Surfaces with hexagonal or trigonal symmetry

The problem of the electron charge-density distribution of a surface with hexagonal symmetry has been treated by Liebsch, Harris, and Weinert (1984). Similar to previous cases, the oo(z) term in Eq. (5.41) comes mainly from the Bloch functions near E, whose lowest Fourier component is  

The Bloch functions near the K points have the longest decay length, which are the dominating contribution to the second term in Eq. (5.41). In general, a surface Bloch function at that point has the form  

This can be easily shown by direct calculation. The charge density is the sum of Eq. (5.43) and the sum of the charge density proportional to I i i ilti 1 + I i i2 i i21 A straightforward calculation gives 

Similarly, 3 = 7 - 2k. The ratio (Ci/Cq) can be determined by comparing Eq. (5.49) with the corrugation amplitudes of the charge-density contours obtained from first-principles calculations. For example, from Fig. 5.7, averaged from five contours ranging from three contours of thinnest densities, we find (C /Co) 5.7 1.0. Following the procedure for the one-dimensional ca,se, the STM image for the p- tip state is 

A comparison of the theory with experiments is shown in Fig. 5.8. For Al(l 11), a = 2.88 A, 4) = 3.5 eV, it follows that k = 0.96 A y = 3.48 A. The slope of the In Az z curve from Equations (5.49) through (5.51) fits well with experimental data. The absolute tip-sample distance is obtained from curve fitting, which gives the shortest average tip-sample distance at / = 40 nA (with bias 50 mV) to be about 2.9 A, consistent with the measured tip-sample distance (Diirig et al, 1986, 1988), about 1 A before mechanical contact. 

---