Conductance Calculations for Small Molecules

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1 Conductance Calculations for Small Molecules Karsten W. Jacobsen CAMP, Dept. of Physics Technical University of Denmark

2 Outline Framework Density Functional Theory Conductance calculations methodology Examples: H2/Pt CO/Pt Bipyridine/Au Nitrobenzene/polyacetylene/Au

electron potential: Plane wave pseudopotential DFT code (Dacapo) Supercell of ''extended

3 Conductance Calculations Landauer Büttiker setup: One electron picture Phase coherent electrons All scattering takes place in S (periodic leads) Semi infinite leads The electron potential: Plane wave pseudopotential DFT code (Dacapo) Supercell of ''extended S'' Separate calculation for lead potential Thygesen, Bollinger, Jacobsen, PRB 67, (2003)

4 Density Functional Theory Ground state energetics in principle exact. The success of approximations based on experience. (Problems with correlations, van der Waals,...) Kohn Sham potential constructed to reproduce electron density with non interacting electrons. Is not meant for excitation spectra but (static) charge transfer should be well described. Attitude: Let us see how far we can go. Try simple explanations first. Starting point for further developments: Time dependent DFT, GW,...

5 One Electron Picture Exactly solvable model (with many names) Newns Anderson model U=0 Anderson model Fano Anderson model The resonant level model Useful for chemisorption, resonant tunneling,... Can be extended with Hubbard U (Anderson model), phonon coupling,...

6 Newns Anderson Model k V ak a a k Density of states projected onto adsorbate site: 1 a = a 2 2 = k V ak k 2 ' 1 = P d ' ' coupling weighted density of states Hilbert transform

7 Wide Band System k 0 a k = k V ak k = 0 2 ' 1 = P d ' =0 ' 0 1 a = a 2 02

8 Two Level System antibonding k a a k bonding = k V ak k = V k 2 2 ' = P d ' = V ' k a =a a

11 Fine tuning reactivity: For a fixed local geometry: Correlation between adsorption energies and activation barriers and the d band center Mavrikakis, Hammer, Nørskov Phys. Rev. Lett. 81, 2819 (1998)

12 The Group Orbital a a =N V k g =N H k ak a N= 1 V k 2 V ag= a H g =N k V ak a H k =N k V ak = 2 ak V k Imaginary part of the self energy: 2 1 = k V ak k = 2 k g k k N 2 1 = 2 g = V ag g N 2 2 ak

The Green's function slide A L R =i [ RL R L R ] Conductance (1,2) 2 2e G= Tr [ G SR L G SA R ] h Central quantity: G =[ S S H S R S R L R 1 R ] S i : Overlap matrix H i : Hamiltonian matrix

13 The Green's function slide A L R =i [ RL R L R ] Conductance (1,2) 2 2e G= Tr [ G SR L G SA R ] h Central quantity: G =[ S S H S R S R L R 1 R ] S i : Overlap matrix H i : Hamiltonian matrix Self-energies due to leads: Left lead: RL = S S0 H SO g 00R S SO H SO Similar expression for right lead Shaded area treated within DFT (1) Meir and Wingreen, PRL 68, 2512 (1992) (2) Y. Xue et al., Chem. Phys. 281, 151 (2002)

14 Conductance Formula Applied to Newns Anderson model 2 2 2e 2e 1 1 R A G= Tr [ G S L G S R ] = 2 L 2 R h h a i a i 2 4 L R 2e = h a 2 2 Symmetric Contact: L = R = e2 2 e2 2 e2 2 G = = a = V ag g a 2 2 h a h h Wide band > G Lorentzian shape peaking at a with value 2 e2 h

15 Choice of Basis Set H KS=T V eff V NL For evaluating Kohn-Sham Hamiltonian: Localized basis set required to separate leads and scattering region Systematically converged i, n r solution to 2D Kohn-Sham equation Localized at 'site' i Basis set Using plane waves i, n = i z i, n r i=0 i z localized wavelet at 'site' i Compact support Non-orthogonal T. A. Arias, Rev. Mod. Phys. 71, 267 (1999) Thygesen, Bollinger, Jacobsen, PRB 67, (2003)

16 Wannier Function Basis Maximally localized Span the set of occupied eigenstates Plane wave accuracy Minimal basis Pt wire: Benzene molecule: X6 X6 X6 Thygesen, Hansen, Jacobsen, PRL 94, (2005) Calzolari, Marzari, Souza, Nardelli, PRB 69, (2004)

17 Partly Occupied Wannier Functions Make linear combinations of eigenstates (found by plane wave code) in order to produce localized wave functions. Unitary transformation to keep orthogonality of states. Include all occupied states + some more to maximize localization. > Bonding antibonding closing. Example: Benzene Marzari, Vanderbilt, Phys. Rev. B 56, (1997) Thygesen, Hansen, Jacobsen, to be published

18 Inclusion of Unoccupied States in WF Maximal localization per WF guides the choice of how many unoccupied states to include. The relevant unoccupied states are not necessarily lowest in energy.

19 Partly Occupied Wannier Functions Pt wire 1 sigma bond + 5 d orbitals Pt wire with Hydrogen molecule

C Au (Theory) (Pulling a wire) Lang, Avouris, PRL 81, 3515, (1998) Au,Pt,Ir (Averaged

20 Conductance of monatomic wires Even-odd oscillations in the conductance of monatomic metal wires. C Au (Theory) (Pulling a wire) Lang, Avouris, PRL 81, 3515, (1998) Au,Pt,Ir (Averaged conductance) Na (Theory) Tsukamoto and Hirose, PRB 66, (2002) Lee, Brandbyge et al., PRB 69, (2004) Smit, Untiedt, Rubio Bollinger, Segers, Ruitenbeek, PRL 91, (2003)

21 Four atom Period in Conductance of Aluminum Wires Three atom basis 4-atom period in conductance p-orbitals carry current Model: Resonance postions obtained from infinite wire Thygesen, Jacobsen PRL 91, (2003) Resonance widths scale as 1/N

22 Conductance of Aluminum Wire The discrete spectrum of an N-atom wire follows from the band of the infinite wire: N n = n N1, n=1,2,, N Thygesen, Jacobsen PRL 91, (2003)

23 Four Atom Period The extended nature of the molecular orbitals on the free wire determines the scaling of the resonance width: 1 / N For short wires (N=1,2) charge neutrality on the wire is important. The oscillations do not decay! Fermi level is determined by fillingfactor of the infinite wire: f=0.25 (Al), f=0.5 (Na,C,Au)

24 Na versus Al wires Na less resonant structure Charge neutrality criterion identical to criterion from infinite wire.

25 How Do Adsorbates Affect Atomic Chains? Passivation closing of conduction channels Al wire+h G F reduced by 1G 0 Al wire+o G F reduced by 2G 0 Thygesen, Bollinger, Jacobsen, PRB 67, (2003)

26 H/Pt Passivation closing of conduction channels H/Pt:

27 High Chemical Activity for Low Metal Coordination Numbers Bahn, Lopez, Nørskov, and Jacobsen, PRB 66, (R) (2002)

28 Adsorption Induced Restructuring CO adsorption Band gap opens up Bahn, Lopez, Nørskov, and Jacobsen, PRB 66, (R) (2002) Oxygen adsorption Flexible Stable Conducting May explain long bond lengths in TEM?

29 H2/Pt Molecular Bridge With H2 Pure Pt Does H2 dissociate? What is the detailed structure? Why is H2 conducting? Smit, Noat Untiedt, Lang, Hemert, Ruitenbeek, Nature 419, 906 (2002)

30 Molecular Stability in Different Contacts H /Pt Usually H2 dissociates on Pt Luntz et al, JCP (1990) Small barrier for some facets Pasteur et al, JCP 106, 8896 (1997) Bonding of H2: f ~ 0.9 Va Cd a E f Vb V 2a a C d b Bonding at both ends essential. Calculations by Kristian Thygesen 2

31 H2/Pt Contacts Dissociated structure suggested byy. Garcia et al., PRB 69, (R) (2004)

32 Vibrations Observation: The observed mode increases its frequency upon stretching! Problem: The longitudinal center of mass mode is expected to decrease. R. H. M. Smit, Y.Noat, C.Untiedt, N. D. Lang, M. C. van Hemert, J. M. van Ruitenbeek, Nature 419, 906 (2002)

33 Calculated Vibrations Calculations by Kristian Thygesen

34 New Measurements Talk Friday! Djukic, Thygesen, Untiedt, Smit, Jacobsen, van Ruitenbeek

35 Conductance of Hydrogen Bridge dh H = 1.0 Å dpt H = 1.76 Å Kristian Thygesen, K.W. Jacobsen, PRL 94, (2005).

Wannier Function Analysis 1 sigma orbital + 5 d orbitals per Pt atom 1 s like orbital for each of the two H atoms Antibonding state at Fermi level: H2 bond weakening.

36 Wannier Function Analysis 1 sigma orbital + 5 d orbitals per Pt atom 1 s like orbital for each of the two H atoms Antibonding state at Fermi level: H2 bond weakening. Transport carried by antibonding state! Bonding state b= 6.4 ev Conductance: Anti bonding state a =0.1 ev 2 2 e2 G = 2 2 V ag g a h Bonding state suggested to carry current by Cuevas et al., Nanotechnology 14, R29 (2003) Garcia et al., PRB 69, (2004)

37 Exact Resonant Level Model Parameters from DFT: a =0.1 ev V ag =1.6 ev (to one side) 2 2 2e 2 G = 2 V ag g a h

38 Benzene dithiol/au Identifying the simplest molecular subspace : Consider S as part of the leads. I Group orbital Other calcs: Di Ventra et al (2000) Emberly, Kirczenow (2001) Stokbro et al (2003) Xue and Ratner (2004) Thygesen, Jacobsen, to be published But experiments? Disagreement 2 3 orders of magn. Lörtscher, Weber, Riel: here Reed et al. Science (1997)

39 CO/Pt Experimental conductance histogram: Clean Pt: Peak around 1.5 G0 With CO: Two peaks around 1.2 G0 and 0.5 G0 Untiedt, Dekker, Djukic, van Ruitenbeek, PRB 69, , 2004

40 CO Molecular States Calculations by Mikkel Strange

41 CO in Pt chains Calculated conductance: 3 channels in wire 1.3 G0 0.5 G0

42 CO in Pt chains Newns-Anderson Model + Two MO determines the conductance

atoms Plateau is due to the central atoms sticking together

43 Clean Pt Contact Clean Pt contact - trace Transmission function Supercell Sum of transmissions d-like-orbitals on the central Pt atoms Plateau is due to the central atoms sticking together Conductance for wire ~1.7 G0 4 Configuration The peak is due to the dorbitals

44 Pt contact + CO CO/Pt Conductance trace and Energy Conductance histogram Binding energy: ~ -2.5eV -> CO prefers to sit in the contact Discontinuous transition: CO tilts 0.5 G0 plateau caused (at least partly) by asymmetry Further work required to identify vibrational modes at 1G0

45 Bipyridine between Gold Contacts Xu and Tao, Science 301, 1221 (2003) Bipyridine conductance: 0.01 G0

Bipyridine between Gold Contacts Octanedithiol: Break force: 1.5 + 0.

46 Bipyridine between Gold Contacts Octanedithiol: Break force: nn Sometimes very long stretching > 1nm chain formation? breaking of Au Au bonds? Conductances in agreement with calculations by Tomfohr and Sankey Bipyridine: Conductance: 0.01 G0 Break force: nn Xu, Xiao, and Tao JACS, 125, (2003)

47 Bipyridine Calculation Geometries Varying coordination number of contacting gold atom. Stadler, Thygesen, Jacobsen Talk by Robert Stadler tomorrow!

2 nn triangle (N=5): 0.8 nn flat surf (N=9): 0.4 nn expt: 0.

48 Calculations for Bipyridine/Au Wire Flat substrate Energy (for two bonds) Break force: chain (N=1): 1.6 nn adatom (N=3): 1.2 nn triangle (N=5): 0.8 nn flat surf (N=9): 0.4 nn expt: 0.8 +/ 0.2 nn stretch (two bonds) (Å) Calculations by Robert Stadler

49 Bipyridine Bonding Trend independent of exchange correlation functional. HOMO hybridizing with metal states (V ~ 1.7 ev)

50 Bipyridine Conductance LUMO Position of LUMO determined by 1) Coordination number of gold contact atom (local field) 2) Work function (non local) Calculations by Robert Stadler

51 Bipyridine Conductance Full calculation Resonant level model (LUMO) Experiment: 0.01 G0

52 Bipyridine Conductance Our result: Calculation by S. Hou et al. (Nanotechnology, 2005) (cluster calculation) Conductance = G0

53 (Nitro )Benzene in Different Contacts Gold contact 1 polyacetylene + gold leads 2 polyacetylene + gold leads polyacetylene leads Robert Stadler, Kristian Thygesen, Karsten W. Jacobsen, to be published

54 Calculated Conductances Rotated nitro group has essentially no effect. Adding one or two PA units increases the conductance.

Fermi level alignment in molecular nanojunctions and its relation to charge transfer

Downloaded from orbit.dtu.dk on: Jan 27, 2018 Fermi level alignment in molecular nanojunctions and its relation to charge transfer Stadler, Robert; Jacobsen, Karsten Wedel Published in: Physical Review