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Published on January 16, 2008

Author: Umberto

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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:  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 Operation of an STM1,2:  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) Modelling an STM:  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 Modelling an STM:  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 Overview:  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 Current theoretical models:  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. B 31, 6207 (1985) [2] L. V. Keldysh, Zh. Eksp. Theor. Fiz. 47, 1515 (1964) [3] C. Caroli et al. J. Phys. C 4, 916 (1971) [4] T. E. Feuchtwang, Phys. Rev. B 10, 4121 (1974) [5] J. Bardeen, Phys. Rev. Lett. 6, 57 (1961) [6] J. Tersoff and D. R. Hamann, Phys. Rev. B 31, 805 (1985) Perturbation theory - starting states:  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 Perturbation theory - in what?:  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) Transfer Hamiltonian method1:  Transfer Hamiltonian method1 [1] J. Pendry et al. J. Phys. Condens Matter 3, 4313 (1991) [2] J. Julian Chen, Introduction to Scanning Tunneling Microscopy Oxford (1993) pp. 65 - 69 Conditions: (never non-zero at same point) Assume  and  each satisfies true Schrödinger eqn to one side of separation surface S S Transfer Hamiltonian method1:  Transfer Hamiltonian method1 [1] J. Pendry et al. J. Phys. Condens Matter 3, 4313 (1991) [2] J. Julian Chen, Introduction to Scanning Tunneling Microscopy Oxford (1993) pp. 65 - 69 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 Transfer Hamiltonian method1:  Transfer Hamiltonian method1 [1] J. Pendry et al. J. Phys. Condens Matter 3, 4313 (1991) [2] J. Julian Chen, Introduction to Scanning Tunneling Microscopy Oxford (1993) pp. 65 - 69 Conditions: (never non-zero at same point) Result: Golden rule with effective matrix element (Off-diagonal element of current density operator) The assumptions:  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) Tersoff-Hamann Theory:  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 Tersoff-Hamman (2):  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 Problems perturbing:  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) Example (1):  Example (1) 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 Conductance Extension Jacobsen et al. (Lyngby) Example (1):  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) Problems perturbing:  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) Example (2):  Example (2) Transport from terminal L… …to terminal P… …requires not just a tunnelling step… …but an additional slow process... Problems with T-H approach:  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 Beyond perturbation theory:  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) The Landauer formula1:  The Landauer formula1 [1] M. Buettiker et al. Phys. Rev. B 31, 6207 (1985) Landauer formulae since Self-consistent potentials far from barrier satisfy: (4-terminal) (2-terminal) General Landauer formula for the STM1,2:  General Landauer formula for the STM1,2 [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. B 31, 6207 (1985) Starting point is the Hamiltonian of the system: The tunnel current for interacting electrons: The tunnel current for non-interacting electrons3: Single-molecule vibrations:  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 Single-molecule vibrations:  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 Inelastic Effects:  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 Inelastic Effects:  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 Overview:  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 Existing numerical codes::  Existing numerical codes: Codes based on the Landauer formula1,2 Codes based on transfer Hamiltonian methods3 Codes based on the Tersoff-Hamann model4-6 [1] J. Cerda et al., Phys. Rev. B 56, 15885 & 15900 (1997) [2] H. Ness and A.J. Fisher, Phys. Rev. B 55, 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. B 58, 16432 (1998) [6] N. Lorente and M. Persson, Faraday Discuss. 117, 277 (2000) Difficulty Implementing Tersoff-Hamann:  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) Bardeen approach1,2: :  Bardeen approach1,2: [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) Issues in bSCAN:  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 Evaluate integral over separation surface analytically for each plane-wave component of tip and surface wavefunctions S Beyond perturbation theory:  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 Justification:  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 Efficient evaluation of G:  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 Alternative approaches:  Alternative approaches 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 Overview:  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 Example 1: benzene on Si(001):  Example 1: benzene on Si(001) Two binding sites with interconversion on lab timescales (Wolkow et al.) Example: benzene on Si(001):  Example: benzene on Si(001) Discriminate between tips on basis of scanlines Example 2: The influence of forces in STM scans1:  Example 2: The influence of forces in STM scans1 [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) Force measurement on Au(111)2 Simulation of forces: Simulation: VASP GGA: PW91 4x4x1 k-points Forces and relaxations: :  Forces and relaxations: Force on the STM tip: 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 Relaxations of tip and surface atoms: Tip-sample distance and currents::  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 Corrugation enhancement:  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 [1] V. M. Hallmark et al., Phys. Rev. Lett. 59, 2879 (1987) Example 3: Atomic scale magnetic imaging:  Example 3: Atomic scale magnetic imaging Anti-ferromagnetic ordering Ferromagnetic ordering Mn layer W(110) surface 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 [1] G. Kresse and J. Hafner, Phys. Rev. B 47, R558 (1993) [2] Ph. Kurz et al. J. Appl. Phys. 87, 6101 (2000) [3] J. P. Perdew et al. Phys. Rev. B 46, 6671 (1992) Surface structure:  Surface structure Surface relaxation: No relaxation effects due to magnetic orientation DOS in the Mn atoms Simulated STM images:  Simulated STM images Paramagnetic STM tip (W): 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) Ferromagnetic STM tip (Fe, Mn): Importance of different effects in STM:  Importance of different effects in STM Overview:  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 Thanks:  Thanks Werner Hofer Hervé Ness Andrew Gormanly £££: EPSRC, HEFCE, British Council

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