Rate Equation Analysis of the Dynamics of First-order Exciton Mott Transition

Abstract

We performed a rate equation analysis on the dynamics of exciton Mott transition (EMT) with assuming a detailed balance between excitons and unbound electron-hole (e-h) pairs. Based on the Saha equation with taking into account the empirical expression for the band-gap renormalization effect caused by the unbound e-h pairs, we show that the ionization ratio of excitons exhibits a bistability as a function of total e-h pair density at low temperatures. We demonstrate that an incubation time emerges in the dynamics of EMT from oversaturated exciton gas phase on the verge of the bistable region. The incubation time shows a slowing down behavior when the pair density approaches toward the saddle-node bifurcation of the hysteresis curve of the exciton ionization ratio.Photocontrol of condensed matter systems has gained continuing interests over decades, [1][2][3] e.g. in semiconductors, 4,5) Mott insulators, 3,6-9) spin crossover complexes, [10][11][12] and superconductors. 13,14) Among diverse material systems, photoexcited electron-hole (e-h) systems in semiconductors offer a unique arena to study the rich variety of phases emergent in many particle systems, and their non-equilibrium dynamics. 4,15) One of the intriguing aspects of e-h systems is that the strength of inter-particle Coulomb interaction can be effectively controlled by changing the density of photoexcited carriers through the screening effect, by simply changing the excitation light intensity. The change of the Coulomb interaction causes a phase transition, or crossover, from the insulating exciton gas phase in the low density regime to the metallic e-h plasma in the high density regime, referred to as exciton Mott transition (EMT). There is a long-standing argument whether EMT is a crossover or a first-order phase transition accompanied by a bistability at sufficiently low temperatures. [16][17][18][19][20][21][22] In general, bistability appears in diverse systems such as electronic circuits, optoelectronic systems, magnetic systems, molecular and biological systems, etc. Figure

The photophysics of Ruddlesden-Popper perovskites: A tale of energy, charges, and spins

“…In typical Saha equation analysis where the EBE is fixed, at a given temperature, typically free carriers dominate at low 𝑛 tot , while it becomes increasingly excitonic with increasing 𝑛 tot .However, when the effect of bandgap renormalizations and screenings as a function of excitation densities are fully incorporated, Cingolani et al demonstrated an inverse transition trend from exciton to free-carrier with increasing 𝑛 tot ,[86]. Similar results were also reported in theoretical studies by Sekiguchi et al[87], where density-dependent EBE could flip the typical Saha trend.The deviation of experimental results of branching ratio values in RPs implies that it is not as simple as a prediction from the Saha equation. Thus, the problem remains: How can we determine the branching ratio between exciton and free-carriers in RP perovskites?…”

Exciton Mott transition revisited

“…However, within a small region in the phase diagram at low exciton binding energy, we also find bistability, which would correspond to a sudden transformation of the exciton gas into the electron-hole liquid without phase-separation. We mentioned that experimentally there are both evidences of continuos exciton Mott transitions, i.e., phase separation [11,13,14,28], as well as of discontinuous ones [6,7,9,10,35,63,65]. The discriminant parameter might well be the exciton binding energy E ex , as we do find, since a discontinuous transition is mostly observed in bulk semiconductors, while a continuous one in confined geometries, like quantum wells, where E ex is supposedly larger.…”

“…However, the concurrent growth of screening might lead to an avalanche effect [3] and thus to an abrupt transition into the EHL. This scenario could reveal itself either by the existence of a Mott transition distinct from the gas-liquid one, as Landau and Zeldovich originally proposed for liquid mercury [5], or through a bistability [6]. Experimentally, the nature of the transition, which can be studied by photoluminescence or optical absorption, is till now rather controversial.…”

An excitonic model for the electron–hole plasma relaxation in proton-irradiated insulators

“…This is due to the extra Coulomb screening which is provided by the excited carriers which reduces the effective attraction between electrons and holes (free or bound). As a consequence, both the band gap and the exciton binding energy tend to decrease as the electron-hole pair density increases [41,42] . The two effects compensate, and it is found experimentally that the exciton absorption frequency does not depend on the electron-hole pair density [38].…”

“…(7) in the limit of k 2 = 0, we obtain that k −1 /k 1 = n 2 /n x . Following Sekiguchi and Shimano [42], we assume that k 1 only depends on the electronic temperature, T n , and the lattice temperature, T L . The assumption is consistent with the socalled columnar model of recombination [48] which uses the Langevin's estimate k 1 ≈ q (M n + M p ) / 0 r , where M n and M p are the electron and hole mobilities, respectively.…”