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.

STM of High Tc Cuprates

According to the BCS theory, the degenerate electron gas can be unstable against pairing of electrons with opposite spin and momentum, if an attractive electron–electron interaction potential exists [1]. The condensation of these Cooper pairs leads to the superconducting ground state. In conventional superconductors, the pairing potential has typically s-wave symmetry. In the superconducting state, a gap forms in the density of states (DOS) of electrons. Virtually no electronic states are available inside of this superconducting gap at zero temperature. Outside of this gap, new excited states called Bogoliubov quasiparticles appear. These ‘dressed-electron’ states near the Fermi level have a DOS and dispersion-relation dramatically different from those of electrons in the normal metallic state. We can use scanning tunneling spectroscopy to measure the directly local density of states of the Bogoliubov quasiparticles. Also, using the STM’s spatial resolution w1 A° , we can get information about local variations of DOS (LDOS). This is possible at low temperature when vibrational and acoustic noise is greatly diminished

Mott Insulators

There are many cuprate high-TC superconductors falling into families of structures with different numbers of immediately adjacent Cu–O planes. La2-xXxCuO4(XZ Ba2C, Sr2C, Ca2C etc.), YBa2Cu3O6Cx, Bi2Sr2CaCu2O8Cx, and Tl2Ba2Ca2Cu3O10, HgBa2Ca3Cu4O, and recent Ca2-x NaxCuO2Cl2 (chronological order of discoveries) are among the most studied [3]. Both experimentalists and theorists agree that superconductivity and charge transport are confined to the CuO2 planes. The Cu atoms are believed to be in the Cu2C3d9 configuration with spin half. As shown in Fig. 1A, the dominating electronic state is an antiferromagnetic Mott insulator due to the fact that strong Coulomb repulsion prevents electron hopping from Cu to Cu, and because the anti-ferromagnetic order is energetically favorable. As we remove some spins from these sites, hopping of electrons between Cu and Cu becomes possible (Fig. 1B). In this fashion, new electronic ordered states emerge as we dope the insulating CuO2 layer with holes or electrons. High temperature superconductivity is one of them. These high-TC superconductors have many physical properties different from the conventional superconductors. Pairing symmetry of cuprates is believed to exhibit dx2Ky2 symmetry [4,5], compared to s-wave like pairing symmetry in conventional superconductors. The highly anisotropic critical current (JC), very high critical field (HC), and the short coherence length (x) are all extremely different from conventional superconductors. Especially significant is that almost all the physical length scales are w1 nm: Cu–Cu distance is around 0.38 nm, superconducting coherence length x is 1–2 nm, inter-dopant atomic distance Lw1.5 nm, and the Fermi wavelength lFw1 nm. These similar length scales imply that different electronic phenomena can interact strongly with each other, unlike in usual metals where those length scales differ by orders of magnitudes. A typical schematic phase diagram of the hole-doped high- TC cuprates is shown in Fig. 2. Around the hole doping level (p) of 0.16, TC reaches its maximum value. One interesting thing about this diagram is that a mysterious ‘pseudogap’ phase [6] exists below the line T*, above the superconducting dome.

SI-STM

We have developed several novel spectroscopic imaging techniques to investigate these highly complicated materials. First, atomic resolution local density of states (LDOS) imaging was introduced to cuprates [7]. To make energy-resolved LDOS maps, we measure differential conductance gððr; VÞhðdI=dVÞjðr ;V between the STM tip and the sample, with spatial (ðr) resolution wA ° /pixel, and energy resolution w1 meV /point. Due to the relation LDOSððr; EZeVÞfgððr; VÞ, this map is actually a spatial image of the LDOS at energy E : LDOSððr; EÞ. Second, we introduced the concept of the gap map where the spatial dependence of the superconducting gap value DððrÞ is mapped out with atomic resolution [8]. Yet another technique we recently developed for cuprate studies is Fourier Transform Scanning Tunneling Spectroscopy (FT-STS) [9]. To achieve FT-STS, one needs large field of view (FOV), as well as high resolution LDOSððr; EÞ map to get good resolution in k-space, and at the same time a jkj-range up to the first brillouin zone. After performing 2D-FFT (Fast Fourier Transform) of LDOSððr; EÞ, we can identify ðq-vectors of spatial modulations in LDOSððr; EÞ by the locations of peaks in LDOSððk; EÞ-the Fourier transform magnitude of LDOSððr; EÞ. This technique has been exceptionally fruitful [10] in relating the atomic-scale ðr-space electronic structure to that in ðk-space. Finally, a fourth technique is atomically registered LDOS subtraction, wherein the perturbations to the background LDOS by an external magnetic field can be determined even in a strongly disordered environment by subtracting the original LDOS measured before application of the field

LDOS Spectrum

Bi2Sr2CaCu2O8Cx single crystals can cleave easily at the weakest link between BiO layers to reveal a relatively flat BiO surface. Therefore, the CuO2 plane is only 5 A ° below from the top surface, separated from the tip by insulating BiO and SrO layers which act to protect the CuO2 plane of interest. Partly due to this remarkable property, most of the studies described here have been performed on Bi2Sr2- CaCu2O8Cx. The Cu atoms are located about 5 A ° below Bi atoms. Fig. 3A shows a topographic image of a 150 A ° square field of view (FOV) of the BiO surface. A clear Bi atomic grid can be seen, but no O atoms are visible due to their LDOS vanishing near EF. The appearance of dark and bright spots is a real effect of the nanoscale inhomogeneity in superconducting Bi2Sr2CaCu2O8Cx [12,13] and not from surface damage. Also along the b-axis, a three-dimensional supermodulation [14] of w26 A ° produces brighter corrugations of the surface, which is due to the vertical displacement of the atoms from their ideal orthorhombic lattice sites. As mentioned, Bi2Sr2CaCu2O8Cx is believed to show dx2Ky2 like pairing potential (gap) symmetry [4,5]. A typical LDOS spectrum measured on the BiO plane of optimally doped Bi2Sr2CaCu2O8Cx is shown in Fig. 3B. This form of function is the central observable of our studies, revealing many of the attributes expected for the dx2Ky2 gap structure. We can immediately point out two energy regions of interest, as colored blue and purple, respectively. The same color scheme is used in Fig. 3C, indicating the regions of the ‘contours of constant energy (CCE)’ in momentum-space associated with the two energy ranges of Fig. 3B: jEj! 25 meV and 25 meV%jEj%75 meV. Also in Fig. 3C, dx2Ky2 superconducting band structure [15] is illustrated in which CCE are plotted in ðk-space.

LDOS modulations

At energies jEj!25 meV, complex patterns of modulations appear in LDOS of Bi2Sr2CaCu2O8Cx. The wavelengths of these modulations are of the order of several unit cells as shown in Fig. 4A. To find the wave vectors associated with these patterns, 2D-FFT analysis proved to be very useful. In Fig. 4B, sets of clearly discernible wave vectors are shown. The brightest spots in the corners in Fig. 4B are the 2D-Bragg peaks. 16 sets of ðq’s have been identified and some of them are illustrated in Fig. 4C [10]. Since we carry out this process as a function of electron energy E, we dubbed this analysis as Fourier Transform Scanning Tunneling Spectroscopy (FT-STS). As we analyze these data within a scattering-induced quasiparticle interference model [16], the ðq-vectors can be consistent [10] within the Fermi-surface and energy gap pictures in agreement with photoemission studies [17] of dx2Ky2 cuprate superconductors. Namely, elastic scattering occurs between the points of high joint density of states at a given energy (red dots), leading to a set of specific interference ðq-vectors responsible for the observed standing waves, as shown in Fig. 4C. Most likely scattering centers responsible for these modulations are related to weak scattering potentials from out-of-plane dopant atoms

nanoscale electronic disorder

Doping dependence of gap maps or DððrÞ [12,13] shows disorder of another kind. Fig. 5 shows DððrÞ maps of Bi2Sr2CaCu2O8Cx in a 60!60 nm FOV at dopings ranging from w0.19 to w0.1. The identical color scale is used for the D values in all four gap-maps. These gap-maps reveal intense electronic disorder, and the gap values range from 20 meV to over 70 meV, with w3 nm big domain-like area of the same D value. The mean gap value rises as the doping falls, in agreement with other probes [15]. All the regions in these disordered images share the same low energy quasiparticles as judged from FT-STS studies [19]. These effects were completely unexpected. By now they have been widely observed [8,10,12,13,20–23]. The source of these electronic disorder phenomena in cuprates is still unknown. To tackle these critical issues, SI-STM studies of the electronic disorder and dopant atom disorder, and their correlations are currently underway.

Checkerboard’ states in vortex core and underdoped region

The STM study of the electronic structure of the cuprate vortex core [11] initiated the interest of the ‘checkerboard’ states which recently became the focus of much theoretical research on under-doped cuprates [24–34]. It was shown that the pseudogap-like conductance spectrum-V shaped spectra without coherence peaks—in the vicinity of the vortex core of the Bi2Sr2CaCu2O8Cx (Fig. 6A) appear with a checkerboard incommensurate low energy LDOS modulations with associated ðq vectors of (G2p/4.3a0,0), (0G2p/ 4.3a0)G15% (Fig. 6B) The w4a0 periodicity in the direction of the Cu–O bonding of the vortex-induced LDOS modulations are believed to be caused by a electronic phase which can become stabilized at the vortex cores when high-TC superconductivity is suppressed. This discovery is very important because it shows one can access the unknown non-superconducting state in high-TC cuprates by suppressing the superconductivity near the vortex core, and also shows clear checkerboard electronic crystal characteristics. The natural question to follow is, therefore, ‘Is the checkerboard state a precursor to the high-TC superconducting phase?’ Recently, a possibly related checkerboard modulation in LDOS is observed in strongly underdoped Bi2Sr2CaCu2O8Cx [19]. To get around the nanoscale disorder of Bi2Sr2CaCu2- O8Cx, we chose another member of high-TC cuprate family to study. Ca2-xNaxCuO2Cl2 is high-TC superconductor, whose parent material Ca2CuO2Cl2 is a canonical Mott insulator [35].We have studied this Ca2-xNaxCuO2Cl2 at the doping level from 8%(TCZ0)!x!12%(TCZ20 K), with the SI-STM technique we described in previous sections. Below about 100 mV, a V-shaped energy gap around the Fermi level emerges in the LDOS, as shown in Fig. 6C. This gap is particle-hole symmetric, and within the gap, strong non-dispersive electronic modulations of an exact commensurate 4a0!4a0 checkerboard are observed (Fig. 6D). We observed this phenomena throughout the doping range studied. This discovery is also consistent with a crystalline electronic state existing at low hole doping in the zero temperature pseudogap regime of Bi2Sr2CaCu2O8Cx [19]. This checkerboard electronic crystal may be a new precursor phase between the Mott insulating phase and the superconductivity.

End Result

Due to the novel spectroscopic imaging STM (SI-STM) techniques, numerous unexpected atomic scale electronic structures have been brought to light. Among these significant discoveries are nanoscale gap disorder, quasiparticle interference modulation and ‘checkerboard’ states. Also, the effects of dopant atoms on the nanoscale disorder, and the relation of the checkerboard state to the superconductivity have become new foci of interest in macroscopic quantum physics of high-TC cuprates.

References

Hook, J.R. and H.E. Hall. Solid State Physics Second Edition. West Sussex: John Wiley & Sons, 2008.

Lee, Jinho, James A. Slezak and J.C. Davis. "Spectroscopic Imaging STM Studies of High-Tc Superconductivity." Journal of Physics and Chemistry of Solids (2005): 1370-1375.