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

Summary What is Molecular Electronics The “gDFTB” method for molecular electronics applications Why use DFTB ? Problems with standard DFT Does DTFB offer any intrinsic advantages ? Is DFTB accurate enough ? Use of gDFTB in interpreting experiment Implementing Symmetry in DFTB Nature of molecular conduction channels

Molecular Electronics: Measuring single molecule conduction Wang et al. PRB 68 (2003) 035416 Nanopore Kushmerick et al. PRL 89 (2002) 086802 Cross-wire STM Break Junction B. Xu & N. J. Tao Science (2003) 301, 1221 Scanning Probe Cui et al. Science 294 (2001) 571 Electromigration H. S. J. van der Zant et al. Faraday Discuss. (2006) 131, 347 Nanocluster Mechanical Break Junction Dadosh et al. Nature 436 (2005) 677 Reichert et al. PRL 88 176804

Single-Molecule Conductivity L ELECTRODE R ELECTRODE MOLECULE

Single-Molecule Conductivity L ELECTRODE R ELECTRODE MOLECULE Molecular Orbitals Fermi energy

Single-Molecule Conductivity L ELECTRODE R ELECTRODE MOLECULE Molecular Orbitals eV V I

Finding a true molecular signature: Inelastic Electron Tunnelling Spectroscopy (IETS) h/e V Elastic V h/e dI/dV Inelastic h/e V d2I/dV2 h/e V

Application to molecules J. GKushmerick, J. Lazorcik, C. H. Patterson & R. Shashidhar Nano Lett. (2004) 4(4) 639 W. Wang, T. Lee, I. Kretzschmar & M. Reed Nano Lett. (2004) 4(4) 643

Shot noise measurements Garcia et al. Phys. Rev. B (2004) 69, 041402 Thygesen & Jacobsen Phys. Rev. Lett. (2005) 94, 036807 Djukic & Van Ruitenbeek Nano Lett. (2006) 6(4), 789 Smit et al. Nature (2002) 419, 906

“gDFTB” Method for Calculating the Current Non-Equilibrium Green’s 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.

Diagonal blocks are the energies of each part 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.

Off- Diagonal blocks are the interaction energies 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.

Landauer Formalism Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994) 6849.

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, Bilić, Hush, Ann. N.Y. Acad. Sci. 1006 (2003) 235.

DFT Failure (1): Dispersion error leads to poor adsorption energies Molecule Surface Observed PW91 Calculated NH3 Au(111) 7.5-10 8 Benzene 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 Bilić, Reimers, Hush & Hafner J. Chem. Phys. 116 (2002) 8981 Bilić, Reimers, Hoft, Ford & Hush J. Theor. Comput. Chem. 2 1093 (2006).

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

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

Electrode cluster – molecule – Electrode cluster MODEL SYSTEM 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

Fermi Level of system is OPEN SHELL Solomon, Reimers and Hush J. Chem. Phys. 112 (2000) 527

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

DFT Failure (3): Partial Electron Removal All modern functionals have an incorrect asymptotic potential Should be Is Vx as a function of nuclear - electron distance r for the H atom Taken from Tozer et al. J. Chem. Phys. 112 (2000) P3507

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 Bilić, Reimers and Hush J. Chem. Phys. 122 (2005) 094708

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

Oligoporphyrins Sendt, Johnston, Hough, Crossley, Hush & Reimers J. Am. Chem. Soc. 124 (2002) 9299 Cai, Sendt & Reimers J. Chem. Phys. 117 (2002) 5543

Can DFTB be better ? Dispersion – yes, via empirical corrections Covalent bond breakage – yes, no singlet/triplet thus no triplet instability ! Partial electron removal/addition (long range electron-transfer processes) ??? Extended  conjugation ???

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  .06 .11 .18 .10 .09 Bond Ang. / ° -1 -2 -9 1 4 14 25 15 3 IP / eV .20 -.13 .27 .33 .23 .22 .34 .66 .51 .65 .44 .50 Hf / kcal mol-1 149 -4 -6 7 246 19 18 23 Incr. Ligand 8 16 6 Binding / kcal mol-1 21 17 Deprotonation -7 2 -17 Energy / kcal mol-1 10 11 9 27 22 Comp. to either experiment or else CBS or else QCISD Cai, Lopez, Reimers, Cui, Elstner in prep.

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

Observed and gDFTB-calculated IETS Reed’s 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

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

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

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

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 a1 a2 b1 b2 a1 a2 b1 b2 H =

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

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

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

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

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

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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ACS Abstract Application of DFTB in molecular electronics Jeffrey R Reimers, reimers@chem.usyd.edu.au1, Gemma C. Solomon, solomon@chem.usyd.edu.au1, Zheng-Li Cai, zlcai@chem.usyd.edu.au1, Noel S. Hush, hush_n@chem.usyd.edu.au2, Alessio Gagliardi, gagliard@phys.upb.de3, Thomas Frauenheim, frauenheim@phys.upb.de4, Alessandro Pecchia5, and Aldo Di Carlo, dicarlo@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.

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