These relaxation steps are shown in Figure 8. On the left, the SPP decay causes the excitation of a single electron-hole pair – the first generation with energy *E*_{1,}_{n} and $\sum _{n=1}^{2}{E}_{1,n}^{\text{\hspace{0.17em}}}}=\hslash \omega $ (assuming that the hole energies are counted down from the Fermi level). Then, either an electron or a hole scatters off the electron residing below the Fermi level, thus creating another electron-hole pair. Once both an electron and a hole scatter once (which, on average, should take time *τ*_{ee}), there are three second-generation holes and electrons each with energies *E*_{2,}_{n} and $\sum _{n=1}^{6}{E}_{2,n}^{\text{\hspace{0.17em}}}}=\hslash \omega .$ Thereafter, on average, another time interval *τ*_{ee} elapses (this interval may be longer than the original as the EE scattering is energy dependent), each of the second-generation carriers engenders three third-generation carriers with energies *E*_{3,}_{n} and $\sum _{n=1}^{18}{E}_{2,n}^{\text{\hspace{0.17em}}}}=\hslash \omega .$ The process continues until the average energy of the *M*-th generation, ${\u3008{E}_{M,n}\u3009}_{n}=\hslash \omega /2\cdot {3}^{M-1},$ becomes comparable to *k*_{B}T, so that there is no distinction between the “hot” and “cool” carriers; therefore, it takes roughly time

Figure 8: Quantum picture of carrier generation and relaxation in metal nanoparticles.

(A) An SPP is excited on a nanoparticle. (B) An SPP has decayed, engendering a primary (first-generation) electron-hole pair. (C) Each of the first-generation carriers decays into three second-generation carriers. (D) Each of the second-generation carriers decays into three third-generation carriers.

$${\tau}_{e,cool}=\mathrm{(}M-1\mathrm{)}{\tau}_{ee}\approx {\mathrm{log}}_{3}\mathrm{(}\hslash \omega /2{k}_{B}T\mathrm{)}{\tau}_{ee}$$(12)

for the electrons to cool down to some kind of equilibrium between themselves. For SPP energies <1.5 eV, it takes no more than three scattering events to cool down the electrons; hence, *τ*_{e}_{,cool}<4*τ*_{ee}.

Thus, while the cool-down time is of the same order of magnitude as the EE scattering time, it is definitely larger than it by a factor of a few. Therefore, the ubiquitous statement stubbornly permeating the literature that a single scattering event is sufficient to establish the equilibrium of the electrons [31], [32], [36], [37], [42] is incorrect. Obviously, during this time interval, there will be electron-phonon scattering events, because, remember, that *τ*_{ee} and *τ*_{ep} are roughly of the same order of magnitude. However, these events cause insignificant loss of energy for each hot carrier and thus can be safely disregarded.

Let us now consider the distribution of the second- through fourth-generation hot carriers in energy space. When the electron of the first-generation hot carriers with energy distribution *f*_{1}(*E*)=*δ*(*E*−*E*_{1}) decays into three new second-generation carriers, their distribution is

$${f}_{2}\mathrm{(}E,\text{\hspace{0.17em}}{E}_{1}\mathrm{)}=2\mathrm{(}{E}_{1}-E\mathrm{)}/{E}_{1}^{2}.$$(13)

Then, these carriers decay into nine third-generation carriers, whose distribution can be found as

$$\begin{array}{l}{f}_{3}\mathrm{(}E,\text{\hspace{0.17em}}{E}_{1}\mathrm{)}={\displaystyle \underset{{E}_{}}{\overset{{E}_{1}}{\int}}{f}_{2}\mathrm{(}{E}_{2},\text{\hspace{0.17em}}{E}_{1}\mathrm{)}{f}_{2}\mathrm{(}E,\text{\hspace{0.17em}}{E}_{2}\mathrm{)}\text{\hspace{0.17em}}\text{d}{E}_{2}^{}}\\ \text{}=\frac{4}{{E}_{1}^{2}}\left[\mathrm{(}{E}_{1}+E\mathrm{)}\mathrm{log}\frac{{E}_{1}}{{E}_{3}}-2\mathrm{(}{E}_{1}-E\mathrm{)}\right],\end{array}$$(14)

and then into 27 fourth-generation carriers, with a distribution of

$$\begin{array}{l}{f}_{4}\mathrm{(}E,\text{\hspace{0.17em}}{E}_{1}\mathrm{)}={\displaystyle \underset{E4}{\overset{{E}_{1}}{\int}}{f}_{3}\mathrm{(}{E}_{3},\text{\hspace{0.17em}}{E}_{1}\mathrm{)}{f}_{2}\mathrm{(}E,\text{\hspace{0.17em}}{E}_{3}\mathrm{)}\text{\hspace{0.17em}}\text{d}{E}_{3}^{}}\\ \text{}=\frac{4}{{E}_{1}^{2}}\left[\mathrm{(}{E}_{1}-{E}_{4}\mathrm{)}\left({\mathrm{log}}^{2}\frac{{E}_{}}{{E}_{4}}+12\right)-6\mathrm{(}{E}_{1}+{E}_{4}\mathrm{)}\mathrm{log}\frac{{E}_{}}{{E}_{4}}\right].\end{array}$$(15)

Note that even though the functions *f*_{m} for *m*>2 diverge near zero energy, they are all perfectly integrable to $\underset{0}{\overset{{E}_{1}}{\int}}{f}_{m}}\mathrm{(}E,\text{\hspace{0.17em}}{E}_{1}\mathrm{)}\text{\hspace{0.17em}}\text{d}E=1.$ The distributions of the total number of carriers in each generation, *N*_{m}(*E*, *E*_{1})=3^{m−1}*f*_{m}(*E*, *E*_{1}), are shown in Figure 9, with energies *E*_{m} scaled relative to energy *E*_{1}. As one can see, the distribution quickly shifts to lower energies; however, when plotted on log scale in Figure 9B, the curves are not linear and, therefore, one cannot ascribe a single electron temperature *T*_{e} to the carriers.

Figure 9: Energy distribution of the second through fourth generation of carriers generated as a result of decay of a single first-generation carrier with energy *E*_{1}.

(A) linear scale; (B) logarithmic scale.

Next, we determine the distribution of all carriers generated by photons with energy *ℏ**ω* as

$${f}_{m}\mathrm{(}E,\text{\hspace{0.17em}}\hslash \omega \mathrm{)}={\displaystyle \underset{{E}_{m}}{\overset{\hslash \omega}{\int}}{f}_{m}\mathrm{(}E,\text{\hspace{0.17em}}{E}_{1}\mathrm{)}\text{\hspace{0.17em}}\text{d}{E}_{1}}$$(16)

and obtain (assuming the original distribution associated with phonon, defect-assisted, or LD process)

$$\begin{array}{l}{f}_{1}\mathrm{(}E,\text{\hspace{0.17em}}\hslash \omega \mathrm{)}={F}_{hot,ph}\mathrm{(}E\mathrm{)}=\frac{1}{\hslash \omega};\text{\hspace{0.17em}}E\le \hslash \omega \\ {f}_{2}\mathrm{(}E,\text{\hspace{0.17em}}\hslash \omega \mathrm{)}=\frac{2}{\hslash \omega}\left(\frac{E}{\hslash \omega}-1-\mathrm{log}\frac{E}{\hslash \omega}\right)\\ {f}_{3}\mathrm{(}E,\text{\hspace{0.17em}}\hslash \omega \mathrm{)}=\frac{2}{\hslash \omega}\left({\mathrm{log}}^{2}\left(\frac{E}{\hslash \omega}\right)+2\mathrm{log}\frac{E}{\hslash \omega}\left(\frac{E}{\hslash \omega}+2\right)+6\left(1-\frac{E}{\hslash \omega}\right)\right)\\ {f}_{4}\mathrm{(}E,\text{\hspace{0.17em}}\hslash \omega \mathrm{)}=\frac{4}{\hslash \omega}\left(-\frac{1}{3}{\mathrm{log}}^{3}\left(\frac{E}{\hslash \omega}\right)+{\mathrm{log}}^{2}\left(\frac{E}{\hslash \omega}\right)\left(\frac{E}{\hslash \omega}-3\right)-4\mathrm{log}\frac{E}{\hslash \omega}\left(2\frac{E}{\hslash \omega}+3\right)-20\left(1-\frac{E}{\hslash \omega}\right)\right).\end{array}$$(17)

The carrier number distributions *N*_{m}(*E*, *ℏ**ω*)=3^{m−1}*f*_{m}(*E*, *ℏ**ω*) for the first four generations of carriers are plotted in Figure 10A and B. As one can see, within roughly time *τ*_{e,cool}~3*τ*_{ee}, the distribution changes dramatically and in fact resembles the distribution one would expect if one used the classical Drude model in which absorption light generates many low-energy carriers via “friction”; however, it is important that in the quantum picture, this does not happen instantly, and hot carriers may depart the metal before they decay. Also, even for the fourth generation of carriers, one cannot introduce equilibrium temperature *T*_{e}, as evident from the Figure 10B where the negative slope of the distribution increases at higher energies, indicating a reduced number of high-energy carriers capable of surpassing the energy barrier.

Figure 10: Energy distribution of the first through fourth generation of carriers generated as a result of decay of a single SPP with energy *ħω*.

(A) linear scale; (B) logarithmic scale. Note that the distribution cannot be defined by a single electron temperature *T*_{e}.

Let us now estimate the chances for the hot carriers of each generation to overcome a potential barrier Φ. Two cases will be considered. In the first case, we assume that the transverse momentum is conserved and the efficiency of carrier extraction is [6], [7]

$${\eta}_{ext,m}^{\mathrm{(}+\mathrm{)}}\mathrm{(}\Phi ,\text{\hspace{0.17em}}\omega \mathrm{)}~\frac{{m}_{s}}{2{m}_{m}{E}_{F}}{\displaystyle \underset{\Phi}{\overset{\hslash \omega}{\int}}\mathrm{(}E-\Phi \mathrm{)}{N}_{m}\mathrm{(}E,\text{\hspace{0.17em}}\hslash \omega \mathrm{)}\text{\hspace{0.17em}}\text{d}E},$$(18)

where *m*_{s} and *m*_{m} are the effective masses of metal. The results are shown in Figure 11A (without the term in front of the integral, as we are only interested in the relative strength of the injection of carriers from different generations). As one can see, the probability of extraction decreases dramatically in each generation for the barrier height that is at least 30% of the photon energy.

Figure 11: Relative extraction efficiency of the primary and secondary generations of non-equilibrium carriers of different generations as a function of barrier height.

(A) momentum conservation is enforced; (B) momentum conservation is violated; (C) impact of the secondary carriers.

For the second case, we assume that the momentum conservation rules are fully relaxed, and, therefore, all we need is to evaluate the total number of the carriers with energy above the barrier [33]

$${\eta}_{ext,m}^{\mathrm{(}-\mathrm{)}}\mathrm{(}\Phi ,\text{\hspace{0.17em}}\omega \mathrm{)}~C{\displaystyle \underset{\Phi}{\overset{\hslash \omega}{\int}}{N}_{m}}\mathrm{(}E,\text{\hspace{0.17em}}\hslash \omega \mathrm{)}\text{\hspace{0.17em}}\text{d}E,$$(19)

where *C* is the band structure-dependent factor to be derived in the next section; however, at this point, we are interested only in the relative impact of the secondary electrons and holes. The results are shown in Figure 11B and are similar to the case of complete momentum conservation, although the secondary carriers become important for the barriers that are less than half of the photon energy. We summarize the impact of extraction of secondary electrons in Figure 11C, where we plot the function ${K}_{\text{sec}}^{\text{\hspace{0.17em}}}={\displaystyle \sum _{m=1}^{4}{\eta}_{ext,m}/{\eta}_{ext,1}}.$ As one can see, for the barrier that is at least half as large as the phonon energy, the impact of the secondary electrons is negligible, no matter what model we assume.

For the IR detectors, which are one of the more promising hot carrier applications, it is desirable to have the barrier height relatively high to reduce thermal noise, and, as shown in Ref. [77], maximum detectivity is achieved at *ℏ**ω*−Φ≈4*k*_{B}T<<*ℏ**ω*. Hence, the impact of secondary carriers can be completely neglected – for all practical purposes, once a single EE scattering event takes place, the carriers are no longer capable of overcoming the barrier. For other cases where the barrier is relatively low, one can simply use the semi-empirical expression to modify the time it takes the primary (first-generation) carriers to decay to the point where they no longer overcome the barrier, as ${\tau}_{ee}^{\text{eff}}={K}_{\text{sec}}{\tau}_{ee},$ where *K*_{sec} is typically <2.

These results are also relevant to the carriers generated via EE-assisted and interband absorption. The holes generated via interband absorption in the d-shell can decay into two holes and one electron, all in the s-band, where they can move relatively fast, and some of those carriers may have energy sufficient to exit across the barrier. The energy distribution of these second-generation carriers is similar to that shown in Figure 9 for *E*_{1}=*E*_{ds}. As long as the barrier is close to the photon energy, one can completely neglect the injection of the intraband-absorption-generated carriers; otherwise, one can simply add their relatively small contribution to *K*_{sec}. At any rate, once interband absorption commences, the *Q* of the SPP mode decreases and so does the field enhancement, thus negating the whole goal of plasmonic-assisted detection or catalysis. Similarly, judging from Figure 4, one cannot expect a large contribution from the carriers generated with the help of EE scattering. Once again, that contribution can definitely be ignored for IR light; for visible light, the contribution can also be incorporated into *K*_{sec}.

Thus, to conclude this section, we state that for all practical purposes, only the primary (first-generation) carriers generated with the phonon/defect assistance or via LD are the ones that can find their way out of the metal. Once these carriers undergo a single EE scattering event, their energies will, for the most part, be way too small to overcome the barrier on the metal/semiconductor (dielectric) interface. We shall refer to these carriers as “quasi-ballistic,” as they are expected to propagate quasi-ballistically (phonon and defect scattering does not reduce energy significantly) toward the interface and then get ejected across the barrier. The distribution into which the secondary electrons created as result of EE scattering eventually settle cannot be characterized by a single electron temperature *T*_{e}, and, practical values of the incident light intensity never contribute to the injection into the semiconductor over a reasonably high barrier. It is harder to speculate whether this conclusion also holds for the process of photocatalysis on the surface of the metal, as these processes are not yet entirely understood. Still, for a reasonably high activation energy, it seems that only the quasi-ballistic carriers have sufficient energy to initiate the chemical reaction.

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