Novel Scattering Mechanisms for

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Novel Scattering Mechanisms for Nanotube-Silicon
Devices
Slava V. Rotkin
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Surface Polariton in SiO2

• 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
  elt0

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

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Maxwell equations in free space
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Maxwell equations in free space are solved by
anzatz
E eikz
algebraic form of Maxwell equations in free space
E
q
H
surface requires that
all field components (but one) can be found from
BC
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"a" for air
E
q
H
"b" for bulk
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E
H
"a" for air
E
J
q
H
"b" for bulk
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last component of the field can be found only
with QM/QED
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Charge ScatteringShort Introduction
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Boltzmann Transport Equation (0)
Equilibrium distribution function is Fermi-Dirac
function
E
e.d.f. is symmetric and thus j 0
k
asymmetric non-e.d.f. provides j gt 0 (both in
ballistic and diffusive model)
Quantum-mechanical calculation of the
conductivity may be reduced to the Drude formula
electron velocity enters the formula
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Conductivity van Hove singularities
Scattering rate is proportional to the
velocity which diverges at the subband edge.
Thus, the Drude conductivity has peculiarities at
vHs.
Prof. T. Ando
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Boltzmann Transport Equation (1)
the current is where s is the conductivity
we obtain thus QM expression for s
where diffusivity is
For Fermi-Dirac degenerate e-gas f0 is step
function, thus
and conductivity reduces to the Drude formula
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Surface Phonon Polariton
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Introduction
q
j
qareanm2
channel heating due to Joule losses and low
thermal coupling to leads
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
Pioneering work by K. Hess and P. Vogl back to
1972 RIP-S in Si.
The mechanism appeared to be ineffective for Si
MOS-FETs and was almost forgotten for decades...
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Remote Polariton Scattering
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
wSO is the SPP frequency
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Surface 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 frequency of LO phonon
• surface wave with a strong decay of the electric
  field in the air appears and interacts with the
  NT charges

for vF108 cm/s and wSO150meV
e 105 V/cm
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Saturation Regime andHeat Dissipation Problem
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Introduction

• Scattering in 1D systems is weak due to
  restricted phase space available for the
  electron k -gt -k.
• However, scattering at high (drift) electric
  field is inevitable due to emission of an optical
  phonon. Which provides a fast relaxation
  mechanism for the hot electrons (and holes).
• Inelastic scattering rates have been calculated
  for SWNTs earlier

However, recent optics experiments showed faster
relaxation rates for the hot electron, which
suggests a new scattering mechanism.
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Introduction
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).
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Remote Phonon Polaron Energy
E(k), eV
19,0 NT RPS
k, 1/A
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Remote Polariton Scattering
T0K - therefore, only SO-phonon emission is
included
Dm0 - intra-subband transitions Dm1 -
inter-subband transitions (neglecting higher
m's) q1/l (forward) and q2k (backward)
scattering
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Remote Polariton Scattering

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

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Conductivity van Hove singularities
Scattering rate is proportional to the velocity
which diverges at the subband edge. Thus, the
Drude conductivity has peculiarities at vHs.
reminder
Prof. T. Ando
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Remote Polariton Scattering

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

inter-subband transitions are negligible due to
non-zero angular momentum transfer
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Remote Polariton Scattering
Scattering rate lifetime 30 meV No sharp
transition could happen Selfconsistent
calculation of the lifetime of the...
RP-polaron
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Surface Polariton Scattering (2)
To correct many-body picture the phonon
renormalization of the electron spectrum was
computed. As a result of Quantum Mechanical
calculation we obtain new scattering rate
scattering is averaged near the vHs but it is
still a fast process.
for vF108 cm/s and wSO140meV l40 nm 2ki
2p/a 1/nm
q1/l forward scattering q2k backward
scattering
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Remote SPP Scattering Rate

• for the SiO2 (quartz) substrate the RPS is
  likely prevailing over inelastic scattering by NT
  (own) optical phonons for the small distance to
  the polar substrate lt l 40 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

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Remote SPP Scattering Rate

• scattering rate increases with the electric
  field strength because of stronger warming of the
  electron distribution function

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Remote SPP Scattering

• IVCs with and without taking into account SPP
  mechanism
• The saturation regime
• is clearly seen at larger
• bias (larger field) for
• SPP scattering
• Inset mobility vs. field

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Remote SPP Scattering

• overheating of the channel neglecting the
  thermal sink in the leads
• where

• two scattering mechanisms SPP phonons take the
  heat directly into bulk substrate NT phonons
  warm the lattice but are inefficient

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

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Remote SPP Scattering

• ratio of "real"-to-expected losses for two tubes
  (R0.5 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)

• different temperature dependence for two
  scattering mechanisms