Boris Shklovskii
A.S. Fine Chair
in Theoretical Physics



Postal Address:

William I. Fine Theoretical Physics Institute
School of Physics and Astronomy
University of Minnesota
116 Church Street SE
Minneapolis, MN 55455


Office: 420 Tate Laboratory of Physics
Phone: (612) 625-0771
Fax: (612) 626-8606




An universal method of calculation of the hopping conductivity based on ideas of the percolation theory was suggested simultaneously by Shklovskii and Efros, Ambegaokar, Halperin and Langer, and Pollak in 1970. In 1974, I put forward a model of "nodes and links" (known now as Shklovskii-de Gennes model) for the geometry of the infinite cluster of the percolation theory[1]. In 1975 Efros and Shklovskii studied the role of Coulomb interaction in the hopping conductivity. They predicted the Coulomb gap in the density of states near the Fermi level of a system with localized states. It was shown that due to the Coulomb gap the power in the exponent of the variable range hopping conductivity should be equal 1/2 both for two and three dimensions (instead of MottÕs 1/3 and _)[2]. Power _ is known as Efros-Shklovskii law and was observed in more than hundred of papers.

In 1982 I studied the variable range hopping conductivity in strong magnetic field taking into account scattering of an electron in the course of tunneling. Using this theory Spivak and I predicted new interference phenomena related to this scattering: negative magnetoresistance, Aharonov-Bohm oscillations with charge 2e, which soon were discovered[3]. In 1986 together with Altshuler we developed statistic theory of spectra of mesoscopic metallic samples[4]. We showed that random matrix theory fails to work at large widths of energy band and predicted new statistics. This work has proven to be central in mesoscopic physics. In 1993 together with H. Shore and B. Shapiro I studied what happens to the level statistics at the metal-insulator transition[5]. We discovered that there is a new universal statistics different from Poisson and Wigner-Dyson ones. We showed that therefore the simplest way to locate transition and to get critical exponents is to study spectrum statistics (one does not need wave functions!). This method became standard (it is used in 70 publications).

In 1996 together with my students M. Fogler and A. Koulakov I studied the ground state of a clean two-dimensional electron liquid in a weak magnetic field in which lower Landau levels are completely filled and the upper spin polarized level is only partially filled. We showed that interaction between electrons in this case leads to instability of the uniform distribution of electrons at the partially filled Landau level. Alternating occupied and empty parallel stripes are formed on the upper level[6]. In 1998 J. Eisenstein discovered anisotropic transport near filling factor \nu = 9/2, 11/2, 13/2, 15/2 and interpreted it as the discovery of the stripe phase. Du, Stormer and Tsui confirmed this discovery. Both groups confirmed our prediction that stripes provide the ground state at all \nu > 4, so that FQHE does not exist at \nu > 4. They discovered that if magnetic field is tilted stripes appear even at 7/2 and 5/2. During next 3 years, approximately 20 experimental and 50 theoretical papers were published on this subject. At filling factor of the top Landau level smaller than 0.3 and larger than 0.7 the second "bubble" phase of electron gas was discovered experimentally, exactly as we predicted. Theorists confirmed our predictions by exact numerical diagonalization. Thus, stripes and bubbles have proven to be the general state of a two-dimensional electron gas in magnetic field.


In 1998 I started to apply experience in the low temperature electronic semiconductor physics to describe screening of macroions such as DNA, proteins, lipid membranes or colloids by multivalent counterions such as short positive polymers (polyelectrolytes) in water solutions. I showed that multivalent counterions form a strongly correlated two- dimensional liquid at the macroin surface. Cohesive energy of this liquid leads to the new, additional attraction of counterions to the surface, which is absent in conventional solutions of Poisson-Boltzmann equation. As a result screening is much stronger than in conventional theories. Screening by multivalent counterions can be so strong that it inverts the sign of the macroion net charge[7]. The absolute value of the inverted charge can be as large as 100% of that of the bare one. For example, macroion with bare charge -100 can get a net charge +100. In an electrophoresis experiment such macroion moves in the direction opposite to that of the bare ion.

Charge inversion was observed in the electrophoresis of several biological and non-biological macroionns before my workwhich gave a theoretical description of this counterintuitive phenomenon. Charge inversion is of special interest for the delivery of genes to the living cell for the purpose of the gene therapy. The problem is that both bare DNA and a cell surface are negatively charged and repel each other, so that DNA does not approach the cell surface. The goal is to screen DNA in such a way that the resulting complex is positive. Multivalent counterions are used for this purpose. For example, DNA can complex with a positive polyelectrolyte, which overcharges it. Such drugs for a cancer and other conditions are already in massive trials.


[1]  A.S. Skal, B.I. Shklovskii, Topology of the infinite cluster of the percolation theory and its application to the theory of the hopping conduction, Sov. Phys.-Semicond. 8, 1586 (1974).
[2]  A.L. Efros, B.I. Shklovskii, Coulomb gap and low temperature conductivity of disordered systems, J. Phys. C8, L49 (1975).
[3]  V.L. Nguyen, B.Z. Spivak, B.I. Shklovskii, Tunnel hops in disordered systems, Zh. Eksp.Theor. Fiz. 89, 1770 (1985) - Engl. transl.: Sov. Phys.-JETP 62, 1021 (1985).
[4]  B.L. Altshuler, B.I. Shklovskii, Repulsion of energy levels and conductivity of small metallic particles, Sov. Phys.-JETP 64, 127 (1986).
[5]  B. I. Shklovskii, B. Shapiro, B.R. Sears, P. Lambrianidis, H.B. Shore, Statistics of spectra of disordered systems near the Metal-Insulator transition, Phys. Rev. B 47, 11487 (1993).
[6]  A.A. Koulakov, M.M. Fogler, and B.I. Shklovskii, Charge density wave in two-dimensional electron liquid in a weak magnetic field, PRL 76, 499 (1996).
[7]  B.I. Shklovskii, Screening of a macroion by multivalent ions: Correlation-induced inversion of charge, Phys. Rev. E 60, 5802 (1999).


Updated: 01/15/02