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Scattering in 1D disordered systems. Landauer-Model and Ohm’s law. Linear growth of resistance in 1D disordered systems.


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If someone is interested in a complete analytical solution of the scattering problem described by the one-dimensional Schrödinger equation with random potentials, one can find it in my diplom thesis. My diplom thesis is available in a German and in a partially translated English version. In my thesis a random equivalent of the famous Kronig-Penney-Model is studied and analytical formulae for all scattering properties are derived. The mathematical method developed to average the product of random matrices might be applied to other problems, which can be described by a product of random matrices, too. If you want to read or to download my thesis just click on the title pages shown below. Because of the restrictions to integrate figures in LaTeX at the time of writing the thesis, the hand drawn potential shapes of figures 1.1 and 2.1-2.4 are currently reproduced correctly only in the German version.





I especially want to call the interested readers attention to the point that a thorough investigation of the problem had shown, that for special cases of disorder (presumably correlated scatterers) a non-exponential growth of the resistance is possible, in fact the resistance might grow even linear with the number of scatterers, i.e. show the common ohmic behaviour of true 3D electrical conductors.


This is also what John B. Pendry in his review article Symmetry and transport of waves in 1D disordered systems (J. B. Pendry, Advances in Physics, 43, 461-542 (1994)) says, when he on page 535-536 writes:


“Many of the problems of waves in 1D systems are not only well understood but also easily accessible to quantitative calculation owing to the transfer matrix. Some problems remain to be addressed. Perhaps the most important of these is what happens if we abandon our assumption that separate slices of the system are statistically independent? Many systems derive their interest from such correlations, and in statistically mechanics it is known that correlations of a long-range nature completely alter the solutions. So far these systems have largely been neglected as we have concentrated on the uncorrelated systems but now is the time to look at them again.”


These were my words. I pointed to this fact several times in my thesis and my thesis contains all tools and formulae to perform the necessary calculations and I guess that correlated systems show a linear growth of resistance as worked out in chapter 4.5 of my diplom thesis, which is why I want to encourage everybody to do further research in this subject.


A Gaussian white noise distribution for the scatterers, which had been assumed by as far as I know all workers on this field in order to obtain analytical results for the mean resistance and its variance, does by no means represent any solid body and especially not any conducting solid, so nobody should wonder, that the resulting mean resistance does not obey Ohm’s law.


Quite contrarily a direct comparison of the resistance of the Kronig-Penney-Model (the ordered periodic case) and the disordered case indicates to my opinion that the oscillating resistance of the Kronig-Penney-Model for electrons inside the energy band might evolve to an ohmic resistance, if the periodicity is perturbed, i.e. a small disorder is introduced. Therefore I refer to tables 2.1 and 4.1-4.3 of my thesis.


To my opinion a quite good idea would also be to investigate the scattering properties of the models of


·    Lazo, E. and Onell, M.E., 2001, Extended states in 1-D Anderson chain diluted by periodic disorder, Physica B 299, 173-179.

·    Dunlap, D.H., Kundu, K. and Phillips, P., 1989,  Absence of localization in certain statically disordered lattices in any spatial dimension, Phys. Rev. B 40, 10999.

·    Dunlap, D.H., Wu, H.L. and Phillips, P., 1990, Absence of localization in a random-dimer model, Phys. Rev. Lett. 65, 88.

·    Flores, J.C., 1989, Transport in models with correlated diagonal and off-diagonal disorder, J. Phys. Cond. Matter 1, 8471.

·    Wu, H.L. and Phillips, P., 1991, Polyaniline is a random-dimer model: A new transport mechanism for conducting polymers, Phys. Rev. Lett. 66, 1366.

·    Phillips, P. and Wu, H.L., 1991, Localization and Its Absence: A New Metallic State for Conducting Polymers, Science 252, 1805.

·    Wu, H.L., Goff, W. and Phillips, P., 1992, Insulator-metal transitions in random lattices containing symmetrical defects, Phys. Rev. B 45, 1623.

·    Sánchez, A. and Domínguez-Adame, F., 1994, Enhanced suppression of localization in a continuous random-dimer model, J. Phys. A: Math. Gen. 27, 3725.  

·    Sánchez, A., Maciá, E., and Domínguez-Adame, F., 1994, Suppression of localization in Kronig-Penney models with correlated disorder, Phys. Rev. B 49, 147.

·    Diez, E., Sánchez, A. and Domínguez-Adame, F., 1994, Absence of localization and large dc conductance in random superlattices with correlated disorder, Phys. Rev. B 50, 14359.

·    Domínguez-Adame, F., Diez, E. and Sánchez, A., 1995, Three-dimensional effects on extended states in disordered models of polymers, Phys. Rev. B 51, 8115.

·    Lazo, E. and Onell, M.E., 1998, Localization in one-dimensional systems with generalized Fibonacci disorder, Revista Mexicana de Física 44, Suplemento 1, 52.

·    de Moura, F.A.B.F. and Lyra, M.L., 1998, Delocalization in the 1D Anderson Model with Long-Range Correlated Disorder, Phys. Rev. Lett. 81, 3735.

·    Izrailev, F.M. and Krokhin, A.A., 1999, Localization and the Mobility Edge in One-Dimensional Potentials with Correlated Disorder, Phys. Rev. Lett. 82, 4062.


which had found extended states in disordered or quasi-periodic 1D systems.


I guess that all of these extended states correspond to a non-exponential increase of the resistance and some of them even correspond to a linear (or quadratic) growth of the resistance, i.e. show an ohmic behaviour. This guess is based on the fact, that I had already shown the possibility of a linear (or quadratic) increase of the resistance for special electron energies in the Kronig-Penney-Model with random potential strengths in my talk hold at the RWTH Aachen (p. 15ff.).


The supervisor of my thesis had been Prof. Dr. Ingo Peschel of the FU Berlin.


During and after the preparation of my thesis I had been invited to talks about my thesis by Prof. P. Erdös (Université de Lausanne), Prof. B. U. Felderhof (RWTH Aachen) and Prof. John B. Pendry (Imperial College London).


Just follow the links for the announcement and the full text of my talk in Lausanne and my talk in Aachen.


In a letter Prof. Felderhof recommended a publication of the results of my diplom thesis, which never took place mainly because I had not been allowed to write my thesis in English and I never had the necessary time for a translation.





In order to ease the understanding of my diplom thesis, I have tried to link as much of the cited literature as possible, so just click on a white or yellow marked article and the article will be opened in Acrobat Reader, if you have access to the appropriate online journal. The yellow marked links are considered to be most important for an understanding of my diplom thesis. According to German copyright law printed matter can be copied freely 70 years after the death of the author (§ 64 UrhG).


     [1]  Abrahams, E., and Stephen, M.J., 1980, Resistance fluctuations in disordered one-dimensional conductors, J. Phys. C, 13, L377.

     [2]  Abramowitz, M., and Stegun, I., 1965, Handbook of Mathematical Functions, (Dover Publications Inc., New York).

     [3]  Abrikosov, A.A., 1981, The paradox  with the static conductivity of a one-dimensional metal, Solid St. Commun., 37, 997.

     [4]  Abrikosov, A.A. and Ryzhkin, I.A., 1978, Conductivity of quasi-one-dimensional metal systems, Adv. Phys., 27, 147.

     [5]  Afriat, S.N., 1959, Analytical functions of finite dimensional linear transformations, Proc.Cambridge.Phil.Soc., 55, 51.

     [6]  Andereck, B. and Abrahams, E., 1980, Numerical studies of inverse localization length in one dimension, J. Phys. C, 13, L383.

     [7]  Anderson, P.W., Thouless, D.J., Abrahams, E., and Fischer, D.S., 1980, New Method for a scaling theory of localization, Phys. Rev. B, 22, 3519.

     [8]  Ashcroft, N.W. and Mermin, D.N., 1976, Solid State Physics, (Holt Saunders International edition, Tokyo).

     [9]  Bargmann, V., 1947, Irreducible Unitary Representations of the Lorentz Group, Ann. of Math., 48, 568.

[10]  Berezinskii, V.L., 1974, Kinetics of a quantum particle in a one-dimensional random potential, Soviet Phys. JETP, 38, 620.

[11]  Büttiker, M., Imry, Y., Landauer, R. and Pinhas, S., 1985, Generalized many-channel conductance formula with application to small rings, Phys. Rev. B, 31, 6207.

[12]  Cvetič, M., and Pičman, L., 1981, Scattering states for a finite chain in one dimension, J. Phys. A, 14, 379.

[13]  Drude, P., “Zur Elektronentheorie der Metalle”, 1900, Annalen der Physik, 1, 566; 3, 369.

[14]  Dunford, N., 1943, Spectral Theory. I Convergence to projections, Trans.Amer.Math.Soc., 54, 185.

[15]  Eberle, G., 1982, Etude théorique de modèles pour la résistance d’un conducteur unidimensionnel à structure désordonnée, (Dissertation, Lausanne).

[16]  Eberle, G. and Erdös, P., 1981, in Erdös & Herndon(1982) this is noted as “unpublished results”. The result V.(1’) for the relative variance could not be found in the dissertation of Eberle [Eberle(1982)] and could not be cleared in my personal communication with Prof. P. Erdös, so it has to be considered unsure.

[17]  Erdös, P., 1967, Quantum Mechanical Electrical Conductivity of the One-Dimensional Landauer Model of an Impure Metal, (unpublished paper).

[18]  Erdös, P., and Herndon, R.C., 1972, Computational Methods for Large Molecules and Localized States in Solids, edited by F. Herman, A.D. Mclean and R. Nesbet, (Plenum, New York), p.275.

[19]  Erdös, P. and Herndon, R.C., 1982, Theories of electrons in one-dimensional disordered systems, Advances in Physics, 31, No. 2, 65-163.

[20]  Felderhof, B.U., 1986, Transmission and Reflection of Waves in a One-Dimensional Disordered Array, J. Stat. Phys., 43,267.

[21]  Felderhof, B.U. and Ford, G.W., 1986, Growth of Resistance with Density of Scatterers in One Dimensional Wave Propagation, J. Stat. Phys., 45, 695.

[22]  Frobenius, G., 1896, Über die cogredienten Transformationen der bilinearen Formen, Berliner Sitzungsberichte, 7.

[23]  Furstenberg, H., 1963, Noncommuting Random Products, Trans. Am. Math. Soc., 108, 377.

[24]  Gantmacher, F.R., 1958, Matrizenrechnung Teil I, (VEB Deutscher Verlag der Wissenschaften, Berlin).

[25]  Gradshteyn, I.S. and Ryzhik, I.M., 1965, Table of Integrals, Series and Products, (Academic Press, New York and London).

[26]  Heinrichs, J., 1986, Invariant-imbedding approach to resistance fluctuations in disordered one-dimensional conductors, Phys. Rev. B, 33, 5261.

[27]  Ishii, K., 1973, Localization of eigenstates and transport phenomena in one-dimensional disordered systems, Supp. of the Prog. of Theor. Phys., 53, 77.

[28]  Jayannavar, A.M., 1987, Resistance fluctuation in a one-dimensional disordered conductor in the presence of an electric field Sol. Stat. Com., 62, 355.

[29]  John, S., 1988, The Localization of Light and Other Classical Waves in Disordered Media, Comments Cond. Mat. Phys., 14, 193.

[30]  Kaufman, B., 1949, Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis Phys. Rev., 76, 1232.

[31]  Kirkman, P.D. and Pendry, J.B., 1984a, The statistics of one-dimensional resistances, J. Phys. C, 17, 4327.

[32]  Kirkman, P.D. and Pendry, J.B., 1984b, The statistics of the conductance of one-dimensional disordered chains J. Phys. C, 17, 5707.

[33]  Knapp, A., 1986, Representation Theory of Semisimple Groups, (Princeton University Press).

[34]  Kree, R. and Schmid, A., 1981, Stochastic properties of the resistivity in a one-dimensional disordered conductor, Z. Phys. B, 42, 297.

[35]  Kronig, R. and Penney, W.G., 1931, Quantum Mechanics of Electrons in Crystal Lattices, Proc. R. Soc., 130, 499.

[36]  Kumar, N., 1984, Resistance fluctuation in a one-dimensional conductor with static disorder, Phys. Rev. B, 31, 5513.

[37]  Landauer, R., 1970, Electrical resistance of disordered one-dimensional lattices, Phil. Mag., 21, 863.

[38]  Landauer, R., 1987, Electrical transport in open and closed systems Z. Phys. B, 68, 217.

[39]  Lappo-Danilewsky, J.A., 1953, Mémoires sur la Théorie des Systèmes Des Équations Différentielles Linéaires , (Chelsea Publishing Company).

[40]  Lorch, E.R., 1942, The Spectrum of Linear Transformations, Trans.Amer.Math.Soc., 52, 238.

[41]  Lorch, E.R., 1962, Spectral Theory, (New York-Oxford University Press).

[42]  Messiah, A., 1962, Quantum Mechanics I, (North-Holland Publishing Company, Amsterdam-Oxford), 5th printing 1975.

[43]  Mello, P.A., 1986, Central-limit theorems on groups, J.Math.Phys., 27, 2876.

[44]  Melnikov, V.I., 1980a, Motion of electrons in finite one-dimensional disordered systems, Soviet Phys. Solid State, 22, 1398.

[45]  Melnikov, V.I., 1980b, Distribution of resistivity probabilities of a finite, disordered system, Soviet Phys. JETP Lett., 32, 197.

[46]  Melnikov, V.I., 1981, Fluctuations in the resistivity of a finite disordered system, Soviet Phys. Solid State, 23, 444.

[47]  Melnikov, V.I., 1982, Exact solution for the resistivity distribution in disordered systems, Soviet Phys. Solid State, 24, 598.

[48]  Oberhettinger, F., and Badii, L., 1973, Tables of Laplace Transforms, (Springer-Verlag, Berlin-Heidelberg-New York).

[49]  Pendry, J.B., 1982, 1D localisation and the symmetric group, J. Phys. C , 15, 4821.

[50]  Pendry, J.B., 1988, Symmetry and transport in disordered systems, IBM J. Res. Develop., 32, 137.

[51]  Pendry, J.B. and Castaño, E., 1988, Electronic properties of disordered materials: a symmetric group approach, J. Phys. C., 21, 4333.

[52]  Peschel, I., 1989, personal communication.

[53]  Rammal, R. and Doucot, B., 1987, Invariant imbedding approach to localization. I. General framework and basic equations, J. Physique, 48, 509.

[54]  Rinehart, R.F., 1955, The Equivalence of Definitions of a Matric Function, Amer.Math.Monthly, 62, 395.

[55]  Sak, J. and Kramer, B., 1981, Transmission of particles through a random one-dimensional potential, Phys. Rev. B, 24, 1761.

[56]  Schwerdtfeger, H., 1938, Les Fonctions Des Matrices I. Les Fonctions Univalentes, (Actualités Scientifiques et Industrielles, No.649, Paris, Herman).

[57]  Stickelberger, L., 1881, Zur Theorie der linearen Differentialgleichungen, (Akademische Antrittsschrift, B.G. Teubner in Leipzig).

[58]  Stone, A.D. and Szafer, A., 1988, What is measured when you measure a resistance? -The Landauer formula revisited, IBM J. Res. Develop., 32, 384.

[59]  Strutt, M.J.O., 1928, Zur Wellenmechanik des Atomgitters, Annalen der Physik, 391, 319.

[60]  Taylor, A.E., 1943, Analysis in complex Banach spaces, Bull.Amer.Math.Soc., 49, 652.

[61]  Tung, Wu-Ki, 1985, Group Theory in Physics, (World Scientific Publishing Co Pte Ltd., Philadelphia-Singapore).

[62]  Vilenkin, N.Ja., 1969, Fonctions spéciales et théorie de la représentation des groupes, (Dunod, Paris).

[63]  Vossen, M., 1989, (Diplomarbeit, Aachen).

[64]  Weller, W. and Kasner, M., 1988, Localization in a System of N Coupled Chains. Recursion Method. I. General Approach, phys. stat. sol.(b), 148, 273.

[65]  West, B.J., Lindenberg, K. and Seshadri, V., 1980, Brownian motion of harmonic systems with fluctuating parameters I. Exact first and second order statistics of a mechanical oscillator, Physica A, 102, 470.



Last updated: 03-25-2009.