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C: Physical Processes in Nanomaterials and Nanostructures

Conductance of Curved 3M-1 Armchair Graphene Nanoribbons Jie Zhang, and Eric P. Fahrenthold J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b06717 • Publication Date (Web): 12 Aug 2019 Downloaded from pubs.acs.org on August 13, 2019

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The Journal of Physical Chemistry

Conductance of Curved 3M-1 Armchair Graphene Nanoribbons Jie Zhang and Eric P. Fahrenthold∗ Department of Mechanical Engineering, University of Texas, Austin, TX 78712 * E-mail: [email protected] Phone: 512-471-3064. Fax: 512-471-8727

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Abstract The electrical properties of deformed graphene nanoribbons (GNRs) are of considerable current research interest, since GNRs have wide potential application in electronic and nanoelectromechanical devices. Examples include flexible carbon based electrical conductors, soft actuators, and chemiresistive sensors. Although considerable experimental and computational research has investigated both zigzag and armchair GNR performance, general analytical descriptions of electromechanical coupling in GNRs are needed. Motivated by questions raised in published experimental research, the new modeling research presented here provides a general description of the combined effects of length and curvature on the current-voltage characteristics of 3M-1 aGNRs. Consistent with experiment, the modeling results show that total length and total rotation (along a circular arc) are orthogonal coordinates which determine the GNR resistance. The ratio of curved GNR current to flat GNR current, at the same length and voltage bias, is a linear function of the rotation over a wide operating range. The proposed model generalizes the well known exponential decay law for the resistance of flat semiconducting nanowires, and has direct application in the design of experiments and the conceptual design of new devices, including high resolution displacement transducers, strain gauges, and nanoresonators.

Introduction Graphene offers an unusual combination of electrical, optical, and mechanical properties, and is therefore the focus of much recent research on a variety of electrical and electromechanical devices, such as touch panels, 1–4 soft actuators, 5 and wearable sensors. 6,7 Understanding the effects of mechanical deformation on graphene properties is often of central interest in device design: examples include electrical conductors, 8–10 high strength fibers, 11,12 nanowall temperature sensors, 13 resonators, 14,15 strain gauges, 16 and ‘sandwich’ structures 17 or ‘bubble’ inclusions 18 in graphene devices. The development of bottom-up synthesis methods and

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atomically precise measurement systems 19–22 has dramatically increased the potential for applying graphene’s novel properties in practical devices. The preceding advances should however be complemented by first principals based modeling research, to facilitate the application of computer aided design methods (well established at the macro and microscale level) in nanoscale electromechanical systems development. An electromechanical structure-property relationship of cross-cutting interest in graphene based device design is the effects of deformation on graphene nanoribbon (GNR) conductance. Previous work has investigated the effects of strain, curvature, and in-plane bending on GNR conductance, in various geometries, and attributed measured or computed conductance changes to bandgap changes, rehybridization, 23,24 and other effects. The most basic previous work has computed conductance changes due to uniaxial loading 25,26 and associated those changes with bandgap effects. Other work has considered bent or folded GNRs , 2,27,28 ‘mapping’ local strain distributions to the aforementioned basic cases in order to explain overall GNR performance. An additional series of papers has considered ‘in-plane’ bending, 29,30 defined as bending about an axis normal to the GNR surface, which is distinct from the deformation mode considered in this paper. Controlled fabrication of ultra-narrow GNR’s has been demonstrated, 22 and the electrical performance of curved 5-aGNRs and 7-aGNRs has been measured in scanning tunneling microscope (STM) experiments. 19,31 In the cited ‘lifting’ experiments, nanoribbons were subjected to large bending deformations and their conductance measured as a function of bias voltage. Such basic test data is well suited to elucidate fundamental physics of wide design interest. In the 7-aGNR case, the experiments showed, as expected, an exponential decay of the current with GNR length, at a fixed bias voltage. In the 5-aGNR case however the test data 31 showed unexpected results, namely a non-monotonic variation in the nanoribbon current, with a peak current observed at low liftoff values followed by an exponential decay of the current at high conducting lengths. Although the measured 5-aGNR charge transport was found to be consistent with ab initio model calculations, the physical basis for the

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observed nonlinearity was described as ‘unknown’ and speculatively attributed to complex coupling of the carbon nanowire with the metal substrate. Recognizing the importance of electromechanical coupling in nanoelectromechanical systems (NEMS) design and the widespread interest narrow bandgap GNRs for electronic devices, 22 this paper describes a new ab initio study of the conductance properties of curved 3M-1 aGNRs, extending work performed to complement the cited aGNR experiments. The numerical results are used to formulate a new analytical conductance model for 3M-1 aGNRs, including both length and curvature effects, which is then evaluated by comparison with published experimental data. An example application of the modeling results to nanoscale displacement transducer design illustrates the utility of the modeling research.

Computational methods The ab initio equilibrium and change transport calculations presented in this paper were performed using the open source code suite SIESTA. 32 The nanoribbon currents, as a function of bias voltage, were computed using the TranSIESTA 33 module, which employs a non-equilibrium Green’s function method. Equilibrium was defined by relaxing the models to meet an atomic force convergence criterion of 0.01 eV per angstrom. A local density approximation (LDA) 34 exchange-correlation functional was employed in the relaxation calculations, and a generalized gradient approximation (GGA) method with the PerdewBurke-Ernzerhof (PBE) 35 exchange-correlation functional was employed in the transmission calculations. Double-zeta polarized basis sets were used for all atoms, and the mesh cutoff energy was set to 300 Ry. Nanoribbon geometry. The generic nanoribbon model described in this paper is defined in Figure 1. Figure 1a shows the notional physical system, a curved nanoribbon in contact with two electrodes. The corresponding computational model is described as follows. The geometry of the curved segment of the GNR, used to initialize the computational

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model, is indicated in Figure 1b: the curved segment of the GNR model has length L and is made up of two circular arcs, defined by a rotation θ and a radius of curvature R. Figure 1c depicts a GNR configuration for the extreme case θ = π, while Figure 1d depicts a configuration for which θ < 0.5π. The vertical separation distance of the ends of the GNR is H, and the slopes of the lines which define the curved portions of the GNR are zero at their end points. Note that for the assumed initial geometry of the curved segment of the GNR, the parameters L, θ, R, and H are related by the constraints

H θ = L (1 − cos θ) ,

L=2Rθ

(1)

so that only two parameters specify the curved geometry. Attached to each end of the curved portions of the GNR are two flat sections, the first a mechanical buffer and the second a computational electrode. The bias applied to the nanoribbon is determined by stipulating the difference between the electrode voltages, and the current carrying length of the modeled GNR is S = L + Lbuf

(2)

where 0.5 Lbuf is the length of the mechanical buffer segment on each end of the GNR. In the equilibrium calculations, the atom positions in the electrode and buffer regions are fixed, while the atoms in the curved portion of the nanoribbon are free. The modeled nanoribbons are in all cases 3M-1 aGNRs (M an integer > 1) with a unit cell (u.c.) length of 0.4278 nm, edge-terminated by hydrogen atoms. Note that the modeled class of nanoribbons offers the smallest aGNR band gap, according to the well known 3-fold periodic pattern, 36,37 hence the modeled aGNR are very close to metallic. The modeling results presented in the next section include geometric parameter variations over the ranges

4 ≤ L ≤ 18 (u.c) , 0 ≤ θ ≤ π , 0 ≤ H ≤ 5.39 (nm) , 6.80 ≤ R ≤ ∞ (nm)

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(3)


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where an infinite radius of curvature denotes a flat nanoribbon. The mechanical buffers and electrodes on each end of the GNR were three unit cells in length. Current-voltage (I-V) performance was computed for bias voltages in the range 0.0-0.8 volts. Nanowire model. The ab initio calculations described in the next section were used to formulate a nanowire model which extends the well known exponential decay model for current flow in flat semiconducting nanowires subjected to a bias voltage. The analytical modeling work presented here seeks a separable, two-parameter description of length and curvature effects on the nanoribbon current I(η, S, V ) in the form

α(η) ≡

I(η, S, V ) , η ∈ {θ, H} I f lat (S, V )

(4)

where α is the ratio of the current in a curved nanoribbon to the current in a corresponding flat nanoribbon, η is a curvature parameter (either H or θ), V is the applied bias voltage, and I f lat (S, V ) is the current in a flat nanoribbon at the same length and bias voltage. Consistent with previous computational 19 and experimental 38,39 work, the current in the flat nanoribbon is represented as

I f lat (S, V ) = Io (So , V ) e−k(V )

(5)

where < x − xo > is the Macualey bracket (defined as ‘x − xo ’ for x > xo and zero otherwise), k(V ) is the voltage dependent inverse decay length, and So is the largest nanoribbon length for which no exponential decay is observed. Voltage dependance of the inverse decay length is illustrated by previous computational work 19 on 7-aGNRs, which indicated a conductance transition from exponential decay with length (at low bias) to near ballistic conductance (at high bias). The functional form of the curved wire model implies that the dependence of the inverse decay length on the bias voltage will also apply to the curved wire case. This is consistent with published work on curved nanowire experiments, 40 in which it was noted that higher bias voltages may compensate for the detrimental effects of increased nanowire 6


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length on conductance.

Results and Discussion The computational results are presented and discussed as follows. First the ab initio modeling results for flat 5-aGNRs are shown to be consistent with the well known exponential decay relation for semiconducting nanowires. Next the ab initio modeling results for curved 5aGNRs are compared to the separable form of equation (4) proposed here, which is shown to extend the existing flat armchair nanowire model to the curved nanowire case. Third, the curved nanowire analysis applied to 5-aGNRs is shown to describe the current response of 8-aGNR and 11-aGNR to bias voltages; the effects of curvature on conductance for all three 3M-1 nanoribbons are attributed to changes in bandgaps and scattering properties induced by the nanowire deformation. The discussion which follows the presentation of the modeling results considers strain distributions in the modeled nanoribbons, a focus of much previous computational work, and finally compares the form of the nanowire model developed in this paper with published experimental data. Current-voltage results for 5-aGNRs. A series of ab initio calculations were performed to compute the current-voltage response of flat 5-aGNR nanowires, as a function of length, at four different bias voltages. The results are shown in Figure 2, which confirms the exponential decay model of equation (5). Next a series of ab initio calculations were performed to compute the current-voltage response of curved 5-aGNR nanowires, as a function of rotation (θ) and vertical electrode separation (H), at four different bias voltages The rotation was varied over the range 0 ≤ θ ≤ π and the separation height was varied from zero to six nm. The models and results are shown in Figure 3. Figures 3(a) and 3(b) illustrate the effects of the parameter variations on the deformed nanoribbon shapes; Figures 3(c) and 3(d) show the computed current ratios (α) for the curved nanoribbon currents to those of corresponding (same length and bias voltage) flat nanoribbons. Note the color consistency

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of the geometry plots shown in Figures 3(a) and 3(b) and the current ratio plots shown in Figures 3(c) and 3(d). Finally Figures 3(e) and 3(f) plot the same current ratios in the form α(H) versus H and α(θ) versus θ respectively, evaluating the postulated separable form of equation (4) as a general description of curvature effects on the conductance of 3M-1 aGNRs. The ab initio results shown in Figure 3 indicate that: (1) the function α(θ) is single valued, while the function α(H) is not, (2) the variation in α(θ) is not monotonic, showing a peak current which significantly exceeds that of the corresponding flat nanoribbon at low rotations, and (3) the same current ratio function α(θ) applies for all bias voltages. Note that the function α(θ) is linear over a wide range of values of the rotation (θ), and that the modeled configurations spanned a very wide range of θ values near the mid-range of H, as indicated in Figure 3(e). The functional dependence of the current ratio α on θ suggests a generalization, in separable form, of the exponential decay model for current flow in semiconducting nanowires, one which includes the effects of both curvature and length on the conductance of 3M-1 aGNRs. Note that the largest deviation of the ab initio data from the average trend depicted in Figure 3(f) occurs for a bias voltage of 0.2V. At this low bias voltage, the current response for both the straight and the curved aGNRs is small, and the variations in α are associated with computing the ratios of two small numbers. An analytical expression for the function α(θ) was obtained by fitting the ab initio 5aGNR data to the polynomials

fa (θ) = 1 +

4 X

ai θ i ,

fb (θ) = bo + b1 θ

(6)

i=1

where the coefficients ai were obtained by fitting over the entire range 0 ≤ θ ≤ π while the coefficients bi were obtained by fitting over the range π/6 ≤ θ ≤ π. The fitted coefficients are listed in Table 1. Figure 4(a) shows the fitted result for fa (θ). The best interpolation of the computed data appears to be provided by the composite interpolation shown in Figure 4(b), using the fit fa (θ) (solid line) at small values of θ and the fit fb (θ) (dotted line) elsewhere.

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The physical bases for the computed effects of rotation on aGNR conductance appear to be as follows: • At low rotations, the HOMO-LUMO (highest occupied molecular orbital-lowest unoccupied molecular orbital) band gap is a sensitive function of rotation (θ). Figure 4(c) plots the band gap for 5-aGNR, computed at zero bias. The result suggests that band gap effects are responsible for the high currents measured at small rotations, currents which exceed those for the corresponding flat nanoribbons. The Supporting Information provides a band diagram and HOMO-LUMO orbital plots for the 5-aGNR, both at zero bias, as well as tabulated data for the computed current at various bias voltages. • At all rotations, the transmission of electron waves is hindered by the need for the flux to follow the curved path of the aGNR surface. Once the band gap stabilizes, this scattering effect dominates and the change in nanoribbon current is approximately proportional to the change in rotation. It is important to emphasize that the preceding description of rotation dependent effects applies at all nanowire lengths and for all bias voltages. General current-voltage results for 3M-1 aGNRs. To evaluate the general applicability of the expression (4) for 3M-1 aGNRs, additional ab initio calculations, like those made for the 5-aGNRs, were performed for 8-aGNRs and 11-aGNRs and additional analytical interpolations were constructed. As indicated by the plotted results shown in Figure 4(c) and Figure 4(d), and the fitted coefficients listed in Table 1, the responses of the three a-GNRs are qualitatively very similar. The ab initio conductance calculations show clear trends with aGNR width: (1) at small rotations, the peak current decreases with aGNR width, while (2) at high rotations, the rate of decrease of the current ratio with aGNR width (fitted parameter b1 ) is also reduced. The physical bases for the computed effects of aGNR width on current are suggested by the band gap plots in Figure 4(c) and the strain distribution plots shown in Figure 5. Note that in the later figure, a ‘bond’ strain (ε) is computed 9



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for every covalently bonded atom pair using

ε=

d − do do

(7)

where d is the separation distance of the nuclei in the computed equilibrium configuration of the curved GNR and do is the corresponding separation distance of the nuclei in the computed equilibrium configuration for a corresponding flat nanoribbon: • With regard to the band gaps: at low rotations, as the aGNR width is increased, the maximum initial HOMO-LUMO zero bias band gaps shown in Figure 4(c) decrease. Hence the initial current enhancing benefits of rotation on band gap closure decline with increasing width. The Supporting Information provides zero bias band diagrams and HOMO-LUMO orbital plots for all three modeled aGNRs, as well as tabulated data for the computed currents in all three modeled aGNRs at various bias voltages. Although the zero bias band gap plots provide important insights, it is the computed current data (hence the current ratio α) which reflects the combined effects of both rotation and voltage bias on the nanowire conductance. Note that for all of the modeled aGNRs, the computed current is small but nonzero at the lowest modeled bias voltage. • With regard to the strains: at all rotations, the variation of the strain fields with nanoribbon width in the deformed aGNR suggest an additional scattering effect on the curved nanoribbon current. All of the modeled GNRs show a strain distribution which is predominantly tensile at the GNR edges and compressive over the GNR interior. Figure 5 shows color plots of the (dimensionless) strain, for 5-aGNRs and 11aGNRs. The narrower the GNR, the more significant is the edge tension in reducing the overall nanoribbon electron transmission. Conversely, the wider the GNR, the more important is the interior compressive strain field in enhancing the overall nanoribbon transmission. The preceding interpretation of strain effects is consistent with previous work 25,26 on the effects of imposed bulk strains on conductance in flat GNRs. It should 10


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be emphasized however that in the present work considers strain fields obtained from equilibrium analysis of curved GNR configurations subjected to end constraints, which are therefore more heterogeneous than those discussed on the last cited works. Comparison with experiment. The analytical model developed in the preceding subsections was evaluated by comparing equation (4) with data from a published nanoribbon ‘lifting’ experiment. 31 The nanoribbon current measurements were made on a test article which corresponds to one half of the system modeled in this paper: the current was measured as a function of the ‘liftoff’ (z) of the tip of a 5-aGNR from a flat gold electrode, where the input voltage was the difference between the applied tip voltage and the voltage of the gold electrode. Hence, on the basis of symmetry alone (and not invoking any linearity assumption), the model calculations are made for a nanoribbon length twice that reported in the experiments and a bias voltage twice that applied in the experiment. The only significant difference between the model and the experiments is in the precise shape of the curved nanoribbons. As noted previously, the model calculations are made by equilibrating nanoribbons whose atomic positions were initialized to follow circular arcs, so that the initial values of θ, H, and L are linked by the constraints (1). By contrast, the experimental nanoribbon shape was determined by equilibrating nanoribbons subjected to a vertical (lifting) force applied at the nanoribbon tip. The experimental equilibrium nanoribbon shape is described here using: (a) published data 19 (see Supplementary Information, Figure 8 of the last cited reference) on the variation of a narrow GNR conductor length as a function of the liftoff, and (b) published data 19 (see Supplementary Information, page S5 of the last cited reference) on the values of a narrow GNR conductor rotation (θ) as a function of the liftoff. Here the model calculations estimate the experimental nanoribbon length and rotation, at all values of the liftoff (z), by linear interpolation of the length (S(z)) and the rotation (θ(z)) using the cited data. The interpolated descriptions of the variation of the 5-aGNR length and rotation with liftoff allow the model of equation (4) for the curved nanoribbon current to be fit to the 11



experimental data, 31 which provides measured nanoribbon current as a function of liftoff. Figure 6 compares the experimental data with the model, for the fitted parameters

a1 = 3.549 , a2 = −6.366 , a3 = 2.514 , a4 = −0.335 , k = 0.338 nm−1 , So = 4.0 u.c. (8)

and suggests that the separable form of equation (4) provides an accurate representation of the combined effects of length and rotation on conductance, over the experimental range 0 ≤ θ ≤ 1.17, extending the well known exponential decay model for semiconducting nanowires to an important class of deformed nanoribbons of significant technological interest.

Conclusions The electromechanical properties of graphene nanoribbons are of central interest in the development of a number of new devices and systems, and have been a focus of much recent research. Analytical design models built on a fundamental understanding of the underlying physics can assist in the detailed design of proposed nanoelectromechanical systems and facilitate the conceptual design of new nanoelectromechanical devices. Although previous experimental and computational work has studied the conductance of deformed GNRs, curvature effects on electronic transport are not yet well understood. The research presented in this paper addresses the very basic question of curvature effects on the electrical performance of semiconducting 3M-1 armchair GNRs, and suggests the following conclusions: • nanowire rotation and length are orthogonal coordinates which determine the current flow, at a fixed voltage bias; • at low rotations, the dependance of the current flow on rotation is non-monotonic and the current in a curved nanowire can exceed that of a flat nanowire of the same length, due to bandgap effects; • at large rotations, the change in current flow with change in rotation is linear and due 12


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to scattering effects as the electrons are forced to flow along the curved nanowire; • as the GNR width increases, bandgap effects are strongly attenuated and the dependance of the current flow on rotation approaches a monotonic decline; and • as the GNR width increases, the rate of reduction in the current flow with change in rotation is reduced (scattering effects are slightly attenuated). Application of the modeling results presented in this paper may be illustrated by the conceptual design of a nanoscale displacement transducer. Figure 7(a) shows the reference and deformed configurations of a displacement transducer which consists of a single 5-aGNR nanoribbon subjected to a bias voltage, applied across two electrodes. Assuming that the kinematics (1) describe the motion, the horizontal separation distance for the electrodes (D) is D=L

sin(θ) θ

(9)

where L denotes the fixed length of the transducer nanoribbon. If the sensor is operated in the broad region θ > π/6 where the current ratio α(θ) varies linearly with θ, then

α − αref = b1 (θ − θref )

(10)

where the subscript ‘ref’ denotes the reference configuration for the transducer.

Since

α/αref = I/Iref the displacement D − Dref may be obtained from the measured current I using D − Dref θref sin(θ) αref = − 1 , θ = θref + Dref θ sin(θref ) b1

I − Iref Iref

!

(11)

where the separation distance (Dref ), rotation (θref ), current (Iref ), and current ratio (αref ) in the reference configuration are known. Note that the preceding expression describes large motions (relative to Dref ) and is not obtained by linearization about some operating point. Figure 7(b) plots the normalized displacement as a function of the measured current for a 5-aGNR nanoribbon with the reference configuration as depicted in Figure 7(a); the plotted 13



response spans a one hundred percent change in the normalized displacement. Note the linear response over a wide range of negative displacements. The nanoribbon length (L) and the reference configuration rotation (θref ) are adjustable parameters; changing L scales the transducer to smaller or larger dimensions, while changing θref will bias the allowable operating range in the positive or negative displacement direction. Alternative applications for the modeling work presented here might include the design of nanoresonators or nanoscale strain gauges.

Acknowledgements This work was supported by the Office of Naval Research (Grant number N00014-16-2357). Computer time support was provided by the Department of Defense High Performance Computing Modernization Program (project number ONRDC40983493) and the Texas Advanced Computing Center (project number G-815029) at the University of Texas at Austin.

Supporting Information: Band diagrams and HOMO-LUMO plots at zero bias, and computed nanoribbon currents at various bias voltages.

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Table 1: Polynomial coefficients for α(θ) nanoribbon 5-aGNR 8-aGNR 11-aGNR

0≤θ a1 a2 0.443 -0.796 0.289 -0.637 0.167 -0.548

≤ π/6 π/6 ≤ θ ≤ π a3 a4 b0 b1 0.314 -0.042 1.224 -0.322 0.285 -0.045 1.190 -0.300 0.302 -0.057 1.107 -0.245

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Figure 1: a, Schematic depicting a curved graphene nanoribbon conductor. b, Geometric model of a curved nanoribbon. c, Oblique view of a curved nanoribbon with a rotation (θ) greater than π/2. d, Oblique view of a curved nanoribbon with a rotation (θ) less than π/2.

Figure 2: a, Current-voltage curves for flat 5-aGNRs of various lengths (length in unit cells). b, Current versus length (in unit cells) for flat 5-aGNRs at various bias voltages.

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Figure 3: a, Changes in the curved nanoribbon geometry for variations in GNR length (in unit cells) at a fixed electrode separation distance (H). b, Changes in the curved nanoribbon geometry for variations in GNR length (in unit cells) at a fixed rotation (θ). c, Current ratio (α) versus bias voltage for curved 5-aGNRs, at a fixed electrode separation distance (H) and various lengths (in unit cells). d, Current ratio (α) versus bias voltage for curved 5-aGNRs, at a fixed rotation (θ) and various lengths (in unit cells). e, Plot of the computed current ratios for 5-aGNRs versus electrode separation distance (H); note that α is not a single valued function of H. f, Plot of the computed current ratios for 5-aGNRs versus rotation (θ); note that α is a single valued function of θ.

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Figure 4: a, Polynomial approximation (solid line) of the computed current ratio function α(θ) as a function of the rotation (θ). b, Two part polynomial approximation of the computed current ratio function α(θ) as a function of the rotation (θ), employing a linear approximation for θ > π/6. c, Variation of the computed zero bias bandgaps with rotation (θ), for 5-aGNRs (model length of ten unit cells), 8-aGNRs (model length of six unit cells), and 11-aGNRs (model length of six unit cells). d, Variation of the computed current ratio with θ, for 3M-1 aGNRs.

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Figure 5: a, Oblique views of the strain distribution in curved 5-aGNRs, as a function of the rotation (θ). b, Planar views of the strain distribution in curved 5-aGNRs, as a function of the radius of curvature (R) and the rotation (θ), for a length of ten unit cells. c, Oblique views of the strain distribution in curved 11-aGNRs, as a function of the rotation (θ). d, Planar views of the strain distribution in curved 11-aGNRs, as a function of the radius of curvature (R) and the rotation (θ), for a length of six unit cells.

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Figure 6: Comparision of experimental data 31 for current versus liftoff in a 5-aGNR under a bias voltage with the fitted analytical model of equation (4); fitted parameters are provided in the text.

Figure 7: Notional displacement transducer, the proposed design is based on the curved nanoribbon model developed in this paper; reference configuration, θ = π/2; deformed configuration, θ > π/2. b, Computed displacement transducer performance for a 5-aGNR.

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Table of Contents Graphic

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Conductance of Curved 3M-1 Armchair Graphene Nanoribbons | The

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