electronic structure of solidsth.fhi-berlin.mpg.de/sitesub/meetings/dft-workshop... · insulators,...
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Electronic structure of solids
Sergey V. Levchenko
Fritz Haber Institute of the Max Planck Society, Berlin, Germany
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Extended (periodic) systems
There are 1020 electrons per 1 mm3 of bulk Cu
x
y
z
a1
a2
a3
Position of every atom in the crystal (Bravais lattice):
332211321 )0,0,0(),,( aaarr nnnnnn
lattice vector:
,2,1,0,,
),,(
321
332211321
nnn
nnnnnn aaaR
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Example: two-dimensional Bravais lattice
1a
2a
primitive unit cells 12 23 aa
The form of the primitive unit cell is not unique
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From molecules to solids
Electronic bands as limit of bonding and anti-bonding combinations of
atomic orbitals:
electronic
band
band
gap
Adapted from: Roald Hoffmann, Angew. Chem. Int. Ed. Engl. 26, 846 (1987)
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Bloch’s theorem
Rr
r
R
0
)()exp()( rkRRr i
In an infinite periodic solid, the solutions of the
one-particle Schrödinger equations must
behave like
)()( rRr UU Periodic potential
(translational symmetry)
Consequently:
)()(),()exp()( rRrrkrr uuui
332211 aaaR nnn
Index k is a vector in reciprocal space
332211 gggk xxx ijji 2ag
1g
2g
nm
laa
g 2 – reciprocal lattice vectors
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The meaning of k
chain of hydrogen atoms
j
sjk ajikx )()exp( 1
Adapted from: Roald Hoffmann, Angew. Chem. Int. Ed. Engl. 26, 846 (1987)
k shows the phase with which the orbitals are combined:
k = 0: )2()()()0exp( 1110 aaaj ss
j
s
k = : )3()2()()()exp( 11110 aaaajji sss
j
s π a
a a a a
k is a symmetry label and a node counter, and also
represents electron momentum
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Bloch’s theorem: consequences
kGkGk krGrkr nn uiiui ~)exp()]exp()[exp(
a Bloch state
at k+G with
index n
a Bloch state at k with
a different index n’
kkk nnnh ˆ
In a periodic system, the solutions of the Schrödinger equations are
characterized by an integer number n (called band index) and a vector k:
For any reciprocal lattice vector
332211 gggG nnn
a lattice-periodic
function u~
1g
2g
21 ggG
Can choose to consider only k within single primitive
unit cell in reciprocal space
)()(),()exp()( rRrrkrr kkkk nnnn uuui
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Brillouin zones
A conventional choice for the reciprocal lattice unit cell
For a square lattice
For a hexagonal lattice
Wigner-Seitz cell
Wigner-Seitz cell
In three dimensions:
Face-centered
cubic (fcc) lattice Body-centered
cubic (bcc) lattice
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Time-reversal symmetry
For Hermitian , can be chosen to be real h kn
)()(ˆ)()(ˆ **rrrr kkkkkk nnnnnn hh
From Bloch’s theorem:
)()exp()()()exp()( **rkRRrrkRRr kkkk nnnn ii
)(*
kk nn kk nn )(
Electronic states at k and –k are at least doubly degenerate
(in the absence of magnetic field)
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Electronic band structure
kn
k0 π/a -π/a
For a periodic (infinite) crystal, there is an infinite number
of states for each band index n, differing by the value of k
Band structure represents dependence of on k
)(k
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Electronic band structure in three dimensions
z
Γ Λ
L U
X W K
Δ Σ y
x
Brillouin zone of the fcc lattice
By convention, are measured (angular-resolved
photoemission spectroscopy, ARPES) and calculated
along lines in k-space connecting points of high symmetry
kn
εn(k
), e
V
Al band structure (DFT-PBE)
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Finite k-point mesh
kk nn , – smooth functions of k
can use a finite mesh, and then
interpolate and/or use perturbation
theory to calculate integrals
Charge densities and other quantities are
represented by Brillouin zone integrals:
occ
BZ
32
BZ
)()(j
j
kdn rr k 8x4 Monkhorst-Pack grid
occ
1
2kpt
)()(j
N
m
jm mwn rr k
xk
yk 1b
2b
H.J. Monkhorst and J.D. Pack, Phys. Rev.B 13, 5188 (1976); Phys. Rev. B 16, 1748 (1977)
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Band gap and band width (dispersion)
a a a … 0.8 Å – hydrogen molecule chain
(DFT-PBE)
a = 10.0 Å
ε(k
)
a = 5.0 Å
ε(k
)
0 a
a
0
a
a
a = 2.0 Å
ε(k
)
a = 1.6 Å
ε(k
)
Overlap between interacting orbitals determines band gap and band width
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Insulators, semiconductors, and metals
Eg>>kBT
Insulators (MgO, NaCl,
ZnO,…)
Eg~kBT
Semiconductors (Si,
Ge,…)
Eg=0
Metals (Cu, Al, Fe,…)
k k
εF
In a metal, some (at least one) energy bands are only partially occupied
The Fermi energy εF separates the highest occupied states from lowest
unoccupied
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Density Of States (DOS)
n
N
j
jnj
n
n wkdgkpt
BZ1
3 ))(())(()( kk
Number of states in energy interval dε per unit volume,
d
d1
1
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Atom-Projected Density Of States (APDOS)
Decomposition of DOS into contributions from different atomic
functions : i
n
nnii kdrdgBZ
32
3 ))(()()()( krr k
Recovery of the chemical interpretation in terms of orbitals
Qualitative analysis tool; ambiguities must be resolved by truncating
the r-integral or by Löwdin orthogonalization of i
O(2s)
O(2p)
Mg(3s)
Mg(3p) Mg(3d)
O(3d)
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Periodic and cluster models of defects
Embedded cluster model Periodic model
+ Higher-level ab initio methods
can be applied
+/- Defects in dilute limit
- Effect of embedding on the
electronic structure and Fermi
level – ?
+ Robust boundary conditions
+ Higher defect concentrations
+/- Higher defect concentrations
- Artificial defect-defect
interactions
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DFT and wave-function methods for solids
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})({extiV r
Ground-state electronic structure problem
}{},{ ii r – many-body wave function, depends on spatial ( )
and spin ( ) coordinates of particles (also on nuclear
coordinates ( ) and )
iri
JR
}{},{
}{},{})({1
2
1 ext
2
2
ii
ii
i i J ji
i
jiJi
J
i
E
VZ
r
rrrrRrr
• already include approximations (Born-Oppenheimer,
non-relativistic, no spin-orbit coupling, no magnetic field)
• wave function depends on 4N variables (spatial + spin)
• electrons interact via Coulomb forces
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Non-interacting fermions
i i J ji
i
jiJi
J
i
VZ
H })({1
2
1ˆ ext
2
2
rrrRrr
i
ii vV )(})({extrr
),()()(2
12
2
rrrRrr
nnn
J J
Jv
Z
),(,),,(det!
1,, 11111 NNNNN
N rrrr
one-particle states
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Non-interacting fermions – periodic system
)()()(2
12
2
rrrRRrr
kkk
R
nnn
J J
Jv
Z
n – band index,
k – k-point
41 ,,,1 kkk n
41 ,,,2 kkk n
41 ,,,3 kkk n
Born-von Karman periodic boundary conditions
finite number of k-points, infinite (macroscopic) system as physical
limit
)()( Rrr kk nn
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),(,),,(det!
1,, 11111 NNNNN
N rrrr
Interacting fermions (electrons)
i i J ji jiJi
J
i
ZH
rrRrr
1
2
1ˆ2
2
0ˆ
ˆmin
i
HH
i
Introduce basis set: ,
variational principle
j
jiji cs )()(),( rr 0ˆ
ijc
H
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Approximations to the electronic problem: Basis set
Widely used basis sets:
gaussians (localized, analytic integrals)
plane waves (delocalized, analytic integrals)
Slater-type (localized, nuclear cusp)
grid-based (localized, analytic integrals)
)exp( 2rzyx kji
)exp( rk i
)exp( rzyx kji
)( irr
Core electrons are often treated separately
(pseudopotentials, plane-wave + localized basis)
Introduce basis set: , j
jiji cs )()(),( rr 0ˆ
ijc
H
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k
kkkR
k
R rr )(e1
)( )(m
m
nmi
n UN
w
From reciprocal to real space
Wannier functions: not unique, can
be chosen more or less localized An arbitrary unitary
transformation within
occupied and virtual
subspaces
)()(det!
111 11 NNww
Nrr RR
Wannier functions in each
unit cell
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The Hartree-Fock (HF) approximation
),(,),,(det!
1,, 11111 NNNNN
N rrrr
rrddnnZZ
hEM
I
M
J JI
JIN
n
nn
33
1 11
tot
)()(
2
1
2
1ˆrr
rr
RR
N
nm
mnnmddrrdd
1,
33** ),(),(),(),(
2
1
rr
rrrr
HF (exact) exchange energy
• No self-interaction
• Coulomb mean-field no dynamic correlation, single determinant
no static correlation
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The Hartree-Fock (HF) approximation
Dynamic correlation:
versus
Coulomb
potential
Non-dynamic (static) correlation:
versus +
HF approximation ≥90% of total energy, overestimates ionicity
smaller e density, large r /
(quasi)degenerate HOMO-LUMO)
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Density functional theory
Density functional theory: Hohenberg-Kohn theorem
rn
H NN rr ,,11
totE
– many-body Hamiltonian
– many-body wave function
– total energy
][)()(
2
1
2
1)(][ XC
33
1 1
3
1
tot nErrddnnZZ
rdn
ZnTEM
I
M
J JI
JIM
I I
I
rr
rr
RRRr
r
Approximations to : Local density approximation (LDA),
generalized gradient approximation (GGA), meta-GGA
][XC nE
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Standard DFT and the self-interaction error
][)()(
2
1
2
1)(][ XC
33
1 1
3
1
tot nErrddnnZZ
rdn
ZnTEM
I
M
J JI
JIM
I I
I
rr
rr
RRRr
r
)(rn -- electron density
LDA, GGA, meta-GGA: ][][][ loc
C
loc
XXC nEnEnE
Standard DFT: (Semi)local XC operator low computational cost
(includes self-
interaction)
exchange-
correlation
(XC) energy
Removing self-interaction + preserving fundamental properties (e.g.,
invariance with respect to subspace rotations) is non-trivial
residual self-interaction (error) in standard DFT
Consequences of self-interaction (no cancellation of errors):
localization/delocalization errors, incorrect level alignment (charge
transfer, reactivity, etc.)
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Hybrid DFT
][][)α1(}][{α}][{ loc
C
loc
X
HF
XXC nEnEEE
-- easy in Kohn-Sham formalism ( ) n
nnfn2
Perdew, Ernzerhof, Burke (J. Chem. Phys. 105, 9982 (1996)): α = 1/N
MP4 N = 4, but “An ideal hybrid would be sophisticated enough to
optimize N for each system and property.”
PBEC
PBEX
HFX
PBE0XC )α1(α EEEE
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Hartree-Fock exchange – the problem
rrddDDE
ljki
lkji
jkil33
,,,
HFX
)()()()(
2
1
rr
rrrr
)(ri
)(rk
)'(rj
)'(rl
rr
1
electron repulsion integrals
Lots of integrals, naïve implementation N4 scaling (storage
impractical for N > 500 basis functions)
• need fast evaluation
• need efficient use of sparsity (screening)
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Hartree-Fock exchange in extended systems
',,
33*
''*
HFX
),(),(),(),(
2
1
kk
kkkk
rr
rrrr
nm
mnnmddrrddE
rrdd
DDE
ljki
lkji
jkil
33
,,,,,
HFX
)()()()(
)()(2
1
rr
RRrRrRrr
RRRR
RRR
R
kRkk Rrr
,
e)()(),(i
iimim cs
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Electron repulsion integrals in periodic systems
)(ri
)( Rr j
Q Q
Q
RQQRQRRRRR
μ ν
ν
,)(μνμ,, lkjilkij CVCI
RI-LVL: Sparse, easier to calculate, linear scaling is possible!
RI-V is impractical for extended systems
)(μ Qr P
0μ, Q
RjiC
Q
QR QrRrr
μ
μμ, )()()( PC jiji
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Hybrid functionals -- scaling with system size
Periodic GaAs, HSE06 hybrid functional
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Hybrid functionals -- CPU scaling
Periodic GaAs, HSE06 hybrid functional
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Beyond mean-field for extended systems
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The concept of mixing excitations
iii EH 00ˆ – non-interacting effective particles (HF, DFT, etc.)
}{ i – a basis set for N-electron wave functions
j
j
ijij
j
ijj
j
iji cEcHHHc )]ˆˆ(ˆ[, 00
ikki
j
jkij cEEHc )(ˆ 0
Project onto equations for : k ijc
configuration interaction
,,, 210
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Configuration interaction – matrix diagonalization
abij
abij
abij
ai
ai
ai cc
,,
0
},{},{,0abij
ai DS
00ˆ H0
0ˆ HSS
D
T
Q
0
0ˆ HD
0
0
S D T Q
SH0
SHS ˆ
SHD ˆ
SHT ˆ
0
DH0
DHS ˆ
DHD ˆ
DHT ˆ
DHQ ˆ
0 0
THS ˆ
THD ˆ
THT ˆ
THQ ˆ
0
QHD ˆ
QHT ˆ
QHQ ˆ
!)!(
!
nnM
M
n-tuple
excitations
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Configuration interaction in periodic systems
n – band index,
k – k-point
41 ,,,1 kkk n
41 ,,,2 kkk n
41 ,,,3 kkk n
0
rqpk
rqpkrpqk
rqpk
qk
qkqk
qk
baji
bjai
baji
ai
ai
ai cc ,
,
,,
,,
,
0
excitations can change not only n but also k-point
momentum conservation
rqpk
qk
baji
ai H ,
,ˆ
RI is a must!
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Hierarchies of GS wave-function methods
Møller-Plesset
perturbation theory
(MP2, MP3, MP4,…)
Coupled-cluster
(CCD,CCSD,CCSDT,…)
0)0()0(
0
2
0)2(
0
ˆ
i i
i
EE
HE
MPm: ~nNm+2
0
ˆ}{
0 e mi
iTm
95
100
CID
CISD
CISDT
CISDTQ
MP2 MP3
MP4 MP5 MP6
CCD
CCSD
CCSD(T) CCSDT
CCSDTQ
R.J. Bartlett, private communication
Truncated CI
(CISD, CISDT,…)
0}{
0ˆ
mi
im T
CI{m}: ~nmNm+2
90
85
80
75
70
CC{m}: ~nmNm+2
% c
orr
ela
tion e
nerg
y
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Ripples in the charged Fermi liquid
0000ˆ EH VHH λˆˆ
00
λ0
1
tadiabatic switching
Long-range correlation (screening) contributions can be
summed up to infinite order (EX+cRPA, rPT2, GW, BSE)
- - - - - - - - -
- - - - -
+ + + + + + + + +
+ + + +
fluctuating charge density
polarization (screening)
i i
i
EE
VE
)0()0(0
2
0)2(0
λ
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Random-phase approximation
Second-order energy correction (MP2, HF orbitals):
rr
rrrr )()()()()|(
**bjai
jbia
abij bajii i
i jaibjbiajbia
EE
VE
,)0()0(
0
2
0)2(0
εεεε
))|()|)((|(2
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Random-phase approximation
correlation exchange
Second-order energy correction (MP2, HF orbitals):
rr
rrrr )()()()()|(
**bjai
jbia
abij bajii i
i jaibjbiajbia
EE
VE
,)0()0(
0
2
0)2(0
εεεε
))|()|)((|(2
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Random-phase approximation
correlation exchange
Second-order energy correction (MP2, HF orbitals):
abij bajii i
i jaibjbiajbia
EE
VE
,)0()0(
0
2
0)2(0
εεεε
))|()|)((|(2
rr
rrrr )()()()()|(
**bjai
jbia
- - - - - - - - -
- - - - -
+ + + + + + + + +
+ + + + cE
kcjbia cakibaji
iakckcjbjbia
,, )εεεε)(εεε(ε
)|)(|)(|(
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Ren, Rinke, Joas, and Scheffler, Invited Review, Mater. Sci. 47, 21 (2012)
Random-phase approximation and beyond
RPAcE
0
μν0μν0RPAc ω)}ω)(χ({Tr))}ω)(χ{1ln(det( dVVE
.c.c)εε(ω
)(ψ)(ψ)(ψ)(ψω),,(χ
occ unocc **
0
i a ia
iaai
i
rrrrrr
rPT2cE
Kohn-Sham polarizability:
Renormalized second-order perturbation theory (rPT2):
single excitations second-order screened exchange RPA
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Conclusions
• Periodic models can be efficiently used to study
concentration/coverage dependence, including infinitely dilute limit
(low-dimensional systems, defects, etc.)
• Practically all beyond-DFT methods involve calculation of the
electron repulsion integrals (ERIs)
• ERIs can be evaluated efficiently by using resolution of identity
(density fitting) and taking into account sparsity
• Wave function methods slowly but steadily enter the arena of
condensed matter physics
• Exploration of the connection between wave-function methods
(coupled-cluster) and many-body perturbation theory (beyond RPA)
continues
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Acknowledgements
Volker Blum
Patrick Rinke
Rainer Johanni
Jürgen Wieferink
Xinguo Ren
Igor Ying Zhang
Matthias Scheffler