Density Functional Theory Methods for Transport and Optical Properties: Application to Warm Dense Silicon

Transcription

1 Density Functional Theory Methods for Transport and Optical Properties: Application to Warm Dense Silicon 2200 Si, T = 62.5 kk K-edge position (ev) DFT (shifted by 50 ev) AOT Significant K-edge shift in dense silicon is predicted Astrophysics opacity table (AOT) data fail to predict it V. V. Karasiev University of Rochester Laboratory for Laser Energetics t (g/cm 3 ) 48th Annual Anomalous Absorption Conference, Bar Harbor, ME 8 13 July

2 Summary Finite-temperature density functional theory (DFT) is a powerful tool to accurately predict properties of matter at a wide range of densities across temperature regimes [warm-dense-matter (WDM) conditions] Exchange-correlation (XC) thermal effects in simulations of WDM are taken into account via use of a temperature-dependent XC functional A set of accurate all-electron pseudo-potentials for the calculation of x-ray absorption near edge structure (XANES) spectra has been constructed Absorption coefficients and opacities of silicon (densities between 0.1 and 500 g/cm 3 and temperature between 0.5 and 1000 ev) have been calculated TC

3 Collaborators S. X. Hu University of Rochester Laboratory for Laser Energetics L. Calderin University of Arizona Acknowledgement: NERSC, supported by the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 Laboratory for Laser Energetics 3

4 Motivation Silicon is important to HED physics, such as planetary science and ICF capsules The high-pressure silicon equation of state (EOS) is crucial to understanding the dynamics of siliconrich planets (i.e., Earth) Planetary science* NIF ICF capsule** 11 to 14 nm CHSi (6%) 1 to 2 nm mid-z (Z = 6 to 14, Si) 6 nm Be 125 nm DT Silicon is used in ICF capsules to reduce fuel preheat and laser plasma instability (LPI) effects 1350 nm OMEGA 4 nm CHSi (6%) 0.5 to 1 nm mid-z 3 nm Be 45 nm DT E25548c * ** V. N. Goncharov et al., Phys. Plasmas 21, (2014). HED: high-energy density ICF: inertial confinement fusion NIF: National Ignition Facility 4

5 Warm dense matter is of interest in planetary science, astrophysics, and ICF log 10 T/K Classical plasma i = 10 C = 1 i = 1 r s = 10 White dwarf HE envelope Metals i = 0.1 XFEL r s = 1 WDM ICF Jupiter core Earth core Diamond anvils log 10 n/cm 3 r s = 0.1 Solar core Shock waves Exoplanets Red giants White dwarfs Ideal Fermi gas 30 C = e 2 / r s k B T $ 1: the Couloumb coupling parameter i = t = T/T F á 1: reduced temperature T F : Fermi temperature Classical plasma approaches work only for weakly coupled nondegenerate systems (C% 1: low density, very high temperature) All regions except left-upper corner require quantum treatment of electronic degrees of freedom Most of (semi-) classical models are inaccurate or fail TC14293 Schematic temp.-density diagram (PR 744, 1 (2018)) HE: high energy XFEL: X-ray free-electron laser 5

6 Thermal DFT coupled with ab initio molecular dynamics (AIMD) has become a standard tool in high-energy-density physics (HEDP) Start Pseudopotential Molecular dynamics m Rp = dv VR ^, R,..., R I I I 1 2 N Born Oppenheimer energy surface: V^" R, h= X^" R, h+ E " ^Rh, h ion ion Ion distribution Electron density t(r) V eff Kohn Sham equation The current best practice uses free-energy DFT with oneelectron Kohn Sham orbitals Molecular dynamics MD step Yes No E min? Thermodynamic data DFT step E [t(r)] f k z k The Kohn Sham scheme replaces the (3N e )-dimensional problem by N e -coupled 3-D problems Free-energy density functionals: # X ^nh= F^nh+ dr6 vext ^rh n@ n^rh: grand potential F^nh= F ^nh+ F ^nh+ F ^nh: free-energy functional s H xc F H ^nh: Hartee energy F xc ^nh: XC energy F s ^nh: non-interacting (Kohn Sham) free energy TC14294 MD: molecular dynamics 6

7 DFT-based AIMD makes it possible to calculate many material properties required for simulations of ICF implosions and provides predictions for HEDP experiments Some of material properties accessible from DFT-based AIMD simulations Equation of state Thermal conductivity Electrical conductivity Reflectivity Absorption coefficients " Rosseland and Planck mean opacities HEDP requires development of new methods and functionals to accurately predict matter properties Temperature-dependent exchange-correlation functionals are required to take into account the XC thermal effects All-electron (i.e., with active 1s-core electrons) pseudo-potentials are needed for x-ray absorption calculations TC

8 A framework for temperature-dependent XC functionals has been developed to address the issue of thermal effects Exchange # FGGA n, T nflda = ^n, TF h 6s ^Th@ dr x x x 2x Generalized gradient approximation (GGA) Correlation F 6 n, T@ = # nf ^n, nt, hdr c GGA c GGA d TC13730 s ^n, dnt, h/ s2 ^n, dnb hu ^th; 2x x flda^n, Th= flda ^nhau ^th; t = T/ T x x x Constraints Reproduce finite-temperature gradient expansion Satisfy Lieb Oxford bound at T = 0 Reduce to correct T = 0 limit Reduce to correct high-temperature limit F GGA correlation energy per particle: fgga ^n, d nt, h= flda ^n, Th+ H8fLDA, q ^ThB c q ^n, dnt, h/ qn ^, dnh Bu ^nt, h c Constraints Reproduce finite-temperature gradient expansion Reduce to correct T = 0 limit c Reduce to correct high-temperature limit * V. V. Karasiev, J. W. Dufty, and S. B. Trickey, Non-Empirical Semi-Local Free-Energy Density Functional for Matter Under Extreme Conditions. Phys. Rev. Lett. 120, (2018). c c c 8

9 Electronic transport coefficients are calculated from ab initio simulations within the framework of the Kubo Greenwood approach (linear response theory) The real part of electrical conductivity v 1 and thermal conductivity l are given in terms of the Onsager coefficients L mn i ; L ; T L L v = v1+ v2 v1= 11 l = d 22 n L11 The frequency-dependent Onsager coefficients are defined in terms of the velocity operator matrix elements: 2 } d } The imaginary part v 2 could be calculated via Kubo Greenwood formula in terms of the velocity operator matrix elements or alternatively from the principal value integral The dielectric function and index of The reflectivity absorption coefficient and refraction (Re and Im parts, n and k): grouped Rosseland mean opacity: n^~ h@ + k2 ^~ h 4 f~ ^ h= f 1^~ h+ if 2^~ h= 6 n^~ h+ ik^~ h@ r^~ h = a ^ r ~ h = v 1 ^~ h n^~ h@ + k2 n ~ c ^~ h ^ h f 1 4 ; 4 1 ~ ~ r ~ 2 ^ h r = v 2 ^ ~ h f 2 ^ ~ h= ~ v 1 ^~ h 2B^~, Th # n2 ^~ h d~ 2T ~ 1 f~ ^ h + f 1 ^~ h f~ ^ h f 1 ^~ h KR ^~ 1: ~ 2h = ~ 2 n^~ h = ; k^~ h = B, T ^~ h # n2 ^~ h d~ a ^ ~ h 2T TC14297 ki a kj ~ 1 L. Calderín, V. V. Karasiev, and S. B. Trickey, Comput. Phys. Commun. 221, 118 (2017). m 9

10 Calculation of optical properties within the x-ray range involves transitions from core states " explicit treatment of 1s electrons in electronic structure calculations is required There is a need for an all-electron pseudo-potential (PP) to treat 1s states Energy (kev/atom) E cut = 5.4 kev Si 64, 2.57 g/cm 3, 2 kk P (GPa) We constructed an all-electron PP for Si, r c = 0.75 bohr Converged cutoff energy: E cut = 400 Ry = 5.4 kev Convergence study of the total energy and pressure with regard to E cut for all-electron PP for Si 64 atoms molecular-dynamics snapshot (t = 2.57 g/cm 3, T = 2000 K) Cutoff energy (kev) 0 TC

11 Convergence study of absorption with regard to pseudopotential cutoff radius for Si 1, t = 9.0 g/cm 3, and T = 62,500 K Absorption (cm 1 ) Si 1 sc, t = 9.0 g/cm 3, T = 62.5 kk r c = 0.75 bohr, E cut = 11 kev r c = 0.60 bohr, E cut = 11 kev r c = 0.40 bohr, E cut = 22 kev r c = 0.20 bohr, E cut = 27 kev r c = 0.12 bohr, E cut = 76 kev ,000 10,000 TC14301 ~ (ev) 11

12 A novel method of extending calculations to the x-ray range was proposed: combine the MD snapshot (Si 32 ) and single-atom (Si 1 ) data for absorption: Si 32 data at ~ < 300 ev and Si 1 data at ~ > 300 ev Absorption (cm 1 ) TC Si 32 : d = 4.0 ev, ~ < 300 ev Si 1 : d = 60.0 ev, ~ < 300 ev Si: t = 9 g/cm 3, T = 62.5 kk, r c = 0.20 bohr ~ (ev) DOS (1/eV) s E F 2s 2p t = 9 g/cm 3, T = 62.5 kk DOS, Si 32, N b = 8192 DOS 32, Si-sc, N b = Energy (ev) MD: molecular dynamics DOS: density of states 12

13 Comparison to the NIST reference data for Si absorption, t = 2.33 g/cm 3, T = 300 K confirms accuracy of the method; some discrepancies are observed at low ~ Absorption (cm 1 ) Si: t = 2.33 g/cm 3, T = 300 K Si 64 MD Si 1 NIST data TC14302 ~ (ev) NIST: National Institute of Standards and Technology 13

14 Comparison between the AOT and DFT data for opacity, silicon, t = 9.0 g/cm 3, T = 62,500 K shows good agreement only at some conditions TC14304 Rosseland mean opacity (cm 2 /g) Si, t = 9 g/cm 3, T = 62,500 K AOT, 10 g/cm 3, T = 58,000 K AIMD, Si 32, = d 1Le = 4 ev, D~ = 8 ev Si-sc, d = 60 ev, D~ = 4 ev ~ (ev) AOT: astrophysics opacity table 14

15 Comparison between the AOT and DFT data for total opacity, silicon t = 50 g/cm 3 K shows discrepancy at T < 30 ev Total Rosseland opacity (cm 2 /g) AOT DFT, d = 40 ev Si, t = 50 g/cm T (ev) AOT data are not accurate at low temperatures The same trend previously was observed in AOT data for other materials D 2 and CH* TC14305 S. X. Hu et al., Phys. Rev. E 90, (2014); S. X. Hu et al., Phys. Rev. B 96, (2017). 15

16 We predicted the K-edge shift of the x-ray absorption spectra in dense silicon as a result of continuum lowering and Fermi surface rising effects* DOS (1/eV) t = 2.33 g/cm 3, T = 62.5 kk 1s E F 2s 2p a m (cm 2 /g) t = 2.50 g/cm DOS (1/eV) t = 50 g/cm 3, T = 62.5 kk 1s 2s 2p E F a m (cm 2 /g) t = 50 g/cm 3 DOS (1/eV) t = 250 g/cm 3, T = 62.5 kk 1s 2s 2p Energy (ev) E F a m (cm 2 /g) ~ (ev) t = 250 g/cm3 TC14306 * S. X. Hu, Phys. Rev. Lett. 119, (2017). 16

17 K-edge shift in dense silicon is significant; AOT data fail to predict the shift K-edge position (ev) Si, T = 62.5 kk DFT (shifted by 50 ev) AOT t (g/cm 3 ) LLE is planning experiments on OMEGA for experimental measurements of the K-edge shift TC

18 Summary/Conclusions Finite-temperature density functional theory (DFT) is a powerful tool to accurately predict properties of matter at a wide range of densities across temperature regimes [warm-dense-matter (WDM) conditions] Exchange-correlation (XC) thermal effects in simulations of WDM are taken into account via use of a temperature-dependent XC functional A set of accurate all-electron pseudo-potentials for the calculation of x-ray absorption near edge structure (XANES) spectra has been constructed Absorption coefficients and opacities of silicon (densities between 0.1 and 500 g/cm 3 and temperature between 0.5 and 1000 ev) have been calculated TC