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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

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Summary 1.What is Molecular Electronics 2.The “gDFTB” 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 gDFTB in interpreting experiment 5.Implementing Symmetry in DFTB 6.Nature of molecular conduction channels

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

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Single-Molecule Conductivity L ELECTRODE R ELECTRODE MOLECULE

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

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

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Elastic Inelastic V h /e V V V I dI/dV d 2 I/dV 2 h /e Finding a true molecular signature: Inelastic Electron Tunnelling Spectroscopy (IETS)

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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

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

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“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.

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Partitioning the Electrode-Molecule-Electrode Hamiltonian Operator for the System Energy LMR Diagonal blocks are the energies of each part Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994) 6849.

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Partitioning the Electrode-Molecule-Electrode Hamiltonian Operator for the System Energy LMR Off- Diagonal blocks are the interaction energies Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994) 6849.

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

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Why DFTB? General Serious Failures of DFT 1.Dispersion 2.Covalent bond breakage 3.Partial electron removal/addition (long range electron-transfer processes) 4.Extended conjugation ALL RELEVANT TO PHOTONICS AND MOLECULAR ELECTRONICS ! Can DFTB do better ??? Reimers, Cai, Bilić, Hush, Ann. N.Y. Acad. Sci (2003) 235.

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DFT Failure (1): Dispersion error leads to poor adsorption energies Bilić, Reimers, Hush & Hafner J. Chem. Phys. 116 (2002) 8981 Bilić, Reimers, Hoft, Ford & Hush J. Theor. Comput. Chem (2006). MoleculeSurfaceObservedPW91 Calculated NH 3 Au(111) BenzeneAu(111)92 Cu(111)141 Cu(110)236 kcal/mol DFT calculations for benzene on a Cu 13 model cluster for (110) = 19 kcal/mol CASPT2 dispersion energy error for DFT = 15 kcal/mol

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DFT Failure (2): Covalent Bond Breakage H 2 : Source of long- range correlation Single bonds break properly if and electrons have different orbitals Cai & Reimers J. Chem. Phys. 112 (2000) 527

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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

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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

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

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Closed-Shell treatments lead to split orbitals Open-shell calculation gives asymptotically correct answer Closed-shell hybrid density fucntionals gives asymptotically very poor result … perceived as strong coupling, resultant currents x 100 too high … useless Closed-Shell GGA density functionals have incorrect asymptotes but maintain double degeneracy … results in additional weak conduction channels … is useful Solomon, Reimers and Hush J. Chem. Phys. 112 (2000) 527

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

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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 lineup error 3.4 eV Band-gap error 5.6 eV Bilić, Reimers and Hush J. Chem. Phys. 122 (2005)

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

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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 PropertyB3LYPSCC-DFTBAM1PM3MNDO-dPM5 Bond length / ÅAve Bond Ang. / °Ave IP / eVAve H f / kcal mol -1 Ave Incr. LigandAve Binding / kcal mol -1 DeprotonationAve Energy / kcal mol -1 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) 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 Alkane chain S head group Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) 124,

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W. Wang, T. Lee, I. Kretzschmar & M. Reed Nano Lett. (2004) 4(4) Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) 124, Observed and gDFTB-calculated IETS Binding site Reed’s experimentCalculations match and enhance experimental assignment

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Effect of the binding site on CH intensity higher energy lower energy J. Kushmerick, Lazorcik, Patterson & Shashidhar Nano Lett. (2004) Wang, Lee, Kretzschmar & Reed Nano Lett. (2004) Opt structureCalculated IETS Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) 124,

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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 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: H = S T H S 4.Diagonalize H to get symmetry-adapted molecular orbitals C 5.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

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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 2v ) eg. Say that H has transforms according to the C 2v point group symmetry operators C 2z, xz, yz, and E irreducible representations a 1, a 2, b 1, and b a1a1 a2a2 b1b1 b2b2 a 1 a 2 b 1 b 2 H =

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What is the point group in gDFTB calculations? Symmetry of entire system H is C 2h (operators are C 2z, xy, and i) Symmetry of molecular component H M is C 2h Symmetry of individual molecule- electrode couplings J L and J R is C s only gDFTB equations use J L and J R 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

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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 2h C 2v Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) submitted

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Conduction split into symmetry channels Total transmission A 2 component E f = Fermi energy of Au, controls low- voltage conductivity … its B 1 ! Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) submitted

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The transmission through each symmetry block can then be partitioned in other ways: Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush Nano Letts (2006) in press 1. Büttiker eigenchannels (shot noise) 2. Junction eignchannels coupled by the molecule 3. Interference between Molecular Conductance Orbitals coupled through the junction

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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

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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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Application of DFTB in molecular electronics Jeffrey R Reimers, Gemma C. Solomon, Zheng-Li Cai, Noel S. Hush, Alessio Gagliardi, Thomas Frauenheim, Alessandro Pecchia5, and Aldo Di Carlo, (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 , Paderborn, , 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. ACS Abstract

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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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