NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Silicon-Interface Scattering in...

NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

Silicon-Interface Scattering in Carbon Nanotube Transistors

Slava V. Rotkin

Physics Department & Center for Advanced Materials

and Nanotechnology

Lehigh University


AcknowledgementsAcknowledgements

Dr. A.G. Petrov (Ioffe)

Prof. J.A. Rogers (UIUC)

Dr. V. Perebeinos and Dr. Ph. Avouris (IBM)

Prof. K. Hess (UIUC) and Prof. P. Vogl (UVienna)


OUTLINEOUTLINE

Introduction: - NT Transistors with "non-monolithic" channel

The old "new" Surface Scattering- Remote Coulomb Impurity scattering- Remote Polariton Scattering

Physics of Surface Phonon Polariton (SPP)

SPP and heat dissipation in NT devices

Conclusions


NT TransistorsNT Transistors


Fabrication of NT-Array TFTs revealed new "old" physics.

• very large gate coupling – too strong if not taking into account intertube coupling

• non-uniformity of the channel – self-screening and "defect healing"

• multi-layer dielectrics and surface E/M modes

• interface scattering

Most of the tubes are parallel, but the distance between neighbor tubes may vary.

Quantum physics of TFT capacitanceQuantum physics of TFT capacitance

For TFT applications only semiconductor tubes are needed. Thus one needs to destroy (burn out) metallic tubes. Which randomizes the channel.

self-consistent modeling (Poisson+Schroedinger eqs) including e/m response


integratedintegratedintegratedintegrated

Physics of NT Devices on SiO2Physics of NT Devices on SiO2

• weak interaction • electr. transport• thermal coupling• alignment

empty spaceempty spaceempty spaceempty space

Weak van der Waals interactions...

For a polar substrate -- such as quartz, sapphire, calcite -- new physics due to evanescent Electro-Magnetic (EM) modes, aka Surface Phonon-Polariton modes


Charge Scattering:Short IntroductionCharge Scattering:Short Introduction


e.d.f. is symmetric and thus j = 0

Transport Theory: What to Forget and What to Remember

Transport Theory: What to Forget and What to Remember

Quantum-mechanical calculation of the conductivity may be reduced to the Drude formula

electron velocity which enters the formula

can be related to m.f.p. vtr

The asymmetric non-e.d.f. provides j > 0 (both in ballistic and diffusive model)

Equilibrium distribution function is Fermi-Dirac function:


Conductivity: van Hove singularitiesConductivity: van Hove singularities

after Prof. T. Ando

Scattering rate is proportional to electron velocity which diverges at the subband edge. Thus, the Drude conductivity has "zeroes" at vHs.

Which holds for both metallic and semiconductor tubes.


Remote impurity ScatteringRemote impurity Scattering


Scattering in 1D systems is weak due to restricted phase space available for electron: k -> -k

Coulomb Center ScatteringCoulomb Center Scattering

on average the Coulomb potential

where e* and nS are the charge and density of impurities

the Coulomb impurities are on the substrate, not within the NT lattice – the remote impurity scattering


Scattering in 1D systems is weak due to restricted phase space available for electron: k -> -k

Coulomb scattering: ResultsCoulomb scattering: Results

Within this model a universal expression for conductance was found

Modeling uses the nonequilibrium solution of the Boltzmann transport equation

where a quantum mechanical scattering rate

is calculated in the Born Approximation and parameterized by the strength of the Coulomb centers' potential

and DoS


RIS Details: Statistical averagingRIS Details: Statistical averaging

starting with the Coulomb potential

then, the scattering rate is

here we used notations:

and

on average isproportional to

Statistical averaging over a random impurity distribution of

scattering form-factor

DoSstrength of potential


Surface Phonon PolaritonSurface Phonon Polariton


Digression: Digression: A tutorial on SPP A tutorial on SPP

Digression: Digression: A tutorial on SPP A tutorial on SPP


Specifics of surface polaritons:• electric field is not normal to the surface (at 45o)

• electric field decays exponentially from the surface (not a uniform solution of Maxwell equations)

• existence of a surface mode essentially depends on existence of the anomalous dispersion region <0

Surface Polariton in SiO2Surface Polariton in SiO2

Surface phonons exist in polar dielectrics:

• due to the dielectric function difference between the substrate and the air, a surface EM wave could exist

• dielectric function of the polar insulator has a zero at LO , at the

LO phonon frequency

• surface wave can be obtained by solving Maxwell equations with proper boundary conditions

q

H

E


Remote Polariton ScatteringRemote Polariton Scattering


Estimates for SiO2-quartz:

• electric field in the air is proportional to decay constant, determined from Mxw.Eq+B.C., and F-factor

• relevant is proportional to the wavelength of hot electron

• electric field ~107 V/m

• finally the scattering time

for vF~108 cm/s and SO~150meV :for vF~108 cm/s and SO~150meV : ~ 105 V/cm~ 105 V/cm

Physics of SPP scattering in SiO2Physics of SPP scattering in SiO2


Conductivity: van Hove singularitiesConductivity: van Hove singularities

Prof. T. Ando

Scattering rate is proportional to the velocity which diverges at the subband edge. Thus, the Drude conductivity has peculiarities at vHs.

rem

inder


Surface Polariton ScatteringSurface Polariton Scattering

inter-subband transitions are negligible due to non-zero angular momentum transfer

• RPS rate varies for intra-subband and inter-subband scattering• RPS has maximum at the van Hove singularities (for semiconductor-SWNT)

At vHs our Born approximation fails which manifests itself as diverging scattering rate

JETP Letters, 2006


Correct many-body picture includes phonon renormalization of the electron spectrum.

Within iterative Quantum Mechanical calculation (aka SCBA) new scattering rate obtained: - averaged near the vHs - still faster than other channels

Surface Polariton Scattering (2)Surface Polariton Scattering (2)

for vF~108 cm/s and SO~140meV : ~40 nm2ki ~ 2/a ~ 1/nm

Forward scattering dominates:

q~1/ : forward scatteringq~2ki : backward scattering

JETP Letters, 2006


• scattering rate increases with the electric field strength because of stronger warming of the electron distribution function• similarly it increases with the temperature • concentration dependence is weak and can be attributed to the tails of distribution function

Remote SPP Scattering RateRemote SPP Scattering Rate

lattice T

T=77; 150; 210; 300; 370;

450 K


• for the SiO2 substrate the SPP channel is likely prevailing over inelastic scattering, such as due to NT (own) optical phonons for the small distance to the polar substrate < ~ 4 nm;

SPP Scattering Rate and MobilitySPP Scattering Rate and Mobility

JETP Letters, 2006 (3V,300K)

• low-field mobility at 100+K is totally dominated by SPP

• the effect is even stronger for high-k dielectrics due to increase of the Froehlich constant : x20 and more;

• RPS has a weak dependence on the NT radius, thus for narrow NTs it will dominate over the other 1/R mechanisms

Nano Letters, 2009

SPP

NT


• for the SiO2 substrate the SPP channel is likely prevailing over inelastic scattering, such as due to NT (own) optical phonons for the small distance to the polar substrate < ~ 4 nm;

SPP Scattering Rate and MobilitySPP Scattering Rate and Mobility

JETP Letters, 2006

• SPP low-field mobility for a large number of various chirality NTs allows to infer empirical scaling on the NT radius

• comparison with other mechanisms: R2 for NT acoustic phonons

• lattice temperature is taken as given

Nano Letters, 2009

lattice T


Saturation RegimeSaturation Regime


Scattering in 1D systems is weak due to restricted phase space available for the electron: k -> -k. However, the strong scattering at high drift electric field is inevitable: saturation regime. The scattering mechanism is an optical phonon emission which results in fast relaxation rates for the hot electrons and holes. Inelastic scattering rates have been calculated for SWNTs earlier:

Saturation Regime: Optical PhononsSaturation Regime: Optical Phonons


What was known so far? Inelastic optical phonon relaxation scattering is likely a factor determining the saturation current in SWNTs :

The hot electron energy is transferred to the SWNT phonon subsystem.The energy dissipation depends on the environment (thermal coupling).

Saturation Regime: Heat GenerationSaturation Regime: Heat Generation


Inverse drain current vs. inverse appliedelectric field

low-F and high-FIs are essentially different, being determined bydifferent scattering mechanisms

SPP and Saturation RegimeSPP and Saturation Regime

[17,0] NT at the doping level 0.1 e/nm

Deviation from Ohm's law: first nonvanishing term in R(Vd)=Ro

+Vd/Io

Kane, PRL, 2000


low-F scattering is due to all phonons (including NT intrinsic phonon modes) and high-F scattering is due to SPP mechanism

SPP and Saturation RegimeSPP and Saturation Regime

Inverse drain current vs. inverse appliedelectric field

low-F and high-FIs are essentially different, being determined bydifferent scattering mechanisms

Deviation from Ohm's law: first nonvanishing term in R(Vd)=Ro

+Vd/Io

Kane, PRL, 2000


Modern Electronics andHeat Dissipation ProblemModern Electronics andHeat Dissipation Problem


ITRS Grand Challenges: The HeatITRS Grand Challenges: The Heat

S. Borkar, “Design challenges of technology scaling,” IEEE Micro, vol. 19 (4), 23–29, Jul.–Aug. 1999.

Among main evaluation parameters for novel semiconductor electronics technologies the power consumption, and in particular the power dissipation

become more and more important

????"Energy in Nature and Society: General Energetics of Complex Systems" by V. Smil (2008)


SPP Heat DissipationSPP Heat Dissipation


It exists, however, a relaxation mechanism which transfers the energy directly to the substrate without intermediate exchange with the SWNT lattice (phonons) which is an inelastic remote optical phonon scattering

The mechanism appeared to be ineffective for Si MOS-FETs and was almost forgotten for decades...

Pioneering work by K. Hess and P. Vogl – back to 1972 – RIP scattering in Si.

Vdq j

q~area~nm2

channel heating due to Joule losses and low thermal coupling to leads

q

jJoule Heat GenerationJoule Heat Generation


• two scattering (NT and SPP) and two coupling (SPP and Kapitsa) mechanisms : • NT phonons warm the NT lattice but the Kapitsa resistance is high

overheating of the channel : we neglect the thermal sink in the leads (area~nm2), then only substrate contributesvia thermal coupling:

where

j

qC

qph

QSPP

SPP and OverheatingSPP and Overheating

Material g=1/, W/(m·K)

Silica Aerogel 0.004 - 0.04

Air 0.025

Wood / wool 0.04 - 0.4

Water (liquid) 0.6

Thermal epoxy 1 - 7

Glass 1.1

Concrete, stone 1.7 – 2.4

Stainless steel 12.11 ~ 45.0

Aluminium 200

Copper 380

Silver 429

Diamond 900 - 2320


• assume for a moment that SPP channel is absent


• Joule losses are NOT the same as the total dissipation: NT phonons take only a small fraction of IdF

where

j

qph

QSPP




• assume for a moment that SPP channel is absent


• Joule losses are NOT the same as the total dissipation: NT phonons take only a small fraction of IdF

where

j

qph

QSPP



12

510

20

50100

200

2 4 6 8 10 12

F (V/mm)

PSPP/PNT

substrate T


• opposite R-dependence for two scattering mechanisms

• ratio of "real"-to-expected losses for two tubes (R~0.5 and 1.0 nm) at two to= 77 and 300K

• inset: data collapse for (linear) dependence on the electron concentration (0.1 and 0.2 e/nm)

• SPP scattering is higher in smaller diameter tubes: simply the SPP field is stronger

SPP and Overheating (2)SPP and Overheating (2)


• different temperature dependence for two scattering mechanisms

• even in case of no other thermal coupling to substrate, SPP channel releases the heat (R~0.5 nm, T=300K)

• inset: same data vs. Joule loss

• NT transport in saturation regime is determined by both channels

SPP and Overheating (2)SPP and Overheating (2)


ConclusionsConclusions

• Theory of NT scattering after 10 years still has new uncovered physics

• Physics of interactions in NTs at the hetero-interface with Si/SiO2 is rich for fundamental research

• Hot electron scattering due to SPP modes is by orders of magnitude faster channel for non-suspended NT

• Remote SPP scattering provides a new and very effective thermo-conductivity mechanism


Nanotube Quantum Nanotube Quantum CapacitanceCapacitanceNanotube Quantum Nanotube Quantum CapacitanceCapacitance


Classical Capacitance: 1D caseClassical Capacitance: 1D caseClassical 1D capacitance: line charge has = 2 log r + const

therefore: Cg-1 = 2 log z/R

where z = min(d, L, lg)

Distance to metal leads around/nearby1D channel defines the charge density

(z) is different for different screeningof 1D, 2D and 3D electrodes.

RR

dd

LL


Quantum Mechanical view: Selfconsistent calculation of the charge density

Rotkin et.al. JETP-Letters, 2002

The transverse size a of nanowires and nanotubes is less than the Debye screening length and other microscopic lengths of the material.

Classic view: Linear connection between electric potential and charge Q=C V ,

in a 1D device: ~ - C ext

which is to be compared with 3D and 2D: ~ - d2/dx2 ~ - d/dx

Atomistic Capacitance of 1D FETAtomistic Capacitance of 1D FET


which is to be compared with 3D and 2D: ~ - d2/dx2 ~ - d/dx

The transverse size a of nanowires and nanotubes is less than the Debye screening length and other microscopic lengths of the material.

Classic view: Linear connection between electric potential and charge Q=C V ,

in a 1D device: ~ - C ext

Quantum Mechanical view: Selfconsistent calculation of the charge density

Rotkin et.al. JETP-Letters, 2002

Atomistic Capacitance of 1D FETAtomistic Capacitance of 1D FET


Capacitance of the NT ArrayCapacitance of the NT ArrayMethod of potential coefficients (or EE circuit analysis): Screening by neighbor NTs in the array – total capacitance is of a bridge circuit

Screening depends on single parameter: 2d/o which has a physical meaning of the number of NTs electrostatically coupled in the array. The tubes that are further apart do not "know" about each other

2d/2d/

Fig. : Gate coupling in array-TFT as a function of the screening by neighbor NTs (top to bottom): same SiO2 thickness = 1.5 um, NT densities = 0.2, 0.4 and 2 NT/um

1 m

1 m

1 m


Three sample distributions of the tubes in the random-tube array (d=160 nm, 80% variance).

d=40 nm

d=600 nm

Current nonuniformity is a deficiency for device production.

Consider due to non-uniform screening.

Random Array Coupling: Self-healingRandom Array Coupling: Self-healing

-0.35

-0.25

-0.15

C/C

One may expect a severe variance in device characteristics because of non-uniform Cg


The capacitance of a random TFT array (a single given realization) as a function of the external screening (insulator thickness).

Correlation vs. RandomnessCorrelation vs. Randomness

C, %

d, nm

25 50 75 100 125 150

2.42.62.8

3.23.4

3.0

The low density TFT array is within a single tube limit...

...in the high density TFT array the inter-NT coupling is very strong and stabilizes the overall device response.


In a single tube FET total capacitance has 2 terms:

geometric capacitance

and quantum capacitance

for NT array geometrical capacitance further decreases:

10 20 50 100 200 5000.5

0.6

0.7

0.8

0.9

1

d, nm

C/Cclass

Quantum Capacitance in NT-Array TFTQuantum Capacitance in NT-Array TFT


Charge TrappingCharge TrappingCharge TrappingCharge Trapping


1. Stacy E. Snyder, and Slava V. Rotkin, “Optical Identification of a DNA-Wrapped Carbon Nanotube: Signs of Helically Broken Symmetry", Small, accepted, 2008.2. Seong Jun Kang, Coskun Kocabas, Taner Ozel, Moonsub Shim, Ninad Pimparkar, Muhammad A. Alam, Slava V. Rotkin, and John A. Rogers, “High performance electronics

using dense, perfectly aligned arrays of single walled carbon nanotubes”, Nature Nanotechnology, vol. 2 (no.4) 230-236 (2007). 3. Vadim Puller, and Slava V. Rotkin, "Helicity and Broken Symmetry in DNA-Nanotube Hybrids", Europhysics Letters 77 (2), 27006--1-6 (Jan 2007). 4. Qing Cao, Ming-Gang Xia, Coskun Kocabas, Moonsub Shim, John A. Rogers, and Slava V. Rotkin, “Gate Capacitance Coupling of Single-walled Nanotube Thin-film Transistors”,

Applied Physics Letters, vol. 90 (2), 023516 (2007). 5. Slava V. Rotkin, Narayan R. Aluru, and Karl Hess, ”Multiscale Theory and Modeling of Carbon Nanotube Nano-Electromechanical Systems”, in "Handbook of Nanoscience,

Engineering and Technology (2nd Edition)", Eds.: W. Goddard, D. Brenner, S. Lyshevski, G.J. Iafrate; Taylor and Francis-CRC Press, Chapter 13, pp. 13.20-13.32 (2007).6. Slava V. Rotkin, Alexander Shik, “Electrostatics of nanowires and nanotubes: Application for field-effect devices”, in the Special Issue Nanowires and Nanotubes, Editor: Peter

Burke, Publ.: World Scientific, Singapore. International Journal of High Speed Electronics and Systems, vol. 16 (no.4), 937-958, (2006).7. Stacy E. Snyder, and Slava V. Rotkin, “Polarization component of the cohesion energy in the complexes of a single-wall carbon nanotube and a DNA", JETP Lett 84, 348, (2006).8. Alexey G. Petrov, Slava V. Rotkin, “Hot carrier energy relaxation in single-wall carbon nanotubes via surface optical phonons of the substrate” JETP Lett 84 (3), 156-160 (2006). 9. Yan Li, Umberto Ravaioli, and SV. Rotkin, "Metal-Semiconductor Transition and Fermi Velocity Renormalization in Metallic Carbon Nanotubes", Phys. Rev. B 73, 035415 (2006). 10. L. Rotkina, S. Oh, J.N. Eckstein, S.V. Rotkin, “Logarithmic behavior of the conductivity of electron-beam deposited granular Pt/C nanowires”, Phys. Rev. B 72, 233407 (2005). 11. Salvador Barraza-Lopez, Slava V. Rotkin, Yan Li, and Karl Hess, "Conductance Modulation of Metallic Nanotubes by Remote Charged Rings", Europhysics Lett 69, 1003 (2005).12. Slava V. Rotkin, “From Quantum Models to Novel Effects to New Applications: Theory of Nanotube Devices”, in “Applied Physics of Nanotubes: Fundamentals of Theory, Optics

and Transport Devices”, Nanoscience and Nanotechnology Series, Ser.Ed.: Ph. Avouris, Springer Verlag GmbH & Co. KG (2005).13. Yan Li, Deyu Lu, Klaus Schulten, Umberto Ravaioli, and Slava V. Rotkin, “Screening of Water Dipoles Inside Finite-Length Armchair Carbon Nanotubes”, Journal of

Computational Electronics, vol. 4, 161-165 (2005).14. Arnaud Robert-Peillard, Slava V. Rotkin, “Modeling Hysteresis Phenomena in Nanotube Field-Effect Transistors”, IEEE Transactions on Nanotechnology, 4 (2), 284-288 (2005). 15. Deyu Lu, Yan Li, Slava V. Rotkin, Umberto Ravaioli, and Klaus Schulten, “Finite-Size Effect and Wall Polarization in a Carbon Nanotube Channel”, Nano Lett 4, 2383-2387 (2004).16. Yan Li, Slava V. Rotkin, and Umberto Ravaioli, "Metal-Semiconductor Transition in Armchair Carbon Nanotubes by Symmetry Breaking", Applied Physics Lett 85, 4178 (2004). 17. Alexey G. Petrov, Slava V. Rotkin, "Transport in Nanotubes: Effect of Remote Impurity Scattering", Phys. Rev. B vol. 70 (3), 035408-1-10, 15 Jul 2004.18. Slava V. Rotkin, and Karl Hess, "Possibility of a Metallic Field-Effect Transistor", Applied Physics Letters vol. 84 (16), p.3139-3141, 19 April 2004.19. Slava V. Rotkin, Harry Ruda, Alexander Shik, "Field-effect transistor structures with a quasi-1D channel", International Journal of Nanoscience vol. 3 (1/2), 161-170, Feb 2004.20. Kirill A. Bulashevich, Slava V. Rotkin, Robert A. Suris, "Excitons in Single Wall Carbon Nanotubes", International Journal of Nanoscience vol. 2 (6), pp. 521-526, Dec 2003.21. Slava V. Rotkin, Harry Ruda, Alexander Shik, "Universal Description of Channel Conductivity for Nanotube and Nanowire Transistors", Applied Physics Letters 83, 1623, 2003.22. Alexey G. Petrov, Slava V. Rotkin, "Breaking of Nanotube Symmetry by Substrate Polarization", Nano Letters vol. 3, No.6, 701-705, 2003.23. Yan Li, Slava V. Rotkin, Umberto Ravaioli, "Electronic response and bandstructure modulation of carbon nanotubes in a transverse electrical field", Nano Letters 3, 183, 2003.24. Slava V. Rotkin, "Theory of Nanotube Nanodevices", in Nanostructured Materials and Coatings for Biomedical and Sensor Applications. Editors: Y.G. Gogotsi and Irina V.

Uvarova. Kluwer, pp. 257-277, 2003.25. Slava V. Rotkin, Vaishali Shrivastava, Kirill A. Bulashevich, and Narayan R. Aluru, "Atomistic Capacitance of a Nanotube Electromechanical Device", International Journal of

Nanoscience vol. 1, No. 3/4, 337-346, 2002.26. Slava V. Rotkin, Ilya Zharov, "Nanotube Light-Controlled Electronic Switch", International Journal of Nanoscience vol. 1, No. 3/4, 347-355, 2002.27. Kirill A. Bulashevich, Slava V. Rotkin, "Nanotube Devices: Microscopic Model", JETP Letters vol. 75 (4), 205-209, 2002.28. Slava V. Rotkin, Yuri Gogotsi, "Analysis of non-planar graphitic structures: from arched edge planes of graphite crystals to nanotubes", Materi. Res. Innovations, 5, 191, 2002.29. Marc Dequesnes, Slava V. Rotkin, Narayan R. Aluru, "Parameterization of continuum theories for single wall carbon nanotube switches by molecular dynamics simulations",

Journal of Computational Electronics 1 (3), 313-316, 2002.30. Slava V. Rotkin, Karl Hess, "Many-body terms in van der Waals cohesion energy of nanotubes", Journal of Computational Electronics 1 (3), 323-326, 2002.31. Marc Dequesnes, Slava V. Rotkin, Narayan R. Aluru, "Calculation of pull-in voltages for carbon nanotube-based nanoelectromechanical switches", Nanotechnology 13, 120, 2002.

List of publications used in this presentation:

downloadable from http://theory.physics.lehigh.edu/rotkin/text/pub-list.html

NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Silicon-Interface Scattering in...

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