The purpose of this chapter is to present basic experimental results regarding the alterability of a surface using a tunneling probe. This chapter will begin with a theoretical overview of quantum mechanical electron tunneling through a square potential energy barrier and its application to STM. Knowledge of the related tunneling parameters are necessary for a better understanding of surface alteration experiments. After this knowledge is reviewed, various surface alteration experiments will be reported.
4.1 Quantum mechanical electron tunneling
An electron has been demonstrated to act not only like a particle (classically) but also like a wave (interference and diffraction). A probability interpretation provides the means for interrelating the particle and wave nature of electrons. Similar to the electric field strength of an electromagnetic wave, an electron with a constant linear momentum, mv, and energy, E, is attributed a simple harmonic wave function, , where 4
44(x,t) sin 2%% x t cos 2%% x t e k x e 66E t 55(x) 55(t) . (4-1)
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x = 0 x = a V = 0
V = Vo
e-
Figure 4:1. Electron incident on a potential
energy barrier Vo greater than electron
energy, E.
66
2 2m d255(x) dx2 E 55(x), for x < 0 and x > a66
2 2m d255(x) dx2 (E Vo )55(x), for 0 <x > a . (4-4)frequency are respectively mvh and Eh , t is the time, and where k2%%
and h is Plank's constant. By using this equation to solve Schrödinger's equation,
66
2 00244(x,t)00x2 V(x,t)44(x,t)
66
0044(x,t)
00t , (4-2)
the probability of an electron to be found at a position inside or beyond a classically forbidden potential barrier, V(x,t), can be determined. This probability, P, is defined by the equation
P(x,t ) 44(x,t )44(x,t). (4-3)
Assuming that an electron is incident on a time independent square potential barrier as shown in Fig. 4:1, then
Schrödinger's equation for the electron reduces to Eqn. (4-4).
After calculating the probability amplitudes of the incident and reflected waves for each of the regions, one can calculate the probability
T
[ probability at x < 0 ]##x < 0 [ probability at x > a ]##x > a 1 e oa e oa 2 16E Vo(1 E Vo) 1 , where o 2m (Vo E ) 662 16 E Vo 1 E Vo e 2 oa, if oa > > 1. (4-5)flux transmitted through the barrier. This transmission coefficient1, T, is given as
The barrier region between a STM tunneling tip and a conducting sample is an example of a potential barrier. The potential barrier, Vo, of this
region is related to the work functions of the sample (1s) and tip (1t), and also
the bias, Vb, between the tip and sample. This work function, 1, is the amount
of energy necessary for an electron (at Fermi energy, EF) to escape from the
surface into a vacuum. Work functions for the elements with clean surfaces range from 2-6 eV. Using Vo-EF=4 eV and no applied bias, we obtain a value
for ko=1.15/Å. This implies that for tip-sample separations greater than an
Angstrom, T is proportional to e 2os, where s is the tip-sample separation and is taken to be equivalent to the barrier thickness. The current, I, which flows between the sample and tip will be proportional to T. The proportionality constant will depend on the electron density at the surface of the electron source.
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J
3e
22h
22m11
0 ½V
bs
e
4%% h (2m110) ½s (4-6)barrier is nonuniform. For eVb<1o=(1t+1s)/2, the barrier height can be
estimated using the WKB2 integral approximation to Vo=1o-eVb/2, where e is
the charge of an electron. For Vb <<1o, Simmons [1963] estimated the current
density, J, to be:
From Eqn. (4-6), there are three interrelated variables which pertain to tunneling microscopy, s, Vb, and I (or J). The current can be calculated from
I=J×A, where A is the effective tunneling area under the tip. Using a tunneling area of 1 Å2, and calculating 1
o=5.3 eV from the work functions of a Pt tip and
a graphite sample, tunneling currents vs. tip-sample separation, I(s), plots were obtained. Fig. 4:2 shows I(s) plots for 1, 10, and 100 mV tip-sample biases. The natural log of the currents of Fig. 4:2 are plotted in Fig. 4:3, showing a slight deviation from a straight line due to the 1/s term in the current.
The three interrelated variables of the tunneling equation yield three possible tunneling configurations. The first is applying a constant current and servoing s to keep Vb constant. This method was used by the topografiner but
is not used for STM imaging. The second is the normal topographic operation of a scanning tunneling microscope, where Vb is fixed and a constant tip-
sample distance, s, is obtained by servoing the probe to maintain a constant current value, I. STM images produced from topographic data will be called Z-images for the purpose of discussion. The third configuration is the current imaging operation of an STM where