Nonlocal Electron Transport Research Articles

Charged particle transport coefficient challenges in high energy density plasmas

High energy density physics (HEDP) and inertial confinement fusion (ICF) research typically relies on computational modeling using radiation-hydrodynamics codes in order to design experiments and understand their results. These tools, in turn, rely on numerous charged particle transport and relaxation coefficients to account for laser energy absorption, viscous dissipation, mass transport, thermal conduction, electrical conduction, non-local ion (including charged fusion product) transport, non-local electron transport, magnetohydrodynamics, multi-ion-species thermalization, and electron-ion equilibration. In many situations, these coefficients couple to other physics, such as imposed or self-generated magnetic fields. Furthermore, how these coefficients combine are sensitive to plasma conditions as well as how materials are distributed within a computational cell. Uncertainties in these coefficients and how they couple to other physics could explain many of the discrepancies between simulation predictions and experimental results that persist in even the most detailed calculations. This paper reviews the challenges faced by radiation-hydrodynamics in predicting the results of HEDP and ICF experiments with regard to these and other physics models typically included in simulation codes.

Coulomb-assisted nonlocal electron transport between two pairs of Majorana bound states in a superconducting island

We investigate the nonlocal transport modulated by Coulomb interactions in devices comprising two interacting Majorana wires, where both nanowires are in proximity to a mesoscopic superconducting (SC) island. Each Majorana bound state (MBS) is coupled to one lead via a quantum dot with resonant levels. In this device, the nonlocal correlations can be induced in the absence of Majorana energy splitting. We find that the negative differential conductance and giant current noise cross correlation could be induced, due to the interplay between nonlocality of MBSs and dynamical Coulomb blockade effect. This feature may provide a signature for the existence of the MBSs.

The fastVFP code for solution of the Vlasov–Fokker–Planck equation

We describe the fastVFP code for solution of the Vlasov–Fokker–Planck equation for non-local electron transport and the generation of magnetic field, especially for application to laser-produced plasmas. We describe the essential features of the code that make it fast and robust and suitable for inclusion as a transport package in a fluid simulation. We present a few sample results that demonstrate the abilities of the code.

Time-dependent density-functional-theory calculations of the nonlocal electron stopping range for inertial confinement fusion applications.

Nonlocal electron transport is important for understanding laser-target coupling for laser-direct-drive (LDD) inertial confinement fusion (ICF) simulations. Current models for the nonlocal electron mean free path in radiation-hydrodynamic codes are based on plasma-physics models developed decades ago; improvements are needed to accurately predict the electron conduction in LDD simulations of ICF target implosions. We utilized time-dependent density functional theory (TD-DFT) to calculate the electron stopping power (SP) in the so-called conduction-zone plasmas of polystyrene in a wide range of densities and temperatures relevant to LDD. Compared with the modified Lee-More model, the TD-DFT calculations indicated a lower SP and a higher stopping range for nonlocal electrons. We fit these electron SP calculations to obtain a global analytical model for the electron stopping range as a function of plasma conditions and the nonlocal electron kinetic energy. This model was implemented in the one-dimensional radiation-hydrodynamic code lilac to perform simulations of LDD ICF implosions, which are further compared with simulations by the standard modified Lee-More model. Results from these integrated simulations are discussed in terms of the implications of this TD-DFT-based mean-free-path model to ICF simulations.

Effect of non-local transport of hot electrons on the laser-target ablation

The non-local heat transport of hot electrons during high-intensity laser interaction with plasmas can preheat the fuel and limit the heat flow in inertial confinement fusion. It increases the entropy of the fuel and decreases the final compression. In this paper, the non-local electron transport model that is based on the improved Schurtz–Nicolaï–Busquet (SNB) algorithm has been embedded into the radiation hydrodynamic code and is benchmarked with two classical non-local transport cases. Then, we studied a 2ω laser ablating a CH target by using the non-local module. It is found that the non-local effect becomes significant when the laser intensity is above 1×1014W/cm2. The mass ablation rate from the SNB model is increased compared to that of the flux-limited model due to the lower coronal plasma temperature. This non-local model has a better agreement with the experimental results compared to that of the flux-limited model. The non-local transport is strongly dependent on the laser frequency, and the thresholds that the non-local transport should be considered are obtained for lasers of different frequencies. The appropriate flux-limiters that should be employed in the flux-limited model for different lasers are also presented. The results here should have a good reference for the laser-target ablation applications.

A nonlocal electron transport model in the diffusion scaling of hydrodynamics

We analyze a model of nonlocal electron transport named P1-diffusion based on a spherical harmonic expansion in velocity space and a diffusion scaling, which makes it compatible with assumptions from magneto-hydrodynamics (MHD). An iterative, fully implicit (CFL-free, as defined by the Courant Friedrich Levy condition) and asymptotic preserving discretization is proposed, which necessitates the inversion of a possibly large number of—but small—linear systems. It is found accurate with respect to reference solutions from a Vlasov–Fokker–Planck–Maxwell code (based on a Polynomial expansion of order N, or PN expansion) on a series of tests, which are representative of the conduction zone in laser-created plasmas. Thereby, the present approach is a good candidate for being embedded in multi-D MHD codes.

Thermal transport modeling of laser-irradiated spheres

Thermal transport of uniformly laser-irradiated spheres of various materials is investigated computationally. One-dimensional simulations of low- to mid-Z materials (Be, Al, and Cu) are performed to evaluate the impact of nonlocal electron transport on experimental observables under laser intensities of relevance to direct-drive inertial confinement fusion. We compare thermal transport models of different levels of fidelity: flux-limited Spitzer–Harm diffusion, the Schurtz–Nicolai–Busquet (SNB) reduced-order nonlocal model, and a Fokker–Planck description. Spitzer–Harm diffusion with different flux-limiter factors are compared with different implementations of the SNB model in the HYDRA radiation hydrodynamics code. Under the conditions of interest, the peak heat flux in the thermal front with the SNB model shows good agreement with Fokker–Planck calculations, with the largest errors below 10% at 1015 W/cm2 laser intensity. From HYDRA-SNB simulations, two experimentally relevant effects are observed from nonlocal heat transport when compared to flux-limited Spitzer–Harm modeling: coronal temperatures are cooler due to reduced heat fluxes in the expanding plasma and (for mid-Z materials) x-ray emissions are enhanced due to preheating in the dense plasma.

Quantum-correlated photons generated by nonlocal electron transport

Since the realization of high-quality microwave cavities coupled to quantum dots, one can envisage the possibility to investigate the coherent interaction of light and matter in semiconductor quantum devices. Here we study a parallel double quantum dot device operating as single-electron splitter interferometer, with each dot coupled to a local photon cavity. We explore how quantum correlation and entanglement between the two separated cavities are generated by the coherent transport of a single electron passing simultaneously through the two different dots. We calculate the covariance of the cavity occupations by use of a diagrammatic perturbative expansion based on Keldysh Green's functions to the fourth order in the dot-cavity interaction strength, taking into account vertex diagrams. In this way, we demonstrate the creation of entanglement by showing that the classical Cauchy-Schwarz inequality is violated if the energy levels of the two dots are almost degenerate. For large level detunings or a single dot coupled to two cavities, we show that the inequality is not violated.

Negative nonlocal and local voltages (resistances) in a quasi-one-dimensional superconducting aluminum structure

To study a nonlocal electron transport in an aluminum superconducting quasi-one-dimensional structure, we measured negative nonlocal (local) direct current voltages in the structure in a magnetic field near the critical temperature. The structure is a normal-superconducting at Tcn<T<Tcw (Tcn and Tcw are the critical temperatures for narrow and wide wires, respectively, making up this structure). Negative voltage arises due to a quasiparticle current flowing through the N-S interface. We plotted the experimental and theoretical temperature and magnetic-field dependences of current, resistance and voltage corresponding to the peak of negative voltage, taking into account either equilibrium or nonequilibrium superconducting fluctuations.

A selection of theoretical results in the context of laser-plasma interaction and inertial fusion

I recall a selection of my results of the last 40 years. I first present a theoretical model of absorption of laser light by a plasma, with an emphasis on its dependence on the laser intensity and wavelength. A second topic concerns the nonlinear and nonlocal electron transport in steep temperature gradients. Thirdly, I present the work on the propagation of a short intense laser pulse in tenuous plasmas. Finally, I discuss the plasma expansion into a vacuum, in particular the structure of the ion front and the prediction of the maximum velocity attained by the ions in the expansion.

Cross-correlated shot noise in three-terminal superconducting hybrid nanostructures

We work out a unified theory describing both non-local electron transport and cross-correlated shot noise in a three-terminal normal-superconducting-normal (NSN) hybrid nanostructure. We describe noise cross correlations both for subgap and overgap bias voltages and for arbitrary distribution of channel transmissions in NS contacts. We specifically address a physically important situation of diffusive contacts and demonstrate a non-trivial behavior of non-local shot noise exhibiting both positive and negative cross correlations depending on the bias voltages. For this case, we derive a relatively simple analytical expression for cross-correlated noise power which contains only experimentally accessible parameters.

Transition from nonlocal electron transport to radiative regime in an expanding blast wave

We have investigated the formation, evolution, and late-time propagation of a laser-generated cylindrical blast wave (BW). The whole blast wave evolution over timescales of several nanoseconds was reconstructed experimentally (via temporally resolved interferometric measurements) and via hydrodynamic simulations that included modeling of nonlocal electron transport and radiation diffusion. Comparison between the experimental results and the simulations indicates that the early expansion phase is characterised by nonlocal electron heat transport causing energy spread on times shorter than the typical timescales for hydrodynamic expansion. Nonlocal electron transport ionizes the gas ahead of the plasma front and gives rise to a smooth radial density gradient. At later times, once the shock is launched and the BW is formed, radiation results in reduced shock velocity compared to the adiabatic case. These investigations provide a suitable and effective platform to benchmark the inclusion of kinetic and radiative effects in fluid modeling of the plasma dynamics over timescales that may be inaccessible to fully kinetic simulations.

Simulations of nonlocal electron transport in cylindrical and spherical thermal waves

In spherical and cylindrical plasmas, the relaxation processes of small amplitude temperature perturbations are simulated with electron Fokker–Planck equation for a wide range of the collision parameter and different charged number Z. The computed electron nonlocal thermal conductivities for spherical and cylindrical plasmas are then compared and analyzed. Based on our sufficient numerical simulations, a practical formula of nonlocal heat flux is presented for a wide range of the collision parameter and different charged number Z, which is in good agreement with the published results.

Non-local electron transport through normal and topological ladder-like atomic systems

We propose a locally protected ladder-like atomic system (nanoconductor) on a substrate that is insensitive to external perturbations. The system corresponds to coupled atomic chains fabricated on different surfaces. Electron transport properties of such conductors are studied theoretically using the model tight-binding Su-Schriffer-Hegger (SSH) Hamiltonian and Green's function formalism. We have found that the conductance of the system is almost insensitive to single adatoms and oscillates as a function of the side chain length with very large periods. Non-local character of the electron transport was observed also for topological SSH chains where nontrivial end states survive in the presence of disturbances as well as for different substrates. We have found that the careful inspection of the density of states or charge waves can provide the information about the atom energy levels and hopping amplitudes. Moreover, the ladder-like geometry allows one to distinguish between normal and topological zero-energy states. It is important that topological chains do not reveal Friedel oscillations which are observed in non-topological chains.

Measurement of Preheat Due to Nonlocal Electron Transport in Warm Dense Matter.

This Letter presents a novel approach to study electron transport in warm dense matter. It also includes the first x-ray Thomson scattering (XRTS) measurement from low-density CH foams compressed by a strong laser-driven shock at the OMEGA laser facility. The XRTS measurement is combined with velocity interferometry (VISAR) and optical pyrometry (SOP) providing a robust measurement of thermodynamic conditions in the shock. Evidence of significant preheat contributing to elevated temperatures reaching 17.5-35eV in shocked CH foam is measured by XRTS. These measurements are complemented by abnormally high shock velocities observed by VISAR and early emission seen by SOP. These results are compared to radiation hydrodynamics simulations that include first-principles treatment of nonlocal electron transport in warm dense matter with excellent agreement. Additional simulations confirm that the x-ray contribution to this preheat is negligible.

An improved and fully implicit multi-group non-local electron transport model and its validations

The combined effect of thermal flux inhibition and non-local electron heat flux in the radiation hydrodynamics (RHD) simulation of laser-driven systems can be accurately predicted by using non-local electron transport (NLET) models. These models can avoid commonly used space and time-independent ad-hoc flux-limiting procedures. However, the use of classical electron collision frequency in these models is rigorously valid for high temperature non-degenerate plasmas. In laser-driven systems, the electron thermal energy transport is important in regions between the critical density and ablation surface where the plasma is partially degenerate. Therefore, an improved model for electron collision frequency in this regime is required to accurately predict the thermal energy transport. Previously, we have reported an improved single group non-local electron transport model by using a wide-range electron collision frequency model valid from warm-dense matter (WDM) to fully ionized plasmas. In this work, we have extended this idea into a two-dimensional multi-group non-local electron transport (MG-NLET) model. Moreover, we have used a fully implicit numerical integration scheme in which the models for multi-group thermal radiation transport, laser absorption, electron-ion thermal energy relaxation and ion heat conduction are included in a single step. The performance of this improved MG-NLET model has been assessed by comparing the simulated foil trajectories with the reported experimental data for laser-driven plastic foils. The results indicate that the improved model yields results that are in better agreement with the experimental data.

A comparison of non-local electron transport models for laser-plasmas relevant to inertial confinement fusion

We compare the reduced non-local electron transport model developed by Schurtz et al. [Phys. Plasmas 7, 4238 (2000)] to Vlasov-Fokker-Planck simulations. Two new test cases are considered: the propagation of a heat wave through a high density region into a lower density gas, and a one-dimensional hohlraum ablation problem. We find that the reduced model reproduces the peak heat flux well in the ablation region but significantly over-predicts the coronal preheat. The suitability of the reduced model for computing non-local transport effects other than thermal conductivity is considered by comparing the computed distribution function to the Vlasov-Fokker-Planck distribution function. It is shown that even when the reduced model reproduces the correct heat flux, the distribution function is significantly different to the Vlasov-Fokker-Planck prediction. Two simple modifications are considered which improve agreement between models in the coronal region.

Improved robustness study of a shock ignited target, with DUED code including non-local electron transport and 3D laser ray-tracing

Accurate descriptions of laser power coupling to the plasma and electron energy transport are crucial for designing shock-ignition targets and assessing their robustness (in particular with regard to laser and positioning errors). To this purpose, the 2D DUED laser fusion code has been improved with the inclusion of a 3D laser ray-tracing scheme and a model for non-local electron transport. 2D simulations with the upgraded code are presented; the dependence of the fusion yield vs target displacement is studied. Two different irradiation configurations are considered.

Laser irradiance scaling in polar direct drive implosions on the National Ignition Facility

Polar-direct-drive experiments conducted at the National Ignition Facility [E. I. Moses, Fusion Sci. Technol. 54, 361 (2008)] performed at laser irradiance between 1 and 2×1015 W/cm2 exhibit increased hard x-ray emission, decreased neutron yield, and reduced areal density as the irradiance is increased. Experimental x-ray images at the higher irradiances show x-ray emission at the equator, as well as degraded symmetry, that is not predicted in hydrodynamic simulations using flux-limited energy transport, but that appear when non-local electron transport together with a model to account for cross beam energy transfer (CBET) is utilized. The reduction in laser power for equatorial beams required in the simulations to reproduce the effects of CBET on the observed symmetry also reproduces the yield degradation consistent with experimental data.

Majorana-assisted nonlocal electron transport through a floating topological superconductor

The nonlocal nature of the fermionic mode spanned by a pair of Majorana bound states in a one-dimensional topological superconductor has inspired many proposals aiming at demonstrating this property in transport. In particular, transport through the mode from a lead attached to the left bound state to a lead attached to the right will result in current cross-correlations. For ideal zero modes on a grounded superconductor, the cross-correlations are however completely suppressed in favor of purely local Andreev reflection. In order to obtain a non-vanishing cross-correlation, previous studies have required the presence of an additional global charging energy. Adding nonlocal terms in the form of a global charging energy to the Hamiltonian when testing the intrinsic nonlocality of the Majorana modes seems to be conceptually troublesome. Here, we show that a floating superconductor allows to observe nonlocal current correlations in the absence of charging energy. We show that the non-interacting and the Coulomb-blockade regime have the same peak conductance $e^2/h$ but different shot-noise power; while the shot noise is sub-Poissonian in the Coulomb-blockade regime in the large bias limit, Poissonian shot noise is generically obtained in the non-interacting case.