Collective excitations and quantum incompressibility in electron-hole bilayers

We apply quantum continuum mechanics to the calculation of the excitation spectrum of a coupled electron-hole bilayer. The theory expresses excitation energies in terms of ground-state intra- and interlayer pair correlation functions, which are available from quantum Monte Carlo calculations. The final formulas for the collective modes deduced from this approach coincide with the formulas obtained in the “quasilocalized particle approximation” by Kalman et al. [G. Kalman, V. Valtchinov, and K. I. Golden, Phys. Rev. Lett. 82, 3124 (1999)], and likewise the theory predicts the existence of gapped excitations in the charged channel, with the gap arising from electron-hole correlation. An immediate consequence of the gap is that the static density-density response function of the charged channel vanishes as $q^2$ for wave vector $q\to 0$, rather than linearly in $q$, as commonly expected. In this sense, the system is incompressible. This feature, which has no analog in the classical electron-hole plasma, is consistent with the existence of an excitonic ground state and implies the existence of a discontinuity in the chemical potential of electrons and holes when the numbers of electrons and holes are equal. It should be experimentally observable by monitoring the densities of electrons and holes in response to potentials that attempt to change these densities in opposite directions.

Figure 1

Schematics of an electron-hole bilayer heterostructure. The densities of electrons and holes are controlled by changing the potentials of the electron and hole layers relative to the top and bottom layers (gates).

Figure 2

Interlayer distance $d=0.3$ a.u. Left panel: Longitudinal modes ${\omega }_{L-}\left(q\right), {\omega }_{L+}\left(q\right)$. Right panel: Transverse modes ${\omega }_{T-}\left(q\right), {\omega }_{T+}\left(q\right)$. The noninteracting single-particle spectrum is shown for reference as a shaded region. Solid lines: Dispersion of the antisymmetric mode in the quadriexcitonic (q) phase (red lines) and in the excitonic (e) phase (orange lines); dispersion of the symmetric mode in the quadriexcitonic phase (green lines) and in the excitonic phase (blue lines). Dashed lines with the same color coding present the dispersion of the modes calculated without the structure factor terms in Eqs. (12, 13, 14, 15).

Figure 3

Interlayer distance $d=0.5$ a.u. Left panel: Longitudinal modes ${\omega }_{L-}\left(q\right), {\omega }_{L+}\left(q\right)$. Right panel: Transverse modes ${\omega }_{T-}\left(q\right), {\omega }_{T+}\left(q\right)$. The noninteracting single-particle spectrum is shown for reference as a shaded region. Solid lines: Dispersion of the antisymmetric mode in the excitonic (e) phase (orange lines) and in the plasma (pw) phase (red lines); dispersion of the symmetric mode in the excitonic phase (blue lines) and in the plasma phase (green lines). Dashed lines with the same color coding present the dispersion of the modes calculated without the structure factor terms in Eqs. (12, 13, 14, 15).

Figure 4

Interlayer distances $d=1.0$ a.u. and $d=1.4$ a.u, plasma phase (pw). Left panel: Longitudinal modes ${\omega }_{L-}\left(q\right), {\omega }_{L+}\left(q\right)$. Right panel: Transverse modes ${\omega }_{T-}\left(q\right), {\omega }_{T+}\left(q\right)$. The noninteracting single-particle spectrum is shown for reference as a shaded region. Solid lines: Dispersion of the antisymmetric mode for interlayer distance $d=1.0$ a.u. (red lines) and $d=1.4$ a.u. (orange lines); dispersion of the symmetric mode for interlayer distance $d=1.0$ a.u. (green lines) and $d=1.4$ a.u. (blue lines). Dashed lines with the same color coding present the dispersion of the modes calculated without the structure factor terms in Eqs. (12, 13, 14, 15).

Figure 5

Plot of $\omega \left(0\right)$ vs $d$ for different values of $r_s$. $\omega \left(0\right)$ is calculated from Eq. (19) and the pair correlation function is obtained from the self-consistent solution of the BCS mean-field theory, according to Eq. (20), as explained in the text.

Figure 6

Plot of $\omega \left(0\right)$ as a function of distance $d$ for $r_s=20,8$ and 4, respectively, in (a)–(c). The solid red dots are calculated from Eq. (19) using the pair correlation function obtained from the self-consistent solution of the BCS mean field theory, according to Eq. (20). The solid black squares are obtained using the pair correlation functions from QMC simulations [6]. The open blue dots are the values of $\omega \left(0\right)$ obtained from the direct comparison of Eqs. (32) and (29).

Figure 7

Plot of $\omega \left(0\right)$ as a function of $r_s$ for various values of the distance $d$. Solid red and blue lines are, respectively, for ${\omega }_{\text{QCM}}\left(0\right)$ and ${\omega }_{\chi }\left(0\right)$. The solid orange line is for ${\omega }_{\text{BF}}\left(0\right)$, which is the $q\to 0$ limit of the Bijl-Feynman frequency, Eq. (36), calculated with the BCS structure factor of Eq. (37), as explained in Sec. 5. The dashed black line is ${\omega }_{\text{QCM}}\left(0\right)$ for a system made of isolated excitons, with the $h_{eh}\left(r\right)$ from Eq. (21). The solid dark-green line is $-2\mu$. (a)–(c) show data for $d=1,0.5$ and $d=0.05$, respectively.