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Évora 17th September 2009

Classical and Many-Body Theory of Image Potentials at Solid-Molecule Interfaces

Juan María García Lastra

Kristian Sommer Thygesen

Ángel Rubio

- Introduction
- Motivation
- Our work
- A simple model to explain the results
- Outlook

q

Semiconductor

C60 on Ag(111)

-q

R. Hesper, L.H. Tjeng and G.A. Sawatzky, Europhys. Lett. 40, 177 (1997)

1.Introduction

Image charge

Metal

z0

Is it possible to reproduce this effect within DFT?

Some definitions and equivalences in DFT

Ionization Potential (IP)

Electron affinity (EA)

Gap (D)

DFT

Vacuum

Exact Vxc

LUMO

C is the derivative discontinuity

HOMO

J.P. Perdew and M. Levy Phys. Rev. Lett. 51, 1884 (1983)

+

-2

1.Introduction

DSCF

Alternative : DSCF

LUMO

IP

HOMO

EA

Problem: EXTENDED SYSTEMS

In practice the KS orbital gap is taken as the gap

DFT vs. GW

DFT + local xc-functionals underestimate HOMO-LUMO gaps

Hartree-Fock is good for small molecules (SI-free), but overestimates the gap for extended systems

GW includes screening in the exchange and this solves the gap problem.

Schilfgaarde, Kotani, and Faleev, PRL 96, 226402 (2006)

Hartree-Fock exchange

Screening correction

Theoretical interest

STM

User’s project

D. G. de Oteyza, J.M. García-Lastra et al., Adv. Func. Mater., accepted

Molecules and layers on surfaces

DIP and F16CuPc on Cu(111)

D. G. de Oteyza, J.M. García-Lastra et al., Adv. Func. Mater., in press

Aromatic molecules on Cu(110)

N. Atodiresei, V. Caciuc et al., PRL 102, 136809 (2009)

Conductance at molecular junctions

Amine-Gold Linked Single-Molecule Circuits

SY Quek et al., Nano Lett 7, 3477 (2007)

Transmission peaks: Resonances at frontier orbitals energies

Resonance at Zero-Bias potential: Tail of the peaks

Error in the position of the peaks

Huge error in the conductance at Zero-Bias

Conductance at molecular junctions

SY Quek et al., Nano Lett 7, 3477 (2007)

Shift due to classical image potential+ Self energy correction

Conductance at molecular junctions: Dielectrics

K. Kaasbjerg and K. Flensberget, Nano Lett 8, 3809 (2008)

S

D

SiO2

er(SiO2) = 3.9

D(SiO2) = 8.9 eV

Authors use a classical model to explain a reduction of 0.5 eV in the gap

Is it correct or a microscopic description is needed?

GW-TB Microscopic model of metal-molecule interface

METAL

Thygesen and Rubio, Phys. Rev. Lett. 102, 046802 (2009)

GW-TB Microscopic model of metal-molecule interface

LUMO

HOMO

Free HOMO

3.Our work

Weak physisorption limit

Energy

Static linear response

Small t Large metal DOS at EF Large density response Efficient screening

Now my work starts…

- DFT vs. GW for image potential
- Bulk dielectric constant: Is a good descriptor?
- Check the GW-TB findings: Image charge proportional to DOS at Fermi Level

First-principles GW calculations: Physisorbed benzene

DFT calculations performed with PWSCF code (#)

G0W0calculations performed with the Yambo code(*).

Yambo:

G0W0 LDA, Plane wave basis, norm-conserving pseusopotentials, plasmon pole approximation.

9 Å >Z>4 Å

(#) S. Baroni et al. (2009), QUANTUM ESPRESSO package, www.quantum-espresso.org/

(*) A. Marini, C. Hogan, M. Grüning, D. Varsano, Comp. Phys. Comm. 180, 1392 (2009).

Benzene Molecule

- Previously obtained by Neaton et al.
- LDA underestimates the gap by a factor of 2 (mainly due to Self-interaction)
- GW HOMO-LUMO gap agrees with experiment (IP-EA)
- LUMO predicted to be above the vacuum level in GW, in agreement with experiment

5.2 eV

10.5 eV

PBE: 5.2 eV

PBE0:7.1 eV

Experiment:

IP = 9.25 eV L. Klasinc et al., Pure Appl. Chem. 55, 289 (1983)

EA = -1.15 eV B.T.Hill, J. Chem. Soc. Perkin Trans. II 1027 (1998)

D = 10.4 eV

CaO(001)

BaO(001)

MgO(001)

BaO(111)

3.Our work

Substrates

Insulator and semiconductor

7.7 eV

6.3 eV

4.0 eV

8.9 eV

- Same structure (fcc)
- Varying the gap
- Varying the surface

Metallic surface!

Substrates

Metals

Li(001)

Pt(111)

Rh(111)

Ti(001)

Al(111)

sd

sp

s

sd

sd

- Different DOS at Fermi Level
- Similar interatomic distances
- Except Li: Electrons outer of the surface

Substrates

Semimetallic

- Benzene on Graphite(0001)
- Previously studied by Neaton, Hybertsen and Louie, PRL 97, 216405 (2006)
- Neaton et al. z = 3.25 Å
- Our work 4 Å < z < 9 Å

GW and LDA benzene HOMO-LUMO gaps

4.5 Å

J.M.G-L, A. R. and K.S.T., submitted

- LDA gaps are independent of substrate and distance
- Same result with other functionals (GGA, hybrid or exact exchange)
- GW gaps show large variation across different surfaces
- GW gap sensitive to atomistic details, e.g. surface plane (BaO)

Classical image charge model

Electrostatic energy of point charge above a polarizable medium:

Classical model describes the physics of the gap reduction qualitatively.

Fitted for the gap: Different values if HOMO or LUMO are fitted independently

Dynamic interaction between benzene orbitals and surfaces: Bulk Dielectric Constant is not a good descriptor

Best-fit values for and z0:

Variation of HOMO and LUMO levels

Vacuum

Vacuum

GW: Symmetric effect on HOMO and LUMO. Exceptions Li and BaO(111)

LDA: HOMO level agrees better with GW than does LUMO

Very good agreement between LDA and GW for HOMO at metallic surfaces (error cancellation in LDA between self-interaction and image charge)

General trends in level shifts

Insulator and semiconductor

Gap reduction increases with decreasing substrate band gap

General trends in level shifts

Metals

Gap reduction increases with increasing substrate DOS at EF

Li and BaO(111) deviate from general trend!

4. A simple model to explain the results

GW S to second order in V

Renormalization of single electronic level, , by non-local interactions with substrate electrons:

Hartree-Fock exchange

Screening correction

We truncate the expansion in the second order term

L

L

L

Substrate joint density of states weighted by particle-hole transitions

4. A simple model to explain the results

Semiconductors

Effective interaction strength

4. A simple model to explain the results

Metals

Assumption: Vkk’ similar for all the systems

L proportional to JDOS

Slope of JDOS at w=0 proportional to DOS at EF

The correction increases if DOS at EF increases

4. A simple model to explain the results

BaO(111) and Li(001)

Li(001)

Rh(111)

Much bigger in Li and BaO(111) than in the other systems

- DFT (local xc-functionals) is not able to reproduce image charge effect
- GW includes dynamic correlation (polarization) and solves the problem
- Classic image potential describes the effect phenomenologically
- However microscopic description is required
- Renormalization of the gap in molecules follows the band gap in semiconductors
- Renormalization of the gap in molecules follows the DOS at Fermi level in metals
- It is possible to understand the results truncating at second order the self energy.

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