Electronic Properties of Graphene Quantum Ring with Wedge Disclination
EElectronic Properties of Graphene Quantum Ringwith Wedge Disclination
Abdelhadi Belouad a , Ahmed Jellal a,b ∗ and Hocine Bahlouli ca Laboratory of Theoretical Physics, Faculty of Sciences, Choua¨ıb Doukkali University , PO Box 20, 24000 El Jadida, Morocco b Canadian Quantum Research Center, 204-3002 32 Ave Vernon,BC V1T 2L7, Canada c Physics Department, King Fahd University of Petroleum & Minerals,Dhahran 31261, Saudi Arabia
Abstract
We study the energy spectrum and persistent current of charge carriers confined in a graphenequantum ring geometry of radius R and width w subjected to a magnetic flux. We consider thecase where the crystal symmetry is locally modified by replacing a hexagon by a pentagon, square,heptagon or octagon. To model this type of defect we include appropriate boundary conditionsfor the angular coordinate. The electrons are confined to a finite width strip in radial direction bysetting infinite mass boundary conditions at the edges of the strip. The solutions are expressed interms of Hankel functions and their asymptotic behavior allows to derive quantized energy levelsin the presence of an energy gap. We also investigate the persistent currents that appear in thequantum ring and how wedge disclination influences different quantum transport quantities. PACS numbers: 81.05.ue, 81.07.Ta, 73.22.PrKeywords: Graphene, quantum ring, wedge disclination, energy levels, persistent currents. ∗ [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Introduction
Graphene is a two-dimensional lattice of carbon atoms that are arranged in a honeycomb patternforming a hexagonal lattice [1]. After its experimental discovery, graphene has attracted a of attention,which is in part is due to its exotic electronic properties. Some of these include high electron mobility,excellent conductivity, peculiar tunneling phenomena and other interesting transport and structuralproperties, which would be too many to list [2]. Graphene is interesting from a fundamental researchperspective, as well as for potential technological applications [3], for example, in the study of liquidcrystals [4], solar cells [5] and high-frequency electronic devices [6]. Particles in graphene near highsymmetry K points are described by a low energy effective model, the Dirac-Weyl Hamiltonian formassless charged particles. One of its striking features is the linear gapless energy dispersion relationrepresented by conic conduction and valence bands [1].Recently, graphene-based quantum rings produced by lithographic techniques have been experi-mentally investigated [7]. These systems have been studied theoretically using a tight-binding model,which does not provide simple analytical expressions for eigenstates and eigenvalues [8, 9]. It has alsobeen shown [10] that it is possible to realize quantum rings of finite width by using electric fieldsto confine electrons in this geometry. The influence of an applied magnetic field has drawn a lot ofattention [11] in the context of the important and interesting physical properties of quantum rings ingraphene. In this respect, one has to mention several recent theoretical studies related to differentproperties of quasiparticles confined in nanostructures such as quantum dots [12–14] and quantumrings [15, 16]. However, some of the most interesting effects on quasi-particles in graphene are dueto the presence of a disclination defect, which can significantly alter the electronic structure, mag-netic and transport properties [2]. The desire to employ graphene for studying curvature effects ismotivated by the simplicity of the Hamiltonian and the important potential applications of graphenein nanoelectronics and potential future quantum computing devices [2]. An individual dislocation infree-standing graphene layers has been imaged using transmission electron microscopy [17]. Topologi-cal defects resulting from either kinetic factors or substrate imperfections have also been reported forepitaxial graphene grown on SiC [18], Ir (111) [19] and polycrystalline Ni surfaces [20].In this paper we follow a similar set of ideas and consider a quantum ring of graphene with a wedgedislocation that can be understood from Volterras cut-and-glue constructions [21] The calculations areperformed in the continuum approximation limit in the vicinity of the Dirac points. After a generalmodel description, we solve the model taking into account the radial and angular degrees of freedom.We find eigenspinors in terms of the Hankel functions showing quantized energy levels in the asymptoticlimit. These results allow us to end up with a gap opening separating the conduction and valencebands. We also find analytical expressions for the persistent currents as a function of ring radius, totalmomentum, magnetic field, width of quantum ring and an integer index n . Such an index is inducedby disclination defect and quantifies the curvature of our geometry. To investigate the behavior of oursystem, we provide numerical studies for a suitable selection of the physical parameters characterizingour system.The manuscript is organized as follows. In section 2, we present our theoretical model based on theDirac Hamiltonian to describe the new geometry obtained via Volterra construction. By introducing amagnetic flux, we give the analytic solutions for eigenvectors and quantized eigenenergies in section 3.1e then explicitly determine the corresponding persistent currents generated by rotating our systemin section 4. In section 5, we discuss different numerical results related to the energy spectrum andthe persistent currents. Section 6 contains a summary of the main results and conclusions. We consider a graphene quantum ring of radius R and width w in the presence of a magnetic fluxas shown in Figure 1a and study its electronic properties. We use the Volterra construction [21] tomodel the disclination defect of our system by the regularized rings of radius R and R around theapex and the removed wedge disclination as presented in Figure 1b. R R R w ( a ) R R R α = φ = α = π , φ = π - n π α ( b ) Δ = ∞ Δ = Δ = ∞ Figure 1 – (color online) (a): The conical ring after Volterra construction. (b): Unfolded plane of lattice where a wedgeof angle nπ/ is removed ( n = 1 here). A potential (sketch on the left part of the ring) confines the electrons in the lowestradial mode on a ring of radius R , avoiding the singularity at the origin. We rescaled the angle ϕ of the unfolded plane to thenew angle α = ϕ/ (1 − n ) . The carbon atoms of the removed sector are denoted by open symbols, those which remain afterthe cut are represented in solid symbols. Particles in graphene have an electronic band structure with low energy band crossings at twoinequivalent high symmetry K and K (cid:48) points, at low energies one can neglect contributions away fromDirac points. Because of translational symmetry the Hamiltonians at the two different Dirac pointscan be described independently as H = v F [ τ z σ x p x + σ y p y ] + ∆( r ) τ z σ z (1)where σ i , τ i are Pauli matrices denoting the sublattice and valley degrees of freedom, respectively, v F is the Fermi velocity and ∆( r ) is the confining potential of the ring as shown in Figure 1, which isdefined by ∆( r ) = (cid:40) , R ≤ r ≤ R ∞ , otherwise . (2)The Hamiltonian (1) acts on the two spinorsΨ τ ( r, ϕ ) = (cid:32) Ψ A ( r, ϕ )Ψ B ( r, ϕ ) (cid:33) (3)where Ψ A/B ( r, ϕ ) is the spinor describing either of the two graphene sublattices A and B . It isknown that deformations of the honeycomb lattice enter the continuum description via fictitious gaugefields [22]. As we review below, topological point defects manifest themselves by spatially well-localizedfluxes of the fictitious fields [23]. 2e mathematically discuss the new geometry obtained via Volterra construction. Indeed, for thenon-rotated system, the eigenvalue equation Hψ (0) = Eψ (0) implies that the new spinor ψ ( ϕ ) = e iϕσ z τ z / ψ (0) (4)fulfils H ( R z ( ϕ )) ψ ( ϕ ) = Eψ ( ϕ ) after rotation where R z ( ϕ ) is a rotation matrix. This is because theHamiltonian (1) transforms as H ( R z ( ϕ )) = e iϕσ z τ z / He − iϕσ z τ z / (5)which can easily be checked. Let us remove a sector from the graphene sheet (e.g. with a pentagonreplacement) and then glue the sheet back together at the edge where we cut it as shown in Figure1b. Now one of the cuts can be chosen by convention to be at angle 0 and the other at angle − nπ/ n is the curvature index that quantifies disclination defects. After gluing both of these anglesbecome the same point and are related by a rotation of angle − nπ/
3. Then, the wavefunction at thispoint has to be single valued and gives directly the boundary conditionΨ( r, α = 2 π ) = − e i π [1 − ( n/ σ z τ z / Ψ( r, α = 0) (6)where we rescale the angle ϕ of the unfolded plane to the new angle α = ϕ/ (1 − n ), α is varyingfrom 0 to 2 π . This, however, is not the complete story and one has to look slightly beyond thecontinuum model to find the full story. Particularly, let us go to momentum space and cut a wedgein the reciprocal lattice and then glue the lattice back together. We then find that for angles 2 nπ/ K points get glued onto each other. For angles (2 n + 1) π/
3, however, K points get gluedonto inequivalent K (cid:48) points (a 60 ◦ rotation connects inequivalent K points). Now one can switch the K and K (cid:48) blocks in the Hamiltonian by the unitary transformation e iπσ y τ y / . Keeping track of bothobservations one finds the boundary conditionΨ( r, α = 2 π ) = − e i π [ − ( nσ y τ y / − ( n/ σ z τ z / Ψ( r, α = 0) . (7)For a general index n we introduce polar coordinates ( r, ϕ ) defined in the unfolded plane according toFigure 1b and perform two singular transformationsΨ( r, α = 2 π ) = λ ( ϕ ) µ n ( α )Ψ( r, α = 0) (8)where λ ( ϕ ) = e iϕσ z τ z / and µ n ( α ) = e inασ y τ y / . The first one λ ( ϕ ) transforms Ψ to a spinor thatis expressed in the local frame ( (cid:126)e r , (cid:126)e ϕ ) with unit vectors along the radial and azimuthal directions,effectively replacing ∂ r by ∂ r + 1 / (2 r ) in the Hamiltonian. The second µ n ( α ) introduces a matrix-valued gauge field into the Hamiltonian, effectively replacing ∂ α by ∂ α + i n σ y τ y . One should note thatthis transformation is performed to simplify the boundary conditions [24]. At this stage, we introduce an external source related to the magnetic field. Indeed, in the effectiveHamiltonian, a magnetic flux Φ crossing through the origin is accounted for by replacing momentum (cid:126)p by the conical momentum (cid:126)p + e (cid:126)A with vector potential (cid:126)A = ΦΩ n Φ r ( − sin ϕ, cos ϕ ) (9)3here Ω n = 1 − n and Φ = he is the magnetic flux quantum. We write the Hamiltonian (1) in polarcoordinate and apply the transformations through λ ( ϕ ) and µ n ( α ), namely˜ H ( r, α ) = λ † µ † n Hµ n λ (10)to obtain the transformed Hamiltonian˜ H ( r, α ) = (cid:126) v F (cid:18) k r − i r (cid:19) τ z σ x + (cid:126) v F (cid:18) k α + ΦΩ n Φ r + n n r τ z (cid:19) σ y + ∆ τ z σ z (11)we have set k r = − i ∂∂r and k α = − ir Ω n ∂∂α . Since ˜ H ( r, α ) commutes with the total angular momentum J z = L z + S z , then we can split the eigenspinors into angular and radial partsΨ( r, α ) = e ijα (cid:32) χ A ( r ) iχ B ( r ) (cid:33) (12)such that j = m + 1 / J and m ∈ Z . Inside the quantum ring, we use H Ψ = E Ψand show that the radial components satisfy (cid:20) ∂∂r + 12 r + 1 r Ω n (cid:16) j + β + nτ (cid:17)(cid:21) χ B ( r ) = E (cid:126) v F χ A ( r ) (13) (cid:20) ∂∂r + 12 r − r Ω n (cid:16) j + β + nτ (cid:17)(cid:21) χ A ( r ) = − E (cid:126) v F χ B ( r ) (14)where we have defined the quantum quantity ν = n (cid:0) m + + β + nτ (cid:1) and β = ΦΦ is the dimension-less flux. Decoupling the above equations, we arrive at the Hankel differential equation for χ A ( r ) (cid:34) r ∂ ∂r + r ∂∂r + (cid:18) Er (cid:126) v F (cid:19) − (cid:18) ν − (cid:19) (cid:35) χ A ( r ) = 0 (15)giving rise to the solution of the radial part of the eigenspinors χ τ ( ρ ) = a τ H (1) ν − ( ρ ) i sgn( E ) H (1) ν + ( ρ ) + b τ H (2) ν − ( ρ ) i sgn( E ) H (2) ν + ( ρ ) (16)where H (1 , ν ( ρ ) are Hankel functions of the (first, second) kind, ( a τ , b τ ) are the normalization constantand we set the variable as ρ = Er (cid:126) v F .Now we look for the eigenvalues associated to the above eigenspinors. To this end, we first de-termine the coefficients a τ and b τ using the boundary condition of the ring induced by ∆( r ) suchthat ∆( r ) → ∞ outside the graphene ring. For this purpose, we adopt the infinite mass boundaryconditions [8,14] at the two points r = R − w/ r = R + w/
2, with R = R + R and w = R − R .Indeed, for χ B ( ρ ) = − iτ χ A ( ρ ) we find a τ b τ = − H (2) ν − ( ρ ) + τ sgn( E ) H (2) ν + ( ρ ) H (1) ν − ( ρ ) + τ sgn( E ) H (1) ν + ( ρ ) (17)and χ B ( ρ ) = iτ χ A ( ρ ) gives a τ b τ = − H (2) ν − ( ρ ) − τ sgn( E ) H (2) ν + ( ρ ) H (1) ν − ( ρ ) − τ sgn( E ) H (1) ν + ( ρ ) (18)4here R is the ring radius and w its width, see Figure 1. We can rearrange (17) and (18) to obtainthe energy eigenvalue equation z (1) = z (2) , with z ( i ) = H ( i ) ν − ( ρ ) − τ sgn( E ) H ( i ) ν + ( ρ ) H ( i ) ν − ( ρ ) + τ sgn( E ) H ( i ) ν + ( ρ ) , i = 1 , . (19)To obtain an analytical approximation of the energies, we use the asymptotic form of the Hankelfunctions for large ρ , including corrections up to order 1 /ρ . That is H (1) ν ( ρ ) = (cid:112) πρe i ( ρ − νπ/ − π/ (1 + δ ν ) (20)where δ ν = − (4 ν − ν − ρ + i ν − ρ + O ( ρ − ) . (21)Now using the fact that (cid:104) H (1) ν ( ρ ) (cid:105) ∗ = H (2) ν ( ρ ) and z ( i ) = z ∗ ( i ) , we obtain the approximate energy levels E mn = E (cid:34) τ π ± (cid:114)(cid:16) τ π (cid:17) + ν (cid:16) wR (cid:17) (cid:35) (22)where we have defined E = (cid:126) v F w . Explicitly, we have E mn = E τ π ± (cid:115)(cid:16) τ π (cid:17) + 1Ω n (cid:18) m + 12 + β + nτ (cid:19) (cid:16) wR (cid:17) (23)which are in fact the quantized energies and reflect the basic characteristics of our system that willbe numerical analyzed. In graphene systems the persistent current (PC) can generated as follows. We wrap up grapheneribbon to form a tube or ring with a magnetic flux passing through the hole, then the helical edgestate provides a robust channel for PC. On the other hand, the existence of PC in a normal metal ringwas first proposed by Buttiker, Imry and Landauer [25]. Recently, with the advent of nano-fabricationtechniques, several experimental investigations have been made to confirm the existence [26, 27] andthe periodicity [28,29] of PC in semiconductor quantum rings. As for graphene rings, there are severalstudies of PC with various geometrical shapes (without spin-orbit interaction). For example, flat rings(similar to Corbino-disks) with various types of boundaries [8, 30, 31] or a tube with zigzag edges [32]and folding a graphene ribbon into a ring [33]. In general the electronic current element between thefirst neighbor sites i and j is given by [34] I ij = 4 e (cid:126) Im (cid:88) n f ( E n ) c ∗ in H ij c jn (24)where H ij is the Hamiltonian of the system, f ( E n ) being the Fermi function, c in are the eigenvectorscorresponding to the eigenenergy E n . The total persistent current of a ring can be calculated by thefollowing formula I pc = − (cid:88) n ∂E n ∂ Φ (25)5ith the magnetic flux Φ.Motivated by the above findings, we calculate the persistent current, carried by a given electronstate, for our model of quantum ring in graphene with a disclination. Then, as far as our system isconcerned we have the relation I = − (cid:88) m,n ∂E mn ∂β . (26)and after some calculation, we end up with I = ∓ I (cid:88) m,n n m + + τn + β (cid:114)(cid:0) τπ (cid:1) + (cid:0) m + + τn + β (cid:1) (cid:16) wR Ω n (cid:17) (27)with I = E Φ (cid:0) wR (cid:1) . This result will be investigated numerically for a suitable choice of physicalparameters. We consider graphene quantum ring, with inner and outer radii R and R , respectively, of width w = R − R and a confining potential ∆( r ) for the ring defined in (2) as shown in Figure 1. Here weassume that an applied magnetic flux Φ only passes through a disk region enclosed by the inner circle(i.e. R ≤ r ≤ R ), allowing us to explore the valley energy entirely due to the magnetic flux effect.We will analyse the energy levels and the persistent currents to extract some informations about oursystem. .................................................................................................................................................................. n = β =- β = β = τ = τ =- . ( a ) - -
10 0 10 20 - - E / E .................................................................................................................................... ( b ) n = β =- β = β = τ =- . τ = - -
10 0 10 20 - - E / E .......................................................................................................................................................................................................................... ( c ) n =- β =- β = β = τ =- τ = . - -
10 0 10 20 - - E / E .............................................................................................................................................................................................................................................. n =- ( d ) β =- β = β = τ =- τ = . - -
10 0 10 20 - - E / E Figure 2 – (color online) Energy level for a graphene with pentagon defect ( n = 1 ), square defect ( n = 2 ), heptagondefect ( n = − ) and octagon defect (n=-2) defect as function of the quantum angular moment m for different values of themagnetic flux: β = 0 (blue curve), β = 0 . (red curve), β = − . (green curve).
6n Figure 2, we present the results for the energy levels as function of the quantum angular moment m for a graphene layer. We show in panel (a) the case of a pentagon defect n = 1, in panel (b) thecase of a square defect n = 2, in panel (c) the case of a heptagon defect n = − n = −
2. We consider different valleys K ( τ = 1)/ K (cid:48) ( τ = −
1) and threevalues of β = − . β = 0 (blue curve) and β = 0 . m = − (cid:18)
12 + β + nτ (cid:19) (28)for a given value for β and is independent of w and R [35]. Comparing our results with those in [36],we notice the creation of an induced pseudo-field β = nτ due to the presence of the disclinationeffect. Such field is measured by a non-zero integer index n that quantifies the curvature and allowsthe energy spectrum to be displaced in opposite directions of the two Dirac points K and K (cid:48) . As aconsequence, in a single valley the dependence of energy, for both the conduction and valence bands,as function of m is symmetric according to ± mE ( τ, m, n, β ) = E ( τ, − m, n, − β ) . (29)However, the energy also has a valley-index τ -dependent symmetries E ( τ, m, n, β ) = − E ( − τ, m, n, β ) , E ( τ, m, n, β ) = − E ( − τ, − m, n, − β ) . (30)Furthermore, in Figures 2(a,b,c,d) we see that there is an energy gap between the conduction andvalence bands, which is given by ∆ E = 2 E (cid:114)(cid:16) τ π (cid:17) + ν (cid:16) wR (cid:17) (31)and depending on different values of the physical quantities such as quantum angular moment m ,curvature index β n = β + nτ , quantum ring parameters w , R and valley index τ .Figure 3 shows the energy levels as a function of the magnetic flux β for different valleys K ( τ = 1)and K (cid:48) ( τ = −
1) while the disclination is defined by the curvature index ( n = − , − , , , β = 0), the states with an quantum angular number m in the valleys K and K (cid:48) have different energy for the valence and conduction bands E ( τ, m, β, n ) = E ( τ, − m, β, n ) (32)that is because of the deformation of graphene. For β (cid:54) = 0, the states with an opposite m in K and K (cid:48) have the same energies E ( τ, m, β, n ) = E ( τ, − m, − β, n ) . (33)However, it is interesting to notice that the degeneration may still possibly remain for a non-zeroflux, which is not the case for graphene with no defect ( n = 0) [8, 36]. This comes from the periodicdependence of the energy spectrum on magnetic flux β n = β + nτ . Physically the periodic dependenceof the energy spectrum on β n originates from the combined effect of graphene defect and the presenceof a magnetic flux, which affects the phase of the wavefunction.7 a ) m = m = m = m = - m = - m = - m = n = ( pentagon ) τ = - - - - β E / E m = m = m = m = - m = - m = - ( e ) n = ( pentagon ) τ =- m = - - - - β E / E m = m = m = m = - m = - m = - n =- ( heptagon ) τ = ( b ) m = - - - - β E / E m = m = m = m = m = - m = - m = - n =- ( heptagon ) τ =- ( f ) - - - - β E / E m = m = m = m = - m = - m = - m = n = ( square ) τ = ( c ) - - - - β E / E m = m = m = - m = - m = - m = n = ( square ) τ =- m = ( g ) - - - - β E / E m = m = m = m = - m = - m = - m = n =- ( octagon )( d ) τ = - - - - β E / E m = m = m = m = - m = - m = m = - n =- ( octagon ) τ =- ( h ) - - - - β E / E Figure 3 – (color online) Energy levels with m = 0 (blue), m = 1 , , (red) and m = − , − , − (green) as function ofmagnetic flux β with R = 30 nm, w = 5 nm for pentagon (n=1) panels (a,e) , heptagon ( n = − ) panels (b,f), square (n=2)panels (c,g), octagon ( n = − ) panels (d,h). The left panels are for τ = 1 and the right ones are for τ = − . The energy levels
E/E as function of ring radius R are shown in Figure 4 for valleys K ( τ = 1)and K (cid:48) ( τ = −
1) with β = 1 / w = 5 nm, m = 0 (black line), m = 1 (blue dashed), m = − n = 1), panel (b): square defect ( n = 2), panel (c): heptagondefect ( n = − n = − E/E have branches converginglike R with an energy gap between the conduction and valence bands, which depends on the curvature8ndex n . This behavior quantitatively differs from that found in standard quantum rings in graphene( n = 0) in the absence and presence of magnetic flux [36]. In the limit of large radii R , we find thatthere is a constant energy gap ∆ E = E π , which does not depend on the index n . As a consequence,the symmetry E ( τ, m = 0 , n ) = − E ( − τ, m = − , n ) (34)holds for all n . We also find the asymmetry E ( τ, n, m ) (cid:54) = E ( τ, − n, m ) is broken. ( a ) n = ( pentagon ) ── τ = - - τ = - - - - ( nm ) E ( m e v ) ( b ) n =- ( heptagon ) ── τ = - - τ = - - - - ( nm ) E ( m e v ) ( c ) n = ( square ) - - τ = - ── τ = - - - ( nm ) E ( m e v ) ( d ) n =- ( octagon ) ── τ = - - τ = - - - - ( nm ) E ( m e v ) Figure 4 – (color online) Energy levels with m = 0 (red), m = 1 (blue) and m = − (green) for β = 0 . and w = 5 nmand different values of the index n = 1 (pentagon), -1 (heptagon), 2 (square), -2 (octagon) for τ = 1 (dashed curve), τ = − (solid curve). ( a ) τ = ( nm ) ( nm ) E ( m e v ) ( b ) τ = - ( nm ) ( nm ) E ( m e v ) Figure 5 – (color online) Energy levels for a graphene layer with a square defect as a function of R for different values ofwidth w ( . , . , . , ) nm and β = 0 . . The left panel is for τ = 1 and the right one is for τ = − . To better understand the dependence of the energy levels
E/E on the confinement of the quantumring of graphene defect, we present in Figure 5 E/E as a function of the ring radius R , at constantflux β = 0 . n = 2 (square) for several values of ring width w . In the regime R < . E ( τ ) = E ( − τ ) = 0, for R = 0 . E ( τ = 1) = 8800 meV9or the valley K and E ( τ = −
1) = 3000 meV for the valley K (cid:48) . We find that the energy spectrumdecreases with the increase in R . We observe that for small values of w , we reach large values of E ,such effect is due to the strong confinement. By increasing R to higher values, we notice that theenergy spectrum changes only slightly and converges towards constant values. It is clearly seen thatthe energy spectrum exhibits an asymmetry E ( τ ) (cid:54) = E ( − τ ). n = n = n = - n = - ── τ = τ = - - - - - β I / I Figure 6 – (color online) Persistent current I as a function of magnetic flux β for several values of the index n (-2, -1, 1,2). The values for the other parameters are wR = 0 . and τ = ± . In Figure 6, we present the persistent current I as function of magnetic flux β . We consider indexvalues n = − , − , , wR = 0 .
5. Note that solid and dashed lines correspond to thevalleys K ( τ = 1) and K (cid:48) ( τ = − n . It shows the degeneration of the valley I ( τ ) = I ( − τ ) in thecases β ≤ − β ≥
4, but this degeneration is broken in the interval − < β <
4. Then, thepersistent current oscillates as a function of magnetic flux with the same period as in the well-knownAharonov-Bohm oscillations [8]. The persistent current displays the following symmetries I ( τ, β, n ) = I ( − τ, − β, n ) , I ( τ, β, n ) = − I ( τ, − β, n ) . (35)Moreover, the amplitude of the current depends on the curvature index n . n = n = n = - n = - ── τ = τ = - - - - / R I / I Figure 7 – (color online) Persistent current I/I as a function of the ratio w/R for valley K ( τ = 1 ) and K (cid:48) ( τ = − )with β = 0 . . The red, green, blue and magenta curves correspond to the curvature index n = 2 , , − , − , respectively. Results for the persistent current I of a graphene ring with disclination defect as a function of w/R are shows in Figure 7. The red, green, blue and magenta curves are for the indices n = 2 , , − , − K and K (cid:48) , respectively.10he curves have a fixed maximum I max for w/R →
0, which can be discussed by taking into accountthe symmetry I max ( τ, n ) = I max ( − τ, − n ) . (36)In addition, it can be noticed that the curves are different for the same valley exhibited by our system.We can clearly see that when w/R increases then the persistent current I decreases up until w/R (cid:39) K and K (cid:48) is clearly lifted. In the regime w/R >
1, the persistent current I becomes almost constant. We have studied a graphene model in the shape of a quantum ring of inner radius R , outer radius R and width w subjected to a magnetic flux. We have considered a disclination defect using a procedureknown as the Volterra process [21]. This transformation can be looked at as a cut-and-glue processwhere we are cut and remove a sector in the quantum ring of a graphene layer. Due to the sixfoldrotational symmetry of graphene honeycomb lattice the removed angular sector must be a multiple of π . We have involved a topological defect in a graphene layer using a fictitious gauge field following theapproach used in [22] where a gauge field is introduced in the Dirac equation in order to reproduce thealready known effect of the disclination on the behavior of the spinor. We have obtained analyticalexpressions for the energy levels and the corresponding eigenspinors as well as the persistent currentusing the infinite mass boundary condition [8, 14]. Our numerical results were exposed in termsvariations in the ring radius R , total moment, magnetic field, ring width and integer-valued curvatureindex n .In particular, we have shown that the disclination modifies the energy spectrum and shifts themagnetic flux in opposite directions for the valleys K and K (cid:48) . Furthermore, the charge current isaffected by the pseudo-magnetic field produced by the disclination having opposite signs at the twoDirac points. We have found an interesting behavior in the presence of a induced magnetic field,which has no analog in quantum rings with no defect ( n = 0). Indeed, for a graphene quantumring with a defect ( n (cid:54) = 0), we have obtained a gap opening in the energy spectrum between theconduction and valence bands that depends on the index n . We also have found that the persistentcurrent has oscillations as a function of the magnetic flux with an analogous period to the famousAharonov-Bohm oscillations [8] and obey the following symmetries I ( τ, β, n ) = I ( − τ, − β, n ) and I ( τ, β, n ) = − I ( τ, − β, n ). Moreover, we have shown that the amplitude of the energy spectrum andthe current depends on the index n of the defects in graphene and on the width w of the quantumring. Acknowledgments
The generous support provided by the Saudi Center for Theoretical Physics (SCTP) is highly appre-ciated by all authors. AJ and HB acknowledge the support of KFUPM under research group projectRG181001. HB also acknowledges discussions with Michael Vogl.11 eferenceseferences