Abstract
Hot carriers (HC) generated by surface plasmon polaritons (SPPs) in noble metals are promising for application in optoelectronics, plasmonics and renewable energy. However, existing models fail to explain key quantitative details of SPPtoHC conversion experiments. Here we develop a quantum mechanical framework and apply firstprinciples calculations to study the energy distribution and scattering processes of HCs generated by SPPs in Au and Ag. We find that the relative positions of the s and d bands of noble metals regulate the energy distribution and mean free path of the HCs, and that the electron–phonon interaction controls HC energy loss and transport. Our results prescribe optimal conditions for HC generation and extraction, and invalidate previously employed freeelectronlike models. Our work combines density functional theory, GW and electron–phonon calculations to provide microscopic insight into HC generation and ultrafast dynamics in noble metals.
Introduction
Surface plasmon polaritons (SPPs) are electron collective excitations generated by light at the interface between a metal and a dielectric^{1}. Modelling of SPPs is dominated by approaches that employ classical electromagnetism and account empirically for the properties of the materials supporting the SPP. Yet, the inherently quantum mechanical nature of SPPs^{2,3} becomes manifest in their scattering and decay processes in bulk and nanoscale materials. A key example is the decay of SPPs to electronhole pairs, a process whose crucial importance is twofold: first, it is a main energy loss mechanism of SPPs in metals, currently limiting the applicability of plasmonics^{4,5}; second, it leads to generation of hot electrons and holes with application in several branches of applied physics^{6}. Recent experiments reported the extraction of such SPPinduced hot carriers (HCs) in metals before they thermalize—typically over nanometer lengths and subpicosecond times—by injecting the HCs over a Schottky barrier into a semiconductor or oxide^{7,8,9}, or by transferring them into surface adsorbates to perform chemical reactions with large activation barriers^{6,10,11}. The vast literature on HCs in noble metals generated by intense light pulses^{12,13,14} (instead of SPPs), together with recent calculations on SPPtoHC conversion^{15,16}, help interpret these recent experiments on SPPinduced HCs.
Lack of predictive and accurate quantum mechanical approaches to study HCs generated by SPPs or photons has led to ambiguity in the microscopic interpretation of experiments involving HCs in Au and Ag, two mainstay materials in plasmonics^{4}. In particular, recent studies indicate that SPP decay excites electronic transitions from occupied states close to the Fermi energy (E_{F})^{10}, implying that most of the SPP energy goes into hot electrons rather than hot holes, and that nanocrystals with <10 nm diameter are necessary to obtain HCs with significant energy^{17} (for example, 1–2 eV away from E_{F}). The mean free paths (MFPs) of HCs in noble metals appear to be short^{10}, though a quantitative estimate of the MFPs and their dependence on crystal direction is not available. In addition, it is often assumed that HC scattering and thermalization are dominated by Auger processes^{10,14}, since thermalization induced by phonons occurs on a slower time scale^{14}. As shown in this work, all these conclusions must be revised.
Here, we develop a quantum mechanical framework to study the energy distribution of HCs generated by SPPs and photons in Au and Ag, and employ ab initio calculations of the electron–phonon (e–ph) and electronelectron (e–e) interactions to study the MFPs and relaxation times of HCs within 5 eV of the Fermi energy. Our approach is free of experimental input, and combines density functional theory (DFT)^{18}, the GW (where G is the Green function, W is the screened Coulomb potential, and GW is the diagram employed for the electron self energy) method^{19}, and ab initio e–ph calculations^{20}. Our ability to use extremely fine grids for Brillouin zone (BZ) integrations allows us to resolve HC scattering with unprecedented accuracy^{21}. We find that the interband transition threshold (between d and s states) defines two regimes for HC generation and transport. The decay of SPPs with energy lower than the interband threshold leads to generation of longlived HCs with long isotropic MFPs of up to 40 nm and energy within 1–2 eV of E_{F}. On the other hand, decay of SPPs with energy higher than the interband threshold leads to generation of shortlived hot holes in d states with anisotropic and short (sub 5 nm) MFPs, and hot electrons with only <1 eV energy above E_{F}. The regime characterized by SPP energy below the interband threshold is better suited to employ HCs in applications requiring long MFPs, and allows one to optimize HC generation by tuning the SPP energy. These results represent an important first step to understand SPP decay and energy loss, and to control HC generation and transport in bulk and nanostructured noble metals.
Results
Theory of SPP—electron coupling
Coupling of SPPs or photons to electrons in materials can be described in the framework of manybody perturbation theory^{22}. The lowest order Feynman diagram for this coupling process, shown in Table 1, describes a boson (here, SPP or photon) coupling to the electron gas through the electronic polarizability χ and a coupling matrix element g. This diagram is analogous to the lowest order coupling of phonons to electrons^{23,24}. The decay rate Γ(q_{p},E_{p}) for a boson of momentum ħ q_{p} and energy E_{P} to electron–hole pairs is proportional to the imaginary part of the selfenergy^{22}:
where −E_{n,k} and are, respectively, the quasiparticle energies of the hot hole in a state with band n and crystal momentum –k, and hot electron in a state with band n′ and crystal momentum k+q_{p}, as generated in the boson decay process. In addition, f_{n,k} and are Fermi occupations, η is a small broadening, and the sum is contributed by electronic transitions from occupied to empty states that differ by ħ q_{p} in crystal momentum and E_{p} in energy. If the matrix elements have a weak band and k dependence, the decay rate of a SPP or photon is proportional to the imaginary part of the electronic polarizability, Imχ(q_{p},E_{P}).
Here, we employ a quantity proportional to Imχ(q_{p},E_{P}), the finite momentum joint density of states (FMJDOS) J(q_{p},E_{P}), defined as the number of states per unit energy separated by momentum ħ q_{p} and energy E_{p} in the quasiparticle bandstructure:
The FMJDOS is a measure of the phase space available for the decay of a SPP or photon with momentum ħ q_{p} and energy E_{p}. Greater FMJDOS values generally would correspond to greater SPP decay and HC generation rates.
The coupling matrix elements in equation (1) can be derived with a range of theoretical approaches. We first discuss the coupling of electrons to plasmons—that is, collective electronic excitations in the absence of light—since theories developed for plasmons have been occasionally applied to SPPs^{25}. The socalled Landau damping^{26} treatment of the electron gas derives the decay rate Γ_{p} of a bulk plasmon in the weak coupling limit via the zeros of the complex macroscopic dielectric function ɛ_{M} of the metal close to the real energy axis, that is, . Due to the bosonic nature of plasmons, the decay rate Γ(q_{p},E_{p}) has the form given in equation (1), with coupling matrix elements^{26} . In spite of its common use, Landau damping is rarely discussed in terms of the diagram in Table 1 and the coupling g^{PL}, though this viewpoint elucidates its physical origin.
The coupling matrix elements for SPPs and photons can be derived by second quantization of the vector potential^{27} (see Methods). For photons, the coupling is given by the wellknown dipole matrix element , where V is the volume, is the polarization vector, and the transition dipole matrix element. For SPPs at a metal–dielectric interface, we apply a vector potential second quantization procedure^{28,29} (see Methods) to obtain the SPP–electron coupling given in Table 1. The SPP–electron coupling has a form analogous to that for photons, with two important differences. First, the volume V is replaced by the SPPfield volume V_{SPP}=A·L_{z}, where A is the metal–dielectric interface area and L_{z} the decay length of the SPP in the metal. Since L_{z} is of order 10 nm at visible wavelength for noble metals^{1} and thus much shorter than the light absorption depth, the electric field associated with the SPP is concentrated to a small volume V_{SPP} at the metal–dielectric interface^{1}, resulting in enhanced local generation of HCs from SPPs compared with light. Second, the SPP wavevector q_{p} is in general comparable to the size of the BZ, while for photons q_{p}≈0. The finite SPP wavevector can significantly alter the phase space available for SPP decay, as dictated by the q_{p} dependence of the FMJDOS. We remark that additional momentum can be transferred by phonons and defects assisting the SPP scattering process.
For both SPP and photon decay, the FMJDOS regulates the phase space for HC generation. Since the experimental conditions and the materialspecific SPP dispersion relation dictate the SPP energy E_{p} and wavevector q_{p}^{1,30,31}, we study the FMJDOS as a function of q_{p} and E_{p} and arbitrarily treat them for the purpose of this work as independent variables; HC generation from light is obtained as the specific limit q_{p}=0. We note that the SPP decay rate Γ_{p}(q_{p},E_{p}) could also be computed by evaluating the coupling matrix elements explicitly for fixed experimental conditions and dispersion relation^{1} . This approach would enable firstprinciples calculations of plasmonic losses and the MFPs of SPPs in materials^{4,5}.
HC generation
To study HC generation from SPPs at a metal–dielectric interface (see Fig. 1a), we compute the FMJDOS for Au and Ag, as shown in Fig. 1b, for SPP energies up to 5 eV and increasing SPP wavevectors along the Γ–L direction in the BZ (that is, for SPPs propagating along the [111] crystallographic direction). Our calculations are carried out using GW bandstructures (see Methods).
Detailed features of the bandstructures of Au and Ag, shown in Fig. 2, regulate HC generation and transport in these materials. The main features of the Au and Ag bandstructures are the presence of relatively dispersionless, occupied d bands with large associated electronic DOS, straddled by a freeelectronlike s band. The top of the d bands is located at energy E_{int} below the Fermi energy, where E_{int} is the threshold for interband (d to s) electronic transitions^{32,33}. From our GW calculations, we find E_{int} values of 2.3 eV in Au and 3.7 eV in Ag (see Fig. 2), in excellent agreement with experiment^{5,32,33}. The freeelectronlike s band spans an energy window between E_{int} below and over 4 eV above E_{F}, and hence dominates the electronic properties of Au and Ag near E_{F}.
The FMJDOS in Fig. 1b follows similar trends in Au and Ag, with deviations due to the different E_{int} values in the two materials. We first discuss the FMJDOS with q_{p}=0, which quantifies the phase space for HC generation for both SPPs with very small wavevectors and photons. The FMJDOS curve with q_{p}=0 (Γ) has a zero value for SPP energies up to E_{int}, indicating the absence of possible SPP decay into HCs in this energy range. For SPP energies greater than E_{int}, the FMJDOS increases linearly with energy, indicating that a large phase space opens up for HC generation. In this regime, SPP damping is strong, and HC generation is intense and dominated by formation of hot d holes collecting most of the energy from the SPP, as discussed below.
For small SPP wavevectors along the Γ–L direction up to ∼0.2L, we find a peak in the FMJDOS corresponding to SPP decay through intraband transitions within the s band. The energy position of the peak increases linearly with SPP wavevector, and for large enough wavevectors merges with the FMJDOS above E_{int}. In Au, the intraband peak is below E_{int} for q_{p} values up to ∼0.2 L, and in Ag for q_{p} values up to ∼0.3 L due to the higher E_{int} value in Ag. The presence of an intraband peak associated with s–s transitions can be understood by examining the energy difference between states with momentum k and k+q_{p} in the freeelectronlike s band:
For small wavevector q_{p}, the term quadratic in q_{p} can be neglected, while the term linear in q_{p} increases by ∼ (a is the lattice constant) for an increase of q_{p} by 1/10 the Γ–L distance. The computed FMJDOS confirms these trends, as seen by the increase in the energy of the intraband peak by ∼1 eV in going from 0.1 L to 0.2 L and from 0.2 L to 0.3 L. In this small wavevector regime, generation of HCs is appreciable only if the SPP energy and wavevector are matched to the intraband FMJDOS peak, namely if E_{p}(q_{p})≈(ℏ^{2}k·q_{p})/m for a set of k points in the BZ. When this condition is met, intense HC generation from intraband transitions can occur, resulting in an almost equal apportioning of the SPP energy between hot electrons and holes, as discussed below.
For SPP wavevectors larger than ∼0.3 L, the FMJDOS resembles that for q_{p}=0 except from a flat tail that extends to low SPP energy, as shown by the dashed lines in Fig. 1. This tail is due to generation of HCs from intraband transitions with large transferred momentum from the decay of SPPs with wavevector comparable to the size of the BZ. The FMJDOS for q_{p}>0.5 L is unchanged and nearly identical to the 0.5 L case.
To summarize, we find two HC generation regimes. For SPPs with energy E_{p}<E_{int}, optimal generation can be achieved using SPPs with relatively small wavevector (less than ∼1/5 the BZ size) that are matched with the FMJDOS peak, while HC generation is relatively weak for SPPs with larger momentum. For SPPs with energy E_{p}>E_{int}, HC generation is intense regardless of the SPP momentum. We find identical trends for SPPs with wavevector in the Γ–X highsymmetry direction, that is, propagating along the [100] direction. Removal of the electron gas approximation used in simplified treatments of SPPs in noble metals^{17} is necessary to capture the delicate interplay found here between the energetics of the s and d states and intraband (s–s) and interband (d–s) transitions.
Energy distribution of HCs
Figure 2 shows the energy distribution of HCs generated by specifically assumed SPPs with energy increasing in small steps from ∼1 to 5 eV and a range of wavevectors in the Γ–L direction, along with the GW bandstructures. From now on throughout the manuscript, all HC energies are referenced to the Fermi energy (E_{F}), and the energy of the hot hole is −E, that is, the direction of increasing energy in the bandstructure is downward for holes and upward for electrons. The energy distribution histograms identify the HCs generated in the two regimes discussed above. For SPPs with energy lower than E_{int}, hot holes and electrons are generated with a roughly uniform energy distribution as a result of the s–s intraband transitions. The HC energy distributions shown in Fig. 2 in this energy range pertain to SPPs with energy and wavevectors matched to the FMJDOS intraband peaks (see Fig. 1). While the details of the energy distribution in this regime are sensitive to the SPP energy E_{p}, the overall trends indicate that HCs are generated with an average energy of ∼E_{p}/2.
For SPPs with energy greater than E_{int}, we find a change to a different HC generation regime. At the onset of the interband transitions for SPP energy of 2.3 eV in Au and 3.7 eV in Ag, on SPP decay the hot holes absorb most of the SPP energy, creating a spike in the hot hole population at the top of the d band. On the other hand, hot electrons have only modest energies in this regime. At higher SPP energies of ∼5 eV, the HC population become approximately uniformly distributed over a wide energy range, although a peak in the hot d hole population is still present in Ag due to its higher E_{int}. These trends are common to both Au and Ag, and we have verified that Cu, a material less commonly used in plasmonics^{5} and not discussed here, follows identical HC generation and energy distribution trends as Au and Ag.
HC scattering processes
While ample experimental^{13,32} and theoretical^{14} data exists on HCs in noble metals, characterization of the MFPs and relaxation times of HCs using ab initio theory has been limited by the absence of accurate e–ph calculations including all phonon modes over the entire BZ. Firstprinciples calculations combining e–ph and e–e interactions are necessary to characterize the anisotropic and energydependent MFPs of HCs in materials, as shown in our recent work on HCs in semiconductors^{21}. We compute the scattering rate (and its inverse, the relaxation time) for the e–ph and e–e interactions from the imaginary part of the respective selfenergies, and , and resolve these quantities for each electronic state with band n and kpoint on very fine BZ grids (see Methods and ref. 21). The total relaxation times τ_{n,k}, combining the e–ph and e–e interactions, are obtained as .
Figure 3 shows the total relaxation times τ_{n,k} for HCs in Au and Ag with energy up to 5 eV. The total relaxation times decay rapidly away from E_{F} following a volcanoshaped trend, with the peak of the volcano centred close to E_{F}. In Ag, we find an additional peak in the relaxation times at the top of the d bands, in agreement with recent photoemission experiments^{34}. Analysis of the e–ph and e–e scattering rates (see the bottom panels in Fig. 3) highlights the differences between the energy windows spanned by the s and d states, and elucidates the origin of the relaxation time trends. In the energy range spanned by the s states, the e–e scattering rates (that is, ) form a parabola with minimum value of zero at E_{F} as predicted by Fermi liquid theory^{14}, while the e–ph scattering rates are relatively constant and exhibit a minimum at the onset of the d states and a maximum 1–2 eV above E_{F}. The e–e scattering rates become greater than the e–ph ∼2 eV away from E_{F}, thus indicating that Auger and impact ionization processes included in the e–e term dominate HC scattering only 2 eV or more away from E_{F}, while e–ph dominates HC scattering within 2 eV of E_{F}. Combining the two scattering mechanisms leads to total relaxation times with a broad maximum centred at E_{F} and a rapid decay 1–2 eV away from E_{F} as the parabolic e–e scattering rate becomes dominant (see Fig. 3). In the energy window spanned by the d bands, the large increase in the electronic DOS causes very strong e–ph scattering, and the localized nature of the d states leads to deviations from the freeelectron parabolic trends for e–e scattering. The total relaxation times show that hot holes arising from d states lose energy on a sub5fs time scale, thus making such hot d holes challenging to extract before thermalization. Finally, our calculations yield comparable time scale and scattering rate for e–ph and e–e scattering, in contrast with the common heuristic assumption that scattering by phonons occurs on a much slower time scale^{14}. We thus conclude that previous models including only e–e scattering^{14} and missing the important e–ph component are incomplete for understanding HC relaxation times and photoemission linewidths in noble metals.
To validate our results, we compare our relaxation times at the Fermi energy in Au and Ag with those inferred from carrier transport measurements. Kopitzki^{35} combined the Drude model with room temperature resistivity measurements, and obtained semiempirical relaxation times of 29 fs for Au and 41 fs for Ag, in close agreement with those obtained by Ashcroft and Mermin^{36} with the same approach. Averaging our relaxation times over a small energy window around E_{F} yields relaxation times of 24 fs for Au and 27 fs for Ag. While the agreement is excellent for Au and good for Ag, our data shows a large spread in the relaxation times at E_{F}, a feature not captured by the Drude model and stemming from the Fermi surface anisotropy. It is therefore puzzling that the Drude relaxation times^{33,35} are widely used in plasmonics^{4,5}, given that their physical meaning is questionable when applied to nonequilibrium situations involving SPPs. We conclude that further investigation on this point is necessary, and remark that ours is the first truly ab initio determination of the relaxation times on the Fermi surface of noble metals, to be discussed in detail elsewhere.
HC MFP
Next, we turn our attention to HC transport. Figure 4 shows the MFP for the three crystallographic directions [100], [110] and [111] within 5 eV of the Fermi energy. The MFPs are obtained by multiplying the total relaxation times by the carrier velocities computed from the slope of the GW bandstructures (see Methods). Our computed MFPs are of the order 10–40 nm in the energy window spanned by the s states, and much shorter (1–5 nm) for the d states. The MFPs of the s states exhibit an energy dependence with the volcano shape also seen in the relaxation times, with deviations coming from the different band velocities along each direction. The MFPs of hot holes with 1–2 eV energy are nearly isotropic, while the MFPs of hot holes with <1 eV energy are longer for the [100] and [110] directions. The MFPs of hot electrons are highly anisotropic due to the absence of electronic states up to 4 eV above E_{F} in the [111] direction and at energy >1–2 eV in the [100] direction. The longest hot electron MFPs are in the [110] direction up to 4 eV above E_{F}. We note that the energy derivative of the MFPs at E_{F} at is negative, consistent with the positive thermopower in Au and Ag^{37}. In the energy window spanned by the d states, the MFPs are of order 5 nm in the [111] direction and 1 nm in the other directions, and thus highly anisotropic with longer MFPs in the [111] direction. The isotropic MFPs for s holes and anisotropic for d holes are consistent with findings in ballistic electron energy microscopy experiments in noble metals, where it was found that d holes injected by a scanning tunneling microscope tip propagate along narrow cones and span small volumes, while s holes span large volumes isotropically^{38}.
Discussion
The data obtained so far allow us to formulate optimal conditions to generate HCs and extract or utilize them before thermalization. Figure 5 summarizes HC generation and transport in Au and Ag by defining two regimes for HC generation from intraband (E_{p}<E_{int}) and interband (E_{p}>E_{int}) transitions induced by SPP decay. The energy distribution, relaxation time and MFP of the generated HCs are dramatically different in these two regimes. For E_{p}<E_{int}, the HCs consist of s holes and s electrons with average energies up to E_{int}/2 (that is, 1–2 eV) generated by weakly damped SPPs due to the small FMJDOS values. HC generation can be optimized by matching the energy and momentum of the SPP to the FMJDOS intraband peak described above and shown in Fig. 1. The HCs in this energy range possess relatively long MFPs of ∼10–40 nm, and can thus be extracted before thermalization over ∼50 nm lengths in ideally pure samples. This regime is thus optimal for HC generation and extraction. For hot electrons, we predict that the (110) surface would be best suited to extract HCs due to the longer MFPs in the [110] direction. For hot holes with ∼1–2 eV energy, the (100) and (110) surfaces are predicted to enable optimal extraction. Our results further suggest that Ag may be better suited than Au for HC generation due to the wider energy window for intraband transitions, thus motivating studies of HC generation from SPPs in Ag. On the other hand, the strong SPP damping regime with E_{p}>E_{int} is nonoptimal for extracting HCs, since in spite of strong HC generation the SPP energy is mostly transferred to shortlived hot d holes with ∼1 nm MFP, while hot electrons possess only low energy of <1 eV. These conditions prevent HC extraction in this regime, unless a nanometerthick thin film or nanostructured metal is employed.
Our data further show that nanostructuring the metal to sub10nm size is not required to obtain energetic HCs, and not needed to extract them unless HCs with >2 eV energy are desired. In addition, for SPP with energy lower than E_{int}, the SPP energy is equally distributed between hot electrons and holes, and energy loss is dominated by e–ph rather than Auger scattering. These findings address the misconceptions emerged in recent work as outlined above, and show that optimal HC generation is possible by carefully tuning the SPP energy and wavevector at noble metal–dielectric interfaces. Finally, we note that our approach is for now limited to SPP at metal–dielectric interfaces, while localized surface plasmons in nanostructures need to be treated differently due to their localized nature^{39}. We believe that the method presented in this work can be extended to study HC generation from localized surface plasmon modes, which will be the subject of future investigation.
In summary, we establish a theoretical framework to study SPP damping and HC generation and transport on the same footing using manybody perturbation theory. Our firstprinciples calculations can accurately describe SPPinduced generation and ultrafast scattering of HCs in noble metals of use in plasmonics, photocatalysis, photovoltaics and optoelectronics. Our work highlights the interplay of the s and d bands in noble metals, and prescribe optimal experimental conditions for generation and extraction of HCs. Our approach paves the way to firstprinciples calculations of SPP losses in materials^{4}.
Methods
Firstprinciples calculations
We carry out ab initio calculations on Au, Ag and Cu in the facecentred cubic structure with lattice parameters of 7.72, 7.71 and 6.82 bohr, respectively. The groundstate electronic structure is computed within the local density approximation (LDA)^{40} of DFT using the QUANTUM ESPRESSO code^{41}. We use normconserving pseudopotentials (which include semicore s and p states) to describe the corevalence interaction^{42}, together with a planewave basis set with kinetic energy cutoff of 60 Ry for Ag and Au, and 240 Ry for Cu. The e–e (that is, GW) and e–ph selfenergies are computed separately, and then combined together.
The real and imaginary parts of the GW selfenergy^{19} are computed using the BerkeleyGW code^{43}. The real part is obtained with a generalized plasmon–pole calculation^{19} on a 12 × 12 × 12 kpoint grid and then interpolated using Wannier functions (see below). Kinetic energy cutoffs of 50 Ry and 120 Ry are used, respectively, for the screened and bare Coulomb interactions, together with ∼1,000 empty bands and a 8 × 8 × 8 qpoint grid for the dielectric screening. The DFT eigenvalues are corrected with GW selfenergies to obtain the quasiparticle bandstructures, , used in all calculations in this work. The imaginary part of the GW selfenergy, , is computed using fullfrequency GW calculations^{43}. Here are evaluated onshell at the LDA eigenvalues , and then plotted versus the corresponding GW eigenvalues E_{n,k}. We use 20 × 20 × 20 kpoint grids to converge within ∼10 meV. In addition, we use kinetic energy cutoffs of 20 Ry and 120 Ry, respectively, for the screened and bare Coulomb interactions, together with ∼100 empty bands and a 20 × 20 × 20 qpoint grid for the dielectric screening.
The imaginary part of the e–ph selfenergy, , is computed using the EPW code^{20}. We compute the selfconsistent potential and Kohn–Sham states on a 12 × 12 × 12 kpoint grid using DFT, and latticedynamical properties with DFPT^{44} on a 4 × 4 × 4 qpoint grid. The e–ph matrix elements are first computed on these coarse grids, and then obtained on significantly finer grids using an interpolation procedure based on Wannier functions as implemented in the EPW code^{20,24}. Maximally localized Wannier functions^{45} are obtained starting from five d orbitals on the metal atoms and two s orbitals, each in a tetrahedral interstitial point of the facecentred cubic lattice, for a total of seven wannierized bands (we skip the semicore states). The Wannier interpolated and DFT bandstructures agree within 5 meV in an energy window up to 5 eV above the Fermi energy. The BlochtoWannier rotation matrices are then used to interpolate the GW bandstructures, which are used to compute . The fine grids for the calculation of consist in a 40 × 40 × 40 kpoint grid, and up to 1 million random phonon qpoints in the BZ. Such fine grids allow us to converge within 1 meV. Further details of our approach to compute the e–ph selfenergy are discussed in ref. 21.
The FMJDOS calculations are carried out using Wannier interpolated GW bandstructures, a 100 × 100 × 100 kpoint grid and a small Lorentzian broadening η≈25 meV. The energy distribution of the generated HCs in Fig. 2 are obtained by counting the momentum and energyconserving transitions in equation (2) as a function of occupied band (for hot holes) and empty band (for hot electrons) quasiparticle energies.
The MFPs are obtained using velocities computed from the slope of the GW bandstructure along the given highsymmetry direction, together with total relaxation times τ_{n,k} combining the e–ph and e–e relaxation times with Matthiessen’s rule. The total relaxation times are obtained from computed on grids of over 100 k points along each highsymmetry direction, and computed on a 20 × 20 × 20 kgrid and then averaged over k points to generate an energydependent evaluated at each energy E_{n,k} for which a calculation of is performed.
Derivation of the coupling matrix elements
The derivation of the plasmonelectron matrix element in Table 1 within the Landau damping approximation is given in ref. 26 and will not be discussed. Here we derive the coupling matrix elements for the photon–electron and SPP–electron interactions given in Table 1. We use SI units throughout the derivations. The photon case is well known^{27} and discussed here as a starting point for the SPP case. The second quantized vector potential for a photon with momentum ħ q_{p}, energy and polarization unit vector can be obtained by associating the amplitude with a destruction operator and its complex conjugate with a creation operator . We use the vector potential with unit amplitude since we are interested in the coupling matrix element, and thus the transition rate per unit of incident power. It reads^{27}:
where is a normalization constant. Using the Coulomb gauge , the second quantized electric field can be obtained from .
The normalization constant is determined by equating the energy of the classical and quantized fields. The classical (cycle averaged) energy for a photon with momentum ħ q_{p} in a homogeneous and isotropic material with dielectric constant is , where V is the volume of the material. This energy needs to be equal to its quantum counterpart, . In addition, can be obtained by averaging the quantized electric field over the quantum state with photons. We get:
where in the last equation we used the time derivative of equation (4) the electric field and the commutation rules for boson creation and annihilation operators, . Equating equations (5) and (6) yields a normalization constant , and a vector potential:
The photon–electron interaction Hamiltonian is , as obtained by expanding the kinetic energy to first order in the vector potential. The second quantized fields associated with the electronic Bloch states are:
where and are, respectively, fermion annihilation and creation operators satisfying the anticommutation rule . The second quantized interaction Hamiltonian can be written as:
Crystal momentum conservation requires k′=±q_{p}, where the plus and minus sign apply for the first and second integral above, respectively. The interacting Hamiltonian becomes:
where h.c. is the hermitian conjugate, and we defined the momentum matrix element as the integral over the unit cell volume Ω, as commonly used in the study of optical absorption^{46}. We obtain the photon decay rate by applying Fermi's golden rule. Photon absorption occurs with a transition rate between an initial state , consisting of the ground electronic state at room temperature and one photon with momentum q_{p}, and a final state S;0〉 with an electron in the Bloch state n′,k+q_{p}〉, a hole in the Bloch state n,k〉 and no photons. The absorption rate is given by:
Only the first term in equation (8) gives a nonzero contribution, and we get:
The inverse process of photon emission from an initial state S;0〉 to a final state has a rate given by:
The net rate is the decay rate of a photon with momentum ħ q_{p} and energy into a hot electron–hole pair. Writing the delta function as a Lorentzian with broadening η→0, namely , we obtain the decay rate as:
This expression is identical to equation (1) with the squared coupling matrix elements in Table 1. Since for photons q_{p}≈0 compared with the size of the BZ, the squared coupling matrix elements are:
as shown in Table 1. We note that for slowly varying matrix elements in the BZ, the photon decay rate is proportional to the zeromomentum JDOS commonly used in the theory of the optical properties of solids ^{46}, that is, .
The derivation of the coupling matrix elements for SPPs follows similar steps to the photon case discussed above. Following Nkoma et al.^{28}, we study the case of a SPP generated at the interface between a metal with frequencydependent dielectric function and a dielectric material with frequencyindependent dielectric constant . A similar second quantization procedure was also developed by Elson et al.^{29}. It is well known that the intensity of the SPP wave decays exponentially in the direction perpendicular to the dielectricmetal interface^{1,30}. To study HC generation, we focus here on the SPP field in the metal, and its associated vector potential. Since SPP possess transverse magnetic polarization, their wavevector can be taken as real in the xy plane—here the plane of the metal–dielectric interface, with area A—and purely imaginary in the z direction normal to the interface. With the metal in the z>0 space and the xaxis parallel to the propagation direction of the SPP wave, the SPP wavevector is q_{p}=(q_{p,x},0,iq_{p,z}). While in the main text, we define q_{p}=(q_{p,x},0,0) as the propagation wavevector in the xy plane, for the purpose of this derivation q_{p} is defined differently. Here q_{p,x} is a real positive number since the electroninduced dissipation effects we aim to find would be encoded in its imaginary part, and q_{p,z} is related to the decay length L_{z} of the SPP field intensity by^{30} q_{p,z}=1/(2L_{z}). The relative magnitude of the components of q_{p} are determined by the dielectric properties of the metal and dielectric^{1,30}, which impose (here, the real parts of and should be employed for consistency). Similarly, the components of the unit polarization vector are related by^{30} .
The vector potential of the SPP wave in the metal, with unit amplitude as in the photon case, can be written in second quantized form as:
where, as in the photon case, the normalization constant can be determined by energy considerations. We write the classical electric field associated with the SPP in the metal as . Using a result in ref. 28, the cycleaveraged energy in the SPP field—which accounts for contributions from both the field in the dielectric and the metal—can be written as , where we defined an effective dielectric constant:
We use the correspondence principle for a quantum state with SPP quanta. Similar to the photon case, the total energy in the SPP mode can be equated to its classical counterpart, while the square modulus of the electric field can be replaced with its quantum mechanical average , with . We obtain the two equations:
By equating equations (12) and (13), and using q_{p,z}=1/(2L_{z}), we obtain the normalization constant , and thus the SPP vector potential:
This vector potential is similar to the photon case, equation (7). The key differences are the decay in the z direction for the SPP field, which introduces an effective SPP volume AL_{z} in the vector potential, the appearance of an effective dielectric constant for the medium and the fact that the polarization is transverse magnetic with a direction imposed by the properties of the metal and dielectric material.
Since the form of the SPP vector potential is completely analogous to the one for photons, the decay rate of SPPs to hot electron–hole pairs can be carried out following the same steps as in the photon case discussed above, with the substitutions V→AL_{z}, , and allowing for a finite momentum^{1} q_{p} with magnitude . This leads to the SPP–electron coupling matrix element in Table 1:
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How to cite this article: Bernardi, M. et al. Theory and computation of hot carriers generated by surface plasmon polaritons in noble metals. Nat. Commun. 6:7044 doi: 10.1038/ncomms8044 (2015).
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Acknowledgements
This research was supported by the SciDAC Program on Excited State Phenomena in Energy Materials funded by the US Department of Energy, Office of Basic Energy Sciences and by the Advanced Scientific Computing Research, under Contract No. DEAC0205CH11231 at Lawrence Berkeley National Laboratory which provided for algorithm and code developments and simulations; and by the National Science Foundation under grant DMR 101006184 which provided for basic theory and formalism. Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences and by the the US Department of Energy under Contract No. DEAC0205CH11231. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US Department of Energy.
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M.B. and J.M. carried out the calculations. All authors conceived the research and wrote the manuscript.
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Bernardi, M., Mustafa, J., Neaton, J. et al. Theory and computation of hot carriers generated by surface plasmon polaritons in noble metals. Nat Commun 6, 7044 (2015). https://doi.org/10.1038/ncomms8044
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DOI: https://doi.org/10.1038/ncomms8044
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