Optoelectronics with Carbon Nanotubes
Optoelectronics with Carbon Nanotubes Optoelectronics with Carbon Nanotubes
Figure III-10. (Main) EL intensity spectra at VDS of 9 V (lowest intensity) to 15 V (highest intensity) in 1 V steps (empty symbols) as a function of energy. Solid lines are fit functions that are linear combinations of Lorentz and black-body distributions. The dotted line is the Lorentz function for the 15 V spectrum without the blackbody. (Inset) “Blackbody” temperatures as a function of applied electrical power (black circles). The blue and red dotted lines are the calculated effective temperatures of Gphonons and RBM phonons respectively, as in Ref. 115. Tube heterogeneity and phonon scattering described so far (i.e., at least ~70 meV of the total 190 meV width at P = 40 W m -1 ) explain the constant floor value of the broadening, but not its increase as a function of applied bias. The increase due to acoustic phonons seems to be modest at best, as explained above; contribution from merging with the broader optical phonon peak is also expected to be small because of the limited weight transfer to this side peak. Other possible broadening mechanisms that increase with applied bias are the effect of longitudinal electric field, the electronic temperature, and shortened lifetime of excitons by exciton-exciton annihilation. Perebeinos et al. theoretically investigated the effects of longitudinal electric field on the absorption spectrum of SWNTs 124 . Using their formalism, we first calculate the broadening from lifetime shortening by exciton ionization due to an external field. We estimate the size of 55
the effect following Perebeinos’ formalism for the exciton dissociation rate as a function of electric field F (motivated by the solution to the hydrogen atom in an electric field), where 0 3/ 2 1/ 2 1.74 b exc / F0 F0 ( F) 4.1Ebexp (Eq. III.4) F F F E m e , and mexc is a reduced exciton mass (mexc -1 = me -1 + mh -1 ). Here we set me = mh, from the symmetrical band structure in single-particle theory. We use Eb = 0.23 eV according to Ref. 18, and mexc = Δ/(2 υF 2 ) where υF is the Fermi velocity (~10 6 m/s), and Δ (~0.42 eV/d) is half of the single particle bandgap energy, yielding the F0 value of 108 V/μm. Using the value of VDS from Figure III-4 and assuming that most of the voltage drop occurs at the Schottky barriers whose length into the channel is in the order of the substrate thickness 125 (100 nm), we obtain Γ values of 54 meV to 171 meV for fields ranging from 25 to 40 V/μm, respectively. Figure III-11 compares the calculated broadening due to the field ionization with the experimental data to show that this effect can account for the change in width. The broadening calculated is actually 12 % greater than the experimental data. However, there is quite a bit of uncertainty in the length of the Schottky barriers, exciton binding energy, and whether the fields are equally distributed between the source and the drain. In our best estimate, we can conclude that the external field is a significant contribution to the spectral broadening. When the recombination region is not subjected to very high fields, as in the case of ambipolar devices or p-n junction devices, an external electric field can mix the wave functions of exciton and the interband continuum states (i.e., free-carrier recombination) and still broaden the spectrum. The wave mixing also transfers the oscillator strength from the first excitonic state to the free-carrier state so that the latter grows as much as 400 % in strength at just 1/3 of the critical field for full ionization, according to the simulation by Perebeinos et al 124 . This type of broadening will be discussed further in the chapter on p-n junction spectra. 56
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Figure III-10. (Main) EL intensity spectra at VDS of 9 V (lowest intensity) to 15 V<br />
(highest intensity) in 1 V steps (empty symbols) as a function of energy. Solid lines<br />
are fit functions that are linear combinations of Lorentz and black-body distributions.<br />
The dotted line is the Lorentz function for the 15 V spectrum <strong>with</strong>out the blackbody.<br />
(Inset) “Blackbody” temperatures as a function of applied electrical power (black<br />
circles). The blue and red dotted lines are the calculated effective temperatures of Gphonons<br />
and RBM phonons respectively, as in Ref. 115.<br />
Tube heterogeneity and phonon scattering described so far (i.e., at least ~70 meV of the<br />
total 190 meV width at P = 40 W m -1 ) explain the constant floor value of the broadening, but not<br />
its increase as a function of applied bias. The increase due to acoustic phonons seems to be<br />
modest at best, as explained above; contribution from merging <strong>with</strong> the broader optical phonon<br />
peak is also expected to be small because of the limited weight transfer to this side peak. Other<br />
possible broadening mechanisms that increase <strong>with</strong> applied bias are the effect of longitudinal<br />
electric field, the electronic temperature, and shortened lifetime of excitons by exciton-exciton<br />
annihilation.<br />
Perebeinos et al. theoretically investigated the effects of longitudinal electric field on the<br />
absorption spectrum of SWNTs 124 . Using their formalism, we first calculate the broadening<br />
from lifetime shortening by exciton ionization due to an external field. We estimate the size of<br />
55
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