1. Novel Scattering Mechanisms for Nanotube-Silicon Devices
Slava V. Rotkin

5. 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 e<0
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

7. Maxwell equations in free space

8. 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:

9. "a" for air E q H
in non-retarded limit
the frequency of SPP mode
is found from
"b" for bulk

10. E H
"a" for air J E q H
in non-retarded limit
the frequency of SPP mode
is found from
"b" for bulk

11. last component of the field can be found only with QM/QED

12. Charge Scattering: Short Introduction
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13. 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 > 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

14. 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

15. Surface Phonon Polariton
_____________

16. Introduction q j Vd
q~area~nm2
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...

17. 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

18. 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 vF~108 cm/s and wSO~150meV :
e ~ 105 V/cm

19. Saturation Regime and Heat Dissipation Problem
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20. Introduction
Scattering in 1D systems is weak due to restricted phase space available for the electron: k -> -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.

21. 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).

22. Remote Polariton Scattering
T=0K - therefore, only SO-phonon emission is included
Dm=0 - intra-subband transitions
Dm=1 - inter-subband transitions (neglecting higher m's)
q~1/l (forward) and q~2k (backward) scattering

23. Remote Polariton Scattering
RPS rate varies for intra-subband and inter-subband scattering
RPS has maximum at the van Hove singularities (for semiconductor-SWNT)

24. 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

25. 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

26. Remote Polariton Scattering
0.02
0.04
0.06
0.08
0.1
0.65
0.75
0.8
0.85
E(k), eV
k, 1/A
Scattering rate = lifetime ~ 30 meV
No sharp transition could happen
Selfconsistent calculation of the lifetime of the...
RP-polaron

27. 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.
q~1/l forward scattering
q~2k : backward scattering
for vF~108 cm/s
and wSO~140meV : l~40 nm
2ki ~ 2p/a ~ 1/nm

28. 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 < 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

29. Remote SPP Scattering Rate
scattering rate increases with the electric field strength because of stronger warming of the electron distribution function

30. 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

31. 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

32. Remote SPP Scattering
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)
different temperature dependence for two scattering mechanisms