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arXiv:cond-mat/0312565 · Physical Review B

Quasiparticle effective-mass divergence in two-dimensional electron systems

Ying Zhang, S. Das Sarma

Published Jan 18, 2005• 42 citations
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Abstract

Within the infinite series of ring (or bubble) diagram approximation for the electronic self-energy as appropriate for the long-range Coulomb interaction, we calculate the density-dependent $T=0$ Fermi-liquid quasiparticle effective-mass renormalization in two-dimensional (2D) electron systems in the random-phase approximation. We find that the quasiparticle effective mass increases very strongly with increasing ${r}_{s}$, exhibiting an unexpected divergence at a critical ${r}_{s}$ value ($\ensuremath{\sim}16$ in the ideal 2D system). We discuss the possibility for and the consequences of such an interaction-induced effective-mass divergence as the electron density decreases and critically comment on the likely theoretical scenarios which could lead to such a low-density effective-mass divergence in electron liquids.

DOI
10.1103/physrevb.71.045322

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Within the infinite series of ring (or bubble) diagram approximation for the electronic self-energy as appropriate for the long-range Coulomb interaction, we calculate the density-dependent T=0T=0T=0 Fermi-liquid quasiparticle effective-mass renormalization in two-dimensional (2D) electron systems in the random-phase approximation. We find that the quasiparticle effective mass increases very strongly with increasing rs{r}_{s}rs​, exhibiting an unexpected divergence at a critical rs{r}_{s}rs​ value (\ensuremath∼16\ensuremath{\sim}16\ensuremath∼16 in the ideal 2D system). We discuss the possibility for and the consequences of such an interaction-induced effective-mass divergence as the electron density decreases and critically comment on the likely theoretical scenarios which could lead to such a low-density effective-mass divergence in electron liquids.
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We theoretically consider Fermi surface anomalies manifesting in the temperature-dependent quasiparticle properties of two-dimensional (2D) interacting electron systems, comparing and contrasting with the corresponding three-dimensional (3D) Fermi liquid situation. In particular, employing microscopic many-body perturbative techniques, we obtain analytically the leading-order and the next-to-leading-order interaction corrections to the renormalized effective mass for three distinct physical interaction models: electron-phonon, electron-paramagnon, and electron-electron Coulomb coupling. We find that the 2D renormalized effective mass does not develop any Fermi surface anomaly due to electron-phonon interaction, manifesting O(T2)\mathcal{O}({T}^{2})O(T2) temperature correction and thus remaining consistent with the Sommerfeld expansion of the noninteracting Fermi function, in contrast to the corresponding 3D situation where the temperature correction to the renormalized effective mass has the anomalous T2logT{T}^{2}logTT2logT behavior. In contrast, both electron-paramagnon and electron-electron interactions lead to the anomalous O(T)\mathcal{O}(T)O(T) corrections to the 2D effective mass renormalization in contrast to T2logT{T}^{2}logTT2logT behavior in the corresponding 3D interacting systems. We provide detailed analytical results, and comment on the conditions under which a T2logT{T}^{2}logTT2logT term could possibly arise in the 2D quasiparticle effective mass from electron-phonon interactions. We also compare results for the temperature-dependent specific heat in the interacting 2D and 3D Fermi systems, using the close connection between the effective mass and specific heat.
We have carried out a comprehensive investigation of the quasiparticle properties of a two-dimensional electron gas, interacting via the long-range Coulomb interaction in the presence of bare mass anisotropy (i.e., with an elliptic noninteracting Fermi surface) by calculating the self-energy, the spectral function, the scattering rate, and the effective mass within the leading-order dynamical self-energy approximation. Our theory is exact in the high-density limit. We find anisotropic features of quasiparticle properties that are not captured by the commonly used isotropic approximation where the anisotropic effective mass is replaced by the isotropic averaged density-of-states mass. Some of these interesting results are as follows: (1) The many-body renormalization of the quasiparticle spectrum becomes highly anisotropic as the quasiparticle energy increases away from the Fermi energy; (2) the interaction-induced inelastic-scattering rate features a strong anisotropy, exhibiting an abrupt jump at different injected energies depending on the momentum direction of the injected electron; (3) the effective-mass enhancement is larger (smaller) for the light (heavy) mass, showing that the anisotropy is reduced by interactions. Our results and analysis show that the unjustified neglect of the mass anisotropy can lead to an incorrect description of quasiparticle properties of the anisotropic system and inaccurate estimates of physical quantities of interest although the use of an equivalent isotropic approximation using the density-of-states effective mass as is commonly and uncritically performed in the literature, works as a reasonable approximation in many situations. In addition to the complete random phase approximation theory for the anisotropic quasiparticles, we also provide a theory using the simpler plasmon-pole approximation, commenting on its validity for anisotropic self-energy calculations. We comment also on the interaction effect on the Fermi-surface topology, finding that the elliptic shape of the bare Fermi surface is preserved with suppressed ellipticity in the interacting system to a high degree of accuracy except in the very strongly interacting limit (and for very high bare mass anisotropy). Our theory provides a complete generalization of the existing isotropic many-body theory of interacting electrons to the corresponding anisotropic systems.
The real-space variation quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) are used to calculate the energy bands and the effective mass of the paramagnetic and ferromagnetic two-dimensional uniform electron liquid (2D-UEL). The many-body finite-size errors are minimized by performing simulations for three system sizes with the number of electrons, N = 146 , 218, and 302, for paramagnetic and N = 91 , 139, and 151 for ferromagnetic systems. We consider 2D-UEL to be within the density range 1 ≤ r s ≤ 10 . The VMC and DMC results predict that the effective mass m * of the paramagnetic 2D-UEL at high density r s = 1 is very close to 1, suggesting that the effective mass renormalization due to electron-electron interaction is negligible. We find that m * of the paramagnetic 2D-UEL obtained by the VMC and DMC methods increases by r s but with different slopes. Our VMC and DMC results for ferromagnetic 2D-UEL indicate that m * decreases rapidly by reducing the density due to the strong suppression of the electron-electron interaction.
We calculate the momentum distribution of the Fermi-liquid phase of the homogeneous two-dimensional electron gas. We show that close to the Fermi surface, the momentum distribution of a finite system with NNN electrons approaches its thermodynamic limit slowly, with leading-order corrections scaling as ${N}^{\ensuremath{-}1/4}$. These corrections dominate the extrapolation of the renormalization factor ZZZ and the single-particle effective mass ${m}^{\ensuremath{\ast}}$ to the infinite system size. We show how convergence can be improved using analytical corrections. In the range $1\ensuremath{\le}{r}_{s}\ensuremath{\le}10$, we get a lower renormalization factor ZZZ and a higher effective mass ${m}^{\ensuremath{\ast}}>m$ compared to the perturbative random-phase approximation values.