Letter Cite This: Nano Lett. 2019, 19, 4146−4150
pubs.acs.org/NanoLett
Theoretical Design of Topological Heteronanotubes Chen Hu,*,† Vincent Michaud-Rioux,† Wang Yao,‡ and Hong Guo† †
Center for the Physics of Materials and Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada Department of Physics and Center of Theoretical and Computational Physics, University of Hong Kong, Hong Kong, China
‡
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S Supporting Information *
ABSTRACT: We propose and investigate the idea of topological heteronanotubes (THTs) for realizing an one-dimensional (1D) topological material platform that can pave the way to low-power carbon nanoelectronics at room temperature. We predict that the coaxial double-wall heteronanotube, a carbon nanotube (CNT) inside a boron nitride nanotube (BNNT), can act as a THT. Dissipationless topological conducting pathways on the THT are protected by a valley-dependent topological invariance that originates from local topological phase transitions of the CNT modulated by the CNT−BNNT interaction. Spiral THTs, where topological current flows spirally around the tube, function as nanoscale solenoids to induce remarkable magnetic fields due to the dense moiré nanopatterning. The generality and robustness of the THT materials are demonstrated by investigating different tube diameters, tube indexes, and tube types as well as topologicalpathway orientations through first principles. KEYWORDS: Topological heteronanotube, carbon devices, low dissipation, topological solenoid
T
structure of THT is protected by a valley topological invariant familiar in 2D.16,17 1D topological materials are fundamentally important for investigating the Majorana zero modes10 and topological quantum computing,18 and they are also practically significant for fabricating low-dissipation 1D nanoelectronic devices due to substantially reduced carrier scattering.19 Therefore, 1D topological materials have attracted a lot of attentions in the past years, including quantum wires20 and heavy-element nanotubes.21 The idea of THT is general and robust, here we demonstrate it by investigating the CNT@BNNT system, which is experimentally feasible.15 We also show that many other types of THTs can be realized (see section 8 of the Supporting Information). Even in the CNT@BNNT platform, various tube indexes and the diversity of relative chiralities between the two tubes can provide a playground for numerous possibilities of functional THTs. In armchair CNT@BNNT THTs, depending on the tube indexes, single-pair or multiple-pair valley topological transport pathways flow along the tube axis (on CNT); especially for spiral THTs, the topological transport pathways goes spirally around the tube functioning as conducting coils of a nanosolenoid. Because such topological coils are produced by the atomic scale moiré patterns, it is extremely dense to induce remarkable magnetic fields.
he discovery of carbon nanotubes (CNT) by Sumio Iijima1 in 1991 opened an exciting field of carbon nanoelectronics where transistors and interconnect wires are entirely made of CNTs.2−4 From the fundamental physics point of view, CNT is of particular interest due to its onedimensional (1D) nature, very long carrier mean free path, and massless Dirac electrons at low energy. Recently, highperformance single-wall CNT field-effect transistor (FET) with 5 nm gate length5 and low sub-60 millivolts per decade switching6 were experimentally realized. With these and other7−12 impressive progresses, CNT may well provide a viable route toward an emerging electronics beyond silicon.13,14 More recently, double-wall heteronanotubes made by a single-wall CNT confined inside a boron nitride nanotube (BNNT) was experimentally synthesized using in situ electron irradiation.15 The original motivation of producing such composite CNT@BNNT tubes was to use the outer BNNT wall as a protective insulating shell against environmental perturbations to the inner CNT. In this work, we shall theoretically show that CNT@BNNT heteronanotubes can provide a new material platform for realizing topological lowdissipation CNT nanoelectronics. The outer BNNT wall, which was considered to have little interaction with the inner CNT, can in fact strongly modulate the electronic structure and carrier transport of the CNT. This is because for a wide range of tube indexes, moiré-like local atomic registries appear on the CNT@BNNT to induce topological physics, these tubes become topological heteronanotubes, THT, an 1D intrinsic topological material. The low-energy electronic © 2019 American Chemical Society
Received: April 22, 2019 Published: May 22, 2019 4146
DOI: 10.1021/acs.nanolett.9b01661 Nano Lett. 2019, 19, 4146−4150
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Nano Letters
Figure 1. (a−c) Geometries of the armchair single-wall CNT, single-wall BNNT, and THT, respectively, where THT is a double-wall CNT(inner) @BNNT(outer) tube. (d−f) Calculated band structures of the three tubes with n = 96 and m = 1. The Fermi level is shifted to energy zero, and the valley points K and K′ are at kz = 1/3 and 2/3, respectively. Bulk bands are the gray lines. In panel f, the linear red and blue bands near the Fermi level are the two valley-dependent topological helical states. a = 2.4595 Å is the lattice constant of the transnational vector T. For armchair CNT and BNNT, the small lattice difference of T (1.8%) is neglected. (g−i) Spatial distribution of modular squared wave functions |ψ|2 on the tube circumference, which are the Bloch states indicated in the inset at chemical potential μ = E. The solid (empty) circles denote carriers with positive (negative) group velocity, i.e., right-movers (left-movers), and solid (dashed) lines are helical states with valley index K or K′, respectively. In panel i, because |ψ|2 is from carbon atoms, only CNT circumference is plotted for clearer vision. (j−l) Schematics of charge transport in these nanotubes. Arrows on the wires denote electron flow, curved white arrows on the nanotubes indicate backscattering from one conducting channel to another.
The geometry and electronic property of nanotubes are uniquely characterized by the chiral vector Ch = n1a1 + n2a2 ≡ (n1,n2), where a1 and a2 are the hexagonal lattice vectors and n1 and n2 are integers (tube index).22 Let us begin by considering an armchair CNT(n,n) in Figure 1a and an armchair BNNT(n + m,n + m) in Figure 1b; together, they form a THT in Figure 1c. Here, m is a nonzero integer denoting the difference in circumferential unit cell number between the two tubes. To make a specific calculation as one example without losing generality, we start from the case of n = 96 and m = 1; thus, the THT is CNT(96,96)@BNNT(97,97). We emphasize that the generality and robustness of THT shall be demonstrated (see below) by investigating different tube diameters, tube indexes, and tube types as well as topological-pathway orientations. For the CNT(96,96)@BNNT(97,97), the optimized interwall distance of this THT is 3.2 Å by structural relaxation (details are given in section 1 of the Supporting Information). The CNT(n,n) is metallic and the BNNT(n + m,n + m) is insulating, as shown by the band structures in Figure 1d,e obtained from density functional theory (DFT) calculations.23
Compared with band structures of individual CNT or BNNT, the band structure of THT in Figure 1f is significantly different. First, there opens a noncryogenic bulk band gap of 150.6 meV due to local sub-lattice symmetry breaking in the THT (details are given in section 6 of the Supporting Information). Second, there are two linear bands (red and blue lines) near the Fermi level at each Dirac point K and K′, which turn out to be valley-dependent helical states (see below). Third, wave functions of these linear bands of THT are concentrated at two locations on the circumference as shown by red and green isosurfaces coded in Figure 1i. This is to be compared to that of the individual CNT, where the wave functions spread over the entire circumference, as shown by the yellow isosurface coded in Figure 1g. In section 3 of the Supporting Information, results of a THT with much smaller diameter, CNT(22,22)@BNNT(23,23), are presented that show the same topological physics. Based on the electronic structure, quantum transport properties of the three nanotubes in Figure 1 can be established. The BNNT is an insulator with a large band gap 4147
DOI: 10.1021/acs.nanolett.9b01661 Nano Lett. 2019, 19, 4146−4150
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Nano Letters shown in Figure 1h; thus, no current flows in the circuit of Figure 1k. Metallic CNT(n,n) is a conductor, but intravalley backscattering at K with positive velocity to the same K with negative velocity can occur due to interaction (the same is true for K’), which decreases conductivity as illustrated in Figure 1j. For the THT of Figure 1l, however, each topological conducting pathway (red or green regions on the tube) supports one pair of counter-propagating helical states: carriers with different valley index (K or K′) have opposite group velocities, the intravalley scattering is significantly suppressed due to the spatial separation of states in the same valley but with opposite velocities (Figure 1i). Meanwhile, because the two valleys are far apart in the Brillouin zone (BZ), the intervalley scattering from K to K′ or vise versa are largely suppressed on each topological edge.16,17 Therefore, THT behaves as a low-dissipation CNT conductor for ballistic transport with little or no backscattering for low-energy 4e 2
excitation, giving a quantized conductance of G = 2G0 = h (spin degenerate), where h is the Planck’s constant and e the electron charge. Analogous to the spin-dependent helical states in a quantum spin Hall system,24 the valley-dependent topological current is robust and holds a high transport efficiency against small disorders and chemical potential variations.25,26 Finally, carriers with K and K′ indices travel in different pathways as shown in Figure 1l, and thus, THT can also be exploited as a perfect valley splitter. In the following, we discuss the physical origin of the topological channels in THT. For 1D systems, one may characterize topological properties by calculating the Zak phase.27 As for the low-energy electronic structure of CNT systems, the valley topological invariant can be employed here.16,17 We build a mapping between the THT and its corresponding 2D double-layer flat sheet, shown in Figure 2a, to obtain an intuitive perspective into the local atomic registries of the circular tube.11 As shown in Figure 2b, the 2D mapping of a THT CNT(n,n)@BNNT(n + 1,n + 1) is a moiré pattern with one moiré period. Along the circumference direction Ch (armchair direction), the atomic registry varies gradually and continuously, giving rise to various local stacking configurations. In a long-period moiré structure with a moiré period much larger than each unit cell of the double-wall tube, the electronic structure and topological nature in a local region can be well-determined by that of the lattice-matched stacking configuration of the corresponding atomic registry, as depicted in the green circles in Figure 2b.28−31 The topological properties of the latter is described by the valley-dependent Chern number CV: CV =
1 2π
∫ Ωxy(k)dk xdk y
Figure 2. (a) Schematics of rolling and unrolling processes, building a mapping between 1D THT and its corresponding 2D flat double-layer moiré structure. (b) 2D moiré mapping with n = 96. Ch is the circumference direction (chiral vector), and T is the tube-axis direction (translational vector). Some representative high-symmetry local atomic registries are shown in green circles, where gray, pink, and blue balls denote carbon, boron, and nitrogen atoms, respectively. (c) Valley-dependent Chern numbers (CV) for valley K of latticematched configurations at the corresponding real-space locations. (d) Circular distribution of CV for valley K rolled up from the flat-sheet in panel c. In panels c and d, purple and light blue regions denote CV = −1/2 and 1/2, respectively. (e) Spatial distribution of modular squared wave functions |ψ|2 along the circumference of the topological helical states in Figure 1f.
Chern number on the THT, CV, shown in Figure 2d; namely, there are two topological phase transitions when the local atomic registry varies by one circumference (one moiré period), corresponding to two groups of topological edges named “topological moiré edge” in the rest of this paper. In Figure 2e, it can be clearly seen that the calculated locations of topological conducting pathways of the THT match well with the topological moiré edges, revealing the moiré related topological bulk-edge correspondence that such helical states are the direct manifestation of the topological phase transition induced by the variation of atomic registry between the CNT and BNNT walls. The great richness of THT is further seen by controllable orientations of topological transport pathways. As a demonstration, we roll up the 2D mapping sheet of the THT with the index of CNT(16,16)@BNNT(17,17) along the particular chiral vector direction (Ch) shown in Figure 3a. It is clear that in this structure the variation of atomic-registry is tilted from T so that the topological moiré edge vector e is not parallel to the transitional vector T. Figure 3b shows that in this spiral THT, topological helical states still appear (albeit with some bandfolding). The calculated wave functions of these helical states, shown in Figure 3c, are located spirally wrapping the tube. Such spiral THTs act as nanoscale solenoids, where the “coil” is the topological conducting pathways. When a current I passes through, the magnitude of the magnetic field B per unit length is obtained by Ampere’s law:
(1)
where Ω(k) is the Berry curvature calculated by:32−34 Ωq , xy(k) = −2Im
∑ p≠q
⟨ψqk|vx|ψpk ⟩⟨ψpk|vy|ψqk ⟩ [ϵpk − ϵqk ]2
(2)
Here, p and q are band indexes, ϵk is the eigenvalue of eigenstate |ψk⟩, and vx and vy are the components of velocity operators. As shown in Figure 2c, various local stackings give rise to different signs of the Chern number CV due to the staggered on-site potential, which varies over the moiré period (see section 2 of the Supporting Information for more details). Therefore, we infer the circular spatial distribution of the
B = μ0 NI 4148
(3) DOI: 10.1021/acs.nanolett.9b01661 Nano Lett. 2019, 19, 4146−4150
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Nano Letters
Figure 3. (a) Illustration for creating spiral THTs with topological moiré edge vector e (indicated by yellow arrow) not parallel to the transitional vector T. The orange rectangle denotes the supercell of the 2D mapping of the spiral THT containing 8448 atoms. (b) Band structure of the spiral THT. (c) Spatial distribution of modular squared wave functions |ψ|2 of the topological helical states, where the plotted isosurfaces (red and green areas) are 1.8 × 10−11 a.u. (d) Schematic demonstration of the spiral THT device in a circuit to serve as a topological solenoid or nanomagnet. Arrows denote the direction of electron flow.
where μ0 is the permeability of free space, N is the number density of the coil turns depending on the transitional vector T, which can be manipulated by various chiral angle and THT index. In the case of Figure 3, T = 7.87 nm; thus, N is as large as 1.27 × 108 per meter, which is extremely large. Such a dense nanocoil would generate a magnetic field of 1.6 G for a 1 μA current, which should be remarkable to experimental observations. Most importantly, the current flowing in the transport pathways are topologically protected against backscattering and such spiral THT realizes a dense moiré nanopatterning technique to produce “topological solenoids”. Finally, we discuss the generality and robustness of the THT systems. (i) The general form of armchair THT is CNT(n,n)@ BNNT(n + m,n + m), where various nanotube indexes (m and n) can provide essentially infinite possibilities for the type of THTs. Different values of n correspond to different tube diameters. In experiments, small-diameter CNTs, typically with diameters below 4−5 nm, are more common due to their stable circular structures.35,36 The geometric curvature effect of the small-diameter THT is much larger than that that of the large-diameter THT of Figure 1. As shown in section 3 of the Supporting Information, by investigating a small-diameter tube, CNT(22,22)@BNNT(23,23) whose diameter is 2.98 nm, it is shown that the same topological physics occurs. This demonstrates the topological robustness of THT against curvature (and tube diameter). The result also reveals that the scheme of THT is general to a broad range of tube diameters. (ii) The value of m equals to the number of the moiré periods on the circumference. As a result, m > 1 gives rise to multiple topological conducting pathways on a THT
(see section 4 of the Supporting Information). The condition of establishing topological physics in THT is that two stable and robust topological phases coexist on the tube circumference. Therefore, each moiré period (n/m) or each topological phase region should be large enough to support a stable topological phase.30 (iii) The THTs discussed so far are formed by inner CNT and outer BNNT (m > 0), while, in fact, another experimentally fabricated system is with a reverse order,37 namely outer CNT and inner BNNT (usually, when m < 0), which also leads to a moiré structure supporting topological helical states in exactly the same manner (see section 5 of the Supporting Information). For the case of m = 0, CNT(n,n)@BNNT(n,n), where a commensurate atomic registry instead of moiré pattern locates along the circumference. In such a case, no topological phase transition occurs (see section 6 of the Supporting Information). (iv) The necessary condition for the appearance of 1D helical channels is the moiré-induced topological phase transition located at topological moiré edges of the THT. In zigzag-type THTs, CNT(n, 0)@BNNT(n + m, 0), the topological moiré edges are oriented along the armchair direction, and thus, the K and K′ are mixed together, resulting in the disappearance of topological helical states (see Figure S6a), while in chiraltype THTs, the moiré edges are terminated by chiral crystal directions rather than armchair, which gives rise to the topological helical channels (Figure S6b), similarly as the armchair THT in Figure 1. (v) Besides the CNT@BNNT system presented here, the general moiré topological physics is found in other double-wall or multiwall nanotubes. For example, for the double-wall CNT,38 we can predict that an 4149
DOI: 10.1021/acs.nanolett.9b01661 Nano Lett. 2019, 19, 4146−4150
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Nano Letters electric field can induce similar topological helical states (see section 8 of the Supporting Information). In summary, we propose a general, robust, and experimentally feasible material platform, the topological heteronanotube, for naturally realizing 1D topological helical states at room temperature. By first-principles calculations on the prototypical CNT@BNNT system, two groups of valleydependent helical states are discovered to form inside a noncryogenic bulk band gap. By analyzing wave functions and valley topological invariance, these helical states are found to locate at two groups of topological moiré edges due to the local topological phase transition associated with the local atomic registry that is varying along the circumference of the tubes. In principle, low-energy carriers in the helical states supported by the topological moiré edges do not suffer intravalley and intervalley scattering; thus, the THT is an ideal 1D topologically protected ballistic conductor, where transport can be dissipationless. The spiral THTs have topological current flowing spirally around the tube to function as topological nanosolenoids with ultrahigh coil density, giving rise to a large induced magnetic field. Our results suggest that 1D topological heterotubes should provide very rich opportunities for both fundamental science of 1D topological physics and practical applications in low-power nanodevices.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.9b01661. Additional details on electronic structure calculation, inter-wall distance optimization, Chern number analysis, various THT characteristics, commensurate double-wall hetero-nanotubes, and topological properties of doublewall CNTs (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Chen Hu: 0000-0003-2333-2182 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS C.H. thanks Xiaodong Xu for discussions on nanotube transport and Peng Kang for the help of building spiral double-wall heteronanotubes models. This work is financially supported by the NSERC of Canada and FQRNT of Quebec (H.G.). We thank Compute Canada and the High Performance Computing Center of McGill University for the substantial computational support that made this work possible.
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REFERENCES
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DOI: 10.1021/acs.nanolett.9b01661 Nano Lett. 2019, 19, 4146−4150
