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arXiv:cond-mat/0606100 · Physical Review B

Excitonic BCS-BEC crossover at finite temperature: Effects of repulsion and electron-hole mass difference

Yuh Tomio, Kotaro Honda, Tetsuo Ogawa

Published Jun 12, 2006• 28 citations
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Abstract

The BCS to Bose-Einstein condensation (BEC) crossover of electron-hole $(e\text{\ensuremath{-}}h)$ pairs in optically excited semiconductors is studied using the two-band Hubbard model with both repulsive and attractive interactions. Applying the self-consistent $t$-matrix approximation combined with a local approximation, we examine the properties of a normal phase and an excitonic instability. The transition temperature from the normal phase to an $e\text{\ensuremath{-}}h$ pair condensed one is studied to clarify the crossover from an $e\text{\ensuremath{-}}h$ BCS-like state to an excitonic Bose-Einstein condensation, which takes place on increasing the $e\text{\ensuremath{-}}h$ attraction strength. To investigate effects of the repulsive interaction and the $e\text{\ensuremath{-}}h$ mass difference, we calculate the transition temperature for various parameters of the interaction strengths, the $e\text{\ensuremath{-}}h$ particle density, and the mass difference. While the transition temperature in the $e\text{\ensuremath{-}}h$ BCS regime is sufficiently suppressed by the repulsive interaction, that of the excitonic BEC is largely insensitive to it. We also show quantitatively that in the whole regime the mass difference leads to large suppression of the transition temperature.

DOI
10.1103/physrevb.73.235108

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The BCS to Bose-Einstein condensation (BEC) crossover of electron-hole (e\ensuremath-h)(e\text{\ensuremath{-}}h)(e\ensuremath-h) pairs in optically excited semiconductors is studied using the two-band Hubbard model with both repulsive and attractive interactions. Applying the self-consistent ttt-matrix approximation combined with a local approximation, we examine the properties of a normal phase and an excitonic instability. The transition temperature from the normal phase to an e\ensuremath-he\text{\ensuremath{-}}he\ensuremath-h pair condensed one is studied to clarify the crossover from an e\ensuremath-he\text{\ensuremath{-}}he\ensuremath-h BCS-like state to an excitonic Bose-Einstein condensation, which takes place on increasing the e\ensuremath-he\text{\ensuremath{-}}he\ensuremath-h attraction strength. To investigate effects of the repulsive interaction and the e\ensuremath-he\text{\ensuremath{-}}he\ensuremath-h mass difference, we calculate the transition temperature for various parameters of the interaction strengths, the e\ensuremath-he\text{\ensuremath{-}}he\ensuremath-h particle density, and the mass difference. While the transition temperature in the e\ensuremath-he\text{\ensuremath{-}}he\ensuremath-h BCS regime is sufficiently suppressed by the repulsive interaction, that of the excitonic BEC is largely insensitive to it. We also show quantitatively that in the whole regime the mass difference leads to large suppression of the transition temperature.
Motivated by recent experimental and theoretical studies of excitonic insulators, we use the variational cluster approximation to calculate the single-particle Green's function and anomalous Green's function of the Falicov-Kimball model extended by including a finite valence bandwidth. We thus determine the single-particle excitation gap due to the bound-state formation of an electron and a hole. We also evaluate the order parameter of the excitonic insulating state indicating quantum coherence of condensed excitons. We thereby discuss the excitonic insulator that typifies either a BCS condensate of electron-hole pairs (weak-coupling regime) or a Bose-Einstein condensate of preformed excitons (strong-coupling regime). A BCS-BEC crossover of the model thus manifests itself as a function of the Coulombic coupling strength.
We study the spontaneous symmetry breaking of the excitonic insulator state induced by the Coulomb interaction UUU in the two-dimensional extended Falicov-Kimball model. Using the variational cluster approximation (VCA) and Hartree-Fock approximation (HFA), we evaluate the order parameter, single-particle excitation gap, momentum distribution functions, coherence length of excitons, and single-particle and anomalous excitation spectra as functions of UUU at zero temperature. We find that in the weak-to-intermediate coupling regime, the Fermi surface plays an essential role and calculated results can be understood in close correspondence with the BCS theory, whereas in the strong-coupling regime, the Fermi surface plays no role and results are consistent with the picture of a Bose-Einstein condensate (BEC). Moreover, we find that HFA works well in both the weak- and strong-coupling regimes, and that the difference between the results of VCA and HFA mostly appears in the intermediate-coupling regime. The reason for this is discussed from a viewpoint of the self-energy. We thereby clarify the excitonic insulator state that typifies either a BCS condensate of electron-hole pairs (weak-coupling regime) or a BEC of preformed excitons (strong-coupling regime).
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We have analyzed the doping and temperature dependencies of the physical properties of a square-lattice narrow-band superconductor with a relevant for systems with local Hubbard repulsion nematic pairing due to effective nearest-neighbor attraction. Our focus was a possible crossover of the system from the Bardeen–Cooper–Schrieffer (BCS) superconductivity to the Bose–Einstein condensation (BEC) regimes with increasing temperature. We have found that in the d-channel the crossover does not take place at realistic values of doping and temperature, and even at a large attraction, i.e., the system is always in the BCS regime, unlike the “usual” d-wave superconductor. Contrary to non-nematic d-wave superconductors, the chemical potential μ always grows with temperature increase that might indicate an unexpectedly large size of Cooper pairs. At the same time, in the s-channel, we have found a BCS–BEC crossover with temperature for a wide range of temperatures even at rather high dopings, nf ≈ 0.2. This result is different from the “usual” s-case, where a narrow-band (i.e., at strong coupling) system is always in the BEC state. We have found for a given large value of attraction that there is a value of doping when for a wide range of temperature the system is in a “pre-crossover” regime with μ ≈ 0, i.e., in a regime of coexistence of local and extended pairs. Thus, in narrow-band systems a “usual” s-superconductor is in a BEC state, but in the nematic case one can expect a BCS–to–BEC crossover with temperature growth at realistic system parameters, i.e., the crossover might be measured experimentally.
The exciton Mott transition and electron‐hole (e‐h) pair condensation are studied using a two‐band Hubbard model with both repulsion and attraction. With the use of the dynamical mean‐field theory, we clarify the phase diagram for the phase transitions at zero and finite temperatures. Away from half‐filling we find an exciton insulating phase, which is followed by the formation of incoherent e‐h pairs, and the first‐order transition between a metallic phase and the exciton insulating phase. We also examine the transition temperature to an e‐h pair condensed phase where BEC‐BCS crossover takes place, and discuss characteristic features of the finite‐temperature phase diagram.
We investigate single-particle properties of a mass-imbalanced Fermi gas in the BCS (Bardeen-Cooper-Schrieffer)--BEC (Bose-Einstein condensation) crossover region. In the presence of mass imbalance, we point out that the ordinary TTT-matrix approximation, which has been extensively used to clarify various BCS-BEC crossover physics in the mass-balanced case, unphysically gives a double-valued solution in terms of the superfluid phase transition temperature Tc{T}_{\mathrm{c}}Tc​ in the crossover region. To overcome this serious problem, we include higher order strong-coupling corrections beyond the TTT-matrix level. Using this extended TTT-matrix theory, we calculate single-particle excitations in the normal state above Tc{T}_{\mathrm{c}}Tc​. The so-called pseudogap phenomena originating from pairing fluctuations are shown to be different between the light-mass component and heavy-mass component, which becomes more remarkable at higher temperatures. Since Fermi condensates with hetero-Cooper pairs have recently been discussed in various fields, such as exciton (polariton) condensates as well as color superconductivity, our results would be useful for the further development of Fermi superfluid physics, beyond the conventional superfluid state with homo-Cooper pairs.