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The resistivity and conductivity of macroscopic conductors with disorder are well described by the Drude theory. This theory is based on the classical picture, that electrons are accelerated on their way from one scattering center to the next. At an impurity they are scattered and have to be accelerated again. Such impurities can be doped atoms or dislocations of the crystal for example. This model leads to a conductance that does only depend on the averaged strength of the disorder and not on the details of the configuration of the scattering centers.
This situation is drastically changed if one considers temperatures below one Kelvin and very small samples: In that case the phase of the wave function which describes the electrons in the sample stays coherent over the whole volume of the sample. Therefore interference effects arise which cannot be described by the classical picture.
In experiments with these kinds of samples Laibowitz et al. [1] found in 1984, that the conductivity of a metal ring with a diameter of some microns shows fluctuations in dependence on the magnetic field in which the ring was situated. These fluctuations look randomly and are of the order of some per mille of the total conductivity. It is not just an experimental noise, because it is totally reproducible, if the magnetic field is increased and decreased several times. Only if the sample is heated up and cooled down again, the form of the fluctuations changes. This suggests, that the observed fluctuations depend on the microscopic configuration of the disorder in the sample, which reorganizes, if the sample is heated up. A parameter which is independent of the sample and its disorder configuration is the amplitude of these fluctuations, which at T=0 is always of the order of e²/h. If the temperature is increased by some tenths of a Kelvin, the structure of the fluctuations remains but the amplitude decreases.
The fact, that there can be interferences over the whole size of a sample was quite a surprise at the time of its first observation. Typical mean free paths in a metallic conductor are of the order of a hundred Ångstrøm. They are therefore much shorter than the typical size of a sample in the micron range. Nevertheless interference effects can be observed because the phase of the electron wave function is changed deterministically in elastic scattering processes of the electrons. So electrons which passed some scattering centers still have a defined relative phase. Only inelastic scattering processes destroy the phase coherence. On length scales larger than the mean free path of the inelastic scattering the phase differences of electrons become random and are averaged out such that the classical description becomes valid again. A conductor, which is smaller than the mean free path of inelastic scattering is called a mesoscopic conductor. In these mesoscopic conductors the conductance fluctuations described above can be observed and explained as quantum mechanical interference effects.
It would be interesting to find theoretical predictions about statistical properties of these conductance fluctuations as for example their correlation function
This ensemble of disorder configurations can be modeled by randomly chosen Hamiltonians. The Hamiltonian inside the sample is chosen as a matrix with random entries. The distribution of these entries is assumed to be Gaussian and as uncorrelated as it is possible respecting the symmetry of the problem. The width of the Gaussian distributions is a measure for the strength of the disorder. The product of two conductivities can then be written as a product of four Greens functions. This product is a random variable and has to be averaged over the ensemble of random Hamiltonians.
A convenient technique consists in representing the Greens functions by a generating functional. If one writes the generating functional using so-called super-integrals, the ensemble average of the generating functional can be performed easily. With the method as described in [3] and [4] the conductance fluctuations correlation function can be transformed into a super-integral over a certain space of super-matrices. This space is determined by the symmetries of the original Hamiltonian. In a perturbative way this super-integral can be solved. ([5], [6]) The expansion parameter is the inverse number of channels by which the sample is coupled to the leads. These results are therefore only valid in the limit of a large number of channels.
In the limit of few or only one channel one has to evaluate the super-integral exactly. This is only possible if one can find a coordinate system for the super-matrix space, which is well adapted to the integral. For the averaging of a product of only two Greens functions such a coordinate system is the polar coordinate system which is given in the references cited above.
This polar coordinate system can be generalized to the case of a product of four Greens functions. But it is a special feature of super-integrals to produce extra terms - the so-called boundary terms - if the coordinate system is changed. In order to calculate such a super-integral one therefore has to know how the boundary terms look like. To find these boundary terms for the polar coordinate and another related coordinate system is the main aim of the thesis.
[1] C.P. Umbach, S. Washburn, R.B. Laibowitz, and R.A. Webb,
"Magnetoresistance of Small, Quasi-one-dimensional, Normal-metal Rings
and Lines", Phys. Rev. B30 (1984), 4048.
[2] P.A. Lee, A.D. Stone, and H. Fukuyama,
"Universal conductance fluctuations in metals: Effects of finite
temperature, interactions, and magnetic field",
Phys. Rev. B35 (1987), 1039.
[3] K.B. Efetov, "Supersymmetry and theory of disordered metals",
Adv. Pys. 32 (1983), 53.
[4] J.J.M. Verbaarschot, H.A. Weidenmüller, and M.R. Zirnbauer,
"Grassmann integration in stochastic quantum physics: The case of
compound-nucleus scattering", Phys. Rep. 129 (1985),
367.
[5] S. Iida, H.A. Weidenmüller, and J.A. Zuk, "Statistical Scattering
Theory, the Supersymmetry Method and Universal Conductance Fluctuations",
Ann. Phys. 200 (1990), 219.
[6] A. Altland, "Conductance and conductance fluctuations of mesoscopic
systems with different symmetries: a statistical scattering approach",
Z. Phys. B82 (1991), 105.
The whole thesis takes some 300kB as a gzipped postscript-file.