Optoelectronics with Carbon Nanotubes
Optoelectronics with Carbon Nanotubes Optoelectronics with Carbon Nanotubes
Yoshikawa et al. subsequently obtained the proportionality constant for many SWNTs in a PL experiment and found that this value is diameter-dependent 120 . They fit the FWHM values to the equation B FWHM 20 AT (Eq. III.2) exp( / kT) 1 where A and B are the coupling constants to low-energy and high-energy phonons, respectively. For the temperature range they used (≤ 300 K), it was found that B is negligible (i.e., same as the finding by Lefebvre above) and that A depends inversely on the tube diameter. They attribute the contribution to the widths in their data to low-energy longitudinal acoustic (LA) and twisting acoustic (TA) modes and rule out RBM phonons because of their higher energies. However, in our case the diameter range is at least 50 % larger, and the RBM phonon of the d = 1.5 nm SWNT has the RBM energy at the Γ point of only about 20 meV (calculated from the diameter- Raman energy relationship), well below room temperature. We do not have an estimate of the coupling constant B, but given the “average” temperature between 650 K and 750 K (see the following discussion) which is significantly above 20 meV, we proceed with the assumption that RBM contribution is similar to that of lower energy phonons and contributes linearly with power to the width. It is reasonable to assume that in Lefebvre and Yoshikawa’s work, the phonon modes were in thermal equilibrium, and that the temperature dependence was dominated by acoustic phonon scattering (ħωac « kBT at room T, which is the highest temperature used in their experiments). High energy phonon such as optical phonons (~180 meV) are not populated at room temperature (see Figure I-6 of the phonon dispersion relation in Introduction). Now we calculate the effective temperatures of acoustic phonons and apply the linear relationship to estimate their effect on broadening. Steiner et al. measured effective temperatures of phonons under electrical bias and found that different phonon modes are not at an equilibrium temperature because of decay bottlenecks, and that each phonon mode’s temperature was proportional to the applied electrical power 112 . They give the phenomenological expression T T P / g (Eq. III.3) i sub i 51 B
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). 52
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Yoshikawa et al. subsequently obtained the proportionality constant for many SWNTs in<br />
a PL experiment and found that this value is diameter-dependent 120 . They fit the FWHM values<br />
to the equation<br />
B<br />
FWHM 20<br />
AT <br />
(Eq. III.2)<br />
exp( <br />
/ kT)<br />
1<br />
where A and B are the coupling constants to low-energy and high-energy phonons, respectively.<br />
For the temperature range they used (≤ 300 K), it was found that B is negligible (i.e., same as the<br />
finding by Lefebvre above) and that A depends inversely on the tube diameter. They attribute<br />
the contribution to the widths in their data to low-energy longitudinal acoustic (LA) and twisting<br />
acoustic (TA) modes and rule out RBM phonons because of their higher energies. However, in<br />
our case the diameter range is at least 50 % larger, and the RBM phonon of the d = 1.5 nm<br />
SWNT has the RBM energy at the Γ point of only about 20 meV (calculated from the diameter-<br />
Raman energy relationship), well below room temperature. We do not have an estimate of the<br />
coupling constant B, but given the “average” temperature between 650 K and 750 K (see the<br />
following discussion) which is significantly above 20 meV, we proceed <strong>with</strong> the assumption that<br />
RBM contribution is similar to that of lower energy phonons and contributes linearly <strong>with</strong> power<br />
to the width.<br />
It is reasonable to assume that in Lefebvre and Yoshikawa’s work, the phonon modes<br />
were in thermal equilibrium, and that the temperature dependence was dominated by acoustic<br />
phonon scattering (ħωac « kBT at room T, which is the highest temperature used in their<br />
experiments). High energy phonon such as optical phonons (~180 meV) are not populated at<br />
room temperature (see Figure I-6 of the phonon dispersion relation in Introduction). Now we<br />
calculate the effective temperatures of acoustic phonons and apply the linear relationship to<br />
estimate their effect on broadening.<br />
Steiner et al. measured effective temperatures of phonons under electrical bias and found<br />
that different phonon modes are not at an equilibrium temperature because of decay bottlenecks,<br />
and that each phonon mode’s temperature was proportional to the applied electrical power 112 .<br />
They give the phenomenological expression<br />
T T P / g (Eq. III.3)<br />
i sub i<br />
51<br />
B
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