Bardeen theory

Fig. 2.10. Different approximate methods for the square harrier problem. (Parameters used W= 2 A = 4 eV Utf= 16 eV.) The original Bardeen theory breaks down when the barrier top comes close to the energy level. The modified Bardeen tunneling theory is accurate with separation surfaces either centered L = lV/2) or off-centered L = VV73). By approximating the distortion of wavefunctions using Green s functions, the error in the entire region is only a few percent.

The surface states observed by field-emission spectroscopy have a direct relation to the process in STM. As we have discussed in the Introduction, field emission is a tunneling phenomenon. The Bardeen theory of tunneling (1960) is also applicable (Penn and Plummer, 1974). Because the outgoing wave is a structureless plane wave, as a direct consequence of the Bardeen theory, the tunneling current is proportional to the density of states near the emitter surface. The observed enhancement factor on W(IOO), W(110), and Mo(IOO) over the free-electron Fermi-gas behavior implies that at those surfaces, near the Fermi level, the LDOS at the surface is dominated by surface states. In other words, most of the surface densities of states are from the surface states rather than from the bulk wavefunctions. This point is further verified by photoemission experiments and first-principles calculations of the electronic structure of these surfaces. 

Superconductivity The physical state in which all resistance to the flow of direct-current electricity disappears is defined as superconductivity. The Bardeen-Cooper-Schriefer (BCS) theoiy has been reasonably successful in accounting for most of the basic features observed of the superconducting state for low-temperature superconductors (LTS) operating below 23 K. The advent of the ceramic high-temperature superconductors (HTS) by Bednorz and Miller (Z. Phys. B64, 189, 1989) has called for modifications to existing theories which have not been finahzed to date. The massive interest in the new superconductors that can be cooled with liquid nitrogen is just now beginning to make its way into new applications. 

The electronic theory of metallic superconduction was established by Bardeen, Cooper and Schrieffer in 1957, but the basis of superconduction in the oxides remains a battleground for rival interpretations. The technology of the oxide ( high-temperature ) superconductors is currently receiving a great deal of attention the central problem is to make windable wires or tapes from an intensely brittle material. It is in no way a negative judgment on the importance and interest of these materials that they do not receive a detailed discussion here it is simply that they do not lend themselves to a superficial account, and there is no space here for a discussion in the detail that they intrinsically deserve. 

J. Bardeen (Urbana), L. N, Cooper (Providence) and J, R. SchriefFer (Philadelphia) theory of superconductivity, usually called the BCS theory. 

Some years later a more thorough discussion of the motion of pairs of electrons in a metal was given by Cooper,7 as well as by Abrikosov8 and Gor kov,9 who emphasized that the effective charge in superconductivity is 2e, rather than e. The quantization of flux in units hc/2e in superconducting metals has been verified by direct experimental measurement of the magnetic moments induced in thin films.10 Cooper s discussion of the motion of electron pairs in interaction with phonons led to the development of the Bardeen-Cooper-Schrieffer (BCS) theory, which has introduced great clarification in the field of superconductivity.2... 

The jump in ce is due to the fact that the superconducting metal has a new degree of freedom, i.e. the possibility of entering the superconducting state. For simple superconductors, such as A1 and Sn, the Bardeen-Cooper-Schrieffer (BCS) theory [18-22] gives ... 

Bardeen, Cooper, and Schrieffer (BCS) theory, 23 804, 836 Bareboat charters, 25 327 Barex, composition of, 3 386t... 

In a superconducting system, when one increases the temperature at a given chemical potential, thermal motion will eventually break up the quark Cooper pairs. In the weakly interacting Bardeen-Copper-Schrieffer (BCS) theory, the transition between the superconducting and normal phases is usually of second order. The ratio of the critical temperature TcBCS to the zero temperature value of the gap AbGS is a universal value [18]... 

Bardeen, Leon Cooper and John Robert Schrieffer, which, from their initials, was called BCS theory. 

Another interesting application of the total energy approach involves superconductivity. For conventional superconductors, the 1957 theory of Bardeen, Cooper and Schrieffer [26] has been subject to extensive tests and has emerged as one of the most successful theories in physics. However, because the superconducting transition temperature Tc depends exponentially on the electron-phonon coupling parameter X and the electron-electron Coulomb parameter p, it has been difficult to predict new superconductors. The sensitivity is further enhanced because the net attractive electron-electron pairing interaction is proportional to X-p, so when these parameters are comparable, they need to be determined with precision. 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

Fig. 1.20. The Bardeen approach to tunneling theory. Instead of solving the Schrddinger equation for the coupled system, a, Bardeen (1960) makes clever use of perturbation theory. Starting with two free subsystems, b and c, the tunneling current is calculated through the overlap of the wavefunctions of free systems using the Fermi golden rule.

In this Chapter, we discuss a perturbation theory for STM, the modified Bardeen approach (MBA). The illustrate the concept of a perturbation approach, let us consider the following four regimes of interactions (zo denotes the microscopic tip-sample distance) ... 

The polarization, or the van der Waals interaction, can be accounted for by a stationary-state perturbation theory, effectively and accurately. The exchange interaction or tunneling can be treated by time-dependent perturbation theory, following the method of Oppenheimer (1928) and Bardeen (1960). In this regime, the polarization interaction is still in effect. Therefore, to make an accurate description of the tunneling effect, both perturbations must be considered simultaneously. This is the essence of the MBA. 

Fig. 2.6. Quantum transmission through a thin potential harrier. From the semi-classical point of view, the transmission through a high barrier, tunneling, is qualitatively different from that of a low barrier, ballistic transport. Nevertheless, for a thin barrier, here W = 3 A, the logarithm of the exact quantum mechanical transmission coefficient (solid curve) is nearly linear to the barrier height from 4 eV above the energy level to 2 eV below the energy level. As long as the barrier is thin, there is no qualitative difference between tunneling and ballistic transport. Also shown (dashed and dotted curves) is how both the semiclassical method (WKB) and Bardeen s tunneling theory become inaccurate for low barriers.

As the semiclassical tunneling theory and Bardeen s original approach become inaccurate for potential barriers close to or lower than the energy level, the validity range of the MBA is much wider. In this subsection, the accuracy of the MBA is tested against an exactly soluble case, that is, the one-dimensional transmission through a. square barrier of thickness W=2 A (see Fig. 2.9). 

As we have discussed in Chapter 2, a direct consequence of the Bardeen tunneling theory (or the extension of it) is the reciprocity principle If the electronic state of the tip and the sample state under observation are interchanged, the image should be the same. An alternative wording of the same... 

Superconductivity has not only been beneficial to science and technology but also has been highly rewarding to its scientists. Thus far, Nobel Prizes in Physics have been awarded on four occasions to scientists working in this area. The first of these was for the discovery of superconductivity by Kamerlingh Onnes, awarded in 1913. In 1972 the prize went to John Bardeen, Leon Cooper, and Robert Schrieffer for the BCS theory. The following year (1973), the Prize was awarded to Brian Josephson, L. Esaki and I. Giaever for the... 

We may now consider the most basic form of the relationship between lattice properties and one of the most important physical properties of the bismuthates, the superconducting Tc, as derived from the theories of Bardeen, Cooper and Schrieffer ... 

In 1957 the research team of Bardeen, Cooper, and Schrieffer produced a theory, now known as the BCS theory, that managed to explain all the major properties of... 

---