tensity for a particular excitation profile with a fixed position relative to the SWNT. This corresponds to the acquisition of the PL signal at one pixel in the experiment. An image is formed by raster scanning the sample, i.e. the SWNT, with respect to the exciting laser focus. The same can be done in the simulation. If the spatial distribution of the laser excitation is Iexc(x, y) and x0, y0 are the coordinates describing the offset between the ex-
citation spot and the SWNT center for each image pixel, the initial exciton distribution
σ(x) is given by
σ(x) =Iexc(x−x0,−y0). (6.19)
The PL intensity at each pixel is then
IP L(x0, y0) = QY ∆t τ X x Nss(x, x0, y0) (6.20)
with a different steady-state exciton distributionNss(x, x0, y0)for each x0, y0.
In the TENOM experiments, the tip strongly modifies the excitation but also the detection and the exciton decay. The enhanced excitation underneath the tip is included as an additional component in the intensity distribution Iexc(x, y). It is modeled as a narrow
Gaussian with a FWHM according to the tip diameter of 15−35nm, and an amplitude which is increased by a factor of f2 compared to the far-field excitation, according to the definition in section 3.3.2.
Iexctip(x, y) =Iexc(x, y) +f2exp −4 ln 2 x 2+y2 FWHM2 (6.21)
The detection and exciton decay is modified by the tip due to its influence on the decay rates krad and knr. Since the non-radiative rate is intrinsically very high for SWNTs, it
is not substantially increased by the tip. This is very different for single dye molecules with a high quantum yield, where significant quenching at nanometer tip-sample distances occurs [100, 148].
The radiative rate can be strongly increased underneath the tip and thus varies along the SWNT. The spatial dependence ofkradis modeled similar to the excitation. It is enhanced
by a factor of frad2 with a narrow Gaussian distribution underneath the tip.
krad(x, x0, y0) =k0rad 1 +frad2 exp −4 ln 2(x−x 0)2+y02 FWHM2 (6.22)
This spatial dependence leads to relative changes of τ = (krad+knr)−1 and QY =kradτ
which are calculated as a function ofx,x0 andy0. The detected PL intensity at each pixel including the tip enhanced rates in a TENOM experiment is then
IP L(x0, y0) = X x QY(x, x0, y0) ∆t τ(x, x0, y0)Nss(x, x 0 , y0). (6.23) 74
6.3. Numerical simulation of the TENOM imaging process
The SWNTs in the simulations are always straight and oriented along the x-direction. Their finite lengthl leads to 0≤x ≤l and the two SWNT ends are regarded as efficient quenching sites, i.e. N(0, t) = N(l, t) = 0 for all time steps. This quenching is due to localized states within the bandgap arising from dramatic band modifications related and confined to the ends [149–152]. The far-field excitationIexc(x, y)is given by the x-polarized
componentEx2 of the focused radially polarized donut mode as shown earlier in Fig. 4.2b and calculated according to ref. 8. Its amplitude is normalized to 1.
Fig. 6.1 shows an experimental near-field PL image for a 200 nm long SWNT and the respective simulated image. The excitation and radiative rate enhancement f2 and frad2
underneath the tip were both set to a moderate value of 4 to reproduce the experimen- tal image contrast with clearly visible far-field contributions. The two weak side-lobes correspond to the field components of the laser focus polarized along the nanotube axis, convoluted with a short luminescent nanotube. The tip diameter used for the calculation was taken as the FWHM of the near-field signal across the SWNT, here 20 nm. The near- field signal induced by the tip is shifted by about 70 nm from the center, because the tip was not ideally centered in the focus. Taking this into account for the excitation intensity distribution Iexctip(x, y), the image can be well reproduced.
Figure 6.1:a) Experimental and b) simulated near-field PL image of a short luminescent nanotube with 200 nm length. Due to the moderate field enhancement of this tip, the two side-lobes from the far-field background are clearly visible. This pattern arises from the polarization components parallel to the nanotube in the radially polarized laser focus (compare Fig. 4.2b). The near-field is slightly off-center, because the tip was not perfectly centered in the focus but shifted by about 70 nm. The experimental image can be well reproduced by the simulation, including the fade-out of PL towards the end which is a direct consequence of exciton mobility.
The fade-out of the near-field PL towards the ends is an indication of exciton mobility. A fraction of the excitons generated near the ends at a distance below their diffusion length can reach the ends and get quenched. This significantly reduces the detected PL intensity already for an excitation 50 nm away from the end. The gradual decrease towards the
6. EXCITON DIFFUSION - ANALYTICAL AND NUMERICAL TREATMENT
ends is apparent in the images and is in good agreement with the simulation based on a diffusion length of 100 nm.
As can be seen from the images in Fig. 6.1, signal enhancement factors derived from far-field to near-field ratios at a single sample position are not well defined. In particular for short 1D structures (<700 nm), the effects of far-field excitation pattern and exciton mobility strongly affect the local ratios. Simulating the imaging process allows to determine more reliable values.