Molecular dynamics study of a classical two-dimensional electron system: positional and orientational orders

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Abstract

Molecular dynamics simulation is used to investigate the crystallization of a classical two-dimensional electron system, in which electrons interact with the Coulomb repulsion. From the positional and the orientational correlation functions, we have found an indication that the solid phase has a quasi-long-range (power-law correlated) positional order and a long-range orientational order. This implies that the long-range 1/r system shares the absence of the true long-range crystalline order at finite temperatures with short-range ones to which Mermin's theorem applies. We also discuss the existence of the “hexatic” phase predicted by the Kosterlitz–Thouless–Halperin–Nelson–Young theory.

Introduction

More than 60 years ago, Wigner pointed out that an electron system will crystallize due to the Coulomb repulsion for low enough densities (Wigner crystallization) [1]. Although quantum effects play an essential role in a degenerate electron system, the concept of Wigner crystallization can be generalized to a classical case where the Fermi energy is much smaller than the thermal energy. A classical two-dimensional (2D) electron system is wholly specified by the dimensionless coupling constant Γ, the ratio of the Coulomb energy to the kinetic energy. Here Γ≡(e2/4πεa)/kBT, where e is the charge of an electron, ε the dielectric constant of the substrate, a the mean distance between electrons and T the temperature. For Γ⪡1 the system will behave as a gas while for Γ⪢1 as a solid. Experimentally, Grimes and Adams [2] succeeded in observing a transition from a liquid to a triangular lattice in a classical 2D electron system on a liquid-helium surface around Γc=137±15, which is in good agreement with numerical simulations [3], [4], [5], [6], [7].

On the theoretical side, two conspicuous points have been known for 2D systems: (i) Mermin's theorem dictates that no true long-range crystalline order is possible at finite T in the thermodynamic limit [8]. To be precise, the 1/r Coulomb interaction is too long ranged to apply Mermin's arguments directly. Although there have been some theoretical attempts [9], [10] to extend the theorem to the Coulomb case, no rigorous proof has been attained. (ii) A theory due to Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY) predicts that a “hexatic” phase, characterized by a short-range positional order and a quasi-long-range orientational order, exists between a liquid phase and a solid phase [11]. Because the KTHNY theory is based on various assumptions and approximations, its validity should be tested by numerical methods such as a molecular dynamics (MD) simulation. Several authors have applied numerical methods to classical 2D electron systems, but they arrived at different conclusions on the KTHNY prediction [4], [5], [6], [7].

In order to address both of the above problems, the most direct way is to calculate the positional and the orientational correlation functions, which is exactly the motivation of the present study.1

Section snippets

Numerical method

A detailed description of the simulation is given elsewhere [12], so we only recapitulate it. We consider a rectangular area with a rigid uniform neutralizing positive background in periodic boundary conditions. The aspect ratio of the rectangle is taken to be Ly/Lx=2/3, which can accommodate a perfect triangular lattice [13]. The Ewald summation method is used to take care of the long-range nature of the 1/r interaction. We have employed Nosé–Hoover's canonical MD method [14], [15] to

Results and discussions

Let us first look at the positional and the orientational correlation functions in Fig. 1 for Γ=200 and Γ=160, for which the system is well in the solid phase. The positional correlation is seen to decay slowly, indicative of an algebraic (power-law) decay at large distances. The round-off in the correlation function around half of the system size should be an effect of the periodic boundary conditions. The algebraic decay of the positional correlation indicates that the 2D electron solid has

References (17)

  • E Wigner

    Phys. Rev.

    (1934)
  • C.C Grimes et al.

    Phys. Rev. Lett.

    (1979)
  • R.C Gann et al.

    Phys. Rev. B

    (1979)
  • R.H Morf

    Phys. Rev. Lett.

    (1979)
  • R.K Kalia et al.

    Phys. Rev. B

    (1981)
  • V.M Bedanov et al.

    Sov. Phys. JETP

    (1985)
  • I.V Schweigert et al.

    Phys. Rev. Lett.

    (1999)
  • N.D Mermin

    Phys. Rev.

    (1968)
There are more references available in the full text version of this article.

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