THz Response of Charge Carriers in Nanoparticles: Microscopic Master Equations Reveal an Unexplored Equilibration Current and Nonlinear Mobility Regimes

Abstract

Herein, the THz mobility of charge carriers in low‐dimensional semiconductors based on a density matrix approach involving master equations for population and polarization dynamics is modeled. Pulsed THz fields induce intraband transitions between quantized subband states, creating polarization and subsequent charge transport that governs the electron mobility. It is shown that an equilibration current emerges—a purely quantum mechanical contribution understood via the Ehrenfest theorem in 1D—reshaping the low‐frequency mobility. Apart from thermal population, the results further demonstrate that the frequency‐dependent mobility becomes THz field strength and spectrum, as well as pulse width and chirp dependent, already at moderate THz probe fields of 1 kV cm−1, e.g., for 1D CdSe or GaAs nanostructures. The parametric nature of the underlying master differential equations for polarization and population, driving the intraband conductivity, results in a nonlinear, third‐order mobility and susceptibility, causing a nontrivial field dependence as well as power broadening even at moderate field strength. The obtained results are in good agreement with experiments. The observed high nonlinearities strongly impact the design and interpretation of THz charge carrier mobility experiments and further allow applications like coherent control, frequency mixing, or synthesis through a field‐controlled nonlinearity or high harmonics generation, especially interesting for future 6G telecommunication.

Challenging Kubo-Greenwood Theory: Nonunitarian Dynamics Reshapes THz Conductivity and Transport in Nanosystems

“…In Figure , we compare our extended Kubo-Greenwood model with the experimental data of different samples. In (a) we contrast the results with four-level master equation modeling based on ref for 18.5 nm long CdSe wires (details are given in the Supporting Information, Section S5). Although both models agree with the data convincingly, we point out that the calculation effort is much smaller with the generalized Kubo-Greenwood model as otherwise for N -levels ( N 2 + N – 2)/2 coupled parametric differential equations need to be solved instead of calculating simply N 2 time-independent matrix elements.…”

“…Although both models agree with the data convincingly, we point out that the calculation effort is much smaller with the generalized Kubo-Greenwood model as otherwise for N -levels ( N 2 + N – 2)/2 coupled parametric differential equations need to be solved instead of calculating simply N 2 time-independent matrix elements. Slight differences between the models arise due to the linear response nature of the extended Kubo-Greenwood model, while the master equations account for slight nonlinear mobility contributions at the presumed THz field strength of 1 kV/cm.…”

“…However, they deliver only limited insights into the basis of the observed transport properties. Quantum mechanical formulations to mobility and polarizability in nanosystems ,− have been demonstrated based on Kubo-Greenwood theory or master equations. In the latter case, a new current contributionthe equilibration currenthas been found, driving the quantum system back to thermal equilibrium and suppressing low-frequency conductivity.…”

THz mobility and polarizability: impact of transformation and dephasing on the spectral response of excitons in a 2D semiconductor

“…A changing dipole moment M⃑ ( t ), caused by interaction with a THz field of polarization e⃑ , induces a relative motion of electron and hole with a velocity v⃑ rel .Hence, transitions occur between states of relative motion, for which we rewrite eqn (4) in terms of a quantum mechanical matrix element representing an intraexcitonic transition .Alonso and De Vincenzo first pointed out that – in the case of a non-Hermitean operator – the temporal derivative of the spatial expectation value no longer obeys the correspondence principle through the Ehrenfest Theorem in its known form (〈 v 〉 = d〈 x 〉/d t = 〈 p 〉/ m ). 34–36 Instead, it has to be extended, implementing an additional term to read in our specific excitonic case . We refer to the ESI,† Section S1, for a detailed evaluation of the matrix element's time derivative.…”

“…j 0 j i. Alonso and De Vincenzo first pointed out that -in the case of a non-Hermitean operator -the temporal derivative of the spatial expectation value no longer obeys the correspondence principle through the Ehrenfest Theorem in its known form (hvi = dhxi/dt = hpi/m). [34][35][36] Instead, it has to be extended, implementing an additional term h vðtÞ g i to read in our specific excitonic case dh rðtÞi=dt ¼ h pðtÞi=m r þ h vðtÞ g i. We refer to the ESI, † Section S1, for a detailed evaluation of the matrix element's time derivative.…”