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The Interpretation of Scanning Tunnelling Microscopy

The Interpretation of Scanning Tunnelling Microscopy. Andrew Fisher and Werner Hofer Department of Physics and Astronomy UCL http://www.cmmp.ucl.ac.uk/. Overview. Operation of an STM: the issues to address Theoretical approaches: Within perturbation theory Beyond perturbation theory

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The Interpretation of Scanning Tunnelling Microscopy

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  1. The Interpretation of Scanning Tunnelling Microscopy Andrew Fisher and Werner Hofer Department of Physics and Astronomy UCL http://www.cmmp.ucl.ac.uk/ SRRTnet/CCP3 Code Training Workshop, November 2001

  2. Overview • Operation of an STM: the issues to address • Theoretical approaches: • Within perturbation theory • Beyond perturbation theory • Inelastic effects • Codes and algorithms • Recent examples: • Molecules • Currents and forces • Magnetic imaging SRRTnet/CCP3 Code Training Workshop, November 2001

  3. Operation of an STM1,2 [1] C. Julian Chen, Introduction to Scanning Tunnelling Microscopy, Oxford (1993) [2] G.A.D. Briggs and A. J. Fisher, Surf. Sci. Rep.33, 1 (1999) SRRTnet/CCP3 Code Training Workshop, November 2001

  4. Modelling an STM • Unknown: • Chemical nature of STM tip • Relaxation of tip/surface atoms • Effect of tip potential on electronic surface structure • Influence of magnetic properties on tunnelling current/surface corrugation • Relative importance of the effects Needed: extensive simulations SRRTnet/CCP3 Code Training Workshop, November 2001

  5. Modelling an STM • Theoretical issues: • Open system, carrying non-zero current • Macrosopic device depends on very small active region • No simple “inversion theorem” to deduce surface structure from STM signal SRRTnet/CCP3 Code Training Workshop, November 2001

  6. Overview • Operation of an STM: the issues to address • Theoretical approaches: • Within perturbation theory • Beyond perturbation theory • Inelastic effects • Codes and algorithms • Recent examples: • Molecules • Currents and forces • Magnetic imaging SRRTnet/CCP3 Code Training Workshop, November 2001

  7. Current theoretical models • Theoretical methods- • Non-perturbative: • Landauer formula or Keldysh non-equilibrium Green’s functions 1-4 • Perturbative: • Transfer Hamiltonian methods5 • Methods based on the properties of the sample surface alone6 [1] R. Landauer, Philos. Mag. 21, 863 (1970) M. Buettiker et. al. Phys. Rev. B31, 6207 (1985) [2] L. V. Keldysh, Zh. Eksp. Theor. Fiz. 47, 1515 (1964) [3] C. Caroli et al. J. Phys. C4, 916 (1971) [4] T. E. Feuchtwang, Phys. Rev. B10, 4121 (1974) [5] J. Bardeen, Phys. Rev. Lett.6, 57 (1961) [6] J. Tersoff and D. R. Hamann, Phys. Rev. B31, 805 (1985) SRRTnet/CCP3 Code Training Workshop, November 2001

  8. Perturbation theory - starting states • If tip and sample are weakly interacting, would like to use tip and sample states as a basis for perturbation theory • Problems: • these states are not orthogonal, as they are eigenstates of different Hamiltonians • cannot add the separate Hamiltonians to get the total, as this double counts kinetic energy SRRTnet/CCP3 Code Training Workshop, November 2001

  9. Perturbation theory - in what? • What is the matrix element? • Two ways of thinking: • Potential of system is what changes when tip and sample are coupled • Kinetic energy is non-local part of Hamiltonian that can couple tip and sample states z V(z) SRRTnet/CCP3 Code Training Workshop, November 2001

  10. [1] J. Pendry et al. J. Phys. Condens Matter3, 4313 (1991) [2] J. Julian Chen, Introduction to Scanning Tunneling Microscopy Oxford (1993) pp. 65 - 69 Transfer Hamiltonian method1 Conditions: (never non-zero at same point) Assume  and  each satisfies true Schrödinger eqn to one side of separation surface S S SRRTnet/CCP3 Code Training Workshop, November 2001

  11. [1] J. Pendry et al. J. Phys. Condens Matter3, 4313 (1991) [2] J. Julian Chen, Introduction to Scanning Tunneling Microscopy Oxford (1993) pp. 65 - 69 Transfer Hamiltonian method1 Conditions: (never non-zero at same point) Proceed by perturbation theory in removal of impermeable barrier (Pendry et al.) or integrate using Green’s theorem (Bardeen) to get matrix element as surface integral over S S SRRTnet/CCP3 Code Training Workshop, November 2001

  12. [1] J. Pendry et al. J. Phys. Condens Matter3, 4313 (1991) [2] J. Julian Chen, Introduction to Scanning Tunneling Microscopy Oxford (1993) pp. 65 - 69 Transfer Hamiltonian method1 Conditions: (never non-zero at same point) Result: (Off-diagonal element of current density operator) Golden rule with effective matrix element SRRTnet/CCP3 Code Training Workshop, November 2001

  13. The assumptions • Validity of perturbation theory: tunnelling sufficiently “weak” that a 1st-order expression is sufficient • Possible to find a separation surface S on which potential is zero (vacuum value) SRRTnet/CCP3 Code Training Workshop, November 2001

  14. Tersoff-Hamann Theory • Assume, in addition to validity of perturbation theory in tip-sample interaction, that we have • Spherically symmetric tip potential; • Initial state for tunnelling that is an s state on tip; • Zero bias • Asymptotic forms for wavefunctions thus SRRTnet/CCP3 Code Training Workshop, November 2001

  15. Tersoff-Hamman (2) • Can now do all the integrals to get The differential conductance probes the density of states of the (isolated) sample, evaluated at the centre of the tip apex Constant of proportionality depends sensitively on (unknown) properties of tip states SRRTnet/CCP3 Code Training Workshop, November 2001

  16. Problems perturbing • Perturbation theory itself will not work when • Tunnelling becomes strong (transmission probability of order 1, e.g. on tip-sample contact). Probably OK for most tunnelling situations, as these are limited by mechanical instabilities (see later) SRRTnet/CCP3 Code Training Workshop, November 2001

  17. Example (1) Conductance • When transmission probability in a particular ‘channel’ is close to unity, get ‘quantization’ of conductance in units of e2/h • Happens in specially grown semiconductor wires grown by e-beam lithography, or in metallic nanowires Extension Jacobsen et al. (Lyngby) SRRTnet/CCP3 Code Training Workshop, November 2001

  18. Example (1) • Such nanowires can be produced by pulling an STM tip off a surface, or simply by a ‘break junction’ in a macroscopic wire Jacobsen et al. (Lyngby) SRRTnet/CCP3 Code Training Workshop, November 2001

  19. Problems perturbing • Perturbation theory itself will not work when • Tunnelling becomes strong (transmission probability of order 1, e.g. on tip-sample contact). Probably OK for most tunnelling situations, as these are limited by mechanical instabilities (see later) • More than one transmission process of comparable amplitude (e.g. in transmission through many molecular systems) SRRTnet/CCP3 Code Training Workshop, November 2001

  20. Example (2) • Transport from terminal L… • …to terminal P… • …requires not just a tunnelling step… • …but an additional slow process... SRRTnet/CCP3 Code Training Workshop, November 2001

  21. Problems with T-H approach • The Tersoff-Hamann approach will, in addition, be suspect • Whenever tunnelling is not dominated by tip s-states (e.g. graphite surface, transition metal tips); • Whenever we are interested in effects of the tip chemistry or geometry; • Whenever we want to know the absolute tunnel current SRRTnet/CCP3 Code Training Workshop, November 2001

  22. Beyond perturbation theory • Must solve a single scattering problem: • Tip  Adsorbate  Substrate • Tools: • quantum mechanical scattering theory • Landauer formula (formally equivalent) • Express current in terms of transmission amplitude (t-matrix) SRRTnet/CCP3 Code Training Workshop, November 2001

  23. [1] M. Buettiker et al. Phys. Rev. B 31, 6207 (1985) The Landauer formula1 since Self-consistent potentials far from barrier satisfy: (4-terminal) (2-terminal) Landauer formulae SRRTnet/CCP3 Code Training Workshop, November 2001

  24. [1] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992) [2] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover, NY (1975) [3] M. Buettiker et al. Phys. Rev. B31, 6207 (1985) General Landauer formula for the STM1,2 Starting point is the Hamiltonian of the system: The tunnel current for interacting electrons: The tunnel current for non-interacting electrons3: SRRTnet/CCP3 Code Training Workshop, November 2001

  25. Single-molecule vibrations • Study vibrations of individual molecules and individual bonds by looking at phonon emission by tunnelling electrons Wilson Ho et al., UC Irvine SRRTnet/CCP3 Code Training Workshop, November 2001

  26. Single-molecule vibrations • Study vibrations of individual molecules and individual bonds by looking at phonon emission by tunnelling electrons • New possibilities for inducing reactions by selectively exciting individual bonds…. Wilson Ho et al., UC Irvine SRRTnet/CCP3 Code Training Workshop, November 2001

  27. Inelastic Effects • Inelastic effects becoming important for • Chemically specific imaging (Ho et al.) • Local manipulations (e.g. selective H desorption, Avouris et al.) • “Molecular Nanotechnology” Tip Sample =vibrational excitation =electronic transition SRRTnet/CCP3 Code Training Workshop, November 2001

  28. Inelastic Effects • Need to make separate decisions about whether to treat red and blue processes perturbatively • e.g. neither for electron transport through long conjugated molecules strongly bonded to two electrodes (Ness and Fisher) • e.g. both for inelastic STM of small molecules (Lorente and Persson) Tip Sample =vibrational excitation =electronic transition SRRTnet/CCP3 Code Training Workshop, November 2001

  29. Overview • Operation of an STM: the issues to address • Theoretical approaches: • Within perturbation theory • Beyond perturbation theory • Inelastic effects • Codes and algorithms • Recent examples: • Molecules • Currents and forces • Magnetic imaging SRRTnet/CCP3 Code Training Workshop, November 2001

  30. Existing numerical codes: Codes based on the Landauer formula1,2 Codes based on transfer Hamiltonian methods3 Codes based on the Tersoff-Hamann model4-6 Difficulty [1] J. Cerda et al., Phys. Rev. B56, 15885 & 15900 (1997) [2] H. Ness and A.J. Fisher, Phys. Rev. B55, 12469 (1997) [3] W.A. Hofer and J. Redinger, Surf. Sci. 447, 51 (2000) [4] K. Stokbro et al. Phys. Rev. Lett. 80, 2618 (1998) [5] S. Heinze et al. Phys. Rev. B58, 16432 (1998) [6] N. Lorente and M. Persson, Faraday Discuss.117, 277 (2000) SRRTnet/CCP3 Code Training Workshop, November 2001

  31. Implementing Tersoff-Hamann • Almost any electronic structure code can be (and probably has been!) adapted to generate STM images in the T-H approximation • Need to take care that • Have adequate description of wavefunction in vacuum region • If a basis set code, have adequate variational freedom for wavefunction far from atoms • Supposed tip-sample separations are realistic (often taken much tool close in order to match experimentally observed corrugation) SRRTnet/CCP3 Code Training Workshop, November 2001

  32. [1] C.J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford Univ. Press (1993) [2] W.A. Hofer and J. Redinger, Surf. Sci. 447, 51 (2000) Bardeen approach1,2: SRRTnet/CCP3 Code Training Workshop, November 2001

  33. Issues in bSCAN • Choice of surface to perform integral: always assume planar separation surface under tip • in practice, cannot check self-consistent potential for each tip position S Evaluate integral over separation surface analytically for each plane-wave component of tip and surface wavefunctions SRRTnet/CCP3 Code Training Workshop, November 2001

  34. Beyond perturbation theory • Must solve a single scattering problem: • Tip  Adsorbate  Substrate • Main difficulty: representation of the asymptotic scattering states • One solution: calculate conductivity instead between localised initial and final states and • Time-averaged measure of conductivity through states of energy E in terms of the Green function SRRTnet/CCP3 Code Training Workshop, November 2001

  35. Occupationnumber Retarded Green function Advanced Green function Embedding potential System Left Lead Right Lead Justification • Compare the most general version of the Landauer formula (Meir and Wingreen 1992): • Reduces to this approach in the wide-band limit of the leads, provided that they are `coupled’ into the system only through the chosen initial and final states SRRTnet/CCP3 Code Training Workshop, November 2001

  36. Efficient evaluation of G • Evaluate Green function efficiently using sparse matrix techniques (e.g. Lanczos algorithm): require only ability to compute H • Can do by post-processing output from a standard total energy code SRRTnet/CCP3 Code Training Workshop, November 2001

  37. Do full scattering calculation in a relatively simple localized orbital basis set (e.g. ESQC - Elastic Scattering Quantum Chemistry - approach: Sautet, Joachim) Find t by transfer matrix approach or from the Green’s function Advantages: Full scattering Relatively simple ‘chemical’ interpretation Disadvantages: Restricted freedom of wavefunction in vacuum No self-consistency Include full self-consistency with open boundary conditions from the outset New self-consistent open-boudary condition codes being developed based on O(N) approaches (e.g. SIESTA, CONQUEST) Advantages Fullest treatment of problem so far Truly self-consistent open system Disadvantages Difficult to do May not be needed for STM tunnel junctions Alternative approaches SRRTnet/CCP3 Code Training Workshop, November 2001

  38. Overview • Operation of an STM: the issues to address • Theoretical approaches: • Within perturbation theory • Beyond perturbation theory • Codes and algorithms • Recent examples: • Molecules • Currents and forces • Magnetic imaging SRRTnet/CCP3 Code Training Workshop, November 2001

  39. Example 1: benzene on Si(001) Two binding sites with interconversion on lab timescales (Wolkow et al.) SRRTnet/CCP3 Code Training Workshop, November 2001

  40. Example: benzene on Si(001) Discriminate between tips on basis of scanlines SRRTnet/CCP3 Code Training Workshop, November 2001

  41. [1] W.A. Hofer, A.J. Fisher, R.A. Wolkow, and P. Gruetter, Phys. Rev. Lett. in print (2001) [2] G. Cross et al. Phys. Rev. Lett. 80, 4685 (1998) Example 2: The influence of forces in STM scans1 Force measurement on Au(111)2 Simulation of forces: Simulation: VASP GGA: PW91 4x4x1 k-points SRRTnet/CCP3 Code Training Workshop, November 2001

  42. Forces and relaxations: Force on the STM tip: Relaxations of tip and surface atoms: The force on the apex atom is one order of magnitude higher than forces in the second layer Substantial relaxations occur only in a distance range below 5A SRRTnet/CCP3 Code Training Workshop, November 2001

  43. Tip-sample distance and currents: The real distance is at variance with the piezoscale by as much as 2A The surplus current due to relaxations is about 100% per A SRRTnet/CCP3 Code Training Workshop, November 2001

  44. [1] V. M. Hallmark et al., Phys. Rev. Lett. 59, 2879 (1987) Corrugation enhancement STM simulation: bSCAN Bias voltage: - 100mV Energy interval: +/- 100meV Current contour: 5.1 nA Due to relaxation effects in the low distance regime the corrugation of the Au(111) surface is enhanced by about 10-15 pm1 SRRTnet/CCP3 Code Training Workshop, November 2001

  45. [1] G. Kresse and J. Hafner, Phys. Rev. B47, R558 (1993) [2] Ph. Kurz et al.J. Appl. Phys.87, 6101 (2000) [3] J. P. Perdew et al. Phys. Rev. B46, 6671 (1992) Example 3: Atomic scale magnetic imaging Surface relaxation: VASP [1] Electronic structure: FLEUR [2] No spin-orbit coupling Film geometry: 7 layer W(110) film 2 Mn adlayers GGA: PW91 [3] k-points: 16 in IBZ Total energy: Antiferromagnetic ordering: - 0.8587 Ferromagnetic ordering: -0.8584 Difference: ~ 10 meV Anti-ferromagnetic ordering Ferromagnetic ordering Mn layer W(110) surface SRRTnet/CCP3 Code Training Workshop, November 2001

  46. Surface structure Surface relaxation: No relaxation effects due to magnetic orientation DOS in the Mn atoms SRRTnet/CCP3 Code Training Workshop, November 2001

  47. Tunneling conditions: Bias voltage: - 3 mV, constant current contour at z=4.5 A Current: from 0.1 nA (Mn tip) to 0.5 nA (W tip) Simulated STM images Paramagnetic STM tip (W): Ferromagnetic STM tip (Fe, Mn): SRRTnet/CCP3 Code Training Workshop, November 2001

  48. Importance of different effects in STM SRRTnet/CCP3 Code Training Workshop, November 2001

  49. Overview • Operation of an STM: the issues to address • Theoretical approaches: • Within perturbation theory • Beyond perturbation theory • Inelastic effects • Codes and algorithms • Recent examples: • Molecules • Currents and forces • Magnetic imaging SRRTnet/CCP3 Code Training Workshop, November 2001

  50. Thanks Werner Hofer Andrew Gormanly Hervé Ness £££: EPSRC, HEFCE, British Council SRRTnet/CCP3 Code Training Workshop, November 2001

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