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.

In the area between the STM and the material,E < U(z) so the wave functions which satisfy this are decaying waves,

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

with

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

measuring the decay of the wave inside the barrier.

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.

Superconductors

Superconductivity is a phenomenon occurring in certain materials generally at very low temperatures, characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field (the Meissner effect). Like ferromagnetism and atomic spectral lines, superconductivity is a quantum mechanical phenomenon. It cannot be understood simply as the idealization of "perfect conductivity" in classical physics.

The electrical resistivity of a metallic conductor decreases gradually as the temperature is lowered. However, in ordinary conductors such as copper and silver, impurities and other defects impose a lower limit. Even near absolute zero a real sample of copper shows a non-zero resistance. The resistance of a superconductor, despite these imperfections, drops abruptly to zero when the material is cooled below its "critical temperature". An electric current flowing in a loop of superconducting wire can persist indefinitely with no power source. [1]

Superconductivity occurs in a wide variety of materials, including simple elements like tin and aluminium, various metallic alloys and some heavily-doped semiconductors. Superconductivity does not occur in noble metals like gold and silver, nor in pure samples of ferromagnetic metals.

In 1986 the discovery of a family of cuprate-perovskite ceramic materials known as high-temperature superconductors, with critical temperatures in excess of 90 kelvin, spurred renewed interest and research in superconductivity for several reasons. As a topic of pure research, these materials represented a new phenomenon not explained by the current theory. In addition, because the superconducting state persists up to more manageable temperatures, past the economically-important boiling point of liquid nitrogen (77 kelvin), more commercial applications are feasible, especially if materials with even higher critical temperatures could be discovered.

High Tc Cuprates

Cuprate superconductors are generally considered to be quasi-two-dimensional materials with their superconducting properties determined by electrons moving within weakly coupled copper-oxide (CuO2) layers. Neighbouring layers containing ions such as La, Ba, Sr, or other atoms act to stabilize the structure and dope electrons or holes onto the copper-oxide layers. The undoped 'parent' or 'mother' compounds are Mott insulators with long-range antiferromagnetic order at low enough temperature. Single band models are generally considered to be sufficient to describe the electronic properties.

The cuprate superconductors adopt a perovskite structure. The copper-oxide planes are checkerboard lattices with squares of O2− ions with a Cu2+ ion at the centre of each square. The unit cell is rotated by 45° from these squares. Chemical formulae of superconducting materials generally contain fractional numbers to describe the doping required for superconductivity. There are several families of cuprate superconductors and they can be categorized by the elements they contain and the number of adjacent copper-oxide layers in each superconducting block. For example, YBCO and BSCCO can alternatively be referred to as Y123 and Bi2201/Bi2212/Bi2223 depending on the number of layers in each superconducting block (n). The superconducting transition temperature has been found to peak at an optimal doping value (p = 0.16) and an optimal number of layers in each superconducting block, typically n = 3. A small sample of the high-temperature superconductor BSCCO-2223.

Possible mechanisms for superconductivity in the cuprates are still the subject of considerable debate and further research. Certain aspects common to all materials have been identified.[4] Similarities between the antiferromagnetic low-temperature state of the undoped materials and the superconducting state that emerges upon doping, primarily the dx2-y2 orbital state of the Cu2+ ions, suggest that electron-electron interactions are more significant than electron-phonon interactions in cuprates – making the superconductivity unconventional. Recent work on the Fermi surface has shown that nesting occurs at four points in the antiferromagnetic Brillouin zone where spin waves exist and that the superconducting energy gap is larger at these points. The weak isotope effects observed for most cuprates contrast with conventional superconductors that are well described by BCS theory.

Similarities and differences in the properties of hole-doped and electron doped cuprates:

   * Presence of a pseudogap phase up to at least optimal doping.
   * Different trends in the Uemura plot relating transition temperature to the superfluid density. The inverse square of the London penetration depth appears to be proportional to the critical temperature for a large number of underdoped cuprate superconductors, but the constant of proportionality is different for hole- and electron-doped cuprates. The linear trend implies that the physics of these materials is strongly two-dimensional.
   * Universal hourglass-shaped feature in the spin excitations of cuprates measured using inelastic neutron diffraction.
   * Nernst effect evident in both the superconducting and pseudogap phases.