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Oct 9, 2015 - Here we report on an extraordinary inductor nanostructure naturally occurring as a screw dislocation in graphitic carbons. Its elegant h...
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Riemann surfaces of carbon as graphene nano-solenoids Fangbo Xu, Henry Yu, Arta Sadrzadeh, and Boris I. Yakobson Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b02430 • Publication Date (Web): 09 Oct 2015 Downloaded from http://pubs.acs.org on October 12, 2015

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Riemann surfaces of carbon as graphene nano-solenoids Fangbo Xu, Henry Yu, Arta Sadrzadeh, and Boris I. Yakobson* Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, USA Abstract Traditional inductors in modern electronics consume excessive areas in the integrated circuits. Carbon nanostructures can offer efficient alternatives if the recognized high electrical conductivity of graphene can be properly organized in space to yield a currentgenerated magnetic field which is both strong and confined. Here we report on an extraordinary inductor nanostructure naturally occurring as a screw dislocation in graphitic carbons. Its elegant helicoid topology, resembling a Riemann surface, ensures full covalent connectivity of all graphene layers, joined in a single layer wound around the dislocation line. If voltage is applied, electrical currents flow helically and thus give rise to a very large (~1 T at normal operational voltage) magnetic field and bring about superior—per mass or volume—inductance, both owing to unique winding density. Such a solenoid of small diameter behaves as a quantum conductor whose current distribution between the core and exterior varies with applied voltage, resulting in nonlinear inductance. *) E-mail address: [email protected] Keywords: Graphene, electronics, nano-device, magnetism, inductance.

In modern microelectronics, the planar spiral inductors on silicon have been recognized as principle passive components in monolithic radio-frequency integrated circuits1-3. However, not only this design exhibits low quality factors, suffering from ohmic loss, but it also consumes excessive area. Perhaps more efficient and much smaller inductors may become possible if one turns from established technologies on silicon to its steadily maturing contender—carbon, in form of either nanotubes or especially graphene. In recent years, graphene, with electron mobility as high as 200,000 cm2/Vs,4 distinguished itself in ballistic transport regime wherein the resistance is no longer a function of the device dimension5. Crystal of graphite is basically a stack of numerous graphene layers, bonded by weak van der Waals forces but covalently disconnected, which endows graphite with highly anisotropic physical properties. Mechanical stiffness and strength, high thermal and electrical conductivity along the basal plane contrast with low modulus, weakness to delamination, reduced thermal, and very low electrical transport 1 ACS Paragon Plus Environment

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in the direction normal to basal plane. This inability to conduct electricity is an obvious corollary of the lacking connectivity between the graphene planes. This however can be altered qualitatively by a simple topological trick essentially transforming the multiple disconnected graphene layers into one atomic plane spiraling continuously around the line perpendicular to the basal plane, closely following a Riemann surface (namely, a log z type). Nature plays this trick routinely, to circumvent the difficulties of nucleating repeatedly the stacked atomic planes, and instead to grow the crystals along the spiral paths. The price is the flaw in the lattice, a screw dislocation line, γ. With each full turn around γ the graphene plane propagates in γ-direction by the same step, a Burgers vector b || γ. Indeed, such graphene screw dislocations (GSD) are quite common in carbon. Abundant screw dislocations with Burger’s vectors of |b| = 3.4 Å can be found in annealed pyrolytic graphite6, 7, and recently we discussed their role in common anthracite8. Each GSD comprises a dense stack of covalently and continuously connected graphene layers, or can also be viewed as a single nanoribbon, spiraled with a pitch b and coil angle b/2r falling with the distance r from the dislocation line. It must conduct electricity if voltage is applied along γ, generating perhaps considerable magnetic fields at the core region, as we demonstrate theoretically below. Studies of quantum inductance in terms of the dynamic complex admittance, using the non-equilibrium Green’s function approach (NEGF)9-11, mainly focused on the response to the high-frequency external field but did not explicitly reveal the connection with the magnetics. An alternative is to apply the Biot-Savart law directly to the current density field, which involves double integration over the entire meshgrid12, computationally unaffordable for GSD. Instead, we calculate the “bond currents” in the GSD and then find the magnetic field they produce. Higher-level first-principle current density calculation is only invoked to avoid the singularities at the atomic bond-lines. Figure 1 a-c shows the structure of a single screw dislocation. It imposes in its vicinity AA-stacking, whose energy is only ~7 meV/Å2 higher than of AB-stacking13-15. To be concrete, and in accord with typically observed graphene crystallites16, we chose exterior shape as hexagon bounded by all-armchair or all-zigzag edges, AGSD or ZGSD. Following the convention17 for graphene nanoribbons (GNR), the width of AGSD is defined by counting the dimer lines (NA) from the inmost atoms to the outmost edge, and the number of zigzag chains (NZ) denotes the width of ZGSD. 2 ACS Paragon Plus Environment

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Figure 1 (a-c) Atomic structure of a graphite screw dislocation, GSD. (a) Axial view with armchair edge, width NA. (b) Sideview, Burgers vector b parallel to the dislocation line γ (red). (c) Axial view of ZGSD, width Nz. (d-f) The calculated band structure of armchair-edge AGSD of widths NA = 12, 13, 14, and of zigzag-edge ZGSD for Nz = 7, 8, and their bandgaps in (e).

Then, for a chosen atomic structure, the Hamiltonian matrices follow the orthogonal πorbital tight-binding (TB) model with interatomic hopping parameters v1 = -2.7 eV for the first neighbors and v2 = -0.069 eV for the second, respectively18-20; both vary exponentially with the bond length, to account for possible deformations21. Since most experiments find the conductivity along the c-axis to be ~104-105 times lower than within the basal plane, in both graphite crystal and pyrolytic graphite22-25, we assume the c-axis tunneling and the interlayer coupling to be negligible; main expected consequence of small crosstalk between the layers is that it can short-circuit too long helical paths, making solenoids of too large exterior 3 ACS Paragon Plus Environment

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diameter > 1 μm less efficient. One should point out that similar to zigzag GNR, the ZGSD may display edge magnetism, which cannot be captured in our TB model, yet its consequences can be assessed after the currents and produced fields are calculated; on the other hand, earlier studies19 suggest that in the presence of currents through the system, the spin-polarization yields higher energy of the ground states relative to the spin-unpolarized states. Therefore, our spin-restricted single-orbital TB model appears to be sufficient for capturing the electronic behavior of GSD, while also affordable for computing these multi-atomic structures. In particular, the model is able to capture the strain-induced changes in the electronic structure, equivalent to the effects of pseudo-magnetic field, like Landau levels, see Supporting Information. Our band structure calculations show that neither the inner nor outer radius alone fully controls whether the GSD is metallic or insulating. However, the width does, in a way similar to GNR and in spite of qualitative difference caused by the spiraling and the fact that the opposite “edges” of such ribbon drastically differ in length. As shown in Fig. 1 d-e, the AGSD is metallic when NA = 3p+1 (p is an integer). This is indeed analogous to armchair-edge GNR which are only metallic for NA = 3p+2 widths17, 26. In case of ZGSD, interesting width-dependence was found: only when NZ is odd, the ZGSD is metallic as shown in Fig. 1 e-f, in contrast to zigzag GNR established earlier to be metallic regardless of their width NZ (with either tight-binding approximation or density functional theory DFT, ignoring spin). For the insulating GSD of either A or Z type, the energy gap tends to zero with the increasing width, approaching the zero-gap of graphene. For evaluating transport through GSD, using the standard NEGF approach one can obtain the conductance G(E) as a function of energy for various structures, as shown in the Fig. 2a for NZ = 9 as example. Total current I(V) is then found as an integral within the energy range -V/2 < E < V/2, which overall describes the conductor and often constitutes sufficient answer.

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Figure 2 (a-b) Metallic zigzag-edge ZGSD, Nz = 9. (a) Transmission coefficient as a function of energy. (b) Spatial current distribution integrated over various energy ranges of panel (a), left for green region (bias V = 0.4 V), middle for green+blue region (V = 0.7 V), right for green+blue+yellow region (V = 2 V). (c) Bright-dark map of current distribution, for Nz = 21 and bias of 0.4 V.

Here however, our interest goes beyond the total current, since the magnetic field B it produces and the GSD inductance L, both depend on how the current is distributed in space. It is instructive to first consider GSD classically, as if made of a continuous ohmic material sheet of conductivity σ (for graphite it would be σ = 105/Ω·m27) which spans from the core to the exterior (radii R` to R), while also spirals with a winding density n = 1/|b|. A voltage v (per each turn) generates concentric currents, dj = vσb·dr/2πr, that is integral current J = (vσb/2π)·ln (R/R`). The magnetic flux through a plane normal to the GSD axis is Φ = (μovσbn/4)·(R2 – R`2), while the field along the axis is B(r) = (μonvσb/2π) ln (R/r). Integrating μo-1 B2(r) πr·dr yields the field energy, and comparing it with ½LJ2 expression, one finds the inductance, L = πμ0R2n2·f(R`/R), with f(c)  (½ + c2 ln c – ½ c2) / ln2 c,

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(1)

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For 0 < c  R`/R < 1, the dimensionless factor f(c) does not exceed one, approaching it when R`  R; very roughly, f(c) ~ c. The low-frequency inductance of this classical-ohmic solenoid is fully defined by geometry, regardless of the material and the operational voltage. For our GSD of nanometer diameter, the simplistic ohmic model must be replaced by quantum analysis of the transport and its spatial distribution. To this end, the local bond currents, per unit energy at a given E, can be calculated within the orthogonal π- tight-binding model as28:

i  E  

4e

Im  H D ( E )  ,

(2)

where α and β represent the atomic sites, H𝛼𝛽 is the Hamiltonian matrix element, and D(E) is energy-resolved density of states. The latter corresponds to the axially propagating Bloch waves of certain energy E and can be evaluated within the standard real-space NEGF approach28, 29. Then the bond currents under a voltage can be obtained by integration: u

i   i  E dE ,

(3)

d

where µu and µd are the Fermi levels at the end-leads, shifted at the non-equilibrium conditions by the applied bias, V = μu – μd. The bond current flow is denoted by a vector i from α to β. The total circuit current through the helix-solenoid is the sum of all bond currents through an arbitrary plane crossing it. For visualization purposes it is more convenient to define19 a current vector at each atom as i 

1  i . For example, in a flat GNR such vectors are nearly parallel 2 

to its axis19. (Note that the magnitude sum of the bond currents at a given atomic site must be zero per Kirchhoff's current law.) Computed current distributions are shown in Fig. 2 b-c. For AGSDs, although the current pattern changes with the voltage, the core hexagon-spiral always plays a dominant role and the currents decrease towards the exterior, with no qualitative difference from the classical ohmic case above. For ZGSD, on the contrary, the zigzag exterior contributes notably in the current distribution. As the operational voltage rises, the current distribution exhibits roughly three stages. First, in single-channel mode (green region in Fig. 2a) the currents flow along the multiple concentric circles, Fig. 2b, left. Although the core hexagon plays a leading role in transport, the currents do not decrease monotonically towards the exterior, but the perimeter zigzag edge carries substantial currents as well. Second, when the rising voltage opens the 6 ACS Paragon Plus Environment

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second double-channel (light-blue in Fig. 2a), the current vectors almost disappear except those around the core and the ones at the outer zigzag edge, Fig. 2b, middle. The phenomenon of current-carrying edge is consistent with the decomposition of wave functions around the Fermi level; previous analytical study also demonstrated that in a semi-infinite graphene sheet the Bloch states around the Fermi level are localized at the zigzag edges17. The third stage, i.e. the energy region up to the yellow in Fig. 2a is engaged, the incremental currents mostly distribute around the core hexagon but attenuate towards the outer edge, as indicated in Fig. 2b, right; this remains the case thereafter for higher voltages. The outer-edge currents contribute significantly within the energy window of the first single-channel mode, which however shrinks for greater outer radius. If the size is sufficiently large, the currents in the outer-edge bonds are negligible even when the bias is quite low, as shown in Fig. 2c. Accordingly, the zigzag-edge transport contributes only for a relatively small cross-section and under low bias. Such change of current distribution pattern with voltage constitutes a nonlinearity which must further lead to non-linear inductance, as discussed below. Having the currents at all the bonds computed, and the magnetic field from each such current-segment known analytically, one can find the resultant magnetic field19, except the singularities at the bond-line vicinity, where B(r )   when r  r '  0 . Such divergence in the field and in the associated flux and energy is not affordable if one wants to assess the overall inductance of the bond-network. To eliminate it we employ an expression providing a given bond current with its finite spatial current density29: J (r ) 

1   '  i (r ') *j (r ) ,  dE  Gij (E ) lim r r ' 2 ij 

(4)

where i ’s are atomic basis sets, and the G  ( E ) is the matrix correlation function. With the Hamiltonian and basis sets (2s, 2p) extracted from the DFT calculations within the SIESTA code30, 31, we obtained the current density distribution of a monoatomic carbon chain32. We found33 that the distribution of current density is axially-symmetric and well fitted by i(r) = Ar3er/

, r here being the distance from the bond’s axis. When the total current varies, i(r) only scales

in its amplitude A while  remains unchanged. Given the value of a single bond current, the amplitude A is thereby determined. Next, a bond current is partitioned into a bundle of thin cylinders and the field from each can be evaluated analytically via the Biot-Savart law; finer partitioning proceeds until the magnetic field converges. 7 ACS Paragon Plus Environment

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Figure 3 The distribution of magnetic vectors of a metallic ZGSD with Nz = 7. Top: red curve shows the magnetic field variation along the diameter; its non-symmetry is caused by the non-equal cross-plane spacing from the adjacent atomic layers (bottom).

Figure 3 shows the field vectors on a longitudinal cross section through the axis of metallic ZGSD, with NZ = 7 and a hollow ~1 nm core. The maximum is reached at the center axis and the magnetic field is almost uniform within the empty cavity. Outside the cavity the strength of the field decays towards the exterior. We also observe that if the cross-section is large enough the field tends to zero inside-near the edges since the currents at edge vanish. Beyond the physical borders the magnetic vectors vanish rapidly, indicating an advantageous zero magnetic leakage. Moreover, in our simulations the field at the axis is proportional to the total current, as in an ideal solenoid. With the magnetic vectors available at any given point one can compute the magnetic flux  = B(r) 2πr·dr through a plane normal to the axis of a GSD, or energy μo-1 B2(r) πr·dr, to determine the inductance L. We find that for some metallic ZGSDs with a small cross-section and minimum core R` = 1.2 Å, the flux is not simply proportional to the total current (Fig. 4a), indicating that the inductance is a function of the total J, not a constant as for classical ohmic inductor. The turning points on the (J) curves (shown example is for NZ = 5) represent the change out of the first single-channel mode (cf. Fig. 2a), when the contribution of the edge lowers while the currents start to concentrate around the core. A nonlinear inductance of metallic ZGSD has a distinct physical origin, as discussed above: the edge effect significantly alters the pattern of the current, whose value at a given lattice site is not simply linearly proportional to the 8 ACS Paragon Plus Environment

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voltage (and the total current), accordingly the distribution of the magnetic field across the solenoid varies and its resulting flux is not proportional to the current. When the ZGSD diameter is larger, the magnetic flux dependence on the total current gradually becomes linear, as shown for NZ = 21 in Fig. 4a. The inset there shows that our computed flux remains lower than that of the analytical model (continuous ohmic), approaching it as the radius grows. More details can be seen from calculated axial magnetic field versus the radial position. In Fig. 4b, the dashed blue line is obtained by simply scaling the curve of 0.5 V (solid blue), to fit its maximum with that of the red curve, as actually computed. Apparently, at smaller cross-sections, the magnetic field fluctuates significantly, and at any point (except the axis) its value grows with the total current not linearly, but more slowly. Hence the inductance is systematically lower than the continuous model. When the cross-section becomes sufficiently large, the fluctuations are smoothed out, the B(J) relation at each point is closer to linear, and so is the magnetic flux. Therefore we can infer that when it comes to practical cases in which the cross-section can be regarded as infinite, the magnetic field at each point will vary linearly with the total current, and the inductance will be consistent with the prediction of Eq. (1). One can mention here that a large magnetic field in wider solenoids can switch antiparallel edge states to parallel34; this interplay of the current and magnetic state can perhaps be useful in spintronics and deserves further more detailed consideration.

Figure 4 (a) The magnetic flux  vs the total current J in a metallic ZGSD of small width NZ = 5 (red) and wider NZ = 21 (green triangles, fitted by the ohmic-solenoid formula, blue line). Inset: computed inductance for the atomistic models (circles) and the red curve from the classical model per Eq. (1). (b) Magnetic field B vs the radial position. The dashed blue lines are obtained by simply scaling the curves of

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0.5 V (solid blue), to highlight the deviations from the red curves, actually computed for 1.2 V. For a wider GSD (NZ = 21) the field rise is closer to proportional, while change in the pattern lessens.

If the cross-section is sufficiently wide to make the edge effects negligible, a GSD is able—due to the high density of turns—to produce considerable inductance while saving a great amount of space, compared with the existing contemporary options. For example, for an integrated planar polygon spiral inductor with 8 turns, 3.0 µm turn spacing, 6.5 µm turn width, and 205 µm outer diameter, the inductance is 7.3 nH2, while a GSD with the minimum inner radius 1.2 Å and a normal thickness 1 µm requires only an outer diameter of 70 nm to produce the same inductance. Its parasitic capacitance, roughly estimated as for serially connected35 numerous turn-to-turn plates capacitors, is only 3.4 × 10−5 fF; the planar spiral inductor in the above example has a much greater 45 fF2. Therefore, the parasitic capacitance of a GSD is negligible compared to spiral inductors, owing to the tiny cross-section and the massive number of turns. Note that according to Eq. (1), larger inner radius will increase the inductance, while parasitic capacitance will be lessened. In summary, we discover an extraordinary inductor performance of a nanostructure naturally occurring, unlike earlier hypothetical examples36-38, as a screw dislocation in graphitic carbons, e.g. in coal8. In an isolated form, its helicoid topology remarkably mimics a Riemann surface and ensures connectivity of all constituent graphene layers, acting essentially as a single layer wound around the dislocation line. If voltage is applied to GSD, its full covalent continuity permits significant electrical current, whose helical trajectories give rise to very large magnetic field near the center. In particular, for small diameters when such solenoid must be treated as quantum conductor, the detailed analysis (affordable within the tight binding model, augmented by DFT computations when intra-bond current spread is needed to avoid singularities) reveals that how the bond-currents are distributed between the core and exterior varies with the voltage. Consequently, the magnetic flux is not simply proportional to applied current, and magnetic energy is not quadratic, rendering such inductor nonlinear (although we do not find any “negative differential inductance”, in principle possible). For larger diameters the behavior approaches that of classical ohmic helicoid with linear characteristics, the magnetic field in the core B = μonJ = μoVσ·ln (R/R`) and the inductance increasing with the core cavity size, up to L = 10 ACS Paragon Plus Environment

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πμ0(R/b)2 per unit length. Due to unique winding density n = 1/b, such quintessential inductor can produce large magnetic fields (~1 T) in its core cavity and offer inductance several orders of magnitude greater (per weight or volume) than any currently achievable in microelectronics. Last but not the least, the feasibility and even manufacturability of the described nano-solenoid can be mentioned: an area containing a screw dislocation in graphite crystal can be cut out by variety of techniques (top-down), or graphene sheet can be enticed to grow spirally39 (bottom up); in the latter case, a stacked graphene/h-BN bilayer40 helicoid, if possible, could add an advantage of better insulation and suppressed crosstalk between the layers-loops and reduced parasitic capacitance, undesirable at higher frequencies.

Supporting Information Available: Comparison of the higher level TB methods with the single orbital TB, demonstrating its ability to capture the strain-induced pseudomagnetic fields and Landau levels, while also show the absence of the latter in case of GSD structures. This material is available free of charge via the Internet at http://pubs.acs.org.

The authors declare no competing financial interest.

Acknowledgements. We thank Chris Bowen for stimulating discussions at the early stage of this work. H.Y. also thanks Alex Kutana and Evgeni Penev for help.

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(11) Wang, J.; Wang, B.; Guo, H. Phys. Rev. B 2007, 75, (15), 155336. (12) Im, J.; Kim, Y.; Lee, C. K.; Kim, M.; Ihm, J.; Choi, H. J. Nano Lett. 2011, 11, (4), 1418-1422. (13) Bichoutskaia, E.; Popov, A. M.; El-Barbary, A.; Heggie, M. I.; Lozovik, Y. E. Phys. Rev. B 2005, 71, (11), 113403. (14) Telling, R. H.; Heggie, M. I. Philos. Mag. Lett. 2003, 83, (7), 411-421. (15) Suarez-Martinez, I.; Savini, G.; Zobelli, A.; Heggie, M. J. Nanosci. Nanotechnol. 2007, 7, (10), 3417-3420. (16) Luo, Z. T.; Kim, S.; Kawamoto, N.; Rappe, A. M.; Johnson, A. T. C. ACS Nano 2011, 5, (11), 9154-9160. (17) Nakada, K.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. Phys. Rev. B 1996, 54, (24), 17954. (18) Reich, S.; Maultzsch, J.; Thomsen, C.; Ordejon, P. Phys. Rev. B 2002, 66, (3), 035412. (19) Areshkin, D. A.; White, C. T. Nano Lett. 2007, 7, (11), 3253-3259. (20) Xu, F.; Sadrzadeh, A.; Xu, Z.; Yakobson, B. I. J. Appl. Phys. 2013, 114, (6), 063714. (21) Pereira, V. M.; Castro Neto, A. H.; Peres, N. M. R. Phys. Rev. B 2009, 80, (4), 045401. (22) Dutta, A. K. Phys. Rev. 1953, 90, (2), 187-192. (23) Ubbelohde, A. R.; Blackman, L. C. F.; Mathews, J. F. Nature 1959, 183, (4659), 454-456. (24) Tsang, D. Z.; Dresselhaus, M. S. Carbon 1976, 14, (1), 43-46. (25) Klein, C. A. Rev. Mod. Phys. 1962, 34, (1), 56-79. (26) Yariv, A. IEEE J. Quantum Electron. 1973, QE-9, (9), 919-933. (27) Deprez, N.; McLachlan, D. S. J. Phys. D: Appl. Phys. 1988, 21, (1), 101-107. (28) Todorov, T. N. J. Phys. Condens. Matter 2002, 14, (11), 3049-3084. (29) Xue, Y. Q.; Datta, S.; Ratner, M. A. Chem. Phys. 2002, 281, (2-3), 151-170. (30) Soler, J. M.; Artacho, E.; Gale, J. D.; Garcia, A.; Junquera, J.; Ordejon, P.; Sanchez-Portal, D. J. Phys. Condens. Matter 2002, 14, (11), 2745-2779. (31) Artacho, E.; Anglada, E.; Dieguez, O.; Gale, J. D.; Garcia, A.; Junquera, J.; Martin, R. M.; Ordejon, P.; Pruneda, J. M.; Sanchez-Portal, D.; Soler, J. M. J. Phys. Condens. Matter 2008, 20, (6), 064208 (32) Liu, M.; Artyukhov, V. I.; Lee, H.; Xu, F.; Yakobson, B. I. ACS Nano 2013, 7, (11), 1007510082. (33) Xu, F.; Sadrzadeh, A.; Xu, Z.; Yakobson, B. I. Comput. Mater. Sci. 2014, 83, (0), 426-433. (34) Kim, W. Y.; Kim, K. S. Nature Nanotechnol. 2008, 3, (7), 408-412. (35) Grandi, G.; Kazimierczuk, M. K.; Massarini, A.; Reggiani, U. IEEE Trans. Ind. Appl. 1999, 35, (5), 1162-1168. (36) Sadrzadeh, A.; Hua, M.; Yakobson, B. I. Appl. Phys. Lett. 2011, 99, (1), 013102. (37) Avdoshenko, S. M.; Koskinen, P.; Sevinçli, H.; Popov, A. A.; Rocha, C. G. Sci. Rep. 2013, 3, 1632. (38) Zhang, D.-B.; Seifert, G.; Chang, K. Phys. Rev. Lett. 2014, 112, (9), 096805. (39) Yan, Z.; Liu, Y.; Lin, J.; Peng, Z.; Wang, G.; Pembroke, E.; Zhou, H.; Xiang, C.; Raji, A.-R. O.; Samuel, E. L. G.; Yu, T.; Yakobson, B. I.; Tour, J. M. J. Am. Chem. Soc. 2013, 135, (29), 1075510762. (40) Haigh, S. J.; Gholinia, A.; Jalil, R.; Romani, S.; Britnell, L.; Elias, D. C.; Novoselov, K. S.; Ponomarenko, L. A.; Geim, A. K.; Gorbachev, R. Nature Mater. 2012, 11, (9), 764-767.

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