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 1. What is Molecular Electronics 2. The “g. DFTB” method for molecular electronics applications 3. Why use DFTB ? 1. Problems with standard DFT 2. Does DTFB offer any intrinsic advantages ? 3. Is DFTB accurate enough ? 4. Use of g. DFTB in interpreting experiment 5. Implementing Symmetry in DFTB 6. Nature of molecular conduction channels
Molecular Electronics: Measuring single molecule conduction Nanopore Wang et al. PRB 68 (2003) 035416 Cross-wire Kushmerick et al. PRL 89 (2002) 086802 Electromigration H. S. J. van der Zant et al. Faraday Discuss. (2006) 131, 347 STM Break Junction Scanning Probe Cui et al. Science 294 (2001) 571 B. Xu & N. J. Tao Science (2003) 301, 1221 Nanocluster Dadosh et al. Nature 436 (2005) 677 Mechanical Break Junction Reichert et al. PRL 88 176804
Single-Molecule Conductivity L ELECTRODE MOLECULE R ELECTRODE
Single-Molecule Conductivity L ELECTRODE MOLECULE R ELECTRODE Molecular Orbitals Fermi energy
Single-Molecule Conductivity L ELECTRODE MOLECULE R ELECTRODE Molecular Orbitals e. V I V
I Finding a true molecular signature: Inelastic Electron Tunnelling Spectroscopy (IETS) Inelastic h /e V 2 h /e 2 V d I/d. V Elastic d. I/d. V 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 Djukic & Van Ruitenbeek Nano Lett. (2006) 6(4), 789 Smit et al. Nature (2002) 419, 906 Thygesen & Jacobsen Phys. Rev. Lett. (2005) 94, 036807
“g. DFTB” 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 “g. DFTB” • 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.
Partitioning the Electrode-Molecule-Electrode Hamiltonian Operator for the System Energy L M Diagonal blocks are the energies of each part Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994) 6849. R
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 1. 2. 3. 4. 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 kcal/mol PW 91 Calculated kcal/mol NH 3 Au(111) 7. 5 -10 8 Benzene Au(111) 9 2 Cu(111) 14 1 Cu(110) 23 6 • DFT calculations for benzene on a Cu 13 model cluster for (110) = 19 kcal/mol • CASPT 2 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 H 2: Source of longrange 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 H 0 + H 1 + H 2 ) CIS is OK (uses H 0 + H 1 ) Cai & Reimers J. Chem. Phys. 112 (2000) 527
Application to the weak electrode-electrode through molecule bonds that drive singlemolecule 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) P 3507
DFT band lineup error for phenylthiol (RSH) on gold(111) RSH Obs RS • PW 91 RS– PW 91 Adsorbate Bridge FCC Gold (111) PW 91 Obs. Band-gap error 5. 6 e. V Band lineup error 3. 4 e. V 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 ? 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 ? ? ?
SCC-DFTB errors for properties of 63 Mg complexes Property B 3 LYP SCC-DFTB AM 1 PM 3 MNDO-d PM 5 Ave . 00 . 02 -. 03 -. 10 -. 05 -. 06 . 02 . 06 . 11 . 18 . 10 . 09 Ave -1 -2 0 -9 1 -1 4 14 4 25 15 3 Ave . 20 -. 13 . 27 . 33 . 22 . 34 . 66 . 51 . 65 . 44 . 50 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 Bond length / Å Bond Ang. / ° IP / e. V Hf / kcal mol-1 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 g. DFTB-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 1. Find all atoms related by the Albelian symmetry operators C 2 (two-fold rotation), (reflection plane), and i (inversion) 2. Construct the transformation S that forces all atomic orbitals (AO), Cartesian tensor components, etc. , to be eigenfunctions of these operators 3. Transform the Kohn-Sham matrix H, force vector, Hessian matrix of second derivatives, etc. from AO basis and Cartesian coordinates into symmetry adapted representations: 4. H = ST H S 4. Diagonalize H to get symmetry-adapted molecular orbitals C 5. Back transformation to get molecular orbitals in AO basis 6. 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 C 2 v) • eg. Say that H has transforms according to the C 2 v point group • symmetry operators C 2 z, xz, yz, and E • irreducible representations a 1, a 2, b 1, and b 2 a 1 H = a 2 b 1 b 2 0 0 0 a 1 a 2 b 1 0 0 0 b 2 0 0 0
What is the point group in g. DFTB calculations? • Symmetry of entire system H is C 2 h (operators are C 2 z, xy, and i) • Symmetry of molecular component HM is C 2 h • Symmetry of individual molecule- electrode couplings JL and JR is Cs only • g. DFTB 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- C 6 H 4 -S All symmetry operators that enforce end-to-end symmetry are lost All other symmetry operators are retained In this case, D 2 h C 2 v Solomon, Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J. C. P. (2006) submitted
Conduction split into symmetry channels Total transmission A 2 component Ef = Fermi energy of Au, controls lowvoltage conductivity … its B 1 ! 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 • g. DFTB 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
The end
ACS Abstract Application of DFTB in molecular electronics Jeffrey R Reimers, reimers@chem. usyd. edu. au 1, Gemma C. Solomon, solomon@chem. usyd. edu. au 1, Zheng-Li Cai, zlcai@chem. usyd. edu. au 1, Noel S. Hush, hush_n@chem. usyd. edu. au 2, Alessio Gagliardi, gagliard@phys. upb. de 3, Thomas Frauenheim, frauenheim@phys. upb. de 4, Alessandro Pecchia 5, and Aldo Di Carlo, dicarlo@ing. uniroma 2. it 5. (1) School of Chemistry, The University of Sydney, 2006, Australia, (2) School of Molecular and Microbial Biosciences, The University of 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
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