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Ab initio Study of the Excitonic Effects on the Optical Spectra of Single-layer, Double-layer, and Bulk MoS2

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(1)

Ab initio Study of the Excitonic

Effects on the Optical Spectra of

Single-layer, Double-layer, and

Bulk MoS

2

(2)

Single-layer semiconductors

From graphene to a new

family

of

materials

Graphene

Hexagonal

boron nitride

Molybdenum

disulfide

No bandgap

E

F

~6 eV

~1,9 eV

Appealing for transistors, tunable photo-thermoelectric effect, field-effect

transistors …

The bandgap and the high mobility makes MoS2 a suitable alternative to

graphene or carbon nanotubes.

See review Nature Nanotechnology 7, 699 (2012) A. Castellanos, Nano Lett.,

(3)

Transition metal dichalcogenides

Single-layer semiconductors

Crystal symmetry

determines the

electronic

structure and

optical properties

Single-layer (D

3h

)

No inversion symmetry

Double-layer and bulk (D

6h

)

Inversion symmetry

SO inter. splits

the valence bands Interlayer inter. and SO splits

(4)

Transition metal dichalcogenides

Single-layer semiconductors

Crystal symmetry

determines the

electronic

structure and

optical properties

Single-layer (D

3h

)

No inversion symmetry

Double-layer and bulk (D

6h

)

Inversion symmetry

SO inter. splits

the valence bands Interlayer inter. and SO splits

the VB A

B

VB splitting is exhibited in the PL and absorption by a two peaks structure Phys. Rev. Lett. 105, 136805 (2010)

Control of the light polarization helicity by tunning the excitation energy (valley physics).

CB

VB

Nature Nanotechnology 7, 494 (2012)

MoS2, SL, DL and BULK - Electronic Structure: LDA and GW methods. - Excitonic effects:

(5)

MoS

2

. Band Structure. LDA and GW method

LDA GW

Correction of LDA bandgap

underestimation by means of the GW method (spin-orbit

interaction is included).

(6)

MoS

2

. Band Structure. LDA and GW method

LDA GW

Correction of LDA bandgap

underestimation by means of the GW method (spin-orbit

interaction is included).

The addition of more layers

changes the dielectric screening, the symmetry and the interlayer interaction.

(7)

MoS

2

. Band Structure. LDA and GW method

LDA GW

Correction of LDA bandgap

underestimation by means of the GW method (spin-orbit

interaction is included).

The addition of more layers

changes the dielectric screening, the symmetry and the interlayer interaction.

Phys. Rev. B 84 , 155413 (2011). LDA calculations made with ABINIT

GW correction:

- Rigid shift of the conduction band

- Smaller correction for increasing number of layers Bandgap extremely sensitive to lattice optimization

For multi-layers: Pushing of the valence band at Γ (indirect)

(8)

MoS

2

. Excitonic Effects. Bethe-Salpeter Equation

e h W Coulomb e-h interaction forms the exciton XA XB

K ≡ (c, v, k)

Energy difference Bethe-Salpeter Kernel

Unscreended short ranged exchange interaction

Ξ

K

1

K

2

=

 iV

K

1

K

2

+ iW

K

1

K

2

H

exc (n1,n2),(n3,n4)

= (E

n2

 E

n1

)

δ

(n1,n3)

δ

(n2,n4)

+ i(f

n 2

 f

n1

)Ξ

(n 1,n2),(n3,n4) Screened coulomb interaction

(9)

MoS

2

. Excitonic Effects. Bethe-Salpeter Equation

e h W Coulomb e-h interaction forms the exciton XA XB

K ≡ (c, v, k)

Energy difference Bethe-Salpeter Kernel

Unscreended short ranged exchange interaction

Ξ

K

1

K

2

=

 iV

K

1

K

2

+ iW

K

1

K

2

H

exc (n1,n2),(n3,n4)

= (E

n2

 E

n1

)

δ

(n1,n3)

δ

(n2,n4)

+ i(f

n 2

 f

n1

)Ξ

(n 1,n2),(n3,n4) Screened coulomb interaction

Unscreended short ranged exchange interaction

(10)

MoS

2

. Excitonic Effects. Bethe-Salpeter Equation

Key issue in reliable results: convergence in number of conduction and valence band states and k-points!

Lack of convergence in k-points overestimate the exciton binding energy.

The k-sampling is directly related to the numbers of unit cells employed to map the exciton wave

functions.

Low k-sampling gives also artifacts in the optical absorption. e h W Coulomb e-h interaction forms the exciton XA XB

K ≡ (c, v, k)

Energy difference Bethe-Salpeter Kernel

Unscreended short ranged exchange interaction

Ξ

K

1

K

2

=

 iV

K

1

K

2

+ iW

K

1

K

2

H

exc (n1,n2),(n3,n4)

= (E

n2

 E

n1

)

δ

(n1,n3)

δ

(n2,n4)

+ i(f

n 2

 f

n1

)Ξ

(n 1,n2),(n3,n4) Screened coulomb interaction

(11)
(12)

MoS

2

. Excitonic Effects. Bethe-Salpeter Equation

e h W XA XB

The exciton binding energy decreases

with the number of layers. Consequence of larger dielectric screening. This

compensates partially the GW correction.

(13)

MoS

2

. Excitonic Effects. Bethe-Salpeter Equation

e h W XA XB

The exciton binding energy decreases

with the number of layers. Consequence of larger dielectric screening. This

compensates partially the GW correction. For single-layers (around 3 eV) the

absorption gains in efficiency (strongly bound exciton).

The theoretical spectra captures nicely the peaks separation for all the cases.

(14)

MoS

2

. Excitonic Effects. Bethe-Salpeter Equation

(c)

(15)

MoS

2

. Excitonic Effects. Bethe-Salpeter Equation

(c)

The intensity of the optical absorption is related with the localization of the excitons.

For bulk, the exciton is confined in only one layer, due to the large interlayer distance.

(16)

Conclusions and ongoing work

●

The bandgap (direct or indirect) depends critically

on the number of layers and lattice optimization.

●

Excitonic effects are stronger in enviroments with

small dielectric constant (single-layers).

●

Further studies will deal with strained layers and

(17)

Acknowledgements

Kerstin Hummer, University of Vienna.

(18)

Acknowledgements

Kerstin Hummer, University of Vienna.

Davide Sangalli and Andrea Marini, Istituto di Struttura della Materia (ISM), Consiglio

Nazionale delle Ricerche (CNR), Rome Italy.

Références

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