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Ab-initio study of new materials: from 2D to
3D-Dirac systems
Olivia Pulci
Department of Physics,
University of Rome Tor Vergata,
European Theoretical Spectroscopy Facility(ETSF),
MIFP, CNR-ISM
MIFP 1
Condensed Matter Theory (Sub)Group
Dipartimento di Fisica Università di Roma Tor Vergata
http://www.fisica.uniroma2.it/~cmtheo-group
Ihor Kupchak Adriano Mosca Conte OP
2
Marco Polimeni
Stella Prete
Davide Grassano
Gianluca Tirimbo’ Valerio Armuzza
•0-D
•1-D •2-D
•3-D
•Nanoclusters
• Surfaces
•Biological
systems
Diamond
water
Ice
graphene
Ab-initio calculations of electronic and
optical properties of complex systems
•Generality, transferability 0D-3D
• Detailed physical informations
• Predictivity
• Complex theory+large computational cost
Ancient Paper
3
•0-D
•1-D •2-D
•3-D
Hamiltonian of N-electron system: •Biological
systems
ji ji
ji
ji ij
iM
I ji jiI
IN
i
ieZZeZe
M
P
m
pH
||2
1
||||2
1
22
2
,
2
1
22
1
2
RRRrrr
4
•0-D
•1-D •2-D
•3-D
Hamiltonian of N-electron system: •Biological
systems
ji ji
ji
ji ij
iM
I ji jiI
IN
i
ieZZeZe
M
P
m
pH
||2
1
||||2
1
22
2
,
2
1
22
1
2
RRRrrr
...not possible to solve it
5
EH
GROUND-STATE
• 1964: Density Functional Theory
E=E[n]
1998 Nobel Prize to Kohn n
•Impossible to solve
for reasonable N
EXCITED STATES
• Many Body Perturbation Theory
Green’s function method
GW (L. Hedin 1965) + Bethe Salpeter
Equation (1965-->today)
• Time Dependent DFT (TDDFT)
(Gross 1984)
G n(t)
6
For a review, see for example G. Onida, L. Reining, A. Rubio, Rev. Mod.Phys (2002)
3 steps:DFT, GW and Bethe Salpeter Eq.
DFT
DIFFICULTY
1) geometry 2) Electronic band structure 3) Optical spectra with excitons
BSE
wcv
hn W
EXC
Ground state N electrons
GW
Photoemission N+1, N-1 electrons
Optical excitation N electrons
hn
TDDFT
7
OUTLINE
-DENSITY FUNCTIONAL THEORY
Examples: Silicene, Cd3As2
-Electronic and optical gaps: GW, BSE
Examples: Silicene, Silicane and other 2D
8
Density Functional Theory (1)
Hohenberg-Kohn Theorem 1964 Kohn-Sham equations 1965 Nobel Prize to Kohn in 1998
),..,(),..,(||2
1)(
22121
,
22
2
NN
ji ji
iext
i
i
i
rrrErrrrr
erV
m
Interacting N-electron system:
][|| nEH “the total energy of the ground state of a system of interacting electrons is a unique
functional of its electron charge density.”
9
][][][][|ˆ| 0 nEEnVextnTnEHE XCHartree
Density Functional Theory (1)
Hohenberg-Kohn Theorem 1964 Kohn-Sham equations 1965 Nobel Prize to Kohn in 1998
),..,(),..,(||2
1)(
22121
,
22
2
NN
ji ji
iext
i
i
i
rrrErrrrr
erV
m
Interacting N-electron system:
][|| nEH “the total energy of the ground state of a system of interacting electrons is a unique
functional of its electron charge density.”
10
][][][][|ˆ| 0 nEEnVextnTnEHE XCHartree
Density Functional Theory(2): Kohn-Sham eqs. (1965)
rVrnrr
erdrVrV xcextKS
'
||
23
rrrVm
iiiKS
2
2
2
where
The interacting N-electron system can be mapped into an
effective single particle equation: Kohn Sham equation:
Exchange
and correlation:
unknown!
N
i
iNII rrnrn1
2||)(
Hartree potential
External potential
(ions)
Interacting electrons+real potential
Non-interacting fictious particles +effective potential
11
Density Functional Theory(2): Kohn-Sham eqs. (1965)
rVrnrr
erdrVrV xcextKS
'
||
23
rrrVm
iiiKS
2
2
2
where
The interacting N-electron system can be mapped into an
effective single particle equation: Kohn Sham equation:
Exchange
and correlation:
unknown!
N
i
iNII rrnrn1
2||)(
Hartree potential
External potential
(ions)
Interacting electrons+real potential
Non-interacting fictious particles +effective potential
12
Approximations for Vxc -Local density approx: LDA The Vxc is taken as the one of the homogeneus electron gas
-GGA: takes better into account the non homogeneity of the system -Hybrid XC… -……………. In general when one talks about DFT, means DFT-LDA or DFT-GGA, since the TRUE Vxc is unknown
NOTE: DFT IS AN EXACT THEORY, DFT-LDA/GGA NOT!
13
3/12
))(3()( rne
rVx
The Ground State
DFT (DENSITY FUNCTIONAL THEORY) well describes:
• Atomic Structure
• Lattice Parameters
• Elastic Constants
• Phonon Frequencies
• ......................................
that is, all Ground State Properties. BUT.....
DFT is also the most used technique to study excited states
14
Common (mis)use of DFT (beyond its realm)
rrrVrnrr
erdrV
mnknknkxcione
'||2
2
32
2
Interpreted as electron wavefunctions
kvc
vkckvck kD,
2
22 )(||1
)(Im ww
w dipole
Fermi golden rule:
Interpreted as electron states
LiF
DFT
Exp
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THE GAP PROBLEM
16 DFT gives gaps in qualitative but not quantitative agreement with experiments
OUTLINE
-DENSITY FUNCTIONAL THEORY
Examples: Silicene, Cd3As2
-Electronic and optical gaps: GW, BSE
Examples: Silicene, Silicane and other 2D
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Everything started with graphene
3D: stacked in graphite 2D: graphene 1D: rolled in nanotubes 0D: wrapped in fullerens
Unique physical properties: High carrier mobility
Ambipolar field effect RT quantum Hall Single molecule detection Special mechanical properties …………………
Novoselov et al. Science 2004
For a review see for example: Castro et al. Rev. Mod. Phys. 81, 109 (2009) Allen et al. Chem. Rev. 110, 132 (2010) 18
SILICENE: the 2D ubiquitous Silicon
Li Tao et al. Nature Nanotech. 2015
P. Vogt et al.
PRL2012
Ag(111)4x4:Si 3x3
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Graphene Silicene!!
No buckling Buckling D= 0.45 A
Buckling D= 0.69 A
germanene
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Graphene! silicene
No buckling
germanene
Buckling=0.63 Angstrom Buckling=0.44 Angstrom
____C ____Si ____Ge
massless fermions
21
a=e2/hc~1/137 Fine structure constant (universal)
=0.02295.. A=
Graphene Absorbance A~1-T~2.3%
What about Silicene and Germanene???
Analytic derivation
2
Fermi Golden rule
+ =conduction band
- =valence band
around K, K’
Simple integral
Universal behavior
a=e2/hc~1/137 23
)()(
|ˆ|
kk
kvpekc
m
eM
vc
cv
Absorbance A(w)=w *L*2(w)/c
Our (numerical) method
• Density Functional Theory Hohenberg Kohn 1964, Kohn and Sham 1965
• Fermi Golden Rule
Ingredients: ab-initio single particle
eigenvalues and eigenstates
rrrVrnrr
erdrV
mnknknkxcione
'||2
2
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2
24
Van Hove singularities:
_____C
---------Si
……….Ge
C Si Ge
M0 M1 M2
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Graphene: 0.02293
Silicene : 0.02290
Germene: 0.02292
Independent on:
•Group IV atoms
•Buckling
•Fermi velocity
Dirac fermions (a):
0.022925
APL 2012; PRB 2013; New J. of Phys. 2014
a
Lars Matthes
AMAZING PROPERTY THAT CAN BE USED TO EXP.
PROVE THE EXSISTENCE OF DIRAC CONE 26
OUTLINE
-DENSITY FUNCTIONAL THEORY
Examples: Silicene, Cd3As2
-Electronic and optical gaps: GW, BSE
Examples: Silicene, Silicane and other 2D
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NEW MATERIALS: Weyl/Dirac 3D fermions
3D Semimetals analogue to Graphene
• TaAs, TaP, NbAs, NbP……3D Weyl semimetal
•Cd3As2 3D Dirac semimetals
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Cd3As2
NEW MATERIALS:
„Small cube“ building block
Anti-fluorite structure
2 Cd vacances
6.46 A
Low T polymorph needle-like
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“Small cube”
DFT BANDS with SOI: no inversion symmetry, no Dirac cone
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Small cube
Small cube is a metal: As(P) character at the Fermi Energy but a Cd(S) crosses the
Fermi level.
No qualitative differences with (continuous black) and whithout (dashed red) SOI.
With spin-orbit
Without spin-orbit
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Small cube: optical properties
Re diverges for small frequencies
(typical of a metal).
Optical properties are almost isotropic.
M
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Building the Body Centered Tetragonal
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•Each corkscrew is surrounded by corkscrews of opposite handedness
•2x2x4 small cubes cell--160 atoms
Electronic bands
BCT structure belongs to I42/acd symmetry group.
Dirac cone!!!
Inversion symmetry, therefore NO spin splitting near the Dirac cone, like graphene 35
BCT optical properties
.
M
Z, N
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37
In 3D 2constant/vF; in 2D (graphene) 2constant/w
Optical conductivity (theory)
Im( ) tends to a plateau.
Consequently the optical conductivity
is almost linear near the 3D Dirac
cone (in 2D is constant)
Im
w 4/)Im()Re(
Fj
jjv
cwa
w
12
1))(Re(
38
EXP. optical conductivity
39
w 4/)Im()Re(
Neubauer et al,
Conclusions (1)
• Infrared absorbance in 2D Dirac systems:
• In 2D honeycomb sheets, as long as a Dirac cone exists, infrared absorption is a UNIVERSAL constant ( a
• Independent of the value of vF , the degree of sp2 and sp3 hybridization, and the sheet buckling.
• New 3 D materials: topological semimetal with Dirac cone Cd3As2 • Dirac nodes near G along the tetragonal axis
• Anisotropic Dirac cone (in contrast with 2D graphene)
• linear Optical conductivity (in 2D graphene tends to a constant)
• Good agreement with experiments
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