Physica B 194-196 (1994) 1055-1056 North-HoUand PHYSiCA

P l a s m o n s o f a n E l e c t r o n G a s o n a T u b u l e

Osamu Sato

Graduate School of Science and Technology, Niigata University, Ikarashi Niigata 950-21, Japan

Yukio Tanaka and Michisuke Kobayashi

Department of Physics, Niigata University, Ikarashi Nfigata 950-21, Japan

Akira Hasegawa

College of General Education, Niigata University, Ikarashi Niigata 950-21, Japan

Plasma dispersion relations of art electron gas on a tubule are obtained in the random phase approximation and the one dimensional limit. Intraband plasmon has a relationship between the plasma frequency wp and the wavevector q of wp --~ q(]ln Rq/21) 112, and the interband plasmon has a relationship wp ,,~ const, in the long wave length limit.

Discovery of graphane tubules [1] gives a new stage of one-dimensional electron sys- tem. The graphane tubules have various elec- tronic properties, some of which are metallic, others semiconductive, etc[2] ,[3]. In this pa- per, a model of an electron gas on a tubule which describes the metallic graphane tubule is investigated and plasma dispersion rela- tions of the system are derived.

Consider a system that electrons confined on a surface of a cylinder which has a ra- dius R and extends infinitely. Introducing the cylindrical coordinates, an electron posi- tion can be expressed by ¢ and z. Having the rotational invariance around the z-axis and continuous traslational symmetry along the z-axis, canonical momenta of ¢ and z ex- pressed by Pz = hq and pc = hl (l =integer) are good quantum numbers. Hamiltonian of this system is given by

H = H o + V

The first term denotes the Hamiltonian of free electron whose energy dispersion is eq,Z = (h2/2m)(q2+(1/R)2) . Where, q is the wavevec- tot along the z -ax i s , l is the integer associ- ated with the angular momentum and m is the electron mass. The second term expresses the coulomb interaction and regarded to be a perturbat ion term. There are remarkable

features of this model. From the boundary conditions of the wave function around the z-axis , free electron energy dispersion rela- tions can be considered to be one-dimensional energy bands splitted into subbands corre- sponding to each angular momentum. In the case, all the electrons in the system occupy the lowest energy subband (l = 0), the sys- tem is regarded to be quasi one-dimensional, that is to say, the Fermi energy EF is be- low the 1 = 1 subband's bottom. We intro- duce a parameter r~ which characterizes areal electron density of the tubule, and represents the radius of the circle per electron measured in units of the Bohr radius a 0 . . T h e condi- tion that the system is quasi one-dimensional can be rewritten in terms of the parameter r, as R < aor~/Tr 112. The following discus- sion is restricted to this one-dimensional case at zero temperature. Matrix elements of the coulomb interaction can be specified by the only remainder of the momentum and the angular momentum between those of initial state and final state. Since the system pos- seses symmetries along the direction of the one-dimensional axis and around of the axis, the momentum and the angular momentum corresponding to these symmetries must be conserved in the process of scattering by such interned force as the coulomb interaction among electrons in the system. In the case of the

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quatum wire model, there is no symmetry ex- cept in the direction of one-dimensional axis. To specify matrix elements of the coulomb interaction, it needs four quantum numbers (i.e. subband induces of two initial electron states and two final electron states) and a remainder of the momentum along the one- dimensional axis between initial state and fi- nal state. The model of an electron gas on a tubule is new type of quasi one-dimensionai model and easy to treat for its symmetry.

The longitudinal dielectric function n(q, l, w) in the random phase approximation is,

: 1 +

where v(q, l) is the Fourier transform of the coulomb interaction and II°(q, l, w) is the po- larization function of the lowest order. In or- der to find plasma dispersion relations, we solve the equation of

~(q, ~, ~) = 0.

The intraband and interband plasma disper- sion is shown in Fig.1. In the long wave length limit, the plasma frequency w~ is

~ ~ (v(q,~)q~-)l/~(q~ + (~_)~)~/~

where, qF is a reduced Fermi wave vector de- fined by qF = ( 2 m E F ) ~ / 2 / h . In the special case of the intraband plasmon (l = 0 band), the plasma dispersion relation of long wave length limit is

qF Rq ~/2 w~ ,~ 2eq(~--~m I ln("-~--)l) .

The first derivative of the plasma disper- sion relation of the intraband plasmon at q = 0 has a singularity which comes from the Fourier transform of the coulomb potential in the case of l = 0 which posseses a logarithmic singu- larity near q = 0. This means that long wave length intraband plasmon transmits with the velocity of infinity. The interband plasmon has a constant dispersion relation in the long

wave length limit similar to that of the ho- mogeneous three-dimensional electron gas.

In conclusion, we investigate plasma dis- persions of an electron gas on a tubule in the one-dimensional limit. Features of plasmons can be divided into two types, intraband plas- mon and interband plasmons. The intraband plasma dispersion behaves as wp = 0 at q = 0 and dwp/dq has a singularity at q = 0. In addition, our obtained intraband plasma dis- persion relation is in good agreement with quantum wire results by Li and Sarma in the long wave length limit[ 4]. The interband plas- mons have a dispersion of wp ,~ const , near q - 0 .

p l a s m a d i s p e r s i o n

. . . . .-. -~,~ ~,~

...... ; l~ ,~,~',~ ~ ~;" i

~- ' ~- ' - ' -~gt~

_~**~ ~ ' ~ ~ ~ * : . . . . . . ..~,~:ff~--":~ - , , ,

0.127 0.255

q/qF

Figure I. P l a s m a dispers ions of the intra- (l --- 0) and inter- (l = 0, 1) band p lasmon. T h e shaded area shows s ingle part ic le excita- t ion region. (R = 0.5a0, r~ = 1) .

K E F E R E N C E S [1] S.Iijima, Nature 354, 56 (1991). [2] R.Saito, M.Fujita, G.Dresselhaus and M.S.Dresselhaus, Phys.l~ev.B 46, 1804 (1992). [3] N.Hamada, S.Sawada and A.Oshiyama, Phys.Rev.Lett. 68, 1579 (1992). [4] Q.P.Li and S.D.Sarma, Phys.Rev.B 43, 11768(1991).