Heterogeneous carbon-based devices: Towards integration with Si technology Slava V. Rotkin Physics...

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Heterogeneous carbon-based devices: Towards integration with Si technology

Slava V. Rotkin

Physics Department & Center for Advanced Materials and Nanotechnology, Lehigh University

USC, May 28, 2009 Slava V Rotkin

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

Motivation: NT array Thin Film Transistors (TFT)- Charge coupling: Classical and Quantum terms- When "dice" has only "6" face

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-Array Thin NT-Array Thin Film TransistorsFilm Transistors

Courtesy Prof. John Rogers (UIUC)Courtesy Prof. John Rogers (UIUC)

NT aligned array : Novel type TFTNT aligned array : Novel type TFT

Novel fabrication technique (Left) allows fabrication of Thin-Film Transistors of parallel NT arrays.

Courtesy Prof. John Rogers (UIUC)Courtesy Prof. John Rogers (UIUC)

X-cut [2-1-10]Z-cut [0001]Y-cut [01-10]

SEM reconstruction ("fake" 3D view) of NT-TFT and gold electrodes.

SEM of NT growth on different quartz facets NTs can be transferred on plastic

Aligned NT for Transparent ElectronicsAligned NT for Transparent Electronics

NTs are so small that absorption in a single layer of well- separated

tubes is negligible

Adapted from Zhou (2008)

Aligned NT GrowthAligned NT Growth

Courtesy J.A. RogersNT alignment is not independent of the gas flow direction:

competition of gas flow and surfacealignment = serpentine growth

Criss-crossed NT arrays

NT transistor as an element of a FM-radio

Physics of NT Field-Effect Transistor (FET):

• NT channel is conducting at Vg=0 (non-intentional p-doping)

• long mean free path (due to 1D symmetry)

• optical phonons limit the high field current

• work function of the electrodes defines the height of the contact Schottky barrier

Single NT FETSingle NT FET

insulatorinsulatorgate @ Vgate @ Vgg

source @ groundsource @ ground drain @ Vdrain @ Vdd

1D channel1D channel

• Gate voltage (charge of the gate electrode and "its vicinity") controls the transport

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

Nanotube Quantum CapacitanceNanotube Quantum Capacitance

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.

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

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

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

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

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

25 50 75 100 125 150

2.42.62.8

3.23.4

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

C/Cclass

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

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

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

electron velocity enters the formula

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:

on average isproportional to

Statistical averaging over a random impurity distribution of

scattering form-factor

DoSstrength of potential

Saturation Regime andHeat Dissipation ProblemSaturation Regime andHeat Dissipation Problem

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:

However, recent optics experiments indicated that the relaxation rates for hot electrons are even faster, which suggests a possibility for a new unknown scattering mechanism.

Saturation Regime: Heat GenerationSaturation Regime: Heat Generation

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

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-S in Si.

q~area~nm2

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

jHeat Generation (2)Heat Generation (2)

Surface Phonon PolaritonSurface Phonon Polariton

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 in polar dielectrics:

• due to the dielectric function difference between the substrate and the air, a surface e.m.w. could exist

• dielectric function of the polar insulator has a singularity at the LO phonon frequency

• surface wave with a strong decay of the electric field in the air appears and interacts with the NT charges

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

Maxwell equations in free space

Maxwell equations in free space are solved by anzatz

algebraic form of Maxwell equations in free space

surface requires that:

additional materials connection:

Maxwell equations in free space

"b" for bulk

"a" for air

all field components (but one) can be found from BC:

frequency of the SPP provides consistency of BC:

Remote Polariton ScatteringRemote Polariton Scattering

Estimates for SiO2-quartz:

• electric field in the air is proportional to decay constant, determined from MEq+BC, 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

Interaction potential (e-dipole)

where the (dipole) polarization is calculated following Mahan et al.

here q is the SPP wavenumber; x is normal to the surface

F is related to Froehlich constant:and SO is the SPP frequency

Details of SPP scattering in SiO2Details 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.

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

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 nm

2ki ~ 2/a ~ 1/nm

Forward scattering dominates:

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

• for the SiO2 (quartz) substrate the SPP scattering is likely prevailing over inelastic scattering by NT (own) optical phonons for the small distance to the polar substrate < ~ 4 nm;• the effect is even stronger for high-k dielectrics due to increase of the Froehlich constant : x20 and more;• the effect is independent of the radius of the NT, thus for narrow NTs it will dominate over the other 1/R mechanisms

Surface Polariton Scattering RateSurface Polariton Scattering Rate

ConclusionsConclusions

• Theory of NT scattering is not complete yet

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

• Hot electron scattering due to SPP modes provides a new and very effective thermo-conductivity mechanism

• Graphenes – another example of nano-hetero-interface where quantum effects may nicely develop into effects useful for applications

overheating of the channel : neglecting the thermal sink in the leads (~nm2)

Remote SPP ScatteringRemote SPP Scattering

• two scattering mechanisms : • NT phonons warm the NT lattice but are inefficient

• SPP phonons take the heat directly into bulk substrate;

• Joule losses - IsF are for the total energy loss; while NT phonons take only a small fraction of that

• different temperature 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)