# Theory of hot-carrier generation in bimetallic plasmonic catalysts

Hanwen Jin  Department of Materials, Imperial College London, South Kensington
Campus, London SW7 2AZ, UK  Matias Herran  Nanoinstitute Munich, Faculty of
Physics, Ludwigs-Maximilians-Universität München, 80539 Munich, Germany
Emiliano Cortés  Nanoinstitute Munich, Faculty of Physics, Ludwigs-
Maximilians-Universität München, 80539 Munich, Germany  Johannes Lischner
Department of Materials and the Thomas Young Centre for Theory and Simulation
of Materials, Imperial College London, South Kensington Campus, London SW7
2AZ, UK<EMAIL_ADDRESS>
###### Abstract

Bimetallic nanoreactors in which a plasmonic metal is used to funnel solar
energy towards a catalytic metal have recently been studied experimentally,
but a detailed theoretical understanding of these systems is lacking. Here, we
present theoretical results of hot-carrier generation rates of different Au-Pd
nanoarchitectures. In particular, we study spherical core-shell nanoparticles
with a Au core and a Pd shell as well as antenna-reactor systems consisting of
a large Au nanoparticle with acts as antenna and a smaller Pd satellite
nanoparticle separated by a gap. In addition, we investigate an antenna-
reactor system in which the satellite is a core-shell nanoparticle. Hot-
carrier generation rates are obtained from an atomistic quantum-mechanical
modelling technique which combines a solution of Maxwell’s equation with a
tight-binding description of the nanoparticle electronic structure. We find
that antenna-reactor systems exhibit significantly higher hot-carrier
generation rates in the catalytic material than the core-shell system as a
result of strong electric field enhancements associated with the gap between
the antenna and the satellite. For these systems, we also study the dependence
of hot-carrier generation rate on the size of the gap, the radius of the
antenna nanoparticle and the direction of light polarization. Our insights
pave the way towards a mechanistic understanding of hot-carrier generation in
heterogeneous nanostructures for photocatalysis and other energy conversion
applications.

##  1 Introduction

There is currently significant interest in harnessing energetic or ”hot”
electrons and holes generated in metallic nanoparticles for applications in
photocatalysis 1, 2, 3, 4, 5, photovoltaics 6, 7, 8 and sensing 10, 11, 12,
13, 14, 15, 16, 17. Metallic nanoparticles feature localized surface plasmons
(LSPs) which give rise to large light absorption cross sections 18. The LSP
has a short lifetime (typically on the order of tens of femtoseconds). Among
the various decay mechanisms, the Landau damping decay plays a prominent role
because it results in the generation of hot carriers 19, 20.

However, photocatalytic hot-carrier devices often have relatively low
efficiencies 21. A possible explanation for this is that standard plasmonic
materials, such as Au and Ag, are generally not good catalysts22. Therefore,
attempts have been made to combine plasmonic materials with catalytic
materials, such as Pt, Pd or Rh, into functional nanoarchitectures. Examples
of such heterostructures include Janus nanoparticles 9, core-shell systems 23,
24 or nanoparticle dimers and trimers 25. Recently, Herran and coworkers
studied different nanoarchitectures of Pd and Au, including core-shell
nanoparticles and antenna-reactor systems in which a large Au nanoparticle is
”decorated” with small Pd nanoparticles (or satellites), for the production of
H2 from formic acid 26. These authors found significant enhancements in H2
production upon illumination of the plasmo-catalyst with the largest increase
in chemical activity for antenna-reactor systems. Despite these advances,
however, there is still no detailed mechanistic understanding of the catalytic
activity in bimetallic nanoarchitectures27.

Insights into microscopic hot-carrier processes, including their generation,
thermalization and extraction, can be gained from theoretical modelling.
Atomistic first-principles techniques, such as ab initio time-dependent
density-functional theory, can be used to investigate hot-carrier processes in
very small nanostructures 28, but are challenging to apply to experimentally
relevant system sizes. On the other hand, non-atomistic approaches, such as
jellium or spherical well models, can be applied to large systems, but do not
capture important aspects, including d-band derived nanoparticle states or
facet-specific surface properties 40, 46, 41, 42, 43, 44, 45. To address this
challenge, Jin and coworkers recently developed a new approach that combines a
solution of Maxwell’s equation with large-scale atomistic tight-binding models
which enables the modelling of hot-carrier processes in nanoparticles
containing millions of atoms 39. So far, however, this approach has only been
applied to spherical nanoparticles.

In this paper, we use the method of Jin and coworkers 39 to study the
enhancement hot-carrier generation in a catalytic metal (Pd) induced by the
presence of a plasmonic metal (Au) in different bimetallic plasmo-catalytic
nanoarchitectures. In particular, we study Au@Pd core-shell nanoparticles and
antenna-reactor systems consisting of a large Au nanoparticle which acts as
antenna and a small satellite nanoparticle. The satellite is either a
spherical Pd nanoparticle or a Au@Pd core-shell nanoparticle. We compare our
results to hot-carrier generation rates in spherical Pd nanoparticles. We find
that the largest hot-carrier generation rates in the catalytic metal are found
in antenna-reactor systems, in particular in those where the satellite
nanoparticle is a core-shell system. This can be explained by the large
enhancement of the electric field arising from the strongly confined plasmon
mode associated with the gap between the antenna nanoparticle and the
satellite nanoparticle. We also explore the dependence of hot-carrier
generation rates on the light polarization, the size of the antenna
nanoparticle and the gap size between the antenna nanoparticle and the
satellite. The resulting insights pave the way towards a mechanistic design of
heterogeneous nanoarchitectures for energy conversion devices. The approach
can readily be applied to other materials.

##  2 Methods

###  2.1 Hot-carrier generation rates

We use the approach developed by Jin and coworkers 39 to calculate hot-carrier
generation rates in Pd-Au nanoarchitectures. In this method, the rate of hot
electrons $N_{e}(E,\omega)$ of energy $E$ excited by light of frequency
$\omega$ is obtained by evaluating Fermi’s golden rule according to 40

$N_{e}\left(E,\omega\right)=\frac{2}{V}\sum_{if}\Gamma_{if}\left(\omega\right)\delta\left(E-E_{f};\sigma\right),$
(1)

where $i$ and $f$ label the initial and final state with energy $E_{i}$ and
$E_{f}$, respectively. Also, $V$ is the volume of the nanoparticle and we
define
$\delta(x;\sigma)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp(-\frac{x^{2}}{2\sigma^{2}})$
with $\sigma=0.05$ eV being standard deviation of the Gaussian. In the above,
$\Gamma_{if}$ is given by

$\Gamma_{if}\left(\omega\right)=\frac{2\pi}{\hbar}\left|\bra{f}\hat{\Phi}_{tot}\left(\omega\right)\ket{i}\right|^{2}\delta(E_{f}-E_{i}-\hbar\omega;\gamma)f\left(E_{i}\right)\left(1-f\left(E_{f}\right)\right),$
(2)

where $f(E)$ is the Fermi-Dirac distribution with temperature $T=298$ K,
$\gamma=0.06$ eV is a broadening parameter and $\hat{\Phi}_{tot}(\omega)$
denotes the total potential inside the nanoparticle. This potential is
calculated using the quasistatic approximation 34, 35, 37, 36, 33, 38. In
particular, we use the finite element method as implemented in COMSOL ® 47 to
solve the Laplace equation

$\nabla\cdot(\epsilon(\mathbf{r},\omega)\nabla\Phi_{tot}(\mathbf{r},\omega))=0,$
(3)

where $\epsilon(\mathbf{r},\omega)$ is dielectric function of the material at
position $\mathbf{r}$. We use experimental dielectric functions for Au 48 and
Pd 48. In the calculations, we first specify the geometry of the
nanoarchitecture and the external potential
$\Phi_{ext}(\mathbf{r},\omega)=-\mathbf{E}_{0}\cdot\mathbf{r}$ with
$\mathbf{E}_{0}$ denoting the corresponding electric field and then solve
Laplace’s equation subject to the far-field condition that

$\lim_{|\mathbf{r}|\to\infty}\Phi_{tot}(\mathbf{r},\omega)=\Phi_{ext}(\mathbf{r},\omega).$
(4)

Once the total potential of the full Au-Pd nanoarchitecture is determined, we
evaluate the hot-carrier generation rate in the Pd subsystem (as only the hot
carriers in the Pd are catalytically active). Note that our approach does not
capture charge transfer processes between the Au and the Pd which can play an
important role in core-shell nanoparticles 30, 31. The electronic states of
the Pd subsystem are obtained using the tight-binding method. For this, we
first determine the atomic positions by carving the desired Pd shape (either a
spherical nanoparticle or a spherical nanoshell) from the bulk material. The
eigenstates of the Hamiltonian are expressed in terms of linear combinations
of atomic orbitals according to
$\ket{i}=\sum_{J,\alpha}C_{J,\alpha}\ket{J,\alpha}$, with $J$ labelling atoms
and $\alpha$ labelling orbitals. For each Pd atom, the basis consists of five
4d orbitals, one 5s orbital and three 5p orbitals. The hopping and onsite
energies of the Pd tight-binding model are taken from the ”Handbook of the
Band Structures of Elemental Solids” 29.

The matrix element in Eq. 1 is evaluated using 32

$\bra{I,\alpha}\hat{\Phi}_{tot}(\omega)\ket{J,\beta}=\Phi_{tot}(\mathbf{r}_{J},\omega)\delta_{I,J}\delta_{\alpha,\beta},$
(5)

where $\mathbf{r}_{J}$ denotes the position of atom $J$ and we have ignored
the transition dipole contribution to the matrix elements (which we have found
to be small in our previous calculations 39).

The total generation rate of hot carriers (both electrons and holes) per unit
volume is given by

$N_{tot}(\omega)=2\int_{E_{F}}^{\infty}N_{e}(E,\omega)\mathrm{d}E$ (6)

with $E_{F}$ denoting the Fermi energy. The hot hole generation rate
$N_{h}(E,\omega)$ is obtained by replacing $E_{f}$ by $E_{i}$ in Eq. 1.

##  3 Results

We have calculated hot-carrier generation rates in four systems, see Fig. 1:
(a) spherical Pd nanoparticles consisting of 3589 atoms (corresponding to a
diameter $D=4.42$ nm), (b) spherical Au@Pd core-shell nanoparticles with a Au
core (of diameter $D_{c}=4.90$ nm) and a Pd shell (of thickness 0.40 nm
containing 3740 Pd atoms), (c) a Au-Pd antenna-reactor architecture consisting
of a small spherical Pd nanoparticle of the same size as in (a) which is
separated by a small gap from a larger Au nanoparticle and (d) a Au-Au@Pd
antenna-reactor architecture in which the small nanoparticle has a core-shell
structure as in (b). The sizes of the Pd nanoparticle and of the Au@Pd core-
shell nanoparticle are similar to the satellite nanoparticles used in the
experiment of Herran and coworkers 26.

Figure 2(a) shows the evolution of the hot carrier generation rate for the
spherical Pd nanoparticle as function of photon energy. Both the hot hole and
the hot electron rates exhibit two peaks. The hot hole rate has a peak near
the Fermi level with the hot electron rate having a corresponding peak at
$\sim\hbar\omega$ above the Fermi level. The other peak of the hot hole rate
is at $\sim-\hbar\omega$ and the hot electron rate has a corresponding peak
just above the Fermi level. As the photon energy is varied, the peaks near the
Fermi level are pinned at their positions, but the other peaks move to higher
(in case of the hot electron rate) and lower (in case of the hot hole rate)
energies. This finding can be understood from an analysis of the electronic
structure of Pd. In particular, the band structure of Pd, see Fig. 1(e),
exhibits flat d-bands just above and below the Fermi energy. These flat bands
give rise to a large density of states which translates into a high hot
carrier generation rate near the Fermi level. The positions of the other peaks
are then determined by energy conservation, i.e. for each electron of energy
$E$ a corresponding hole of energy $E-\hbar\omega$ must be created. Finally,
we do not observe a dramatic enhancement of the hot-carrier rates at a
particular photon energy which reflects the absence of a strong LSP resonance
in spherical Pd nanoparticles.

e Figure 1: Schematic illustration of (a) a spherical Pd nanoparticle, (b) a
Au@Pd core-shell nanoparticle, (c) a Au-Pd antenna-reactor system and (d) a
Au-Au@Pd antenna-reactor system, (e) band structure of bulk fcc Pd, the zero
of energy is set to the Fermi level.

Figure 2(b) shows the dependence of the hot carrier generation rate of a Au@Pd
core-shell nanoparticle on the photon energy. Again, the rates exhibit peaks
near the Fermi level. Some differences in the shapes of the hot-carrier
generation rates can be observed compared to the spherical Pd nanoparticle:
these are caused by the increase of surface area of the thin Pd shell which
enhances the generation of hot carriers from intraband transitions 39. In the
core-shell system, an enhancement of the hot-carrier generation rate at a
photon frequency of 2.4 eV can be observed. This energy is close to the LSP
energy of a spherical Au nanoparticle. In other words, at this photon energy
the field enhancement of the Au core increases the hot-carrier generation in
the Pd shell 31.

ab Figure 2: Dependence of hot-carrier generation rate on photon energy for
(a) a spherial Pd nanoparticle with a diameter of $D=4.42$ nm and (b) a Au@Pd
core-shell nanoparticle with a core diameter of $D_{c}=4.90$ nm and a shell
thickness of 0.40 nm. Hot hole (electron) generation rates are red (blue). The
zero of energy is set to the Fermi level.

Next, we investigate hot-carrier generation in the Au-Pd antenna-reactor
system. Adding the Au nanoparticle to the spherical Pd nanoparticle lifts the
rotational symmetry of the Pd system. As a consequence, the hot-carrier
generation rate now depends on the polarization vector of the electric field.
We assume that both the centre of the Au and the centre of the Pd nanoparticle
lie on the z-axis, see Fig. 1, and describe the electric field through its
polar angle $\theta$. Fig. 3(a) shows the dependence of the hot carrier rate
on $\theta$. The hot-carrier generation rate is largest when
$\theta=0^{\circ}$ and then decreases as $\theta$ is increased. In particular,
we find that the hot-carrier generation rate is proportional to $\cos\theta$,
i.e. $N_{e}(\omega,E,\theta)=N_{e}(\omega,E,0)\cos\theta$. When $\theta=0$,
the charge carriers in the Au nanoparticle are pushed towards the Pd
nanoparticle creating a strongly enhanced electric field. The field is further
enhanced by the confinement effect of the small gap between between the Au and
the Pd nanoparticles giving rise to a so-called gap plasmon 49. We note that a
large number of satellites is attached to the Au nanoparticles in the
experiments of Herran and coworkers 26. Also, the light used in experiments is
not polarized. Therefore, the experimentally measured hot-carrier generation
rate is an average over $\theta$.

Figure 3 shows the same results for a Au-Au@Pd antenna-reactor system. Again,
the hot-carrier generation rate of the core-shell system has a different shape
than the pure Pd system, but it exhibits a similar dependence on the light
polarization.

ab Figure 3: Dependence of hot-carrier generation rate on light polarization
for (a) a Au-Pd antenna-reactor system with a diameter of 4.42 nm, (b) Au-
Au@Pd reactor system, with a core diameter of 4.9 nm and a total diameter of
5.7 nm. The diameter of the Au nanoparticle is 49 nm. The size of the gap
between the Pd and Au nanoparticles is 0.40 nm and the photon energy is is
2.28 eV for Au-Pd system and 2.20 for Au-Au@Pd system.

Next, we study the dependence of the hot-carrier generation rate of the two
antenna-reactor systems on the size of the gap $d$ between the Au nanoparticle
and the satellite, see Fig. 4. In practice, this parameter is determined by
the size of the ligands on the surface of the nanoparticles and is difficult
to control experimentally (in the experiment of Herran et al., the gap is
estimated to be approximately 1 nm 26). We find that the largest hot-carrier
rates are obtained for the systems with the smallest gaps. The insets show
that the total number of hot carriers decreases quickly as the gap size is
increased. For example, increasing the gap from $0.49$ nm to $1.6$ nm reduces
the total number of hot carriers by a factor of 0.38 (Au-Au@Pd) and 0.42 (Au-
Pd) and further increase to $d=3.2$ nm gives rise to an additional reduction
by a factor of 0.58 (Au-Au@Pd) and 0.63 (Au-Pd) in the total hot carrier
generation rate. This result demonstrates the importance of the confinement
effect associated with the gap between the Au and the satellite particles on
the strength of the electric field.

ab Figure 4: Dependence of the hot-carrier generation of Au-Pd antenna-reactor
systems on the size of the gap $d$ between the Au nanoparticle and the
satellite. (a) Results for a Pd satellite. (b) Results for a Au@Pd core-shell
satellite. The diameter of the Au antenna nanoparticle 49 nm, the photon
energy is 2.28 eV for Au-Pd system and 2.20 for Au-Au@Pd system and the
calculation is averaged over all polarization angles. The insets show the
total hot-carrier generation rate

Next, we study the dependence of the Pd hot-carrier generation rate of the
reactor systems on the size of the Au nanoparticle, see Fig. 5. We find that
the hot-carrier generation rate increases as the size of the Au nanoparticle
increases. Increasing the Au nanoparticle size while keeping the distance
between the Au nanoparticle and the satellite fixed reduces the volume
available in the gap between the two nanoparticles and thus enhances
confinement effects. This in turn leads to electric field enhancement. As the
Au nanoparticle size increases, the additional reduction of the gap volume
becomes less and less and only results in a small increase in hot carrier
production.

ab Figure 5: Dependence on the hot-carrier generation of Au-Pd antenna-reactor
systems on the size of the Au nanoparticle. (a) Results for a Pd satellite.
(b) Results for a Au@Pd core-shell satellite. The size of the gap is 0.40 nm,
the photon energy is 2.28 eV for Au-Pd system and 2.20 eV for Au-Au@Pd system
and the calculation was averaged over all polarization angles. The insets show
the total hot-carrier generation rates.

Finally, we study the dependence of the hot-carrier generation rates of the
reactor systems on the photon energy, see Fig. 6. For both systems, a dramatic
enhancements of the rates is observed at a photon energy of 2.24 eV compared
to the other photon energies. This photon energy is close to the LSP energy of
the Au nanoparticle (note that the presence of the satellite modifies the LSP
energy of the Au nanoparticle) so the increase of the hot-carrier generation
rates reflects the electric field enhancement caused by the plasmon mode. The
LSP acts as an optical lens which efficiently funnels energy towards the
catalytic material.

ab Figure 6: Dependence of hot-carrier generation rate on the photon energy
for (a) a Au-Pd antenna-reactor system with a diameter of 4.42 nm, (b) Au-
Au@Pd antenna-reactor system, with a core diameter of 4.9 nm and a total
diameter of 5.7 nm. The diameter of the Au nanoparticle is 49 nm. The size of
the gap between the Pd and Au nanoparticles is 0.40 nm, and the calculation is
averaged over all polarisation angles. For visual aid, the hot-carrier
generation rates at $\hbar\omega=2.80$ eV and 3.40 eV were multiplied by a
factor of 5.

We compare the total Pd hot-carrier generation rates of the 4 different
systems (spherical Pd nanoparticle, Au@Pd core-shell nanoparticle, Au-Pd
antenna-reactor system and Au-Au@Pd antenna-reactor system) in Fig. 7(a). It
can be seen that the two antenna-reactor systems produce significantly more
hot carriers in the Pd than the other two systems. In particular, a dramatic
increase in the generation rate is observed near the plasmon frequency of
$\sim 2.2$ eV. As discussed above, the large generation rates are a
consequence of the gap plasmon mode, which gives rise to large electric field
enhancements. In the antenna-reactor system with a core-shell satellite, the
electric field is more strongly confined because of the presence of the Au
inside the core-shell satellite and therefore this system gives rise to the
largest hot-carrier generation rate overall.

In their experiments, Herran and coworkers measure the increase in H2
production from formic acids by bimetallic plasmonic catalysts upon
illumination with a solar simulator (see Figure 2(a) of Ref. 26). In Fig. 7(b)
we show the measured difference in H2 production in the dark and upon
illumination for a Au@Pd core-shell nanoparticle, a Au-Pd antenna-satellite
system and a Au-Au@Pd system. We compare this difference to the total number
of hot holes $N_{sol}$ generated by the solar spectrum $S(\omega)$ (using the
Air Mass 1.5 spectrum) obtained via

$N_{sol}=\int d\omega S(\omega)N_{tot}(\omega).$ (7)

In excellent agreement with experiment, we find that the Au-Au@Pd antenna-
satellite system is the best bimetallic plasmonic photocatalyst while the
Au@Pd system performs much worse.

ab Figure 7: (a) Total hot carrier generation rate as a function of photon
energy (averaged over all polarisation vectors) for a spherical Pd
nanoparticle, a Au@Pd core-shell nanoparticle, a Au-Pd reactor and a Au-Au@Pd
reactor. The power (per unit volume) absorbed by the Pd subsystem is shown in
dashed lines. Note that the curves for the Pd nanoparticles and the Au@Pd have
been multiplied by a factor of 5 to improve visibility. (b) Comparison of the
measured difference in H2 production upon illumination by a solar simulator
and in the dark to the calculated number of hot holes excited by solar
illumination.

##  4 Conclusion

We have studied hot-carrier generation in Au-Pd nanoarchitectures using an
atomistic quantum-mechanical modelling approach. We have found that Au-Pd
antenna-reactor systems exhibit significantly higher hot-carrier generation
rates than core-shell nanoparticles. This is caused by the large electric
field enhancements due to the localized plasmon mode associated with the gap
between the antenna and the satellite nanoparticles. In particular, the
largest overall hot-carrier generation rate is found for an antenna-reactor
system in which the satellite is a core-shell nanoparticle. For the antenna-
reactor systems, we also studied the dependence of the hot-carrier generation
rates on the size of the gap, the radius of the antenna nanoparticle and the
light polarization direction. We find that the largest rates are found when
the electric field is parallel to the axis connecting the centres of the
antenna and satellite nanoparticles. Also, small gaps and large antenna sizes
favor hot-carrier generation. The insights from our work can guide the
development of highly efficient heterogeneous hot-carrier nanodevices for
energy conversion applications.

##  5 Acknowledgments

HJ acknowledges financial support from his parents. M.H. and E.C. acknowledge
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
e-conversion Germany´s Excellence Strategy – EXC 2089/1 – 390776260, the
Bavarian program Solar Energies Go Hybrid (SolTech), the Center for
NanoScience (CeNS) and the European Commission for the ERC-STG Catalight
(802989). JL acknowledges funding from the EPSRC programme grant EP/W017075/1.

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