Theoretical Background

Before starting this experiment, you must be familiar with the following concepts:

• The Schrödinger equation
• Tunneling phenomena
• The electronic structure of metals (density of states, Fermi level)
• Work function
• Physical properties and crystal structure of graphite
• Piezo-driven actuators

Note that the following section is intended as introductory reading only. To carry out the experiment you must read the literature listed at the end of this document.

The principle of the STM is based on the tunneling of electrons through a potential barrier. Quantum mechanics predicts that there is a certain probability for particles to pass through a potential barrier even if their kinetic energy $E_{kin}$ is lower than the barrier height $V_{0}$. It is said that some particles tunnel through the barrier. Not only electrons have the ability to undergo tunnelling: another well-known example is the emission of $\alpha$-radiation from radioactive nuclei.

Qualitatively, the tunneling of electrons from the surface to the scanning probe tip is based on the application of the one-dimensional stationary Schrödinger equation for an electron with a total energy $E_{tot}$ which shall be assumed to be lower than the height a potential barrier $V_{0}$ (see fig. 2). It can be shown [3] that the wave functions in the regions labelled A, B and C satisfy the expressions

Figure 2: Further information concerning the tunneling of electrons

$$\psi_{A}(x) = A \cdot e^{ik_{1}x} + B \cdot e^{-ik_{1}x}$$ (1)

$$\psi_{B}(x) = C \cdot e^{k_{2}x} + D \cdot e^{-k_{2}x}$$ (2)

$$\psi_{C}(x ) = E \cdot e^{ik_{1}x} + F \cdot e^{-ik_{1}x}$$ (3)

with $k=\frac{1}{\hbar}\sqrt{2mE}$ and $l=\frac{1}{\hbar}\sqrt{2m(V_{0}-E)}$. The constants A to F are determined by the boundary conditions that the wavefunction and its first derivative must be continuous at $x=0$ and $x=d$. $\psi_A$(x) and $\psi_C$(x) represent the oscillatory solutions of the free electron( $e^{ikx} = cos(kx) + isin(kx)$) while the solution in the classically forbidden region ($\psi_B$(x)) consists of an (exponentially) increasing and decreasing component. The transmission coefficient T giving the probability that the potential barrier is penetrated is then given by

$$T = \frac{\vert F\vert^{2}}{\vert A\vert^{2}} \simeq e^{-\frac{2d}{\hbar}\sqrt{2m(V_{0}-E)}}$$ (4)

Assuming that the applied voltage between the STM tip and the sample is small, i.e.,

• $eU_{T} \ll V_{0} = \phi$ and
• D(E) (the density of states)

  being constant in the range from E$_{f}$ to E$_{f}$-eU$_{T}$

,

the tunneling current $I_{T}$ can be approximated by

$$I_{T} \sim U_{T} \cdot e^{-\frac{2d}{\hbar}\sqrt{2m\phi}}$$ (5)

with d being the distance between tip and sample and $\phi$ being the ``effective work function'' $\phi = \frac{1}{2}(\phi_{1}+\phi_{2})$.

Fig. 3 shows a schematic energy diagram for the tunneling of electrons between two metals.

Figure 3: Tunneling of Electrons