Diffusivity and Mobility Quantization. in Quantum Electrical Semi-Ballistic. Quasi-One-Dimensional Conductors

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1 Advaces i Applied Physics, Vol., 014, o. 1, 9-13 HIKARI Ltd, Diffusivity ad Mobility Quatizatio i Quatum Electrical Semi-Ballistic Quasi-Oe-Dimesioal Coductors M. A. Grado-Caffaro ad M. Grado-Caffaro Scietific Cosultats, C/ Julio Palacios 11, 9-B 809-Madrid, Spai Copyright 014 M. A. Grado-Caffaro ad M. Grado-Caffaro. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract We show that both the diffusio ad drift mobility coefficiets i a geeric quatum electrical (semi-ballistic) quasi-oe-dimesioal coductor are quatized. As a matter of fact, we derive expressios for the above coefficiets ad their associated quatum operators relative to highly excited quatum states. I parallel, the mai aspects of the ivolved quatum trasport are discussed i relatio to metallic aowires. PACS: 6.3.Hj; b; Rt; 84.3.Ff Keywords: Quatum electrical coductor; Diffusio coefficiet; Drift mobility; Semiballistic; Quatum trasport; Naowires

2 10 M. A. Grado-Caffaro ad M. Grado-Caffaro 1. Itroductio Electro trasport through quatum coductors is a subject of great relevace i aophysics. I practice, quatum coductors are aodevices which coduct electricity. I order to exemplify, let us cosider metallic ad semicoductor aowires as well as carbo aotubes. I this respect, ote that quatum electrical coductors are, by defiitio, electrical coductors where quatum effects occur. Cosequetly, sice these effects appear at aometric scale, it is clear that electrical coductors at aoscale are quatum coductors. O the other had, the fasciatig area of aophysics offers a wide variety of potetial applicatios [] but a umber of ope questios remai ope o essetial aspects of quatum trasport i aowires, quatum dots (aodots), ad carbo aotubes. With respect to theoretical research, importat research efforts are eeded istead to do o-trasparet ad o well-grouded formulatios as, for example, ref.[3]. The preset paper presets relevat ew ideas ad is devoted to semi-ballistic electro trasport i geeric quatum coductors but, say, implicitly, we will refer primarily to both metallic ad semicoductor aowires. I particular, we will show that both diffusivity ad drift mobility i these wires are quatized ad we shall determie their associated operators. Our results will be referred to semi-ballistic regime.. Theory.- Cosider a quatum electrical coductor as, for istace, a metallic aowire. I fact, we will focuse o metallic aowires. A coductor of this type ca be simulated by a ideal oe-dimesioal potetial well [1] withi the oe-electro approximatio; withi this framework, the coductio electros (a Fermi gas) are assumed as o-iteractig particles [1]. Uder these coditios, for a sigle electro i the above well, the electro kietic eergy equals (approximately) the quatized electro eergy for very high values of the ivolved quatum umber which will be deoted by so we have: 1 mv π h (1) ml where m is the free-electro mass, v is the magitude of the quatized electro velocity, L is the legth of the wire, ad h is, of course, the reduced Plack costat. Moreover, we recall that >> 1 which correspods to highly excited states.

3 Diffusivity ad mobility quatizatio 11 The diffusio coefficiet for the wire equals the squared electro mea free path divided by the relaxatio time ad, sice the relaxatio time is the electro mea free path divided by the magitude of the electro velocity, the the diffusio coefficiet is equal to the electro mea free path multiplied by the magitude of the electro velocity. Sice this velocity is quatized, the it is obvious that the aforemetioed diffusio coefficiet is also quatized. O the other had, we are iterested i regardig wires with some defects so that we are iterested i the semi-ballistic regime which, i our cotext, meas that L l where l stads for the electro mea free path i the wire. Notice that the above approximate mathematical coditio for defiig the semiballistic regime is clearly reasoable because ballistic regime meas that l > L (or eve l >> L ) while diffusive behaviour meas that l < L (or eve l << L ) so that the case i which l L, as, say, itermediate case betwee ballistic ad diffusive behaviours, the may be viewed as semi-ballistic (or semi-diffusive) regime. O the other had, ote that this type of regime ivolves partially elastic scatterig betwee the coductio electros ad defects or disorder i geeral. Therefore, by takig ito accout formula (1) ad that the quatized diffusio coefficiet reads D = lv as well as that L l, the it follows: h D () m where >> 1 ad, of course, h = πh. Relatioship () is cosistet with the approximate equatio amely: D ˆ ψ ψ (3) D which expresses that { } >>1 D lvˆ lpˆ m ( ihl m)( d dx) D is the eigevalue spectrum of the self-adjoit operator ˆ (diffusio-coefficiet operator) whose eigestates (boud quatum states) are represeted by the cartesia-coordiate- x depedet eigefuctios amely: 1 iπx ψ ( x) = exp (4) L L where >> 1 ad L l. Of course, the wavefuctios (4) are the solutios of the correspodig o-relativistic, time-idepedet, Schrödiger equatio with the ψ 0 = ψ L =. boudary coditios amely ( ) ( ) 0

4 1 M. A. Grado-Caffaro ad M. Grado-Caffaro Fially, aother issue to be treated is the quatized electro drift-mobility whose eigevalues (desigated by μ ; >> 1) obey: D kt = (5) μ e where k,t ad e deote Boltzma costat, absolute temperature, ad absolute value of the electro charge, respectively. The cojuctio of formulae () ad (5) gives: eh μ (6) ktm The correspodig operator (obviously self-adjoit) satisfies: ˆ μ ψ μ ψ (7) where, by usig the expressio of Dˆ, the aforemetioed drift-mobility operator reads: 3. Coclusio ihle d μˆ (8) ktm dx We have developed ew ideas givig rise to formulas (), (3), (6), (7) ad (8) as mai results. Although, i order to fix ideas, we have referred our study to electro trasport i metallic aowires, our results ca be extrapolated immediately, i a first approximatio, to -type semicoductor aowires. I additio, replacig the freeelectro mass by the carrier effective masses (for electros ad holes) ad itroducig the correspodig reduced effective mass, the derivig a formulatio for both electro ad hole coductio is ot hard. Refereces [1] M.A. Grado-Caffaro, M. Grado-Caffaro, A potetial-well based formulatio to calculate the quatized coductace of a oe-atom costrictio, Phys. Lett. A 37 (008),

5 Diffusivity ad mobility quatizatio 13 [] D.P.E. Smith, Quatum poit cotact switches, Sciece 69 (1995), [3] Y.-G. Yoo, P. Delaey, S.G. Louie, Quatum coductace of multiwall carbo aotubes, Phys. Rev. B 66 (00), Received: November 15, 013