Tunnelling conduction

Whereas a film formed in dry air consists essentially of an anhydrous oxide and may reach a thickness of 3 nm, in the presence of water (ranging from condensed films deposited from humid atmospheres to bulk aqueous phases) further thickening occurs as partial hydration increases the electron tunnelling conductivity. Other components in contaminated atmospheres may become incorporated (e.g. HjS, SO2, CO2, Cl ), as described in Sections 2.2 and3.1. 

Furthermore, it was shown the unpaired spin S = 1/2, which is delocalized over the two Pc rings, still remained in the Jt-orbitals after absorption on Au(lll). Consequently, STS measurements also provided direct observation ofthe S = 1/2 radical on the TbPc2 molecules on Au(lll) whereby the indicative Kondo-peak could be switched off by tunnelling current pulses [215]. Indeed the tunnelling conductance (dl/dV) was analysed from STS experiments of TbPc2 on Au(lll) near the Fermi level showed a zero-bias peak (ZBP) in the spectra, which could be assigned as a Kondo resonance. Clear Kondo features for the molecules with 9 = 45° were observed when the tip was positioned over one ofthe lobes of TbPc2. 

The effective tunneling barrier may be quantified by measuring the distance dependence of the tunnel conductance or tunneling current [Eq. (1) or (2), respectively]. Experimentally, the decay constant, k, may be derived from dc or ac measurements. The more accurate ac modulation... 

STM images of molecules are often a sensitive function of the tunnel conductance. For example, when examining TMPyP on Au( 111 )-I, the iodine underlayer was imaged at a tunnel resistance of 5 X 10 ohms while the electron density of TMPyP was observed at 8 X 10 ohms [484], Similar effects have been noted for protoporphyrins [488] and purines [489,490] adsorbed on graphite. 

Fig. 13 Temperature dependence of the tunneling conductance for a 3.8 nm Al/6 nm mbrene/ 15 nm Co junction and 3.8 nm Al/0.5 nm AI2O3/5.5 nm mbrene/15 nm Co Junction. Ths solid line is a fit the VHR model to the data at high temperatures for the mbrene-only barrier. The inset shows I V curves for Junctions at 4.2 K. Taken from [57] with permission...

Butler WH, Zhang XG, Schulthess TC, MacLaren JM (2001) Spin-dependent tunneling conductance of Fe/MgO/Fe sandwiches. Phys Rev B 63 054416... 

Brinkman WF, Dynes RC, Rowell JM (1970) Tunneling conductance of asymmetrical barriers. J Appl Phys 41 1915-1921... 

Thus, with a free-electron metal tip, the dynamic tunneling conductance is proportional to the DOS of the sample. 

Fig. 2.1. Correlation between tunneling conductance and force. The tunneling conductance and the force gradient are measured simultaneously. The origin of the abscissa is the equilibrium point, where the net force is zero. It corresponds to a tunneling conductivity of 10 O. The experimental setup is schematically shown in the inset. (Reproduced from Diirig et al., 1988, with permission.)...

There are two consequences of the i-wave model on graphite that can be tested experimentally. First, from Eq. (5.67), the average tunneling conductance varies with z as... 

In this section, we study the images of atomic states. Starting with an analytic expression of the tunneling conductivity distribution, we will eompare the theoretical predictions and experimental findings of an easily measurable quantity the radius of the STM image of an atomic state near its peak. 

To determine the image, the first step is to determine the distribution of tunneling current as a function of the position of the apex atom. We set the center of the coordinate system at the nucleus of the sample atom. The tunneling matrix element as a function of the position r of the nucleus of the apex atom can be evaluated by applying the derivative rule to the Slater wavefunctions. The tunneling conductance as a function of r, g(r), is proportional to the square of the tunneling matrix element ... 

A convenient quantity for comparing with the STM image profile for a single atomic state is the apparent radius of the image near its peak, R, or the apparent curvature, K = l/R. Considering the w = 0 state only, these quantities are related to the tunneling conductance distribution by... 

Intuitively, we will expect that for p. or d- states on the tip and on the sample, the images should be sharper that is, the apparent radius should be smaller. This is indeed true. For example, for a 4- tip state and an. r-wave sample state, the tunneling conductance distribution is... 

Lang s result from the numerical theory is in excellent agreement with the general theory in the previous section. If both tip state and sample state are. 9-wave states, the tunneling conductance should have a spherical distribution, and the apparent radius should equal the nominal distance between the two nuclei. The radius obtained from the figure fits well with this expectation. 

A tetragonal lattice is the combination of two perpendicular one-dimensional lattices with the same lattice constant and having a reflection symmetry, as shown in Fig. 6.5. Therefore, we start with a one-dimensional case. The tunneling conductance from the nth atom is g(x — na, z). The total tunneling conductance from all the atoms is... 

Therefore, the problem of calculating the STM images reduces to the problem of evaluating the Fourier coefficients for the tunneling conductance distribution of a single atomic state, Eq. (6.23). 

This treatment can be extended immediately to the case of surfaces with tetragonal symmetry. The Fourier coefficients for the tunneling conductance of a tetragonal lattice is... 

This fundamental relation can be extended to the many-body case, and a correlation between the interatomic force in the attractive-force regime and the tunneling conductance can be established. For metals, an explicit equation between two sets of measurable quantities is derived. Of course, the simple relation between the measured force and measured tunneling conductance is not valid throughout the entire distance range. First, the total force has three components, namely, the van der Waals force, the resonance force, and the repulsive force. Second, the actual measurement of the force in STM and... 

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