Introduction to DFT and its Application to Defects in Semiconductors
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1 Introduction to DFT and its Application to Defects in Semiconductors Noa Marom Physics and Engineering Physics Tulane University New Orleans
2 The Future: Computer-Aided Materials Design Can access the space of materials not experimentally known Can scan through structures and compositions faster than is possible experimentally Unbiased search can yield unintuitive solutions Can accelerate the discovery and deployment of new materials Accurate electronic structure methods Efficient search algorithms
3 Dirac s Challenge The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems... P. A. M. Dirac Physics Nobel Prize, P. A. M. Dirac, 1929
4 The Many (Many, Many) Body Problem Schrödinger s Equation: But The physical laws are completely known There are as many electrons in a penny as stars in the known universe!
5 Electronic Structure Methods for Materials Properties Structure Ionization potential (IP) Absorption Mechanical Electron Affinity (EA) spectrum properties Fundamental gap Optical gap Vibrational Defect/dopant charge Exciton binding spectrum transition levels energy Ground State Charged Excitation Neutral Excitation DFT + electronic relaxation GW + electron-hole interaction BSE TDDFT
6 DFT- Density Functional Theory Use what you can measure! The density is used as the basic variable instead of the many-body wave function with no loss of information The Hohenberg & Kohn theorem (1964): One-to-one correspondence between the ground state density and the external potential Walter Kohn Nobel Prize in Chemistry, 1998 For DFT 3 n ( r) Vext ( r) ( r1, r2,..., rn) 3N Electron Density ion-e Many-body wave function
7 DFT- Density Functional Theory Exact mapping to a single particle problem with the many-body effects contained in the exchange-correlation functional Kohn & Sham, V ion r + V Hartree n r + V xc n r φ i r = ε i φ i (r) kinetic ion-e e-e many-body n r = φ i (r) 2 occupied states The Kohn-Sham equation is a Schrödingerlike eigenvalue equation, solved selfconsistently to find the ground state density V xc is unknown and must be approximated
8 DFT- Density Functional Theory The success of DFT calculations hinges on a good choice of approximation for the exchangecorrelation functional
9 DFT Functionals: Local and Semi-Local Functionals The local density approximation (LDA) E xc is approximated by its value per particle in a uniform electron gas weighted by the local density Perdew s ladder of DFT functionals D. M. Ceperley, B. J. Alder, Phys. Rev. Lett. 45, 566 (1980) Generalized gradient approximation (GGA) E xc includes a dependence on the density gradient to account for density variations (PBE) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996); 78, 1396 (1997) Meta-GGA E xc also depends on the kinetic energy density (TPSS) J. Tao, J.P. Perdew, V.N. Staroverov, G.E. Scuseria Phys. Rev. Lett. 91, (2003) J. P. Perdew, MRS Bulletin 38, 743 (2013)
10 E x HF = 1 2 i,j DFT Functionals: Hybrid Functionals Hybrid Functionals A fraction,α, of exact (Fock) exchange is mixed with GGA exchange and correlation. PBE0 has 25% EXX φ i (r 1 )φ j (r 1 ) 1 r 12 φ i (r 2 )φ j (r 2 )dr 1 dr 2 C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 (1999) Range-Separated Hybrids The Coulomb potential is split into short-range and long-range parts 1 r 12 = erfc (ωr 12) r 12 + erf (ωr 12) r 12 1/ω is a characteristic length scale for SR-LR transition HSE reduces to PBE0 in the SR and to PBE in the LR (α=0.25; ω=0.11 Bohr -1 Perdew s ladder of DFT functionals E HSE xc = αe HF,SR x ω + 1 α E ωpbr,lr PBE x ω + E c J. Heyd, G. E. Scuseria, M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003); 124, (2006) J. P. Perdew, MRS Bulletin 38, 743 (2013)
11 Pathologies of Semi-Local Functionals The (lack of) Derivative Discontinuity: The chemical potential is supposed to jump discontinuously when going through an integer particle number The Self-Interaction Error: Spurious Coulomb repulsion of an electron from itself due to incomplete cancellation of the self-interaction in the Hartree term by the approximate exchange term Effects: Destabilization of localized states and severe gap underestimation Hybrid functionals mitigate (but not completely correct!) these deficiencies
12 The quasiparticle equation: The GW Approximation 1 2 QP QP V (,, ) ( ) ( ) 2 ion VHartree r r Ei i r Ei i r The GW approximation (Hedin, 1965): The self-energy is approximated by the first order term in a perturbative expansion in the screened Coulomb interaction igw G 0 W 0 (Hybertsen and Louie, 1986): Assume that the KS wave-function and eigenvalues are good approximations for the many-body wave-function and QP energies Calculate the QP energies non-self-consistently as perturbative corrections to the KS energies: E self-energy many-body V G0W0 KS G0W0 i i i xc i Use KS orbitals and energies to evaluate G 0 and W 0
13 DFT in Practice: Common Basis Sets and Codes To solve the Kohn-Sham eigenvalue equation in practice, it must be discretized in a basis set. Different DFT codes use different basis sets: Plane-waves (PW) are the most convenient basis set for periodic systems: VASP, Quantum Espresso, ABINIT, CASTEP Other types of basis sets: Real space: PARSEC, OCTOPUS Wavelets: BigDFT, MADNESS Gaussian orbitals (GTO): NWChem, Q-Chem, Gaussian, CRYSTAL Slater orbitals (STO): ADF Numeric atom-centered orbitals (NAO): FHI-aims, SIESTA CONQUEST Full potential linearized augmented plane-waves (FP-LAPW): EXCITING, FLEUR, ELK, WIEN2k See also:
14 DFT in Practice: Pseudo Potentials Core electrons do not contribute significantly to chemical bonding and physical phenomena of interest (except for X-ray spectroscopies) but do contribute high frequency terms in the wave-function that are numerically difficult to deal with all-e The wave-function in the core region is replaced by a smooth function and the 1/r potential by a slowly-varying pseudo-potential Common types of pseudo-potentials: Norm-conserving Ultra-soft Projector augmented waves (PAW) pseudo
15 Keeping Supercomputers Busy Hopper (#34) Cray XE6 153,216 cores 1.28 petaflops Edison (#18) Cray XC30 133,824 cores 2.77 petaflops Mira (#5) BlueGene/Q 786,432 cores 8.6 petaflops The computational cost of quantum mechanical simulations increases with the accuracy of the method and with the system s size and complexity Configuration space exploration may require running thousands of trial structures New at Tulane: Cypress (#271) Dell cluster 2,480 Intel Ivy Bridge+ 15,128 Xeon Phi cores 0.7 petaflops
16 Computational Cost of Different Methods (FHI-aims) Local and semi-local functionals -> low computational cost, good scaling Non-local functional -> better accuracy, higher computational cost, worse scaling Scaling of semi-local DFT: O(N 3 ) -> O(N) Hybrid DFT: O(N 4 ) -> O(N)*large prefactor G 0 W 0 : O(N 4 ) -> O(N 3 ) (post SCF) PBEh PBE G 0 W
17 Applying DFT to Defects in Semiconductors Formation energies: Neutral defect: Charged defect: Freysoldt et al., Rev. Mod. Phys. 86, 253 (2014) The chemical potential: Determined by the conditions (T,P) of the experiment and used to simulate different conditions A consistent choice of reference must be made, either to the total energy of the elemental phases at 0K or at STP Chemical potentials are bound by the existence or appearance of secondary phases
18 Applying DFT to Defects in Semiconductors The supercell approximation: Defects in diamond The defect is placed in a large simulation cell with periodic boundary conditions Higher effective concentration of defects Interactions between periodic replicas may broaden the defect state For charged defects a compensating background charge is added The slow decay of the Coulomb potential leads to spurious interactions Various correction schemes: Makov and Payne (MP) Freysoldt, Neugebauer, and Van de Walle (FNV) Lany and Zunger (LZ) Potential alignment (PA) H. P. Komsa, T. Rantala, & A. Pasquarello PRB 86, (2012)
19 Applying DFT to Defects in Semiconductors Comprehensive review: Freysoldt et al., Rev. Mod. Phys. 86, 253 (2014) Things to worry about: (In)accuracy of the exchange-correlation functional Supercell artifacts Best practices: Start by reproducing known results Check carefully the sensitivity of the results to the parameters of the calculations (DFT functional, supercell size, etc.) Cross-validate using different methods Qualitative trends are more reliable than absolute numbers
20 Case Study: Designing a Shallow Donor in Diamond UT-Austin, 2010 Jonathan E. Moussa, Noa Marom, Na Sai, James R. Chelikowsky PRL 108, (2012) Motivation: Diamond has desirable properties for high-power, hightemperature electronics Boron is a good p-type dopant The lack of appropriate n-type dopant limits its applications P. W. May A New Diamond Age? Science 319, 1490 (2008)
21 Case Study: Designing a Shallow Donor in Diamond Shallow donor proposals: The broken C-N bond near a substitutional N impurity forms a deep, localized donor state Preserving the bond may create a shallower and more delocalized donor A series of XN n impurity complexes is generated by reducing the valence of the central atom and compensating with neighboring N atoms For LiN 4 there are no broken bonds at the equilibrium geometry Is it shallow? Can it be synthesized? J. E. Moussa, N. Marom, N. Sai, J. R. Chelikowsky PRL 108, (2012)
22 Case Study: Designing a Shallow Donor in Diamond Method: Decomposition of the donor activation energy: Relaxation energy (the energy difference between the charged and neutral geometries) calculated with a total energy method: PBE Vertical ionization energy (at the fixed geometry of the ionized donor) calculated with an accurate quasi-particle method: PBE0-ε and corrected for finite size effects M. Jain, J. R. Chelikowsky, S.G. Louie PRL 107, (2011) J. E. Moussa, N. Marom, N. Sai, J. R. Chelikowsky PRL 108, (2012)
23 Case Study: Designing a Shallow Donor in Diamond Step 1: Pick a reliable method PBE0-ε: A PBE-based hybrid functional with a systemdependent fraction,α, of exact (Fock) exchange, determined by the dielectric constant: α = 1/ε Marquez et al. PRB 83, (2011) For diamond: ε = 5.7; α= 0.18 J. E. Moussa, N. Marom, N. Sai, J. R. Chelikowsky PRL 108, (2012)
24 Case Study: Designing a Shallow Donor in Diamond Defect band dispersion model: Step 2: Finite size correction Interactions between periodic replicas cause broadening of the defect band D ionize was calculated using supercells of 5x5x5 to 8x8x8 diamond unit cells The impurity band becomes shallower and narrower with size Define: 6x6x6 supercell A least square fit is performed with x as a free parameter to minimize the size dependence of D ionize around an asymptotic value This gives D ionize =0.27 ev (x = 0.74) J. E. Moussa, N. Marom, N. Sai, J. R. Chelikowsky PRL 108, (2012)
25 Case Study: Designing a Shallow Donor in Diamond The marker method: Step 3: Cross-validation Activation energies are calculated based on PBE total energy differences and referenced to an experimentally known marker impurity fit to experiment same simulation conditions same material environment same defect type same charge state A. Resende et al., Phys. Rev. Lett. 82, 2111 (1999) The applicability of the marker method is limited to cases with: Activation energies obtained with the marker method are overestimated due to the self-interaction (delocalization) error in PBE J. E. Moussa, N. Marom, N. Sai, J. R. Chelikowsky PRL 108, (2012)
26 Case Study: Designing a Shallow Donor in Diamond LiN 4 is a promising shallow donor in diamond J. E. Moussa, N. Marom, N. Sai, J. R. Chelikowsky PRL 108, (2012)
27 Case Study: Designing a Shallow Donor in Diamond Possible CVD precursor: 1,7-diazacyclododecane-4,10-diamine can strongly bind a Li atom The Li atom is not bound to the B-center in diamond but kinetically stabilized by a high energy barrier E=0.00 ev E= ev E= 0.91 ev J. E. Moussa, N. Marom, N. Sai, J. R. Chelikowsky PRL 108, (2012)
28 Applying DFT to Defects in Semiconductors Comprehensive review: Freysoldt et al., Rev. Mod. Phys. 86, 253 (2014) Things to worry about: (In)accuracy of the exchange-correlation functional Periodic boundary conditions artifacts Best practices: Start by reproducing known results Check carefully the sensitivity of the results to the parameters of the calculations (DFT functional, supercell size, etc.) Cross-validate using different methods Qualitative trends are more reliable than absolute numbers
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