Principles and Techniques
I describe a brief summary of quantum mechanics and principles of scanning tunneling microscopy (STM) in this chapter. Quantum tunneling is an important concept to describe the transportation of small particles, like electron and hydrogen. In Sect. 2.1 I
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Principles and Techniques
Abstract I describe a brief summary of quantum mechanics and principles of scanning tunneling microscopy (STM) in this chapter. Quantum tunneling is an important concept to describe the transportation of small particles, like electron and hydrogen. In Sect. 2.1 I focus on H-atom tunneling in a double minimum potential that is the simplest but a ubiquitous system in physics and chemistry. Quantum tunneling of electron directly relates to the principle of the STM. In Sect. 2.2 the principles and applications of STM are described. I focus on the inelastic electron tunneling process in an STM junction, which can be applied to the vibration spectroscopy and reaction control of single molecules.
Keywords Quantum tunneling Double minimum potential Scanning tunneling microscope/spectroscopy Inelastic electron tunneling
2.1 Quantum Mechanics 2.1.1 Basics of Quantum Tunneling Quantum tunneling is derived from the concept of quantum mechanics, where a particle can pass through a barrier that is classically insurmountable. This is a result of the wave-particle duality of matter. Quantum tunneling becomes pronounced in light particles such as electron and H atom. Figure 2.1 shows the simplest model in which a particle tunnels through a onedimensional rectangular barrier. The time-independent Schrödinger equation of the particle is written by
2 d 2 h wðxÞ þ VðxÞwðxÞ ¼ EwðxÞ 2m dx2
T. Kumagai, Visualization of Hydrogen-Bond Dynamics, Springer Theses, DOI: 10.1007/978-4-431-54156-1_2, Ó Springer Japan 2012
ð2:1Þ
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2 Principles and Techniques
Fig. 2.1 Schematic diagram of one-dimensional rectangular potential barrier
d2 h2 ðVðxÞ EÞwðxÞ wðxÞ ¼ dx2 2m
ð2:2Þ
where m is the mass, w(x) is the wave function, E is the energy of the particle and V(x) is the potential. In the region of V(x) \ E wðxÞ ¼ Aekx þ Bekx
ð2:3Þ
2mðVðxÞ EÞ h2
ð2:4Þ
k2 ¼
The wave function shows the oscillation characterized by the wave vector k. On the other hand, in the region of V(x)[ E wðxÞ ¼ Aejx þ Bejx
ð2:5Þ
2mðVðxÞ EÞ h2
ð2:6Þ
j2 ¼
The wave function no longer shows the oscillation, and it monotonically decreases with the decay constant j. For light particles we have a chance to find the particle beyond the classically impenetrable barrier, and the wave function starts oscillating again. The transmission probability T of a particle can be analytically-derived in this simple model, which is given ( )1 ðejL ejL Þ2 T ¼ 1þ ð2:7Þ 16eð1 eÞ where e = E/V, and L = (x2-x1) is the barrier width of a one-dimensional rectangular potential. If we assume jL 1, corresponding to a barrier of sufficient height and width, T 16eð1 eÞe2jL
ð2:8Þ
2.1 Quantum Mechanics
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Fig. 2.2 Schematic diagram of one-dimensional smooth potential barrier
The transmission probability decreases exponentially with an increase of the mass, m1/2, and the barrier width, L. Thus a significant isotope effect between H and D (deuterium) is expected in the dynamical processes. The one-dimensional rectangular
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