Possibility of an excitonic insulator at the semiconductor-semimetal transition

We calculate the critical temperature below which an excitonic insulator exists at the pressure-induced semiconductor-semimetal transition. Our approach is based on an effective-mass model for valence and conduction band electrons interacting via a statically screened Coulomb potential. Assuming pressure to control the energy gap, we derive, in the spirit of a crossover from a Bose-Einstein (BE) to a BCS condensate, a set of equations that determines, as a function of the energy gap (pressure), the chemical potentials for the two bands, the screening wave number, and the critical temperature. We (i) show that in leading order the chemical potentials are not affected by the exciton states, (ii) verify that on the strong-coupling (semiconductor) side the critical temperatures obtained from the linearized gap equation coincide with the transition temperatures for a BEC of noninteracting bosons, (iii) demonstrate that mass asymmetry strongly suppresses BCS-type pairing, (iv) investigate the composition of the environment of the excitonic insulator, and (v) discuss in the context of our theory recent experimental claims for exciton condensation in $\mathrm{Tm}{\mathrm{Se}}_{0.45}{\mathrm{Te}}_{0.55}$.

Figure 1

Qualitative sketch of the phase boundary of an EI with equal band masses (adapted from Ref. 11.)

Figure 2

(a) Exchange self-energy ${\Sigma }^X$ to be used in the mean-field approximation, (b) ladder self-energy ${\Sigma }^L$ additionally used in the $T$-matrix approximation, and (c) ladder equation for the $T$ matrix. Solid (dashed) lines denote matrix propagators (statically screened Coulomb potentials).

Figure 3

Phase boundary of an EI with equal band masses $(\alpha =m_1∕m_2=1)$. The solid line (full circles) denotes the critical temperature obtained from the linearized gap equation (Thouless criterion for BEC). The dotted line shows part of the phase boundary on the SC side when no screening due to thermally excited charge carriers is taken into account. The thick (thin) dashed line indicates the SC-SM transition, defined by the vanishing of the renormalized (screened) binding energy, taking screening and Pauli blocking (only screening) into account.

Figure 4

Phase boundaries for an EI as a function of $\alpha =m_1∕m_2$. For $\alpha ⪡1$, the EI is strongly suppressed on the SM side because the finite-temperature chemical potentials for CB electrons and VB holes are different for asymmetric band masses, leading to breaking of Cooper-type electron-hole pairs. On the SC side, on the other hand, transition temperatures are only moderately reduced because of the $1∕M$ dependence typical for BEC.

Figure 5

Bound state fraction of the exciton matter above an EI with $\alpha =0.0125$ as a function of the energy gap and for various temperatures. The interceptions with the dotted line indicate the energy gaps for which the bound state fraction is for the given temperature 50%.

Figure 6

Phase boundary of an EI with $\alpha =0.0125$ and contours where the bound state fraction of its halo is 99.9%, 96%, 70%, and 50%, respectively. The temperature-dependent position of the maximum of the bound state fraction is shown by the thin solid line. Thin dashed lines indicate the temperatures for which in Fig. 7 the densities are plotted.

Figure 7

Bound (solid line) and free (dashed line) parts of the CB electron density as a function of the energy gap and for two temperatures above the EI (dotted lines in Fig. 6). Notice the very sharp decrease of the bound part of the density when the SC-SM transition is approached.

Figure 8

Comparison of the experimental data for $\mathrm{Tm}{\mathrm{Se}}_{0.45}{\mathrm{Te}}_{0.55}$ (Refs. 13, 15) with the phase boundary and the 50% contour calculated for an effective model applicable to this substance (see text for a description). Open circles, squares, and triangles denote the onset of the resistivity anomaly, the maximum of the resistivity, and the SC-SM transition, respectively (Ref. 13). The full diamond on the dotted line indicates the point where the linear increase of the diffusivity has been found in Ref. 15. The inset shows the bound part of the density $n_1^b$ along the thick solid line, which denotes the $(E_g,T)$ points for which $n_1^b$ is maximal. At the full circle $n_1^b\approx 1.3\times {10}^{21}{\mathrm{cm}}^{-3}$.