Finite-temperature spectral density for carbon nanotubes with a magnetic impurity

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Abstract

Finite-temperature spectral density for metallic single-walled carbon nanotubes (SWNTs) with a magnetic impurity is studied theoretically. A single orbital Anderson model and a rapidly convergent perturbation method for dilute magnetic alloy are applied to investigate the effect of magnetic impurities on density of states (DOS) of both thin and thick SWNTs at various temperatures. It is found that for nanotubes of different diameter the DOS exhibit different resonance properties and different changing trend with increase of temperature. Rules behind the behavior are manifested.

Introduction

Single-walled carbon nanotubes (SWNTs) are ideal systems for investigating fundamental properties and applications of one-dimensional electronic systems [1], [2], [3], [4]. Recently, the research on metallic carbon nanotubes with magnetic impurities has aroused both experimental and theoretical interests [5], [6], [7], [8]. A scheme is proposed to employ the magnetic impurity/SWNT system to generate quantum entanglement states [7] for quantum computer, a currently fashionable area. The local density of states (LDOS) of SWNTs with a magnetic impurity has been investigated with a low-temperature scanning tunneling microscope (STM) [5]. Spectroscopic measurements performed on and near the magnetic clusters of nanometer and subnanometer diameter exhibit a narrow peak near the Fermi level, which has been identified as a Kondo resonance. Motivated by the experiment, Fiete et al. [8] study theoretically how the STM spectra and Kondo temperature depend on size of magnetic clusters. They find that the Kondo temperature is rapidly suppressed as size of the cluster increases. Furthermore, in a previous paper [9], we calculated and discussed the Kondo resonance of the magnetic impurity/SWNT system at T=0. With a single orbital Anderson model and a rapidly convergent perturbation method applied, we testified that a narrow Kondo resonance peak arises near the Fermi energy which is very localized in real space, as observed in the experiment. The analytical relationships between the position and width of resonance and the nanotube parameter are given. We predicted that the Kondo temperature decreases with the increase of nanotube diameter. However, an important fact, that the Kondo resonance disappears at temperatures above the Kondo temperature (Tk) [5], is not explained in the previous work. Dependence on temperature is a remarkable characteristic distinguishing the Kondo resonance from resonances caused by non-magnetic impurities. Therefore, in this paper, we investigate the effect of magnetic impurities on DOS of SWNTs at various temperatures. The armchair (4,4) and (10,10) nanotubes are studied as examples of thin and thick nanotubes, separately. It is found that for nanotubes of different diameter the DOS exhibit different resonance properties and different changing trend with increase of temperature. Explanation relies on division of the influence into two parts, the unperturbed Hartree–Fock (HF) part and the part describing the fluctuations above the mean field. Resonance peaks caused by HF part remain nearly unchanged while many-body effect is rapidly suppressed as temperature increases.

Section snippets

Formulation

Metallic SWNTs with one magnetic impurity at an A or B sublattice site are studied. A single orbital Anderson model [10] is applied to describe the magnetic impurity and electronic states of nanotubes are described with the tight-binding model [11]. The total Hamiltonian for the magnetic impurity/SWNT system is given by H=kCk+εkCk+σεdCd+Cd+k(Cd+VkCk+Ck+VkCd)+Und,↑nd,↓,where the first term describes the conduction electrons of nanotubes in which Ck+ and Ck are

Results and discussion

The case εd+(1/2)U=0 is called “symmetrical case” [15]. If εd+(1/2)U≠0, it is called “asymmetrical case”. The parameter Ed0=Ed(T=0) can be calculated self-consistently through the Friedel sum rule for T=0 [14]: Ed0Δ+utan−1Ed0Δ=1Δεd+U2.It is obvious that for the symmetrical case Ed0=0 while for asymmetrical case, Ed0≠0.

The self-energy Σ(ω) and the Green function Gdσ(ω) depend on temperature both explicitly, through β, and implicitly, through the temperature dependence of Ed [14]. The parameter Ed

Acknowledgements

The financial supports from China’s ‘973’ program and NSF China (Grant No. 90101993) are gratefully acknowledged.

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