Simulating STM images: a theoretical perspective

A theoretical understanding of the STM images can be obtained in the framework of the perturbation theory. According to this theory, if the sample-tip distance is large enough (more than 4 ˚A) such that the characteristic wavefunctions of the two electrodes do not overlap strongly then one can write the tunneling probability of an electron from a sample state µ (with the energy Eµ) to a tip state ν (with the energy Eν) as being equal to [35]:

w = 2π

~ |Mµν|²δ(Eµ− E^ν) (4.17) This equation is known also as the Fermi golden rule, and Mµν is the tunneling matrix element between the states µ and ν. The presence of the delta Dirac function in (4.17) guarantees that we are dealing with an elastic tunneling process. Inelastic tunneling pro-cesses with the quasi-particles of the electrodes, i.e. phonons, plasmons, etc, could be also taken into account but then the matrix elements would become much more complicated.

Taking only elastic processes makes also sense as the inelastic part of the tunneling current can be estimated to represent about 1 % of the total tunneling current [46].

The total tunneling current can be calculated by taking into consideration all the pos-sible initial and final states of the tip and the sample, inside an energy window established by the presence of a bias voltage V . Thus [46]:

I = 4πe

~ Z _∞

−∞

[f (EF − eV + ε) − f(E^F+ ε)] · ρ^s(EF − eV + ε) · ρ^t(EF+ ε) · |M|²· dε (4.18)

where the Fermi distribution functions f (E) have been used in order to cope with the states occupancy at non-zero temperatures. ρs and ρt are the sample and the tip density of states.

In the limit of small temperatures (i.e. when kBT is smaller than the energy resolution) the Fermi functions in (4.18) can be replaced by their zero-temperatures values, i.e. the

4.4 Scanning tunneling microscopy simulations 135 unit step functions, in which case the term in the square brackets becomes either zero or one:

The equations obtained this way are relatively simple. The only real difficulty one would still have is in the evaluation of the tunneling matrix M .

Bardeen [47] solved this problem by showing that, under certain assumptions, the matrix elements from (4.19) can be expressed as a surface integral, i.e.

M = ~ 2m

Z

Σ

(ν^∗(r)∇µ(r) − µ(r)∇ν^∗(r)) · dS (4.20) where the integral is over any surface Σ lying entirely within the vacuum barrier region which separates the two systems (the sample and the tip).

By using equations (4.19) and (4.20) one can calculate now the tunneling current between two electrodes (e.g. a conductive surface and a tip) provided that an explicit form for the electrodes wavefunctions (ν and µ) is known. When trying to simulate STM experiments however, this might be a problem as the tunneling tip geometric structure and electronic properties are typically poorly known. For the theorist this means that one will have to adopt a somehow arbitrary - yet reasonable - model for the tip, when simulating the STM images. One of the simplest possible model for the tip was proposed by Tersoff and Hamann [48] [49] who assumed the tip to have a spherical geometry (i.e. the tip is approximated by a sphere or a single atom) and its wavefunctions to be symmetric about the center of the sphere, i.e. s-wavefunctions.

The approximation of the whole STM tip through a single atom makes sense due to the fact that the exponential decay of the tunneling current with the tip-sample distance essentially leads to an image of the surface mainly obtained from a single atom, i.e. the most protruded atom of the tip. Therefore this is a reasonable approximation. The further assumption that from the apex atom total wavefunction the main contribution to the tunneling current will come from the spherical symmetric part (i.e. the s-wave) is justified by the simplicity of such an approach which allows one to reduce the STM tip to a mathematical point source of current (i.e. to obtain an ideal tip). This utilization of only the s-wave component to characterize the electronic structure of a real tip can occasionally lead to theoretical conclusions which do not really agree with the experiment.

Thus, for example, the s-orbital tip approximation fails to explain the observed atomic resolution for the close-packed elemental metal surfaces (Al(111), Au(111), Cu(100), etc) [50] [51] [52] [53] [54]. Considering the actual electronic states of a real tip, Chen [55]

[56] has managed to explain this failure through the presence of the localized pz or d_z² states on the real tip. Only through the presence of this non-s-wave components STM can achieved its highly praised atomic resolution.

Adding to the approximation of the spherical symmetry those of small temperature and small voltage (∼ 10 meV) Tersoff and Hamman [TH] have found that the tunneling current between a conductive surface and a tip can be approximated only through the density of states of the sample, evaluated at the center Rt of the spherical tip, i.e.

I(Rt) ∼ const ·

Z EF+eV EF

ρs(Rt, ε) · dε (4.21)

≈ eV · ρs(R_t, E_F) for very small V (4.22) where Rtcorresponds to the center of the outermost atom of an experimental tip (”loca-tion of the tip”). The lineariza(”loca-tion of the integral from (4.21) in order to obtain (4.22) is valid only in the limit of very small voltages.

Thus, according to the TH model, the equi-current contours observed in a constant-current STM experiment correspond to constant contours of the partial electron density of the sample ρs(Rt, E) integrated over a energy interval eV around the Fermi level and evaluated at the tip position Rt. The fact that in the final formulation of the TH method the tip is completely neglected (in the simulations, the density of states of the sample is evaluated at the position of the tip but in the absence of any tip) has both advantages and disadvantages. On the positive side, we can directly relate our STM images with the properties of the bare surface (which is also what the experimentalists want to measure) without having to be concerned with the accuracy of a complicated convolution as it can be the one between the tip and the sample electronic spectra. On the negative side, by excluding from the very beginning any effects which could come from the various states of a real tip, a quantitative interpretation of the STM images in terms of the obtained experimental resolution or of some other properties of the tip-sample system is obviously impossible within this model.

An important drawback of TH approximation is also the fact that the eq.(4.21) can strictly be used only in the limit of small voltages, i.e. of eV ≪ Φ^sample. While this might not be a problem in the studies of simple metallic surfaces, for most semiconductor surfaces or organic layers adsorbed on metallic surfaces the states of interest are many times 2-3 eV below the Fermi level making thus necessary an applied bias voltage which is comparable in magnitude with the sample workfunction. However, as one increases the energy window below the Fermi level a binding energy weighting factor should be taken into account in equation (4.21) because the further a state is from the Fermi level the larger will be the energy barrier seen by it ( Ebarrier−height = (E_state−EF)+Φ_sample = E_binding+Φ_sample). To cope with this exponential decay of the tunneling current with the barrier height Seloni et al. [57] has suggested the introduction of a transmission coefficient weighting factor into the equation (4.21), i.e. :

I(R_t, eV ) ∼ const ·

Z EF+eV EF

ρ_s(R_t, ε) · T (eV, ε) · dε (4.23) where T (eV, ε) has the meaning of the probability of an electron with the energy ε to tunnel across the energy barrier, in the presence of an applied bias voltage equal with eV

An explicit form for the T (eV, ε) factor can be obtained within the WKB theory for the planar tunneling. According to this theory [34] [29] [58]:

T (eV, ε) = e^−2d

√2m

~

qΦsample+Φtip

2 +^eV₂ −ε (4.24)

where d is the sample-tip distance and m the mass of an electron.

The absence of the T (eV, ε) factor from the TH approximation will lead, for large voltages, to a more clear visualization of the low lying electronic states than one can

4.4 Scanning tunneling microscopy simulations 137 obtain in a standard STM experiment. As discussed earlier, in the experiments one can also improve the visibility of the low lying states through the use of the so called modulation techniques. Essentially these involve superimposing a small high-frequency sinusoidal modulation voltage on top of the constant DC bias between the sample and the tip [29]. For low to modest voltages, however, the TH approximation can provide - in spite of its seemingly crude assumptions - an excellent starting point for the interpretation of the STM data.