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  • Charge Density Waves: Models, Current Characteristics and Applications

    Abstract

    This project paper mainly focuses on the physical models and the current characteristics of charge density waves (CDW) in the niobium triselenide (NbSe\(_{3}\)) with anisotropic band structures. The models discussed in this paper are classical single particle transport model (a.k.a. sliding model) and quantum tunneling model. The classical model gives a threshold electric field beyond which the CDW material is a conductor. Finally, CDW effects could be modulated by applying gate voltage, resulting in field-effect CDW transistors.

    Introduction

    The idea of charge density waves was first proposed by Frohlich (Frohlich 1954) in 1954. It was not until the late 1970s that CDW phenomenon was investigated and physical models were constructed (Bardeen 1979). Later on, the CDW material was found to have interesting non-linear current response and frequency dependence when applied dc or ac electric fields (Thorne 1996). The electric properties of CDW could be engineered by applying gate voltage to build field-effect CDW electronic devices, which can function as gate-controlled switch (Adelman 1995) and current oscillator (Thorne 1986).

    The CDW is produced in some quasi-one-dimensional materials such as niobium triselenide (NbSe\(_3\)) (Fig.1). Electrons in the conduction band of the quasi-one-dimensional structure will undergo phase transition due to the electron-phonon interaction. Pointed out by Peierls in 1955 (Peierls 2001), the conduction electron state is unstable below certain temperature \(T_P\), also known as Peierls temperature. The ground state is the CDW state which reduces the total electric energy at the cost of small amount of deformation energy of the quasi-one-dimensional lattice. Conducting electrons pile up near the deformation, which might be due to phonon or impurities in the lattice. The charge density varies periodically according to the deformation, producing non-linear current characteristics.

    (Jonge 2011)

    Fig.1 Quasi-one-dimensional NbSe\(_3\) crystal structure. The crystal is composed of many 1D chains along which electrons can move easily. The conductance along the chain direction is \(10^{-3}\) less than the conductance of copper. The conductance for electrons flowing perpendicular to the chains is \(10^{-3}\) less than that in the parallel direction.

    The Peierls Transition

    The Fermi surface of one-dimensional lattice is not stable at zero temperature due to its geometry and lattice deformation. Considering an one-dimensional lattice with spacing a at zero temperature. Without the electron-phonon interaction, the electrons will fill half of the band to Fermi energy \(\epsilon_F\) with Fermi wave vector \(k_F = \frac{\pi}{2a}\) (Fig.2a). The empty states above the Fermi energy suggest the metallic conductance of the lattice. This state is unstable in the presence of lattice deformation and phonon. If we now introduce lattice distortion with \(\lambda = 2a = \frac{\pi}{k_F}\), transforming the lattice spacing from a to 2a, the period of electric potential doubles and the strength of the potential increases accordingly. This distortion opens a band gap at Fermi surface (Fig.2b) and the lattice becomes insulator. The total energy of the distorted lattice is determined by the increase of elastic deformation energy in the lattice and the decrease in electric energy of electrons. The piled up electrons form an energetically favorable state corresponding to the CDW as an analogy to the bonding state of molecules.

    When the lattice is cooled down below certain critical temperature, also called Peierls temperature \(T_P\), the transition from the state (a) to state (b) will take place, which is known as the Peierls transition (Peierls 2001). The Peierls temperatures for NbSe\(_3\) are \(T_P = T_1 = 142K\) and \(T_P = T_2 = 58K\), below which the CDW as a whole is pinned on the sites of atoms, resulting in increase in resistance at these two temperatures. It is clearly seen that the externally applied electric field should be large enough to depin the CDW from the pinning sites, giving a threshold electric field \(E_{th} \approx 0.1 V/cm\) for NbSe\(_3\). The diode-like behavior and detailed physical models will be discussed later. On the other hand, above the Peierls temperature, the band gap is obscured by the excitation of electrons, thermal vibrations in the lattice and decoupling of the 1D chains.

    The CDW state is the ground state of quasi-1D materials below Peierls temperature. In an pure 1D crystal with no degree of freedom in the other two directions , there is no CDW state, i.e. \(T_P \rightarrow 0K\) since the effects of random distortion cancel within the crystal. If the distortion in the lattice is coherent, for example, with displacement of the ith atom \(\delta u_i\) being \[\delta u_i = (-1)^{i} \delta a \sin{\omega t}\] where \(\delta a\) represents the displacement amplitude, the CDW state will then take place. Such coherence can be achieved in 2D or 3D materials with 1D conducting chains of weak coupling among neighboring chains.