Optoelectronics with Carbon Nanotubes

where i is the phonon mode, Tsub is the ambient (substrate) temperature, the applied
electrical power per unit length is P = IVDS/L (L is the channel length) and g is a parameter that
depends on the phonon dynamics and the nanotube-substrate interaction. From a linear fit to
their data, Steiner’s group obtained the value gRBM = 0.11 W m -1 K -1 for RBM phonons. Note
that this is a phenomenological value and therefore includes all the mechanisms that determine
the phonon temperature. Using this value and Tsub = 300K, and applying it to Equation III.2 with
an A value of 0.018 meV K -1 (extrapolated from the data by Yoshikawa et al. 120 ), we obtain the
widths of 13 meV at VDS = -5 V up to 45 meV at VDS = -8 V, or 7 % to 15 % of the total width,
respectively.
Note that “temperature” in this case is derived from the occupation number of a phonon
mode as measured by the anti-Stokes to Stokes Raman intensity ratio. Since temperatures of
different phonon modes are not at thermal equilibrium, there is no well defined lattice
temperature at a given power, but this is a more intuitive and useful construct when discussing
the energy exchange between hot carriers and phonons.
The acoustic phonon temperatures calculated as indicated above yield values ranging
from 710 K to 2500 K, the higher end of which is unreasonably high for carbon nanotubes. In
electrically-driven light emission experiments with suspended metallic or quasi-metallic SWNTs,
Mann et al. extracted the hot optical phonon temperature of up to 1200 K in the negative
differential conductance (NDC) regime 87 . Our devices are on the substrate and NDC behavior is
never observed, so they should have a more efficient heat sink. Since hot optical phonon is the
dominant scattering mechanism as evidenced by the NDC of their device, it can serve as the
ceiling for the acoustic phonon temperature, meaning that it is very unlikely that our acoustic
phonon temperatures are as high as 2500 K.
There are also indications that Equation III.2 significantly overestimates the acoustic
phonon temperature under very high bias. In the study on phonon populations by Steiner et al.,
the high end of the power range was 40 W m -1 , and the linear fit to the RBM phonon temperature
works only below 16 W m -1 . The temperature for electrical powers above 25 W m -1 shows a
sign of saturation (Figure III-9).
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