Landau quantization, Aharonov-Bohm effect and two-dimensional pseudoharmonic quantum dot around a screw dislocation
Cleverson Filgueiras, Moisés Rojas, Gilson Aciole, Edilberto O. Silva
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Landau quantization, Aharonov-Bohm effect and two-dimensional pseudoharmonicquantum dot around a screw dislocation
Cleverson Filgueiras, ∗ Moisés Rojas, † Gilson Aciole, and Edilberto O. Silva ‡ Departamento de Física, Universidade Federal de Lavras,Caixa Postal 3037, 37200-000, Lavras-MG, Brazil Unidade Acadêmica de Física, Universidade Federal de Campina Grande,POB 10071, 58109-970, Campina Grande-PB, Brazil Departamento de Física, Universidade Federal do Maranhão, 65085-580, São Luís-MA, Brazil (Dated: August 28, 2018)In this paper, we investigate the influence of a screw dislocation on the energy levels and thewavefunctions of an electron confined in a two-dimensional pseudoharmonic quantum dot under theinfluence of an external magnetic field inside a dot and Aharonov-Bohm field inside a pseudodot.The exact solutions for energy eigenvalues and wavefunctions are computed as functions of applieduniform magnetic field strength, Aharonov-Bohm flux, magnetic quantum number and the param-eter characterizing the screw dislocation, the Burgers vector. We investigate the modifications dueto the screw dislocation on the light interband absorption coefficient and absorption threshold fre-quency. Two scenarios are possible, depending on if singular effects either manifest or not. Wefound that as the Burgers vector increases, the curves of frequency are pushed up towards of thegrowth of it. One interesting aspect which we have observed is that the Aharonov-Bohm flux canbe tuned in order to cancel the screw effect of the model.
PACS numbers: 73.43.Cd,73.43.Qt
I. INTRODUCTION
The study of quantum dynamics for particles in con-stant magnetic and Aharonov-Bohm (AB) flux fields ,which are perpendicular to the plane where the particlesare confined, has been carried out over the last years.The existence of other potentials are also included, de-pending on the purpose of the investigation. For exam-ple, in Ref. an exactly soluble model to describe quan-tum dots, anti-dots, one-dimensional rings and straighttwo-dimensional wires in the presence of such fields wasproposed. It is an ideal tool to investigate the AB ef-fects and the persistent currents in quantum rings, forinstance. In , the exact bound-state energy eigenvaluesand the corresponding eigenfunctions for several diatomicmolecular systems in a pseudoharmonic potential wereanalytically calculated for any arbitrary angular momen-tum. The Dirac bound states of anharmonic oscillator and the nonrelativistic molecular models under externalmagnetic and AB flux fields were investigated recently.Other examples can be found elsewhere.On the other hand, the investigation on how a screwdislocation affects quantum phenomena in semiconduc-tors has received considerable attention. In the contin-uum limit (low energy), such works are based on the geo-metric theory of defects in semiconductors developed byKatanaev and Volovich . In this approach, the semi-conductor with a screw dislocation is described by aRiemann-Cartan manifold where the torsion is associ-ated to the Burgers vector. In this continuum limit, ascrew dislocation affects a quantum system like an iso-lated magnetic flux tube, causing an AB interferencephenomena . The energy spectrum of electrons aroundthis kind of defect shows a profile similar to that of the AB system . These works describe the effect dueto the geometric electron motion only. A second in-gredient plays an important role in these quantum sys-tems. It is an additional deformed potential induce bya lattice distortion . It is a repulsive scalar poten-tial(noncovariant) and shows pronounced influences inthe physical quantities in such systems. The impactof this potential was first addressed in Ref. , wherethe scattering of electrons around a screw dislocationwas investigated. Recently, it was showed that a sin-gle screw dislocation has profound influences on the elec-tronic transport in semiconductors . Both contribu-tions, the covariant and noncovariant terms, were takeninto account. For the electronic device industry, thesedefects represents a problem since they interfere in theelectronic properties of the materials by way of scatter-ing, due to such repulsive potential. Therefore, researchon screw dislocation and how it may influence the dy-namics of carriers is important for the improvement ofelectronic technology, the discovery of new phenomenaand better control of transmission processes .In this paper, we investigate how the quantum dotsand antidots, with the pseudoharmonic interaction andunder the influence of external magnetic and AB fluxfields, are influenced by the presence of a screw dislo-cation. We obtain exact analytical expressions for theenergy spectrum and wavefunctions. The modificationdue such topological defect in the light absorption co-efficient is examined and its influences in the thresholdfrequency value of absorption coefficient are addressed.Two scenarios are possible, depending on if singular ef-fects are taking into account or are not. It is found thatwhen the Burgers increases, the curves of such frequencyare pushed up towards its growth. It is also noted thatthe AB flux can be tuned in order to cancel the influenceof the screw dislocation in those physical quantities.The plan of this work is the following. In Sec. II, wederive the Schrodinger equation for an electron around ascrew dislocation in the presence of an external magneticfield, an AB field and in the presence of a two-dimensionalpseudoharmonic potential. This case can find applica-tions in the context of quantum dots and anti-dots. InSec. III, we investigate how the screw dislocation affectssuch energy levels and we investigate the impact on themdue to the deformed potential. We consider the electronconfined on an interface so that we can discus our re-sults in the context of a (quasi) two dimensional electrongas (2DEG). In Sec. IV, we investigate the modificationsdue to the screw dislocation on the light interband ab-sorption coefficient and absorption threshold frequency.The conclusions remarks are outlined in Section V. II. THE SCHRODINGER EQUATION FOR ANELECTRON AROUND A SCREW DISLOCATION
Consider the model consisting of a non interacting elec-tron gas around an infinitely long linear screw dislocationoriented along the z -axis. The three-dimensional geome-try of this medium is characterized by a torsion which isidentified with the surface density of the Burgers vectorin the classical theory of elasticity. In order to under-stand the dynamics of this system in a more consistentmanner, we must take into account the existence of adeformed potential, which is induced by elastic deforma-tions on the 3D crystal. The metric of the medium withthis kind of defect is given (in cylindrical coordinates)by ds = ( dz + βdϕ ) + dρ + ρ dϕ , (1)with ( ρ, ϕ, z ) → ( ρ, ϕ + 2 π, z ) and β is a parameter re-lated to the Burgers vector b by β = b/ π . The inducedmetric describes a flat medium with a singularity at theorigin. The only non-zero component of the torsion ten-sor is given by the two form b Figure 1.
Cylindrical portion of a 3D solid showing the dislocation. T = 2 πβδ ( ρ ) dρ ∧ dϕ, (2)with δ ( ρ ) being the two-dimensional delta function inthe flat space. Figure 1 illustrates the formation of ascrew dislocation in the bulk of a 3D crystal. Since we consider electrons on common semiconduc-tors, we have to introduce a deformed potential whichdescribes the effects of the lattice deformation on theelectronic properties in such materials . For a screwdislocation, it is found to be V d ( ρ ) = ~ ma b π ρ " a (cid:18) ∂∂z (cid:19) , (3)where a is the lattice constant.The Hamiltonian for a quantum charged particle in abackground g ij in the presence of the potential describedabove and in the presence of magnetic fields is given by H = 12 m ( p i − eA i ) g ij ( p j − eA j ) + V d ( ρ ) , (4)where g ≡ det g ij , with i, j = ρ, ϕ, z and e is the electriccharge. For the field configuration, we consider the exis-tence of a constant magnetic field along the z -direction, B = B ˆz , which is obtained from the potential (in theLandau gauge), A = Bρ ˆ ϕ. (5)We also consider in the model the presence of the ABpotential, A = φ AB πρ ˆ ϕ, (6)which provides the magnetic flux tube B = φ AB π δ ( ρ ) ρ ˆz , (7)and a scalar pseudoharmonic interaction defined by V conf = V (cid:18) ρρ − ρ ρ (cid:19) , (8)where ρ and V are the zero point (effective radius) andthe chemical potential . As pointed out in Ref. , thepresence of a screw dislocation causes an effective vectorpotential A eff defined by A eff ≡ ˆ ϕ ~ b πρ ∂∂z . (9)The magnitude of screw dislocation, b , plays a similarrole to φ AB in the AB system, but A eff is a differentialoperator instead.Our goal is to solve the problem of an electron gasinteracting with A + A + A eff . This model is describedby the Schrödinger equation − ~ m " ∂ ∂z + ∂ ∂ρ + 1 ρ ∂∂ρ + 1 ρ (cid:18) ∂∂ϕ − β ∂∂z + δ (cid:19) ψ + (cid:20) ieB ~ m (cid:18) ∂∂ϕ − β ∂∂z + δ (cid:19) + e B ρ m (cid:21) ψ + V d ( ρ ) ψ + V conf ( ρ ) ψ = Eψ, (10)where δ ≡ eφ AB / π ~ is the flux parameter.In the next section, we solve the Schrodinger equationabove and discuss the impact of such deformed potentialon the energy levels around this kind of defect. III. INFLUENCE OF A SCREW DISLOCATIONON THE ENERGY LEVELS
In this section, we start by investigating the influenceof the screw dislocation on the energy levels of electronson a 3D solid taking into account the deformed potential(3). The equation (10) is solved by considering the ansatz ψ ( ρ, ϕ, z ) = Ce ilϕ e ik z z R ( ρ ) , where l = 0 , ± , ± , ± , ... , k z ∈ ℜ and C is a normalization constant. Equation (10)leads to − ~ m (cid:20) d dρ + 1 ρ ddρ − ρ ( l − βk + δ ) − k (cid:21) R ( ρ )+ " e B ρ m − eB ~ m ( l − βk + δ ) + V (cid:18) ρρ − ρ ρ (cid:19) R ( ρ )+ ~ ma b π ρ (cid:0) a k (cid:1) R ( ρ ) = ER ( ρ ) . (11)This differential equation can be rewritten as (cid:20) d dρ + 1 ρ ddρ − ν ρ − Ω ρ + ǫ (cid:21) R ( ρ ) = 0 , (12)where ǫ = 2 m ~ (cid:20) E − k + eB ~ m ( l − βk + δ ) + 2 V (cid:21) ,ν = ( l − βk + δ ) + 2 β a (cid:18) − k a (cid:19) + 2 mV ρ ~ , and Ω = e B ~ + 2 mV ~ ρ . The general solution of the eigenvalue equation (12) isgiven by R ( ρ ) = C e − Ω2 ρ (Ω ρ ) + | ν | M (cid:0) d ν , | ν | , Ω ρ (cid:1) + C e − Ω2 ρ (Ω ρ ) + | ν | U (cid:0) d ν , | ν | , Ω ρ (cid:1) , (13)with d ν = 12 + | ν | − ǫ . (14)The functions M and U in Eq. (13) denote the confluenthypergeometric functions of the first and second kind, re-spectively. Unlike M( a, b, z ) , which is an entire functionof z , U ( a, b, z ) may show a singularity at zero. If our sys-tem does not show singularity, then we can make C ≡ . However, if the wavefunction couples with singular po-tentials, we can instead, make C ≡ . Remember thatour electron gas is in a region where exist a topologicaldefect, which may introduce a singularity in our problem.We first consider the case for regular wavefunctions andafter that we discuss what changes whenever we have theirregular ones. Therefore, we are left with R ( ρ ) = C e − Ω2 ρ (Ω ρ ) + | ν | M (cid:0) d ν , | ν | , Ω ρ (cid:1) . (15)A necessary condition for R ( ρ ) to be square-integrableis lim ρ →∞ R ( ρ ) = 0 , which is fulfilled if d ν = − n , with n = 0 , , , , ... . In this way, the eigenvalues of Eq. (12)are given by E n = (2 n + | ν | + 1) Ω ~ m − ( l − βk + δ ) ~ ω c ~ k m − V , (16)where ω c ≡ eB/m is the cyclotron frequency. Equation(16) are the modified Landau levels. For δ = 0 and V =0 , we have E n ≡ E deff n = (cid:18) n + | ν | − l − βk (cid:19) ~ ω c + ~ k m . (17)Notice that the deformed potential has a pronounced in-fluence on the energy levels. In the absence of such in-teraction, the energy levels are given by E n = (cid:18) n + | l − βk | − l − βk (cid:19) ~ ω c + ~ k m . (18)For δ = 0 , V = 0 , β = 0 and ignoring the motion alongthe z − direction, we have E n = (cid:18) n + 12 (cid:19) ~ ω c , which are the usual Landau levels for electrons on a flatsample.At this point, we consider the electrons on an flat inter-face, with thickness d , around a screw dislocation. Theyare confined by an infinite square well potential in the z -direction ( ≤ z ≤ d ). This way, we have k z = ℓπ/d ,where ℓ = 1 , , , ... We will consider just the first trans-verse mode ℓ = 1 filled. Then, the parameter ν in Eq.(12) can be put in the following way, ν = (cid:18) l − b d + δ (cid:19) + b a π − b d + 2 mV ρ ~ . (19)In the case where δ ≡ and V ≡ , we have ν = (cid:18) l − b d (cid:19) + b a π − (cid:18) b d (cid:19) . (20)As pointed out above, the presence of a deformed poten-tial has a pronounced influence on the energy levels, sincein its absence, we have just ν = (cid:16) l − a d (cid:17) . (21)In the absence of both the magnetic fields, we find E n = ~ s V mρ (cid:18) n + 12 + | µ | (cid:19) + ~ k m − V , (22)where µ = (cid:18) l − b d (cid:19) + b a π − b d + 2 mV ρ ~ . In the absence of the deformed potential, we have µ = (cid:18) l − b d (cid:19) + 2 mV ρ ~ . (23)We now turn our attention to the case considering theirregular solution which is achieved by considering C ≡ in Eq. (13), that is R ( ρ ) = C e − Ω2 ρ (cid:0) Ω ρ (cid:1) + | ν | U (cid:0) − n, | ν | , Ω ρ (cid:1) . (24)As showed in Ref. , this solution diverges as ρ → butis square integrable when − < | ν | < . (25)The eigenvalue of (24) are the same given by Eq. (16)but with the constraint (25) above, which can be achievedonly if l = 0 . In Eq. (16), any value of l is allowed. IV. INFLUENCE OF THE SCREWDISLOCATION ON THE INTERBAND LIGHTABSORPTION COEFFICIENT AND IN THETHRESHOLD FREQUENCY VALUE OFABSORPTION
In this section, we calculate the direct interband lightabsorption coefficient K ( ω ) and the threshold frequencyof absorption in a quantum pseudodot under the influ-ence of external magnetic field, AB flux field and thescrew dislocation. The light absorption coefficient canbe expressed as K ( ω ) = N × X n,l,ν X n ′ ,l ′ ,ν ′ (cid:12)(cid:12)(cid:12)(cid:12)Z ψ en,m,ν ( ρ, φ ) ψ hn ′ ,l ′ ,ν ′ ( ρ, φ ) ρdρdφ (cid:12)(cid:12)(cid:12)(cid:12) × δ (cid:0) ∆ − E en,m,ν − E hn ′ ,l ′ ,ν ′ (cid:1) , (26)where ∆ ≡ ~ ̟ − ε g , ε g is the width of forbidden en-ergy gap, ̟ is the frequency of incident light, N is aquantity proportional to the square of dipole momentmatrix element modulus, ψ e ( h ) is the wave function ofthe electron(hole) and E e ( h ) is the corresponding energyof the electron (hole). Considering the solution (15), the Eq.(26) becomes K ( ω ) = N X n,l,ν X n ′ ,l ′ ,ν ′ Ω | ν | + | ν ′ | +2 ( n + | ν | )!( n ′ + | ν ′ | )! π n ! n ′ ! ( | ν | !) ( | ν ′ | !) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z πe i ( l + l ′ ) Z ∞ ρdρe − (Ω+Ω ′ ) ρ ρ | ν | + | ν ′ |× M (cid:0) − n, | ν | , Ω ρ (cid:1) M (cid:0) − n ′ , | ν ′ | , Ω ′ ρ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × δ (cid:0) ∆ − E en,m,ν − E hn ′ ,l ′ ,ν ′ (cid:1) . (27)Following Ref. , the light absorption coefficient is givenby K ( ω ) = N X n,l,ν X n ′ ,l ′ ,ν ′ P νnn ′ Q νnn ′ δ (cid:0) ∆ − E en,m,ν − E hn ′ ,l ′ ,ν ′ (cid:1) , (28)where P νnn ′ = 1( | ν | !) (ΩΩ ′ ) | ν | +1 (cid:18) Ω + Ω ′ Ω − Ω ′ (cid:19) ( n + n ′ ) × ( n + | ν | )! ( n ′ + | ν | )! n ! n ′ ! , and Q νnn ′ = " A | ν | , Ω M n, n ′ , | ν | + 1; − ′ (Ω − Ω ′ ) ! , where A | ν | , Ω = ( | ν | )! (cid:18)
2Ω + ω ′ (cid:19) | ν | +1 . For the case considering the irregular solution above, wemust consider the expressions above but changing thefunction M (the confluent hypergeometric functions of thefirst kind) by U (the confluent hypergeometric functionsof the second kind). For this last case, only l = 0 isallowed.From Eqs. (16) and (27), the threshold frequency ofabsorption will be given by ~ ̟ = (2 n + | ν | + 1) (cid:18) Ω h m h + Ω e m e (cid:19) ~ − (cid:18) m h + 1 m e (cid:19) ( l − βk + δ ) ~ eB + 12 (cid:18) m h + 1 m e (cid:19) ~ k + ε g − V , (29)where m e ( m h ) are the electron effective mass(hole effec-tive mass) and Ω e = s e B ~ + 2 m e V ~ ρ , Ω h = s e B ~ + 2 m h V ~ ρ . By ignoring the motion along z -direction and in theabsence of both the defect( β ≡ ) and the AB fluxfield( δ ≡ ), we recover the threshold frequency of ab-sorption found in Ref. . Further, taking ( n, l ) = (0 , in the presence of the fields (5) and (6), we find ~ ̟ = ( | ν l =0 | + 1) (cid:18) Ω h m h + Ω e m e (cid:19) ~ + 12 (cid:18) m h − m e (cid:19) ( δ − βk ) ~ eB + ε g + 12 (cid:18) m h + 1 m e (cid:19) ~ k − V . (30)In the absence of screw dislocation, we recover the ex-pressions of Ref. , that is ~ ̟ b ≡ = r δ + 2 m h V ρ ~ ! Ω h m h ~ + r δ + 2 m e V ρ ~ ! Ω e m e ~ + ε g + 12 (cid:18) m h − m e (cid:19) ~ eBδ − V . (31)Due to the presence of the defect, we now have the fol-lowing expression for the the threshold frequency, ~ ̟ screw00 = r δ − δbd + b a π + 2 m e V ρ ~ ! Ω e m e ~ + r δ − δbd + b a π + 2 m h V ρ ~ ! Ω h m h ~ + ε g + 12 (cid:18) m h − m e (cid:19) (cid:18) δ − b d (cid:19) ~ eB − V , (32)if the deformed potential is taken into account. When δ ≡ , Eq. (32) become ~ ̟ screw00 = r b a π + 2 m e V ρ ~ ! Ω e m e ~ + r b a π + 2 m h V ρ ~ ! Ω h m h ~ + ε g − d (cid:18) m h − m e (cid:19) ~ eBb − V . (33)Let us now investigate the influence of the screw disloca-tion on the threshold value of absorption for transition → , comparing the two cases, one with and theother without the noncovariant deformed potential. Theargument of Dirac delta function allows one to definesuch threshold value of absorption as ~ ̟ ε g = 1 + E e + E h ε g . Let us now set the following parameters: η = ρ ε g ~ r m e V , η ′ = ρ ε g ~ r m h V ,κ = e ~ Bm e ε g , κ ′ = e ~ Bm h ε g . For a quantum dot, we take V ρ → . In this case, weobtain E e ε g = 12 (1 + ξ ) r κ + 8 η − κ (cid:18) δ − b d (cid:19) − V ε g , (34) E h ε g = 12 (1 + ξ ) r κ ′ + 8 η ′ + κ ′ (cid:18) δ − b d (cid:19) − V ε g , (35)where ξ = r δ − δbd + b a π . On the other hand, for a quantum anti-dot, we take thelimit (cid:0) V /ρ (cid:1) → . In this case, by maintaining the de-formed potential, we have Figure 2. The variations of threshold frequency of absorption ̟ (in unit of ǫ g ) as function of B for several values of the r parameter. In (a) we display ̟ for absence of a flux field δ = 0 and in (b) we display ̟ for δ = 1 . . E e ε g = κ (cid:18) ς + 1 − δ + b d (cid:19) − V ε g , (36) E h ε g = κ ′ (cid:18) ς + 1 + δ − b d (cid:19) − V ε g , (37)where ς = r δ − δbd + b a π + 2 m e V ρ ~ . Now we study the effect of the AB flux field, the pres-ence and absence of Burgers vector, quantum dot poten-tial and quantum anti-dot potential on the threshold fre-quency of absorption ̟ for the typical 2D structure ofGaAs, with the following parameters: V = 0 . meV, a = 5 Å, d = 15 nm, m e = 0 . m and m h = 0 . m ,where m is the free electron effective mass. In Fig. 2, weshow the variations of the threshold frequency of absorp-tion ̟ (in units of ε g ) for a quantum dot as a functionof magnetic field (in units of Tesla) for different values ofthe ratio λ = b/ d , which characterize the influence of ascrew dislocation and the parameter r = b/a . Figure 3. The variations of threshold frequency of absorption ̟ (in unit of ǫ g ) as function of ρ for several values of the r parameter. In (a) we depicted ̟ for δ = 0 and in (b) ̟ is plotted for δ = 1 . . In Fig. 2(a) shows the variations of the threshold fre-quency of absorption ̟ at a fixed energy gap ε g , thechemical potential V and AB flux δ = 0 , for four valuesof positive λ and r . It is seen that ̟ increases whenthe applied magnetic field increases. It is easily seen fromthe figure that the dependence of ̟ on B is nonlinearfor small applied magnetic fields. On the other hand, byincreasing the magnetic field the lines remain linear. It isalso noted that when r increases, the curves of frequency ̟ are pushed up towards of the growth r .The Fig. 2(b) illustrates the behavior of ̟ for dif-ferent values of the parameter r and fixed AB flux δ = 1 .From this figure, we can see that the threshold frequency ̟ displays a minimum for weak magnetic field. On theother hand, for strong magnetic field behavior is linear.One interesting aspect is that the magnetic flux δ con-tributes to the cancellation of screw effect of the model.In the figure this cancellation occurs when δ = 1 and r = 1 .In Fig. 3, we have plots of the variations of the thresh-old frequency of absorption ̟ with quantum dot size(in units of ρ ). It is seen in Fig. 3(a) that ̟ decreases Figure 4. The anti-dot frequency of absorption ̟ as func-tion of B for several values of the r parameter. In (a) wedisplay ̟ for absence of a flux field δ = 0 . In (b) we display ̟ for δ = 1 . when the quantum dot size increases in absence of theAB flux δ = 0 . In Fig 3(b), we can see that there is anoverlap of the curves for r = 0 and r = 1 when δ = 1 , thisindicates to us that the AB flux field help to be canceledthe screw dislocation.The effect of the AB flux field on the anti-dot thresholdfrequency of absorption ̟ are shown in Fig. 4. InFig. 4(a), we plot the variation of the anti-dot thresholdfrequency of absorption ̟ in the absence of the ABflux field δ = 0 as a function of magnetic field. We findthat the dependence of ̟ on B is linear. Fig 4(b)demonstrates the dependence of the anti-dot thresholdfrequency on B at δ = 1 . with different values of r . Thebehavior of ̟ is similar frequency of absorption of Fig.4(a), but the value of ̟ for δ = 1 is always less thatfrequency of absorption ̟ when δ takes the value . V. CONCLUDING REMARKS
In conclusion, we have investigated the energy levelsfor a 2DEG around a screw dislocation under the pseu-doharmonic interaction consisting of quantum dot andanti-dot potentials in the presence of an uniform strongmagnetic field B and AB flux.It is known that such a defect on elastic media gen-erates a torsion field that acts on the particle as if anexternal AB flux were being applied to it. In the usualAB effect, the charged particles in the presence of anuniform magnetic field may be confined to a plane per-pendicular to the field lines. This is not possible in thecase of torsion, which needs motion in three-dimensionalspace in order to show up its effects. Because of this fact, we have considered a quasi two-dimensional electron gasconfined on a thin interface in such a way that the effectsof torsion can manifest. Due to the existence of a screwdislocation, we have considered the effects of two contri-butions: a covariant term, which comes from the geomet-ric approach in the continuum limit and a noncovariantrepulsive scalar potential. Both appear due to elasticdeformations on a semiconductor with such kind of topo-logical defect. We have found that this noncovariant termchanges significantly the energy levels of electrons in thissystem. As we have said above, these defects representsa problem since they interfere in the electronic propertiesof the materials by way of scattering, as we can note byanalyzing the non covariant potential in Eq. (3). There-fore, investigations on how this kind of defect influencesthe dynamics of carriers in common semiconductors areimportant for the improvement of electronic technology.In our case, the modification introduced by such topolog-ical defect in the light absorption coefficient is due to aneffective angular momentum induced by torsion. For thethreshold frequency value of absorption, we have foundthat, as the Burgers vector increases, the curves of suchfrequency are pushed up towards its growth. Moreover,it was noted that the singular effects can take place aswell. It is also noted that the AB flux can be tuned inorder to cancel the influence of the screw dislocation inthese physical quantities. ACKNOWLEDGMENTS
This work was supported by the Brazilian agenciesCNPq, FAPEMA and FAPEMIG. ∗ cleverson.filgueiras@dfi.ufla.br † moises.leyva@dfi.ufla.br ‡ [email protected] L. D. Landau and E. M. Lifschitz,
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