Exact solution for the density of electronic states in a model of a disordered system Institute of the Physics of Metals, Ural Scientific Center, Academy of Sciences of the USSR (Submitted 24 May 1979) Zh. Eksp. Teor. Fiz. 77, 2070-2080 (November 1979) A onedimensional system of electrons is considered, in a Gaussian random field with a correlator whose form (in the momentum representation) is a Lorentzian with its center at Q = 2p,. This can be considered as a Gaussian model of the Peierls transition in the fluctuation region. An exact summation of all Feynman diagrams is carried out, and a representation of the averaged one-electron Green's function as a continued fraction is obtained. A density of states with a characteristic pseudogap is found. It is shown that when the correlation range of the short-range order is decreased there is a gradual filling in of the pseudogap and a transition to a "metallic" state.

PACS numbers: 71.20. + c , 71.30. + h, 71.25.C~

INTRODUCTION

There i s a limited number of models of the electronic structure of one-dimensional disordered systems that admit of exact solution.' Interest in such models i s due both to the general problem of studying the electronic properties of disordered systems and to questions of the physics of quasi-one-dimensional systems, the majority of which display some s o r t o r other of properties associated with their disorder. In the last few years several important new results have beenobtained, casting considerable light on the situation of an electron in a one-dimensional random field.2-4 This work is also mostly characterized by the use of specific methods of solution, specially adapted to the solution of one-dimensional problems, and a s a rule not capable of further generalization because they a r e s o cumbersome. Only in a very few cases is it possible to obtain an exact solution of a problem about the electron in a one-dimensional random field by means of standard methods of present-day many-particle theory.5 One model of this s o r t was proposed some time ago by the present writer (see Ref. 6). In the framework of this model it could be shown now the scattering of the electron by a random field with a definite type of short-range order leads to the formation of a peculiar "band structure" of the energy spectrum, which appears in the form of a characteristic pseudogap in the density of electronic states, in the absence of any sort of long-range order. It was also possible to consider high-frequency conductivity and optical absorption in terms of the pseudogap. This model was used to describe the fluctuation region of quasi-one-dimensional systems that undergo a Peierls t r a n ~ i t i o n with , ~ the result that the predictions of this model a r e in good quantitative agreement with optical experiments on KCP and TTF-TCNQ,' a t least a t sufficiently high temperatures. A form of this model was considered i n Ref. 9 a s an extension7 to the fluctuation region of a commensurable Peierls transition. The exact was obtained in the limit of large range of the close-order correlation, and qualitative criteria were indicated for the applicability of this treatment for a finite correlation 989

Sov. Phys. JETP 50(5), Nov. 1979

length. In the present paper an exact solution for the one-electron Green's function is obtained in the form of a continued fraction, and also for the density of electron states, for arbitrary values of the correlation length for shortrange order; this permits us to trace a smooth transition to the "metallic" state (pseudogap filled in) as the correlation length is decreased and t o justify the qualitative criteria given earlier7 for the use of the asymptotic form for large correlation lengths. 1. FORMULATION OF THE MODEL AND ANALYSIS OF THE FEYNMAN DIAGRAMS

We consider an electron in a Gaussian random field ~ ( x with ) the correlation function po) analytic formulas are proposed for the calculation of T, and Uc with account of the nonequilibrium pitch of the cholesteric helix. Detailed experimental investigation of the instability threshold and the deformation period for the cases mentioned above have shown excellent agreement of experimental results and theoretical calculations.

PACS numbers: 61.30.Gd

1. INTRODUCTION

mations, respectively.

A field (zero-current) instability is observed in the planar texture of a cholesteric liquid crystal (CLC) with positive dielectric anisotropy (A&=En C, >O) at some threshold voltage U, upon application of an electric field parallel to the helix axis. This instability appears in the form of a spatially periodic deformation of the initial orientation of the director of the liquid crystal and is due to the destabilizing moment, which is proportional to E'AG' (E is the intensity of the electric field). In nematic liquid crystals (NLC) the threshold voltage of the analogous instability (Freedericsz transition') is determined by the formula

The dependences U,m ( ~ / p , ) ' / ' and T,.o (p&)"' that follow from (2) and (3) have been verified experimentally.415 Formulas (2) and (3) were obtained, however, under the assumption that L >>Poand without account of the difference of the real (induced) helical pitch p , which a r i s e s a s a result of the orienting influence of the from the equilibrium value Po, walls of the vessel, and therefore cannot be used directly for the estimation of the instability threshold in the case of a thickness of the CLC layer that is comparable with the helical pitch (L -0,). In the case L -Po, only the electrohydrodynamic instability has been investigated experimentally in detail.6

-

U,=Zn (nK,,lAe)

Ih,

where KI1 is the elastic modulus for a transverse flexure deformation; the wave vector of the deformation is equal to zero in this case. F o r CLC the theoretical value of the threshold U, and the period of the deformation T0=2n/k (k is the wave vector of the deformation) were obtained by elfr rich' and refined by ~ u r a u l t : ~ 8nS U."- - ( 6 K l X s , ) Ae 3Kss T: = 2Kz2

(-)

L

'"-,

(2)

Po

"p

~

(3)

;

here L is the thickness of the CLC, p, is the equilibrium helical pitch, K , , and K z z a r e the elastic moduli for deformations of longitudinal bending and torsional defor994

Sov. Phys. JETP 50(5), Nov. 1979

.

The purpose of the present work is a systematic theoretical and experimental study of the field instability of planar texture of CLC in the case of arbitrary relations between the layer thickness L and the helical pitch Po, with account of the real pitch and the boundary conditions. 2. THEORETICAL CALCULATION

Theoretical consideration of the field instability in planar texture of CLC has been carried out under the assumption of a rigid connection of the CLC molecules with the surface of the cell a t the boundaries of the layer. Two cases a r e considered: the directions of orientations of the molecules on the boundaries of the surface a r e parallel (planar orientation) o r perpendic-

0038-5646/79/110994-06$02.40

O 1980 American Institute of Physics

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