DFT investigations of the piezoresistive effect of carbon nanotubes for sensor application
pphysica status solidi, 8 October 2018
DFT investigations of thepiezoresistive effect of carbonnanotubes for sensor application
Christian Wagner *,1 , J ¨org Schuster and Thomas Gessner Center for Microtechnologies, Chemnitz University of Technology, 09126 Chemnitz Fraunhofer Institute for Electronic Nanosystems (ENAS), Technologie-Campus 3, 09126 ChemnitzReceived 30 April 2012, revised 7 September 2012, accepted 14 September 2012Published online 29 October 2012
Key words:
Acceleration sensor, Piezoresistance, Bandgap, Carbon nanotube, Density functional theory, Electronic structure, Nan-otechnology, Nanoelectronics. ∗ Corresponding author: e-mail [email protected] , Phone: +49-371-531 38699, Fax: +49-371-531 838699
We investigate the piezoresistive effect of carbon nano-tubes (CNTs) within density functional theory (DFT)aiming at application-relevant CNTs. CNTs are excellentcandidates for the usage in nano-electromechanical sen-sors (NEMS) due to their small band gap at zero strainleading to a finite resistivity at room temperature. Theapplication of strain induces a band gap-opening leadingto a tremendous change in the resistivity. DFT with theLDA approximation yields reasonable results for pure carbon systems like CNTs and is applied to calculatethe electronic structure of experimentally relevant CNTs.For the transport part, a simple ballistic transport modelbased on the band gap is used. We compare our DFTresults for the band gaps of strained CNTs to results oftight binding (TB) models. By introducing a scaling fac-tor of √ , an excellent agreement of the the DFT datawith TB model published in [1] is obtained. Copyright line will be provided by the publisher
Carbon Nanotubes (CNTs) are veryinteresting systems from both, the physical and the tech-nological, point of view. In addition to their outstandingmechanical properties like high stiffness and an enormousYoung’s modulus of about 1 TPa [2], their electronic struc-tures show a rich variety of interesting effects.One of them is the huge piezoresistive effect (up toone order of magnitude per percent strain), which is thechange of the resistivity due to a deformation, relying onthe change of the band gap. This behavior is highly desir-able for NEMS sensor applications. The effect is stronglydependent on the chiral angle – in the same way the elec-tronic structure depends on the chirality. Therefore, at-moistic models are needed to build up device simulations.The modeling of the piezoresistive effect of CNTs us-ing TB models is pioneered by Yang and Han [1] as well asKleiner and Eggert [3]. The latter one tries to include cur-vature effects of CNTs by introducing another TB constant.Because its corrections are made using curvature effects,this model is most likely applied for semimetallic CNTs, which have a tiny band gap. The approximations made inthese models are not yet fully confirmed by a more ad-vanced theory e.g. density functional theory (DFT).TB theory only assumes nearest-neighbor coupling oflocalized electronic states. Therefore, coupling of oppositecarbon atoms in the tube is not straight forward, though itis possible to find a corrected band gap formula used in [3]being valid for small-gap CNTs (where [ n − m ] = 0 ).However, these effects may introduce more additional fea-tures in the band structure, which are not covered by a TBmodel. They may be important for transport calculations,because not only the band gap, but also other details likeeffective mass and higher-order bands are important forcharge transport. Therefore, it is beneficial to use DFT as abenchmark.Since the detailed geometry of a CNT determines itselectronic structure very sensitively, the understanding ofmechanics is explored in advance. The obtained results arecompared to literature data. In a next step, electronic prop- Copyright line will be provided by the publisher a r X i v : . [ c ond - m a t . m e s - h a ll ] J un Christian Wagner et al.: DFT investigations of the piezoresistive effect of CNTs erties are investigated. Between the different models, wefind deviations that need explanation.
Atomistix ToolKit (ATK)from QuantumWise is used [4,5] to calculate the elec-tronic structure of the carbon nanotubes within the DFTframework. For the exchange functional, local density ap-proximation (LDA) is chosen, because it is the one withthe lowest computational cost providing sufficient accu-racy for pure carbon systems like CNTs [6]. The Perdew-Wang version of this functional has been applied [7]. Thebasis set is a double-zeta-polarized (DZP) one, which isrecommended for the simulation of CNTs [8]. The CNTsare considered as geometry optimized, when the remain-ing forces fall below 0.01 eV/ ˚A. Geometry optimizationis performed at each mechanical stretch step induced bylinear scaling of the atomic unit cell (including the atoms)along the tube-axis.For the calculations of the nanotubes, periodic bound-ary conditions are used and the k-grid contains (1x1x20)points (Monkhorst-Pack grid). CNTs are strained and com-pressed linearly. No buckling is observed due to the peri-odic boundaries, which are cutting off long-range deforma-tions. Geometry optimization gives insight into a realisticstructure of the deformed CNTs providing the Poisson’sratio.At each deformation state, the band structure (resolvedby 201 k-points) is evaluated, analyzed and compared toother data with partly different approaches [1,3,9,10].
The mechanical data is pre-sented in figure 1. It shows the total energy of three differ-ent zigzag-CNTs (10,0), (11,0) and (12,0) to illustrate thedependency of the total energy on the strain. It can be seenthat a third order polynomial is sufficient to describe thedata over the whole deformation range allowing a simpleextraction of the Young’s modulus. Besides the Young’smodulus, nonlinear moduli are proposed in literature [11,12] and therefore, η from [11] has been determined. Thevalues are shown in table 1. The Young’s moduli are ingood agreement with the values of about 1000 GPa oftenfound in literature [2]. In addition, the data fully agree with[10], where a DFT plain wave approach was used in orderto calculate the mechanical and electronic structure of car-bon nanotubes.More details about the modeling of the mechanicaldata, the range of the linear regime for those sensor-relevant CNTs can be found in [13]. The electronic structure ofdeformed carbon nanotubes has already been analyzedbased on TB models [1,3]. From the application point ofview, one can ask if these approaches provide realisticdata in combination with a ballistic transport model whichis often used [14,15]. Therefore it is examined, if those T o t a l E ne r g y E t o t [ e V ] (10,0) CNT(11,0) CNT(12,0) CNT DFT dataPolynomial 2nd orderPolynomial 3rd order Figure 1
The total energy of a (10,0), (11,0), and (12,0)CNT under strain. A third order polynomial fits the datareasonably over the whole deformation range. The data forthe different CNTs are shifted by 1 eV for clarity.
Table 1
The Young’s modulus as well as the nonlinearity( η as introduced in [11]) of different CNTs obtained coef-ficients of the polynomial fit in figure 1. CNT Young’s modulus [GPa] η [GPa](10,0) . ± . . ± . (11,0) . ± . . ± . (12,0) . ± .
25 982 . ± . models hold against DFT calculations. Before doing this,a detailed understanding of the models is required.The theory in [3] is more formal than [1] and also in-cludes curvature effects in an empirical way, which areknown to generate tiny band gaps of semimetallic CNTs.The approach of [1] is presented in figure 2: The left side ofthis figure shows the k-space of graphene with the accord-ing k-lines representing the periodic boundary conditionsin a (6,3)-CNT. The color indicates the difference betweenthe valence- and the conduction band of graphene in thek-space according to the TB formula derived in [16] withthe parameters s = 0 . and t = 2 . eV. t is taken from[1] and s introduces the asymmetry in the bandstructure ofgraphene within DFT data (e.g. shown in [17]). These k-lines are shifted by the depicted vector ∆ k [1] due to 4%strain, which is the consequence of the coordinate transfor-mation imposed by the deformation. This shift influencesthe band structure strongly, as it is illustrated in the rightfigure, which represents DFT results. The pristine (6,3)-CNT is shown by straight lines and the strained state bythe dashed red lines. The deformation gives rise to a bandgap opening, leading to an enormous increase of the resis-tivity of the CNT.This behavior can now be traced over the whole de-formation range by subsequent stretch steps. The resulting Copyright line will be provided by the publisher ss header will be provided by the publisher 3 k E [ e V ] G k [1/Ang] x k [ / A ng ] y KK K KKK -1.5-1-0.500.51 E G relaxedstrained Figure 2
The Brillouin-zone (left) and the band structure (right) of the (6,3)-CNT without (straight lines) and with appliedstrain (dashed lines). The shift ∆ k of the k-lines relative to the k-points (left) is responsible for the observed bandgapopening, in the right figure.band gaps of three different CNTs are shown in figure 3(left) in comparison to the available theories [1,3]. Dotsrepresent DFT data, straight lines are only to guide theeyes, and dashed lines represent the theory by [3] as well asdash-dotted lines are obtained from the theory in [1]. It canbe seen that there is a significant deviation of the DFT dataand the analytical formula. A similar difference has beenobserved in literature for the (10,0) CNT [10], where ourDFT results completely agree with the published ones de-spite of the different DFT approach. Furthermore, our workagrees well with another work presenting DFT data [9].The remaining differences in the compressive part can beattributed to the coarse k-grid-mesh (1x1x4 points), there.The main impact on the ascend of the band gap-strainrelation is the relative shifting of the k-lines to the K-pointsin the Brillouin-zone. As the k-lines are close, the bandgap-opening (and also closing) due to strain is affected bythe dispersion relation of graphene in the vicinity of theK-points. There, DFT band structure, TB results, and thecommon quantification of the linear dispersion ( v F , Fermivelocity) agree nicely, as can be seen e.g. in [17]. This isalso true for our DFT calculations. Thus, the difference ofthe DFT and the analytical models remains unclear.For a better understanding of this difference, the zone-folding scheme is applied to calculate the electronic struc-ture in combination with the shift of the k-lines from [1].In order to reproduce our DFT results, this shift has beenscaled by a factor of √ . This procedure yields a perfectagreement for all CNTs studied so far. Thus, we enteredthe √ into the analytical model of Yang and Han: ∆E gap = sgn (2 p + 1) √ · t [ (1 + ν ) σ cos 3 θ ] , (1)where p = [ n − m ] with p = 2 (cid:55)→ p = − , t isthe hopping-parameter, ν stands for the Poisson’s ratio, σ denotes the strain, and θ represents the chiral angle. Thisequation holds as long as the same k-line stays the closestone to a K-point. As we did not consider torsional strain,the γ in the second term of the original equation has beenneglected.The fact that such a deviation has not been detected sofar is most likely due to a lack of systematic comparisonof TB results to reference data, which might be either ob-tained by DFT or experiment. From the experimental side,e.g. in [18], data quality is not good enough to preciselydetermine the ascend of the band gap-strain relation. Thedifference to DFT data is not that obvious in the literaturefound [9,10] and this misfit was either overseen or beingattributed to the general known weaknesses of DFT, whenband gaps are predicted. However, band gaps are usuallyunderestimated [19] by DFT which is not the case in thepresent study.All the mentioned facts make us confident that our DFTresults are reliable and they are able to correct the analyt-ical formula published by [1]. This factor √ should thenalso come out of the calculation performed in [1].The change of the band gap due to strain is very inter-esting for sensor applications and NEMS-devices becausethe resistivity R of a CNT is exponentially dependent onthe band gap [14] in the simplest transport approximation R = R S + h e | t | (cid:18) (cid:18) E G k B T (cid:19)(cid:19) , (2)where h e is the quantum resistivity in graphene, E G stands for the band gap, and T represents the temperature.We did not consider contact effects or scattering, so | t | istaken as 1 and R S equals zero. Copyright line will be provided by the publisher
Christian Wagner et al.: DFT investigations of the piezoresistive effect of CNTs (7,4)(6,3) (8,4)
TB Kleiner/EggertBand gap using DFTTB Yang/Han Analytical theoy (improved)Band gap using DFT (7,4) (6,3) (8,4)
Figure 3
Comparison of the DFT band gap with the analytical approaches (left): Dots represent DFT data, straight linesare a guide to the eye, and dashed lines represent the theory by [3]. Dash-dotted lines are obtained from the theory in[1]. Differences between the analytical approach in [1] and DFT can be overcome by introducing a factor of √ in theanalytical formula (right).It should be noted that the zone-folding approach andthe TB scheme presented here only holds for CNTs witha diameter larger than ≈ ˚A. For CNTs with lower di-ameters, the curvature (leading to σ ∗ − π ∗ − hybridization,[20]) becomes important, which cannot be easily taken intoaccount in a simple, general model for the electronic struc-ture. DFT is expected to provide reliable results also forthat kind of CNTs, but it is not expected to find a simple,analytical expression for this behavior. We presented calcula-tions of sensor-relevant CNTs based on DFT. Our mechan-ical data show a good agreement with current literature. Itis found that DFT data and present analytical models dif-fer significantly and could be brought into agreement by aprefactor of √ in the analytical model.In future work, the analytical calculation is repeated towatch out for this √ . Further on, transport simulationsbased on the DFT data will be accomplished and realistic(defective) and functionalized CNT will also be taken intoaccount. Acknowledgements
This work has been done within theResearch Unit 1713 which is funded by the German Research As-sociation (DFG). We gratefully acknowledge the ongoing supportby the group of Michael Schreiber (TU Chemnitz).
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