Generation of plasmonic hot carriers from d-bands in metallic nanoparticles

We present an approach to master the well-known challenge of calculating the contribution of dbands to plasmon-induced hot carrier rates in metallic nanoparticles. We generalize the widely used spherical well model for the nanoparticle wavefunctions to flat d-bands using the envelope function technique. Using Fermi’s golden rule, we calculate the generation rates of hot carriers after the decay of the plasmon due to transitions from either a d-band state to an s-band state or from an s-band state to another s-band state. We apply this formalism to spherical silver nanoparticles of a wide range of sizes (from 5 nm to 65 nm) and find that d-to-s and s-to-s transitions have contributions of the same order of magnitude near the plasmon energy. For small nanoparticles with radii less than 65 nanometers s-band state to s-band state transitions are larger while for larger nanoparticles, d-band state to s-band state transitions give the greater contribution.


INTRODUCTION
There is currently significant interest in the properties of plasmon-induced hot carriers in metallic nanostructures. Such nanostructures absorb sunlight via generating localized surface plasmons (LSPs) which subsequently decay into electron-hole pairs via the Landau damping mechanism. The resulting energetic carriers can be harnessed in new applications for sensing [1,2], catalysis [3,4] or solar energy conversion [5][6][7][8].
Most experiments employ traditional plasmonic metals, such as Ag or Au, as these materials exhibit strong plasmonic resonances in their absorption spectrum. The electronic structure of these materials is characterized by a dispersive band of mixed s-and p-state character (despite its mixed character we will refer to this band as the s-band for simplicity) which crosses the Fermi level and multiple occupied dispersive d-bands with a comparably flat dispersion [9,10]. While the d-bands are quite far from the Fermi level for the case of Ag, they are much closer in the cases of Au and Cu and it is possible to excite electrons from the d-bands into s-band states above the Fermi level [11]. However, it is not clear how important this process is compared to excitations within the s-band.
Theoretical modelling of hot electron processes allows valuable insights into experimental observations. To describe the electronic structure of noble metal nanoparticles, several groups have solved the Schrödinger equation of electrons in a spherical well [12][13][14][15]. While this approach allows the description of experimentally relevant nanoparticles with radii of the order of several tens of nanometers, it does not capture the contribution of dbands. On the other hand, first-principles methods, such as density-functional theory (DFT), allow the description of d-bands [10,16], but can only be applied to very small nanoparticles (typically only a few hundred atoms) [17].
In this paper, we present an approach that bridges atomistic and continuum electronic structure theories of metallic nanoparticles via the envelope function tech-nique [18] and allows the description of nanoparticle states derived from d-bands in metallic nanoparticles with large radii. We apply this approach to silver nanoparticles and show that s-band to s-band transitions give the dominate contribution to hot carrier rates for nanoparticles with radii less than approximately 65 nanometers. However, as the rate of d-band state to sband state transitions increases more quickly with the nanoparticle radius than the rate of s-band state to sband state transitions, we predict that d-to-s transitions dominate for larger nanoparticles.

Hot carrier generation rates
We employ the framework of Dal Forno, Ranno and Lischner to calculate hot carrier generation rates in spherical nanoparticles [13]. In this approach, the total number of plasmon-induced hot electrons generated in a nanoparticle illuminated by light (polarized along the z-direction) of frequency ω is determined using Fermi's golden rule according to where M if = ψ f |Φ pl (ω)|ψ i denotes the matrix element of the total potential Φ pl which is often calculated using the quasistatic approximation. In this approximation, the potential inside the spherical nanoparticle is Φ pl (ω) = −eE 0 (ω)−1 (ω)+2 z, where e denotes the electron charge, E 0 is the strength of the externally applied electric field and (ω) denotes the bulk dielectric function of the material. For the latter, we use a Drude model where γ P denotes the plasmon linewidth, ω 0 is the bulk plasmon frequency and b is the dielectric constant due arXiv:1912.00460v1 [cond-mat.mtrl-sci] 1 Dec 2019 to the background screening by the polarizable d-bands. Also, ψ i and ψ f denote quasiparticle wavefunctions of occupied and empty states with energies E i and E f , respectively. Note that we have neglected effects arising from finite quasiparticle lifetimes in Eq. (1) which give rise to anti-resonant contributions [13]. Such transitions only give rise to low-energy carriers and therefore do not play an important role in applications, such as photodetection and photocatalysis. A factor of two arising from spin is taken into account.

Envelope function method
We calculate the electronic wavefunctions of the nanopartice using the envelope function method originally developed by Kohn and Luttinger to describe the electronic structure of charged defects in semiconductors [18,19]. In this approach, one assumes that the electronic structure of the defect-free material is known, i.e. that the eigenstates φ nk and eigenvalues nk (with n and k denoting the band index and the crystal momentum, respectively) of the crystal Hamiltonian H crys have been determined by solving the Schrödinger equation Nowadays, this task can be carried out routinely using first-principles methods, such as density-functional theory or the more advanced GW approach. Next, a perturbation δV is considered which breaks the discrete translational invariance of the infinite crystal. For charged defects, the perturbation is the screened Coulomb potential induced by the defect. For a nanoparticle, this is the spherical well potential which confines the electrons inside the nanoparticle. The eigenstates ψ (with corresponding eigenenergies E) of the perturbed Hamiltonian H crys +δV can be constructed as linear combinations of the crystal states according to with c nk being complex coefficients. Inserting this ansatz into the Schrödinger equation and multiplying from the right with φ * n k results in a matrix equation for the coefficients. Assuming that the perturbation does not mix different bands leads to Inserting the Fourier transform c nk = d 3 rc n (r) exp(−ik · r) yields np c n (r) + δV (r)c n (r) = Ec n (r), with p = −i ∇ denoting the momentum operator.
For a band with a parabolic dispersion, i.e. nk = 2 k 2 /(2m * ) (with m * denoting the effective mass of the band), the equation for the envelope function c n (r) has the same form as the Schrödinger equation for an electron in the potential δV . This demonstrates how the spherical well approximation for a nanoparticle can be derived from an atomistic model.
Often, an additional approximation is invoked for the wavefunctions [19]. Specifically, it is often found that the linear combination in Eq. (4) is dominated by states near a specific crystal momentum k 0 . For semiconductors, this is typically the crystal momentum corresponding to the valence and conduction band extrema. In this case, the integral over k can be carried out analytically and one finds where u nk0 (r) is a lattice-periodic function. Assuming that c n (r) is normalized, normalization of ψ n requires that For metals, it is much less clear that this simplified form for ψ is justified and which crystal momentum k 0 should be chosen. However, this form is highly advantageous for analytical treatment and will be adopted in this work. Future work will aim to assess the accuracy of this approximation.

Electronic states of spherical nanoparticles
For a parabolic band and a perturbation with spherical symmetry, we can express the kinetic energy operator in Eq. (6) in spherical coordinates and use a separation of variable ansatz for c n (r, θ, φ). The angular part of the envelope function is described by the spherical harmonics Y lm (where l and m denote the orbital and magnetic quantum numbers, respectively) and the radial part R νl (with ν denoting the principal quantum number) is the solution of (9) Analytic solutions of Eq. (9) exist for the infinite square well potential, i.e. a potential that is zero inside the nanoparticle and infinite outside. In this case, the radial solutions are proportional to the spherical Bessel functions of the first kind, i.e. R νl (r) ∝ j l (k νl r) with corresponding eigenenergies E νl = 2 k 2 νl /(2m * ). The allowed values of k νl are determined by the boundary condition R νl (r 0 ) = 0 with r 0 denoting the radius of the nanoparticle. For large nanoparticles, we can approximate j l (x) by sin(x − lπ/2)/x and find with ν ≥ 1 and l ≥ 0 being integers. For later use, we also calculate the corresponding angular momentum resolved density of states, g l ( ) = ν δ( − E νl ) for a parabolic band and find with E min l = 2 π 2 (l + 2) 2 /(8m * n r 2 0 ) denoting the smallest energy for a fixed value of l. Here, we have assumed that the spacing between energy levels is sufficiently small that the sum over ν can be replaced by an integral.

D-band states in spherical nanoparticles
The envelope function method is not limited to parabolic dispersion relations. For any dispersion relation nk a corresponding equation for the envelope function can be derived by replacing k by −i ∇, see Eq. (6). In practice, however, it is difficult to carry out this procedure because the dependence of the band structure on crystal momentum is not known analytically. As a first step towards understanding the role of d-bands in hot carrier generation, we invoke a drastic approximation and consider the limit of a perfectly flat band, i.e. nk = d . For this dispersion, it is straightforward to solve the envelope function equation for an infinite square well. In fact, the solutions of the parabolic case are also solutions of the flat band case, but all have the same eigenvalue d .
As all d-band states have the same energy and there are in principle infinitely many solutions to the spherical well equation, some regularization procedure is required to obtain meaningful results. This is achieved by imposing that the envelope function should not oscillate faster in the radial direction than the spacing between atoms. This leads to the condition that only solutions with quantum numbers ν < ν max (l) = r0 a0 − l 2 are allowed where and a 0 is the lattice constant of the crystal. The maximum allowed value of l is then determined by the condition that ν ≥ 0, i.e. l max = 2r 0 /a 0 . Note that our final results for the hot carrier rates do not depend on a 0 . No regularization procedure is needed for a parabolic band as envelope functions with unphysically fast oscillations have very high energies and do not influence our results.

Matrix elements
Having determined the quasiparticle energies and wavefunctions, we are now in a position to evaluate Eq. (1). We replace the summations over intial and final states by sums over the quantum numbers ν, l and m for each occupied and empty band. In the following, we will restrict ourselves to the case of a single occupied flat bands and a single parabolic band that crosses the Fermi level. Using the Eq. (6) for the nanoparticle wavefunctions, the matrix elements are given by = −eE(ω) d 3 rc * nνlm (r)u * nk0 (r)zc n ν l m (r)u n k0 (r) (13) with E(ω) = E 0 ( (ω) − 1)/( (ω) + 2). Splitting the integral into a sum over all unit cells in the nanoparticle and integrals over each unit cell yields where the orthonormality of the u n 's and the c n 's was used. Here, z crys nn = d 3 ru * nk0 (r)zu n k0 (r)/V uc (where the prime on the integral sign indicates integration over the unit cell with volume V uc ) denotes the transition dipole moment of the vertical transition between the bands n and n in the bulk material and z env nνν ll mm = d 3 rc * nνlm (r)zc nν l m (r) = [A lm δ mm δ l+1,l + B lm δ mm δ l−1,l ]z env νlν l .
Here, we defined and also [20,21] z env The last equality is valid for large nanoparticles [20,22]. The structure of Eq. (14) is interesting: for transitions between nanoparticle states originating from the same band, the matrix element is determined entirely by the envelope functions and the well-known angular momentum selection rules are recovered. In contrast, only the microscopic transition dipole moment determines the strength of inter-band transitions.
To evaluate z crys , an expression for u nk0 (r) is needed. In principle, this can be obtained straightforwardly from a first-principles band structure calculation, but it is not a priori clear how k 0 should be chosen. To make progress, we choose k 0 as the Γ-point of the first Brillouin zone, but interpret u nΓ as an effective function that represents the average behaviour of the band n in the whole Brillouin zone. Using a tight-binding form, we use the following ansatz for the s-band (20) where φ s and φ p denote atomic s-and p-orbitals and R is a lattice vector. The coefficients α p and α s describe the relative contributions of s-and p-states and can be determined from a Mulliken analysis of the Bloch states obtained from a first-principles calculations as explained below. Note that the true u nΓ obtained from a band structure calculation would only have s-contributions by symmetry.
If a similar tight-binding ansatz is used for the d-band (but without any admixture of non-d states), we find z crys ds = α p z at dp (21) with z at dp denoting the dipole moment for an atomic d-to-p transition which can be obtained from a first-principles calculation. Note that the atomic dipole selection rule forbids transitions from d-states to s-states.

Final expressions for the hot carrier rates in spherical nanoparticles
The total number of hot electrons N (ω) can be written as a sum of d-band state to s-band state transitions N d→s and s-band state to s-band state transitions N s→s (there are no transitions among d-band states as they are fully occupied). Using our results for the matrix element for the d-to-s transitions, we find where we defined˜ = 2 π 2 /(2m * r 2 0 ) and we transformed the sum over ν into an integral. Also, E F denotes the Fermi level. For a large nanoparticles, ( d + ω) 1 and the last two terms in the brackets can be neglected resulting in For the s-to-s transitions, we express the summations over initial and final states as integrals over the s-band density of states according to where the angular momentum selection rules were used and the resulting summation over m was carried out.

Material parameters
To evaluate the above expressions for the hot carrier generation rates, the values of several materialspecific parameters are required. Here, we focus on silver nanoparticles. In the Drude model for the bulk dielectric constant, we use γ P = 60 meV 23, ω 0 = 9.07 eV and b = 4.18 [12]. The Fermi energy is set to E F = 5 eV and d = 2 eV [24]. For simplicity the effective mass is set to the bare electron mass.
To determine the atomic transition dipole moment z at dp , we carry out a first-principles density-functional theory (DFT) calculation for an isolated silver atom. We use the all-electron code FHI-aims [25], the PBE exchangecorrelation functional [26] and the default "tight" atomcentered numerical basis sets [25]. We average the square of the transition dipole moments of all possible d-state to p-state transitions and take the square root of this value. This results in z at dp =0.13Å. We also carry out a first-principles DFT calculation for a fcc silver crystal (with a 0 = 4.145Å) using the same parameters as for the silver atom. A 12 × 12 × 12 k-point grid was used to sample the first Brillouin zone. From a Mulliken analysis of the s-band we determine its average p-state content in the k-space region where the s-band lies above the Fermi level (as this region is of relevance to d-to-s transitions). This procedure yields α p = 0.81. Figure 1 shows the contribution of d-to-s transitions to the total hot carrier generation rate in Ag nanoparticles with different radii as function of photon energy. No transitions occur when the photon energy is smaller than the energy difference between E F and d . The rate exhibits a peak at the localized surface plasmon energy, ω LSP = 3.65 eV, as a result of the strong LSP-induced enhancement of the electric field. The leading contribution to N d→s , see Eq. (24), is proportional to the cube of the radius, which explains the rapid growth of the d-to-s excitation rate with increasing nanoparticle size. Figure 2 shows the contribution of s-to-s transitions. For small photon energies, the hot carrier rate diverges. This is a consequence of the large matrix elements for transitions between states that are close to the Fermi energy, see Eq. (19) which is also consistent with the trends observed in time-dependent DFT calculations [27].   Similarly to the d-to-s case, the curves exhibit a peak when the photon energy is sufficiently large to excite a localized surface plasmon. For the small nanoparticles considered in Figs. 1 and 2 the s-to-s hot carrier rate is larger than the d-to-s rate. However, the rate of s-to-s transitions increases only with the square of the radius and therefore we expect that d-to-s transitions will dominate for large nanoparticles. This is expected as only d-to-s transitions occur in bulk silver. The crossover takes place at a radius of approximately 65 nm. Fig. 3 shows that the s-to-s and d-to-s contributions are of comparable magnitude for this nanoparticle size.

RESULTS
Finally, we explore the dependence of the d-to-s hot carrier rate on the energy of the d-band d . Fig. 4 shows results for d = 3 eV and d = 1 eV. For the lower d-band energy, the onset of d-to-s transitions occurs at energies larger than the LSP energy. As a consequence, the dto-s rate does not "benefit" from the LSP electric field enhancement and the overall d-to-s contribution is relatively small.

CONCLUSIONS
We have presented a new approach for calculating the contribution of d-bands to the hot carrier generation rate in metallic nanoparticles. In particular, we have derived an equation for the envelope function of a nanoparticle state that derives from a d-band and solved this equation in the limit of perfectly flat bands. Evaluating Fermi's golden rule for d-band state to s-band state transitions and also for s-band state to s-band state transitions we find for silver nanoparticles that both contributions are of the same order of magnitude near the plasmon resonance. In particular, s-to-s transitions dominate hot carrier rates for small nanoparticles with radii less than approximately 65 nanometers. For larger nanoparticles, dto-s contributions give the larger contribution. It should be straightforward to extend this approach to other materials and nanostructures allowing to refine the design of nanoplasmonic devices with optimised hot carrier rates.