Magnetic impurity effects in metallic carbon nanotubes: local non-Fermi liquid theory

Magnetic impurity effects on metallic carbon nanotubes are studied theoretically. The resolvent method for the multi channel Kondo effect is applied to the band structure of the kp perturbation hamiltonian in the limit of the infinite onsite repulsion at the impurity site. We discuss the local non-Fermi liquid behavior at temperatures lower than the Kondo temperature T_K. The density of states of localized electron has a singularity |omega|^1/2 which gives rise to a pseudo gap at the Kondo resonance in low temperatures. The temperature dependence of the electronic resistivity is predicted as T^1/2, and the imaginary part of dynamical susceptibilities has the |omega|^1/2 dependence. Possible experimental observations are discussed.


I. Introduction
Recently, carbon nanotubes with cylindrical graphite structures have been intensively investigated. Many interesting experimental as well as theoretical researches have been performed (see reviews [1,2] for example), and the fundamental metallic and semiconducting behaviors of single wall nanotubes predicted by theories [3][4][5][6][7][8] have been clarified in tunneling spectroscopy experiments [9,10].
In the magnetoresistance study of metallic nanotubes [11], disorder gives rise to the positive differential resistance at low temperatures due to the weak localization effects [12]. Some samples are known to show the negative resistance around the weak gate voltage region, which is interpreted possibly due to the presence of the Kondo resonance as the result of the magnetic impurity effect. Therefore, a magnetic impurity might change the electronic structures of metallic carbon nanotubes.
The second candidate of the magnetic impurity is the fluorine (F) adatoms on the nanotubes.
The recent experiment [13,14] of the nanographite doped with F shows appearance of electronic spin during the fluorine concentration, [F]/[C] = 0 ∼ 1.2. The computer simulation of F-doping effects on nanographite [15] supports the presence of a local spin at the fluorine atom bonded to the carbon atom.
Thirdly, it is known that pristine carbon nanotubes are found to coexist with amorphous carbon soots [1,2]. Their effects are not of prime interests usually. However, we are aware that amorphous metals are modeled well by the two level system [16,17]. The low temperature behavior of resistivity in the two level system is similar to that of the Kondo model of the magnetic impurity [18,19,20]. So, the metallic nanotube with amorphous carbon soots may show behaviors like effects of magnetic impurities at low temperatures. This viewpoint is very interesting as a possible realization of the results of low temperature properties of the two level system, or equivalently, magnetic impurity effects.
In this paper, we will study effects of a magnetic impurity in metallic carbon nanotubes at low temperatures. The nonmagnetic impurity effects [21,22] have been studied by the present author, too. However, magnetic impurity effects are attractive also, when we look at our experience of the research of f -electron systems [23]. There are two channels of electronic states at the Fermi energy in metallic carbon nanotubes. As known in the magnetic systems, such as, heavy fermion systems [24], the non-Fermi liquid behaviors, i.e., the singular density of states and the power law temperature dependence of the electric resistivity, have been observed experimentally and explained theoretically by using the Kondo model or the Anderson model with multi channel scatterings. The similar effects can occur in the carbon nanotubes when the presence of the two scattering channels plays an important role.
We will use the k · p method [25,26] for the electronic states of carbon nanotubes. The method well describes electronic states around the Fermi energy. The valence and conduction band states have the linear dispersion at the Fermi energy in metallic nanotubes. It is assumed that there is one magnetic impurity at a carbon site. The magnetic impurity is modeled by the Anderson model where there is one localized electronic state and the strong onsite repulsion is assumed. For the treatment of the Kondo effect, we use the resolvent method [27,28] to discuss the infinite repulsion case. This is sufficient for the discussion of the low temperature and low energy behaviors. We will solve spectral functions of the resolvents, and derive analytic formulas of electronic density of states, resistivity, and dynamical susceptibilities. We will discuss the local non-Fermi liquid behavior which could be observed at low temperatures. The density of states of localized electron has a singularity ∼ |ω| 1/2 . This singular behavior gives rise to a pseudo gap at the Kondo resonance in low temperatures. The temperature dependence of the electronic resistivity is predicted as T 1/2 , and the imaginary part of dynamical susceptibilities has the |ω| 1/2 dependence. Possible experimental observations are discussed. This paper is organized as follows. In the next section, we introduce the model and explain theoretical formulations. In Sec. III, we report the low temperature solution of resolvents. In Sec. IV, electronic density of states is derived. In Secs. V and VI, resistivity and dynamical susceptibilities are explained. Section VII is devoted to discussion, and the paper is closed with summary in Sec. VIII.

II. Model
We will study the metallic carbon nanotubes with one Anderson impurity at the A or B sublattice site. In the total hamiltonian, H tube is the electronic states of the carbon nanotubes, and the model based on the k · p approximation [25,26] represents electronic systems on the continuum medium. The second term H imp is the Anderson impurity where the onsite Coulomb interaction strength is taken as infinite in the course of calculation, and the resolvent formalism [27,28] is introduced in order to take into account of the infinite repulsion.
The hamiltonian by the k · p approximation [25,26] in the secondly quantized representation has the following form: where E k is an energy matrix: k = (k x , k y ), and Ψ k,σ is an annihilation operator with four components. In the operator Ψ, the fist and second columns indicate an electron at the A and B sublattice points around the Fermi point K of the graphite, respectively. The third and fourth columns are an electron at the A and B sublattices around the Fermi point K ′ . The quantityγ is defined asγ ≡ ( where a is the bond length of the graphite plane and γ 0 (≃ 2.7 eV) is the resonance integral between neighboring carbon atoms. When the above matrix is diagonalized, we obtain the where k x is parallel with the axis of the nanotube, κ νφ (n) = (2π/L)(n + φ − ν/3), L is the circumference length of the nanotube, n (= 0, ±1, ±2, ...) is the index of bands, φ is the magnetic flux in units of the flux quantum, and ν (= 0, 1, or 2) specifies the boundary condition in the y-direction. The metallic and semiconducting nanotubes are characterized by ν = 0 and ν = 1 (or 2), respectively. Hereafter, we consider the case φ = 0 and the metallic nanotubes ν = 0.
The second term in Eq. (1) is the impurity model: where d is an operator of the localized electron with spin σ; its site energy is E d ; U d is the onsite repulsive interaction and n d,σ = d † σ d σ ; V is the mixing interaction between the electrons on the nanotube and the localized orbital; N s is the total number of lattice sites; η is the angle between the chiral vector of the present nanotube and the tube axis direction of the armchairtype nanotubes; and Ψ is the ith component of the operator Ψ k,σ , assuming that the Anderson impurity is located at the site r 0 of the A sublattice set. When the impurity is located at one of the B sublattice sites, the second and fourth components of Ψ k,σ appear in H imp . However, the result of the single impurity system is the same altogether.
The propagator of the electrons on the nanotube is defined in the matrix form: where T τ is the time ordering operator with respect to the imaginary time τ and Ψ k,σ (τ ) = exp(Hτ )Ψ k,σ exp(−Hτ ). The Fourier transform of G is calculated as: where ω n = (2n + 1)πT is the odd Matsubara frequency for fermions. The components of G are written explicitly: and There are several theoretical characteristic parameters. We will explain them as follows: • The total carbon number N s is given by where A is the length of the nanotube, and ( √ 3/2)a 2 is the area of the unit cell. There are two carbons in one unit cell, so the factor 2 is multiplied.
• The density of states near the Fermi energy E = 0 is calculated as Because two sites in the discrete model correspond to one site in the continuum k · p model, the density of sites in the continuum model is given by: • In the theory of Kondo effect, it is necessary to define the band cutoff D in order that ρ(E = 0) = 1/2D. Thus, the quantity D is written explicitly In the resolvent method [27,28], we mainly consider the limit U d → ∞ so that double whereΓ ≡ πρNV 2 , f (ǫ) = 1/[exp(ǫ/T ) + 1], is the resolvent for the singly occupied d-state, and is the resolvent for the empty d-state. Here, we denote the even Matsubara frequency as ν n .
The validity of the multi channel problem for the application of metallic nanotube can be understood easily by looking at the formula of the self energies. In the case U d = 0, the self energy of the d-electron is where G K is the first diagonal component of the propagator Eq. (6) and G K ′ is the third component. After inserting their explicit forms, we obtain where the scattering channel number is expressed as M and −D < ǫ k < D with the density of states ρ = 1/2D (D is defined by Eq. (12)). Only the two bands which cross the Fermi energy are retained in the sum of the last line, because the characteristic energy E 1 is fairly large as discussed above. Similarly, in the limit U d → ∞, the self energy of the resolvent Φ d is The difference from the U d = 0 case is that the resolvent Φ b appears in the right hand side.
After taking the frequency sum, we obtain the integral form Eq. (13). Therefore, we have shown that the set of integral equations of the multi channel Kondo problem is valid for an magnetic impurity in metallic carbon nanotubes.

III. Low temperature solution of resolvents
The set of integral equation Eqs. (13,14) can be solved analytically in the limit of low frequency and low temperatures [27,28]. Before discussing physical quantities, we derive analytic forms of resolvents. As the resolvents mean singly occupied states and empty state physically, the analytic property is not the same as that of the usual propagators. We define spectral functions for the empty states at ω > E 0 , and spectral functions for the occupied states at ω < E 0 , where E 0 is the ground state energy of the magnetic impurity at zero temperature, δ is positive infinitesimal, and β = 1/T .
After some calculations following Ref. [27], we find the following formula at low frequency: and Here, (24) with γ = M/N, and T K is the Kondo temperature: For the special case of interests of metallic nanotubes, M = 2 and N = 2. Therefore, we find the singular frequency dependence around the ground state energy E 0 : This singular dependence is the main conclusion of this section. The frequency dependence is shown schematically in Fig. 1.

IV. Density of states
The local density of states of d-electron is calculated by the convolution of spectral functions [27,28]. The explicit form of the density of states for spin σ and the channel α at T = 0 becomes where with Here Specially for metallic carbon nanotubes, we know that ∆ sp = ∆ ch = 1/2. This implies the singularity around the Fermi energy ω = 0: Such the singular functional form implies that a pseudo gap develops at the top of the Kondo resonance peak which appear at temperatures much lower than T K . There appears a dip in the density of states at the Fermi energy. The dip structure of the density of states is shown schematically in Fig. 2. This is the local non-Fermi liquid behavior discussed in detail in the literatures [24,29]. If it is possible to measure the local density of states of a metallic atom attached to the carbon nanotubes, for example, by scanning tunneling microscope, we could observe such the pseudo gap behavior when role of the multi channel scatterings is dominant.

V. Resistivity
In this section, we consider the electric resistivity in order to look at how the singular behavior will be observed. The scattering rate τ is calculated from the scattering t matrix: The relation with the electronic resistivitȳ gives the low temperature behavior: where c is a constant, but it is difficult to obtain its explicit form only from the information of ω-dependence of ρ σ,α . Here, we stress that the above singular functional form agrees with that obtained from the conformal field approach [30,31]. Therefore, the temperature dependence of the leading term (T /T K ) min(∆sp,∆ ch ) is a general result which is independent from method of theoretical treatment.
For the metallic carbon nanotubes, we already know ∆ sp = ∆ ch = 1/2. Therefore, the low temperature behavior is expected from the above general formula.

VI. Magnetic susceptibility
Spin and channel susceptibilities are calculated by the linear response function. The spin susceptibility is the magnetic susceptibility in other words. The imaginary parts of dynamical susceptibilities are defined as The first term ofχ ′′ sp at T = 0 is calculated to bẽ where The second correction gives the ω dependence: In the similar way, the dominant term ofχ ′′ ch at T = 0 becomes where C ch = W 2 ch ∆ sp sin(π∆ ch )B(∆ ch , ∆ ch ).
The second term has the ω dependence: The above general formulas Eqs. (40-45) reduce to that of metallic carbon nanotubes. The result of singular behavior is common for spin and channel susceptibilities: where A and B are constants. We find |ω| dependence at low frequencies. We note that the term sgn ω is related with the real part:

VII. Discussion
We have discussed the solutions of the integral equations in the limit M, N → ∞ in this paper.
On the other hand, the higher order corrections of O(1/N) [27,28] are known to change the coefficients of the leading terms of singular behaviors, but they do not affect the power of singularities. Therefore, the non-Fermi liquid behaviors are not affected by the higher order corrections by the O(1/N) expansion.
In the course of the investigation, we have neglected the valence and conduction band states which do not cross the Fermi energy. This approximation applies well at low energies |ω| ∼ T K ≪ E 1 . Typical magnitudes of E 1 have been shown in Table I. However, at higher energies or at high temperatures, such the deep bands will give some corrections to low energy singular behaviors. Furthermore, functional forms of spectral functions at entire frequencies can be affected by the presence of deep bands. In order to treat the deep band effects, we have to solve the resolvent equations numerically at least, even though it is beyond the scope of this paper.
There are two scattering channels when the impurity interacts with the single wall metallic nanotube. Is it possible that there are more scattering channels? When two metallic carbon nanotubes are present and one impurity interacts with both nanotubes, four scattering channels are realized as long as the interactions between electrons of two nanotubes are negligible. The powers of the singularities of the density of states and physical quantities become different from those of the case of the single nanotube. By putting M = 4 and N = 2, the general formulas of this paper can be applied to one magnetic impurity which interacts with two aligned metallic nanotubes. If the impurity interacts with more nanotubes, such the general formulas are useful in order to predict singular behaviors which depend on the scattering channel number.
In the literature, for example in [32], the power law temperature dependences of electronic resistivity have been interpreted by the Luttinger liquid picture of the correlated one-dimensional systems. The Kondo effect in the Luttinger liquid has been studied by Furusaki and Nagaosa