3. Experimental view of plasmons
In the past several decades, new synthetic approaches have produced hundreds of distinct atomically precise noble metal nanoclusters coated with a ligand shell, ranging in size from tens to hundreds of metal atoms. Depending on the size, shape, metal composition, and ligands, these nanoclusters may exhibit either plasmonic or excitonic properties. Spectroscopic studies have shown that there are features that distinguish plasmonic from excitonic behavior in two main areas: (1) the instantaneous optical properties and (2) the dynamics following absorption. This section does not cover every experimental distinction between plasmonic and excitonic systems, which have been reviewed more extensively elsewhere. Instead, we focus on two main goals: (1) relating the key features that are seen experimentally to their classical electrodynamics explanation, and (2) laying the necessary groundwork so that it is clear in the next section what similarities and differences exist between the properties used experimentally vs. in quantum mechanical models to distinguish between plasmonic and excitonic behavior.
The first distinguishing feature of plasmonic behavior is the absorption spectrum, reflecting the instantaneous response of the nanocluster to light. As seen from the classical electrodynamics model, because plasmons are collective oscillations, a (nearly) spherical plasmonic nanocluster should have only one peak in its absorption spectrum, as seen for 22 nm Au nanoparticles in Fig. 1a. In contrast, excitonic nanoclusters may have several or many absorption peaks involving transitions between different molecular orbitals, such as that of the Au6(PPh3)62+ nanocluster in Fig. 1b. In larger nanoclusters, the transition from an excitonic multi-peaked absorption spectrum to a single-peaked plasmonic absorption spectrum is visible as the nanocluster size increases from Au246(SR)80 to Au279(SR)84, though overlap of the intraband and intraband transitions somewhat obscures the intraband absorption peaks (Fig. 1c).
Several spectroscopic techniques that probe the dynamics following absorption also reveal differences between excitonic and plasmonic behavior. To understand the reasons for these differences, we must first examine the typical decay processes. When plasmonic metal nanoparticles absorb light, the initially coherent plasmon dephases into individual hot charge carriers (hot electrons and hot holes) within a few femtoseconds (Fig. 2). The initial distribution of hot carriers is relatively uniform over a broad range of energies. Because the electron–electron scattering rate is higher for charge carriers farther from the Fermi energy, the higher-energy charge carriers lose energy more quickly than the lower-energy carriers. This scattering transforms the initial non-thermal hot carrier distribution into a thermal distribution within a few hundred femtoseconds, in a process called thermalization. On the scale of a few ps, the hot carriers relax, and the excess energy is transferred into phonons.
In contrast, the decay of excitonic systems depends on the specific electronic excited states in the nanocluster. In the prototypical Au nanocluster Au25(SR)18−, three distinct decay time scales have been measured, though the details of the decay depend on the exact ligand structure, the excitation energy, and other experimental factors. First, the hot carriers within the metal core relax within several hundred fs. This is followed by transfer of energy from the metal core to states with ligand-to-metal charge-transfer character within a few ps84 and relaxation to vibrational modes on the order of 50–200 ns. These time scales are several orders of magnitude larger than for similar decay processes in plasmonic systems. Unlike in typical plasmonic systems, higher-energy excited states in excitonic nanoclusters may have longer lifetimes than the low-energy excited states of the same nanocluster. Because there are discrete excited states, decay from one state to another occurs on a distinctive time scale, and that time scale varies between different nanocluster isomers88 or sizes.
Because of the differences in the excited-state dynamics, plasmonic and excitonic noble metal nanoclusters have distinct emission mechanisms and time scales. Both classes of nanoclusters typically have two emissive transitions: a short-lived visible emission attributed to the decay of an excitation in the nanocluster core, and a longer-lived near-infrared emission attributed to ligand states. For the shorter-lived visible emission, plasmonic gold nanoparticle have fluorescence lifetimes <50 fs, which is faster than can be resolved experimentally; this rapid decay reflects the fast dephasing and thermalization processes (Fig. 3a). In contrast, excitonic gold nanoclusters have visible fluorescence time scales on the order of 250–350 fs, reflecting the slower decay from the initial excited states into ligand states.
The differences in the decay processes of excitonic and plasmonic gold nanoclusters can also be observed in differences in the pump-power dependence of their decay time scales, as observed by various time-resolved spectroscopic techniques. For plasmonic nanoclusters and nanoparticles, the time scale of the relaxation of electronic energy into phonons depends on the intensity of the pump laser used to excite the system: because a high pump power can cause one nanoparticle to absorb multiple photons, the initial hot carriers have a higher effective temperature, and these higher-energy hot carriers decay more quickly. In contrast, in excitonic nanoclusters, the relaxation time scale is independent of the pump power because the excitation and decay processes involve discrete excited states that decay with their own distinctive time scales. In thiol-protected Au nanoclusters, a sharp transition from excitonic to plasmonic dynamics is seen as the size increases from Au246(SR)80 to Au279(SR)84 (Fig. 3b). However, the transition from excitonic to plasmonic behavior is not simply a function of nanocluster size: the smaller nanocluster Au144(SR)60 has plasmonic pump-power dependence.
4. Quantum mechanical view of plasmons
Atomistic quantum mechanical methods can practically be applied to systems on the scale of tens to hundreds of atoms. Thus, unlike metal nanoparticles with diameters larger than 2–3 nm, many atomically precise nanoclusters are within a size range that can be studied using quantum mechanical methods to understand their geometries, optical properties, and dynamics with atomistic detail. The majority of the quantum mechanical calculations in the field have used density functional theory (DFT) approaches, but other computational methods like INDO97–99 and density functional tight binding (DFTB)100–103 give results that are largely consistent with DFT.
Early calculations using time-dependent DFT (TD-DFT) showed that bare tetrahedral silver nanoclusters have relatively sharp absorption peaks similar to those characteristic of plasmonic nanoparticles that red-shift with increasing size, and extrapolation of the absorption energy to larger sizes yields results consistent with electrodynamics for plasmonic systems. Based on these early observations, there has been great interest in using quantum mechanical calculations to understand the emergence of plasmonic properties and in developing techniques to characterize whether nanoclusters have plasmonic or excitonic properties. These different characterization methods focus on various aspects of the definition of plasmons from classical electrodynamics, and in some cases multiple characterization approaches attempt to quantify similar aspects of the electrodynamics definition. Here, we summarize the main characterization techniques and then focus on their application to prototypical linear metal nanowires, which are widely used as model systems.
We note that the characterization methods described here all focus on the static excited states computed by frequency-domain methods like TD-DFT or the absorption spectra obtained from time-domain methods like real-time TD-DFT (RT-TDDFT). There have been some computational studies of the excited-state dynamics using RT-TDDFT on the time scale of dephasing106–108 and non-adiabatic molecular dynamics (NAMD) on the time scale of relaxation. However, criteria to distinguish between excitonic and plasmonic states based on the dynamics are much less developed at this point.
4.1. Oscillatory behavior
As discussed earlier, the electrodynamics description of plasmons involves an oscillation of the electron density. However, when excited states are computed using frequency-domain quantum mechanical methods, the states that are obtained are stationary states, meaning that the electron density is constant over time. At a first glance, it may appear that quantum mechanical excited states could never represent the oscillatory behavior inherent to plasmons. However, this apparent contradiction is straightforward to resolve by instead considering time-domain quantum mechanical methods. Since the electrodynamics derivation of plasmonic behavior involves a time-dependent electric field, a time-dependent quantum mechanical method involving an electric field with explicit time dependence allows for a more direct comparison.
Frequency-domain excited-state methods like linear-response TD-DFT compute a list of excited eigenstates ψk, along with their corresponding energy eigenvalues Ek and transition dipole moments equation. The absorption intensity from the ground state ψg to excited state ψk is proportional to equation. This yields a "stick spectrum" where each excited state is represented by a stick with a height corresponding to the absorption intensity. To obtain an absorption spectrum that more closely resembles an experimental spectrum, each stick is broadened using a Lorentzian or Gaussian function, and the intensities are summed to give one total absorption spectrum.
Time-domain quantum mechanical methods apply an electric field as a time-dependent perturbation to the system and compute the evolution of the wavefunction over time. The time-dependent electric field may be a wavepacket with a sinusoidal shape over a finite time; it may also be a delta function where an electric field is applied for only a single time step that, via a Fourier transform, is equivalent to exciting the system with a broad distribution of frequencies. We will focus here on excitation with a sinusoidal wavepacket with a small enough intensity that the system remains mostly in the ground state. In the time domain, absorption can be described using first-order time-dependent perturbation theory, which we describe briefly here; the full derivation can be found in many quantum mechanics textbooks. Within this framework, the Hamiltonian is: equation where Ĥ(0) is the Hamiltonian of the unperturbed system, Ĥ′(t) is the time-dependent perturbation, F0 is the maximum intensity of the electric field of light, and [small mu, Greek, circumflex]; is the dipole operator. The time-dependent wavefunction can be written as a linear combination of the eigenfunctions ψ(0)k of the unperturbed system: euqation with coefficients ck(t) and energies E(0)k = ℏωk. If the system starts in the ground stateΨ(0)g(0) and the perturbation is turned on from time t = 0 until t = t′, the wavefunction at time t′ has the coefficients equation where ωkg = ωk − ωg. If ω is nearly resonant with a transition (such that the denominator ωkg − ω is nearly zero) that has a non-zero transition dipole moment equation, the coefficient ck(t′) will be non-zero, indicating that the system has absorbed light to transition from Ψ(0)g to Ψ(0)k. The probability of a transition from the ground state to state ψk is proportional to |ck(t′)|2 and thus is also proportional to equation. If there are no decay processes, the coefficients ck(t′) remain constant in magnitude after the electric field is turned off. If ω is not close to resonance with the transition from Ψ(0)g to Ψ(0)k, the coefficient ck(t′) remains small.
The resulting time-dependent wavefunction is a superposition of the ground-state wavefunction with a small coefficient for the resonant excited-state wavefunction Ψ(0)k: equation
The time-dependent dipole moment of the system is: equation
Thus, in a time-dependent framework, absorption of light produces an oscillating dipole moment, and the magnitude of the oscillations is directly proportional to the transition dipole moment. If the system is excited using an electric field in the shape of a delta function, the absorption spectrum over a wide energy range can be computed based on a Fourier transform of the time-dependent dipole moment, and the spectra are typically consistent with the results of frequency-domain calculations at the same level of theory.
The crucial result is that any strongly absorbing state, whether plasmonic or excitonic, has a large transition dipole moment, leading to large oscillations of the instantaneous dipole moment in the time domain that continue until a decay process occurs. An example of these dipole oscillations for the icosahedral Ag55 nanocluster is shown in Fig. 4; in this case, the electric field is on for approximately the first 10 fs, and decay processes lead to significant damping of the oscillations between 20 and 25 fs. A large transition dipole moment is necessary for an excited state to yield the large time-dependent charge oscillations characteristic of plasmons. However, large charge oscillations cannot be the only characteristic used to determine whether an excited state is plasmonic or excitonic, since every strongly absorbing state will induce large charge oscillations in the time domain. Thus, more criteria are needed to distinguish strongly absorbing plasmonic excited states from strongly absorbing excitonic excited states.