Cathodoluminescence, or CL, emission has several fascinating properties that can be used for characterizing (nano) materials. In this regard, CL emission from a material is a dynamic process and therefore plenty of information can be obtained by analyzing its behavior in the time domain.
The g(2) function, eminent from quantum optics, serves as an extremely useful tool in this context as will be demonstrated in this article.
Second-Order Autocorrelation Function: g(2)(τ)
Photons in a light beam are distributed in time in a specific way. This distribution can be described by the normalized second-order autocorrelation function g(2)(τ), which denotes the probability to view a pair of photons spaced in time by a delay τ.
g(2)(0) is used to signify the amplitude of the g(2) function at zero delay, and the shape of the g(2) function strongly relies on the type of light source. For a fully coherent Poissonian source that has a single frequency like a laser, g(2)(τ) = 1, and for low-coherence light (also called chaotic light) arising from a discharge lamp, for instance, g(2)(0) = 2.
This effect is known as bunching since photons are grouped together in time. Typically, the bunching process for disordered light will only be visible at femtosecond timescales and is not easily observed in the majority of experiments due to restrictions in the time resolution.
As will be demonstrated in this article, bunching can be observed in CL experiments on longer timescales for varied reasons. On the other hand, a single-photon emitter, for example, a single quantum dot, also possesses a characteristic g(2). As the name indicates, a single-photon emitter can emit only a single photon at a time, thus making it impossible to view coincident photons.
Consequently, the g(2) function falls below 1 with g(2)(0) = 0, which is known as antibunching. This signature can be applied to detect and define such quantum sources of light. For more elaborate theoretical background on the g(2) function and (anti)bunching, refer to References 1 and 2. Figure 1 schematically shows the various characteristic g(2)(τ) curves described above.
Figure 1. Characteristic g(2) curves for different cases. The red balls represent a schematic photon distribution in time for that particular g(2) curve.
(Anti) Bunching in Cathodoluminescence
As mentioned, with regards to a single-photon emitter, the g(2) will exhibit an antibunching effect. This effect is also present, provided such a source is excited with electrons as first shown by Tizei et al. on NV0-center emission in diamond . Since CL has a high spatial resolution, it can be used to exactly localize, identify, and define single-emitters [3,4].
On the other hand, a number of extended solid-state systems can be considered as an ensemble of light emitters (a series of quantum wells or a bulk semiconductor for instance), and here, pronounced bunching can be observed [5–9] in the CL emission — an effect which is not present in photoluminescence. The high initial energy (≥1 keV) of the incoming primary electrons causes this bunching effect in CL. A single primary electron carries sufficient energy to produce multiple excitations in the material, usually resulting in the emission of more than a single photon for each electron. As a result, these photons are bunched together in time within a typical time-window established by the lifetime τe of the excited state. The bunching peak for a single exponential decay is described by:
This expression that contains the lifetime can be used for extracting lifetimes from bunching peaks, through numerical fitting [5, 6]. Moreover, multi-exponential decay can be taken along in this expression, even though decay-trace acquisitions are known to be more accurate for probing intricate decay dynamics (see References 7 and 9 for more information). A similar method can be used in antibunching curves. While the excited-state lifetime exclusively determines the bunching peak width, a number of several factors determine the peak amplitude g(2)(0) [5,7,8]:
1. The lifetime — if the lifetime is shorter, the bunching peak will be higher because the probability of detecting a pair of photons that are closely spaced in time increases.
2. The beam current — in the case of low-beam currents, the different photon bunches are well isolated in time, resulting in a higher possibility of detecting a pair of photons close in time. With regards to higher current, the photon bunches begin to overlap, resulting in a reduction in g(2)(0) and a higher number of uncorrelated events at longer delays. To put this in simpler terms, the normalized peak amplitude at zero delay (g(2)(0)) reduces because the baseline increases in the delay histogram in comparison to zero delay.
3. The excitation probability γ (fraction of interacting primary electrons, the value between 0 and 1) — the reasoning here is analogous to the 2nd point above. If the interaction occurs only in part of the electrons (γ < 1), the bunches become more isolated in time and therefore the effect is analogous to an effective reduction in beam current, resulting in an increased g(2)(0) amplitude.
Shown in Figure 2 is a calculation of the g(2) amplitude at zero delay as a function of γ and τe for a specified beam current of 51 pA. Here, the behavior described in points 1 and 3 is apparently clear. Since the beam current can be determined in the electron microscope and the lifetime can be exclusively extracted from the numerical fit, the excitation probability γ can be retrieved.
As this analysis is mainly based on the g(2) function, γ can be extracted without any a priori know-how on the sample’s composition or geometry. In complex (3D) geometries (near edges of nanostructures for instance), the excitation probability may considerably differ and therefore can have a major influence on the quantified CL maps. A simple means of measuring this effect is found in g(2) mapping. In addition, intensity variations in CL maps can be separated because of the electron excitation probability and also because of the intrinsic material variations .
The most standard method for determining the g(2)(τ) is in a Hanbury Brown and Twiss (HBT) interferometer that consists of a 50/50 beam splitter with a pair of ultrafast single-photon detectors (SPDs), for example, photomultiplier tubes (PMTs) or avalanche photodiodes (APDs). These SPDs are capable of producing one electrical TTL or NIM pulse for each detected photon, which is subsequently registered and timed by time-correlator electronics in a time-correlated single-photon scheme (TCSPC).
Figure 2. Amplitude of g(2) at zero delay (g(2)(0)) as a function of lifetime τe and excitation probability γ, for a beam current of 51 pA, calculated using a Monte Carlo calculation (see Refs. 5,7 for details on calculation). Figure courtesy of Dr Sophie Meuret (AMOLF, Amsterdam) .
When it comes to building a histogram of delay values, it is the time correlator that initiates the clock as soon as a photon arrives at one of the detectors, and halts when a photon arrives at the second detector, providing one count on the delay histogram. When this process is repeated many times, sufficient counts are gathered to achieve a good delay histogram. Since two separately detected photons are required to produce a single count on the delay histogram, acquisition of this histogram for a specified photon flux/beam current is quite slow when compared to the acquisition of a decay trace in which only one detector is employed .
g(2)(τ) is obtained by normalizing the quantified delay histogram with the average number of events recorded per time bin at long delay, which, in turn, relates with the Poissonian statistics of an uncorrelated electron beam. While a better signal-to-noise ratio is achieved for longer integration times, the intrinsic g(2) shape will not change unless the sample or beam position/conditions change at the time of acquisition. However, the jitter in the time-correlator and SPDs limits the time resolution that can be realized in the delay histogram. Based on the detectors used, this corresponds to approximately ~70–260 ps.
In the SPARC, the CL is relayed to the LAB Cube time-resolved detector unit with the help of the fiber coupling module. This detector unit includes suitable coupling optics, two SPDs, a beam splitter, and neutral density/color filters for controlling the intensity and analyzed wavelength band. Schematic illustration of this setup is shown in Figure 3.
Figure 3. Schematic overview of the Delmic LAB Cube system which can be used for g(2) acquisition. CL is coupled into an optical fiber using the fiber coupler module, which sends the light to the LAB Cube system containing a HBT system with a beam splitter (BS) and two ultrafast SPDs. The events recorded by the detectors are read out and turned into delay values by the time correlator. Motorized neutral density and color filters are present for intensity and color filtering, respectively. Bunching g(2) curve courtesy of Dr Sophie Meuret (AMOLF, Amsterdam) .
A major advantage of the g(2) imaging technique is that it can be used to gather information on dynamics both with a pulsed and continuous electron beam. In the latter case, a conventional SEM set to normal beam conditions can be employed and as a result no modifications to the SEM are required. In addition, since the LAB Cube is combined with the SPARC spectral system via the fiber module, it can be easily retrofitted or installed on any SPARC spectral without any further alternations. Therefore, you can easily switch from g(2) detection to other CL imaging modes, for example, lifetime (requires a pulsed electron microscope), hyperspectral, and angle-resolved CL imaging [7,8].
As stated above, the g(2) imaging technique can be used to detect and define single-photon emitters at the nanoscale. As such, it can be used in the context of major studies focused on quantum systems and their interaction with electron beams. There is also a potential for industrially oriented use, where the g(2) imaging technique is applied as a metrology tool for investigating nanoscale quantum-optical devices.
On the other hand, the bunching effect is quite useful for inspecting a host of bulk and mesoscopic materials systems including optoelectronic instruments, for example, solar cells and LEDs. An example of a g(2) measurement on a range of GaN nanorods with InGaN quantum wells inside is shown in Figure 4 . In this example, the electron beam is raster-scanned over the array and a g(2) curve is obtained for each position.
The bunching effect is clearly visible in (b), where g(2)(0) is obviously larger than 1. For each scanning pixel, the bunching peak is applied to extract the excitation probability γ and local lifetime τe. This data can be directly applied to infer and measure electron beam interaction with such an intricate 3D semiconductor structure, and at the same time, it also gives a better understanding of the material’s homogeneity and quality.
Figure 4. Example of g(2) mapping data on InGaN/GaN nanorods shown in the SEM image in (a). (b) The g(2) data recorded at three colored squares as indicated by the arrows in panel (c) which shows SE intensity recorded together with the g(2) data set. Maps of (d) lifetime τe, (e) amplitude g(2)(0)-1, and (f) the probability of excitation γ are also shown. If the data was too noisy to extract these parameters, the pixel was left white in the map. The contours of the nanorods are indicated by the black lines. Figure courtesy of Dr. Sophie Meuret (AMOLF, Amsterdam) (see Ref.  for details).
 M. Fox. Quantum Optics: An Introduction. Oxford Master Series in Physics, (2006).
 Z. Ficek and S. Swain, Quantum Interference and Coherence: Theory and Experiments, Springer Science (2005).
 L. H. G. Tizei and M. Kociak, Phys. Rev. Lett. 110, 153604 (2013).
 R. Bourrellier, S. Meuret, A. Tararan, O. Stéphan, M. Kociak, L. H. G. Tizei, and A. Zobelli, Nano Lett. 16, 4317-4321 (2016).
 S. Meuret, L. H. G. Tizei, T. Cazimajou, R. Bourrellier, H. C. Chang, F. Treussart, and M. Kociak, Phys. Rev. Lett. 114, 197401 (2015).
 S. Meuret, L. H. G. Tizei, T. Auzelle, R. Songmuang, B. Daudin, B. Gayral, and M. Kociak, ACS Photonics 3, 1157-1163 (2016).
 S. Meuret, T. Coenen, H. Zeijlemaker, M. Latzel, S. Christiansen, S. Conesa-Boj, and A. Polman, Phys. Rev. B 96, 035308 (2017).
 S. Meuret, T. Coenen, S. Y. Woo, Y.-H. Ra, Z. Mi, and A. Polman, Nano Lett. 18, 2288-2293 (2018).
 S. Meuret, M. Solà-Garcia, T. Coenen, E. Kieft, H. Zeijlemaker, M. Latzel, S. Christiansen, S. Y. Woo, Y-H. Ra, Z. Mi and A. Polman (submitted) (2018).
This information has been sourced, reviewed and adapted from materials provided by Delmic B.V.
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