The Bardeen Transfer Hamiltonian Approach to Tunneling and its Application to STM and Carbon Nanotubes

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The Bardeen Transfer Hamiltonian Approach to
Tunneling and its Application to STM and Carbon
Nanotubes
Peter Albrecht, Kyle Ritter, and Laura
Ruppalt ECE 497 May 5, 2004
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Outline
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Bardeen Transfer Hamiltonian (BTH) Theory
Origin BTH theory introduced in 1961 to explain
Giaevers observations of tunneling in systems
of superconducting electrodes separated by thin
oxide barriers Bardeen, J. PRL, vol. 6, no.2,
pp.57-59, 1961.
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Separability of System
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Fermis Golden Rule and the Matrix Element
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The Model Hamiltonian
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The Matrix Element, M
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Example 1D Square Barrier
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Example 1D Square Barrier
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Why STM?

• Real-space atomic resolution imaging
• Atomic manipulation
• Electronic structure (local density of
  electronic states)
• Macroscopic realization of quantum mechanics

Carbon Nanotube on GaAs
Carbon Nanotube on Si
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Tersoff-Hamann Theory

• Since the tip and sample are only weakly coupled,
  perturbation theory is appropriate for the
  junction.
• Predominant tip state in tunneling is s-orbital

J. Tersoff and D.R. Hamann. Phys. Rev. B. 31, 805
(1985). G.A.D. Briggs and A.J. Fisher. Surf.
Sci. Rep. 33, 1 (1999).
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Tersoff-Hamann Theory
Substitute Eq. (2)
Substitute Eq. (3)
Introduction to Scanning Tunneling Microscopy.
C.J. Chen. (Oxford University Press, New York,
1993).
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Limitations of Tersoff-Hamann

• Experimental verification of T-H theory Au(110)
  surface

• Tip-sample forces ? modification of sample
• wavefunctions

• Most STM tips are transition metals ? dominant
• d-orbital character

• Poor understanding of tip structure

Tersoff-Hamann Theory Simple model for
fundamental understanding of STM images
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Field Emission ? Nanotechnology
Carbon nanotube field emission display Samsung
Research (Korea) W.B. Choi et al., Appl. Phys.
Lett. 75, 3129 (1999).
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Fowler-Nordheim Theory Field Emission from a
Metal
F electric field
VACUUM
METAL
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Limitations of Fowler-Nordheim Theory
vacuum potential barrier

• W(z,F) valid only asymptotically (z gt few Å)
• Surface-specific electronic structure is absent

R. Ramprasad, L.R.C. Fonseca, and P. Von Allmen,
Phys. Rev. B 62, 5216 (2000)
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Bardeen Transfer Hamiltonian

• Left side ? Jellium metal slab
• Electron gas moving in a uniform positive
  background
• At the surface, electrons can spill out into
  the vacuum

CHARGE DENSITY
?
?e
Phys. Rev. 49, 653 (1936)
METAL
VACUUM
Bardeen used self-consistent field (Hartree-Fock)
method Quantum-mechanical many-body effects
captured
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Jellium Wavefunctions and Energies
Self-consistent solution of the Hohenberg-Kohn-Sha
m (HKS) eqns.
Free electron behavior in the plane
perpendicular to the JELLIUM-VACUUM interface
N.D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970)
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Exchange and Correlation
Incorrect asymptotic behavior in
vacuum Exponential decay rather than 1/4z
local density approximation (LDA)
interpolated local density approximation
(ILDA)
Preserves LDA result in jellium bulk Smoothly
interpolates to classical image potential
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Total self-consistent effective potential
Step potential (work function)
LDA
ILDA
veff(z) (Hartrees)
classical image potential
JELLIUM METAL
VACUUM BARRIER
z (Å)
R. Ramprasad et al., Phys. Rev. B 62, 5216 (2000)
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Field Emission Current Density
Sum over all jellium states indexed by l
What about field emission from a one-dimensional
carbon nanotube?
R. Ramprasad, L.R.C. Fonseca, and P. Von Allmen,
Phys. Rev. B 62, 5216 (2000)
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Modeling field emission from a carbon nanotube

• High-aspect-ratio emitter
• Small radius of curvature (nm)
• Low extracting fields at CNT tip
• Chemical stability
• Large current density

Carbon nanotube cathode
z
V. Filip, D. Nicolaescu, and F. Okuyama, J. Vac.
Sci. Technol. B 19, 1016 (2001)
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Field Emission Current from a CNT
Fermi distribution
group velocity
tunneling probability (WKB)
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(9,9) metallic SWCNT
E(k) for a SWCNT sensitive to chirality and
diameter (n,0) zigzag and (n,n) armchair
metallic SWCNTs are the best field emitters
L. Zinhui, Z. Changchun, and L. Yukui, Physica B
344, 243 (2004)
J.W. Mintmire and C.T. White, Appl. Phys. A 67,
65 (1998)
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Summary
Bardeen Transfer Hamiltonian is a general and
powerful method for tunneling
Applications of BTH include scanning tunneling
microscopy and field emission

• BTH has a simple form for 1-d problems
• carbon nanotubes, nanowires, DNA, etc.

4.5 in
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Supplemental Slides
Great Resources Duke, C.B. Tunneling in Solids.
Academic Press New York, 1969. Hess, Karl.
Advanced Theory of Semiconductor Devices.
Wiley-IEEE Press New York, 2000
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Time-Dependent Perturbation Theory
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Simplification of M
Notice that in RR, (HM-HR)(HR-HL)0, so we can
write M symmetrically
Since were interested only in elastic tunneling,
EL,0ER,v, and note that in RR,
, use Greens Theorem,
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Single-Walled Carbon Nanotubes
Cleaved GaAs(110)
Hydrogen-Passivated Si(100)
2 nm
P. M. Albrecht and J. W. Lyding, Appl. Phys.
Lett. 83, 5029 (2003).
L. B. Ruppalt, P. M. Albrecht and J. W. Lyding,
unpublished.
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Scanning Tunneling Microscopy
Piezoelectric Scanner
Tip
Sample Holder