# Electron Gas Dynamic Conductivity Tensor on the Nanotube Surface in Magnetic Field

Authors: A. M. Ermolaev; S. V. Kofanov; G. I. Rashba
Journal: Advances in Condensed Matter Physics
 (2011)

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## Abstract

Kubo formula was derived for the electron gas conductivity tensor on the nanotube surface in longitudinal magnetic field considering spatial and time dispersion. Components of the degenerate and nondegenerate electron gas conductivity tensor were calculated. The study has showed that under high electron density, the conductivity undergoes oscillations of de Haas-van Alphen and Aharonov-Bohm types with the density of electrons and magnetic field changes.


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## Body

## 1. Introduction

Electromagnetic waves in the systems with cylindrical geometry have been studied for a long time [1–3]. The authors of the work [4] had solved the Maxwell’s equation taking into account the retardation for the conducting cylinder dipped in a dielectric medium. They considered time dispersion of the dielectric permittivity of conductor and dielectric. Also they had solved the dispersion equation for the wave spectrum and obtained new branches in the spectrum of surface polaritons. The paper [5] deals with polaritons in the magnetic wire with uniaxial anisotropy, dipped into a dielectric medium. Wire's dielectric and magnetic permittivity tensor frequency dependence also was studied. Polaritons spectra were found. Constant, parallel to the magnetic wire axis, external magnetic field effect was studied in paper [6]. In this paper, how wave characteristics allow one to obtain data about the material structure was shown.A necessary of semiconductor nanotubes waveguide characteristics researching is caused by growing interest [7–10] to these systems. Plasma and magnetoplasma waves propagation on the nanotube surface were studied in [11–18]. They focused on cylindrical geometry systems calculations. These calculations were made by means of matter dielectric permittivity simple models. The problem is that authors of [11–18] have not used dielectric permittivity and conductivity tensors exact expressions. Also they have not taken into account the time and spatial dispersion. However, these exact formulas may significantly influence the tube waveguide characteristics expressions.In this paper, we present conductivity tensor components calculation and their wave vector and frequency dependencies of the following system: electron gas on the nanotube surface, affected by a parallel to the tube axis external magnetic field. In Section2, Kubo formula was obtained for the conductivity tensor. In Sections 3 and 4 we considered degenerate electron gas, and in Section 5, nondegenerate gas.


## 2. Dynamic Conductivity Tensor

Confined by the nanotube’s cylindrical surface, electron gas imbedded into a parallel to the cylinder axis magnetic field has axial symmetry. Therefore, it is suitable to characterize the electron state by the conserved quantities: angular momentum projectionm=0,±1,…, impulse projection k of electron on tube axis z, and spin quantum number σ=±1. Electron stationary state |mkσ〉  wave function and its energy spectrum are equal [11–14] to(1)Ψmkσ(φ,z,α)=1Sei(mφ+kz)χσ(α),(2)εmkσ=ε0(m+ΦΦ0)2+k22m*+σμBB,
where m* and μB are the effective mass and the electron spin magnetic moment, φ and z are cylindrical coordinates, χσ(α) is a spin wave function, ε0=1/2m*a2 is a rotational quantum, a is the tube radius, Φ=πa2B is the magnetic induction B flux through the tube section, Φ0=2πc/e is flux quantum, S=2πaL is the nanotube lateral surface area, and the length of nanotube is L. Hereinafter, the quantum constant is considered to equal unity. The spectrum (2) is a set of one-dimensional subzones, whose boundaries εm0σ are not equidistant.One can expand the current electron density operator in fieldB into series of cylindrical harmonics (1):(3)J(φ,z)=∑mkJ(m,k)Ψmk(φ,z),
where (4)J(m,k)=e∑m1k1m2k2σ〈m1k1|V(-m,-k)|m2k2〉am1k1σ+am2k2σ.
Here amkσ and amkσ+ are annihilation and creation operators of electrons in state |mkσ〉,and 〈m1k1|V(-m,-k)|m2k2〉 are operator V matrix elements (5)V(-m,-k)=12m*[(-i∇-ecA)Ψmk*(φ,z)+Ψmk*(φ,z)×(-i∇-ecA)]
in the functional basis (1), B=rotA. The operator’s (4) cylindrical components are equal to(6)Jφ(m,k)=em*aS∑m′k′σ(m′+ΦΦ0+m2)am′k′σ+a(m′+m)(k′+k)σ,Jz(m,k)=em*S∑m′k′σ(k′+k2)am′k′σ+a(m′+m)(k′+k)σ.Within the framework of electron gas linear response theory [19, 20] to the weak electric field E, changes with wave vector q and frequency ω, it is possible to obtain the current density(7)jμ(m,q,ω)=∑νσμν(m,q,ω)Eν(m,q,ω),
where(8)σμν(m,q,ω)=ie2nm*ωδμν+1ω∫0∞dteiωt×〈[Jμ(m,q,t),Jν(-m,-q,0)]〉,
Kubo formula for conductivity tensor. Here Jμ(m,q,t) is a component of Heisenberg operator (6), n is a surface electron density, [a,b]=ab-ba, and angle brackets mark Gibbs average.We have substituted (6) in (8) and obtained the relation between the conductivity tensor and Fourier time component of two-electron Greens function [19, 20]:(9)Kt(1,2;3,4)=-iθ(t)〈[a1+(t)a2(t),a3+(0)a4(0)]〉,
where 1=(m1,k1,σ1),…, θ(t) is a Heaviside’s function. In case of free electron gas, function (9) is expressed by one-electron Green function. In the result, the conductivity tensor components (8) are equal to(10)σφφ(m,q,ω)=ie2nm*ω+ie2m*2a2ωS⋅∑m′kσf(εm′kσ)[(m′+Φ/Φ0+m/2)2εm′kσ-ε(m′+m)(k+q)σ+ω+i0-(m′+Φ/Φ0-m/2)2ε(m′-m)(k-q)σ-εm′kσ+ω+i0],σφz(m,q,ω)=σφz(m,q,ω)=ie2m*2aωS∑m′kσf(εm′kσ)⋅[(m′+Φ/Φ0+m/2)(k+q/2)εm′kσ-ε(m′+m)(k+q)σ+ω+i0-(m′+Φ/Φ0-m/2)(k-q/2)ε(m′-m)(k-q)σ-εm′kσ+ω+i0],σzz(m,q,ω)=ie2nm*ω+ie2m*2ωS∑m′kσf(εm′kσ)⋅[(k+q/2)2εm′kσ-ε(m′+m)(k+q)σ+ω+i0-(k-q/2)2ε(m′-m)(k-q)σ-εm′kσ+ω+i0],
where f(ε) is Fermi function. These components are included into the dispersion equation for electromagnetic waves, propagating along the tube.In case ofa→∞, one can obtain from (10) the well-known Drude-Lorentz expression, which describes two-dimensional electron gas with the magnetic field absence: σφφ=σzz=ie2n/m*ω, σφz=0. The real part of conductivity tensor in this case is equal to zero. It is caused by not taking into account the electrons collisions. That effect can be considered by means of ω→ω+iνc replacement in (10). Here νc is the collisions frequency.


## 3. Degenerate Gas

At zero temperature, integrals overk, included into (10), can be calculated   exactly. Thus, real and imaginary parts of conductivity (10) are equal to (11)Reσφφ(m,q,ω)=-e24πm*|q|a3ω×∑m′σ{(m′+ΦΦ0+m2)2×[θ(A+-+)-θ(A--+)]-(m′+ΦΦ0-m2)2×[θ(A++-)-θ(A-+-)](m′+ΦΦ0+m2)2},Reσφz(m,q,ω)=-e24π(qa)2ω×∑m′σ{(m′+ΦΦ0+m2)(ω-Ω+)×[θ(A+-+)-θ(A--+)]-(m′+ΦΦ0-m2)(ω-Ω-)×[θ(A++-)-θ(A-+-)(m′+ΦΦ0+m2)]},Reσzz(m,q,ω)=-m*e24π|q|3aω×∑m′σ{(ω-Ω+)2[θ(A+-+)-θ(A--+)]-(ω-Ω-)2[θ(A++-)-θ(A-+-)]},(12)Im⁡σφφ(m,q,ω)=e2nm*ω-e24π2m*qa3ω×∑m′σ[(m′+ΦΦ0+m2)2ln⁡|A+-+A--+|-(m′+ΦΦ0-m2)2ln⁡|A++-A-+-|],Im⁡σφz(m,q,ω)=-e2mnm*qaω-e24π2(qa)2ω×∑m′σ[(m′+ΦΦ0+m2)(ω-Ω+)×ln⁡|A+-+A--+|-(m′+ΦΦ0-m2)×(ω-Ω-)ln⁡|A++-A-+-|],Im⁡σzz(m,q,ω)=-2e2ε0m2nq2ω-m*e24π2q3aω×∑m′σ[(ω-Ω+)2ln⁡|A+-+A--+|-(ω-Ω-)2ln⁡|A++-A-+-|],
where(13)A+-+=qυm′σ-ω-+Ω+,A--+=-qυm′σ-ω-+Ω+,A++-=qυm′σ-ω++Ω-,A-+-=-qυm′σ-ω++Ω-,υm′σ=2m*(εF-εm′σ)1/2
is the electron motion along the tube axis velocity maximum in subzone (m′,σ), εm′σ=εm′0σ is the subzone boundary, εF is the Fermi energy, ω±=ω±q2/2m*, and(14)Ω±(m,m′)=ε0[±(m′+ΦΦ0±m)2∓(m′+ΦΦ0)2]
is the frequency of the “vertical” electron transitions between the spectrum (2) subzones. While obtaining (12) out of (10), the identity below was used (15)B2B-C=B+C+C2B-C.


## 4. Longitudinal Conductivity

This section concentrates on the analysis of degenerate electron gas longitudinal conductivity, (11) (12), assuming m=0. The numerical calculations were carried out also for m=1 case.  In this case, electrons make only intra-subzone transitions without spin change. Longitudinal conductivity is included into the propagating along the nanotube axis intra-subzone magnetoplasma waves dispersion equation [13], because it is related to the polarization operator P by the relation below(16)Im⁡σzz(0,q,ω)=e2ωq2ReP(0,q,ω).
In case of m=0, we obtain the following equations from (11) and (12): (17)Reσzz(q,ω)=-e2m*ω4π|q|3a×∑m′σ[θ(qυm′σ-ω-)-θ(-qυm′σ-ω-)-θ(qυm′σ-ω+)+θ(-qυm′σ-ω+)],Im⁡σzz(q,ω)=-e2m*ω4π2q3a∑m′σ(ln⁡|qυm′σ-ω--qυm′σ-ω-|-ln⁡|qυm′σ-ω+-qυm′σ-ω+|).
Suppose that the flux ratio in (2) should be equal to Φ/Φ0=M+η, where M=0,1,… is an integer part of Φ/Φ0and η is a fractional part (0≤η<1). One can use the achievable in experiments values of carbon and semiconductor nanotube radii and obtain that the value of M in the accessible magnetic fields is low, unless the tube radius is too large. Then the energy of electron (2) is minimal in the subzone (m′=-M, σ=-1):(18)εmin⁡=ε0η2-μBB.
In this case, the electron rotation energy is nearly compensated by the rotation in the magnetic field. Usually spin level splitting (2) is small, that is,(19)2μBB<ε0(1+2η),
and the electron density corresponds to inequality(20)n<m*μBBπ2a.
Then electrons partially fill only the lower subzone (-M,-1). Therefore, (17) leads to the following:(21)Reδσ=-π2k0qx2,Im⁡δσ=-k0x22qln⁡|(1+q/2k0)2-x2(1-q/2k0)2-x2|,
where(22)δσ=σzz(q,ω)m*ωe2n,x=ωqυ0,υ0=k0m*=2m*(εF+μBB-ε0η2)1/2
is the electrons velocity maximum in subzone (-M,-1). Figures 1 and 2 show the x=ω/qυ0 dependencies of Reδσ and Imδσ under η=0.01,q=k0,and k0a=1 in quantum limit, when Fermi energy is located in subzone (-M,-1).Figure 1
Dependence ofReδσ on x=ω/qυ0 under η=0.01, q=k0,  k0a=1,  m=0 (solid curve), and m=1 (dashed).Figure 2
Dependence ofImδσ on x=ω/qυ0 under η=0.01, q=k0, k0a=1, m=0 (solid curve), and m=1 (dashed).One can calculate what included into (17) sum by m′ using a Poisson formula, assuming εF≫ε0, that is, a large quantity of subzones are filled. Within long wavelength assumption qυF≪ω (υF is a Fermi velocity), we obtain the following formula from (17): (23)Im⁡σzz=e2nm*ω[1+2π2(ε0εF)3/4∑l=1∞1l3/2×sin⁡(2πlεFε0-π4)cos⁡⁡(2πlΦΦ0)].
Here we neglected the spin level splitting (2). Monotonous part of conductivity (23) under the given electron density does not depend on the tube radius and coincides with the two-dimensional electron gas conductivity [21]. It is within the a→∞, when the tube is cut along its generatrix and unrolled onto a plane. Conductivity (23) experiences the Aharonov-Bohm and de Haas-van Alphen oscillation types. The reason of the de Haas-van Alphen oscillation types existance is that the electron states density has root peculiarities on the subzone bounaries. Change of the tube radius or electron density causes these peculiarities to pass throw Fermi energy and de Haas-van Alphen oscillation types to appear. If we study conductivity dependence (23) on εF1/2=(n/ν)1/2 (ν is the two-dimensional electron gas states density), then the oscillation period is 1/2m*a. If we study the radius dependence, the period of conductivity oscillations is equal to 1/2m*εF. Aharonov-Bohm oscillations are caused by the magnetic flux through the tube cross-section change. Their period is equal to the flux quantum. Figure 3 shows the dependence of the (23) first conductivity harmonic on (εF/ε0)1/2 under Φ/Φ0=0.01.Figure 3
Dependence of the first harmonicImδσ (23) on (εF/ε0)1/2 under Φ/Φ0=0.01.


## 5. Nondegenerate Electron Gas

Nondegenerate electron gas emerges if inequalityβ|μ|≫1 is true on the semiconductor nanotube surface. Here β is a reverse temperature and μ is a chemical potential. Using Boltzmann’s statistics, the chemical potential is related to the electron density as (24)n=1π3/2am*2βeβμch(βμBB)∑m=-∞∞e-βεm,
where εm=ε0(m+Φ/Φ0)2. The included here summation may be transformed by means of formula [22] (25)∑m=-∞∞e-x(m+υ)2=πx∑l=-∞∞e-π2l2/xcos⁡(2πlυ),x>0.
Real and imaginary tensor (10) parts in the studied case assuming m=0 are equal to(26)Reσφφ(q,ω)=e24πm*(βε0)3/2|q|a3ωeβμch(βμBB)×{exp⁡[-m*β2(ω-q)2]-exp⁡[-m*β2(ω+q)2]}×∑l=-∞∞exp⁡(-π2l2βε0)(1-2π2l2βε0)cos⁡(2πlΦΦ0),Reσφz(q,ω)=πe22(βε0)3/2(qa)2eβμch(βμBB)×∑l=-∞∞lexp⁡(-π2l2βε0)×{exp⁡[-m*β2(ω-q)2]-exp⁡[-m*β2(ω+q)2]}×sin(2πlΦΦ0),Reσzz(q,ω)=e2nω|q|3πm*β2×{exp⁡[-m*β2(ω-q)2]-exp⁡[-m*β2(ω+q)2]},Im⁡σφφ(q,ω)=e2nm*ω+e24πm*(βε0)3/2qa3ωeβμch(βμBB)×[F(m*β2ω-q)-F(m*β2ω+q)]×∑l=-∞∞exp⁡(-π2l2βε0)(1-2π2l2βε0)cos⁡(2πlΦΦ0),Im⁡σφz(q,ω)=e22(βε0)3/2(aq)2eβμch(βμBB)×∑l=-∞∞lexp⁡(-π2l2βε0)×{F(m*β2ω-q)-F(m*β2ω+q)}×sin(2πlΦΦ0),Im⁡σzz(q,ω)=e2nωq3m*β2×[F(m*β2ω-q)-F(m*β2ω+q)],
where(27)F(x)=1πp∫-∞∞dye-y2x-y.
Components σφφ and σφz in (26) are expressed through β and μ, that is, they refer to an open system of tube surface electrons. They experience only Aharonov-Bohm oscillations with the magnetic field changes. Aharonov-Bohm oscillations are not taking place in σzz component while it is described in β and n terms.


## 6. Conclusion

Simplified conductivity models are usually used for studying electromagnetic waves propagation in the cylindrical geometry systems, for example, in nanotubes. The metal cylinder conductivity is often believed to be endless, and the dielectric permittivity of the matter, where cylinder is dipped, is considered to be constant or only frequency dependent. Conductivity’s spatial dispersion as a rule is not taken into account. Nonetheless, the electromagnetic field’s nature in the tube and its waveguide characteristics are sensitive to the surface currents. Therefore, the electron gas conductivity tensor components calculation problem, taking into account spatial and time dispersion, is worth consideration. It is given in this work conductivity tensor components may be used in obtaining a dispersion equation for electromagnetic waves in the tube. The example of longitudinal conductivity showed what data of electron gas is possible to obtain by measuring conductivity. Particularly, imaginary part of conductivity experiences de Haas-van Alphen and Aharonov-Bohm oscillations types with the electron density and magnetic field changes. The oscillation periods measurement enables one to identify the electron effective mass and the combination of universal constants included into flux quantum.



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