Exciton Dynamics in Carbon Nanotubes
In this work we show that the absorption spectrum in semiconducting nanotubes can be determined using the bosonization technique combined with mean-field theory and a harmonic approximation. Our results indicate that a multiple band semiconducting nanotube reduces to a system of weakly coupled harmonic oscillators. Additionally, the quasiparticle nature of the electron and hole that comprise an optical exciton emerges naturally from the bosonized model.
Singlet fission is a process, discovered almost 40 years ago in organic semiconductors, where a singlet localized on one chromophore cleaves into two independent and separate triplets (Fig.1). This process may be a key to enabling photovoltaic devices that break the Shockley-Queisser limit.[1-3] It is a canonical example of a multielectron process that may lie outside the scope of conventional condensed phase energy and electron transfer theories. The relative roles of electronic coupling, nuclear motion, and environmental dephasing in singlet fission are poorly understood. Our group recently developed a theory of the quantum dynamics of singlet fission to focus on the role of environmental fluctuations in the condensed phase. We show that a simple frontier molecular orbital picture captures some essential features of energy ordering in the states of singlet fission chromophores and illustrates the importance of selection rules. We derive a formally exact multilevel quantum master equation for the time-dependent populations of the relevant (diabatic) states in the problem and analyze it using perturbation theory in the electronic coupling. These results depict the process as a state to state one (Fig. 1), where dynamics obey Foerster-Dexter and Marcus theory at long times and high temperature. Our theory, (Fig. 2) however, is non-Markovian. Because singlet fission takes place on sub-picosecond timescales, a microscopic description based strictly on rate equations is dubious.
|Figure 1: Conceptual diagram of singlet fission and the role of fluctuations. (A) Two chromophores, left and right, have energy levels where the triplet state is nearly half the energy of the singlet S1 state. The electron (solid blue circle) and hole (empty circle) undergo singlet fission, which is an internal conversion process that leaves the system with two excitations, one on each chromophore. (B) Electronic configurations that participate in singlet fission. A singlet (top) can go through a charge transfer state (middle) and then to a doubly excited state (bottom). The states in both A and B are schematic because if taken literally these states are not eigenstates of the total electron spin angular momentum. In addition to satisfying spin selection rules, interconversion must satisfy energy conservation. An initial state, |i⟩, cannot transition to another state, |f⟩, unless energy is conserved. For a system in relative isolation (C), a large energy difference between states forbids transitions. In (D), the solvent induces energy level fluctuations that broaden the relevant densities of states between |i⟩ and |f⟩. Provided that selection rules are satisfied, transitions can occur when two states have the same energies (purple overlap).|
||Figure 2: Non-Markovian rate and golden rule rate from state |j⟩ to |k⟩ as a function of time. Ek − Ej = −0.05 eV at T = 300 K, 1/Γ = 100 fs, and γ =0.1 eV (see text). The stationary phase approximation (SPA) to the non-Markovian rate shows good agreement with the approximate form. The non-Markovian rate converges to the golden rule rate but only for times greater than about 600 fs.|
Carrier Transport Across Strained Heterojunctions on the Nanoscale
An electronic excitation in a semiconductor promotes an electron to an excited state and leaves behind a hole. The electron and hole must be spatially separated in a photovoltaic or photocatalytic device, and a heterostructure is a way to accomplish this task . A heterojunction is the interface between two semiconductors with different electronic properties and lattice constants. They are ubiquitous in bulk semiconductors, like the famous p-n junction, and they can be fabricated on the nanoscale. The mechanism for charge separation, however, is likely different on the nanoscale than in bulk. In a recent paper, we developed a physically-motivated, but exactly solvable model for carrier dynamics and showed how lattice strain and energy level alignments can create favorable conditions for electron and hole transport. Importantly, this mechanism does not rely on a space-charge region, as in traditional bulk heterojunctions. The dynamics do suggest, however, that an analogous mechanism may be at play on the nanoscale. Interfacial dipole-like states may have long lifetimes that support a dynamical space charge region, stabilized by the nonequilibrium currents rather than by equilibrium thermal fluctuations. While simple, our theory does suggest how the nonequilibrium Green function methods developed for molecular electronics may be applied to study the nanoscale heterojunction problem . The peaks and dips in the differential conductance are a signature of the junction size, and are a prediction of the theory that should be observable in future experiments.
|Figure 1: Schematic of the considerations that lead to the model Hamiltonian. Two semiconductors that have different lattice constants (top panel) form bonds at the interface. Each site has two orbitals, one bonding and one antibonding. The bonding−antibonding energy gap is EL for the left side and ER for the right. The wave functions at each site (blue curves) are sufficiently localized that one need only consider nearest-neighbor interactions. The difference in lattice constants applies a strain in the vicinity of the interface so that the equilibrium positions shift by an amount δq that is a function of the distance away from the interface. The strain field displaces the energies of the junction sites, shifting the uncoupled levels (dotted black lines) to new levels (red lines). In this cartoon, the junction consists of six sites.|
||Figure 2: Current and conductance as a function of the driving force Δμ = μL − μR with junction widths of one to five sites. For the electrons (B,D) μR is fixed at 2.7 eV and μL is varied, whereas for holes (A,C) μ̃L is fixed at −0.6 eV and μ̃R is varied. Conductances are in units e^2/ℏ, currents are in units of (e/ℏ)t, and the driving force is in ̅ units of t. The mean hopping term ̅ t̅≡ (tR + tL)/2 is defined separately for holes and electrons. It is inversely proportional to the effective mass, and the electron-to-hole ratio for t̅is ∼1.6. The inset plots (C,D) have a conductance range 0.025 ± 0.005 and illustrate peaks and dips in the conductance relative to the background. The peaks in the conductance are related to the resonances in the junction Green Function.|
Figure 3: Local density of states (LDOS) for electrons and holes for a two site junction. R and L correspond to the right and left sites of the junction. The red dashed line indicates the nanorod left side conduction minimum at 2.7 eV. The blue dashed line is the nanorod right side valence maximum at −0.6 eV. The two-site junction Green function exhibits narrow resonances for the electrons on the left, and holes on the right. These widths are inversely related to the lifetime of a carrier localized to either left or right junction site.http://jcp.aip.org/resource/1/jcpsa6/v121/i23/p11965_s1
Recent experimental work has shown that therapeutically inert molecules become potent antibacterial agents when attached to gold nanoparticles. The nature of the molecules used; such as composition, length, and charge, strongly affect the morphology of the self-assembled layer on the exterior of the gold particle. We use theory and computation to model these complex systems and understand new paradigms, phases, and phenomena.
Our analytical model consists of two types of ligands denoted A and B. Ligand A is longer than ligand B and has a net charge while ligand B is neutral. Both ligand types interact at short range through a competition between attractive van der Waals and repulsive entropic interactions.
To inform our analytical model we have done atomistic molecular dynamics simulations of gold-ligand complexes to study the energetics of specific morphologies. This will give us the coupling constants for our analytical model. We will be able to tune the length, charge, and composition of the ligands to determine how these parameters will affect the morphologies that are physically realizable on the nanoparticles and make predictions for experiments conducted in the Feldheim Lab at CU.
Figure 1: (A) Phase diagram of a coarse-grained lattice model for the morphologies of a ligand‐coated gold nanoparticle in terms of the temperature and long range (screened Coulomb) energy normalized to the short-range interaction energy. (B) Configurations from atomistic simulations with morphologies similar to those appearing in the phase diagram.