Application of DFTB in molecular electronics

1
Application of DFTB in molecular electronics
Jeffrey R Reimers, Gemma C. Solomon, Zheng-Li
Cai, Noel S. Hush, School of Chemistry, The
University of Sydney, Australia Alessio
Gagliardi, Thomas Frauenheim, Department of
Theoretical Physics, Paderborn University,
Germany, Theoretical Physics Department,
University of Bremen, Germany Alessandro
Pecchia, and Aldo Di Carlo Department of
Electronic Engineering, University of Rome "Tor
Vergata", Italy
2
Summary

1. What is Molecular Electronics
2. The gDFTB method for molecular electronics
   applications
3. Why use DFTB ?
4. Problems with standard DFT
5. Does DTFB offer any intrinsic advantages ?
6. Is DFTB accurate enough ?
7. Use of gDFTB in interpreting experiment
8. Implementing Symmetry in DFTB
9. Nature of molecular conduction channels

3
Molecular ElectronicsMeasuring single molecule
conduction
Nanocluster
Mechanical Break Junction
Dadosh et al. Nature 436 (2005) 677
Reichert et al. PRL 88 176804
4
Single-Molecule Conductivity
L ELECTRODE
R ELECTRODE
MOLECULE
5
Single-Molecule Conductivity
L ELECTRODE
R ELECTRODE
MOLECULE
Molecular Orbitals
Fermi energy
6
Single-Molecule Conductivity
L ELECTRODE
R ELECTRODE
MOLECULE
Molecular Orbitals
eV
V
I
7
Finding a true molecular signatureInelastic
Electron Tunnelling Spectroscopy (IETS)
I
h?/e
V
Elastic
h?/e
V
dI/dV
Inelastic
h?/e
V
d2I/dV2
h?/e
V
8
Application to molecules
W. Wang, T. Lee, I. Kretzschmar M. Reed Nano
Lett. (2004) 4(4) 643
J. GKushmerick, J. Lazorcik, C. H. Patterson R.
Shashidhar Nano Lett. (2004) 4(4) 639
9
Shot noise measurements
Djukic Van Ruitenbeek Nano Lett. (2006) 6(4),
789
Smit et al. Nature (2002) 419, 906
10
gDFTB Method for Calculating the Current

• Non-Equilibrium Greens Function (NEGF)
  formalism
• Implementation developed at Tor Vergata
• Reduces to Landauer Formalism in some instances
  (eg., coherent current but not for IETS)
• DFTB implementation developed at Paderborn /
  Dresden
• called gDFTB
• calculates the system Hamiltionain H for
  electrode-molecule-electrode system
• requires an optimized geometry
• requires vibrational analysis for IETS
• See Poster COMP 300 by Gagliardi et al.

11
Partitioning the Electrode-Molecule-Electrode
Hamiltonian Operator for the System Energy
L
M
R
Diagonal blocks are the energies of each part
Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994)
6849.
12
Partitioning the Electrode-Molecule-Electrode
Hamiltonian Operator for the System Energy
L
M
R
Off- Diagonal blocks are the interaction energies
Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994)
6849.
13
Landauer Formalism
Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994)
6849.
14
Why DFTB? General Serious Failures of DFT

• Dispersion
• Covalent bond breakage
• Partial electron removal/addition (long range
  electron-transfer processes)
• Extended ? conjugation
• ALL RELEVANT
• TO PHOTONICS AND
• MOLECULAR ELECTRONICS !
• Can DFTB do better ???

Reimers, Cai, Bilic, Hush, Ann. N.Y. Acad. Sci.
1006 (2003) 235.
15
DFT Failure (1) Dispersion error leads to poor
adsorption energies
Molecule Surface Observed PW91 Calculated
NH3 Au(111) 7.5-10 8
Benzene Au(111) 9 2
Cu(111) 14 1
Cu(110) 23 6
kcal/mol
kcal/mol

• DFT calculations for benzene on a Cu13 model
  cluster for (110) 19 kcal/mol
• CASPT2 dispersion energy error for DFT 15
  kcal/mol

Bilic, Reimers, Hush Hafner J. Chem. Phys. 116
(2002) 8981 Bilic, Reimers, Hoft, Ford Hush J.
Theor. Comput. Chem. 2 1093 (2006).
16
DFT Failure (2) Covalent Bond Breakage
H2 Source of long-range correlation
Single bonds break properly if ? and ? electrons
have different orbitals
Cai Reimers J. Chem. Phys. 112 (2000) 527
17
Time-Dependent DFT (TDDFT) collapses for excited
states
The triplet instability has a profound effect for
TDDFT and its analogue RPA (use H0 H1 H2
) CIS is OK (uses H0 H1 )
Cai Reimers J. Chem. Phys. 112 (2000) 527
18
Application to the weak electrode-electrode
through molecule bonds that drive single-molecule
conductivity experiments
Electrode cluster molecule Electrode
cluster MODEL SYSTEM

• Typical pair of weakly coupled orbitals
• Actually there are 2 such pairs !

Solomon, Reimers and Hush J. Chem. Phys. 112
(2000) 527
19
Fermi Level of system is OPEN SHELL
Solomon, Reimers and Hush J. Chem. Phys. 112
(2000) 527
20
Closed-Shell treatments lead to split orbitals
Closed-Shell GGA density functionals have
incorrect asymptotes but maintain double
degeneracy results in additional weak
conduction channels is useful
Closed-shell hybrid density fucntionals gives
asymptotically very poor result perceived as
strong coupling, resultant currents x 100 too
high useless
Open-shell calculation gives asymptotically
correct answer
Solomon, Reimers and Hush J. Chem. Phys. 112
(2000) 527
21
DFT Failure (3) Partial Electron Removal
All modern functionals have an incorrect
asymptotic potential
Vx as a function of nuclear - electron distance r
for the H atom Taken from Tozer et al. J. Chem.
Phys. 112 (2000) P3507
22
DFT band lineup error for phenylthiol (RSH) on
gold(111)
RSH RS RS
Adsorbate Gold (111) Obs PW91
PW91 PW91 Bridge FCC PW91
Obs.
Band-gap error 5.6 eV
Band lineup error 3.4 eV
Bilic, Reimers and Hush J. Chem. Phys. 122 (2005)
094708
23
DFT Failure (4) Conjugated ? Systems
Examples . overestimation of metallic-like
properties Collapse of band-gap in
oligoporphyrin molecular wires Appearance of
charge-transfer bands in porphyrins and
chlorophylls Loss of band gap in polyacetylene,
very high NLO properties
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Oligoporphyrins
Sendt, Johnston, Hough, Crossley, Hush Reimers
J. Am. Chem. Soc. 124 (2002) 9299
Cai, Sendt Reimers J. Chem. Phys. 117 (2002)
5543
25
Can DFTB be better ?

1. Dispersion yes, via empirical corrections
2. Covalent bond breakage yes, no singlet/triplet
   thus no triplet instability !
3. Partial electron removal/addition (long range
   electron-transfer processes) ???
4. Extended ? conjugation ???

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SCC-DFTB errors for properties of 63 Mg complexes
Property B3LYP SCC-DFTB AM1 PM3 MNDO-d PM5
Bond length / Å Ave .00 .02 -.03 -.10 -.05 -.06
? .02 .06 .11 .18 .10 .09
Bond Ang. / Ave -1 -2 0 -9 1 -1
? 4 14 4 25 15 3
IP / eV Ave .20 -.13 .27 .33 .23 .22
? .34 .66 .51 .65 .44 .50
?Hf / kcal mol-1 Ave 0 149 0 -4 -6 1
? 7 246 19 19 18 23
Incr. Ligand Ave 0 1 14 8 16 6
Binding / kcal mol-1 ? 6 3 23 16 21 17
Deprotonation Ave -1 7 -7 2 -1 -17
Energy / kcal mol-1 ? 10 11 9 27 9 22
Comp. to either experiment or else CBS or else
QCISD
Cai, Lopez, Reimers, Cui, Elstner in prep.
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SCC-DFTB geometries of thiols on Au(111)
Alkane chain
S head group

• p(5 ? 5) Au surface cell
• optimized geometry has S on a top site
• DFT calculations predict either FCC or
  bridge-distorted FCC site
• experiments indicate top site but may involve Au
  adatom instead

Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) 124, 094704
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Observed and gDFTB-calculated IETS
Reeds experiment
Calculations match and enhance experimental
assignment
W. Wang, T. Lee, I. Kretzschmar M. Reed Nano
Lett. (2004) 4(4) 643-646
Binding site
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) 124, 094704
29
Effect of the binding site on CH intensity
Wang, Lee, Kretzschmar Reed Nano Lett. (2004)
4 643
Opt structure Calculated IETS
lower energy
higher energy
J. Kushmerick, Lazorcik, Patterson Shashidhar
Nano Lett. (2004) 4 639
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) 124, 094704
30
Importance of molecular symmetry

• Vibrations are characterized by their symmetry.
• What are the selection rules for IETS?
• What is the nature of the conduction channels
  through the molecule?
• How many are there?
• What is the role of the junction region?
• What is the role of the molecule and its
  molecular orbitals

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Implementing symmetry in SCC-DFTB

• Find all atoms related by the Albelian symmetry
  operators C2 (two-fold rotation), ? (reflection
  plane), and i (inversion)
• Construct the transformation S that forces all
  atomic orbitals (AO), Cartesian tensor
  components, etc., to be eigenfunctions of these
  operators
• Transform the Kohn-Sham matrix H, force vector,
  Hessian matrix of second derivatives, etc. from
  AO basis and Cartesian coordinates into symmetry
  adapted representations
• H? ST H S
• Diagonalize H? to get symmetry-adapted molecular
  orbitals C?
• Back transformation to get molecular orbitals in
  AO basis
• C S C?

Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) submitted
32
Numerical Advantages

• Numerical error is removed (numbers that should
  be zero ARE zero)
• Force optimization of transition states and
  saddle points
• Block diagonalization gives speedup (eg, ? 4 for
  C2v)
• eg. Say that H has transforms according to the
  C2v point group
• symmetry operators C2z, ?xz, ?yz, and E
• irreducible representations a1, a2, b1, and b2

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What is the point group in gDFTB calculations?

• Symmetry of entire system H is C2h (operators
  are C2z, ?xy, and i)
• Symmetry of molecular component HM is C2h
• Symmetry of individual molecule- electrode
  couplings JL and JR is Cs only
• gDFTB equations use JL and JR explicitly hence
  there a new quantity is needed, the
• MOLECULAR CONDUCANCE POINT GROUP

Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) submitted
34
Determining the Molecular Conductance Point Group
Eg., for chemisorbed 1,4-benzenedithiol S-
C6H4-S All symmetry operators that enforce
end-to-end symmetry are lost All other symmetry
operators are retained In this case, D2h ? C2v
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) submitted
35
Conduction split into symmetry channels
Total transmission A2 component Ef Fermi
energy of Au, controls low-voltage conductivity
its B1 !
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) submitted
36
The transmission through each symmetry block can
then be partitioned in other ways
1. Büttiker eigenchannels (shot noise)
2. Junction eignchannels coupled by the molecule
3. Interference between Molecular Conductance
Orbitals coupled through the junction
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush Nano Letts (2006) in press
37
Harnessing the power of DFTB
Au atoms per electrode Black- 3 Red- 25
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) submitted
38
Conclusions

• gDFTB formalism provides powerful application
  areas to molecules coupled to solid-state devices
• implementation of symmetry into SCC-DFTB code
• provides faster and more stable central
  algorithm
• provides key information for understanding
  molecular systems
• must be careful to use DFTB only for suitable
  properties
• initial applications in molecular electronics
  encouraging
• conduction channels
• IETS vibrational spectroscopy
• basic behaviour of method not yet fully
  characterized
• ready for testing on large systems

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The end
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ACS Abstract
Application of DFTB in molecular
electronics Jeffrey R Reimers, reimers_at_chem.usyd.e
du.au1, Gemma C. Solomon, solomon_at_chem.usyd.edu.au
1, Zheng-Li Cai, zlcai_at_chem.usyd.edu.au1, Noel S.
Hush, hush_n_at_chem.usyd.edu.au2, Alessio
Gagliardi, gagliard_at_phys.upb.de3, Thomas
Frauenheim, frauenheim_at_phys.upb.de4, Alessandro
Pecchia5, and Aldo Di Carlo, dicarlo_at_ing.uniroma2.
it5. (1) School of Chemistry, The University of
Sydney, Sydney, 2006, Australia, (2) School of
Molecular and Microbial Biosciences, The
University of Sydney, Sydney, 2006, Australia,
(3) Theoretical Physics Department, University of
Bremen, Germany, Vogeliusweg 25.2.1.14,
Paderborn, 330918, Germany, (4) Bremen Center for
Computational Materials Science, Bremen
University, Bibliothekstrasse 1, Bremen, 28359,
Germany, (5) Department of Electronic
Engeneering, University of Rome "Tor Vergata",
Rome, Italy Molecular electronics involves the
passing of current between two electrodes through
a single conducting molecule. Calculations in
this area require not only the ability to handle
large systems including metal-electrode fragments
but also require accurate positioning of
molecular and metallic energy bands and must
treat occupied and virtual orbitals on an
equivalent footing. Each of these requirements
presents difficulties for standard DFT
calculations, making DFTB an attractive
alternative proposition. We present enhancements
to the SCC-DFTB program that allow it to diagnose
and utilize molecular symmetry, increasing
computational speed and accuracy whilst providing
important information concerning molecular
orbitals and molecular vibrations. Optimized
geometries are then obtained for molecules
sandwiched between gold electrodes, leading to
Green's-function based calculations of
steady-state through-molecule electrical
conductivity and incoherent inelastic tunnelling
spectroscopy (IETS) arising from electrical
current activation of molecular vibrational modes.
41
When the junction symmetry is less than that of
the Molecular Conductance Point Group
Black- 3 Au, exact Green- 3 Au, using higher
symmetry Red- 25 Au, exact
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo,
Reimers, Hush J.C.P. (2006) submitted