Magnetic and geometric effects on the electronic transport of metallic nanotubes
Felipe Serafim, Fernando A. N. Santos, Jonas R. F. Lima, Sébastien Fumeron, Bertrand Berche, Fernando Moraes
MMagnetic and geometric effects on the electronic transport ofmetallic nanotubes
Felipe Serafim † , Fernando A. N. Santos (cid:63) , Jonas R. F. Lima †(cid:107) ,S´ebastien Fumeron ‡ , Bertrand Berche ‡ , and Fernando Moraes †† Departamento de F´ısica, Universidade Federal Ruralde Pernambuco, 52171-900, Recife, PE, Brazil (cid:63)
Departamento de Matem´atica, Universidade Federalde Pernambuco, 50670-901, Recife, PE, Brazil ‡ Laboratoire de Physique et Chimie Th´eoriques,UMR Universit´e de Lorraine - CNRS 7019,54506 Vandœuvre les Nancy, France and (cid:107)
Institute of Nanotechnology, Karlsruhe Instituteof Technology, D-76021 Karlsruhe, Germany (Dated: January 12, 2021)
Abstract
The investigation of curved low-dimensional systems is a topic of great research interest. Suchinvestigations include two-dimensional systems with cylindrical symmetry. In this work, we presenta numerical study of the electronic transport properties of metallic nanotubes deviating from thecylindrical form either by having a bump or a depression, and under the influence of a magneticfield. Under these circumstances, it is found that the nanotube may be used as an energy high-passfilter for electrons. It is also shown that the device can be used to tune the angular momentum oftransmitted electrons. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n . INTRODUCTION Low-dimensional carbon-based materials have long fascinated chemists and physicistsalike. From fullerene [1], which was the object of the year 1995 Chemistry Nobel prize, topolyacetylene [2, 3] (2000 Chemistry Nobel prize), to graphene [4] (2010 Physics Nobel prize),these zero-, one- and two-dimensional forms of carbon certainly deserve all the attention theyhave had due to their unique properties and numerous technological applications. Carbonnanotubes (CNTs) [5], even though are not directly attached to a Nobel award, are of noless importance for similar reasons. Since the first proposals of CNTs [6], there is a greatinterest in the investigation of quantum systems with cylindrical symmetry. The list ofnanotubes synthesized and theoretically proposed of different materials increases every yearand they show a variety of electronic properties [7–16]. For instance, WS nanotubes aresemiconducting, but with doping, it is possible to turn them metallic or superconducting[17, 18]. We can also cite boron nitride nanotubes, which are insulators with a constantband-gap independent of their radius and helicity [19]. On the other hand, depending onthe radius and helicity, CNTs can be metallic or semiconductors [20].The investigation of two-dimensional electron gases with cylindrical symmetry (C2DEGs)also attracted great deal of attention [21–26]. For instance, one of the first theoreticalcalculations of the quantum Hall effect in two-dimentional electron gases (2DEGs) considereda system with cylindrical symmetry [27]. Such C2DEG was already obtained experimentallyby different methods [28–35], with a radius that can varies from tens of nanometers up toseveral microns. Even though the typical diameter of single walled CNT is in the fewnanometers range, the outer nanotube of wide multiwalled CNTs can have a radius of thesame order of magnitude of those obtained for C2DEGs.This work deals with electronic transport in nanotubes. We investigate how one can usea combination of geometry and magnetic field to fine-tune electronic transmission across thelength of the tube. Previously, we investigated [36] how geometry alone can influence theelectronic transport on nanotubes. In that work, we found that simple deformations, likethe ones shown in Figs. 2 and 3 below, can have a deep impact in the transmittance profile.Moreover, periodic deformations clearly induce a gap in the energy spectrum, suggestingthe use of deformed nanotubes as electronic filters. This geometric tuning of electronicproperties of 2D materials has been explored in different contexts by different authors,237–40], including ourselves [41], to cite a few. With the inclusion of the magnetic fieldand the angular momentum states, the results presented here extend and complement theones reported in Ref. [36] and offer a possibility of experimental verification of the theoryproposed in [42]. The generic features of the results, as the shape of the transmittanceand geometric potential curves, apply directly to C2DEGs, but also to metallic nanotubesin general (carbon or not), even though we used CNT parameters. Furthermore, eventhough the electronic properties of semimetallic CNTs are usually obtained from the 2DDirac equation, metallic CNTs own they high conductivity to ballistic (nearly free) electrontransport, which justifies our use of the Schr¨odinger equation.The effect of an applied magnetic field on the electronic transport of nanotubes haslong been studied (see for instance [43–45]). The combination of curvature and electromag-netic field on the properties of charged particles bound to a surface was studied by Ferrariand Cuoghi [42] who derived the corresponding Schr¨odinger equation. They extended theda Costa approach [46], which reveals an effective geometric potential due to curvature,to include an externally applied electromagnetic field. Their approach has been appliedto the study of charged particles on a variety of surfaces under applied electromagneticfields [47–50]. Here, we study the effects of a magnetic field in the electronic transport ofdeformed nanotubes. We solve numerically the Ferrari-Cuoghi Schr¨odinger equation withopen boundary conditions (see [36] or [51] for details of the methodology respectively appliedto corrugations or single constrictions) in order to find the transmittance as a function ofinjection energy and orbital angular momentum. Then, we analyze the data noting that thedeformations allied to the magnetic field make the nanotube a selective tool for the controlof the outgoing electron energy and angular momentum. II. DERIVATION OF THE FERRARI-CUOGHI SCHR ¨ODINGER EQUATIONFOR A CYLINDRICAL SURFACE
In this section, we follow the steps of Ref. [42] and find a Schr¨odinger equation for aspinless charged particle bound to a cylindrical surface in the presence of a magnetic field.Let us first write the Schr¨odinger equation with the covariant derivatives coupled with the3lectromagnetic four-potential. The spatial gauge covariant derivative is defined as D j = ∇ j − iQ (cid:126) A j , (1)where Q is the charge of the particle and A j is the j component of the vector potential, with j = 1 , ,
3. The covariant derivative ∇ j acting on a vector field v i is given by ∇ j v i = ∂ j v i + Γ ijk v k , (2)where Γ ijk are the Christoffel symbols and ∂ j is the derivative with respect to the spatialvariables q j . The temporal gauge covariant derivative is given by D = ∂ t − iQ (cid:126) A , (3)where A = − V , with V being the scalar potential (note that we use the signature( − , + , + , +) for the Minkowski metric). So, the Schr¨odinger equation becomes i (cid:126) D ψ = − (cid:126) m G ij D i D j ψ, (4)where G ij is the inverse of the metric tensor. The gauge invariance can be demonstratedwith respect to the following gauge transformations: A j → A (cid:48) j = A j + ∂ j γ ; A → A (cid:48) = A + ∂ t γ ; ψ → ψ (cid:48) = ψe iQγ/ (cid:126) , (5)where γ is a scalar function of space and time.Substituting Eq. (1) into Eq. (4) we get i (cid:126) D ψ = − (cid:126) m G ij (cid:20)(cid:18) ∇ i − iQ (cid:126) A i (cid:19) (cid:18) ∇ j − iQ (cid:126) A j (cid:19)(cid:21) ψ. (6)Applying each term in parentheses on ψ leads us to i (cid:126) D ψ = 12 m [ − (cid:126) ∇ i ( G ij ∇ j ψ ) + iQ (cid:126) G ij ( ∇ i A j ) ψ + iQ (cid:126) G ij A j ( ∇ i ψ )+ iQ (cid:126) G ij A i ( ∇ j ψ ) + Q G ij A i A j ψ ] . (7)Substituting the covariant derivative defined in Eq. (2) and with some algebra we obtainthat i (cid:126) D ψ = 12 m (cid:20) − (cid:126) √ G ∂ i ( √ GG ij ∂ j ψ ) + iQ (cid:126) √ G ∂ i ( √ GG ij A j ) ψ + 2 iQ (cid:126) G ij A j ∂ i ψ + Q G ij A i A j ψ (cid:21) , (8)where G = det( G ij ). This is the covariant Schr¨odinger equation for any three-dimensionalcurvilinear coordinate system when an electric field and a magnetic field are applied. It isworth noting that Eq. (8) with (cid:126)A = ( A , A , A ) is valid for any gauge choice.4 . Application of the da Costa procedure Let us now apply in Eq. (8) the procedure first proposed by da Costa [46]. Considera surface S of parametric equations (cid:126)r = (cid:126)r ( q , q ), where (cid:126)r is the position vector of anarbitrary point on the surface. The three-dimensional space in the neighborhood of S canbe parametrized as (cid:126)R ( q , q , q ) = (cid:126)r ( q , q ) + q (cid:126)n ( q , q ) , (9)where (cid:126)n is a vector normal to the surface. The relation between the three-dimensional metrictensor G ij and the two-dimensional one g ab is given by G ab = g ab + [ αg + ( αg ) T ] ab q + ( αgα T ) ab q ; G a = G a = 0; G = 1 , (10)where g ab = ∂ a (cid:126)r∂ b (cid:126)r, (11)and α ab is the Weingarten curvature matrix of the surface.We now introduce a potential V λ ( q ) that confines the particle to the surface S , where λ is the parameter that measures the strength of the confinement. This means that in thelimit λ → ∞ the wave function will be nonzero only in the vicinity of q = 0. The goal ofthis procedure is to obtain a wave function that depends only on q and q , which are thecoordinates on the surface. With this purpose, we introduce a new factorized wave function χ ( q , q , q ) = χ s ( q , q ) χ n ( q ) , (12)where the index s indicates components tangent to the surface, and n the normal one. Thetransformation ψ → χ is given by ψ ( q , q , q ) = [1 + Tr( α ) q + det( α ) q ] − / χ ( q , q , q ) . (13)So, taking into account the effects of the potential V λ ( q ), we can apply the limit q → i (cid:126) D χ = 12 m (cid:20) − (cid:126) √ g ∂ a ( √ gg ab ∂ b χ ) + iQ (cid:126) √ g ∂ a ( √ gg ab A b χ ) + 2 iQ (cid:126) g ab A a ∂ b χ + Q ( g ab A a A b + ( A ) ) χ − (cid:126) ( ∂ ) χ + iQ (cid:126) ( ∂ A ) χ + 2 iQ (cid:126) A ( ∂ χ ) − (cid:126) (cid:32)(cid:20)
12 Tr( α ) (cid:21) − det( α ) (cid:33) χ (cid:35) + V λ ( q ) χ. (14)5ote that in Eq. (14), there is no term mixing A j ( j = 1 , ,
3) and the curvature matrix α ab . This is an evidence that the magnetic field does not couple with the curvature of thesurface, regardless of the shape of the surface, the magnetic field, and the gauge choice. Wecan see in Eq. (14) the appearance of the known geometric potential [46] V geo ( q , q ) = − (cid:126) m (cid:32)(cid:20)
12 Tr( α ) (cid:21) − det( α ) (cid:33) , (15)where the first term is the square of the mean curvature and the second is the Gaussiancurvature. The term between parentheses can be written in terms of the principal curvatures κ and κ as ( κ − κ ) , which is always positive [46].Defining a new metric tensor ˜ G as˜ G = g g g g
00 0 1 , (16)we can rewrite Eq. (14) in the compact form i (cid:126) D χ = 12 m ˜ G ij ˜ D i ˜ D j χ + V geo χ + V λ ( q ) χ. (17)We can see that Eq. (14) can not be separated into two equations, one that would dependonly on the tangent coordinates ( q , q ) and other that would depend on the normal coor-dinate q , because the term A ( q , q , ∂ χ couples the dynamics along q with that along( q , q ). However, we can choose a gauge that cancels A , thus eliminating this term. FromEq. (5), we can see that the most suitable choice for γ is γ ( q , q , q ) = − (cid:90) q A ( q , q , z ) dz, (18)which makes A (cid:48) = 0, ∂ A (cid:48) = 0 when the limit q → A and A remain unchanged. Thus, we can now separate Eq. (14) into the equations i (cid:126) ∂ t χ n = − (cid:126) m ( ∂ ) χ n + V λ ( q ) χ n , (19) i (cid:126) ∂ t χ s = 12 m (cid:20) − (cid:126) √ g ∂ a ( √ gg ab ∂ b χ s ) + iQ (cid:126) √ g ∂ a ( √ gg ab A b ) χ s + 2 iQ (cid:126) g ab A a ∂ b χ s ++ Q g ab A a A b χ s (cid:21) + V geo χ s + QV χ s . (20)6q. (19) is the one-dimensional Schr¨odinger equation for a particle bound by the potential V λ ( q ). Eq. (20) is the Schr¨odinger equation describing the dynamics of a particle withmass m and charge Q attached to the surface under the effects of electromagnetic fields.These equations demonstrate that the uncoupling between the tangential and normal surfacedynamics is only possible with an appropriate gauge choice.In the next section we will construct Eq. (20) for a surface with cylindrical symmetry,since we are interested in nanotubes. It is important to mention that in Ref.[42], the Eq.(20) was already written in spherical and toroidal coordinates. B. Application to the cylindrical surface
For a given coordinate system ( θ, q , ρ ), where our coordinate q is the coordinate alongthe axis of revolution, and θ is the angular component, a uniform magnetic field −→ B appliedto a cylinder of radius ρ can always be decomposed into a component B in the directionof q , parallel to the axis of the cylinder, and a perpendicular component B . This lattercomponent defines the direction from which the azimuthal angle θ is defined. It is alongthe ρ -axis at θ = 0, as shown in Fig. 1. This way, the magnetic field is given by (cid:126)B =( B θ , B q , B ρ ) = ( − B sin θ, B , B cos θ ). FIG. 1. Decomposition of the magnetic field.
Using (cid:126)B = (cid:126) ∇ × (cid:126)A , it is possible to obtain a vector potential (cid:126)A (cid:48) = (cid:126)B × (cid:126)r(cid:126)A = ( A θ , A q , A ρ ) = 12 ( ρB − B q cos θ, ρB sin θ, − B q sin θ ) , (21)but in order to simplify the following equations, it is more convenient to proceed to agauge change (cid:126)A = (cid:126)A (cid:48) + (cid:126) ∇ f ( θ, q , ρ ) with f ( θ, q , ρ ) = B ρq sin θ which kills one of the7omponents, (cid:126)A = ( A θ , A q , A ρ ) = (cid:18) ρB , ρB sin θ, (cid:19) . (22)We can see clearly that (cid:126) ∇ × (cid:126)A gives the correct magnetic field.The metric tensor g ab and its inverse g ab are given by g ab = ρ ( q )
00 1 + ρ (cid:48) ( q ) (23)and g ab = ( ρ ( q ) ) −
00 (1 + ρ (cid:48) ( q ) ) − , (24)where ρ (cid:48) = dρ/dq . We also have that √ g = (cid:112) det g = ρ ( q ) (cid:112) ρ (cid:48) ( q ) . (25)We now have the indexes a = q and b = θ . So, χ s becomes a function of q and θ . ReplacingEqs. (24) and (25) in Eq. (20) gives Eχ s = 12 m (cid:20) − (cid:126) ρ (1 + ρ (cid:48) ) / ∂ q (cid:18) ρ (1 + ρ (cid:48) ) / ∂ q χ s (cid:19) + iQ (cid:126) ρ (1 + ρ (cid:48) ) / ∂ q (cid:18) ρ (1 + ρ (cid:48) ) / A q (cid:19) χ s − (cid:126) ρ (1 + ρ (cid:48) ) / ∂ θ (cid:18) (1 + ρ (cid:48) ) / ρ ∂ θ χ s (cid:19) + iQ (cid:126) ρ (1 + ρ (cid:48) ) / ∂ θ (cid:18) (1 + ρ (cid:48) ) / ρ A θ (cid:19) χ s + 2 iQ (cid:126) A q (1 + ρ (cid:48) ) ∂ q χ s + Q A q χ s (1 + ρ (cid:48) ) + 2 iQ (cid:126) A θ ρ ∂ θ χ s + Q A θ χ s ρ + V geo χ s + QV χ s , (26)where, for the sake of simplicity, we suppress the dependence of ρ on q . Furthermore, wedo not take into account the temporal character of Eq. (20).Eq. (26) is the general equation for a charged particle confined to a cylindrical surfaceunder the presence of magnetic and electric fields in any direction. This equation becomesquite complicated to solve for a magnetic field in any direction.In Ref.[47] this equation was solved for a magnetic field transverse to the axis of a straightcylinder (in this case, ρ is constant). In our case, we are interested in deformed nanotubes.Therefore, ρ can not be kept constant and this adds an extra difficulty to the problem. Wesimplify the “electromagnetic configutation”, contemplating V = 0 (absence of electric field)and a magnetic field along the cylinder axis, i.e. only in the direction of q . As a remarkableresult, Eq. (26) becomes separable. This orientation of the magnetic field would not affectthe dynamics of a charged particle if the radius of the nanotube were constant. However, in8ur case, the radius of the nanotube is varying, and as a consequence the charged particlesare affected by the magnetic field as they pass through the deformations.With this choice, we have that B = 0. So, the vector potential simplifies further (cid:126)A = (cid:18) ρB , , (cid:19) (27)and Eq. (26) becomes Eχ s = 12 m (cid:20) − (cid:126) (1 + ρ (cid:48) ) ∂ q χ s − (cid:126) ρ (cid:48) ρ (1 + ρ (cid:48) ) (cid:18) − ρρ (cid:48)(cid:48) (1 + ρ (cid:48) ) (cid:19) ∂ q χ s − (cid:126) ρ ∂ θ χ s + iQ (cid:126) B ∂ θ χ s + Q ρ B χ s (cid:21) + V geo χ s . (28)Eq. (28) as mentioned above is separable. Therefore, we assume that the solution is of thetype χ s ( q , θ ) = χ q ( q ) χ ( θ ) , (29)which consists of the product of two functions, one that depends only on q and other thatdepends only on θ . Using this ansatz, the angular part will be given by ∂ θ χ ( θ ) − il∂ θ χ ( θ ) + 2 l χ ( θ ) = 0 . (30)The solutions of this equation are the eigenfunctions of the angular momentum l (cid:126) along theaxis q , and are of the type χ ( θ ) = e ilθ . (31)The axial part can be written as ∂ q χ q + F ∂ q χ q + G (cid:20) m (cid:126) ( E − V geo ) − l ρ + QB l (cid:126) − Q ρ B (cid:126) (cid:21) χ q = 0 , (32)where G ≡ ρ (cid:48) e F ≡ ρ (cid:48) ρ (cid:20) − ρρ (cid:48)(cid:48) (1 + ρ (cid:48) ) (cid:21) . (33)Eq. (32) is of the form χ (cid:48)(cid:48) q + V ( q ) χ (cid:48) q + V ( q ) χ q = 0 , (34)where V = F (35)and V = G (cid:20) m (cid:126) V geo − l ρ + QB l (cid:126) − Q ρ B (cid:126) (cid:21) . (36)9riting χ q = φ ( q ) λ ( q ), we obtain that φ (cid:48)(cid:48) + (cid:18) λ (cid:48) λ + V (cid:19) φ (cid:48) + (cid:18) λ (cid:48)(cid:48) λ + V λ (cid:48) λ + V (cid:19) φ = 0 . (37)Considering 2 λ (cid:48) λ + V = 0, we get λ ( q ) = e − P ( q ) , where P ( q ) is the primitive of V , whichmeans that P (cid:48) ( q ) = V ( q ). At this way, Eq. (37) becomes φ (cid:48)(cid:48) ( q ) + (cid:18) − V ( q ) − V (cid:48) ( q ) + V ( q ) (cid:19) φ ( q ) = 0 . (38)Here, we will call the term in parentheses an effective potential V eff = (cid:18) − V ( q ) − V (cid:48) ( q ) + V ( q ) (cid:19) . (39)We consider the injection of electrons with energy E k from the negative part of the axis q . Thus, we have that the general solutions outside the deformation are plane waves givenby χ q = a e ik q + b e − ik q , q ≤ a L e − ik L ( q − L ) + b L e ik L ( q − L ) , q ≥ L, (40)with a L = 0. We will consider that V geo (0) = V geo ( L ), so k = (cid:114) m (cid:126) ( E k − V geo (0)) = k L . (41)Note that V ( q ) = 0 for q out of the range 0 < q < L , which makes Eq. (34) and Eq.(38) identical. Then, Eq. (40) is also valid for φ ( q ), which means that χ q (0) = φ (0)and χ q ( L ) = φ ( L ). As a consequence, the reflection and transmission coefficients will notdepend on λ ( q ) and can be obtained directly from φ ( q ).With the boundary conditions a = 1 and a L = 0, we have that a = 12 (cid:104) φ (0) − iφ (cid:48) (0) /k (cid:105) = 1 (42)and a L = 12 (cid:104) φ ( L ) + iφ (cid:48) ( L ) /k L (cid:105) = 0 . (43)Following the calculations done in Ref. [36], we obtain that the transmission and reflectioncoefficients are given, respectively, by T = k L k | φ ( L ) | (44)10nd R = | φ (0) − | . (45)Thus, the problem reduces to finding φ (0) and φ ( L ). This is done by solving the coupleddifferential and algebraic equations (37), (42) and (43) in the range 0 ≤ q ≤ L . The factthat the magnetic field might be present outside this domain does not affect our results.There, we have an ordinary cylinder and the axial magnetic field effect is to add a phase tothe plane wave solution, which is the same on both sides of the deformation. Nevertheless,we point out that this does not apply in the case of a varying magnetic field [52].With the above equations, we implemented a code in MAPLE to find, for each injectedenergy, φ (0) and φ ( L ) and, consequently, the transmission and reflection coefficients [36, 53].We use a mixed unit system in which we have the incident energy in meV, and the distancesin nm. At this way, the electron mass is given by m e = 5 . × − meV · s /nm and thePlanck constant (cid:126) = 6 . × − meV · s. Also, for the magnetic field, we have the unitmeV · s/C · nm .In the next section, we will present our results, where we investigate the influence ofa magnetic field in the transport properties of various deformed nanotubes, where we willmake a comparison between the cases with and without magnetic field. III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we analyze the effects of the presence of the magnetic field in the transportproperties of deformed nanotubes. We consider that the deformation in the nanotubes arecorrugations generated by the curve ρ ( q ) = R + R(cid:15) (cid:104) − cos (cid:16) nπq L (cid:17)(cid:105) , (46)where R is the initial and final radius of the nanotube, (cid:15) gives the strength of the increase(positive values) or decrease (negative values) of the radius of the nanotube, L is the lengthof the corrugated region, and n is the number of corrugations. It is important to mentionthat such corrugations were already investigated in a plane [38] and nanotubes [36, 41], butthe analysis of the influence of a magnetic field has not been addressed yet.The theoretical approach considered here describes a free particle constrained to a cylin-drical surface. It can be used, for instance, to describe the electronic transport in a metallic11ylinder. The most direct application of the model is in C2DEGs. However, it can alsobe applied, for instance, for metallic WS and carbon nanotubes. Due to the low-energylinear dispersion of metallic CNT, they are well described by an effective Dirac equation.However, in the ballistic limit, the wavefunction for the scattering by a one-dimensionalpotential can be obtained from the Schr¨odinger equation [20]. So, our approach can alsobe used in such systems. In what follows, we will consider R = 75 nm, which is a radiusthat can be realized in C2DEGs and also in CNTs, as in the CNT NT145 obtained in Ref.[54]. It is important to mention that, in contrast to graphene, the spin-orbit coupling playsan important role in the transport properties of CNTs [55]. However, the magnitude of thespin-orbit coupling is inversely proportional to the diameter of the CNT. Therefore, sincewe are considering a CNT with a very large diameter, the spin-orbit interaction can be ne-glected. For the sake of clarity, we will show in the results the magnetic field in tesla, where1 T= 6 . × meV · s/C · nm . We will also consider that the charge carriers are electrons,which means that Q = − e . FIG. 2. (b) The transmittance and (c) the effective potential for one bump with (cid:15) = 0 .
5. Weconsider here B = 0 (blue lines), B = 2 T (green lines) and B = 4 T (red lines). We alsoconsider l = 0 (continuous lines) and l = 3 (dashed lines). In Figs. 2 and 3, we analyze the influence of the magnetic field in the transmittance ofa nanotube with one bump and one depression, respectively. The parts ( b ) and ( c ) of thefigures show the transmittance and the effective potential, respectively. Note the similaritybetween the effective potential of the bump and the one of the depression, even though theyhave opposite Gaussian curvatures. This is due to the fact that V geo (see Eq. (15) ) is alwaysattractive. We consider three different values for the magnetic field ( B = 0 T, 2 T, 4 T) andtwo values for the angular momentum ( l = 0 , IG. 3. (b) The transmittance and (c) the effective potential for one depression with (cid:15) = − . B = 0 (blue lines), B = 2 T (green lines) and B = 4 T (red lines). We alsoconsider l = 0 (continuous lines) and l = 3 (dashed lines). scale ( (cid:126) /eB ) / is of the order of 10 nm for the values of the magnetic field considered here,therefore of the typical scale of the corrugation length (cid:39)
30 nm or of the cylinder radius,75 nm. We can clearly note that the transmittance is sensitive to the magnetic field. Infact, the transmittance curve is shifted to the right with the introduction of the magneticfield: the device behaves as a high-pass filter. This displacement is because the effectivepotential is now deeper, causing the less energetic electrons to be reflected or trapped in thepotential. The same occurs when we increase the angular momentum. However, the shiftin the transmission observed when l increases is greater as the B increases. The deeperquantum wells seen in the case of bumps is a consequence of the fact that the term in theeffective potential that contains the geometrical potential becomes positive for the case ofdepressions, reducing the depth of the wells.Note also the appearance of resonance peaks, which correspond to quasi-bound states.These states are associated with a quantum well where a particle is primarily confined, buthas a finite probability of tunnelling and escaping. In the nanotube, the effective potential(containing the geometric potential), if deep enough, there may be similar states. In thecases where the energy of the charged carriers coincides with that of a quasi-bound state,it easily tunnels into the potential region, and so tunnel to the opposite side. Even thoughthe bumps induce deeper wells, a higher oscillation in the transmission is observed for thedepressions. It occurs because a potential barrier appears in the effective potential in themiddle of the depression ( q = 37 . FIG. 4. From left to right: Z , Z and Z as a function of l and B (blue surfaces) with ρ = 75 nm.The red planes reveal when Z , Z and Z are equal to zero. In order to clarify the influence of the magnetic field in the transport properties of thedeformed carbon nanotubes, let us analyze the three terms in the effective potential thatdepend on l and/or B . These terms are shown in the equation below: Z = − l ρ − eB l (cid:126) − e ρ B (cid:126) . (47)In Fig. 4 Z is plotted as a function of l and B . We can see that the angular momentumand the magnetic field can only induce a deeper effective potential, since Z can not havea positive value. Also, for each value of l there is a value of B that Z = 0. It meansthat the magnetic field can be used to choose which angular momentum will cross moreeasily the deformations, since a higher transmission is obtained for Z = 0. In other words,there is a value of B that makes an electron with a specific angular momentum cross thedeformations as if it had no angular momentum and there is no magnetic field applied. Itoccurs when B = − (cid:126) l/ ( eρ ). It is very important to remember here that ρ depends onthe position. So, this last relation has to be used carefully. As we will see below, there is aconstant value of ρ between its maximum and minimum values in the deformed region thatsatisfies this relation. Then, in fact, the value of ρ that can be used in this relation dependson the deformation of the surface.We also define Z = − eB l (cid:126) − e ρ B (cid:126) , (48)14hich consider the two terms in the effective potential that depend on the magnetic field. Z as a function of l and B can be seen in Fig. 4. When Z = 0, the charge carriers do notsee the magnetic field. So, it is possible to choose a specific angular momentum that willnot feel the influence of the magnetic field. It occurs for a given l when B = − (cid:126) l/ ( eρ ).Again, remember that ρ depends on the position.Finally, we define Z = − l ρ − eB l (cid:126) , (49)which is the combination of the two terms in the effective potential that depend on theangular momentum. The plot of Z in terms of l and B is also shown in Fig. 4. As canseen, for a given value of l , there is a magnetic field that cancels the influence of the angularmomentum in the effective potential. It happens when B = − (cid:126) l/ ( eρ ). B = 0 "contour3_fixed.dat" u 1:2:3 -40 -30 -20 -10 0 10 20 30 40 l E ( m e V ) B = 2 T "contour2T3B_fixed.dat" u 1:2:3 -40 -30 -20 -10 0 10 20 30 40 l T FIG. 5. Contour plot of the transmittance as a function of l and E with B = 0 and B = 2 T for3 bumps with (cid:15) = 0 . The influence of the angular momentum and the magnetic field in the transmission thatwere discussed here can be seen more clearly in Figs. 5 and 6, where we have the transmit-tance as a function of the energy and the magnetic field for different values of B for thecases of three bumps and three depressions, respectively. We can see a parabolic contour ofthe transmittance. For B = 0 the vertex of the parabola is at l = 0, which is the angularmomentum that crosses the deformation more easily. When l changes, the transmittance is15 B = 0 "contour0T3D1_fixed.dat" u 1:2:3 -40 -30 -20 -10 0 10 20 30 40 l E ( m e V ) B = 2 T "contour2T3D1_fixed.dat" u 1:2:3 -40 -30 -20 -10 0 10 20 30 40 l T FIG. 6. Contour plot of the transmittance as a function of l and E with B = 0 and B = 2 T for3 depressions with (cid:15) = − . shifted up, opening regions with no transmission for low energies. When B increases, thewhole contour is shifted to the left. With a negative value of B , which means a magneticfield in the negative direction of the axis q , this shift would occur to the right side.Comparing the cases of bumps and depressions, we can see that each one has its advan-tages. The shift induced in the transmittance when l changes is more significant for the caseof depressions. It reveals that deformations with negative values of (cid:15) are more suitable tofilter low energy charge carriers. However, the magnetic field can be used to select whichangular momentum will cross the deformations more easily. And the deformations withpositive values of (cid:15) are more sensitive to a change of B . In fact, the shift induced by achange in the magnetic field ∆ B is given by ∆ l = − eρ ∆ B / (2 (cid:126) ). Since the radius ρ ofthe nanotubes is reduced in case of depressions, more significant shifts of the transmittancecontour are expected in comparison with the case of bumps. IV. CONCLUSIONS
In this work, we investigated some properties of electronic transport on a device consistingof a corrugated metallic nanotube submitted to an external magnetic field. The magnetic16eld is oriented along the cylinder axis for simplicity. The corresponding Schr¨odinger equa-tion was built for a spinless charge, confined to an axisymmetric shell through an effectivepotential (da Costa procedure). With or without the magnetic field, the solution is sep-arable. The motion on the circumference is given by Eq. (31) and the motion along theaxis by the solutions of Eq.(38), which we have found numerically. Our model showed thatthe magnetic field and the curvature could be adequately encompassed in a global effectivepotential to which electrons strongly couple.Numerical simulations were run to compute the longitudinal transmittance of the nan-otube in the presence of bumps and depressions. They reveal that the device acts as ahigh-pass filter which inhibits the flow of low-energy electrons. Moreover, the effective po-tential also favours the transmission of electrons endowed with a tunable value of theirorbital angular momentum l . An additional coupling between the magnetic field and theelectron angular momentum was also mentioned, which enhance transmittance levels when l is negative.In summary, B can thus be used to tune the properties (energy, orbital angular momen-tum) of the transmitted electron flux. Electrons also posses a spin degree of freedom, andit is now recognized that spin-orbit interactions can be significantly high on curved carbonnanotubes [56]. Hence, the possibility to tune spin from the external magnetic field makesour device promising for many applications, such as the generation of carbon nanotube spinqubits. This will be the subject of further investigations. A further development of this workcould be the application of an electric field to the system. The field (cid:126)E = − (cid:126) ∇ V contributesto the Schr¨odinger equation (28) with a term QV χ s ( q , θ ) (see Eq. (13) of Ref. [42]). Inthe simplest case, where the field is parallel to the tube, Eq. (28) remains separable andthe electric field contribution appears only in the motion along q . That is, QV ( q ) willcontribute to V in Eq. (38), changing thus the effective potential V eff . Depending on thefunctional form of V ( q ) new minima may appear in the effective potential leading to newtransmittance peaks, since the peaks are associated to quasi-bound states. More likely, thealready present minima due to the geometric and magnetic contributions will be shifted bythe introduction of the electric potential, which will also shift the transmittance peaks, thusdirectly affecting the conductance.A more subtle development of the model used here, the study of the Aharonov-Bohm-related effects in a deformed nanotube, is certainly worth pursuing. To do this, the magnetic17eld must be restricted to the interior of the tube (a threading Aharonov-Bohm flux) as stud-ied in [52]. As described in this reference, for straight nanotubes, there appear oscillationsin the longitudinal ballistic and persistent currents. It would be interesting to study theconsequences on these properties of the geometric potential introduced by a deformation inthe tube. ACKNOWLEDGMENTS
F.M. thanks CNPq. JRFL thanks Capes, CNPq and Alexander von Humboldt Founda-tion.
Data Availability Statement
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