ANALISIS DE CLIENTES

For a mathematical model of the tunneling of electrons through the gap between tip and sample here the classical Bardeen formalism [18, 19] is used. This is based on a perturbation approach to the two systems, tip and sample, that are initially considered isolated. The time-independent Schrödinger equations for single particles belonging to the sample and to the tip are written in equations 2.5 and 2.6, respectively. T is the electron kinetic energy, US and UT are the potential energies and ψµ and χν are the

two wavefunctions. Given the isolation of the two systems, US assumes finite values

only within the sample, while it is defined null elsewhere. Similarly,UT takes non-zero

values in the tip only.

(T+US)ψµ=Eµψµ (2.5)

(T+UT)χν =Eνχν (2.6)

For one electron, the transition probability per time unitωµν from the sample stateψµ

to the tip state χν is then provided via the Fermi golden rule: ωµν = 2

π ~ |Mµν|

2_δ(E

ν −Eµ) (2.7)

where the matrix element is given by

Mµν =

Z

Figure 2.3: Potential energy for a conductor-vacuum-conductor junction. The descriptors used for the two conductive materials, S and T, indicate thetipand thesample, respectively. a) The

two conductors are held at the same potential. b) Conductor S is kept at a positive voltage. c) Conductor S is kept at a negative voltage. Image adapted from [20]

It is worth noting that theδfunction in equation 2.7 implicitly assumes that the electron

tunnels from the tip to the sample between states with equal energy, meaning that it does not account for inelastic tunneling processes. The total current originating from elastic tunneling can then be calculated by multiplying the electron charge e by the

summation over all the possible tip and sample states of the transition probabilityωµν.

Furthermore, the discrete sum over the states can be alternatively expressed with an integral over all the density of statesρ(E), as follows:

X

µν

|Mµν|2δ(Eν −Eµ) = 2

Z

f()ρ() (2.9)

where f() is the Fermi-Dirac function at the energy and the factor 2 accounts for

the spin degeneracy predicted by Pauli’s exclusion principle. Hence, approximating the Fermi-Dirac distribution of electron energies to a step function, assumption that is reasonable at low temperatures (kBT eV), it is possible to write the total current in

terms of the sample and tip densities of states at the shifted Fermi levels, ρS and ρT,

the bias voltageV and the transition matrix elementMµν: I = 4πe ~ Z eV 0 ρT E_FT −eV +ρS E_FS+|Mµν|2d (2.10)

The integration interval chosen in 2.10 is valid when positive voltages for the sample are chosen, while for negative voltages one would integrate over the interval [−e|V|,0]. The

matrix elementMµν is still unknown, and, in order to simplify 2.10, it is assumed that

only the electrons belonging to the extreme atom of the tip contribute to the tunneling process towards the surface atom that lies directly underneath (inset of figure 2.2), which is acceptable for voltage intensities smaller than 2 Volts [21].

In this hypothesis it is possible to proceed with the Wentzel-Kramers-Brillouin (WKB) semiclassical approximation [22], that permits to calculate the tunneling prob- ability, D, of a particle possessing energy through an unknown energetic barrier U

along the coordinate Z, provided that U. The latter condition is generally true

for metallic materials, which possess workfunctions of the order of a few eV.

D() = exp −2 ~ Z d 0 [2 m(U(z)−)]12_dz (2.11) Where dis the distance separating the tip and the sample. In the tunneling process,

the shape of the barrier U(z) is unknown. Generally, it possesses a trapezoidal shape

(figure 2.3), but could be further approximated to a square barrier, defining an effective work function that depends on the applied bias voltage:

φeff=

φS+φT +eV

2 (2.12)

where φS is the work function of the sample and φT is the work function of the tip.

The introduction of an effective work function enables us to reduce equation 2.11 to:

D(, V, d) = exp " −2d r2 m ~2 (φeff −) # = exp (−2kd) (2.13)

with the constantk defined as: k= r 2m ~2 (φeff −) (2.14)

Finally, replacing the tunneling matrix |Mµν|2 in 2.10 with the tunneling probability D, from 2.13, we obtain the following expression for the total tunneling current:

I = 4πe ~ Z eV 0 ρT E_FT −eV +ρS E_FS+exp (−2kd)d (2.15)

From 2.15, the total current flowing from the tip to the sample is determined by the applied voltage V, the distance dbetween the two conductors and the total densities

of states ρS and ρT. Remembering the semiclassical approximation that assumes the

involvement of only two atoms in the tunneling phenomenon, one in the sample and one in the tip [21], as in equation 2.11, the densities of states in equation 2.15 pertain to single atoms and for this reason they are termedlocal density of states(LDOS). It is also

clear, from equation 2.15, that the total tunneling current decreases exponentially with the distance between sample and tip, with an inverse decay length equal to 2k. For a

quantitative estimation of the STM vertical sensitivity, the constantkis approximated

to: k∼= p m(φS+φT) ~ = 0.51 q φS+φT ( eV) Å−1 (2.16) In the approximation 2.16 the bias voltage is assumed small compared to the two work functions (eV φS,φT). Since typical values for metallic work functions are 4-5 eV,

a change of 1 Å in the tip-sample distance provokes an order of magnitude variation in the tunneling probability and, consequently, in the detected tunneling current. This explains the extremely high vertical resolutions achieved in STM experiments , but also the necessity of mechanical isolation systems in a typical STM apparatus, since small vibrations could cause significant oscillations in the flowing tunneling current.