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Carbon nanotubes for biomolecular sensing and photovoltaics Mohamamd Ali, Mahmoudzadeh Ahmadi Nejad 2008

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Carbon Nanotubes for Biomolecular Sensing and Photovoltaics by Mohammad Ali Mahmoudzadeh Ahmadi Nejad B.Sc., Sharif University of Technology, 2006 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October, 2008 c Mohammad Ali Mahmoudzadeh Ahmadi Nejad 2008  Abstract A computational investigation of some optoelectronic applications of carbon nanotubes (CNT) is presented, including CNT-based solar cells and biosensors. The results could be used to evaluate the performance of CNT devices and clarify the necessity of further experimental research in this area. A coaxially-gated CNT field-effect transistor (CNFET) forms the basic structure of the devices modeled in this thesis. Diffusive transport is present in long-channel devices, as in our case, while the quantum mechanical effects are mainly present in the form of tunneling from Schottky-barrier contacts at the metal-CNT interfaces. Band-to-band recombination of electron-hole pairs (EHP) is assumed to be the source of electroluminescence. In a first-order approximation, protein-CNT interactions are modeled as the modification of the potential profile along the longitudinal axis of CNTs due to electrostatic coupling between partial charges, in the oxide layer of the CNFET, and the nanotube. The possibility of electronic detection is evaluated. The electroluminescence of the CNT is proposed as an optical detection scheme due to its sensitivity to the magnitude and the polarity of the charge in the oxide. The validity of the model is argued for the given models. A value for the minimum required size of a computational window in a detailed simulation is derived. The structure of an electrostatically gated p-i-n diode is simulated and investigated for photovoltaic purposes. The absorbed power from the incident light and the interaction between the nanotubes is modeled with COMSOL. The results are interpreted as a generation term and introduced to the Drift-Diffusion Equation (DDE). We have observed behavior similar to that in an experimentally-realized device. The performance of CNT-based ii  Abstract solar cells under standard AM 1.5 sunlight conditions is evaluated in the form of an individual solar cell and also in an array of such devices.  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  Dedication  xi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . 1.2 Carbon Nanotube Field-Effect Transistor . . . . . . . . 1.3 Carbon Nanotube p-i-n diodes as Photovoltaic Devices . 1.4 A Model for Coaxial Carbon Nanotube FETs and Diodes 1.5 Carbon Nanotubes as Sensors . . . . . . . . . . . . . . . 1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . 2 Modeling CNT Devices 2.1 Electrostatics . . . . 2.2 Charge Transport . 2.3 Electroluminescence 2.4 Photovoltaics . . . .  . . . .  . . . . . . . . .  . . . . . . .  1 3 4 5 6 7 8  for Electronics and Optoelectronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10 11 15 19 19  iv  Table of Contents 3 Molecular Sensing Using CNFET Electronics . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . 3.2 Simulation Parameters . . . . . . . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Effect of Molecular Dipoles on Isat . . . . 3.3.2 Effect of Changing Charge Position on Isat 3.3.3 Gate Voltage Sweep . . . . . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . 4 Molecular Sensing Using 4.1 Introduction . . . . . 4.2 Results . . . . . . . . 4.2.1 Voltage Sweep 4.2.2 Dipole Length 4.3 Summary . . . . . . .  . . . . .  . . . . . . . . . .  . . . . . . . .  . . . . . . . .  CNFET Electroluminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5 Photovoltaic Action in CNT p-i-n diodes . . 5.1 Introduction . . . . . . . . . . . . . . . . . . 5.2 Simulation Parameters . . . . . . . . . . . . . 5.3 Results . . . . . . . . . . . . . . . . . . . . . 5.3.1 Modeling the Experimental Device . . 5.3.2 PV Device Under Sunlight Illumination 5.4 Summary . . . . . . . . . . . . . . . . . . . .  . . . . .  . . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  24 24 25 27 27 28 29 30  . . . . .  . . . . . .  . . . . . .  32 32 33 33 36 36  . . . . . . .  . . . . . . .  . . . . . . .  39 39 39 40 40 42 46  6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 References  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49  Appendices A Formulation and Implementation of Arbitrary Charge Profile in SPDD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 v  Table of Contents B Modification of SPDD for Double Gate Structures . . . . . 62  vi  List of Tables 3.1 3.2  Device Parameters. . . . . . . . . . . . . . . . . . . . . . . . . 25 Model Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26  5.1 5.2 5.3 5.4  Specifications of the simulation box . . . . . . . . . CNT model used in COMSOL simulations . . . . . Split gate device parameters. . . . . . . . . . . . . . Frequency dependent parameters of the simulation .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  39 40 40 43  A.1 De Mari scaling factors for DDE and CE . . . . . . . . . . . . 58 B.1 Poisson operator matrix coefficients . . . . . . . . . . . . . . . 62  vii  List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7  3-D structure of a coaxial CNFET. . . . . . . . . . . . . . Computational window after reducing the axial dimension. 3-D treatment of oxide charges. . . . . . . . . . . . . . . . 2-D charge distribution of a dipole in the dielectric. . . . . Thermionic and tunneling current . . . . . . . . . . . . . . p-i-n Diode . . . . . . . . . . . . . . . . . . . . . . . . . . COMSOL computational window . . . . . . . . . . . . . .  . . . . . . .  10 11 14 15 17 20 21  3.1 3.2 3.3 3.4 3.5 3.6  Modification of the device characteristics . . . . . . . . . . . . Peptide Formation . . . . . . . . . . . . . . . . . . . . . . . . I-V Characteristics in the presence of a lateral dipole. . . . . . Potential profile in the presence of a lateral dipole. . . . . . . Current variation with longitudinal position of a lateral dipole ID vs. VGS at VDS = 25 mV , no change in threshold voltage is observable. . . . . . . . . . . . . . . . . . . . . . . . . . . .  24 26 27 28 29  4.1 4.2 4.3  . . . . . . .  4.4 4.5 4.6  Electron and hole concentrations and their product. . . . . . . Recombination profile at different gate voltages. . . . . . . . . A lateral dipole is placed at the center of the device. Electron and hole concentration at two different bias points. . . . . . . A fixed bright spot is observed at the position of a single charge. Recombination profile for different charge polarities . . . . . . Variation in the luminescence profiles as dipole length varies. .  5.1  I-V characteristics under illumination.  30 33 34 35 35 37 38  . . . . . . . . . . . . . 41 viii  List of Figures 5.2 5.3 5.4 5.5 5.6  The fourth quadrant I-V characteristics . . . . . . . . . . . . . CNT permittivity and AM 1.5 spectrum for terrestrial sunlight The fourth quadrant I-V under AM 1.5 sunlight for (7,0) tube. CNTs interaction under illumination. . . . . . . . . . . . . . . Absorbed power for an array of 5 CNTs. . . . . . . . . . . . .  42 43 44 45 45  A.1 Modification of the meshing profile . . . . . . . . . . . . . . . 58 A.2 Meshing numbering system. . . . . . . . . . . . . . . . . . . . 60  ix  Acknowledgements I am tremendously grateful to Professor David Pulfrey for giving me the opportunity to pursue my research in the area of my interest. I will be indebted to him for his unique mentorship which trained me to think as an insightful researcher. His distinguishing teaching method will always be inspiring to me for the rest of my academic career. In addition I want to express my deepest gratitude to Professor Bizhan Rashidian who motivated me to pursue my graduate studies in physics of electronic devices. I would also like to thank my colleagues and friends George Abadir, Li Chen and Soroush Karimzadeh for making my stay at UBC more enjoyable.  x  To my family, whose presence always cleared the cloudy sky of this city for me, and I have never spent a moment without them.  xi  Chapter 1 Introduction Over the past few years, there has been a remarkable increase in interest in the subject of nanoelectronics. Nanoelectronics has resulted from the continual effort to realize smaller transistor sizes corresponding to Moore’s Law, which predicts that the number of transistors on a chip will double about every two years [1]. Maintenance of this trend needs discovery of new materials and manufacturing techniques that will allow design and development of nanometer-sized devices. The carbon nanotube (CNT) and its synthesis form one of the main centers of attention in this new area. The richness and variety of the properties of carbon nanotubes (electronic, mechanical, thermal, and chemical) have persuaded scientists to investigate them both from an academic point of view and for their potential applications. In 1952, Radushkevich and Lukyanovich published their discovery of the hollow, nanometer sized, carbon-based tubes in the Soviet Journal of Physical Chemistry [2], but the discovery stayed unnoticed until 1991 when Sumio Ijima identified multi-walled carbon nanotubes at the NEC Research Laboratory using high resolution transmission electron microscopy [3]. After a couple of years, he and Donald Bethune synthesized the first single-walled nanotubes (SWCNTs) [4, 5]. The first derivations of the electrical properties of carbon nanotubes were done by Saito et al. in 1992. A tight-binding model was applied to a rolled sheet of graphene in order to model CNTs [6]. Since then, many theoretical and experimental reports have dealt with the properties of CNTs, and have demonstrated several unique electrical properties such as: diameter-dependent bandgap, ballistic transport in short tubes, high mobility, and high current-carrying capacity. All these features poten1  Chapter 1. Introduction tially allow for further miniaturization of high-speed and high-power circuits. Several near-ballistic, high-performance transistors have been realized experimentally [7, 8]. The optical properties of semiconducting carbon nanotubes are currently under study. CNTs are direct bandgap materials, and strong electron-hole interactions due to their quasi-one-dimensional nature results in strongly bound excitons. Great interest in CNT optoelectronic devices started after Misewich et al. demonstrated the first CNT infrared emitter [9]. Light emission in this device was based on the simultaneous injection of electrons and holes and their radiative recombination. Photoconductivity of a single carbon nanotube, incorporated as the channel of an ambipolar field-effect transistor, was also reported by Fujiwara et al. [10] and Freitag et al. [11]. The latter’s photoconductor was basically a single nanotube ambipolar fieldeffect transistor and had an internal quantum efficiency of 10%. One serious obstacle to the further separation of electron-hole pairs was due to the low field in the middle of the device. This problem was overcome by the design on an electrostatically doped p-i-n diode by Lee in 2005 [12] . He suggested an efficiency of 5% for his fabricated device, but expected significant improvements by proper bandgap engineering of the device, and design of a network of devices where higher absorption might be predicted due to strong electromagnetic coupling between nanotubes [13]. In this thesis, we present our computational investigation of some optoelectronic applications of CNTs. The simulation tool is based on the SPDD (Schr¨odinger-Poisson-Drift-Diffusion) Solver, by Dylan McGuire at the UBC Nanoelectronics Group [14], the original version of which was developed in order to describe experimentally observed fixed and mobile electroluminescence. For the first part of the thesis, we summarize the potential of CNTbased transistors as biosensors. The code was expanded to be able to model any arbitrary charge profile in the oxide layer of the transistor, and several electronic and optical characteristics were studied. Secondly, we model  2  1.1. Carbon Nanotubes a p-i-n diode structure for photovoltaic applications. This electrostatically doped diode is modeled by proper adaptation of the original code for a split gate structure.  1.1  Carbon Nanotubes  Carbon nanotube molecules have a cylindrical structure and can be considered to be formed by the concentric rolling of one or more hexagonal-lattice sheets of graphite (often called graphene layers) into seamless tubes. They are named single- or multi-walled nanotubes in accordance with the number of concentric layers [3, 4]. Being formed from graphene layers and having a short length of radial confinement that causes a periodic boundary condition, carbon nanotubes have electrical band structure as subbands of the graphene band structure, with a quantization of wavevectors in the direction of the chiral vector, Ch = nˆ a1 + mˆ a2 , where [ˆ a1 , a ˆ2 ] are the graphene lattice unit vectors. This periodic boundary condition is a real one and it is due to physical structure not a conceptual one [15] for easing the calculation. The dispersion relation of the CNT can be derived by quantizing that of graphene as Ecnt (kz , p) = ±t 1 + 4 cos γ1 cos γ2 + 4 cos2 γ2  1/2  ,  (1.1)  where kz is the wavevector in the longitudinal direction, p indicates the quantized mode, t is the transfer integral, acc is the distance between to adjacent carbon atoms and γ1 and γ2 are defined in 1.2 and 1.3 using the chiral indices of the CNT. γ1 =  π n + 3m 3acc m − n kz + p 4 η 2 η2  (1.2)  γ2 =  3acc m + n π 3n − m kz + p 4 η 2 η2  (1.3)  3  1.2. Carbon Nanotube Field-Effect Transistor Graphene has an energy gap of zero at six corners of its Brillouin zone. Correspondingly, if one of CNT subbands passes one of these points, it will have zero energy gap and thus metallic properties. This situation happens when (n − m) is a multiple of 3. It is important to keep in mind that this distinction is not valid for tubes with diameters smaller than 0.5 nm. In which case, the σ and π orbitals start to re-hybridize and do not follow those of graphene any more [16]. For a semiconducting CNT, the band gap can be expressed as Eg =  |t| acc , 2Rt  (1.4)  where Rt , the radius of CNT, is given in terms of n and m as Rt =  acc |Ch | = 2π  3 (n2 + m2 + nm) . 2π  (1.5)  We used this dependency to engineer the device; we substitute a smaller CNT in order to get a larger energy gap and reduce the transistor’s off-current.  1.2  Carbon Nanotube Field-Effect Transistor  From their early days of discovery, CNTs showed potential application for novel molecular devices [17]. They have near-ballistic, one-dimensional transport as well as a complete crystalline structure that leaves no dangling bonds and hence no surface scattering. In a carbon nanotube field-effect transistor (CNFET) structure, source and drain are usually metal electrodes and their potential is set by the applied bias while conductance of the channel is modulated via its capacitive coupling to the third contact (the gate). Unlike in its silicon base counterpart, switching in CNFET is mainly controlled by Schottky-barriers (SB) at the metal-CNT interface [18, 19]. The p-type char4  1.3. Carbon Nanotube p-i-n diodes as Photovoltaic Devices acter of the early CNFETs was not an intrinsic property of CNT but was a result of oxygen adsorption that works as a p-type dopant for CNTs [20]. For an intrinsic channel CNFET, the polarity of the transport is determined by the electron- and hole-Schottky barrier height. Engineering of metal contact work functions with respect to that of the CNT, results in so called n-type and p-type devices, or, specifically in the case of equal barriers, an ambipolar device [21]. Unipolar characteristics can also be achieved by using chemical doping for the channel and replacing metal contacts by chemical or electrostatically doped CNTs [22]. The ambipolar characteristics are undesirable in digital circuitry, since the channel conducts by electrons for positive gate voltages and by holes in case of negative gate voltages, and one can never turn-off the device. The CNFET structure is the basis for almost all the reported experiments on carbon nanotube optoelectronics. At certain bias conditions, simultaneous injection of electron and holes from the source and drain contacts, respectively, increases the probability of their recombination. Because CNTs are direct bandgap materials, this recombination is mainly radiative and leads to electron-excited emission (electroluminescence). The location of this emission depends on the carrier profiles along the CNT and therefore, depends on the applied bias [9, 23].  1.3  Carbon Nanotube p-i-n diodes as Photovoltaic Devices  SWNTs have several advantages for photovoltaic applications. They have a wide range of band gaps [6] that can be matched to the solar spectrum, high mobility and high optical absorption [24]. Several CNT-based photon detection devices have been realized recently. Photoconductors [11], Schottkybarrier diodes [22], and p-n diodes [12] are some of the experimentally proved devices in this area. In all these devices, the absorption of a photon generates 5  1.4. A Model for Coaxial Carbon Nanotube FETs and Diodes an exciton which is then separated to a free electron-hole pair (EHP). Photocurrent in CNTs was first observed by Zhang and Ijima [25] in SWCNT filaments. Soon after that, photocurrent was also observed in CNFETs [11], SWNT sheets [26] and films [27]. It has been shown that the existence of Schottky-barriers is responsible for the photocurrent in these devices [28]. Zhang et al. proposed that the poor photoconductive performance of CNT films and sheets is due to the difficulty in preparing CNTs with similar properties. Therefore, devices based on individual SWNTs would perform better [29]. Photoconductivity of a single CNT in an ambipolar CNFET structure was studied by Freitag et al., but the device performance was mainly degraded because of the low electric field near the center of the device to separate the carriers [11]. A carbon nanotube p-n junction diode [30] can solve this problem. Formerly, a p-n junction diode through chemical doping had been reported, but its electrical properties were not satisfying due to the abrupt junction formation [31]. Lee et al. reported an electrostatically doped p-i-n diode. An ideal diode behavior was observed from that device and its photovoltaic effects were examined [12]. The power conversion efficiency, η, was estimated to be larger than 5%, and larger values were predicted for a network of tubes with electromagnetic coupling between tubes. In this work, we model the electrostatically-doped, split-gate structure of Ref. [12], and examine both single devices and network properties to optimize its performance.  1.4  A Model for Coaxial Carbon Nanotube FETs and Diodes  Most fabricated CNFET devices are planar ones. Their silicon based counterparts are shifting towards all-around gating, such as in Fin-FET structures and tri-gates, in order to maximize gate-channel coupling and reduce short channel effects like drain induced barrier lowering (DIBL). The same trend 6  1.5. Carbon Nanotubes as Sensors has been implemented in CNFETs by using double gate [24] and all-around gate structures [18]. Other than better performance, the coaxial structure, has many advantages in the simulation area: it allows us to reduce the simulation dimensions by one because of its azimuthal symmetry and dramatically reduce the computational cost. Analysis of a coaxial device can lead to qualitative insight about the behavior of CNFETs in general. The model used here is based on previous work from the UBC Nanoelectronics Group by D.L.McGuire [14], and relevant changes have been made in order to deal with oxide charges and an electrostatically gated p-i-n diode. In the current model, we included electrostatics and charge transport through CNT. Quantum mechanics is employed directly close to the contacts where tunneling through the Schottky-barrier is the charge transport mechanism, and indirectly through the semiconductor bandgap and the electron and hole effective masses along the CNT where the drift diffusion equation defines the transport. Elsewhere in the device, transport is dissipative, and is modeled by the Drift-Diffusion relations.  1.5  Carbon Nanotubes as Sensors  Most methods for sensing biological species conventionally rely on optical detection. Despite their high sensitivity, these techniques suffer from complexity and long operation time; they usually involve multiple preparation steps and advanced data processing. Therefore, looking for a simpler substitute technique is of great importance [32]. Transistors, as the building blocks of active electronics, are the first candidates for being the electronic detector. Conventional transistors have been previously used for this purpose [33, 34], but their sensitivity was not comparable to optical detection techniques. Using a carbon nanotube as the channel in a FET structure makes the device more sensitive. CNFET-protein interactions have been also widely explored [33, 35, 36]. Utilizing CNT-based nanoscale devices has several ad-  7  1.6. Thesis Outline vantages: size compatibility is the first one. Individual proteins have lengths of a few tens of nanometers [37], and the diameter of a DNA duplex is approximately 1 nm [32]. These figures show compatibility with the e-beam lithography limit, and the diameter of a CNT, which are 30 and 1 nm respectively. The second advantage relies on the fact that most biological processes involve electrostatic interactions and charge exchange between atoms [38], which could be well detected by a nano-electronic device. Having all the atoms on the surface in contact with the environment, the carbon nanotube is a very promising nano-material for biosensing applications. Detection of a specific molecule is also available by using functionalization methods [39]. Kong et al. [40] were the first to use CNTs as chemical sensors for the detection of NO2 and NH3 . Since then, CNTs have been used in more complex detection devices [33, 35, 36]. Both mobile and localized electroluminescence, have been observed in semiconducting CNTs [9, 41]. It has been shown that these light emissions are very sensitive to the presence of any defect in the oxide layer [42]. This effect could also be used as an optical detection scheme. Since our emission model is restricted to band-to-band recombination, no spectral analysis is possible, and we mainly focus on the intensity of emitted light. In this work we present our observations on some carbon nanotube-peptide interactions, assuming there are only electrostatic interactions between the CNT and these biomolecules, i.e., all quantum mechanical interactions between those two are ignored [43].  1.6  Thesis Outline  Based on the given model, the optoelectronics of CNFETs is studied. A CNFET based biosensor and a photovoltaic device are two major devices we study in this work. The modeling scheme is summarized in chapter 2, where the emphasis is on the changes we applied to SPDD. Some details on the modeling of oxide charges and the split-gate structure are mentioned  8  1.6. Thesis Outline in this chapter. In the third chapter, we present our expectations about how the existence of a bio-molecule will affect the electrical characteristics of a CNFET. The optical characteristics are studied in chapter 4. Possible schemes for electronic and optical detection are examined. Chapter 5 presents some simulations for an experimentally proven CNT photovoltaic device. Then we present a comparison between our simulation and the experiment work and we discuss possible improvements in the device performance. We used COMSOL1 simulations to measure the amount of absorbed power from the incident light and also to find an optimum separation between nanotubes in an array of CNT-based photovoltaic devices.  1  FEMLAB, see  9  Chapter 2 Modeling CNT Devices for Electronics and Optoelectronics This thesis has two main foci; the first one is on modeling CNFETs as biosensors, and the second one is the simulation of a CNT-based photovoltaic device. Both these devices are based on a coaxially gated CNFET, which is shown in Figure 2.1. The borders of modeling are defined by the physics involved in each case. The Schr¨odinger Wave Equation (SWE), Poisson’s equation, and a semi-classical transport equation describe different aspects of our model. In this chapter, we present the details of the framework of the model, and mention the modeling assumptions that were applied. lg  G  D  tg tins rsd  S  lsd  lCN  Figure 2.1: 3-D structure of a coaxial CNFET. The major device geometric parameters are shown in the figure.  10  2.1. Electrostatics  2.1  Electrostatics  Poisson’s equation is used to model the potential profile throughout the device. In cylindrical coordinates, it is defined as: ∂ 2V 1 ∂ 2V 1 ∂V ∂ 2V ρ3D (r, θ, z) + . + + =− 2 2 2 2 ∂r r ∂r r ∂θ ∂z ε  (2.1)  Since we apply axial symmetry to the device, there is no angular change in = 0 and Poisson’s equation reduces to: electrostatic potential, i.e. ∂V ∂θ ∂ 2V 1 ∂V ∂ 2V ρ2D (r, z) + + = − , ∂r2 r ∂r ∂z 2 2πrε  (2.2)  where ρ2D (r, z) is the circumferential summation of all the charges at radius r and distance z from the source end of the CNT. Using this assumption, we can reduce the computational window to the two dimensions of r and z, as shown in Figure 2.2.  Figure 2.2: Computational window after reducing the axial dimension. Proper boundary conditions must be applied at each interface between two different materials in the system. Since there is no Fermi-level pinning 11  2.1. Electrostatics at metal-CNT interfaces [44], we applied Dirichlet boundary conditions, and the potential on the surfaces of metal contacts are defined as:  VS = −φS  z, r ∈ ∂S  (2.3)  VD = VDS − φD  z, r ∈ ∂D  (2.4)  VG = VGS − φG  z, r ∈ ∂G,  (2.5)  where φ is the metal work function in electron volts, VDS and VGS are drainsource and gate-source voltages, respectively. This results in a Schottkybarrier between CNT and metal contact of height: φB = φM − χcnt = φM − φcnt + (EC − EF ) .  (2.6)  Here, χcnt is the electron affinity of the CNT and since we are using non-doped CNTs, it is higher than φcnt by the energy Eg /2. Null-Neumann boundary conditions are applied on all the other open boundaries with the consideration of their geometric limitation [45]. Other than CNT-metal, there are CNTinsulator and metal-insulator interfaces that must also be treated properly. Due to the discontinuity in ε at the CNT-insulator interface, a matching condition is applied to the electric flux: D1n − D2n = ρS  ⇒  εins  ∂V ∂r  − εcnt + Rcnt  ∂V ∂r  = − Rcnt  ρ1D (z) 2πRcnt ε0  (2.7)  where ρ1D (z) is the sum of all charges over the CNT circumference at distance z from the source-end, including electron- and hole-concentrations and any possible charge defect on the surface of CNT. Since there are no dangling bonds at the CNT surface, the restrictions over choosing the dielectric are less than those in conventional silicon-based FETs. Various dielectrics have been proposed in the literature from silicon dioxide (εr 3.9) [9] to zirconium oxide (εr 25) [46]. Recent works on atomic layer deposition (ALD) of high12  2.1. Electrostatics κ dielectrics on SWCNTs allow conformal, precise and uniform deposition of a variety of materials with a benign CNT-insulator interface that preserves the CNT properties [47]. For biosensing applications, target species are usually dissolved in water, therefore, a relative permittivity of 80, close to that of water, is used for the device dielectric. It is noteworthy that, although solving Poisson equation gives us the electrostatic potential in the entire computational window, we are mainly interested in the longitudinal potential on the CNT surface, VCS , where the charge transport occurs. Through the length of the CNT, the vacuum potential is inflexibly shifted by the local potential difference at the CNT surface, such that Evac = −qVCS . The conduction and valence bands are determined from this by EC = Evac + χcnt and EV = EC − Eg , assuming EC and EV are rigidly shifted by VCS [48]. This assumption might not be valid for CNTs with diameters as large as 1.5 nm [49]. It is shown that for CNTs with large diameter, bandgap alters a lot as a result of transverse electric field. The basic 3-D Poisson equation allows us to deal with any 3-D charge profile in space, ρ3D . In this work, we have simplified the simulation into two dimensions, so any charge in our model should be in the form of a 2-D, angular-independent charge and, therefore, we have to replace ρ3D with its azimuthally symmetric representation, ρ3D , as illustrated in Figure 2.3. This new charge profile must satisfy :  2π  2π  ρ3D dθ = 0  ρ3D dθ  and  ∂ρ3D = 0, ∂θ  (2.8)  0  which simply says the angular-independent version must represent the same amount of charge. The next step is to define the 2-D equivalent of this distribution, ρ2D , which is ρ3D (r, z, θ) =  ρ2D (r, z) . 2πr  (2.9) 13  2.1. Electrostatics  (a)  (b)  Figure 2.3: 3-D treatment of oxide charges. A point charge in space (a) is replaced by an azimuthally symmetric charge profile (b) in our simulation. While in the real situation the partial charge of each atom is concentrated at one point, in our azimuthally symmetric model, the same charge is distributed over the circumference. One might argue that this representation would underestimate any effect of the charge. However, it is important to realize that, in the real case, only a fraction of the current is in direct interaction with the imposed-charge electrostatics, while here all the current is involved in the detection process. The later effect can compensate the reduction in the charge density and, therefore, this treatment is appropriate for our desired qualitative results. Our simulation tool is based on the SPDD solver developed by Dylan McGuire of the UBC Nanoelectronics Group [14] and proper changes have been applied though the work. The original SPDD solver was not initially designed to model oxide charges, so, it basically solved Laplace’s equation in the entire device, with the charge and the nanotube interacting only through the CNT-dielectric matching boundary condition, see (2.7) [14]. While studying the CNT for biosensing applications in the next two chapters, we need the flexibility to include fixed charges anywhere in the oxide. Therefore, this part of the code is replaced by a complete Poisson solver, and a variable, re-  14  2.2. Charge Transport fined mesh is applied to the surrounding area of the charges. A refined mesh size of 0.05 nm is applied in the vicinity of any fixed charge and the a 2-D raised cosine distribution, (2.10), is used to represent the charge distribution. Figure 2.4 shows a sample charge profile for ρ2D (r, z). f (x; µ, s) =  1.5 2 1  1 1 + cos 2s  x−µ π s  for µ − s ≤ x ≤ µ + s (2.10)  r[nm]  z[nm]  1 0  0.5  4999  4999.5  5000  5000.5  5001  5001.5  2  ρ2D[C/nm ] 0  CNT surface −0.5 center line −1 13  x 10 −1.5  Figure 2.4: 2-D charge distribution of a dipole in the dielectric. Positive and negative values represent positive and negative charges respectively.  2.2  Charge Transport  In the early works on modeling the CNFETs, ballistic charge transport was the major assumption. In those works, the 1D potential, VCS (z), obtained 15  2.2. Charge Transport from Poisson’s equation, was used in the Schr¨odinger Wave Equation (SWE) in order to acquire information regarding charge density and transmission probability. This method is applicable to short channel devices where the device length is smaller than the mean free path of carriers in the CNT, and a quantum mechanical (QM) treatment of charge transport is possible over the full length. However, a long channel length makes a full quantum treatment impractical in terms of computational cost. Also, for long channel CNFETs, carrier transport is dominated by scattering, and a semi-classical treatment like the Drift-Diffusion Equation (DDE) is more suitable for describing transport. Light emission and absorption can also be modeled by coupling an appropriate continuity equation to the DDE. Carrier injection from contacts is divided into two parts; the thermionic emission current from carriers with energies above the height of the Schottky-barrier that is used as the Neumann boundary condition for the current, and the tunneling current from carriers with energies below the SB height which is treated as a generation term in our model. Since the transmission coefficient is significantly smaller than unity, it should be considered in the calculation of the thermionic emission current; IT E = −  4q h  Emax  T (E) (fµS (E) − fφN (E)) dE  (2.11)  EC (0)  where T (E) is the transmission coefficient, µS is the source Fermi level and φN represents the electron quasi Fermi level. The length of the QM regions is determined by the electric field in such a way that the electric field reaches a predefined fraction (∼ 0.001) of the maximum field at the edge of the QM region. This results in regions of length between 100 − 300 nm, while a QM treatment is only necessary in those parts of the device where the transmission probability is appreciable, i.e., only few tens of nanometer from the tube ends. A comparison between these two lengths validates our model for transport in the CNFETs. For those biases that produce ambipolar  16  2.2. Charge Transport conduction, carrier injection is assumed to be unidirectional. The height and the thickness of the Schottky-barriers prevent reverse injection, so a boundary condition in terms of the thermionic and tunneling current is applicable. The tunneling current represents itself in our semi-classical DDE as a generation term [50] such that: Gtun,e (z) = −  dψ 4q T (Ec (z)) [fµS (Ec (z)) − fφN (Ec (z))] . h dz  (2.12)  The electron concentration at the drain end of the device and the hole conE Thermionic Emission  Tunneling Schottky-barrier Contacts  Ec  fs(E) 0  0 Ev  Figure 2.5: Energies related to the thermionic emission current and tunneling current. A Fermi distribution is assumed for carriers in both contacts, and a quasi-Fermi level is used in the bulk of the CNT. The inset shows the potential in the QM region compared to the whole device potential.1 1 Adopted from D.L. McGuire, ” A Multi-Scale Model for Mobile and Localized Electroluminescence in Carbon Nanotube Field-Effect. Transistors” (MASc thesis, University of British Columbia, 2006), 15.  17  2.2. Charge Transport centration at eh source end are set to their equilibrium values by using a Dirichlet boundary condition, which implies an infinite recombination velocity at those contacts. Here we used a scaling scheme similar to the one proposed by De Mari [51], and variables are scaled to be unit-less. An appropriate scaling factor for the introduced 2-D charge defect was not mentioned in the original work and we calculated it to be the intrinsic carrier concentration, ni . Considering the exponential dependency of carrier concentration on the CNT potential, a variable substitution is applied to DDE and CE in order to reduce numerical errors in such a way that u = ne−ψ  (2.13)  v = pe+ψ .  (2.14)  where ψ indicates the scaled electrostatic potential at the CNT surface, and n, p are electron- and hole-concentrations, respectively. A set of equations can be extracted using scaled variables and applying the techniques mentioned so far: 1 ∂ r ∂r  ∂V r ∂r  ∂ 2V ρ2D (r, z)/ni ψ −ψ + = ue − ve δ (r − r ) − (2.15) cnt ∂z 2 εins (r/rcnt )  −  d dz  Dn  du ψ e dz  −  d dz  Dp  dv −ψ e dz  + Buv = B − Gn + Buv = B − Gp .  (2.16) (2.17)  Dn and Dp represent diffusivities of electron and holes and vary through the device with electric field, Gn and Gp are electron and hole generation rates and are defined by (2.12), B is the direct radiative recombination velocity and is derived by fitting to the experimental results. The first term in the RHS of (2.15), Poisson’s equation, is the surface charge density and the second term represents charge in the oxide.  18  2.3. Electroluminescence  2.3  Electroluminescence  Band-to-band recombination of electron-hole pairs is assumed to be the source of electroluminescence in this work. It should be noted that since no trapping sites exist in a defect-free semiconducting CNT, Shockley-ReadHall (SRH) recombination is ignored. Exciton relaxation is also not included in this model. We will discuss the validity of this assumption in chapter 3, when we analyze the results. Direct recombination is a spontaneous phenomenon; i.e., the probability of an electron and hole recombination is not time dependent [52]. Therefore, the decay rate of electrons at any time t is proportional to the number of remaining electrons and holes with some proportionality constant for recombination, see (2.18). For a net rate of change, we must also include the thermal generation term as in (2.19). −  −  dn (t) dt dn (t) dt  = Bn (t) p (t)  (2.18)  Rec  = Bn (t) p (t) − Bn2i  (2.19)  net  where the direct radiative recombination coefficient, B, is determined by fitting the width of experimental luminescence profile to the simulated one.  2.4  Photovoltaics  The electrostatically gated CNT-based p-i-n diode (Figure 2.6) was first suggested by Lee et al. in 2004 [30]. A year later they examined the photovoltaic effects in the device after they observed ideal diode behavior from it [12]. A proper model for their experiment can help us in optimizing the p-i-n diode structure, both individually and in interaction with similar devices in a batch. Here, the SPDD solver’s code is modified to model the split gate  19  2.4. Photovoltaics structure. Most changes were applied to meshing and to the Poisson-solver modules. Dirichlet boundary conditions are applied to both gates and a Neumann boundary condition is specified for the gap between two gates. Optical absorption shows its effect in the generation term in the RHS of (2.16) and (2.17) after measuring this quantity through simulation. In order to estimate  Figure 2.6: A split gate CNFET can work as an ideal diode and a photovoltaic device with the proper biasing of the gates. the photocurrent in carbon nanotube-based solar cells, we need an approximate value for optical generation rate, Gop . Evaluating the absorbed power is a useful approach to approximate Gop . We used COMSOL2 to simulate carbon nanotube-wave interaction, where SWNTs were modeled as solid cylinders with homogeneous complex permittivity of εcnt (ω) = εr (ω) + iεi (ω). After measurement of absorbed power, 1D optical generation term-number of EHPs generated per sec per unit length-in the carbon nanotube is defined as: Gop [m−1 s−1 ] = = = 2  # of EHPs generated per sec [s−1 ] L [m] η×# of photons absorbed per sec [s−1 ] L [m] Pa 1 η × ¯hω × L [m−1 s−1 ].  (2.20)  see  20  2.4. Photovoltaics Figure 2.7 shows the cubic computational window of our COMSOL simulations. We placed the CNT in parallel with the top and bottom surfaces. A port boundary condition produces the incoming plane wave from the top side. Since sunlight has an arbitrary polarization, we simulate waves with parallel and perpendicular polarizations (with respect to the CNT longitudinal axis), and add the results to model the sunlight. However, in the perpendicular case the tube is nearly transparent [53]. At the bottom we applied a scattering boundary condition to ensure transparency to the incoming plane wave. Perfect electric conductor- and perfect magnetic conductor-boundary conditions are properly applied to the side-walls and result in a propagating plane wave in the box. The presence of a CNT in the path of an electromagnetic  Figure 2.7: A polarized plane wave propagating from top surface to the bottom. wave, not only alters the electromagnetic field distribution, but also results in some energy absorption by the CNT. One can measure the absorbed energy using electric and magnetic power densities. These quantities are related to absorbed energy through Poynting’s theorem as in (2.21). It states that the net power flowing out of a given volume V is equal to the time rate of 21  2.4. Photovoltaics decrease in energy stored within V minus the conduction losses. (E × H).ndS  =−  S Total power leaving the volume  δ δt  V  1 2 1 εE + µH 2 dV − 2 2  σE 2 dV V Ohmic power dissipated  rate of decrease in energy stored in electric and magnetic filed  (2.21) We are looking for the power loss (not power stored) and that part comes from imaginary part of complex permittivity, εi and can be specified as Pa,opt (r, t) =  δ 1 εi E 2 , δt 2  (2.22)  which is the imaginary part of the electric power density in an isotropic linear material. A more practical quantity than the instantaneous power density would be the average power density, Pa,ave (r), which is the averaged integral of the above equation over the period T = 2π/ω, that is T  1 Pa,ave (r) = T  Pa,opt (r, t) dt,  (2.23)  0  or equivalently, in phasor representation3 ,  1 1 Pa,ave (r) = Re(jωEs .D∗s ) = Re(+ωεi Es .E∗s ) 2 2  (2.24)  where Es and Ds are the phasor forms of E and D. Equation (2.24) is used in COMSOL in order to evaluate the absorbed power. This power is translated 3  Pa,ave (r) = =  Since  dD dt  =  1 T 1 T  T 0 T 0  E. dD dt dt = 1 2  1 T  T  Re(Es ejωt ).Re(jωDs ejωt ) dt 0  Re(jωEs Ds ej2ωt ) + Re(jωEs D∗s ) dt = Re(jωEs D∗s )  d jωt ) dt Re(Ds e  = Re(jωDs ejωt ) and Re(z1 ).Re(z2 ) =  1 2  [Re(z1 .z2 ) + Re(z1 z2∗ )]  22  2.4. Photovoltaics into generation through (2.20) and is used in our device solver to get the photocurrent. Some notes about this translation are necessary at this point: Avouris et al. have shown that in general, most of the optical absorption in CNTs is in the form of excitons [54]. In our device, however, the high electric field will alter the CNT’s optical properties significantly. These effects have been calculated by Perebeinos et al. in 2007 [55]. They have shown that the high electric field : a) shifts the absorption peak from the excitonic absorption energy to that for the band-to-band absorption; b) increases the band-to-band absorption spectral weigh, and c) dissociates the bound exciton. Considering these effects, it is reasonable to assume all the absorption takes place in the form of band-to-band absorption.  23  Chapter 3 Molecular Sensing Using CNFET Electronics 3.1  Introduction  There are several advantages in electronic sensing using CNFETs. Electronic detection is straightforward compared to optical detection. CNFETs are size compatible with biological species and very sensitive since all the current is in interaction with the detection objective. Detection of a specific molecule is also possible using functionalization [39]. Analogy to the situation involving impurities in a conventional FET structure, these are two different effects on the device characteristics. A charge transfer between the CNT and the molecule can shift the threshold voltage, Vth , as shown in Figure 3.1(a), however, the molecule may also induce a scattering potential without any charge transfer. In the latter, we would observe a current suppression rather than a shift in characteristics [32], see Figure 3.1(b). In this work we essentially  ID  ID  VG  (a)  VG  (b)  Figure 3.1: (a) Change in threshold voltage shows itself in Isd Vs Vg . (b)An increase in scattering causes reduction in the current. 24  3.2. Simulation Parameters examine the electrostatic interactions between molecules and the CNT. Any change in the threshold voltage is found to be insignificant but our simulations do show a the current drop in the presence of molecules in the dielectric layer of CNFETs.  3.2  Simulation Parameters  As a first step towards modeling protein interactions with carbon nanotubes, we investigate CNFET characteristics in the presence of a peptide (which is basically a short protein). Peptides are formed by the linking of two or more amino acids through a dehydration synthesis. The characteristic chemical bond is called a peptide bond. Existence of sites with partial charges motivates us to model these molecules, in a first-order approximation, as a series of partial charges. The usual way to deal with such a problem is to model a set of basic elements and then predict the effect of the complete series by superposition of several basic elements. The simplest form of a charge sequence is a dipole moment. Simulation results help us in defining possible sensing mechanisms using carbon nanotubes. A standard cylindrical carbon nanotube field effect transistor forms the basic device of our simulation. Major device parameters are shown in Table 3.1. Geometric parameters have been illustrated in Figure 2.1 in the previous chapter, and φB is the height of the Schottky-barrier. Chirality Eg [eV ] φB [eV ] lCN [nm] rCN [nm] Tins [nm] rSD [nm] κins (25,0) 0.4 0.2 10000 1 100 30 80 Table 3.1: Device Parameters.  The structure of a typical peptide is shown in Figure 3.2. Considering the model dipole as the building block of a peptide, we settled on the parameters of Table 3.2 for the sensing objective of our simulation. All the model 25  3.2. Simulation Parameters values are chosen close to the actual ones, except the dipole length which is higher than the actual value due to the numerical restrictions. Later on this chapter, we will discuss that the dipole length does not decisively effect the sensitivity of our device. As discussed in the previous chapter, in this work each charge-site is represented by an azimuthally symmetric charge distribution (see Figure 2.3).  Figure 3.2: The condensation of two amino acids to form a peptide bond.  Actual Value Model Value Atom radius [nm] 0.1-0.2 0.15 Peptide bond [nm] 0.2 0.5 Partial charge [C] 0.1-0.4 0.3 Table 3.2: Model Values.  26  3.3. Results  3.3 3.3.1  Results Effect of Molecular Dipoles on Isat  The drain I-V characteristic of the CNFET in the presence of various dipole moments are shown in Figure 3.3. After the placement of the dipole in the dielectric 5 ˚ A above the CNT surface, we still observe standard transistor behavior but with some shift in the saturation current. For the following part of our simulations, we used VGS = 0.75 V and VDS = 0.9 V as our bias point5 to make sure that the device was working in the saturation region. The drain current decreases slightly as we increase the dipole moment. 40  35  p=0 q x0.5 nm e  30  p=0.3 qex0.5 nm  ID [nA]  25  p=0.5 qex0.5 nm p=0.7 qex0.5 nm  20  p=0.9 qex0.5 nm  10  ID [nA]  15  5  34 37 36 35 34 0  0.2 0.4 0.6 0.8 Charge [x q ] e  0  0  0.2  0.4  0.8  0.6  1  1.2  1.4  VDS [V]  Figure 3.3: I-V Characteristics at VGS = 0.75V . Dipole of mentioned charge is placed at the middle of CNT length parallel to its longitudinal axis. ID,sat Vs dipole charge is shown in the inset. As shown in Figure 3.4, fixed charges result in high electric fields in the vicinity of defects. On the left side of the dipole, electrons are blocked by the electric field barrier that results from negative charge-induced band bending 27  3.3. Results and we have a huge accumulation of electrons which pushes the band up. On the right side we have holes accumulation that pulls the band down. It is clear that the new potential profile will reduce electron tunneling from the source and hole tunneling from the drain, both phenomena reduce total current. Since at the bias condition of our simulation the electron current provides a large fraction of the total current, here, current reduction is due to electron accumulation and a reduction in electron tunneling from the source. 0.5  p = 0.3 q x 0.5 nm e  Energy [eV]  0  p=0 −0.5  −1  −1.5  0  2000  4000  6000  8000  10000  z [nm]  Figure 3.4: Potential profile in the presence of a lateral dipole, VGS = 0.75V and VDS = 0.9V  3.3.2  Effect of Changing Charge Position on Isat  In order to have a reproducible sensing scheme, the above variations must be position independent, i.e., we should observe no significant change in current while sweeping the charge along the CNT longitudinal direction (z). 28  3.3. Results −8  2.5  x 10  2  Id [nA]  1.5 q=0.3 qe q=0.5qe  1  no charge 0.5  0 0  1  2  3  4  5  6  7  8  9  10  Dipole distance from Source [µm]  Figure 3.5: Current variation with longitudinal position of a lateral dipole Position dependency of the drain current with respect to dipole location is shown in Figure 3.5. The sensitivity of drain current to position of dipole is less than one percent. In short, the insertion of a dipole moment adds a valley and a hill to the potential profile. The hill will block electrons, and the valley is an obstacle for hole transport. Simulation of dipoles with different lengths shows that these two effects occur almost separately. Variation of dipole length to larger values (×10, ×100) results no significant change in the measured current. We can conclude from these two effects that electron and hole blockage are two separate phenomena in these devices.  3.3.3  Gate Voltage Sweep  Variation in the channel conduction as a function of gate voltage is shown in Figure 3.6. We expected to observe some shift in threshold voltage like  29  3.4. Summary  −9  2  x 10  q= 0 q  e  q= 0.3 qe  1.9  q= 0.5 qe  Id [nA]  1.8  1.7  1.6  1.5  1.4  1.3 0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Vgs [V]  Figure 3.6: ID vs. VGS at VDS = 25 mV , no change in threshold voltage is observable. Ref. [56]. However, the small amount of charge has negligible effect on VT ∆VT =  ∆Q 0.3 × 1.6 × 10−19 ∆Q −16 = = 2 × 10 V, = −4 Cox 2.4 × 10 2πεL ln(b/a)  (3.1)  and we observe only some scattering-like deterioration of the current.  3.4  Summary  A position-independent charge sensitivity is observed in long CNFETs. This effect is different from what was previously declared by Guo et al. [56] due to quantum effects in a short CNT. Here, the change in conductance was only due to the electrostatic gating of the channel by the partial charges of molecules in the oxide layer. Unlike the experimental results, the sensitivity was not enough for a good sensing. It is now clear that the sensitivity of those 30  3.4. Summary devices is not only due to these electrostatic coupling but it could be due to the permittivity alteration at the vicinity of biomolecules or change in the electrical properties of the CNT itself. The observed position independency is necessary for effective sensing applications. In the next chapter, we will examine if the electroluminescent properties of CNTs can produce a more effective sensing scheme.  31  Chapter 4 Molecular Sensing Using CNFET Electroluminescence 4.1  Introduction  Photoluminescence, as well as both mobile and localized electroluminescence, have been observed in semiconducting CNTs [9, 41, 42]. Photoexcitation, electron-hole recombination, and hot carrier excitation have been suggested as radiation mechanisms. Chen et al. have shown that the impact excitation process plays a major role in emission from CNTs under hight electric fields [57], but the impact excitation energy threshold for a (25,0) tube has been measured to be 430 meV [55, 58]. This energy is higher than the amount an electron could gain in the high electric field regions of the devices considered in this work, e.g., a 0.3 C charge at the distance 3 ˚ A from the CNT surface, would induce a 250 meV barrier in the potential profile. Therefore, not much impact excitation is expected in our devices. Absence of photoexcitation and exciton generated luminescence in our device geometry allows us to model radiation by just free-carrier band-to-band recombination, equation 4.1. CNTs are direct band gap materials, and radiative recombination of electron-hole pairs can be described by R = B np − n2i .  (4.1)  In equation 4.1, B is the bimolecular radiative recombination velocity and its value is assigned by fitting to experimental results [14]. In short devices, the 32  4.2. Results light emission covers the entire length of the device, but in a long-channel device, light is emitted just from a small part where higher concentration of electrons and holes coexist [23], as shown in Figure 4.1. The emission process is very similar to that in a p-n junction but here injection of carriers happens from tunneling though Schottky-barrier contacts instead of diffusion from chemically doped regions. It is interesting that the emitted light is strongly polarized along the CNT axis [9]. 10  14  10  −1  12  10 8  10  10  10 n(z)p(z) n(z) p(z)  6  10  8  10  n−p product  Carrier concentration [#m ]  10  6  10 4  10  4  1000  2000  3000  4000  5000  6000  7000  8000  9000  10  z [nm]  Figure 4.1: Electron and hole concentrations and their product. VGS = 1.5 V and VDS = 2.3 V . It is clear that maximum np occurs at the center of the device, so, we expect maximum electroluminescence there.  4.2 4.2.1  Results Voltage Sweep  Electron and hole profiles overlap the most when the gate voltage is set at a value halfway between that of source and drain. At this bias device emits its maximum light while working at its minimum current [9]. As we saw in the previous chapter, the presence of some charge defects in the gate dielectric will change the potential profile, (see Figure 3.4). The blocking  33  4.2. Results of carriers results in an accumulation of electron and holes which increase the recombination rate and hence light emission at the location of charged defects. Figure 4.2 shows sweeping of the light spot along CNT, there is an accumulation of electrons on the left side of the defect but it does not increase emission unless the hole concentration reaches a certain level by adjusting the bias condition. Figure 4.3 clearly shows this effect at the presence of a dipole 11  4.5  x 10  Recombination Rate [µm−1s−1]  4 3.5 3 2.5  Vgs=Vds/2=1.15 V sharp peak due to the presence of some fixed charges  Vgs=1.1510 V Vgs=1.1515 V Vgs=1.1520 V V =1.1525 V  2  gs  modulation of recombination profile 1.5 1 0.5 0 Charge location −0.5 0  1000  2000  3000  4000  5000  6000  7000  8000  9000  10000  z[nm] length of CNT  Figure 4.2: Recombination profile at different gate voltages. VGS = 1.5 V and VDS = 2.3 V . A single charge of value 0.3 × qe is placed at 3 ˚ A distance from the nanotube surface with charges of 0.3 × qe . In Figure 4.3(a) VDS is slightly smaller than than 2 × VGS , although we have a large electron accumulation, not much change in recombination profile is observed due to small value of hole concentration at the defect location. When the hole concentration increases by adjusting the bias, a major change occurs in the recombination profile, see Figure 4.3(b), and therefore the electroluminescence as shown in Figure 4.4. Single charge detection is possible in this way. The amount of charge will change the intensity of emitted light. The polarity of the charge can be found by investigating the light spot behavior as it reaches the defect. The light spot shows an abrupt fall in intensity on one side and a gradual drop on 34  4.2. Results 10  Electron concentration 3.5 Recombination Profile 3  3 2.5  6  10  2 1.5  4  10  1  Carrier concentration [# m−1]  3.5  Recombination Profile  8  10  2.5 6  10  2 1.5  4  10  1  0.5 2  10  0  2000  4000  6000 z [nm]  (a)  8000  0 10000  4  Recombination rate[arb. units]  Hole concentration Recombination rate[arb. units]  Carrier concentration [# m−1]  Electron concentration 8  x 10  10  10  4  Hole concentration  10  10  x 10  10  10  0.5 2  10  0  2000  4000  6000  8000  0 10000  z [nm]  (b)  Figure 4.3: Electron and hole concentration at two different bias points, (a) VGS = 0.75 V and VDS = 1.4 V and (b) VGS = 0.75 V and VDS = 1.5 V . One can observe a 25% increase in height of recombination profiles.  Figure 4.4: A fixed bright spot is observed at the position of charge. VGS = 1.14 ∼ 1.16 V and VDS = 2.3 V , q = 0.3 × qe .  35  4.3. Summary the other side. If the abrupt end faces the drain, it means that we have an electron accumulation; therefore the barrier should be a result of a positive charge. On the other hand, a further extension of the light spot towards the source represents a blockage of holes, and the existence of a negative charge in the dielectric, Figure 4.5(a).  4.2.2  Dipole Length  As the oxide charge departs more from the CNT surface, its electrostatic coupling to the CNT decreases and we observe less effect on the bright spot. Figure 4.5(b) shows that this device is not sensitive to defects that are located more than 15 nm from the CNT surface. Since most of desired samples are smaller than 15 nm , this result might not be of any help in reduction of computational cost of a CNT-protein system but the case is different when we allow the screening effect to take place. That situation happens when we model a vertical dipole in the gate dielectric, Figure 4.6. Here the nearer charge screens the electrostatic interactions of the further one so the effects of the second charge vanish rapidly as it gets further from the CNT surface. As the dipole length increases to 1.5 nm, we almost observe a recombination profile similar to that of a single charge (see two rightmost brightnesses of Figure 4.6(b)). Since ab. inito. simulations shows that there is rather no change in the electrical properties of a CNT in the presence of a particle at this distance [59], and we know electrostatic interactions are screened, we can exclude any particle beyond this distance from the CNT surface, from a more detailed atomic level simulation when there are some particles at the surface of the CNT.  4.3  Summary  Various possible detection techniques have been investigated in order to facilitate the process of bio-molecular sensing. The sensitivity of the CNFET’s 36  4.3. Summary 11  6  x 10  Recombination Rate [µm−1s−1]  5  Positive charge Negative Charge  4  3  2  1  0  −1 0  1000  2000  3000  4000  5000  6000  7000  8000  9000  10000  z[nm] length of CNT  (a)  (b)  Figure 4.5: (a) Recombination profile for a single negative charge vs. a single positive charge of value 0.3 × qe . (b) Charge effect fades away as it goes further from CNT surface. optical response to charge defects in the dielectric shows some promising detection behaviors. The maximum effective dipole length has been estimated.; We can use this value in order to reduce computational cost in more complex simulations and ignore any particle farther than this distance from the CNT  37  4.3. Summary  (a)  (b)  Figure 4.6: (a) A Dipole perpendicular to CNT surface. (b) Variation in the luminescence profiles as dipole length varies. The first profile shows electroluminescence in a bare tube and in the last one the dipole is replaced with a single negative charge. surface.  38  Chapter 5 Photovoltaic Action in CNT p-i-n diodes 5.1  Introduction  Photovoltaic behavior in CNT p-i-n diodes has been experimentally observed by Lee [12]. He observed a maximum power conversion efficiency, η, of 0.2% for a circularly polarized light source. A short-circuit current in the range of picoamperes and an open-circuit voltage of a few tenths of a volt were measured in his work. He suggested that much better performance would be achievable by proper choice of the device parameters. Here we try to model his device, and then investigate further improvements of the CNT photovoltaic diodes.  5.2  Simulation Parameters  Here we tried to use the same simulation parameters as the experimental work of Lee [12]. Specifications of the simulation box (Figure 2.7) and CNT model are shown in Table 5.1 and Table 5.2. Input power density  Input frequency  1kW /cm2  2 × 1014 Hz  Simulation-box dimensions x y z 0.5 µm 0.5 µm 0.5 µm  Table 5.1: Specifications of the simulation box  39  5.3. Results radius length 1 nm 100 nm  Permittivity 20 − 20i  Position(center) Conductivity (0.25, 0.25, 0.25) µm 3 × 103 Sm−1  Table 5.2: CNT model used in COMSOL simulations The values of the incident wave’s power density and the CNT permittivity, εcnt , are taken from Ref. [11] since the power density was not mentioned in Ref. [12]. However, it is clear from the results that a smaller power density was used in the experimental work. The nanotube’s conductivity, σcnt , was estimated from its average mobility and carrier concentration. The absorbed energy then was translated to a generation term and used in the modified SPDD Solver. The parameters of the split gate device are listed in Table 5.3, where lgg is the spacing between two gates and has been chosen according to Chirality Eg [eV ] φB [eV ] lCN [nm] lgg [nm] rCN [nm] Tins [nm] rSD [nm] κins (16,0) 0.62 0.31 10000 1000 0.626 100 30 3.9 Table 5.3: Split gate device parameters.  Ref. [12].  5.3 5.3.1  Results Modeling the Experimental Device  Using these parameters an absorbed power of Pabs = 0.85 pW is calculated. Putting this value in (2.20) results in a generation rate of ∼ 5 × 1014 m−1 s−1 . Here we assumed a quantum efficiency of 10% as in Ref. [11]. Figure 5.1 shows the I-V characteristics of the diode under the above conditions. Generally, they are in good agreement with the experimental work of Ref. [12], although there is one order of magnitude difference between the short-circuit currents in two cases. This is probably due to the lower power density of the circularly polarized light source in the experminents of Ref. [12] compared to that 40  5.3. Results −10  10  x 10  8  Simulation G=5x1014 Experiment 13  Simulation G=5x10  IDS [A]  6 4 2 0 −2 −0.2  −0.1  0  0.1  0.2  0.3  VDS [V]  Figure 5.1: I-V characteristics under illumination from Ref. [12] and from our model. The shift towards the fourth quadrant is observable. ISC of Ref [12] is ∼ 1 × 10−12 A and is one order smaller than what we calculated. of the plane wave in our model. In fact, the short-circuit current varies almost linearly with the generation term and hence input power density. It is estimated that the power density used in the expriment was ∼ 100W /cm2 . Figure 5.1 also contains the I-V characteristic under this iluumination. One can observe a smaller open-circuit voltage in our simulation, which is a result of a higher dark current. It indicates lower doping concentration (lower gate voltage) in our work than that in Ref. [12]. Some important figures of merit for the CNT-based photodiode are shown in Figure 5.2. The fill factor (FF) of the PV diode is defined as FF =  IM VM , ISC VOC  (5.1)  where IM and VM are the current and voltage of the point where the device produces the highest power, ISC is the short-circuit current and VOC is the open-circuit voltage. FF is calculated to be 0.554 which is also in agreement with the range of 0.33 − 0.52, mentioned in Ref. [12] for different devices.  41  5.3. Results V  −11  0  VOC  M  x 10  −12  x 10 1.5 −0.5  Power (W)  [A]  1 I  DS  −1  IM  0.5 −1.5  ISC  −2 0  0.05  VDS [V]  0.1  0 0.15  Figure 5.2: The fourth-quadrant I-V characteristics under the light from Ref [12] and from our model.  5.3.2  PV Device Under Sunlight Illumination  After successfully modeling the fabricated device, we can use the same procedure in order to estimate the performance of CNT-based solar cells under one solar equivalent illumination. Here we used a (7,0) CNT with a 1.4 eV band gap, which is the optimal band gap for a solar cell [60]. Values for the frequency dependent permittivity of this CNT was obtained from the analytical expression for optical absorption of carbon nanotubes given in Ref [61], and then scaled to the experimentally observed values. Equation (2.24) is used in order to calculate the absorbed power. The AM 1.5 Spectraum [62] is used to model sunlight and then the frequency range was divided into several intervals with almost constant values of power-intensity and permittivity, as in Figure 5.3. Absorbed power is calculated separately for each interval and an optical generation term is calculated for each one (see Table 5.4). Summation of these generation terms results in a total generation rate of 3×1011 m−1 s−1 . The generation under the sunlight is almost 2 orders of magnitude smaller than that under the intense laser illumination used by Lee [12]. The I-V characteristic using this value in a device with lgg = 0.5 µm is shown in Fig42  5.3. Results  30  1.5  1  2  3  4  5  6 7  8  9  10 11  ε"  20  1  15  10  0.5  Solar irradiance [W m−2 nm−1]  25  5  0 0  0.5  1  1.5  2  2.5  3  3.5  4  0 4.5  Energy [eV]  Figure 5.3: Imaginary part of permittivity on the left axis and AM 1.5 spectrum for terrestrial sunlight on the right. Interval 1 2 3 4 5 6 7 8 9 10 11  Energy [eV ] Plight [W m−2 ] 20 1.4 103.3 9 1.6 113.4 5.5 1.85 117.5 3.25 2.15 100.9 2.5 2.4 77.9 10 2.6 29.1 1.5 2.8 42.5 1.1 3.1 31.9 1 3.3 16.1 1.1 3.6 6.1 8 3.75 1.9  Pabs [W ] Gopt [m−1 s−1 ] 2.77E-14 1.24E+11 1.57E-14 6.13E+10 1.15E-14 3.89E+10 6.80E-15 1.98E+10 4.57E-15 1.19E+10 7.53E-15 1.81E+10 1.73E-15 3.85E+09 1.40E-15 2.82E+09 5.60E-16 1.06E+09 2.30E-16 3.99E+08 5.80E-16 9.67E+08  Table 5.4: Permittivity, power density, absorbed power and generation in each wavelength interval  43  5.3. Results ure 5.4. A great improvement in VOC is observed due to the larger band gap of the (7,0) tube, which results in a lower dark current. However, the effect of the lower illumination intensity is clear in the short circuit current. 0  V  ´ 10−14  VOC  M  −14  x 10  −1  IDS [A]  −2 1 −3 IM ISC−4  −5 0  Power [W]  1.5  0.5  0.1  0.2  0.3 0.4 VDS [V]  0.5  0.6  0 0.7  Figure 5.4: The fourth quadrant I-V characteristics under AM 1.5 sunlight using a (7,0) nanotube as the channel. In order to evaluate the performance of an array of nanotubes for PV purposes, electromagnetic interaction between tubes should be considered. It is expected that electrostatic interactions between induced-charges on CNTs would alter the absorbed power. Five nanotubes were placed close together as shown in Figure 5.5 to find an optimum nanotube spacing. Variation of the absorbed power with respect to the spacing, d, is shown in Figure 5.6. It is clear that for d ≥ 3 nm there is almost no reduction in Pabs due to the interaction of the tubes. Unlike the suggestion of Ref. [12], no improvement in η is observed due to electromagnetic coupling. Taking 3 nm as the minimum separation of two adjacent tubes results in a maximum CNT density of 2.5 × 109 [cm−2 ]. An estimation of the absorbed power from an array of CNTbased PV devices would be 42.5 µW cm−2 . Multilayer structures could be used in order to increase current and voltage. A more efficient conversion of the solar spectrum is also possible by applying different band gap nanotubes 44  5.3. Results to cover a larger range of frequencies.  (a)  (b)  Figure 5.5: CNTs interaction under illumination. In (a) a propagating wave is shown perpendicular to the plane of nanotubes. A closer view is shown in (b), from which it can be seen that the CNTs have severely changed the incident electromagnetic wave close to their ends. 2.2 2.1  Pabs [nW]  2.0 1.9 1.8 1.7 1.6 0  5  10  15  d [nm]  Figure 5.6: Absorbed power for an array of 5 CNTs. After the spacing increases beyond 3 nm, Pabs stays almost constant.  45  5.4. Summary  5.4  Summary  Successful modeling of PV devices allows us to evaluate their performance. The SPDD solver was modified in order to model p-i-n diodes with different specifications. Our simulations show that although the efficiency of a single CNT-based PV device might be considerable [12], not much return can be expected from a batch of CNT-based solar cells. Their nanoscale size hinders proper covering of a large surface. Having a small thickness, they need to be stacked in order to absorb a considerable portion of the incident power but that structure involves a very complex fabrication process. Gating of the multilayer nanotube structure is also another concern. Since diodes are electrostatically doped in our design, proper gating is of a great importance, which is hard to achieve if one wanted to use the device in multiple layers.  46  Chapter 6 Conclusions Carbon nanotubes have been proposed to form the base of a variety of electronic devices, e.g., transistors, sensors and diodes. Different academic and industrial groups around the world are working on the fabrication and optimization of these devices, which are complex and high-cost procedures. Device modeling allows a qualitative prediction of the performance of the devices and can first of all, clarify the necessity of such fabrication and, secondly, evaluate devices that are not currently realizable. Such information is valuable for defining research roadmaps. The nanoscale size of CNTs results in different outcomes for different applications. While a single atomic layer structure is helpful in sensing applications, it severely depreciates the photovoltaic performance of CNTs. The longitudinal electric field in the PV structure modeled in this thesis greatly improves the performance of a single solar cell. It is indicated that the diameter-dependent band gap of CNTs could be of great use in photovoltaic applications, but making multiple layers from an organized array of CNTs does not seem practically possible at the moment, and a single layer of nanotubes cannot absorb significant optical power. The UBC Nanoelectronics Group has been working on modeling CNTbased devices and has developed different solvers to evaluate the performance of CNT-based devices. SPDD (Schr¨odinger-Poisson-Drift-Diffusion) solver has successfully modeled and explained experimentally observed fixed and mobile electroluminescence in CNFETs. We have found that electroluminescence is possible not only due to the presence of single charges in the oxide layer, but also due to sets of partial charges, which may be net neutral. This 47  Chapter 6. Conclusions feature could possibly be utilized in the detection of biological species via CNTs. Chapter 4 shows the details of this detection method. We also tried to explain the electronic detection of proteins by CNTs in Chapter 2 though a pure electrostatic model. Our results show a much smaller sensitivity than that in the experimental reports, which possibly means that a quantum mechanical treatment of the CNT-protein system is necessary for understanding the phenomena. A limit of 1.5 nm is achieved for the largest distance from nanotube surface that a particle can be detected. Since ab inito simulations also show that nanotube electrical properties are not affected by atoms at this distance, this value is helpful for simplification of more detailed simulations in a way that one can omit any further atom from this distance from the simulation without losing any information. An experimentally realized electrostatically gated p-i-n diode is successfully modeled in Chapter 5. The model allows us to evaluate an optimized CNT-based solar cell device under air mass 1.5 (AM 1.5) illumination. Considering the small diameter of CNTs, an enormous current density of ∼ 1 A/cm2 is achieved from a single CNT-based solar cell, which is 40 times larger than the best thin film nanocrystalline silicon or silicon nanowire solar cells. But the problems in batch production of these devices make this outstanding result of doubtful practical value. In short, interesting properties of CNTs such as high mobility, single atomic layer structure and variable band gap, are very hard to utilized. Expensive, narrow-field applications such as biosensing is one open area for using CNTs in which our model shows promising performance. Coupling optical detection with the current electronic detection methods, make CNFETs promising but expensive biosensors.  48  References [1] G.E. Moore. Cramming more components onto integrated circuits. Proceedings of the IEEE, 86(1):82–85, 1998. 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University of British Columbia, Vancouver, BC, Canada, 2006.  56  Appendix A Formulation and Implementation of Arbitrary Charge Profile in SPDD Formulation of the drift-diffusion and continuity equations is presented in details in Appendix B of Ref. [63]. In our work, the possibility of positioning an arbitrary charge profile in the oxide layer of CNFET was added to the existing code. In the following pages, we present a complete description about the modeling of charge profiles and its implementation. The first step is to modify the meshing scheme. Due to the small size of any individual partial charge, we have to refine the meshing in the vicinity of these charges in the file buildMesh.m. After defining the center position and the dimensions of the charge set, we replace the original mesh of that area with a fine mesh of h, k ≤ 0.l nm. This value is chosen to restrict the total number of mesh points while still having the numerical convergency. Figure A.1 shows mesh size variation with respect to position. Meshing indices should also be updated to be in agreement with the new system. Any molecule is presented by a set of partial charges in our work, and the presence of these oxide charges affects the electrostatics of our device through the Poisson Equation 1 ∂ r ∂r  r  ∂V ∂r  +  ∂ 2V ρ2D (r, z)/ni = ueψ − ve−ψ δ (r − rcnt ) − . (A.1) 2 ∂z εins (r/rcnt )  The first term on the RHS of ( A.1) represent carrier concentration which 57  Appendix A. Formulation and Implementation of Arbitrary Charge Profile in SPDD 10  6  9 5 8 7 4 10  h [nm]  hz [nm]  6  8  5  6  4  3  4  2  2  3  0  2  4800  4900  5000  5100  5200  1  1 0 0  1000  2000  3000  4000  5000  6000  7000  8000  9000  10000  0 0  20  40  z [nm]  60  80  100  120  r [nm]  (a)  (b)  Figure A.1: Modification of the meshing profile in (a) longitudinal direction and (b) radial derection. happens only on the CNT surface at r = rcnt . The second term, which is radius-dependent, corresponds to any charge in the oxide. As we described in chapter 2, any 3-D charge profile should be replaced with equivalent azimuthally symmetric one. ρ  (r,z,θ)dθ  ρ3D (r, z, θ) = 3D2πr ρ2D (r, z) = ρ3D (r, z, θ) × 2πr  (A.2)  De Mari scaling factors, Table A.1, are used to normalize variables in our work. Since it was not clear what scaling value should be used for 2-D Variable z, r V n, p Dn , D p B J  Scaling factor ld = 2πrcnt εkB T qe−2 n−1 i Vth = kB T qe−1 ni D0 = 1m2 S −1 B0 = D0 ld−2 n−1 i J0 = qD0 ni ld−1  Unit m V m−1 m2 S −1 mS −1 CS −1  Table A.1: De Mari scaling factors for DDE and CE charge densities, we derived it by ensuring it is consistent with the rest of quantities. Starting with equation (A.1) while neglecting the surface charges  58  Appendix A. Formulation and Implementation of Arbitrary Charge Profile in SPDD we have ∂ 2V qρ (r, z) ∂ 2V 1 ∂V Q2D (r, z) + = − 2D . + =− 2 2 ∂r r ∂r ∂z 2πrε 2πrε then we multiply the right hand side by LV  Rt KB T /(q 2 ni ) . Rt KB T /(q 2 ni )  (A.3)  The result will be  ∂ 2V ∂ 2V 1 ∂V qρ2D (r, z) Rt KB T /(q 2 ni ) + + = − ∂r2 r ∂r ∂z 2 2πrεr ε0 Rt KB T /(q 2 ni ) KB T /q ρ (r, z)/ni vth ρ (r, z)/ni = − 2D = − 2D 2 (r/Rt )εr 2πRt ε0 KB T /(q ni ) (r/Rt )εr ld  =  ⇒ (ld L)(V /vth ) == −  ρ2D (r, z)/ni εr (r/Rt )  (A.4)  Equation ( A.4) suggest that we have to normalize our 2-D charge by same charge as the 1-D charge, i.e. ni . The next step is discretizing the system of equations using Finite Difference (FD) method on an inhomogeneous mesh. Left hand side of 2-D Poisson’s Equation was desctritized to 1 rj  Vi,j+1 − Vi,j−1 kj + kj−1  2 kj + kj−1 2 + hj + hj−1 +  Vi,j − Vi,j−1 Vi,j+1 − Vi,j − kj kj−1 Vi,j − Vi−1,j Vi+1,j − Vi,j − hj hj−1  Qi,j , ε (A.5)  =  where h and k are mesh spacing variables in the longitudinal and radial directions respectively. Coefficients of the left hand side of (A.5) are already in a matrix operator form and the relevant Poisson operator is built through the file build2DCylindricalPoissonOp.m. We have to make a charge distribution matrix and add it to the previously coded system of equations in SPDD. Although solving Poisson’s Equation by using direct methods are much faster than the iterative methods, they are more complex to implement and need more computational resources. Due to the numbering scheme 59  Appendix A. Formulation and Implementation of Arbitrary Charge Profile in SPDD of this method, Figure A.2, charge matrix is a one dimensional vector that covers all the mesh points. This matrix, called Cnew in the code, is defined in the following lines of the spdd7.m for i c =1:atomCnt CnewTemp = sparse ( length (CMask ) , 1 ) ; IndAtom ( i c , : ) = IndAtomDefect ( i c )+ i ∗ length ( z ) − . . . Sz : IndAtomDefect ( i c )+ i ∗ length ( z)+Sz ; IndR ( i c )= c e i l ( IndAtom ( i c , 1 ) / length ( z ) ) ; CnewTemp( IndAtom ( i c ,:))= −1∗ p a r t i a l C h a r g e ( i c ) ∗ . . . ( f x y ( i+Sr + 1 , : ) . / n i ) / ( r ( IndR ( i c ) ) . / f e t . Rt ) ; Cnew=Cnew+CnewTemp ; c l e ar CnewTemp ; end  (Nr-1)xNz (Nr,1)  Nz+1 Nz+2 Nz+3 Nz+4 Nz+5 . . . (2,1) (2,2) (3,3) (4,4) (5,5) ... 1 2 3 4 5 ... (1,1) (1,2) (1,3) (1,4) (1,5) ...  NrxNz (Nr,Nz)  2Nz-1 2Nz (2,Nz-1) (2,Nz)  Nz-1 Nz (1,Nz-1) (1,Nz)  Figure A.2: Meshing numbering system, coordinate of each node in mentioned at the bottom-right while the actaul index in presented in bold on the top-right side of the node. Here, we swept over all atoms and added its pre-defined charge profile to the proper indices of the matrix. Sz and Sr are parameters of raised cosine distribution, fxy contains the indivual charge profile and IndR shows the 60  Appendix A. Formulation and Implementation of Arbitrary Charge Profile in SPDD index of the radius at which the partial charge is placed. It is noteworthy that the permittivity is already included in the Poisson operator, so, we have droped it from the charge vector. This matrix is then used in the procedure of solving poisson’s equation as described in Appendix B of Ref. [63]. LV = C  (A.6)  61  Appendix B Modification of SPDD for Double Gate Structures An electrostatically gated p-i-n diode can be realized by making a structure similar to a CNFET with two gates separated by a gap. By applying opposite potentials to these two gates, two different regions of the device can behave like doped regions of a diode. There will be a high electric field region in the center similar to intrinsic region of a p-i-n diode. Modifications are applied in different files of SPDD in order to model these structures. At first, buildMesh.m has been modified. Gate lengths and spacing between two gaps describe two different gates in this file with the indices H.nsg1, H.nsg1, H.neg2 and H.neg2 representing start points and end points of first and second gate respectively. Then we have to modify the Poisson operator in the file build2DCylindricalPoissonOp.m. Poisson operator matrix coefficients are given in Table B.1. Location i, j i, j + 1 i, j − 1 i + 1, j i − 1, j  Coefficient −2  1 hi hi−1  1 kj +kj−1 1 kj +kj−1  + 1 rj  1 kj kj−1  +  − r1j +  2 kj 2 kj−1  1 2 hi +hi−1 hi−1 1 2 hi +hi−1 hi  Table B.1: Poisson operator matrix coefficients All of our changes are in the box-1 section of build2DCylindricalPoissonOp.m 62  Appendix B. Modification of SPDD for Double Gate Structures that contains CNT and the dielectric layer mesh points. Dirichlet boundary condition, (B.1), is applied to both gates. Vi,j |G1 = VG1  and  Vi,j |G2 = VG2  (B.1)  This BC is forced in the code by setting the respective elements of the operators equal to one cij1G1 cij1G2  = ones ( 1 ,H. neg1−H. nsg1 +1); = ones ( 1 ,H. neg2−H. nsg2 +1);  and use zero coefficient for the neighbor elements cim1j1G1 = zeros ( 1 ,H. neg1−H. nsg1 +1); cip1j1G2 = zeros ( 1 ,H. neg2−H. nsg2 +1); cijm11G1 = zeros ( 1 , length ( c i j 1 G 1 ) ) ; cijm11G2 = zeros ( 1 , length ( c i j 1 G 2 ) ) ; then a Neumann BC is used for the gap between two gates. Since Vi,j−1 = Vi,j+1 , the coefficient of (i, j − 1) element will be 1 kj + kj−1  1 2 1 + + rj kj kj + kj−1  −  1 2 + rj kj−1  =  1 kj + kj−1  2 2 2 + = . kj kj−1 kj kj−1 (B.2)  Each element is then built according the relevant coefficient in Table B.1 and equation (B.2). c i j 1 G g = −2∗(1./(H. hz1 (H. neg1 :H. nsg2 − 2 ) . . . . ∗H. hz1 (H. neg1 +1:H. nsg2 −1))+ 1 /(H. hr1 ( k−1)∗H. hr1 ( k − 1 ) ) ) ; cim1j1Gg = 2 . / ( (H. hz1 (H. neg1 +1:H. nsg2 − 1 ) . . . + H. hz1 (H. neg1 :H. nsg2 − 2 ) ) . ∗H. hz1 (H. neg1 :H. nsg2 − 2 ) ) ; c i p 1 j 1 G g = 2 . / ( (H. hz1 (H. neg1 +1:H. nsg2 − 1 ) . . . + H. hz1 (H. neg1 :H. nsg2 − 2 ) ) . ∗H. hz1 (H. neg1 +1:H. nsg2 − 1 ) ) ; cijm11Gg = ( 2 / ( (H. hr1 ( k − 1 ) ) . ∗ (H. hr1 ( k − 1 ) ) ) ) . . . 63  Appendix B. Modification of SPDD for Double Gate Structures . ∗ ones ( 1 , length ( c i j 1 G g ) ) ;  %Neumann BC  Finally mask vectors should be updated. Mask vectors contains the coordinates of different parts of the device (Source, Drain, Channel , Gate1 and Gate2) and are used in spdd7.m to force or extract respective values of potential before or after solving the system.  gte1Mask ( nPt+H. nsg1 −1:nPt+H. neg1 −1) = ones ( length ( c i j 1 G 1 ) , 1 ) ; gte2Mask ( nPt+H. nsg2 −1:nPt+H. neg2 −1) = ones ( length ( c i j 1 G 2 ) , 1 ) ; The last modification was in spdd7.m where the ohmic contact BC is applied to the DDE.  64  


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