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