System Level Modelling of CNT based Mechanical

11. Chemnitzer Fachtagung Mikrosystemtechnik, 23./24.10.2012

System Level Modelling of CNT based Mechanical Sensor Kolchuzhin, Vladimir1; Mehner, Jan1; Shende, Milind2; Markert, Erik2; Heinkel, Ulrich2; Wagner, Christian3; Gessner, Thomas3,4 1 Chemnitz University of Technology, Department of Electrical Engineering and Information Technology, Chair of Microsystems and Precision Engineering, Reichenhainer Str. 70, 09126 Chemnitz, Germany, [email protected] 2 Chemnitz University of Technology, Department of Electrical Engineering and Information Technology, Chair for Circuit and System Design, Reichenhainer Str. 31-33, 09126 Chemnitz, Germany 3 Chemnitz University of Technology, Center for Microtechnologies (ZfM), Reichenhainer Str. 70, 09126 Chemnitz, Germany 4 Fraunhofer Institute for Electronic Nano System (ENAS), Technologiecampus 3, 09126 Chemnitz, Germany

Abstract This article presents a system level model of carbon nanotube (CNT) based mechanical sensor, which consists of micro-electro-mechanical (MEMS) platform that is served to actuate single-axis force and single CNT is placed between the movable and fixed electrodes. The change in conductivity of CNT, when strain is applied, is used to measure force or displacement. The simulation of the whole sensor element will be achieved by using the framework of hardware description languages (VHDL-AMS and SystemC-AMS) that uses compact models to describe sub-elements performing heterogeneous functions. The CNT piezoresistive compact model presented in the article is based on the analytical model and the simulations results from density functional theory (DFT). The macromodel of the MEMS platform is build using mode superposition technique. The results of the system simulations are presented and discussed in the article.

1

Introduction

Microelectromechanical systems, actuated by electrostatic force are widely used in various applications such as the pressure sensors, microphones, switches, resonators, accelerometers, gyroscopes, RF-filters, tunable capacitors, and micromirrors. But the accuracy of geometry and therefore the characteristics (sensitivity, signal-tonoise ratio, and frequency range) of MEMS based sensors are limited by the technology tolerances. Today, the nanoscale is explored to look out for new sensing principles and one of them is the piezoresistive effect of carbon nanotubes, useful for a nano-scaled acceleration sensor. Recently, several groups have reported on techniques for making networks of CNTs that can be placed onto metallic pads [1]. The NEMS/MEMS designing is a very challenging multilevel and interdisciplinary task. The levels in design include synthesis of lumped elements, process sequence development and mask layout drawing, component and system simulations. Different algorithms and software tools are used for this purpose. Numerical simulations are used both as a design tool and for understanding complex device behavior. Commonly used numerical simulation methods (FDM, FEM and BEM) provide results for a given set of geometrical and material parameters. For CNTs, classical simulation approaches based on finite elements do not hold, because their electronical properties directly depend on the atomic positions of the carbon atoms. Therefore, quantum-mechanical

simulations of CNTs are carried out using density functional theory (DFT), which is a standard technique for simulations of systems consisting up to hundreds of atoms. This article presents a system level model of CNT based mechanical sensor and is subdivided into six chapters. Chapter 2 presents a short description of CNT based mechanical sensor. In Chapter 3 the CNT compact model, its implementation using hardware description languages and the benchmarks are described. Chapter 4 presents the implementation process of MEMS macromodel using mode superposition technique. In Chapter 5 the developed models of CNT and MEMS has been integrated and co-simulated. The paper closes with conclusions drawn from the performed studies and suggestions for future work.

2

Design concept of the sensor

The virtual prototype of CNT based mechanical sensor consists of MEMS platform that is served to actuate single-axis force. The single CNT is placed between the movable and fixed electrodes. The change in conductivity of CNT, when strain is applied, is used to measure force and displacement. A schematic view on the sensor layout and interfaces is shown in Figure 1. It should be noted, that gauge factor of CNT decreases as the chiral angle increases. Zigzag tubes (n,0) have a maximum magnitude of gauge factor [1,4]. The length LCNT of the tube is defined by minimal available air gap between the movable and fixed electrodes (2µm in BDRIE process). In order to avoid

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11. Chemnitzer Fachtagung Mikrosystemtechnik, 23./24.10.2012

damaging the CNT, a measurement is limited in the relative deformation range of 0 to +5 %. The electrostatic comb drive is used to set the operating point (OP) on the strain-displacement curve of CNT.

u

spring

For our results, we use Atomistix ToolKit (ATK) from QuantumWise [5, 6]. The detailed parameters of the calculations are shown in Table 1.

output

Table 1: Overview of the calculation parameters used in DFT calculations for the nanotubes

Figure 1: The 2D layout of mechanical sensor

3

CNT compact model

3.1

Analytical model

The change in electrical resistance of ideal CNT due to strain can be calculated by an empirical transport formula based on band-gap changes [2, 3]:

1

h R = RC + 2 2 t 8e

  E gap (ε )   1 + exp  kT    

DFT modelling

DFT is a quantum mechanical theory, which solves the electronic structure problem in an efficient way. In contrast to the analytical model, where different approximations enter in, it provides more reliable results and helps to confirm analytical models. In this special case, the analytical formula (3.1) has been corrected in comparison to the original publication [4]. Nevertheless, its computational cost is very high and only up to hundreds of atoms can be calculated.

CNT

input

3 0 E gap . If CNTs are strained further, the band 2

gap closes again. Such behaviour can be understood by the underlying theory presented in [4, 10].

3.2

anchor

electrostatic comb drive

max E gap =

(3.1)

Here, RC denotes the contact resistivity, which is neglected; t2 is the transmission probability of the nanotube, usually taken as 1; h, e and k are physical constants; T denotes the temperature and Egap is the bandgap of the nanotube. The bandgap of a strained CNT is linearly changing with the relative deformation. An analytical formula for CNTs with a large radius is given in [4] with a slight modification to fit DFT results: 0 E gap (ε ) = E gap + sgn(2 p + 1)3t 0 2 (1 + v)ε cos 3θ ,

Basis set

Double zeta polarized [7]

k-points

1x1x20, Monkhorst-Pack grid [8]

Functional

Local density appr., Perdew and Wang [9]

Smearing

Fermi-Dirac, 300K = 25 meV

Max. forces

0.01 eV/Å = 0.01602 nN

As the amount of simulated atoms is very small, periodic boundary conditions have been applied to obtain the results for infinite CNTs. These results can be applied to ideal systems on the nano- and microscale, as the length of experimentally fabricated CNTs is long with respect to the electron wavelength. Thus, there will be no quantization effects in this direction in the experiments and infinite CNTs may be assumed for simulation. The results obtained by such kind of simulations are the mechanical and electronical properties of strained CNTs. For the mechanical part, the procedure is as following: the atoms of the CNTs are placed in a unit cell, which is periodic into all special directions, Figure 2.

(3.2) where p is a value determined by the chiral indices (n,m) of the nanotube in a way that p = [n-m]3 : p = 2 a p = −1; t0 = 2.66 eV is the hopping parameter; ν ≈ 0.2 stands for the Poisson’s ratio; θ represents the chiral angle of the CNT and ε denotes the relative 0 strain. Here, E gap is the bandgap of the unstrained state that can be written for semiconducting CNTs as 0 E gap =

3 a0 t0 , 2 r0

where a0 = 1.42 Å is the lattice constant of graphene and r0 is the radius of the CNT. The maximum bandgap realized by straining CNTs is

Figure 2: The geometry of a (12,0)-nanotube and its unit cell. Here, two repetitions of that unit-cell are shown

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11. Chemnitzer Fachtagung Mikrosystemtechnik, 23./24.10.2012

Then, a geometry optimization is performed to relax all internal forces. At each stretch step, the unit cell and the according atomic coordinates are scaled linearly, followed by another geometry relaxation. The total quantum-mechanical energy is saved and traced over the deformation range. The results for different CNTs are depicted in Figure 3, where DFT values and third order fits are shown. A third order polynomial approximation describes the deformation over the whole range and the resulting stress-strain relation as well as the Young’s modulus can be derived. The moduli obtained for the CNTs that are plotted in Figure 3 are shown in Table 2. 50.0

Total energy Etot, eV

CNT (14,0) 40.0

CNT (15,0)

30.0 20.0 10.0

CNT (13,0)

0.0 −10.0 −0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

Relative deformation ε

Figure 3: The total energy of the (13,0), (14,0) and (15,0) CNTs traced over the relative deformation. Squares represent DFT values and lines the fit curves Table 2: Overview of the Young’s moduli of the depicted carbon nanotubes CNT

Young’s modulus E, GPa

(13,0)

983.5 ± 1.1

(14,0)

985.6 ± 0.6

(15,0)

987.2 ± 0.3

The conductivity of ideal CNTs is obtained by an empirical transport formula (3.1), where the band gap of the system is of interest. The band gap is evaluated by DFT calculations as well as the analytical formula (3.2). This is shown in Figure 4. Resistance R, Ohm

1020

CNT (13,0)

CNT (14,0)

These models are extremely helpful, when it comes to compact models for lumped elements simulations for the sensor itself. DFT calculations take hours of time, whereas an analytical model is computed within a fraction of second. In contrast to DFT, analytical models are not that transferable, e.g. they break down at CNTs whose diameter is smaller than the Van-derWaals distance. This is shown in [12]. Within fabrication, such small-diameter CNTs are rarely found and therefore, the analytical model is practically relevant.

3.3

Implementation in VHDL-AMS

The VHDL-AMS model is similar to a black-box model with terminals relating electrical and mechanical quantities based on the generalised Kirchhoff's laws. The CNT at the system level is presented as a four terminals device (two mechanical and two electrical terminals), which consist of the nonlinear mechanical spring unidirectional coupled with the strain dependent resistor. The behavioural model of spring is defined by:

f (t ) = k CNT u (t ) in linear case f (t ) =

∂E tot ∂u

in nonlinear case,

where CNT axial stiffness kCNT can be given as:

k CNT = A ⋅ E / LCNT (0) .

1018

The Van-der-Waals distance (6.3 Å) is the distance of two neighbouring carbon sheets in graphite and is used to define the effective cross section area A, Figure 5. For more details see [11].

1016

CNT (15,0)

1014

CNTs, which are geometrically very similar, show a strongly different behaviour. For the (13,0) and (14,0) CNT, a band gap exists at zero strain. They differ in the ascend of the band gap at positive strain: for the (14,0) CNT, it decreases and for the (13,0) it increases. The behaviour of the (15,0) CNT deviates, as its band gap is approximately zero in the original state and its resistivity increases in both tensile and compressive directions. After reaching the maximum resistivity, it decreases again. The reason for this awkward behaviour is described in [4, 10]. Figure 4 shows only minor deviations of the DFT and the analytical model. Thus, the simplified model can be used for sensor simulations for such kind of CNTs.

1012 1010 108

DVan-der-Waals

106 104 −0.15

−0.10 −0.05

0.00

0.05

0.10

0.15

0.20

RCNT

Relative deformation ε

Figure 4: DFT values of the resistivity of the (13,0), (14,0) and (15,0) CNTs in comparison to the TB-approximation

Figure 5: The effective cross section area A of a CNT, determined by the Van-der-Waals distance

The (n,0)-CNTs with chiral angle θ=0° are the most sensitive ones. The resistivity is shown on a logarithmic scale, as it exponentially depends on the bandgap. It is highlighted that different kinds of

The next step is to define the behavioural model of the resistor:

i(t ) = v(t ) RCNT (ε ) ,

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11. Chemnitzer Fachtagung Mikrosystemtechnik, 23./24.10.2012

where ε = u (t ) LCNT (0) and RCNT is determined by formula (3.1). After development the VHDL-AMS code, the CNTmodel is tested using the simulator Simplorer from ANSOFT (now part of the ANSYS). It is necessary to specify the external circuitry (voltage source E1=1V, and controller units AM1, VM1), loads (mechanical force source F1 and controller unit FM1) and solver parameters (time step size, total simulation time). The schematic of this measurement setup is depicted in Figure 6. The simulation results are shown in Figure 7. It can be seen that for linearly changing input force, CNT shows nonlinear change in resistivity. The current I flowing through the CNT can be calibrated to get percentage change in applied strain. F1

AM1

FM1 ctn1

F

A

F

V

VM1

E1

Current I, µA

Rel. deformation ε

Figure 6: Testbench circuit for VHDL-AMS model of CNT in Simplorer 0.05

b) Linear signal flow (LSF) c) Electrical linear network (ELN). For the simulation of time data flow modelling strategy static scheduling is used, which is activated in discrete time steps. For linear signal flow and electrical linear network modelling styles linear DAE solver is used [16]. MEMS based CNT strain gauge sensor is possibly be the ideal sensor to exploit the strengths of SystemCAMS modelling language. As shown in Figure 1, the sensor has MEMS comb-drive, CNT tube deposited between fixed and moving mass etc. Apart from that the sensor has digital signal processing part. CNT strain gauge schematic model is shown in Figure 6. The model is divided into three sub-modules a) excitation system that provides input force

0.00

-0.05

0

0.5

1

150

1.5

time, ms

2

b) theoretical model of CNT given by equation (3.1), that emulates the piezoresistive property of a single walled carbon nano tube

2.5

RC = 0

100

c) top level module, which combines different submodules to form a complete sensor system.

RC = 10 kOhm

50 0

0

0.5

1

1.5

2

2.5

1.5

2

2.5

1.5

2

2.5

time, ms

20 15

Resistance R, GOhm

a) Timed data flow (TDF)

3.4.2 CNT strain gauge model

+ s

The concept of analog mixed signal extension for SystemC version2.0 has been proposed by Einwich et al. (members of OSCI working group) [14]. The proposed extension is to be called as SystemC-AMS. In 2004, the first prototype of the SystemC-AMS syntax is presented in [15]. In the process of SystemCAMS language evolution, later in 2010 SystemC-AMS 1.0 standard has been released by OSCI. This standard is mainly focusing on three major modelling aspects

R=

10

E1 I

5 0

0

0.5

1

0

0.5

1

1025

time, ms

1020 1015 1010 105

time, ms

Figure 7: Results of an analysis for the semi-metallic CNT (15,0), E1=1V

3.4. Implementation in SystemC-AMS 3.4.1 Open SystemC-AMS SystemC-AMS extension is proposed on existing SystemC system modelling platform to enable modelling of the analog-mixed signal systems. Originally SystemC is a hardware description language dedicated to design digital systems [13].

For mechanical input excitation TDF model is used. TDF module is used to define discrete time or to embed continuous time behaviour. Theoretical model of CNT also uses TDF model. In both the TDF modules, set_timestep attribute is used in member function set_attribute to define the time step between two consecutive samples. Sampling period for both the modules is set to 10 ns. In a top-level module the predefined electrical primitives are defined for resistance, voltage source and current measurement. Child TDF modules like cnt and tri_wave are instantiated and interconnected to form a TDF cluster as in SystemC modules with the help of the macro SC_MODULE.

4

MEMS platform

A number of numerical techniques to handle MEMS macromodel has appeared over the last decade. The two major ones are: mode superposition technique and Krylov subspace based methods. The presented macromodel of the MEMS platform is based on mode superposition technique [17, 18]. The governing equation describing an electrostatically actuated

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11. Chemnitzer Fachtagung Mikrosystemtechnik, 23./24.10.2012

(4.1)

mi q&&i + d i q& i + f i r = f i e + f i m

The modal masses mi and modal damping constants di are calculated from the eigenfrequencies of the modes i and the entries of the modal stiffness matrix kij; ∂E m f i r = strain is the modal reaction force, f i is the ∂qi modal external force, Estrain is the strain energy.

∂C 1 ∑ kl (Vk − Vl ) 2 is the modal electro-static 2 r ∂qi force; r is the number of capacitances involved between the multiple electrodes. The capacitance Ckl between the electrodes k and l provides the electromechanical coupling.

fie =

example, in case of k modal amplitudes in each mode direction, the number of orthogonal sampling points would be km. At each point the r = n(n−1) / 2 linear simulations are performed to compute lumped capacitances Cr(q1,q2,…,qm) in the deformed electrostatic domain, where n is the number of conductors.

5

The created macromodel of the MEMS platform was integrated in a testbench of the complete sensor for verification. Results of an analysis are shown below:

In the electrical domain, the current Ij through the electrode j can be calculated from the stored charge:

 ∂C  I j = ∑  ∑ kl q& m (Vk − Vl ) + C kl V&k − V&l r    m ∂q m

(

) 

(4.2) The structural displacements u are calculated from modal amplitudes qi by m

u ( x, y, z , t ) = u ref + ∑ φi ( x, y , z ) qi (t ) ,

(4.3)

i =1

Results

Displacement u, nm

structure in terms of modal coordinates qi in the mechanical domain is defined by:

20 15 10 5 0

0

20

40

60

80

100

120

140

160

180

Voltage Vs, V

Figure 9: Voltage-to-displacement curve of the MEMS platform

A typical rectangular shaped comb drive design has a parabolic voltage-to-displacement relationship, which is a function of the change in electro-static force. Linearity of the sensor characteristics can be achieved by an optimization of the comb capacitors using FE analysis and optimization algorithms [19]. Relative deformation ε, %

where uref are initial displacements, φi are the eigenvectors involved in the reduced order model. The equations (4.1), (4.2) and (4.3) define the ROM macromodel, which fully describes the static, harmonic and dynamic nonlinear behaviour of the flexible structure. The equivalent circuit of CNT based sensor for the first mode is shown in Figure 8.

Resistance R, kOhm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

16

18

20

30 25

OP

RC = 10kOhm

20 15

RC = 0

10 5

0

2

4

6

8

10

12

14

Displacement u, nm

RCNT

kCNT

s

di

ki

Figure 10: The output signal of the sensor with the CNT (15,0)

mi C

qi

fi

Beyond this simple test case, one can analyse the CNT based mechanical sensor together with any other system environment or complex electronic circuit.

Figure 8: Equivalent circuit model of CNT based sensor for the first mode with modal contribution factor 97%

6

The strain energy, the mutual capacitances, the damping coefficients and the modal load forces are the parameters characterizing the coupled electromechanical system. The ROM macromodel is generated by numerical data sampling and subsequent fit algorithms. Each data point must be obtained by a set of separate FE run in the structural, electrostatic and fluid domains. At each point (q1,q2,…,qm) the microstructure is displaced to a linear combination of m selected mode shapes in order to calculate the strain energy Estrain(q1,q2,…,qm) in the structural domain. For

In the paper, the concept of CNT based MEMS strain gauge sensor is presented. Density function theory model is described which is used for quantum mechanical simulation of CNTs. DFT modelling is important because it deals with the electronic properties of CNT that directly depend on the atomic positions of the carbon atoms in the tube. High level behavioural and structural abstract model of CNT strain gauge sensor is discussed using hardware description languages like VHDL-AMS and SystemCAMS. The simulation results are shown. The library of

Conclusions

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11. Chemnitzer Fachtagung Mikrosystemtechnik, 23./24.10.2012

developed components is available for download and testing. The presented sensor project is in concept development stage. Therefore a simple abstract model of a sensor is demonstrated in this paper. Future work consists of finalizing the detailed specifications of the sensor, implementing precise and possibly every minimal electro-mechanical effects of the sensor in HDL models, considering other environmental effects on sensor such as temperature, humidity, pressure etc. Automation of model extraction and co-simulation using different modelling tools [20] are also the part of future work.

7

References

Culpepper, M. L.: Carbon [1] Cullinan, M. A.; nanotubes as piezoresistive microelectromechanical sensors: Theory and experiment, Physical Review B82, 115428, 2010. [2] Maiti, A.; Svizhenko, A.; and Anantram, M. P.: Electronic transport through carbon nanotubes: Effects of structural deformation and tube chirality, Phys. Rev. Lett., Vol. 88, no. 12, p. 126805, Mar 2002. [3] Minot, E. D.; Yaish, Y.; Sazonova, V.; Park, J.Y.; Brink, M.; McEuen, P. L.: Tuning Carbon Nanotube Band Gaps with Strain. Phys. Rev. Lett., 2003, Vol. 90, p. 156401. [4] Yang, L.; Han, J.: Electronic Structure of Deformed Carbon Nanotubes. Phys. Rev. Lett., 2000, Vol. 85, p. 154-157. [5] Atomistix ToolKit version 12.2, QuantumWise A/S, www.quantumwise.com [6] Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P. ; Sánchez-Portal, D.: The SIESTA method for ab initio order- N materials simulation. J. Phys.: Condens. Matter, 2002, Vol. 14, p. 2745. [7] Abadir, G. B.; Walus, K.; Pulfrey, D. L.: Basisset choice for DFT/NEGF simulations of carbon nanotubes. J. Comput. Electronics, 2009, Vol. 8, p. 1-9. [8] Monkhorst, H. J.; Pack, J. D.: Special points for Brillouin-zone integrations. Phys. Rev. B, American Physical Society, 1976, Vol. 13, p. 5188-5192. [9] Perdew, J. P.; Wang, Y.: Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B, 1992, Vol. 45, p. 13244-13249.

[11] Jeong, B.-W.; Sinnott, S. B.: Atomic-Scale Simulations of the Mechanical Behavior of Carbon Nanotube Systems, Trends in Computational Nanomechanics, Bd. 9, T. Dumitrica, Hrsg., Springer Netherlands, 2010, p. 255. [12] Blase, X.; Benedict, L. X.; Shirley, E. L.; Louie, S. G.: Hybridization effects and metallicity in small radius carbon nanotubes Phys. Rev. Lett., 1994, Vol. 72, p. 1878. [13] Black, D. C.; Donovan, J.: SystemC: From the Ground Up. Springer, 2004. [14] Einwich, K.; Grimm, C.; Vachoux, A.; MartinezMadrid, N.; Moreno, F.R.; Meise, C.: Analog mixed signal extensions for SystemC. White paper and proposal for the foundation of an OSCI Working Group (SystemC-AMS working group), June, 2002. [15] Vachoux, A.; Grimm, C.; Einwich, K.: Towards analog and mixed-signal soc design with SystemC-AMS. Electronic Design, Test and Applications, DELTA 2004. Second IEEE International Workshop on, p. 97-102, 2004. [16] SystemC AMS extensions User’s Guide by the Open SystemC Initiative (OSCI), March 2010. [17] Gabbay, L.: “Computer Aided Macromodeling for MEMS“, PhD Thesis, Massachusetts Institute of Technology, 1998. [18] Schlegel, M.; Bennini, F.; Mehner, J.; Herrmann, G.; Mueller D.; Doetzel, W.: Analyzing and Simulation of MEMS in VHDL-AMS Based on Reduced Order FE-Models, IEEE Sensors Journal, 2005, Vol. 5, No. 5, p. 1019-1026. [19] Scheibner, D.; Wibbeler, J.; Mehner, J.; Brämer, B.; Gessner, T.; Dötzel, W.: A Frequency Selective Silicon Vibration Sensor with Direct Electrostatic Stiffness Modulation. Analog Integrated Circuits and Signal Processing, 2003, p. 35-43. [20] Bansal, R.; Clark, J. V.: Lumped modeling of carbon nanotubes for M/NEMS simulation. Microsystem Technologies, May 2012.

Acknowledgments This work has been done within the Research Unit 1713 which is funded by the German Research Association (DFG).

[10] Wagner, C.; Schuster, J.; Geßner, T.: DFT investigations of the piezoresistive effect of carbon nanotubes for sensor application. Phys. Stat. Sol. C, 2012, in press.

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