ab-initio study of new materials: from 2d to 3d dirac systems

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

15

THE GAP PROBLEM

16 DFT gives gaps in qualitative but not quantitative agreement with experiments

OUTLINE





17

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

19

Graphene Silicene!!

No buckling Buckling D= 0.45 A

Buckling D= 0.69 A

germanene

20

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

32

2

24

Van Hove singularities:

_____C

---------Si

……….Ge

C Si Ge

M0 M1 M2

25

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





27

28

NEW MATERIALS: Weyl/Dirac 3D fermions

3D Semimetals analogue to Graphene

• TaAs, TaP, NbAs, NbP……3D Weyl semimetal

•Cd3As2 3D Dirac semimetals

Cd3As2

NEW MATERIALS:

„Small cube“ building block

Anti-fluorite structure

2 Cd vacances

6.46 A

Low T polymorph needle-like

30

“Small cube”

DFT BANDS with SOI: no inversion symmetry, no Dirac cone

31

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

32

Small cube: optical properties

Re diverges for small frequencies

(typical of a metal).

Optical properties are almost isotropic.

M

33

Building the Body Centered Tetragonal

34

•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

36

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

40

ab-initio study of new materials: from 2d to 3d dirac systems

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