Experimental Techniques
2.1 The Theory of STM
2.1.1 Quantum Tunnelling
STM requires a sharp conducting tip with an applied potential bias, posi-tioned very close to a conducting surface so an electrical current will flow between
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vacuum.
Figure 2.1: Diagram showing the tunnelling of an electron through a 1D potential barrier, for UHV-STM the potential barrier is due to the vacuum gap between the tip and the sample. Left axis is the wavefunction or energy for the
electron or potential barrier respectively.
The quantum-mechanical tunnelling example illustrated in Figure 2.1 can be analysed using the 1D (Z direction in this case) time-independent form of the Schr¨odinger Equation, shown in Equation 2.1.
HΨ(z) = −b ~2 2m
d2Ψ(z)
dz2 + V (z)Ψ(z) = EΨ(z) (2.1) H is the Hamiltonian operator, Ψ is the electron wavefunction, m is the massb of the electron, −2m~2 ddz2Ψ2 is the kinetic energy of the electron, V (z) is the potential and E is the total energy of the electron. The Schr¨odinger Equation can be solved for the three regions shown in Figure 2.1 when the solution takes the form of an
Chapter 2. Experimental Techniques 21
exponential wave [52]. In the ‘tip’ region there are two solutions, for the electron moving in either a positive or negative direction, as shown in Equation 2.2.
Ψ(z) = Ae±ikz (2.2)
The wave vector of the electron, k, can be described as shown in Equation 2.3.
k =p
2m | E |/~ (2.3)
The solution of Equation 2.1 for the classically forbidden ‘vacuum’ region is described in Equation 2.4 for an electron penetrating through the potential barrier in the +z direction, where Equation 2.5 is the decay constant, κ.
Ψ(z) = Be−κz (2.4)
κ =p
2m | V0− E |/~ (2.5)
The probability of an electron being transmitted from the ‘tip’ region and through the barrier to the ‘sample’ region is termed the transmission coefficient, T , and is determined by the width of the barrier, a, as shown in Equation 2.6.
T ∝ e−2κa (2.6)
The flow of electrons through the barrier decreases exponentially with increas-ing separation, a – as a increases, the amplitude of the wavefunction in the third region decreases. Therefore, the ‘tunnel current’ that occurs when an STM tip and sample (with an applied potential between them) are separated by less than 1 nm also decreases exponentially with increasing separation.
The theoretical model illustrated in Figure 2.1 can be envisioned as a metal-vacuum-metal junction. Therefore the potential barrier, V0, can be approximated as the work function, φ, of a metal surface. The work function is the minimum energy required to remove an electron from the bulk of a metal to the vacuum level.
If we neglect thermal excitation, the Fermi level is the upper limit of the occupied states in a metal and becomes equal in magnitude to the work function, Ef = −φ.
Figure 2.2 illustrates this relation between the work function and Fermi energy for a metal-vacuum-metal junction, where the tip and sample are treated as the same metal and therefore their work functions are equal. Electrons can tunnel from the tip to the sample or from the sample to the tip under these conditions. However, with an applied bias voltage a net tunnelling current occurs – in Figure 2.2 this is
Figure 2.2: The effect of a positive bias voltage applied to a metal sample positioned close to a metal tip with the same work function, φ, in vacuum. The
value En is the energy level of an unoccupied sample state.
As shown in Figure 2.2, when an unoccupied sample state with energy En lies between Ef−eV and Ef, an electron may tunnel into the sample from an occupied state in the tip. If eV is very small compared to φ, then φ ≈| En | and the decay constant can be arranged in terms of the work function (which is a function of the applied bias) as shown in Equation 2.7.
κ =
√2mφ
~ = 5.1p
φ(eV ) (2.7)
If we substitute Equation 2.7 into Equation 2.6, the transmission co-efficient, T , is given in another form – as shown in Equation 2.8, where the units of a are nanometres.
T ∝ e10.2√
φ(eV )a (2.8)
The tunnel current is therefore determined by the two work functions of the materials involved, the applied bias and the separation distance between the STM tip and sample. However, in this idealised treatment of STM the tunnelling mech-anism is assumed to be 1D.
Chapter 2. Experimental Techniques 23
Bardeen/Tersoff-Hamann Model
A more complex treatment of the STM tip and sample states can be derived for large tip-sample separations. Under these conditions, the wavefunctions of the tip and sample may be considered as weakly coupled. The Bardeen theory of tunnelling initially treats the 1D model where Equation 2.1 becomes Equation 2.9 for the combined tip-sample system and Vµ and Vν denote the potentials of the tip and surface states respectively.
The wavefunctions ψµ and ψν originate from the Hamiltonian for the tip and surface respectively and therefore neither wavefunction is an eigenfunction of the Hamiltonian of the combined system. Using the Bardeen tunnelling theory, the two sets of wavefunctions are approximately orthogonal [52].
Z
ψµ∗ψνd3r ∼= 0 (2.10) If the Z direction is defined as in Figure 2.1 and zoas the centre of the potential barrier between the tip and the surface (where z > zo describes the sample side of the barrier) a tunnelling matrix element, Mµν, can be constructed by substituting Equation 2.9 into Equation 2.10.
This is assuming the condition of elastic tunnelling of electrons from a state in one system and through the barrier, into a state of another system of the same energy value, Eν = Eµ. Bardeen has shown that this can be approximated as Equation 2.12 [52, 53].
For the tip, the number of available states is defined by the density of states of the tip at energy E, ρµ(E), and the energy interval defined by the bias voltage V . If the density of states of both the tip and sample do not vary appreciably near the Fermi level, EF, and the limits of small voltages and temperature are taken, the tunnelling current, It, can be defined as in Equation 2.13 [52, 53].
the Tersoff-Hamann model, STM topography images would reveal the probability density of the sample states rather than a convolution of the sample and tip states.
Figure 2.3: The Tersoff-Hamann model of STM, where the tip is modelled as a locally spherical potential well with radius of curvature R, centred on ro.
As shown in Figure 2.3, the Tersoff-Hamann model defines an STM tip centred on ro = (0, 0, zr) with a radius of curvature of R (the Z direction extends perpen-dicularly from the surface). If the tip wavefunctions are arbitrarily localised, then the matrix element is simply proportional to the amplitude of ψν at the position ro of the probe and Equation 2.13 reduces to Equation 2.14 [54].
It ∝X
ν
|ψν(ro)|2δ(Eν− EF) = ρν(ro, EF) (2.14) Thus the tunnel current is proportional to the surface LDOS at the position of the point probe, and STM images are a contour map of constant surface LDOS.
There are limitations to this model, because in reality the surface probed by an STM tip is not solely imaged by the s-wavefunction of the atom at the STM tip apex. This is because transition metals with associated d-orbitals are commonly
Chapter 2. Experimental Techniques 25
used for the STM tip. However, for metals with a work function of approximately 5 eV, the contribution of non-spherical tip wavefunctions to the tunnel current can be neglected for surface feature sizes greater than 0.3 nm [52]. The Tersoff-Hamann model also breaks down if the weak coupling limit is not valid, as is the case when the tip and sample become too close and the tunnel current exceeds several nanoamps.
This model is a very powerful tool for interpreting STM images of metal sur-faces, and when the length scale of features on a flat surface is approximately 1 nm or greater there is a good agreement with experimental results.