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### The Interpretation of Scanning Tunnelling Microscopy

Overview

Andrew Fisher and Werner Hofer

Department of Physics and Astronomy

UCL

http://www.cmmp.ucl.ac.uk/

SRRTnet/CCP3 Code Training Workshop, November 2001

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

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

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

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

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

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

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

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

[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 method1Conditions:

(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

[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 method1Conditions:

(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

[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 method1Conditions:

(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

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

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

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

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

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

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

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

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

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

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

[1] M. Buettiker et al. Phys. Rev. B 31, 6207 (1985)

The Landauer formula1since

Self-consistent potentials far from barrier satisfy:

(4-terminal)

(2-terminal)

Landauer

formulae

SRRTnet/CCP3 Code Training Workshop, November 2001

[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 STM1,2Starting 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

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

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

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

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

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

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

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

[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

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

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

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

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

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 approachesSRRTnet/CCP3 Code Training Workshop, November 2001

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

Example 1: benzene on Si(001)

Two binding sites with interconversion on lab timescales (Wolkow et al.)

SRRTnet/CCP3 Code Training Workshop, November 2001

Example: benzene on Si(001)

Discriminate between tips on basis of scanlines

SRRTnet/CCP3 Code Training Workshop, November 2001

[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 scans1Force measurement on Au(111)2

Simulation of forces:

Simulation: VASP

GGA: PW91

4x4x1 k-points

SRRTnet/CCP3 Code Training Workshop, November 2001

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

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

[1] V. M. Hallmark et al., Phys. Rev. Lett. 59, 2879 (1987)

Corrugation enhancementSTM 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

[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 imagingSurface 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

Surface structure

Surface relaxation:

No relaxation effects due

to magnetic orientation

DOS in the Mn atoms

SRRTnet/CCP3 Code Training Workshop, November 2001

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 imagesParamagnetic STM tip (W):

Ferromagnetic STM tip (Fe, Mn):

SRRTnet/CCP3 Code Training Workshop, November 2001

Importance of different effects in STM

SRRTnet/CCP3 Code Training Workshop, November 2001

- 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

Thanks

Werner Hofer

Andrew Gormanly

Hervé Ness

£££: EPSRC, HEFCE, British Council

SRRTnet/CCP3 Code Training Workshop, November 2001

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