density functional theory methods for transport and
TRANSCRIPT
Density Functional Theory Methods for Transport and Optical Properties: Application to Warm Dense Silicon
V. V. KarasievUniversity of RochesterLaboratory for Laser Energetics
48th Annual Anomalous Absorption Conference,
Bar Harbor, ME8–13 July 2018
1
• Significant K-edge shift in dense silicon is predicted
• Astrophysics opacity table (AOT) data fail to predict it
t (g/cm3)
0 100 200 300 400 5001800
1900
2000
2100
2200
K-e
dg
e p
osi
tio
n (
eV)
Si, T = 62.5 kK
DFT (shifted by 50 eV)AOT
TC14292
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/cm3 and temperature between 0.5 and 1000 eV) have been calculated
2
Summary
Collaborators
S. X. Hu
University of Rochester Laboratory for Laser Energetics
L. Calderin
University of Arizona
3
Acknowledgement:NERSC, supported by the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231Laboratory for Laser Energetics
E25548c
Silicon is important to HED physics, such as planetary science and ICF capsules
Motivation
4
Planetary science* ICF capsule**NIF
OMEGA
11 to 14 nm CHSi (6%) 1 to 2 nm mid-Z (Z = 6 to 14, Si) 6 nm Be125 nm DT
4 nm CHSi (6%) 0.5 to 1 nm mid-Z 3 nm Be45 nm DT
1350 nm
• The high-pressure silicon equation of state (EOS) is crucial to understanding the dynamics of silicon-rich planets (i.e., Earth)
• Silicon is used in ICF capsules to reduce fuel preheat and laser–plasma instability (LPI) effects
* http://www.nasa.gov/sites/default/files/images/607068main_world-unlabeled.jpg ** V. N. Goncharov et al., Phys. Plasmas 21, 056315 (2014).
HED: high-energy density ICF: inertial confinement fusion NIF: National Ignition Facility
TC14293
Warm dense matter is of interest in planetary science, astrophysics, and ICF
• Classical plasma approaches work only for weakly coupled nondegenerate systems (C% 1: low density, very high temperature)
5
• All regions except left-upper corner require quantum treatment of electronic degrees of freedom
• Most of (semi-) classical models are inaccurate or fail
C = e2 / rskBT $ 1: the Couloumb coupling parameteri = t = T/TF á 1: reduced temperatureTF: Fermi temperature
182
4
6
8
10
20 22
Cla
ssic
alp
lasm
a
IdealFermi gas
White dwarfHE envelope
White dwarfs
Earth coreDiamond anvils
Metals
XFEL
ICFr s =
10
r s =
1
24 26 28
C = 1
i = 1i = 10
i = 0.1
30
log
10 T
/K
log10 n/cm–3
Shock wavesShock wavesSolar coreSolar core
WDM
r s =
0.1
r s =
0.1
Red giantsRed giants
ExoplanetsExoplanets
Jupiter coreJupiter core
HE: high energyXFEL: X-ray free-electron laser
Schematic temp.-density diagram (PR 744, 1 (2018))
TC14294
Thermal DFT coupled with ab initio molecular dynamics (AIMD) has become a standard tool in high-energy-density physics (HEDP)
Molecular dynamics
Born–Oppenheimer energy surface:
6
The current best practice uses free-energy DFT with one-electron Kohn–Sham orbitals
The Kohn–Sham scheme replaces the (3Ne)-dimensional problem by Ne-coupled 3-D problems
Thermodynamic data
Start
Ion distribution
Pseudo-potential
Veff
E [t(r)]Moleculardynamics
Electrondensityt(r)
E min?
MD step
No
Yes
DFT step
Kohn–Sham equation
fk – zk
:dn n r v r n r–F ext nX = +^ ^ ^ ^h h h h6 @#:n n n nF F F FH xcs= + +^ ^ ^ ^h h h h
:nFH ^ h:nFxc ^ h
:nFs ^ h
V R R E Rion ion–X= +^ ^ ^h h h" " ", , ,
MD: molecular dynamics
grand potential
free-energy functional
Hartee energy
XC energy
non-interacting (Kohn–Sham) free energy
Free-energy density functionals:
– , , ....,m R V R R RI I I N1 2d=p v ^ h
TC14295
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
7
TC13730
A framework for temperature-dependent XC functionals has been developed to address the issue of thermal effects
Exchange
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
Constraints
• Reproduce finite-temperature gradient expansion
• Reduce to correct T = 0 limit
• Reduce to correct high-temperature limit
Correlation
8
, , dF n T n f n T F s T rGGA LDAx x x x2= ^ ^h h6 6@ @#
, , , ;s n n T s n n B tx x22d d/ u^ ^ ^h h h
, ; /f n T n A t t T TLDA LDAFx x xf= =u^ ^ ^h h h
, , , dF n T n f n n T rcGGA
cGGA d= ^ h6 @ #
GGA correlation energy per particle:
, , , ,f n n T f n T H f q TcGGA
cLDA
cLDA
cd = +^ ^ ^h h h8 B, , , ,q n n T q n n B n tc cd d/ u^ ^ ^h h h
* 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, 076401 (2018).
Generalized gradient approximation (GGA)
• The real part of electrical conductivity v1 and thermal conductivity l are given in terms of the Onsager coefficients Lmn
• The frequency-dependent Onsager coefficients are defined in terms of the velocity operator matrix elements:
• The imaginary part v2 could be calculated via Kubo Greenwood formula in terms of the velocity operator matrix elements or alternatively from the principal value integral
TC14297
Electronic transport coefficients are calculated from ab initio simulations within the framework of the Kubo–Greenwood approach (linear response theory)
9
; ;i L T L LL1 –1 2 1 11 22
11
122
v v v v l= + = = d n
ki kj2
d} }a
The dielectric function and index of refraction (Re and Im parts, n and k):
The reflectivity absorption coefficient and grouped Rosseland mean opacity:
i n ik1 22f ~ f ~ f ~ ~ ~= + = +^ ^ ^ ^ ^h h h h h6 @ –
rn k
n k
1
12 2
2 2~
~ ~
~ ~=
+ +
+^ ^^
^^h h
hhh6
6 @@
n c4
1~ ~a~r v=^ ^ ^h h h
;n k2 2–1 1+
~f ~ f ~
~f ~ f ~
= =^ ^ ^ ^ ^ ^h h h h h h;1 4 4–1 2 2 1f ~ ~
rv ~ f ~ ~rv ~= =^ ^ ^ ^h h h h
:,
,
d
d
K
n TB T
n TB T
1R
m
1 22
2
1
21
2
2
2
2
2
~ ~
~a ~
~~
~~
~
=
~
~~
~
^^
^
^
^
^hh
h
h
h
h#
#
L. Calderín, V. V. Karasiev, and S. B. Trickey, Comput. Phys. Commun. 221, 118 (2017).
TC14300
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
• We constructed an all-electron PP for Si, rc = 0.75 bohr
• Converged cutoff energy: Ecut = 400 Ry = 5.4 keV
10
Convergence study of the total energy and pressure with regard to Ecut for all-electron PP for Si64 atoms molecular-dynamics snapshot (t = 2.57 g/cm3, T = 2000 K)
–7.878
–7.876
–7.874
–7.872
En
erg
y (k
eV/a
tom
)
0
1
2
2 4 6 8 10
3
4
P (
GP
a)Cutoff energy (keV)
Ecut = 5.4 keV
Si64, 2.57 g/cm3, 2 kK
TC14301
Convergence study of absorption with regard to pseudopotential cutoff radius for Si1, t = 9.0 g/cm3, and T = 62,500 K
11
106
105
104
103
10210 100
Si1–sc, t = 9.0 g/cm3, T = 62.5 kK
1,000 10,000
~ (eV)
Ab
sorp
tio
n (
cm–1
)
rc = 0.75 bohr, Ecut = 11 keVrc = 0.60 bohr, Ecut = 11 keVrc = 0.40 bohr, Ecut = 22 keVrc = 0.20 bohr, Ecut = 27 keVrc = 0.12 bohr, Ecut = 76 keV
TC14303
A novel method of extending calculations to the x-ray range was proposed: combine the MD snapshot (Si32) and single-atom (Si1) data for absorption: Si32 data at ~ < 300 eV and Si1 data at ~ > 300 eV
12
100
101
102
103
104
105
106
107
Si: t = 9 g/cm3, T = 62.5 kK, rc = 0.20 bohr
Energy (eV)
–2000 0
EF
2s–2p
1s
2000 4000 6000 80000.01
0.1
1
10
100
1000
DO
S (
1/eV
)
Ab
sorp
tio
n (
cm–1
)
100 101 102 103 104
~ (eV)
Si32: d = 4.0 eV, ~ < 300 eV Si1: d = 60.0 eV, ~ < 300 eV
t = 9 g/cm3, T = 62.5 kK
DOS, Si32, Nb = 8192DOS × 32, Si-sc, Nb = 8192
MD: molecular dynamicsDOS: density of states
TC14302
Comparison to the NIST reference data for Si absorption, t = 2.33 g/cm3, T = 300 K confirms accuracy of the method; some discrepancies are observed at low ~
13
105
104
103
102
101100 101 102 103 104
~ (eV)
Ab
sorp
tio
n (
cm–1
)
106
107
Si: t = 2.33 g/cm3, T = 300 K
Si64 MDSi1NIST data
NIST: National Institute of Standards and Technology
TC14304
Comparison between the AOT and DFT data for opacity, silicon, t = 9.0 g/cm3, T = 62,500 K shows good agreement only at some conditions
14
105
104
103
102
101
Ro
ssel
and
mea
n o
pac
ity
(cm
2 /g
)
100
10–1
106
101 102 103 104 105
~ (eV)
Si, t = 9 g/cm3, T = 62,500 K
AOT, 10 g/cm3, T = 58,000 KAIMD, Si32, = d1Le = 4 eV, D~ = 8 eV Si-sc, d = 60 eV, D~ = 4 eV
AOT: astrophysics opacity table
TC14305
Comparison between the AOT and DFT data for total opacity, silicon t = 50 g/cm3 K shows discrepancy at T < 30 eV
15
• AOT data are not accurate at low temperatures• The same trend previously was observed in AOT
data for other materials D2 and CH*
Tota
l Ro
ssel
and
op
acit
y (c
m2 /
g)
103
104
105
101 102
T (eV)
AOTDFT, d = 40 eV
Si, t = 50 g/cm3
S. X. Hu et al., Phys. Rev. E 90, 033111 (2014);S. X. Hu et al., Phys. Rev. B 96, 144203 (2017).
TC14306
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*
16
10
1.0
0.1
0.01
DO
S (
1/eV
)
1.0
0.1
0.01
0.001
DO
S (
1/eV
)
Energy (eV)
–1500–2000 –1000 –500 0 500 1000 1500 3500
1.0
0.1
0.01
0.001
DO
S (
1/eV
)105
104
103
102
am
(cm
2 /g)
105
104
103
102
am
(cm
2 /g)
0 500 1000 1500 2000 2500 3000
105
104
103
102
am
(cm
2 /g)
~ (eV)
EF
2s 2p
1s
1s
1s
EF
2s 2p
EF
2s 2p
t = 2.33 g/cm3, T = 62.5 kK
t = 50 g/cm3, T = 62.5 kK
t = 250 g/cm3, T = 62.5 kK t = 250 g/cm3
t = 50 g/cm3
t = 2.50 g/cm3
* S. X. Hu, Phys. Rev. Lett. 119, 065001 (2017).
TC14307
K-edge shift in dense silicon is significant; AOT data fail to predict the shift
17
t (g/cm3)
0 100 200 300 400 5001800
1900
2000
2100
2200
K-e
dg
e p
osi
tio
n (
eV)
Si, T = 62.5 kK
DFT (shifted by 50 eV)AOT
• LLE is planning experiments on OMEGA for experimental measurements of the K-edge shift
TC14292
18
Summary/Conclusions
• 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/cm3 and temperature between 0.5 and 1000 eV) have been calculated
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]