Recent conductance measurements on multi-wall carbon nanotubes (CNTs) reveal
an effective behavior similar to disordered single-wall CNTs. This is due to
the fact that electric current flows essentially through the outermost shell
and is strongly influenced by inhomogeneous electrostatic potential coming from
the inner tubes. Here, we present theoretical studies of spin-dependent
transport through disorder-free double-wall CNTs as well as single-wall CNTs
with Anderson-type disorder. The CNTs are end-contacted to ferromagnetic
electrodes modelled as fcc (111) surfaces. Our results shed additional light on
the giant magnetoresistance effect in CNTs. Some reported results concern
realistically long CNTs, up to several hundred nanometers.Comment: 9 pages, 5 figures, presented at the European Conference PHYSICS OF
  MAGNETISM 2005, Poznan, Polan

Baibich

Krompiewski

Lake

Ohno

Prinz

Roche

Sahoo

Stojetz

Triozon

Tsukagoshi

Wolf

Xiong

Zutic

English

arXiv

Crossref

Spin transport in disordered single-wall carbon nanotubes contacted to ferromagnetic leads

Recent conductance measurements on multi-wall carbon nanotubes (CNTs) reveal an effective behavior similar to disordered single-wall CNTs. This is due to the fact that electric current flows essentially through the outermost shell and is strongly influenced by inhomogeneous electrostatic potential coming from the inner tubes. Here, we present theoretical studies of spin-dependent transport through disorder-free double-wall CNTs as well as single-wall CNTs with Anderson-type disorder. The CNTs are end-contacted to ferromagnetic electrodes modelled as fcc (111) surfaces. Our results shed additional light on the giant magnetoresistance effect in CNTs. Some reported results concern realistically long CNTs, up to several hundred nanometers

Krompiewski, Stefan

Nemec, Norbert

Cuniberti, Gianaurelio

University of Regensburg Publication Server

arXiv:cond-mat/0509148v1  [cond-mat.mes-hall]  6 Sep 2005Spin transport in disordered single-wall carbon nanotubescontacted to ferromagnetic leadsS. Krompiewski,1 N. Nemec,2 and G. Cuniberti21Institute of Molecular Physics, Polish Academy of Sciences,ul. M. Smoluchowskiego 17, PL-60179 Poznan´, Poland2Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany(Dated: April 17, 2007)AbstractRecent conductance measurements on multi-wall carbon nanotubes (CNTs) reveal an effectivebehavior similar to disordered single-wall CNTs. This is due to the fact that electric current flowsessentially through the outermost shell and is strongly influenced by inhomogeneous electrostaticpotential coming from the inner tubes. Here, we present theoretical studies of spin-dependenttransport through disorder-free double-wall CNTs as well as single-wall CNTs with Anderson-typedisorder. The CNTs are end-contacted to ferromagnetic electrodes modelled as fcc (111) surfaces.Our results shed additional light on the giant magnetoresistance effect in CNTs. Some reportedresults concern realistically long CNTs, up to several hundred nanometers.PACS numbers: 85.35.Kt, 85.75.-d, 81.07.De, 73.63.-b, 79.60.Ht1INTRODUCTIONOver the last two decades, the magneto-electronics, based on all-metal multilayers, hasproven to be very successful indeed [1]. The most important effect which should be invokedin this context is giant magnetoresistance (GMR) discovered in 1988 [2]. This effect makesit possible to control electric current flowing through magnetic materials by means of amagnetic field. In other words, the essence of the GMR effect lies in taking advantage of notjust the electronic charge alone but also of its spin counterpart. Quite naturally researchersinvolved so far in physics of semiconductors, as well as those studying molecular systemshave intensified their efforts in search for possible analogous effects in all-semiconducting [3]and/or hybrid systems (combinations among metals, semiconductors and molecules) [4, 5].Consequently, a new field of science and technology has been triggered, under the name ofspintronics [6, 7]. Here we report our results on the GMR effect in perfect and disorderedcarbon nanotubes sandwiched between ferromagnetic electrodes. There is no doubt nowa-days that miniaturization requirements imposed on the emerging spintronics will be met byapplying the so-called bottom-up approach as far as designing of new electronic devices isconcerned. From this point of view carbon nanotubes are surely excellent candidates.DOUBLE-WALL CNTSWe start our studies with carbon nanotubes (CNTs) end-contacted to metal electrodes.Our present approach is essentially that described in detail in [8] with an improved simu-lation method of CNT/metal-electrode interface developed in [9]. Spin-polarization of theferromagnetic leads is defined as P = 100(n↑ − n↓)/(n↑ + n↓), where nσ stands for a num-ber of σ-spin electrons per lattice site. It is noteworthy that the structures in question arerelaxed under the Lennard-Jones potential in order to find energetically favorable relativepositions of CNTs’ and electrodes’ interface atoms. During the relaxation process the ex-ternal electrodes are allowed to rotate and shift independently of each other, similarly theinner tube is also free to rotate. As regards the inter-tube hopping integrals, they are takenin the form as proposed in [10], i.e. set to tint = −(t/8)cos θljexp[(d l j − b)/δ], where θ isthe angle between the pi orbitals, d is a relative distance, t stands for the nearest neighborhopping integral (chosen as energy unit), δ = 0.45 A˚ and b = 3.34 A˚. The GMR coefficient2is defined in terms of the conductances, G, as GMR=1 − G↑↓/G↑↑, with ↑↑ (↑↓) denotingaligned (antialigned) magnetization orientation of the electrodes.Most of the hitherto existing experiments on electronic transport suggest that currentflowing through MWCNTs goes mostly through the outermost shell (see e.g. [11]). A preciserole of the inner shells is still hardly known. Here we show the results on the GMR effectin two double-wall (DW) CNTs which have the same outer shell but different - thoughnon-conductive in each case - inner shells. Specifically the DWCNTs in question are: (i)the zigzag at armchair (45-(5,0)@39-(8,8)) and (ii) the armchair at armchair (38-(3,3)@ 39-(8,8)), using a short-hand notation L-(n,m), for the lenghth (in carbon rings) and the chiralvector, respectively. In the former case the corresponding lengths are roughly the same (ca.5 nm each) so both the inner tube and the outer one are well contacted to the magneticelectrodes. In the latter case, in turn, the inner shell is artificially shortened and forcedthereby to be out of contact to the drain electrode. Fig.1 presents giant magnetoresistancefor the two DWCNTs. Despite the fact that both the systems are formally similar (atleast in the presented ”energy window” (|E/t| < 0.2), which falls into the zigzag-tube gap),the GMR curves are clearly different. We attribute these differences to intertube-quantuminterferences which are present owing to the non-vanishing tint. To mimic a possible effectof some additional disorder we present also GMR curves (thin lines) calculated from theenergy-averaged conductances, where the averaging has been made over the most obviousenergy scale in this context, namely over an energy bin equal to the inter-level spacing ofthe outer shell ∆E = pi√3/L (in the present units). In the following subsection we presenta more direct approach to the disorder issue.SINGLE-WALL CNTS WITH ANDERSON DISORDERThe simplest way to include the effect of disorder in a system described in terms of thetight-binding model is to allow all on-site (atomic) potentials to take random values withina given energy interval. Such an approach has been already applied to CNTs [12], but toour knowledge, only for tubes with non-magnetic leads. In the case of disordered systems,it is necessary to perform statistical (ensemble) averaging of the results corresponding toparticular sets of on-site potential distributions (to be referred to as samples hereafter).This is a purely technical problem easy to overcome at the expense of the computation3time. Another more serious problem is to develop a recursive procedure, which wouldmake it possible to deal with big systems approaching macroscopic sizes of the order ofseveral hundred nanometers. Recursive algorithms based on Dyson-type equations for theGreen’s function are well-known [13, 14]. Here however we modify those methods in orderto make them work in the case of highly non-homogenous systems composed of disorderedcarbon nanotubes and two adjacent atomic layers from each electrode (to be referred toas the extended molecule). While conductance computations are usually rather fast, thecomputations of electronic charge at all atoms of a big system are extremely computertime consuming and expensive. In order to surmount this difficulty we impose a globalcharge neutrality condition only on a disorder-free ”parent” system and self-consistentlydetermine detailed values of all its on-site potentials (ca. 4000 and 40000 atoms for theSWCNT(8,8), 30 and 300 nm long). On introducing disorder, these on-site potentials aremodified by random corrections fluctuating around zero within an interval [-W/2, W/2]. So,on the average the global charge of the Anderson-disordered extended molecule might beregarded as roughly close to that of the neutral parent system. Our computations proceedaccording to the following protocol: First the surface Green functions are found (see [9]for details). Second, the set of on-site potentials which ensure the charge neutrality ofthe parent system is found. Third, conductance calculations along with the correspondingGMR coefficients are performed for 100 different samples with random on-site corrections.Finally the results are ensemble-averaged. The main results of this study is presented inFig.2, for the SWCNT (8,8) consisting of 240 carbon rings (120 unit cells ∼ 30 nm). It isseen that although disorder suppresses the GMR, it happens to be of about the right valueas compared to recent experiments on MWCNTs with transparent ferromagnetic contactsmade from Pd0.3Ni0.7 (device resistances are then as low as 5.6 kΩ at 300K) [15]. Othernoteworthy points are: (i) on the average the GMR remains positive, and (ii) there existsome extra features in the GMR spectrum at energies close to ±0.4 and ±0.7 correspondingto higher sub-band onsets in the pristine (ideal) SWCNT.For smaller W , GMR increases and eventually oscillates with the amplitude of roughly±0.2 in the disorder-free (parent) case, as shown in Fig.3 (l.h.s). Additionally the right-handside of this figure highlights the length-effect on the period of oscillations. For the sake ofsimplicity this is shown for the paramagnetic leads. It is clearly seen that in the absenceof disorder the observed periodicity reflects length-dependent interferences, as expected for4a (quasi) ballistic transport. In the magnetic case the peaks are split due to the lifting ofspin-degeneracy. The quasi-periodic behavior does always occur when there is no disorder,regardless of whether the electrodes are ferromagnetic or not (c.f. the inset in Fig.3 withthe thick solid curve on the r.h.s). From the present results one sees that the process ofaveraging of conductance and GMR spectra leads to a subtle interplay between the lengthand the amount of disorder in the CNTs.SUMMARYIn this work it has been shown theoretically that the GMR effect in ferromagneticallycontacted carbon nanotubes is quite considerable and may reach a few tens percent. Ideally,the GMR coefficient oscillates as a function of energy (gate-voltage) with a quasi-periodclose to the inter-level spacing of the CNT, which scales inversely proportional to the nan-otube length. Yet, such a picture is to some extent too detailed if the system at hand isimperfect, e.g. due to some impurities, dopants or a presence of incommensurate inner shellsin a MWCNT. The disorder-averaged GMR rages from 6% down to 2% in the vicinity ofthe charge neutrality point, in conformity with recent experiments on MWCNTs with trans-parent ferromagnetic contacts. Furthermore, the aforementioned periodicity gets nearlycompletely suppressed, and there is no more tendency for the GMR to become negative.S. K. thanks the KBN project (PBZ-KBN-044/P03-2001), the Centre of Excellence (con-tract No. G5MA-CT-2002-04049) and the Poznan Supercomputing and Networking Centerfor the computing time. This work was partially funded by the Volkswagen Foundation andthe Vielberth Foundation.[1] G. Prinz, Science 282, 1660 (1998).[2] M. N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988).[3] H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani,Nature, 408, 946 (2000).[4] K. Tsukagoshi, B. W. Alphenaar, and H. Ago, Nature 401, 572 (1999).5[5] Z. H. Xiong, Di Wu, Z. Valy Vardeny, and Jing Shi, Nature, 427, 824 (2004).[6] I. Zutic, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).[7] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes,A. Y. Chtchelkanova, and D. M. Treger8, Science 294,1488 (2001).[8] S. Krompiewski, R. Gutierrez, and C. Cuniberti, Phys. Rev. B 69, 155423 (2004).[9] S. Krompiewski, physica status solidi (b) 242, 226 (2005), 121401 (2001).[10] S. Roche, F. Triozon, A. Rubio, and D. Mayou, Phys. Rev. B 64, 121401 (2001).[11] B. Stojetz, C. Miko, L. Ferro, and C. Strunk, Phys Rev. Lett., 94, 186802 (2005).[12] F. Triozon, S. Roche, A. Rubio, and D. Mayou, Phys. Rev. B 69, 121410 (2004).[13] R. Lake, G. Klimeck, R. Bowen, D. Jovanovic, J.Appl.Phys. 81, 7845 (1997).[14] S. Krompiewski, J. Martinek, and J. Barnas, Phys. Rev. B 66, 073412 (2002).[15] S. Sahoo, T. Kontas, C. Scho¨nenberger, and C. Su¨rgers, Appl. Phys. Lett. 86, 112109 (2005).6-0.2 -0.1 0.0 0.1 0.2-1.0-0.50.00.5 45(5,0)@39(8,8) 38(3,3)@39(8,8)GMREFIG. 1: GMR for the double-wall carbon nanotubes 45-(5,0)@39-(8,8) (solid thick line) and 38-(3,3)39-(8,8) (thick dashed line) attached to ferromagnetic leads with 50% spin-polarization. Tomimic a possible effect of disorder, there are also shown the GMR curves computed from ∆E-averaged conductances (thin curves of the same style), where ∆E is the inter-level spacing of theouter shell.7-0.04 -0.02 0.00 0.02 0.04-0.050.000.050.10GMR=1-<G > < G  >GMREFIG. 2: Left hand side: GMR for individual SWCNTs (8,8), ca. 30 nm in length (points), alongwith the GMR (white curve) averaged over 100 samples with disorder-induced corrections to theon-site potentials (within [-W/2, W/2] for W=0.2). On the right hand site the GMR computedfrom the disorder-averaged conductances together with the standard-deviation error bars are shownin the vicinity of the charge neutrality point.8-1.0 -0.5 0.0 0.5 1.0-0.4-0.20.00.20.40.60.8-0.1 0.0 0.1-0.2-0.10.00.10.20.3GMR0.00 0.05 0.10 0.150.00.51.01.52.0G [e2 /h] per spin L=300 nm L=30 nm L=5 nmSWCNT (8,8)FIG. 3: Left hand side: GMR for disorder-free SWCNT (8,8), P=50%, W=0, L=240 carbon rings(∼ 30nm). Right hand side: visualization of the length-dependent periodicity of the conductancefor the case of paramagnetic leads, P=0, and L=5, 30 and 300 nm. Compare the inset with thethick solid curve to see that the quasi-period of oscillations is roughly maintained, but in themagnetic case the peaks are spin-split.9

Spin transport in disordered single-wall carbon nanotubes contacted to
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Spin transport in disordered single-wall carbon nanotubes contacted to ferromagnetic leads

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