2009). 4 M. Jain, J. R. Chelikowsky and S. G. Louie, Phys. Rev. Lett. 107, (2011).

32 Defect charging energy U 23/46 U, defect charging energy, is dened as the energy of the reaction: 2V + V 2+ + V 0 U = E f +2 ( R +2 ) + E f 0 ( R 0 ) 2E f +1 ( R +1 ) = ε +2/+1 + ε +1/0

33 Defect charging energy U 23/46 U, defect charging energy, is dened as the energy of the reaction: 2V + V 2+ + V 0 U = E f +2 ( R +2 ) + E f 0 ( R 0 ) 2E f +1 ( R +1 ) = ε +2/+1 + ε +1/0 U elec + U relax = {E f +2 ( R +1 ) + E f 0 ( R +1 ) 2E f +1 ( R +1 )}+ {[E f +2 ( R +2 ) E f +2 ( R +1 )] + [E f 0 ( R 0 ) E f 0 ( R +1 )]}

34 Defect charging energy U 23/46 U, defect charging energy, is dened as the energy of the reaction: 2V + V 2+ + V 0 U = E f +2 ( R +2 ) + E f 0 ( R 0 ) 2E f +1 ( R +1 ) = ε +2/+1 + ε +1/0 U elec + U relax = {E f +2 ( R +1 ) + E f 0 ( R +1 ) 2E f +1 ( R +1 )}+ {[E f +2 ( R +2 ) E f +2 ( R +1 )] + [E f 0 ( R 0 ) E f 0 ( R +1 )]} U elec 0 and U relax 0

35 Defect charging energy U 24/46 2V + V 2+ + V 0 U elec U relax U V O(3) V O(4)

36 High-κ dielectric materials Conclusions 25/46 DFT+GW method for calculating the stability of oxygen vacancies. Qualitative agreement with previous hybrid functional calculations on the vacancies. Quantitative disagreement with previous hybrid functional calculations - vacancies near a Si/HfO 2 interface. DFT+GW HSE V O +1,0 +2,+1

37 Outline 26/46 Motivation. Computational methods. Defects in oxides. Why are defects challenging? Defects in high-κ dielectric HfO 2. Defects in technologically important material TiO 2. Defects for quantum computing.

38 Vacancies in TiO 2 Computational details 27/46 Oxygen vacancies (V O ) in rutile in charge states {0, +1, +2} 72 atom super cells. 200 Ry energy cuto for wavefunctions. HSE functional for structural energies. `One shot' GW done within generalized plasmon pole model done with BerkeleyGW. Bulk structural parameters and band gap (3.13 ev) in good agreement with experiment.

39 Schematic quasiparticle level diagram - VO 28/46 3 Energy (ev) V +2 V +1 O O V 0 O

40 Polaron formation V +1 O 29/46 Relaxation of the atoms : Two minima

41 Polaron formation V +1 O Relaxation of the atoms : Two minima A state in the gap polaron. V +1 O V+2 + polaron. O Lower in energy by 1.2 ev. Not relevant for charge transition level A state in the gap defect state. Charge density on the defect site. Local minima. Relevant for charge transition level 29/46

42 TiO 2 Formation energy of vacancies 30/46 Oxygen rich Titanium rich

43 Comparison with previous studies 31/46 All values are in ev. V O ε +2/+1 ε +1/0 U GGA HSE GW Negative U defect. 1 H. Iddir, S. Ogut, P. Zapol, and N. D. Browning, Phys. Rev. B 75, (2007). (value from plot) 2 A. Janotti, J. B. Varley, P. Rinke, N. Umezawa, G. Kresse, and C. G. Van de Walle, Phys. Rev. B (2010). (value from plot) 3 A. Malashevich, M. Jain and S. G. Louie, Phys. Rev. B 89, , (2014).

44 TiO 2 Conclusions 32/46 DFT+GW method for calculating the stability of oxygen vacancies. Negative U defect. Qualitative agreement with previous hybrid functional calculations. Quantitative disagreement with previous hybrid functional calculations. DFT+GW HSE V O +2,0 +2

45 Outline 33/46 Motivation. Computational methods. Defects in oxides. Why are defects challenging? Defects in high-κ dielectric HfO 2. Defects in technologically important material TiO 2. Defects for quantum computing.

46 Defects for spin-qubit applications Room temperature individually addressable spin systems in the solid-state for quantum computing.

47 Defects for spin-qubit applications 34/46 Room temperature individually addressable spin systems in the solid-state for quantum computing. NV center in diamond is the leading candidate.

48 Defects for spin-qubit applications 34/46 Room temperature individually addressable spin systems in the solid-state for quantum computing. NV center in diamond is the leading candidate. Possible room-temperature qubit with long coherence time ( 1ms) for quantum computing. High sensitivity, high-spatial resolution magnetometry.

49 NV center in diamond Motivation 35/46 Optical initialization at room-temperature provides initial spin pure state for spin-qubit operation. Notation : 2S+1 Λ Λ : Irreducible representation of the orbital symmetry S : Total Spin

50 NV center in diamond Motivation 35/46 Optical initialization at room-temperature provides initial spin pure state for spin-qubit operation. Notation : 2S+1 Λ Λ : Irreducible representation of the orbital symmetry S : Total Spin

51 NV center in diamond Unsolved problem 36/46 Identication of singlet-level structure. Eective optical initialization path between the two triplet levels.

52 NV center in diamond Single particle levels NV is a deep level center in a band gap of diamond with multiple localized, interacting electrons. CBM Open-shell problem 1. Correlation due to degenerate ground-states. VBM 1 J. Lischner, J. Deslippe, M. Jain and S. G. Louie, Phys. Rev. Lett. 109, (2012) 37/46

53 NV center in diamond Dangling orbitals 38/46 8 dangling spin-orbitals in and near band-gap of diamond Good basis for low energy excitations.

54 NV center in diamond Extended Hubbard model i, j :Atomic sites (C or N) Ĥ = E i n iσ + t i,j c i,σ c j,σ i,σ i j,σ + Un i, n j, + Vn i,σ n j,σ i i j,σ,σ σ, σ :Spin direction E i :On-site energy t i,j :Hopping integral U :On-site Coulomb repulsion V :Nearest neighbor Coulomb repulsion Eective Coulomb interaction parameters Strong electron-electron correlation Screening from diamond Geometry dependent parameter sets Large structural relaxation Hindered by the diculty in getting physically grounded model parameters. 39/46

55 NV center in diamond Model parameters from GW 40/46 Use ab initio GW to get model parameters, incorporating realistic electron-electron interactions. Initial sets of parameters Ab initio GW Extended Hubbard model GW Defect state quasiparticle energy Defect state quasiparticle energy E ab qp = E H qp? No Yes Tune model parameters Done

56 NV center in diamond Model parameters from GW 41/46 U/t > 3 strongly correlated system. Reasonable values of parameters. Geometry dependent parameters. All parameters are in ev. E C -E N t NC t CC U V Ground-state Excited-state

57 NV center in diamond Level diagram Level diagram 1 from exact diagonalization of model Hamiltonian all many-electron correlation eects within our Hilbert space. All energies in ev. 1 S. Choi, M. Jain and S. G. Louie, Phys. Rev. B 86, (2012) 42/46

58 NV center in diamond Proposed optical pathway 1 43/46 Spin triplet Spin singlet 1 E 3 E m s = ±1 SO coupling 1 A 1 3 A 2 m s = 0 SO coupling 1 E Qualitatively and quantitatively consistent with available experimental data. 1 S. Choi, M. Jain and S. G. Louie, Phys. Rev. B 86, (2012)

59 NV center in diamond Conclusion 44/46 Constructed extended Hubbard Hamiltonian from ab initio GW calculations. Through exact diagonalization, many-electron eects strongly aect the energy level diagram qualitatively and quantitatively. Computed ground- and excited-state energy surfaces and transition rates between them provided a consistent picture with experiments. Proposed an optical initialization pathway in which inter-sytstem crossing plays a crucial role. S. Choi, M. Jain and S. G. Louie, Phys. Rev. B 86, (2012)

60 Conclusion 45/46 Ab initio methods to understand and predict properties of defects in materials. Oxygen vacancies in high-κ material HfO 2 and photocatalytic material TiO 2. NV center in diamond for quantum computing application.

61 Acknowledgements 46/46 Prof. Steven Louie Prof. James Chelikowsky Dr. Andrei Malashevich Dr. Johannes Lischner Dr. Sang-Kook Choi Dr. Jack Deslippe Dr. Georgy Samsonidze Prof. Marvin Cohen

Defects in Semiconductors

Defects in Semiconductors Mater. Res. Soc. Symp. Proc. Vol. 1370 2011 Materials Research Society DOI: 10.1557/opl.2011. 771 Electronic Structure of O-vacancy in High-k Dielectrics and Oxide Semiconductors