Carbon NanoTube Noise Characterization Harold ...

Carbon NanoTube Noise Characterization Harold Szu and Bassam Noaman, Digital Media RF Lab Dept. ECE, GWU, Washington DC [email protected], 202-994-0880 (lab) ABSTRACT Without relying on the cumbersome liquid Nitrogen coolant, necessary for the conventional mid IR (3~5 µm wavelength) cameras, we designed a new mid wave IR camera, according to biomimetic human vision 2 color receptor system. We suspended over the non-cryogenic long wave IR (HgCdTe) CCD backplane with Single Wall Carbon NanoTubes (SWNT) pixels, which have the band gap energy εBG ~1/d tuned at the few nanometer diameter d for the mid wave. To ascertain noise contribution, in this paper, we provided a simple derivation of frequency-dependent Einstein transport coefficient D(k) = PSD(k), based on KuboGreen (KG) formula, which is convenient to accommodate experimental data. We conjectured a concave shape of convergence1/k α at α = -2 power law at optical frequency against the overly simplest 1-D noise model about ½ KBT, and the ubiquitous power law 1/k α where α =1 gave a convex shape of divergence. Our formula is based on the Cauchy distribution [1+(k d)2]-1 derived from the Fourier Transform of the correlation of charge-carrier wave function been scattered against lattice phonons spreading over the tubular surface of the diameter d, similar to the Lorentzian line shape in molecular spectral exp(-|x|/d). According to the band gap formula of SWNT, a narrower tube of SWNT worked similarly as Field Emission Transistor (FET) can be tuned at higher optical frequencies revealing finer details of lattice spacing, a and b. Experimental determination of our proposed multiple scales responses formula remained to be confirmed. KEYWORDS: CNT, Carbon Nano Tube, Kubo-Green formula, Lorentzian line shape. Mid IR spectral detector, non-cryogenic cooling 1.

INTRODUCTION

The thermal management of electronic devices remains to be an issue ever since Moose’s quest of device miniaturization and better yields. Recently, the integration of Micrometer (10-6 m) Electro-Mechanical Systems (MEMS) with Nanometer (10-9 m) devices, known as NEMS, becomes a major thrust of nanoscience and nanotechnology. In this paper, a hybrid methodology of classical & quantum physics shall help design a better performance of NEMS involving a high Signal-to-Noise Ratio (SNR) of the Carbon NanoTubes (CNT). Among their remarkable properties, such as a great strength, a light weight, and a high stability of pure Carbon lattices, we are especially interested in the hybrid electronic properties, namely its classical pseudo-one-dimensional field emission transistor (FET) diode based on Einstein’s photoelectrical effect where the work function is the band gap near the Fermi surface with a reduced thermal noise in its scattering interaction with the quantum mechanical lattice vibration phonons, which makes CNT an ideal

Independent Component Analyses, Wavelets, Unsupervised Smart Sensors, and Neural Networks IV, edited by Harold H. Szu, Proc. of SPIE Vol. 6247, 62470S, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.670047

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material for non-cryogenic Infrared (IR) sensory applications. Because of the versatility of CNT, a great deal of effort has been devoted to the understanding and characterization of their properties [1-4], since their discovery by Ijima [5] in 1991 following the Nobel Laureate Smalley 1989. 1.1 Mid Wave IR Sensor Motivation:

For early detection of malign tumors, Szu et al. have applied

modern the precision Satellite multiple spectral imaging cameras for remote sensing [1] to passively screening breast cancers in 1999 [2,3] using a single-pixel unsupervised neural network fusion algorithm [5]. This algorithm was based on the necessary and sufficient condition of unsupervised learning such as the power of pairs, say the Mid IR and Long IR satellite-grade cameras, together with the physics of isothermal equilibrium at the minimum of Helmholtz free energy [4]: H = E − T0 S a tradeoff between the analytical information energy E and the Shannon-Boltzmann entropy S. This capability is now fully understood, because the physics of human grey-body radiation matches uniquely with the oncology knowledge of tumor physiology, called the angiogenesis effect, i.e. blood vessel generation of any rapidly growing tumor producing warmer thermal radiation. This mid wave IR is believed to become appreciable from an angiogenesis malign tumor, amid those originally dominating long IR radiation at 8~12 µm. According to the Planck grey-body distribution law manifested by an increased capillary blood supply temperature by 3oC, i.e. To = 40oC+273oK. We estimated the Planck radiation energy at 13o C above a warm ambient ( at the temperature 27oC or 300oK) the ratio implied the total signal energy is about (313/300)x(1/40) eV, since KBT of 300oK is (1/40) eV. In fact, the ratio of two color IR intensity per pixel becomes independent of the tumor depth and radiation propagation damping characteristics. The ratio is directly proportional to the percentage for arbitrary tumor characteristics. Then, the time rate of IR spectral colors ratio can measure the growth rate of malign tumor in breasts and elsewhere. Furthermore, we proposed a non-cryogenic dual-color IR camera, employing SWNT mid IR pixels suspended over the long wave IR CCD backplane. This combination means affordable for every household private screening without the cumbersome coolant of liquid Nitrogen.. According to the design, the high sensitivity pixel detection mechanism is based on the Einstein photoelectrical effect of Ijima’s Carbon NanoTube. Since SWNT is linearly polarized, so that for random thermal irradiation we need measure all three Stokes vector directions. Since three polarized SWNT diodes must cross one other to form a heterogeneous overpass junction, such a semiconductor device can only be easily tailor made by lithographic means. Thus, Szu and Xi has applied the biomimetic muscle principle to modify the Cantilever arm of the Atomic Force Microscope (AFM) of Nobel Laureate Binning, which allows us a real-time multiplexing operation of the piezoelectric material from a soft mode to hard stiff state. This is called a NanoRobot manipulator[7]. They further measured the thermal noise of SWCNT along the longitudinal propagation of the p-n junction diode which is smaller than the ideal gas law (3/2) KBTo at the equivalent temperature. Thus, in principle, one may apply the specific diameter of SWNT for the Mid IR pixel at near the room temperature operation with electrical cooling. Therefore, we investigate theoretically the noise floor of SWNT in this paper


1.2 Thermal Conductivity of SWNT (Jianwei Che*, Tahir Cagin, and William A. Goddard III, Materials Center, Caltech) CNT has a unique electronic property. It can be either metallic or semiconductor depending on its chirality (i.e. conformational variation). A lot of experiments and theoretical investigations have focused on electronic structures of CNT in order to understand the origin of the remarkable phenomena. In addition, large effort has been given to characterize their mechanical properties, such as Young's modulus, energetics, etc. However, to finally assemble fully functional NEMS/MEMS, the thermal management has to be addressed. To date, there is little progress made to characterize and to understand the thermal conduction in nanoscale materials. Our approach to understand the lattice thermal transport properties for CNT follows the Kubo-Green correlation method [6] which is generally valid near the equilibrium linear response. KG generalized the Einstein’s fluctuation-dissipation theorem relating the Diffusion D and Viscosity η at the Kelvin Temperature Do =

K BT

η

Due to technological difficulties of synthesizing and characterizing CNT diode, it is challenging to experimentally measure the thermal conduction in a controlled environment. Thus, it is desirable for theory to predict the formula of thermal conductivity influenced by point defects. In general, there are two directions to calculate transport properties of materials. One is based on phenomenological Boltzmann transport equation (BTE), and the other is based on fluctuation-dissipation relation from linear response theory. Usually, the parameters in BTE are deduced from experimental measurements. For a novel material CNT without collision model of phonons and charge carriers, BTE may not be directly predict its transport properties. On the other hand, due to the development of empirical Leonard-Jones potentials for a wide range of systems, it is relatively easy to obtain high quality empirical potentials for ab initio supercomputer calculations of CNT so that such a Molecular Dynamics (MD) computation could be of interests at a microscopic level. The motions of atoms are completely governed by inter-atomic interactions. In MD, the Einstein fluctuation-dissipation relationship can be utilized to calculate the transport coefficients in the linear response regime. Therefore, MD has the unique advantage in predicting thermal transport properties of novel materials and materials that are difficult to perform experiments on. 1.2

Surprise: As one expected, the thermal conductivity decreases as the vacancy concentration

increases. However, the rate of decreasing in thermal conductivity is quite unexpected. Caltech had calculated the vacancy influence in thermal conductivity for diamond crystal previously [7], which showed inverse proportionality between thermal conductivity and vacancy concentration. It is natural to expect that defects & vacancies should have a less severe effect in 3-dimensional materials than in 1-dimensional ones. However, on the contrary to one intuition, the MD calculation results show that vacancies in CNT are even less influential than in 3-dimensional diamond. This is probably due to the fact that the strong double


valence bond Carbon-six network provides effective additional channels for charge carriers to bypass the vacancy sites. 1.3 1/k noise (E. S. Snow, J. P. Novak, M. D. Lay, and F. K. Perkins, NRL) The scaling behavior of 1/k noise in single-walled carbon nanotube devices was studied by twodimensional carbon nanotube networks to explore the geometric scaling of 1/k noise. NRL found that for devices of a given resistance the noise scales inversely with device size. NRL established an empirical formula that can be used to assess the noise characteristics of CNT-based electronic devices and sensors. Nevertheless, the simplicity of scale-less correlation might be support by the preliminary experimental measurement done by Collins et al reveals only 1/f under K Hz radiation (phillip Collins, “1/k noise in cabon nanotube”). No result at the optical IR frequency which is believed to be relevant to SWNT tube diameter in proportional to the bang gap at the Fermi surface.

SWNT ID Sample

10'

T=3410K

nA

S

10

I nA

10_I I

10 '3 0.2 mA

10.13

10_ '4

10

4kTR

10.2

'O

iO°

io

101

f. Hz

FIG- I - Voltage noise power S. vs frequency for a single SWNT. for three

values of applied bias current At low frequency, the noise greatly exceeds the thermal noise Innit S4kTR The SWNT has a two probe resistance of 335 kO. 1.4 Surprises:

We conjectured that, rather than

1 , where 1 ≤ a ≤ 2 to be exactly 1 for ka

all frequency responses. This was demonstrated subsequently the ubiquitous case of the exact inversely linear power

1 behavior is related to the Heaviside step correlation function, which says an identical k

strength of the stationary correlation of any scale length that is greater than zero, namely a scale-less or all scale correlation. This is based on the classical Wiener-Khintchine relationship between Fourier transform stationary pair correlation function that equals to the power spectral density (PSD) as proved in Sect. 3. Thus, in terms of pseudo-one-dimensional geometry of quantum mechanical SWNT, we conjectured the Lorentz atomic spectral line shape to be appropriate to the mixture of response behaviors to be a linear


combination of Lorentzian line shape functions: we assume a linear combination of various Lorentzian line shapes among lattice constants a & b and the tube diameter d.

FT{ ∑ ci exp(i =1,2,3

|x| 1 )} = ∑ 4ci a i ≅ ∑ 4ci a i (1 − (ka i ) 2 ) 2 ai 1 + ( k a ) i =1, 2 , 3 i =1, 2 , 3 i

(3)

where we can have a polynomial fit by the linear combination of Laplacian density of various unique scale lengths which are two lattice spacing a=1.41Ao & b= & the diameter d=10 Ao ?, etc. 1.5 Working Hypothesis SWNT applications have two classes, being conductors as nano wire or semiconductors as sensors. A specific application of our interest is a mid IR sensor hopefully without liquid nitrogen coolant. If we can build it with only electrical cooling, the application domain can be used in every household for early tumor screening. The question arises, is SWNT really suffering less thermal noise in the IR regime than a normal semiconductor? We expect that the generalized Green-Kubo correlation approach to all response frequency can experimentally determine that a single wall Carbon NanoTube (SWNT) to be better than the conventional cryogenic charged coupled camera, made of Compound (HgCdTe) in a higher dimensionality. This might be due to a narrower radius for Mid IR detector, which will behave more like an ideal 1-D p-n junction IR detector, as demonstrated recently by GE using 1.5µm LED to test a CNT for photovoltaic cell with high 5% conversion efficiency and extended by Szu et al. to solar photovoltaic cells with 4x4= 16 SWNTs array giving 80% efficiency per lenslet.[11] . 2. Review current understanding of SWNT Single wall carbon nano tubes (SWNTs) are among the most exciting possible ingredients of future nanoelectronic devices. It can be a semiconductor or metallic, depending on the exact atomic geometry of the seamless molecular cylinder. Among the most robust conductors known, it can sustain electrical current densities of more than 100 times larger than conventional metallic interconnect without failure. It appears to have higher carrier mobility than any other semiconductor at room temperature [1]. 3-Einstein and Kubo-Green relation Transport coefficients such as the diffusion coefficient D are shown to be related to velocity correlation function by Kubo- Green a la Einstein diffusion. Einstein noticed that the viscous friction of the Brownian motion must relate to the diffusion constant of particles by the equation

Do = Where Do is the diffusion coefficient constant and

KT mγ

mγ is the friction coefficient. With the electric field,

the particles flow with a drift velocity: v d = − dV / dx . This flow of particles will be opposed by the mγ


diffusion flow, so the net current flow is given by j ( x ) = − Do ∂f ( x ) + vd f ( x ) , where f ( x ) is the ∂x concentration at location x. In thermal equilibrium f

(x ) ∝

exp

(− V

/ KT

) , the drift current is

equal to the diffusion current. It can be easily seen that the vanishing net current in thermal equilibrium requires Einstein relation D o =

KT . mγ

Diffusion process can be observed in a system (powder in a liquid, smoke from a chimney) where the concentration of particles is non uniform Do =

KT . Particles undergo random motion due to collision mγ Plums grows later like

sqLiaIs mct olaisla,00

Do,M,d dta,ce

Bsii flp€d

prbsb!ly t su,,bSg

FIG. 2. The spread of smoke from a chimney. With the liquids molecules. The probability distribution of the location x of a particle at a time t , can describe the microscopic transport. The probability distribution function is the solution of Einstein diffusion equation.

∂ t p( x, t | x0 , t0 ) ∂ 2 x p( x, t | x0 , t0 ) =D ∂t ∂2x p ( x, t → t0 | x0 , t0 ) = δ ( x − x0 ) p ( x → ∞, t | x0 , t0 ) = 0

⎛ ( x − x0 )2 ⎞ 1 ⎟⎟ p ( x, t | x0 , t0 ) = exp⎜⎜ − 4 D(t − t0 ) ⎝ 4 D(t − t0 ) ⎠ The above equation produces the diffusive behavior of particles. We begin the definition of mean square displacement of particles in terms of the probability density function p: ⎛ x (t ) − x ⎛ t ⎞ ⎞ ⎜ ⎟⎟ ⎜ ⎝ 0 ⎠⎠ ⎝

2

= ∫ d 3 x ⎛⎜ x (t ) − x ⎛⎜ t ⎞⎟ ⎞⎟ ⎝ 0 ⎠⎠ ⎝ Ω

2

p

(x , t | x 0 , t 0 )

Integrating over the diffusion equation (1)


(2)

(x (t ) − x (t0 ))2

d dt

(

After simple calculation for 1-D:

(

)

1 2t

lim t→ 0

⎛ x (t ) − x ⎛ t ⎞ ⎞ ⎜ ⎟⎟ ⎜ ⎝ 0 ⎠⎠ ⎝

2

( 3)

= Do

(4)

is taken over an ensemble.

Where the average We also have: x (t

( ))

= Do ∫ d 3 x x (t ) − x t 2 ∇ 2 p x , t | x , t 0 0 0 Ω

)−

x (t 0

t

)= ∫

v d dt

(5 )

'

0

From ( 4) and (5) : we get an expression for Do in terms of the Green-Kubo [12] velocity-velocity correlation formalism

Do =

D

( ) ( )

t lim 1 t ∫ dt 1 ∫ dt 2 v d t1 v d t 2 t → 0 2t 0 0

(6 )

t − t 1 lim 1 t = dt ' v t v ⎛⎜ t + t ' ⎞⎟ ∫ ∫ dt 1 1 d ⎝ 1 d ⎠ t → 0 t 0 0

( )

o

(

∞ D o = ∫ dt v (t 0 )v t + t d d 0 0

(7 )

)

(8 )

Since the process is a stationary, it is a function of the time or space difference along the longitudinal direction x of SWNT. ∞

D

o

=

∫

( x ).ν

ν

(0 )

dx

( 9 )

0

Since the charge current of holes and electrons may be modeled as charge q and local density gradient ∆n(x) as J(x) = q . Wiener Khintchine relationship: The effective current-current correlation of SWNT could be measured as If and only if C(x) = is stationary, then PSD =FT{C(x)} Proof: ) J (k ) = FT {J ( x)} =

∞

∫ dx J ( x) exp( − jkx)

−∞

∞ L ) ) ) PSD ==< J ( k ) J ( k )* >= lim L→∞ (1 / 2 L) ∫ ∫ d ( x − x' )dx ' ' < J ( x − x' ) J (0) > exp( − jk ( x − x' )) −∞ − L

= FT {C ( x − x ' )}.

Corollary: Ubiquitous 1/k power law is correspondent to all scale step(x) correlation function, or scaleless correlation in the sense of identical correlation intensity as long as the stationary correlation length in temrs of Heaviside step(x), for x= >0.


∞

1

1

1

∫ dk exp( jkx) k = ∫ dz exp( jzx) z = ∫ dz exp( jk x) exp(−k x) z 1

2

−∞

Proof: Let a complex variable z= k1+jk2. Then, applying the Cauchy contour integral, we can augment the infinite limits of FT with a large semi-circle in the upper half plane of the complex domain z =k1+jk2 and a small semi-circle around the origin of complex domain z. Then, it is readily shown that when the difference magnitude of the correlation function C(z) is non-zero, i.e. |x| > 0, the upper semi-circle contour damps away approaching the infinite and will not contribute to the complex contour integral. Then, the Cauchy residual theorem of closed contour can be deformed to the origin and gives the 1/k power law. Cauchy contour integral takes the advantage of an analytical continuation of FT into the upper complex domain where the damping exponential Fourier kernel vanishes as the upper semicircle approaches the infinity and thus this added new path integral makes no contribution. Similarly the lower semicircle vanishes as the radius becomes infinitesimal small. Then, the Cauchy residue theorem for an equivalent deformable small contour near the origin

∫

dz = 2πj giving the equivalent two semi-circle & real axis z

contours. α

The experimental data of SWNT confirmed up to k Hz a slightly faster decay: 1 / k at 1< α< 2 with a convex shape power law at the divergence. We conjectured at optical frequency a convergent concave shape power law, the details of phonon scattering of current over the tubular structure might reflect the single wall diameter d. Using the Weiner-Khintchin theorem we derived a generalized Kubo-Green Theorem that relates the frequency dependent transport coefficient to the experimentally measured Power Spectral Density PSD(k) : Generalized Fluctuation Dissipation Theorem: It is straightforward, after Szu & Uhlenbeck [7], that we can generalize the long-time equilibrium or equivalently the zero frequency formula to any frequency formula of dimensionless conductivity σ(k)

σ

∞

SU

(k ) = ∫

J ( x ). J (0 ) * exp

(o ) =

=

( jkx )dx

(10 )

0

σ

σ

SU

SU

σ

(k ) =

∞ o

∫

J ( x ). J (0 ) dx

(11 )

0

PSD (k )

(12 )

We can replace the experimental measurement of current J (t, xo) at a fixed point xo over time t as illustrated into two point product separated by x, and averaged over many stationary segments of time pieces as follows.


Current J(t, xo) time t

Then, Fourier Transform of these experimentally measured two-point correlation function over many difference value x, we can deduce the PSD according to the above definition, and obtained the frequency dependent conductivity of SWNT to be fitted by our dimensionless PSD (k) of multiple relaxation scales, Eq(3). These results remained to be repeated in a control chamber of the environment of AFM/NanoRobot measurements of a pin-read-out setup of SWNT under various intensity of mid IR LED. 4. Transport Property of CNT Emerging need for decrease in a device size, the use of molecular level theories in device design and modeling becomes more important. A specific application of this pseudo-one dimensional Carbon NanoTube (CNT) is to design a “non-cryogenic” Infrared spectrum detectors array. One particular issue needs to be addressed is the thermal transport property of the charge carriers: holes and electrons of the Einstein photo-electric band gap effect near the room temperature. Hence, the previous studies of the thermal conductivity of nanotubes need to be extended to the infrared frequency in order to elucidate the structure, defects, stress and strain effect upon (SWNT).

1 Curling up with a nanotube b

E

y x

if k

Fermi

EF

energy /\

aa#:.t;...t!a.

.-fi _k_metallic

if Fermi

EF

energy /

sem icond uctin g

FIG. 3. (a) The lattice structure of graphene – the two-dimensional material that is rolled up to form a nanotube. The lattice is made up of a honeycomb of


carbon atoms. (b) The energy of the conducting states in graphene as a function of the wavevector, k, of the electrons. The material does not conduct, except along certain, special directions where “cones” of states exist. (c) If the graphene is rolled up around the y axis, the nanotube is a metal (upper .gure), but if it is rolled up around the x axis, the nanotube is a semiconductor (lower .gure). The band structure of the nanotube is then given by one-dimensional slices through the two-dimensional band structure shown in (b). The permitted wavevectors are quantized along the axis of the tube. The 1-D anisotropic character of the Carbon hexagon crystalline in different chirality’s for conductor or bang-gap semiconductor, which reflects in the electro-thermal conductivity property of CNT. However, when the tube diameter decreases, the change from 2-D planar structures to a quasi 1-dimensional tube plays a crucial role in the thermal conductivity.

E2 D = ±γ o [1 + 4 cos(

k a k a kxa 3 ) cos( y ) + 4 cos 2 ( y )]1 / 2 2 2 2

Where γo is the nearest-neighbor overlap integral, eliminating kx and ky using periodic boundary condition, c.k = 2πm for integer m, weget 1D enegy bands for general chiral structure. Rakitin derived the bang gap formulae of SWNT

ε BG ≅

3 a3 M iϖ 2 d 2

Where Mi = 2 10-23g, ω=1600 cm-1, and a= 11.41 Ao. We found for 3 ~5 µm mid IR regime, d ranges about 45 nm to 55 nm. Caltech has taken Molecular Dynamic (MD) approach to compute the fluctuation dissipation relationship to relate the transport Diffusion coefficient of a system in the linear regime to predict the transport coefficients for better thermal management in nano size devices. In the simulation of equilibrium MD, CNT’s thermal conductivity turns out to be depending on the vacancies and the defects. The thermal conductivity in CNTs is expected to decrease with the local defect & point vacancies concentration. It is natural to think that vacancies should have less severe effects in 3-D materials than 1-D ones. However, vacancies in 1-D CNT are even less influential than 3-D materials. This surprise is due to the fact that the tightly double bound of Carbon Chains can bridge over local defects for the passing of charge carriers. To understand this fact quantitatively, modern QM approach without the detail of Boltzmann transport equation is the Green-Kubo relation. One can extract the heat conductivity from the heat current correlation function. In fact, in analogous to Eq(8) for classical Brownian particles, Grern & Kubo had derived for QM correlation formula

ς = Where

1 K

B

T 2V

∞

∫ 0

j

q

(τ );

j

q

(0 )

dτ


( 13 )

j (τ ); j (0 ) Is a quantum canonical correlation function, and a; b is defined as: q q a; b =

1

β

β

∫ d ξ .Tr [ρ exp (ξH )a. exp (− ξ H )b ] , where ρ is the density matrix of the system at 0

equilibrium. The classical equivalence is the correlation function given by the phase space averaging as a ; b =

∫ d Γ exp (− β H )ab ∫ d Γ exp (− β H )

. Both classical and canonical correlation functions are symmetric. The

quantum effects are not so important when T >> TD where TD is the Debye temperature characterizing the lattice phonon vibration. For the purpose of this paper it is important to note that the major contribution of thermal conductivity is from phonon modes that have wavelengths equal to or longer than the microscopic scale at the lattice constants a & b, and the tube diameter d, while the macroscopic length is along the longitudinal current propagation direction z. The higher optical modes do not contribute to the thermal conductivity in a significant way. . 5-Conclusion Single Wall Carbon NanoTube (SWNT) had been successfully made an ideal p-n-junction Photovoltaic Cell (PC) having 5% conversion efficiency per single infrared (IR) LED. Recently, Szu et al. designed four bundles of four multispectral SWNT’s into Solar PC with a potential 16 x 5%=80% efficiency. Whether SWNT abides a universal 1/k

α

noise characteristics at α =1, according to the Wiener Kinchine Theorem

that the Fourier Transform (FT) of a stationary correlation equals to the power spectral density (PSD): a scale-less correlation: On the other hand, according to a classical fluctuation-dissipation theorem, the pseudo-one-dimensional SWNT is bounded by 1-D thermal noise about ½ KBT. In this paper, we provide a statistical mechanical approach to the thermal characteristics of SWNT at optical regime, and suggested Thus, in terms of peudo-one-dimensional geometry of quantum mechanical SWNT, the Lorentz atomic spectral line shape may be appropriate to the multiple scale response behaviors to be a linear combination of Lorentzian line shape functions: we assume a linear combination of the Lorentzian line shapes among lattice constants a & b and the single tube diameter d.

FT{ ∑ ci exp(i =1,2,3

|x| 1 )} = ∑ 4ci a i ≅ ∑ 4ci a i (1 − (ka i ) 2 ) 2 k ai 1 + ( a ) i =1, 2 , 3 i =1, 2 , 3 i

(3)

where we can have a polynomial fit by the linear combination of Lorentz density of various unique scale lengths which are two lattice spacing a=1.41Ao & b=1.5Ao & the diameter d=45~55 nm, 1 Ao=10 nm REFERENCES [1] H. Szu, C. Hsu, “Landsat Spectral Demixing a la Super-resolution of Blind Matrix Inversion by Constraint MaxEnt Neural Nets,” SPIE Proc. 3078, Wavelet Applications IV, pp.147-160, Orlando April 1997 [2] H. Szu, “Progresses in unsupervised artificial neural networks of blind image demixing, “IEEE Ind. Elec. Soc. Newsletter, June 1999, pp. 7-12.


[3] H. Szu, “ICA-an enabling tech for Intelligent Sensory Processing”, IEEE Circuits and Systems Newsletters December, 1999, pp. 14-41 [4] H. Szu, “Thermodynamics Energy for both supervised and unsupervised learning neural nets at a constant temperature,” Int’l J. Neural Sys. Vol.9, pp. 175-186, June 1999 [5] “Progress in unsupervised learning of artificial neural networks and real world applications,” H. Szu, published by Russian Academy of Nonlinear Sciences, S.A.B. (ISBN 0-620-23629-9), pp.1-277, 1998. [6] Szu, H, "Unsupervised learning Artificial Neural Networks," Chapter 16 In: "Computational Intelligence" book edited by Farber John Wiley & IEEE Press, 2003. [7] Hwa-Ling. Harold Szu, “Contributions to the kinetic theory of dilute gases,” Ph D Thesis with G.E. Uhlenbeck of the Rockefeller University, New York, New York, 1971. Chapter The Fluctuation Problem pp. 50-66: Appendix: A Generalized Langevin Equation and Einstein Relation, pp.67-69. [7] ONR Press Release Oct 2005 In: Nanowire Technology web [8] Jianwei Che, Tahir C¸ ag˘ ın and William A Goddard II, “Thermal conductivity of carbon nanotubes” Nanotechnology 11 (2000) 65–69. [9] A. Rakitin, C. Papadopoulos, and J. M. Xu, “Electronic properties of amorphous carbon nanotubes,” Phys. Rev. 61, pp 5793-5796 [10] R. Saito, M. Fujita, G. Dresselhaus, and M. Dresselhaus, “Electronic structure of chiral grapheme tubules,” Appl. Phys. Lett. 60 (18), 4 May 1992, pp. 2204-2206. [11] H-C. Ou & H. Szu, “Designs of Solar Voltaic Cells based on Carbon Nano-Tubes” SPIE Proceeding V. 6247 “ICA Wavelets Unsupervised Smart Sensors and NN,” April 17-21, 2006 Orlando
