Performance analysis of Carbon Nano Tubes

Transcription

1 IOSR Journal of Engineering (IOSRJEN) ISSN: Volume 2, Issue 8 (August 2012), PP Performance analysis of Carbon Nano Tubes P.S. Raja, R.josep Daniel, Bino. N Dept. of E & I Engineering, Annamalai University, Annamalainagar, , Tamilnadu, India. ABSTRACT: - Carbon nanotubes (CNTs) ave received muc attention for teir unique caracteristics as a possible alternative to Cu interconnect. Cu interconnect dimensions begin to come into te range of mean free pat of electron typically 40nm. Tis results in surface and grain boundary scattering. Owing to tese scattering penomena resistivity of Cu begins to increase. Tis drives us to look for new materials for future very large scale integration (VLSI) interconnects. Carbon nanotubes are alternative to copper because of teir remarkable conductive, mecanical and termal properties. Tey are effectively long, tin cylinders of grapite wic is made up of layers of carbon atoms arranged in a exagonal lattice wit stiffest and strongest fibers. Carbon nanotubes ave superior properties like current carrying capacity and conductivity compared to Cu interconnect. It exibits metallic or semi-conducting property depending on cirality. A matlab coding as been developed to evaluate te performance of various carbon nano tubes (CNT). Te various parameters evaluated are Resistance, Capacitance and Inductance. Te evaluations of tese parameters are very important since tey affect te performance of te device. I. Introduction Carbon nanotubes are widely used in nanoelectronics because of its mecanical and electrical properties [1]. Wit one undred times te tensile strengt of steel, termal conductivity better tan all but te purest diamond, and electrical conductivity similar to copper, but wit te ability to carry muc iger currents, tey seem to be a wonder material. Metallic carbon nanotubes ave been identified as a possible interconnect material of te future tecnology generations and be eir to Al and Cu interconnects. It can be metallic or semiconducting depending upon cirality. It as a wide range of application in te fields like big markets, flat panel display, ligting, fuel cells, electronics, biosensors, electrostatic painting of composites in products (car parts). Te canges in resistance and capacitance during interconnect scaling poses a serious callenge to te interconnect delay. Terefore te analysis of various parameters like Resistance, Capacitance and Inductance as a significant impact on te performance and reliability of VLSI circuits. II. Carbon nanotubes Carbon nanotubes were discovered in 1991 by SUMIO IIJIMA of NEC. Carbon nanotubes are alternative to copper because of teir remarkable conductive, mecanical and termal properties [2], [4]. Tey are effectively long, tin cylinders of grapite. Grapite is made up of layers of carbon atoms arranged in a exagonal lattice, like cicken wire. Carbon nanotubes are molecular-scale tubes of grapitic carbon wit stiffest and strongest fibres. Wit one undred times te tensile strengt of steel, termal conductivity better tan all but te purest diamond, and electrical conductivity similar to copper, but wit te ability to carry muc iger currents, tey seem to be a wonder material. Tere are six types of carbon nanotubes, of wic mostly used single-walled and Multi-walled nanotubes. Tey are Long, Sort, Single-walled, Multi-walled, Open and Closed types. Te structure of carbon nanotubes consist of one or more coaxial cylindrical seets of grapite wit an aspect ratio typically greater tan 100 and outer diameter measuring tens of nanometer and closed at ends wit two semi domes. Essentially two families of carbon nanotubes exist: (1) Single walled nanotubes (SWNT), made up of only one straigt tubular unit [5]. It is fundamental form of carbon nanotube. Single-walled nanotubes are 1 to 2 nm in diameter and combine to form rope-like structures. Te lengts of te ropes are difficult to determine but are approximately 1 μm long, wit diameters ranging from 10 to 30 nm. (2) Multi walled nanotubes (MWNT), made up of series of coaxial tubes 0.34 nanometer apart. Any carbon nanotubes can be represented wit indices (n,m) were n,m are two integers. CNTs can eiter be metallic or semiconducting. Depending on weter te tube is wrapped as a simple cylinder or a tigter elical form, te nanotube material can possess te properties of a semiconductor or a metallic material. Te electrical beaviour of CNTs can be determined from te following simple rule wic states tat te difference between te indices of te tube sould be an integral multiple of tree. Matematically, it can be stated tat if (n-m) =3q ten it is metallic and if (n-m) 3q ten it is semiconducting were q is an integer. 54 P a g e

2 Carbon nanotubes also exist in te form of bundles. It can be single walled carbon nanotubes bundle or mixed CNT bundle. SWNT bundle consists of a number of SWNT closely packed witin a seat wile mixed CNT bundle consist of bot SWNT and MWNT closely packed witin te same seat. Te igest quality carbon nanotube films we ave developed acieve rougly 90% visible-ligt transmittance and about 200 Ω/ seet resistance RS. Te nanotubes also exibit oter useful properties suc as electrical conductivities as ig as100 Ω -1 /cm -1, plus ig strengt, termal conductivity and flexibility. Fig.1.(a) Single walled CNT Fig.1.(b) Multi walled CNT Fig.1.(c) CNT bundle III. Analysis of different CNTs 3.1 Isolated single-walled carbon nanotube Resistance of Single walled CNT Te conductance of a carbon nanotube is evaluated using te two-terminal LANDAUER- BUTTIKER formula [1]. Tis formula states tat, for a 1-D system wit N cannels in parallel, te conductance is G= (Ne²/) T, were T is te transmission coefficient for electrons troug te sample. Due to spin degeneracy and sub lattice degeneracy of electrons in grapene, eac nanotube as four conducting cannels in parallel (N=4). Hence te conductance of a single ballistic single-walled CNT (SWNT) assuming perfect contacts (T=1), is given by 4e²/ = 155µS, wic yields a resistance of 6.45kΩ. Tis is te fundamental resistance associated wit a SWCNT, tis fundamental resistance (R f ) is equally divided between te two contacts on eiter side of te nanotube. R F = /4e²= 6.45kΩ, were is Planck s constant (= J-S), e is te carge of an electron (e= coulomb).te mean free pat of electrons (te distance across wic no scattering occurs) in a CNT is typically 1000nm. For CNT lengts less tan 1µm, electron transport is essentially ballistic witin te nanotube and te resistance is independent of lengt (6.45kΩ). However, for lengts greater tan te mean free pat, resistance increases wit lengt as R CNT = (/4e²) L/Lo were L is te lengt of te CNT, Lo is te mean free pat (1000nm) Capacitance of a Single walled CNT Te capacitance of a CNT arises from two sources. Te Electrostatic capacitance(c E ) is calculated by treating te CNT as a tin wire, wit diameter d, placed at a distance y away from a ground plane, and is given by, C E = (2πε)/ (ln(y/d)) = 30aF/µm [1] were ε is te permittivity of te dielectric (ε= ε o ε r ), ε o = For 22nm tecnology, te relative permittivity ε r =2. Fig.2. Isolated conductor, wit diameter d, over a ground plane at a distance y below it. Te Quantum capacitance (C Q ) accounts for te quantum electrostatic energy stored in te nanotube wen it carries current. Due to te Pauli Exclusion Principle, it is only possible to add electrons into te nanotube at an available quantum state above te Fermi energy level. By equating tis energy to an effective capacitance, te expression for te quantum capacitance (per unit lengt) is obtained as C Q = 2e²/v F were is Planck s constant and v F is te Fermi velocity (v F= m/s). As a CNT as four conducting cannels, te effective quantum capacitance resulting from four parallel capacitances C Q is given by 4C Q. Te same effective carge resides on bot tese capacitances (C E and 4C Q ) wen te CNT carries current, as is true for any two capacitances in series. Hence tese capacitances appear in series in te effective circuit model Inductance of a Single walled CNT Te inductance associated wit an isolated SWCNT can be calculated from te magnetic field of an isolated current carrying wire some distance away from a ground plane [1]. Te magnetic inductance (L M ) is as, 55 P a g e

3 L M = (µ/2π)*ln (y/d) = 1.4pH/µm, Were µ is te permeability (µ=µ o µ r ),µo=4π 10-7,te relative permeability µ r = In addition to tis magnetic inductance (L M ), te kinetic inductance is calculated by equating te kinetic energy stored in eac conducting cannel of te CNT to an effective inductance L K = () / (2e²v F ) = 16nH/µm. Te four parallel conducting cannels in a CNT give rise to an effective kinetic inductance of L K /4, wic gives 4nH/µm. Since L K >>L M, te inclusion of L K can ave a significant impact on te delay model for interconnects. Fig.3.(a) Resistance of SWCNT Fig.3.(b) Capacitance of SWCNT Fig.3.(c) Inductance of Single walled CNT 3.2 Bundle of CNT S CNT-bundle interconnect is assumed to be composed of exagonally packed identical metallic singlewalled carbon nanotubes Resistance of a CNT-bundle In order to calculate te effective resistance of a CNT-bundle, it is assumed tat all CNT S packed into te interconnect structure are metallic and conducting [1]. Te fact tat it is, in general, difficult to control te conductance properties of all CNT S in te bundle is accounted for by considering reduced packing densities. Eac CNT is surrounded by six immediate neigbors, teir centers uniformly separated by a distance x. Te expressions to calculate te number of CNTs in te bundle are n w = w d d and n H = [ x ( 3/2)x ]+1 were n H is te number of rows in te interconnect bundle, n W is te number of columns, w and are te widt and eigt of te bundle, ncnt is te total number of CNTs, given by n CNT= n W n H - nh if n H is even 2 nh 1 and n CNT = n W n H if n H is odd. Te CNT-bundle resistance is given by, R bundle = R isolated /n CNT. 2 It must be noted tat it is implicit in tis formulation tat te coupling between adjacent CNT S of a bundle is weak. However, tis is a fair assumption because it as been sown tat tere exists a large tunneling resistance (~ 2-140MΩ) between te CNT S forming a bundle Capacitance of a CNT-bundle It is observed, as expected, tat te CNT S completely surrounded by oter nanotubes ave a very small electrostatic coupling capacitance to ground compared to tose along te edges of te bundle (tree orders of magnitude smaller) [1].Teir contribution to te total electrostatic capacitance is tus neglected. Using tese relations, we calculate te electrostatic capacitance contributed by eac edge CNT based on te corresponding value of C E. Te total electrostatic capacitance of te bundle is given by te sum of te contribution from eac of tese CNT S. i.e., C E = 2C En + (n w 2)/2* C Ef + 3(n H -2)/5* C En were C En and C Ef are te intrinsic plate capacitance for an isolated CNT over a ground plane. C En = 2 log(y d) is calculated assuming te ground plane to be at a distance equal to te separation distance from te "near" adjacent interconnect. C Ef = 2 log(2 y d) is calculated assuming te ground plane to be at a distance equal to te separation distance from te "far" adjacent interconnect. Since te quantum capacitances of all te CNT S forming a CNT bundle appear in parallel, te effective quantum capacitance of te bundle is te sum of te individual quantum capacitances. C Q bundle = C Q CNT * n CNT were C Q is te quantum capacitance of an isolated CNT and η is te total number of CNT S forming te bundle. Te effective capacitance (C bundle ) of te series combination of a quantum and electrostatic capacitance is given by 1/C bundle = 1/C Q bundle + 1/C E bundle. 56 P a g e

4 3.2.3 Inductance of a CNT-bundle Te inductance [1] of a CNT bundle is given by te parallel combination of te inductances corresponding to eac CNT forming te bundle, L bundle =L CNT /n CNT. Fig.4.(a) Resistance of CNT bundle Fig.4.(b) Capacitance of CNT bundle Fig.4.(c) Inductance of CNT Bundle 3.3 Multi walled CNT Multi walled nanotubes (MWNT), made up of series of coaxial tubes 0.34 nanometer apart Resistance of multi-walled CNT Te resistance of a sell consists of tree parts: quantum contact resistance RQ, scattering-induced resistance RS, and imperfect contact resistance Rmc [3]. RS only occurs if te lengt of te nanotube (sell) is larger tan te electron MFP. RQ and RS are intrinsic, and Rmc is due to fabrication process. Te total resistance is determined by R=R Q +R S *L= + L were 2e 2 N 2e 2 N λ /2e2 ~ 12.9 kω, and L, λ, and N are te lengt, MFP, and number of conducting cannels of te sell, respectively. Te imperfect contact resistance Rmc can range from zero to undreds of kilo-oms for different growt processes. λ = D vf /(αt) were vf is te Fermi velocity of CNTs (~8 105 m/s), α is te coefficient of scattering rate, and T is te temperature. Nsell(D) a D + b, were D is te diameter of te sell, a = nm 1, and b = Capacitance of multi-walled CNT Te capacitance of a CNT also as two parts: quantum capacitance CQ and electrostatic capacitance CE [3]. Quantum capacitance per unit lengt of a sell can be derived as CQ/cannel =2 2e af/μm and vf CQ/sell =CQ/cannel (a D + b) Inductance of multi-walled CNT For a CNT, tere are magnetic inductance and kinetic inductance [3]. Te magnetic inductance of eac sell is a weak function of te factor H/D. For 1 < H/D < 100, te magnetic inductance ranges from 0.2 to 1.2 ph/μm, wic is muc smaller tan te kinetic inductance. Hence, te magnetic inductance as been ignored. Kinetic inductance is given by LK/cannel = 8 nh/μm and LK/sell =LK/cannel/(a D + b). 2e 2 vf 1 2 Fig.5.(a) Resistance of MWCNT Fig.5.(b) Capacitance of MWCNT Fig.5.(c) Inductance of MWCNT 57 P a g e

5 IV. CONCLUSION Te various results obtained from te matlab simulation reveals te following. For a SWCNT, as te lengt is reduced, resistance decreases and as te diameter is reduced, capacitance decreases wile te inductance increases. For a SWCNT bundle, as te widt is reduced, resistance increases, capacitance decreases and inductance increases. For a MWCNT, as te diameter is reduced, resistance decreases, capacitance decreases and inductance increases. REFERENCES [1] Navin Srivastava and Kaustav Banerjee, Performance Analysis of Carbon Nanotube Interconnects for VLSI Applications. [2] Arijit Raycowdury and Kausik Roy A Circuit Model for Carbon Nanotube Interconnects: Comparative Study wit Cu Interconnects for Scaled Tecnologies. [3] Hong Li, Kaustav Banerjee Circuit Modeling and Performance Analysis of Multi-Walled Carbon Nanotube Interconnects IEEE JUNE [4] A. Naeemi,et al., Performance comparison between Carbon Nanotube and Copper Interconnects for Giga scale integration(gsi), IEEE [5] P.L.McEuen,et,al., Single-Walled carbon Nanotube Electronics, IEEE, P a g e