The electronic spectra of long carbon nanotubes (CNTs) can, to a very good
approximation, be obtained using the dispersion relation of graphene with both
angular and axial periodic boundary conditions. In short CNTs one must account
for the presence of open ends, which may give rise to states localized at the
edges. We show that when a magnetic field is applied parallel to the tube axis,
it modifies both momentum quantization conditions, causing hitherto extended
states to localize near the ends. This localization is gradual and initially
the involved states are still conducting. Beyond a threshold value of the
magnetic field, which depends on the nanotube chirality and length, the
localization is complete and the transport is suppressed.Comment: 5 pages, 3 figure

del Valle, Miriam

Grifoni, Milena

Jhang, Sung Ho

Marganska, Magdalena

Strunk, Christoph

English

arXiv

Magdalena Margańska

Miriam del Valle

Sung Ho Jhang

Christoph Strunk

Milena Grifoni

Crossref

Physical Review B

Localization induced by magnetic fields in carbon nanotubes

The electronic spectra of long carbon nanotubes (CNTs) can, to a very good approximation, be obtained using the dispersion relation of graphene with both angular and axial periodic boundary conditions. In short CNTs one must account for the presence of open ends, which may give rise to states localized at the edges. When a magnetic field is applied parallel to the tube axis, it modifies both momentum quantization conditions, causing hitherto extended states to localize near the ends. We study analytically and numerically the appearance and evolution of this peculiar localization phenomenon in CNTs of any nonarmchair chirality, including the electron spin. Conductance calculations show different evolution of spin up and down states in increasing magnetic field

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PHYSICAL REVIEW B 83, 193407 (2011)Localization induced by magnetic fields in carbon nanotubesMagdalena Margan´ska, Miriam del Valle, Sung Ho Jhang, Christoph Strunk, and Milena GrifoniUniversity of Regensburg, D-93053 Regensburg, Germany(Received 4 March 2011; revised manuscript received 21 March 2011; published 23 May 2011)The electronic spectra of long carbon nanotubes (CNTs) can, to a very good approximation, be obtained usingthe dispersion relation of graphene with both angular and axial periodic boundary conditions. In short CNTsone must account for the presence of open ends, which may give rise to states localized at the edges. When amagnetic field is applied parallel to the tube axis, it modifies both momentum quantization conditions, causinghitherto extended states to localize near the ends. We study analytically and numerically the appearance andevolution of this peculiar localization phenomenon in CNTs of any nonarmchair chirality, including the electronspin. Conductance calculations show different evolution of spin up and down states in increasing magnetic field.DOI: 10.1103/PhysRevB.83.193407 PACS number(s): 73.63.Fg, 73.23.Ad, 75.47.−m, 85.75.−dThe existence of geometry-induced localized states at thezigzag edge of graphene nanoribbons has been predictedsome years ago,1,2 recently seen experimentally and shownto influence the transport in graphene quantum dots.3 Similarstates at the ends of single-wall nanotubes have been observed4and studied.5–8 In zigzag-armchair nanotube junctions, theinterface states calculated to appear at the junction were iden-tified with the end states of the zigzag nanotube fragment.9,10In this work we study another type of edge states, namelythose which arise when initially extended states becomelocalized in a parallel magnetic field.11,12 We present adetailed analytical and numerical study of this effect in CNTsof arbitrary nonarmchair chirality, including the spin-orbitcoupling and the Zeeman effect. We find that this phenomenonoccurs also in those chiral CNTs which have no localizedend states when the magnetic field is absent. Numericalcalculations of the conductance of finite CNT devices in acontinuously varying magnetic field suggest the possibility ofspin-selective transport.The model. Our starting point is the tight-binding Hamilto-nian for a honeycomb lattice with one pz orbital per atom andwith the interatomic potential ˆV . If we calibrate our energyscale so that the on-site energies vanish, the Hamiltonian isgiven byˆH =∑i =jtij |zj 〉〈zi |, (1)where i and j are the lattice site indices, |zj 〉 is a pz orbital atsite j , and tij = 〈zj | ˆV |zi〉 is the hopping integral between thesites. This Hamiltonian nicely captures the properties of flatgraphene and CNTs. In order to properly describe finite sizenanotubes in the magnetic field it is necessary to include thePeierls phase and curvature effects in the hopping elementstij . We follow here the approach of Ando.13 For the sake ofclarity we shall initially neglect the spin-orbit coupling andthe Zeeman effect, as they do not change our main conclusion.The spin-dependent effects will be addressed later.The graphene coordinate system and the relevant real spacevectors are shown in Fig. 1(a), while the graphene Brillouinzone with K and K ′ points is shown in Fig. 1(b). In order to findthe appropriate boundary conditions and eigenstates of CNTswe use an approach based on the Dirac equation treatment.2,14Parallel magnetic field. The magnetic field modifies allhopping integrals by a Peierls phase factor. Its form can bederived using the substitution p → p − eA and readstij (B) = tij (0) exp{ieh¯∫ rjriA(r) · dr}. (2)In the cylindrical coordinates (r,ϕ,z), with the z directionaligned with the axis of the nanotube, a parallel magneticfield has coordinates B = (0, 0, B). In the tangential gaugethis gives A = (0, Br/2, 0). The Peierls phase then becomesieh¯∫ rjriA(r) · dr = i φφ0(ϕj − ϕi), (3)where φ is the magnetic flux threading the nanotube, φ0 = h/ethe flux quantum, and ϕj − ϕi is the difference between theangular coordinates of site j and site i.Curvature. In a nanotube the σ bonds are not orthogonal tothe pz orbitals and the hopping integral tij can be expressed astij (0) = 〈zj | ˆV |zi〉 = Vπ nin · njn + Vσ nit · nj t, (4)whereVπ andVσ are hopping parameters for the correspondingbonds.13 In our calculations we shall use the parameters fromRef. 15, Vσ = 6.38 eV and Vπ = −2.66 eV. The vector ni isa unit vector normal to the nanotube surface at the site i. Thecomponents nin (normal) and nit (tangential) are defined withrespect to a plane containing the σ bond between i and j andparallel to the CNT axis. The hopping integral tij (0) then reads〈zj | ˆV |zi〉 = Vπ cos(ϕi − ϕj )−(Vσ − Vπ )R2a2c[1 − cos(ϕi − ϕj )]2, (5)where R is the nanotube radius and ac = 1.42 A˚ is the bondlength in graphene.In order to find the CNT spectrum it is convenient to expressthe Hamiltonian in the Bloch wave basis. The Bloch waves forthe CNT sublattice p are given by16|p(k)〉 = 1√NN∑i=1eik·rpi |zpi〉, (6)193407-11098-0121/2011/83(19)/193407(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW B 83, 193407 (2011)(a) (b)(c) (d)FIG. 1. (Color online) (a) Fragment of a graphene lattice. Whenconsidering a CNT with chiral angle θ , we shall use the system ofcoordinates defined by directions perpendicular (x⊥) and parallel (x‖)to the tube axis. (b) Brillouin zone of graphene. (c) Real and imaginarysolutions of Eq. (17), determining the quantization of κ ′‖ as a functionof κ ′⊥. (d) Some of the real and imaginary solutions of Eq. (17) forthe Fermi subband (κ⊥ = 0) of an (18,0) CNT with 100 unit cells,with κ ′⊥ and κ ′‖ as functions of the magnetic flux φ.where N is the number of the unit cells. The Bloch wave onthe whole lattice is a linear combination of Bloch waves onindividual sublattices and can be written as|(k)〉 =∑p=A,Bηp(k)|p(k)〉 =(ηA(k)ηB(k)). (7)In this basis the Hamiltonian acquires the form,ˆH (k) =(0 HAB(k)H†AB(k) 0), (8)where HAB(k) =∑3i=1 ti eik·di, di are the vectors connectingan A sublattice atom with its neighbors, as shown in Fig. 1(a),and ti are the hopping integrals between an atom on sublatticeA and its neighbors. Because both the magnetic field and thecurvature are uniform along the whole nanotube, the ti do notdepend on the position of the initial A atom. This Hamiltoniancan be further expanded around the K (τ = 1) and K ′ (τ =−1) points [see Fig. 1(b)], yieldingˆHτ (κ) = h¯vF(0 eiτθ (τκ ′⊥ + iκ ′‖)e−iτθ (τκ ′⊥ − iκ ′‖) 0), (9)where θ is the CNT chiral angle, indices (⊥ , ‖) denote thecomponents perpendicular and parallel to the nanotube axis,respectively, h¯vF = 3|Vπ |ac/2, κ = k − τK, andκ ′⊥ = κ⊥ + τkc⊥ +1Rφφ0, (10a)κ ′‖ = κ‖ + τkc‖. (10b)The last term in (10a) is the Aharonov-Bohm contributionwhile kc⊥ and kc‖ are due to the curvature,kc⊥ =ac4R2(1 + 38Vσ − VπVπ)cos(3θ ), (11a)kc‖ = −ac4R2(1 + 58Vσ − VπVπ)sin(3θ ). (11b)In this derivation we used a small angle approximation,sin(ϕi − ϕj )  ϕi − ϕj , which is good already for CNTswith R  5 A˚. The energy eigenvalues of the HamiltonianareE± = ±h¯vF√(κ ′⊥)2 + (κ ′‖)2, ˜E± := E±/(h¯vF ). (12)Eigenfunctions of the Hamiltonian. The energy eigenstatesare a linear combination of Bloch waves. Since we haveexpanded the Hamiltonian around the K and K ′ points, thecorresponding Bloch waves and the coefficients ηp acquirethe index τ . We shall be usingτp(r,κ) = 〈 r |p(τK + κ) 〉. (13)Angular boundary condition. The wave function in theangular direction must be periodic. This imposesτp((2πR,x‖),κ) != ei2πn τp((0,x‖),κ),⇒ (τK⊥ + κ⊥) = nR, (14)which is the standard quantization condition.16Axial boundary condition. The wave function at the endsof the nanotube must satisfy open boundary conditions. Weshall derive them for a zigzag nanotube (θ = 0◦), but they arevalid for any other chirality except armchair,17 provided thenanotube edge is a so-called minimal boundary (there are noatoms with only one neighbor).The Hamiltonian (9) acting on the wave functions |τ (κ)〉,(7) with (13), gives two equations:eiτθ [τκ ′⊥ + iκ ′‖] ητB(κ) = ˜E± ητA(κ),e−iτθ [τκ ′⊥ − iκ ′‖] ητA(κ) = ˜E± ητB(κ).We choose then, up to a normalization factor,ητA(κ) = [τκ ′⊥ +iκ ′‖] and ητB(κ) = ±|ητA| e−iτθ . We can see from (10b) and(12) that the energies of states with κ‖ and −(κ‖ + 2τkc‖)are the same. The energy eigenstate is therefore a linearcombination of both:ψτ (r,E±) = a1 τ (r,(κ⊥,κ‖))+ a2 τ (r,(κ⊥, − (κ‖ + 2τkc‖))). (15)From the structure of the lattice in Fig. 1(a) we see thatwhen the graphene patch is rolled in order to create a zigzagnanotube, the lower CNT edge is formed entirely by Bsublattice atoms while the upper edge only by A sublatticeatoms. Therefore the wave function on this patch must vanishat the “missing” A atoms below the lower edge (x‖ = 0) andB atoms above the upper edge (x‖ = L). The conditions forthe sublattice components of ψτ (r,E±) areψτA((x⊥,0),E±) != 0,→ a1(τκ ′⊥ + iκ ′‖) + a2(τκ ′⊥ − iκ ′‖) = 0, (16a)193407-2BRIEF REPORTS PHYSICAL REVIEW B 83, 193407 (2011)ψτB((x⊥,L),E±) != 0,→ a1 eiκ ′‖L−iτkc‖L + a2 e−iκ ′‖L−iτkc‖L = 0. (16b)These equations lead to a constraint on the values of κ ′‖,τκ ′⊥ + iκ ′‖τκ ′⊥ − iκ ′‖= e2iκ ′‖L. (17)Thus the allowed values of κ‖ depend on κ ′⊥, and in particularon the Aharonov-Bohm flux φ. The quantity κ ′‖ can beeither real or imaginary. If it is real, the wave functiondescribes an extended state. If κ ′‖ is imaginary, then κ‖ mustbe complex, with its real part equal to the second term in(10b). Equation (17) has then one trivial (κ ′‖ = 0) and twonontrivial solutions. The latter describe evanescent waveslocalized near the ends of the nanotube,11 because the factorexp[i(τK + κ) · rpi] from (6) acquires a damping real part.The regions where κ ′‖ is real or imaginary are determinedby the value of τκ ′⊥ [see Fig. 1(c)]. The two localized statesolutions exist iffor K : κ ′⊥ > 1/L, for K ′ : κ ′⊥ < −1/L. (18)The spectrum of the CNT is then determined by the value ofthe magnetic field, which enters into κ ′⊥ via (10a). In order tocalculate the energy levels, the allowed values of κ‖ must befound from (17) for each value of κ ′⊥ separately, as shown inFig. 1(d) for an (18,0) zigzag CNT.The analytical method described above gives a remarkableagreement with the spectra obtained by the numerical diag-onalization of the nanotube Hamiltonian (1) [see Figs. 2(a)and 3(a)]. The energy of the decaying states tends to 0 withincreasing magnetic flux because for φ → ∞, |iκ ′‖| → κ ′⊥ forthe K point solutions [see Fig. 1(d)]. The CNT spectrummay contain localized (E = 0) states even at φ = 0, as canbe seen in Fig. 2(a) for an (18,0) CNT. If the neighboring(κ⊥ = 0) subbands lie on the Dirac cone and the condition(18) is fulfilled, then the lowest κ‖ states in the neighboringsubbands are localized, while the remaining ones have energiesin a higher range, appropriate for their subband. Whether theother subbands lie on the Dirac cone depends on the chiralityand diameter of the CNT.Spin effects. With spin, the Bloch waves (7) become four-component spinors and both the spin-orbit coupling (SOC) andthe Zeeman effect must be considered. They will be treated indetail elsewhere. Here we just note that SOC can be taken intoaccount by yet another shift of κ⊥, while the Zeeman effectsplits the energy:κ ′⊥ → κ ′⊥ + σ kSO, E± → E± + σμBφπR2, (19)kSO = 2δR(1 + 38Vσ − VπVπ), (20)where σ = +1/ − 1 for spin parallel/antiparallel to the CNTaxis, μB is the Bohr magneton, and δ is a parameter definingthe SOC strength. In our calculations we take δ ∼ 2.8 × 10−3,as, for example, in Ref. 18. The resulting spectra of (18,0) and(12,9) CNTs are shown in Figs. 2(c) and 3(b).Localization. Equation (18) defines the localization fluxφloc, at which the extended solutions morph into localized(a) (b)(c)(d)FIG. 2. (Color online) Magnetic-field-induced localization in a(18,0) zigzag CNT with 100 unit cells (L = 42.6 nm, θ = 0◦).(a) The spectra close to the Fermi level obtained by a numericaldiagonalization of the real space Hamiltonian (1) and analyticallyfrom the Dirac-like dispersion (12) with κ⊥ = 0 (E = 0 states) andκ⊥ = ±1/R (E ≡ 0 states), where κ‖ is defined by (17), neglectingthe spin. The black dashed line marks the onset of the localization.(b) The setup used for the conductance calculation. (c) Spectraobtained analytically with the electron spin included through (19).(d) Grayscale plot of conductance, in units of conductance quantumG0 = e2/h, as a function of φ and the chemical potential E, includingspin effects.states. This threshold flux depends on the spin via (19) andusing it together with (10a) we obtainφloc = τR(1L− kc⊥)φ0, φσloc = φloc − σRkSO φ0.(21)The value of φloc depends on the length of the nanotube. Forsufficiently long CNTs the spectra are very close to those ofthe infinite nanotubes.The localization induced by the magnetic field is gradual,in principle, allowing the involved states to conduct as long asthe two sublattice wave functions overlap. The evolution of aneigenstate in increasing magnetic field is shown in Fig. 3(c),through a sequence of plots of the wave function amplitudeat each atom, projected onto x‖. The apparent continuity ofthe curves is due to the overlap between close plot points;near the CNT ends the wave function oscillates with theazimuthal angle ϕ and the individual points can be seen clearly.Initially (φ = 10−3φ0) the state is extended; when the magneticflux reaches φloc, its wave function begins to be describedby an imaginary solution of (17). With the magnetic fieldincreasing further, the wave function decays exponentially193407-3BRIEF REPORTS PHYSICAL REVIEW B 83, 193407 (2011)(a) (b)(c) (d)FIG. 3. (Color online) Localization in a (12,9) chiral CNT with16 unit cells (L = 41.5 nm, θ = 25◦). (a) Comparison of numericaland analytical spectra, neglecting the spin. In this case there are nolocalized states at φ = 0. The dashed line marks the onset of thelocalization. (b) Analytical spectrum including spin effects. (c) Theamplitude of the highest valence eigenstate (obtained numerically) ateach atom, projected onto x‖, for different values of φ and neglectingthe spin. (d) Conductance as a function of magnetic flux φ andchemical potential E, including spin effects.with the distance from the CNT ends, the localization becomescomplete, and the state ceases to conduct.The above analysis is confirmed by conductance cal-culations, with the CNT in a setup shown in Fig. 2(b).We derive the elastic linear response conductance via theFisher-Lee formula for the quantum mechanical transmission:G = e2hTr {LGRG†}, where L/R = i(L/R − †L/R), L/Ris the self-energy of the left or right lead, respectively, andG is the Green function of the central region dressed by theelectrodes. For simulating bulk metal electrodes we considerwide band leads, that is, WB(E) = −i Im(EF ). The resultsshown in Figs. 2(d) and 3(d) were obtained with WB =−i 0.22 eV. In both we see a gradual drop of the conductanceof the highest valence and lowest conduction spin states, asthey become localized in the increasing magnetic field. The“native” end states of the (18,0) CNT, localized also at φ = 0,can be seen in the spectrum in Fig. 2(c), but do not contributeto the conductance, as we expect. The good matching ofanalytical spectra of isolated CNTs and the conductance peaks[compare Fig. 2(c) with 2(d) and Fig. 3(b) with 3(d)] impliesthat even with rather strong coupling the CNT is sufficientlydistinct from the leads for the transport to be determined bythe spectrum of an isolated nanotube.The magnetic field Bloc corresponding to φloc depends onthe nanotube length and radius. For our choice of Vπ andVσ , Bloc of CNTs with R = 7 A˚ and L = 40 nm rangesfrom 50 T (θ ≈ 30◦) to 85 T (θ = 0◦). However, for thesame nanotubes with L = 500 nm the value of Bloc drops to4–42 T. Hence the localization induced by the magnetic fieldmight be detected in currently accessible transport experimentsor by spin-resolved STM spectroscopy revealing localizedspin states at the nanotube ends. Moreover, the localizationinduced by a magnetic flux appears to be a chirality-independent phenomenon, to which only armchair CNTs areimmune.The authors acknowledge the support of the DeutscheForschungsgemeinschaft under GRK Grant No. 1570.1K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys.Rev. B 54, 17954 (1996).2L. Brey and H. A. Fertig, Phys. Rev. B 73, 235411 (2006).3K. A. Ritter and J. W. Lyding, Nat. Mater. 8, 235 (2009).4P. Kim, T. W. Odom, J.-L. Huang, and C. M. Lieber, Phys. Rev.Lett. 82, 1225 (1999).5S. Ryu and Y. Hatsugai, Phys. Rev. B 67, 165410 (2003).6K. Sasaki, K. Sato, R. Saito, J. Jiang, S. Onari, and Y. Tanaka, Phys.Rev. B 75, 235430 (2007).7A. Man˜anes, F. Duque, A. Ayuela, M. J. Lo´pez, and J. A. Alonso,Phys. Rev. B 78, 035432 (2008).8J. Wu and F. Hagelberg, Phys. Rev. B 79, 115436 (2009).9L. Rosales, M. Pacheco, Z. Barticevic, C. G. Rocha, and A. Latge´,Phys. Rev. B 75, 165401 (2007).10H. Santos, A. Ayuela, W. Jasko´lski, M. Pelc, and L. Chico, Phys.Rev. B 80, 035436 (2009).11K. Sasaki, S. Murakami, R. Saito, and Y. Kawazoe, Phys. Rev. B71, 195401 (2005).12K. Sasaki, M. Suzuki, and R. Saito, Phys. Rev. B 77, 045138(2008).13T. Ando, J. Phys. Soc. Jpn 69, 1757 (2000).14S. Koller, L. Mayrhofer, and M. Grifoni, New J. Phys. 12, 033038(2010).15D. V. Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B 77, 235301(2008).16R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Proper-ties of Carbon Nanotubes (Imperial College Press, London, 1998).17A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B 77, 085423(2008).18S. H. Jhang, M. Marganska, Y. Skourski, D. Preusche, B. Witkamp,M. Grifoni, H. van der Zant, J. Wosnitza, and C. Strunk, Phys. Rev.B 82, 041404 (2010).193407-4

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Magnetic field induced localization in carbon nanotubes

Magnetic field induced localization in carbon nanotubes

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