Attractive atomic force and tunneling conductance

To assess the relative importance of the van der Waals and the resonance contributions occurring within the normal operational distances in STM, it is instructive to compare the quantities occurring in real experiments versus those quantities in the exactly soluble H2+ problem. 

the range of separation. The experimentally determined tip-sample separation is 1-4 A before a mechanical contact (Diirig et al. 1988). For most metals, the normal nucleus-nucleus distance of a mechanical contact is 2.5-3 A. Therefore, on the absolute tip-sample scale, it is 3.5-7 A. By comparing with the case of H2, we notice that this is the range where the resonance interaction dominates. In other words, under normal STM operation conditions, over a distance range of about 3 A, resonance energy is almost solely responsible to the atomic force, and the distance dependence of the force should be approximately exponential. 

Second, the sensitivity of AFM. In a typical AFM (Binnig et al., 1986), the force sensitivity is about 0.01 nN. In the range of 4-10 a.u., the resonance force in the hydrogen molecular ion is 4 nN to 0.01 nN. Therefore, the resonance force (attractive atomic force) of a single chemical bond, extended over a distance of 3 A, can be detected. On the other hand, the van der Waals force of a pair of neutral atoms, when it is distinguishable from the total force. 

To make a quantitative treatment, we define a system including a tip and a sample, as shown in Fig. 7.6. Independent electron approximation is applied. The Schrbdinger equation is identical to Eq. (7.6), with the potential surface shown in Fig. 7.6. Similar to the treatment of hydrogen molecular ion, a separation surface is drawn between the tip and the sample. The exact position of the surface is not important. Define two subsystems, the sample S and the tip T, with potential surfaces Hs and Ut, respectively, as shown in Fig. 7.6 (c) and 

To account for the attractive force in the normal range of STM operation, we consider surface states on the tip near the Fermi level. Because the 

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