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Physical review. B, Condensed matter

Quasiparticle lifetime in a two-dimensional electron system in the limit of low temperature and excitation energy

David Menashe, Boris Laikhtman

Published Oct 15, 1996• 25 citations
Publisher

Abstract

We study theoretically the quasiparticle lifetime, due to Coulomb scattering, in a two-dimensional electron system. It has long been established that in the limit of low excitation energy and temperature, the inverse lifetime behaves as ${\mathrm{\ensuremath{\tau}}}_{\mathrm{ee}}^{\mathrm{\ensuremath{-}}1}$(p)\ensuremath{\propto}(\ensuremath{\Delta}/${\mathrm{\ensuremath{\epsilon}}}_{\mathit{F}}$${)}^{2}$ln(${\mathrm{\ensuremath{\epsilon}}}_{\mathit{F}}$/\ensuremath{\Delta}), where \ensuremath{\Delta}\ensuremath{\equiv}max(|${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\epsilon}}}_{\mathit{F}}$|,${\mathit{k}}_{\mathit{BT}}$). This result has been obtained in the leading order of the Coulomb interaction. We show that higher-order terms in the interaction contribute higher-order logarithmic factors, meaning that a correct theory must sum up all orders of the interaction. After performing this summation, using the Keldysh diagram technique, we find that the renormalized lifetime still has the same general form as above. On the other hand we find that only forward scattering contributes to the lifetime, whereas, in leading-order theory, both forward and backward scattering make a contribution (a similar result has already been obtained for the two-dimensional Hubbard model). This leads to a different proportionality constant for the lifetime. Because we employ the Keldysh diagram technique, we are also able to give an expression for the collision operator, which can be used in the study of near-equilibrium processes. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.11561

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We study theoretically the quasiparticle lifetime, due to Coulomb scattering, in a two-dimensional electron system. It has long been established that in the limit of low excitation energy and temperature, the inverse lifetime behaves as \ensuremathτee\ensuremath−1{\mathrm{\ensuremath{\tau}}}_{\mathrm{ee}}^{\mathrm{\ensuremath{-}}1}\ensuremathτee\ensuremath−1​(p)\ensuremath{\propto}(\ensuremath{\Delta}/{\mathrm{\ensuremath{\epsilon}}}_{\mathit{F}}$${)}^{2}ln(\ensuremathϵF{\mathrm{\ensuremath{\epsilon}}}_{\mathit{F}}\ensuremathϵF​/\ensuremath{\Delta}), where \ensuremath{\Delta}\ensuremath{\equiv}max(|\ensuremathϵp{\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}\ensuremathϵp​-\ensuremathϵF{\mathrm{\ensuremath{\epsilon}}}_{\mathit{F}}\ensuremathϵF​|,kBT{\mathit{k}}_{\mathit{BT}}kBT​). This result has been obtained in the leading order of the Coulomb interaction. We show that higher-order terms in the interaction contribute higher-order logarithmic factors, meaning that a correct theory must sum up all orders of the interaction. After performing this summation, using the Keldysh diagram technique, we find that the renormalized lifetime still has the same general form as above. On the other hand we find that only forward scattering contributes to the lifetime, whereas, in leading-order theory, both forward and backward scattering make a contribution (a similar result has already been obtained for the two-dimensional Hubbard model). This leads to a different proportionality constant for the lifetime. Because we employ the Keldysh diagram technique, we are also able to give an expression for the collision operator, which can be used in the study of near-equilibrium processes. \textcopyright{} 1996 The American Physical Society.
We calculate the inelastic lifetime of an electron quasiparticle due to Coulomb interactions in an electron liquid at low (or zero) temperature in two and three spatial dimensions. The contribution of ``exchange'' processes is calculated analytically and is shown to be non-negligible even in the high-density limit in two dimensions. Exchange effects must therefore be taken into account in a quantitative comparison between theory and experiment. The derivation in the two-dimensional case is presented in detail in order to clarify the origin of the disagreements that exist among the results of previous calculations, even the ones that only took into account ``direct'' processes.
The electron self-energy for long-range Coulomb interactions plays a crucial role in understanding the many-body physics of interacting electron systems (e.g., in metals and semiconductors) and has been studied extensively for decades. In fact, it is among the oldest and the most-investigated many-body problems in physics. However, there is a lack of an analytical expression for the self-energy $Re{\mathrm{\ensuremath{\Sigma}}}^{(R)}(\ensuremath{\varepsilon},T)$ when energy $\ensuremath{\varepsilon}$ and temperature kBT{k}_{\mathsf{B}}TkB​T are arbitrary with respect to each other (while both being still small compared with the Fermi energy). We revisit this problem and calculate analytically the self-energy on the mass shell for a two-dimensional electron system with Coulomb interactions in the high density limit ${r}_{s}\ensuremath{\ll}1$, for temperature ${r}_{s}^{3/2}\ensuremath{\ll}{k}_{\mathsf{B}}T/{E}_{\mathsf{F}}\ensuremath{\ll}{r}_{s}$ and energy ${r}_{s}^{3/2}\ensuremath{\ll}|\ensuremath{\varepsilon}|/{E}_{\mathsf{F}}\ensuremath{\ll}{r}_{s}$. We provide the exact high-density analytical expressions for the real and imaginary parts of the electron self-energy with arbitrary value of $\ensuremath{\varepsilon}/{k}_{\mathsf{B}}T$, to the leading order in the dimensionless Coulomb coupling constant rs{r}_{s}rs​, and to several higher than leading orders in kBT/rsEF{k}_{\mathsf{B}}T/{r}_{s}{E}_{\mathsf{F}}kB​T/rs​EF​ and $\ensuremath{\varepsilon}/{r}_{s}{E}_{\mathsf{F}}$. We also obtain the asymptotic behavior of the self-energy in the regimes $|\ensuremath{\varepsilon}|\ensuremath{\ll}{k}_{\mathsf{B}}T$ and $|\ensuremath{\varepsilon}|\ensuremath{\gg}{k}_{\mathsf{B}}T$. The higher-order terms have subtle and highly nontrivial compound logarithmic contributions from both $\ensuremath{\varepsilon}$ and TTT, explaining why they have never before been calculated in spite of the importance of the subject matter.
A model for nonequilibrium dynamical mean-field theory is constructed for the infinite-dimensional Hubbard lattice. We impose nonequilibrium by expressing the physical orbital as a superposition of a left (L)(L)(L)-moving and right (R)(R)(R)-moving electronic state with the respective chemical potentials ${\ensuremath{\mu}}_{L}$ and ${\ensuremath{\mu}}_{R}$. Using the second-order iterative perturbation theory we calculate the quasiparticle properties as a function of the chemical potential bias between the LLL and RRR movers, i.e., $\ensuremath{\Phi}={\ensuremath{\mu}}_{L}\ensuremath{-}{\ensuremath{\mu}}_{R}$. The evolution of the nonequilibrium quasiparticle spectrum is mapped out as a function of the bias and temperature. The quasiparticle states with the renormalized Fermi-energy scale ${\ensuremath{\epsilon}}_{QP}^{0}$ disappear at $\ensuremath{\Phi}\ensuremath{\sim}{\ensuremath{\epsilon}}_{QP}^{0}$ in the low-temperature limit. The second-order perturbation theory predicts that in the vicinity of the Mott-insulator transition at the Coulomb-parameter U=UcU={U}_{c}U=Uc​, there exists another critical Coulomb-parameter Ud{U}_{d}Ud​ (<Uc)(<{U}_{c})(<Uc​) such that, for Ud<U<Uc{U}_{d}<U<{U}_{c}Ud​<U<Uc​, quasiparticle states are destroyed abruptly when ${({\ensuremath{\epsilon}}_{QP}^{0})}^{2}\ensuremath{\sim}a{(\ensuremath{\pi}{k}_{B}{T}_{c})}^{2}+b{\ensuremath{\Phi}}_{c}^{2}$ with the critical temperature Tc{T}_{c}Tc​, the critical bias ${\ensuremath{\Phi}}_{c}$, and the numerical constants aaa and bbb on the order of unity.
We consider the problem of relaxation in a one-dimensional system of interacting electrons. In the limit of weak interactions, we calculate the decay rate of a single-electron excitation, accounting for the nonlinear dispersion. The leading processes that determine the relaxation involve scattering of three particles. We elucidate how particular forms of Coulomb interaction, unscreened and screened, lead to different results for the decay rates and identify the dominant scattering processes responsible for relaxation of excitations of different energies. Interestingly, temperatures much smaller than the excitation energy strongly affect the rate. At higher temperatures the quasiparticle relaxes by exciting copropagating electron-hole pairs, whereas at lowest temperatures the relaxation proceeds via excitations of both copropagating and counterpropagating pairs.
We examine the validity of Fermi-liquid theory in the low-density two-dimensional Hubbard model. From the ground-state energy we obtain the quasiparticle energy and the Landau interaction function. Then we prove that the quasiparticle-interaction function is finite and that the quasiparticle residue is nonzero for repulsive interactions. We also find, as a function of the excitation energy \ensuremath{\epsilon}, that the quasiparticle lifetime \ensuremath{\sim}1/${\mathrm{\ensuremath{\epsilon}}}^{2}$ln\ensuremath{\epsilon}. Hence, we conclude that Fermi-liquid theory holds in our model.
We consider a system of charged one-dimensional spin-1/2 fermions at low temperature. We study how the energy of a highly excited quasiparticle (or hole) relaxes toward the chemical potential in the regime of weak interactions. The dominant relaxation processes involve collisions with two other fermions. We find a dramatic enhancement of the relaxation rate at low energies, with the rate scaling as the inverse sixth power of the excitation energy. This behavior is caused by the long-range nature of the Coulomb interaction.