Silicon Nanocrystals
2.5 Photoluminescence Time Dynamics
Figure 2.7. The depth-resolved differential peak photoluminescence intensity (drawn in red) suggests that the optically active nanocrystals are distributed in a region that is narrow in comparison to the implantation profile.
given by Ledoux [55]. Note that the spectra do not show a shift to shorter wavelengths until after the large nanocrystals at the peak of the distribution have been removed. The dashed line is drawn to suggest the actual depth distribution for nanocrystal diameters which may be symmetrical.
This optical characterization technique could easily be used in a comprehensive experi- ment correlating implantation or annealing conditions to the nanocrystal depth distribution or applied in the study of nanocrystals fabricated in thin films by other methods. Improve- ments that could be made include the use of ultraviolet ellipsometry that might allow the silicon concentration to be detected or the use of improved surface chemistry to limit pos- sible oxide regrowth after the staircase etching procedure.
Figure 2.8. The peak wavelength for photoluminescence as a function of remaining ox- ide thickness suggests a depth distribution for the average nanocrystal size created by ion implantation. The wavelength shifts after the peak of the differential photoluminescence in- tensity (red curve) is etched away. The dashed curve shows the nanocrystal size distribution suggested by the wavelength data [55].
contrast to much faster photoluminescence decay rates associated with oxide defects states (∼10–100 MHz) and can provide strong supporting evidence when attempting to identify nanocrystal luminescence against a background signal. At much slower time scales, it is possible to observe “blinking” effects in single nanocrystal measurements, in which the nanocrystal switches between dark and emitting states [115]. The blinking phenomena is believed to be caused by the intermittent trapping of an electron or hole from a photoexcited nanocrystal within the surrounding oxide matrix. The charge remaining behind completely quenches photoluminescence through Auger recombination until the trapped carrier returns to the nanocrystal [110]. The time scale for Auger recombination is assumed to be several orders of magnitude faster than radiative recombination, but to our knowledge has not been quantitatively measured in silicon nanocrystals [119]. Under intense pulsed excitation in dense nanocrystal ensembles, it is additionally possible to observe stimulated emission processes on nanosecond time scales in silicon nanocrystals [66–69].
μ
τ 37 μ β 0.7
Figure 2.9. A typical time resolved photoluminescence measurement for silicon nanocrystals recorded at an emission wavelength of 750 nm. The pump beam (488 nm; ∼100 W/cm2) is abruptly turned off in∼10 ns using an acousto-optic modulator. The excited population of nanocrystals decays with a characteristic microsecond scale time constant.
Figure 2.9 shows a typical example of a time-resolved photoluminescence measurement for silicon nanocrystals formed by ion implantation. The data correspond to a sample in which a 100 nm thick oxide layer was implanted at 5 keV to a total implanted fluence of 1.2 × 1016 ions/cm2, corresponding to a 20% atomic excess Si concentration at the peak of the implanted ion distribution. This sample was annealed in a tube furnace at Caltech once to form nanocrystals (1100 ◦C; 10 min; 2000 ppm O2:Ar) and again to passivate implantation damage (450◦C; 30 min; 10% H2:N2). The data clearly show the characteristic microsecond time scale for photoluminescence decay in silicon nanocrystals. The drawn line is the least-squares best fit to the “stretched exponential” or Kohlrausch decay equation,
I(t) =I0e−(τt)β,
where the photoluminescence intensityI(t) decays from an initial value I0,τ is the 1/eex- perimental decay time, and the dimensionless parameterβis an ideality factor that accounts
for deviation from the single exponential decay curves (β = 1) that would be expected for an isolated two-level optical system. The stretched exponential decay function is nearly universally used to report photoluminescence decay measurements for silicon nanocrystal ensembles. Our data are well fit by a stretched exponential with β = 0.7, a typical value for ion implanted silicon nanocrystals.
Despite the widespread use of the stretched exponential function in reports of time- resolved silicon nanocrystal photoluminescence decay, the underlying physical mechanism is not well understood. Measurements of samples with varying nanocrystal density strongly suggest that concentration effects involving internanocrystal interactions are responsible for the “stretched” decay behavior [27, 100]. The β parameter tends to be closest to unity in sparse ensembles at low emission energies (long wavelengths) that correspond to large nanocrystals [109]. This suggests that the shape of the decay is related to energy transfer relaxation processes, which require the nanocrystals to be in close proximity and are most pronounced for small nanocrystals that are “uphill” in the energy landscape of the ensemble.
However, a consensus model for this phenomenon has not yet emerged.
Recently it has been proposed that the decay should be expressed as a distribution of single exponential decay components [120]. This decomposition procedure is essentially equivalent to transforming the decay data from the time domain into the Laplace domain and adds little in the way of new physical insight by itself. However, one can propose that energy transfer causes the deviation from single exponential decay by introducing a distribution of additive decay paths to the ensemble. In this case, the distribution of single exponential decay is caused by the distribution of energy transfer rates. The average rate found in the calculated distribution should be interpreted as the most likely total decay rate, rather than be associated directly with the radiative decay component. Mathematically, the exact transform of the stretched exponential must include decay rate components that are arbitrarily slow, and the average decay rate is undefined [121]. This underscores the importance of considering the stretched exponential decay as a phenomenological description of silicon nanocrystal photoluminescence.
We would like to be able to determine the radiative rate for spontaneous emission from our ensemble decay measurement. The radiative rate effectively sets a lower bound on the decay dynamics because any additional decay pathways can only increase the overall decay rate. Therefore the radiative rate for isolated nanocrystals must be identified with the
rate of an ensemble measurement, we are forced to model the energy transfer process and predict the shape of the distribution for energy transfer rates. This is problematic because the energy transfer process is not well understood, and many different models correspond to the available data.
Some insight into the difficulty of modeling the energy transfer contribution to photolu- minescence decay in dense ensembles can be gained by considering the simple rate equation that describes a two level optical system:
d
dtN∗(t) =σ·φ·(Ntotal−N∗(t))−Γdecay·N∗(t), (2.1) where N∗(t) and Ntotal are the excited state and total nanocrystal populations, σ is the cross section for excitation,φis the incident pump photon flux, Γdecayis the total relaxation rate, typically taken to be the sum of radiative and nonradiative contributions. In order to model nonsingle exponential decay we must add physically meaningful complexity to this equation. It is reasonable to adjust the nonradiative decay component to reflect a concentration dependent energy transfer process:
Γdecay(N∗) = Γradiative+ Γnonradiative+ Υ (Ntotal−N∗),
where Υ describes a energy transfer decay component that is explicitly dependent on the population of ground state nanocrystals and the time dependence of the excited state na- nocrystal population has been suppressed for clarity. Applying this modification to equa- tion (2.1) immediately results in complicated nonlinear system dynamics.
Following the method of Inokuti and Hirayama [122] we can introduce a simplifying assumption that the energy transfer decay component is independent of the excited state population and instead assume that each arrangement of nanocrystals in the ensemble can
be described independently:
{Γdecay}i = Γradiative + Γnonradiative+ Υ (Ntotal)i,
leading to an expression for the decay of each individual nanocrystal arrangement from steady state when the pump is turned off:
{N∗(t)}i = N∗steady−state,i·e−t(Γradiative+Γnonradiative+Υ(Ntotal)i)
= N∗steady−state,i·e−t(Γradiative+Γnonradiative)·e−tΥ(Ntotal)i,
where the linear decay components have been separated from the energy transfer term. In our experiments we measure the ensemble average decay curve:
N∗(t)=N∗steady−state ·e−t(Γradiative+Γnonradiative)·
e−tΥ(Ntotal)i
,
in which the ensemble average energy transfer decay component
e−tΥ(Ntotal)i
must be evaluated by assuming a particular model for the energy transfer rate as a function of nano- crystal separation distance, Γ (r), and the distribution of nanocrystal separation distances in the sample. By assuming that the nanocrystal locations are random and uncorrelated and that the energy transfer rate is a decreasing function of separation distance we find:
e−tΥ(Ntotal)i
= lim
Vsphere(r)=4π3 r3→∞
4π V
r
0
e−tΓ(r)r2dr.
This expression can be evaluated numerically for several different energy transfer mod- els, including the exponentially varying exchange mechanism described by Dexter [123], dipole–dipole interaction described by F¨orster [124], and higher-order electromagnetic cou- plings (dipole-quadrupole, quadrupole-quadrupole, etc.) [122]. The decay dynamics are sufficiently rich to fit silicon nanocrystal photoluminescence decay curves that are more commonly reported in terms of the stretched exponential function.
The important result of this model is the prediction of a slowest decay rate that we can associate with the radiative emission of an isolated nanocrystal. Stretched exponential decay curves withβ = 0.7 can be fit using several different expressions for the transfer rate Γ (r). For the case of dipole-dipole energy transfer (Γ (r)∝1/r6)), we find that our data are
fit to Inokuti’s model yields an ensemble rate that is ∼3.6 times faster than the suggested radiative recombination rate. Energy transfer occurs at a critical length scale of ∼3.8 nm and remains an∼1μs process even for closely spaced nanocrystals. These model predictions match well with experimental evidence suggesting that the stretched exponential decay rate for a well passivated dense ensemble is ∼3 times faster than the single exponential decay rate observed at the same wavelength in sparse nanocrystal samples [27, 100].
This model has many apparent weaknesses and the decay dynamics for silicon nanocrys- tals in ensembles should be considered an open problem. The assumed uniform separation distance probability distribution fails to properly consider the characteristic separation dis- tances that should arise during the nanocrystal formation anneal as a consequence of the Ostwald ripening process. The size distribution of the silicon nanocrystals in the ensem- ble is ignored, as is the dependence of the energy transfer rate on the size of the donor and acceptor nanocrystals. The probability that the acceptor nanocrystal is already in the excited state is not included, although this should be less important in the low pump power regime. Perhaps most importantly, the model predicts that energy transfer is more likely than radiative decay, but fails to properly account for energy transfer as an excitation mechanism.