Scanning tunneling microscopy (STM) of high Tc cuprates

What is a Scanning Tunneling Microscope?

A scanning tunneling microscope (STM) is an incredibly useful device that uses the concept of quantum tunnelling to investigate surfaces using electrons. STM are able to produce a three dimensional profile of the surface of a metal on the atomic level. The STM, which was invented in the early 1980s by Gerd Binnig and Heinrich Rohrer, was of such importance that it was worthy of the Nobel Prize in Physics. For the STM to work, the sharp tip must be moved just slightly above the area of interest. At this point, electrons tunnel quantum mechanically across the gap from the tip to the area, investigating the density of states (DOS) of the object.

Instrumentation

The most important elements of the STM would have to be the scanning tip and the transducers. Tungsten is generally used when constructing the tip, although Platinum-Iridium and Gold are also used. It is crucial that the tip of the STM be extremely sharp (ending in a single atom)to provide the most accurate imaging when used. The piezoelectric transducers play an important role as they allow for independent motion in the x, y, and z planes. Also, spring systems are often used in STM in order to minimize the vibrations caused by the tunnel current. Without these spring systems, it would be very difficult to interpret the data.

Tunneling

As was previously stated, the basic principles of the STM are those of quantum mechanics; specifically tunneling. The idea behind tunneling is that objects of very small mass have a quantifiable probability of passing through other objects due to the wavelike nature in which it moves.

In a one-dimensional case, with the presence of a potential U(z), one can find that the electrons energy levels ψ_n(z) are found by solutions to Schrödinger's Equation,

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\psi _{n}(z)}{\partial z^{2}}}+U(z)\psi _{n}(z)=E\psi _{n}(z)$,

ħ = reduced Planck’s constant, z = position, and m = mass of an electron.

The wave function of the electron may be a traveling wave solution if the electron is incident upon an energy barrier of height U(z)

$\psi _{n}(z)=\psi _{n}(0)e^{\pm ikz}$,

with

$k={\frac {\sqrt {2m(E-U(z))}}{\hbar }}$

if E > U(z), which is true for a wave function inside the tip or inside the sample. Inside a barrier, such as between tip and sample, E < U(z) so the wave functions which satisfy this are decaying waves,

$\psi _{n}(z)=\psi _{n}(0)e^{\pm \kappa z}$,

where

$\kappa ={\frac {\sqrt {2m(U-E)}}{\hbar }}$

quantifies the decay of the wave inside the barrier, with the barrier in the +z direction for $-\kappa$ .

Knowing the wave function allows one to calculate the probability density for that electron to be found at some location. In the case of tunneling, the tip and sample wave functions overlap such that when under a bias, there is some finite probability to find the electron in the barrier region and even on the other side of the barrier. Let us assume the bias is V and the barrier width is W, as illustrated in Figure 1. This probability, P, that an electron at z=0 (left edge of barrier) can be found at z=W (right edge of barrier) is proportional to the wave function squared,

$P\propto |\psi _{n}(0)|^{2}e^{-2\kappa W}$ .

If the bias is small, we can let U − E ≈ φM in the expression for κ, where φM, the work function, gives the minimum energy needed to bring an electron from an occupied level, the highest of which is at the Fermi level (for metals at T=0 kelvins), to vacuum level. When a small bias V is applied to the system, only electronic states very near the Fermi level, within eV, are excited. These excited electrons can tunnel across the barrier. In other words, tunneling occurs mainly with electrons of energies near the Fermi level.

However, tunneling does require that there is an empty level of the same energy as the electron for the electron to tunnel into on the other side of the barrier. It is because of this restriction that the tunneling current can be related to the density of available or filled states in the sample. The current due to an applied voltage V (assume tunneling occurs sample to tip) depends on two factors: 1) the number of electrons between E_f and eV in the sample, and 2) the number among them which have corresponding free states to tunnel into on the other side of the barrier at the tip. The higher density of available states the greater the tunneling current. When V is positive, electrons in the tip tunnel into empty states in the sample; for a negative bias, electrons tunnel out of occupied states in the sample into the tip.

Mathematically, this tunneling current is given by

$I\propto \sum _{E_{f}-eV}^{E_{f}}|\psi _{n}(0)|^{2}e^{-2\kappa W}$.

One can sum the probability over energies between E_f − eV and eV to get the number of states available in this energy range per unit volume, thereby finding the local density of states (LDOS) near the Fermi level. The LDOS near some energy E in an interval ε is given by

$\rho _{s}(z,E)={\frac {1}{\epsilon }}\sum _{E-\epsilon }^{E}|\psi _{n}(z)|^{2}$,

and the tunnel current at a small bias V is proportional to the LDOS near the Fermi level, which gives important information about the sample. It is desirable to use LDOS to express the current because this value does not change as the volume changes, while probability density does. Thus the tunneling current is given by

$I\propto V\rho _{s}(0,E_{f})e^{-2\kappa W}$

where ρ_s(0,E_f) is the LDOS near the Fermi level of the sample at the sample surface. By using equation (6), this current can also be expressed in terms of the LDOS near the Fermi level of the sample at the tip surface,

$I\propto V\rho _{s}(W,E_{f})V$

The exponential term in (9) is very significant in that small variations in W greatly influence the tunnel current. If the separation is decreased by 1 Ǻ, the current increases by an order of magnitude, and vice versa.

This approach fails to account for the rate at which electrons can pass the barrier. This rate should affect the tunnel current, so it can be accounted for by using Fermi’s Golden Rule with the appropriate tunneling matrix element. John Bardeen solved this problem in his study of the metal-insulator-metal junction, MIM. He found that if he solved Schrödinger’s equation for each side of the junction separately to obtain the wave functions ψ and χ for each electrode, he could obtain the tunnel matrix, M, from the overlap of these two wave functions. This can be applied to STM by making the electrodes the tip and sample, assigning ψ and χ as sample and tip wave functions, respectively, and evaluating M at some surface S between the metal electrodes at z=zo, where z=0 at the sample surface and z=W at the tip surface.

Now, Fermi’s Golden Rule gives the rate for electron transfer across the barrier, and is written

$w={\frac {2\pi }{\hbar }}|M|^{2}\delta (E_{\psi }-E_{\chi })$,

where δ(Eψ-Eχ) restricts tunneling to occur only between electron levels with the same energy. The tunnel matrix element, given by

$M={\frac {\hbar }{2\pi }}\int _{z=z_{0}}(\chi *{\frac {\partial \psi }{\partial z}}-\psi {\frac {\partial \chi *}{\partial z}})dS$,

is a description of the lower energy associated with the interaction of wave functions at the overlap, also called the resonance energy.

Summing over all the states gives the tunneling current as

$I={\frac {4\pi e}{\hbar }}\int _{-\infty }^{+\infty }[f(E_{f}-eV)-f(E_{f}+\epsilon )]\rho _{s}(E_{f}-eV+\epsilon )\rho _{T}(E_{f}+\epsilon )|M|^{2}d\epsilon$,

where f is the Fermi function, ρ_s and ρ_T are the density of states in the sample and tip, respectively. The Fermi distribution function describes the filling of electron levels at a given temperature T.