Remembering the Work of Phillip L. Geissler: A Coda to His Scientific Trajectory.

Phillip L. Geissler made important contributions to the statistical mechanics of biological polymers, heterogeneous materials, and chemical dynamics in aqueous environments. He devised analytical and computational methods that revealed the underlying organization of complex systems at the frontiers of biology, chemistry, and materials science. In this retrospective we celebrate his work at these frontiers.

Polaritonic Bottleneck in Colloidal Quantum Dots.

Controlling the relaxation dynamics of excitons is key to improving the efficiencies of semiconductor-based applications. Confined semiconductor nanocrystals (NCs) offer additional handles to control the properties of excitons, for example, by changing their size or shape, resulting in a mismatch between excitonic gaps and phonon frequencies. This has led to the hypothesis of a significant slowing-down of exciton relaxation in strongly confined NCs, but in practice due to increasing exciton-phonon coupling and rapid multiphonon relaxation channels, the exciton relaxation depends only weakly on the size or shape. Here, we focus on elucidating the nonradiative relaxation of excitons in NCs placed in an optical cavity. We find that multiphonon emission of carrier governs the decay, resulting in a polariton-induced phonon bottleneck with relaxation time scales that are slower by orders of magnitude compared to the cavity-free case, while the photon fraction plays a secondary role.

Size-Dependent Lattice Symmetry Breaking Determines the Exciton Fine Structure of Perovskite Nanocrystals.

The order of bright and dark excitonic states in lead-halide perovskite nanocrystals is debated. It has been proposed that the Rashba effect, driven by lattice-induced symmetry breaking, causes a bright excitonic ground state. Direct measurements of excitonic spectra, however, show the signatures of a dark ground state, bringing the role of the Rashba effect into question. We use an atomistic theory to model the exciton fine structure of perovskite nanocrystals, accounting for realistic lattice distortions. We calculate optical gaps and excitonic features that compare favorably with experimental works. The exciton fine structure splittings show a nonmonotonic size dependence due to a structural transition between cubic and orthorhombic phases. Additionally, the excitonic ground state is found to be dark with spin triplet character, exhibiting a small Rashba coupling. We additionally explore the effects of nanocrystal shape on the fine structure, clarifying observations on polydisperse nanocrystals.

The Effect of Monomer Size on Fusion and Coupling in Colloidal Quantum Dot Molecules.

The fusion step in the formation of colloidal quantum dot molecules, constructed from two core/shell quantum dots, dictates the coupling strength and hence their properties and enriched functionalities compared to monomers. Herein, studying the monomer size effect on fusion and coupling, we observe a linear relation of the fusion temperature with the inverse nanocrystal radius. This trend, similar to that in nanocrystal melting, emphasizes the role of the surface energy. The suggested fusion mechanism involves intraparticle ripening where atoms diffuse to the reactive connecting neck region. Moreover, the effect of monomer size and neck filling on the degree of electronic coupling is studied by combined atomistic-pseudopotential calculations and optical measurements, uncovering strong coupling effects in small QD dimers, leading to significant optical changes. Understanding and controlling the fusion and hence coupling effect allows tailoring the optical properties of these nanoscale structures, with potential applications in photonic and quantum technologies.

Energy Funneling in a Noninteger Two-Dimensional Perovskite.

Energy funneling is a phenomenon that has been exploited in optoelectronic devices based on low-dimensional materials to improve their performance. Here, we introduce a new class of two-dimensional semiconductor, characterized by multiple regions of varying thickness in a single confined nanostructure with homogeneous composition. This "noninteger 2D semiconductor" was prepared via the structural transformation of two-octahedron-layer-thick (n = 2) 2D cesium lead bromide perovskite nanosheets; it consisted of a central n = 2 region surrounded by edge-lying n = 3 regions, as imaged by electron microscopy. Thicker noninteger 2D CsPbBr3 nanostructures were obtained as well. These noninteger 2D perovskites formed a laterally coupled quantum well band alignment with virtually no strain at the interface and no dielectric barrier, across which unprecedented intramaterial funneling of the photoexcitation energy was observed from the thin to the thick regions using time-resolved absorption and photoluminescence spectroscopy.

Composition-Defined Optical Properties and the Direct-to-Indirect Transition in Core-Shell In1-xGaxP/ZnS Colloidal Quantum Dots.

Semiconductors are commonly divided into materials with direct or indirect band gaps based on the relative positions of the top of the valence band and the bottom of the conduction band in crystal momentum (k) space. It has, however, been debated if k is a useful quantum number to describe the band structure in quantum-confined nanocrystalline systems, which blur the distinction between direct and indirect gap semiconductors. In bulk III-V semiconductor alloys like In1-xGaxP, the band structure can be tuned continuously from the direct- to indirect-gap by changing the value of x. The effect of strong quantum confinement on the direct-to-indirect transition in this system has yet to be established because high-quality colloidal nanocrystal samples have remained inaccessible. Herein, we report one of the first systematic studies of ternary III-V nanocrystals by utilizing an optimized molten-salt In-to-Ga cation exchange protocol to yield bright In1-xGaxP/ZnS core-shell particles with photoluminescence quantum yields exceeding 80%. We performed two-dimensional solid-state NMR studies to assess the alloy homogeneity and the extent of surface oxidation in In1-xGaxP cores. The radiative decay lifetime for In1-xGaxP/ZnS monotonically increases with higher gallium content. Transient absorption studies on In1-xGaxP/ZnS nanocrystals demonstrate signatures of direct- and indirect-like behavior based on the presence or absence, respectively, of excitonic bleach features. Atomistic electronic structure calculations based on the semi-empirical pseudopotential model are used to calculate absorption spectra and radiative lifetimes and evaluate band-edge degeneracy; the resulting calculated electronic properties are consistent with experimental observations. By studying photoluminescence characteristics at elevated temperatures, we demonstrate that a reduced lattice mismatch at the III-V/II-VI core-shell interface can enhance the thermal stability of emission. These insights establish cation exchange in molten inorganic salts as a viable synthetic route to nontoxic, high-quality In1-xGaxP/ZnS QD emitters with desirable optoelectronic properties.

Stochastic Vector Techniques in Ground-State Electronic Structure.

We review a suite of stochastic vector computational approaches for studying the electronic structure of extended condensed matter systems. These techniques help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. While their scope is vast, we limit our study to ground-state and finite temperature density functional theory (DFT) and second-order many-body perturbation theory. More advanced topics, such as quasiparticle (charge) and optical (neutral) excitations and higher-order processes, are covered elsewhere. We start by explaining how to use stochastic vectors in computations, characterizing the associated statistical errors. Next, we show how to estimate the electron density in DFT and discuss effective techniques to reduce statistical errors. Finally, we review the use of stochastic vectors for calculating correlation energies within the second-order Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods.

Structure optimization with stochastic density functional theory.

Linear-scaling techniques for Kohn-Sham density functional theory are essential to describe the ground state properties of extended systems. Still, these techniques often rely on the localization of the density matrix or accurate embedding approaches, limiting their applicability. In contrast, stochastic density functional theory (sDFT) achieves linear- and sub-linear scaling by statistically sampling the ground state density without relying on embedding or imposing localization. In return, ground state observables, such as the forces on the nuclei, fluctuate in sDFT, making optimizing the nuclear structure a highly non-trivial problem. In this work, we combine the most recent noise-reduction schemes for sDFT with stochastic optimization algorithms to perform structure optimization within sDFT. We compare the performance of the stochastic gradient descent approach and its variations (stochastic gradient descent with momentum) with stochastic optimization techniques that rely on the Hessian, such as the stochastic Broyden-Fletcher-Goldfarb-Shanno algorithm. We further provide a detailed assessment of the computational efficiency and its dependence on the optimization parameters of each method for determining the ground state structure of bulk silicon with varying supercell dimensions.

Hybridization and deconfinement in colloidal quantum dot molecules.

The structural, electronic, and optical properties of CdSe/CdS core-shell colloidal quantum dot molecules, a new class of coupled quantum dot dimers, are explored using atomistic approaches. Unlike the case of dimers grown by molecular beam epitaxy, simulated strain profile maps of free-standing colloidal dimers show negligible additional strain resulting from the attachment. The electronic properties of the relaxed dimers are described within a semiempirical pseudopotential model combined with the Bethe-Salpeter equation within the static screening approximation to account for electron-hole correlations. The interplay of strain, hybridization (tunneling splitting), quantum confinement, and electron-hole binding energies on the optical properties is analyzed and discussed. The effects of the dimensions of the neck connecting the two quantum dot building blocks, as well as the shell thickness, are studied.

Simulations of nonradiative processes in semiconductor nanocrystals.

The description of carrier dynamics in spatially confined semiconductor nanocrystals (NCs), which have enhanced electron-hole and exciton-phonon interactions, is a great challenge for modern computational science. These NCs typically contain thousands of atoms and tens of thousands of valence electrons with discrete spectra at low excitation energies, similar to atoms and molecules, that converge to the continuum bulk limit at higher energies. Computational methods developed for molecules are limited to very small nanoclusters, and methods for bulk systems with periodic boundary conditions are not suitable due to the lack of translational symmetry in NCs. This perspective focuses on our recent efforts in developing a unified atomistic model based on the semiempirical pseudopotential approach, which is parameterized by first-principle calculations and validated against experimental measurements, to describe two of the main nonradiative relaxation processes of quantum confined excitons: exciton cooling and Auger recombination. We focus on the description of both electron-hole and exciton-phonon interactions in our approach and discuss the role of size, shape, and interfacing on the electronic properties and dynamics for II-VI and III-V semiconductor NCs.

Interplay of Surface and Interior Modes in Exciton-Phonon Coupling at the Nanoscale.

Exciton-phonon coupling (EXPC) plays a key role in the optoelectronic properties of semiconductor nanocrystals (NCs), but a microscopic picture of EXPC is still lacking, particularly regarding the magnitude and scaling with NC size, the dependence on phonon frequency, and the role of the NC surface. The computational complexity associated with accurately describing excitons and phonons has limited previous theoretical studies of EXPC to small NCs, noninteracting electron-hole models, and/or a small number of phonon modes. Here, we develop an atomistic approach for describing EXPC in NCs of experimentally relevant sizes. We validate our approach by calculating the reorganization energies, a measure of EXPC, for CdSe and CdSe-CdS core-shell NCs, finding good agreement with experimental measurements. We demonstrate that exciton formation distorts the NC lattice primarily along the coordinates of low-frequency acoustic modes. Modes at the NC surface play a significant role in smaller NCs while interior modes dominate for larger systems.

Universal Inverse Scaling of Exciton-Exciton Annihilation Coefficient with Exciton Lifetime.

Be it for essential everyday applications such as bright light-emitting devices or to achieve Bose-Einstein condensation, materials in which high densities of excitons recombine radiatively are crucially important. However, in all excitonic materials, exciton-exciton annihilation (EEA) becomes the dominant loss mechanism at high densities. Typically, a macroscopic parameter named EEA coefficient (CEEA) is used to compare EEA rates between materials at the same density; higher CEEA implies higher EEA rate. Here, we find that the reported values of CEEA for 140 different materials is inversely related to the single-exciton lifetime. Since during EEA one exciton must relax to ground state, CEEA is proportional to the single-exciton recombination rate. This leads to the counterintuitive observation that the exciton density at which EEA starts to dominate is higher in a material with larger CEEA. These results broaden our understanding of EEA across different material systems and provide a vantage point for future excitonic materials and devices.

Structural relaxation in quantum supercooled liquids: A mode-coupling approach.

We study supercooled dynamics in a quantum hard-sphere liquid using quantum mode-coupling formulation. In the moderate quantum regime, classical cage effects lead to slower dynamics compared to the strongly quantum regime, where tunneling overcomes classical caging, leading to faster relaxation. As a result, the glass transition critical density can become significantly higher than for the classical liquids. A perturbative approach is used to solve time dependent quantum mode-coupling equations to study in detail the dynamics of the supercooled liquid in the moderate quantum regime. Similar to the classical case, the relaxation time shows the power-law increase with the increase in the density in the supercooled regime. However, the power-law exponent is found to be dependent on the quantumness; it increases linearly as the quantumness is increased in the moderate quantum regime.

Stochastic density functional theory: Real- and energy-space fragmentation for noise reduction.

Stochastic density functional theory (sDFT) is becoming a valuable tool for studying ground-state properties of extended materials. The computational complexity of describing the Kohn-Sham orbitals is replaced by introducing a set of random (stochastic) orbitals leading to linear and often sub-linear scaling of certain ground-state observables at the account of introducing a statistical error. Schemes to reduce the noise are essential, for example, for determining the structure using the forces obtained from sDFT. Recently, we have introduced two embedding schemes to mitigate the statistical fluctuations in the electron density and resultant forces on the nuclei. Both techniques were based on fragmenting the system either in real space or slicing the occupied space into energy windows, allowing for a significant reduction in the statistical fluctuations. For chemical accuracy, further reduction of the noise is required, which could be achieved by increasing the number of stochastic orbitals. However, the convergence is relatively slow as the statistical error scales as 1/Nχ according to the central limit theorem, where Nχ is the number of random orbitals. In this paper, we combined the embedding schemes mentioned above and introduced a new approach that builds on overlapped fragments and energy windows. The new approach significantly lowers the noise for ground-state properties, such as the electron density, total energy, and forces on the nuclei, as demonstrated for a G-center in bulk silicon.

Nonuniqueness of generalized quantum master equations for a single observable.

When deriving exact generalized master equations for the evolution of a reduced set of degrees of freedom, one is free to choose what quantities are relevant by specifying projection operators. However, obtaining a reduced description does not always need to be achieved through projections-one can also use conservation laws for this purpose. Such an operation should be considered as distinct from any kind of projection; that is, projection onto a single observable yields a different form of master equation compared to that resulting from a projection followed by the application of a constraint. We give a simple example to show this point and give relationships that the different memory kernels must satisfy to yield the same dynamics.

Frequency-dependent specific heat in quantum supercooled liquids: A mode-coupling study.

Frequency-dependence of specific heat in supercooled hard sphere liquid is computed using quantum mode-coupling theory (QMCT). Mode-coupling equations are solved using a recently proposed perturbative method that allows us to study relaxation in the moderate quantum regime where quantum effects assist liquid to glass transition. Zwanzig's formulation is used to compute the frequency-dependent specific heat in the supercooled state using dynamical information from QMCT. Specific heat shows strong variation as the quantumness of the liquid is changed, which becomes more significant as density is increased. It is found that, near the transition point, different dynamical modes contribute to specific heat in classical and quantum liquids.

Tempering stochastic density functional theory.

We introduce a tempering approach with stochastic density functional theory (sDFT), labeled t-sDFT, which reduces the statistical errors in the estimates of observable expectation values. This is achieved by rewriting the electronic density as a sum of a "warm" component complemented by "colder" correction(s). Since the warm component is larger in magnitude but faster to evaluate, we use many more stochastic orbitals for its evaluation than for the smaller-sized colder correction(s). This results in a significant reduction in the statistical fluctuations and systematic deviation compared to sDFT for the same computational effort. We demonstrate the method's performance on large hydrogen-passivated silicon nanocrystals, finding a reduction in the systematic deviation in the energy by more than an order of magnitude, while the systematic deviation in the forces is also quenched. Similarly, the statistical fluctuations are reduced by factors of ≈4-5 for the total energy and ≈1.5-2 for the forces on the atoms. Since the embedding in t-sDFT is fully stochastic, it is possible to combine t-sDFT with other variants of sDFT such as energy-window sDFT and embedded-fragmented sDFT.

Stochastically Realized Observables for Excitonic Molecular Aggregates.

We show that a stochastic approach enables calculations of the optical properties of large 2-dimensional and nanotubular excitonic molecular aggregates. Previous studies of such systems relied on numerically diagonalizing the dense and disordered Frenkel Hamiltonian, which scales approximately as O(N3) for N dye molecules. Our approach scales much more efficiently as O(Nlog(N)), enabling quick study of systems with a million of coupled molecules on the micrometer size scale. We calculate several important experimental observables, including the optical absorption spectrum and density of states, and develop a stochastic formalism for the participation ratio. Quantitative agreement with traditional matrix diagonalization methods is demonstrated for both small- and intermediate-size systems. The stochastic methodology enables the study of the effects of spatial-correlation in site energies on the optical signatures of large 2D aggregates. Our results demonstrate that stochastic methods present a path forward for screening structural parameters and validating experiments and theoretical predictions in large excitonic aggregates.

Efficient Langevin dynamics for "noisy" forces.

Efficient Boltzmann-sampling using first-principles methods is challenging for extended systems due to the steep scaling of electronic structure methods with the system size. Stochastic approaches provide a gentler system-size dependency at the cost of introducing "noisy" forces, which could limit the efficiency of the sampling. When the forces are deterministic, the first-order Langevin dynamics (FOLD) offers efficient sampling by combining a well-chosen preconditioning matrix S with a time-step-bias-mitigating propagator [G. Mazzola and S. Sorella, Phys. Rev. Lett. 118, 015703 (2017)]. However, when forces are noisy, S is set equal to the force-covariance matrix, a procedure that severely limits the efficiency and the stability of the sampling. Here, we develop a new, general, optimal, and stable sampling approach for FOLD under noisy forces. We apply it for silicon nanocrystals treated with stochastic density functional theory and show efficiency improvements by an order-of-magnitude.

Range-separated stochastic resolution of identity: Formulation and application to second-order Green's function theory.

We develop a range-separated stochastic resolution of identity (RS-SRI) approach for the four-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic RI and the remaining terms are treated using a SRI. The approach is implemented within a second-order Green's function formalism with an improved O(N3) scaling with the size of the basis set, N. Moreover, the RS approach greatly reduces the statistical error compared to the full stochastic version [T. Y. Takeshita et al., J. Chem. Phys. 151, 044114 (2019)], resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains and water clusters.