Extremely imbalanced two-dimensional electron-hole-photon systems

We investigate the phases of two-dimensional electron-hole systems strongly coupled to a microcavity photon field in the limit of extreme charge imbalance. Using variational wave functions, we examine the competition between different electron-hole paired states for the specific cases of semiconducting III-V single quantum wells, electron-hole bilayers, and transition-metal dichalcogenide monolayers embedded in a planar microcavity. We show how the Fermi sea of excess charges modifies both the electron-hole bound state (exciton) properties and the dielectric constant of the cavity active medium, which in turn affects the photon component of the many-body polariton ground state. On the one hand, long-range Coulomb interactions and Pauli blocking of the Fermi sea promote electron-hole pairing with finite center-of-mass momentum, corresponding to an excitonic roton minimum. On the other hand, the strong coupling to the ultra-low-mass cavity photon mode favors zero-momentum pairs. We discuss the prospect of observing different types of electron-hole pairing in the photon spectrum.

Figure 1

Schematic representation of the system. (a) Two quantum wells, labeled by the index $\sigma =1,2$ and separated by a distance $d$, form an electron-hole bilayer in the extremely imbalanced limit. The minority species belongs to the $\sigma =2$ layer, while the majority species at $\sigma =1$ forms an interacting Fermi sea—the mass ratio $m_2/m_1$ establishes which layer is populated by either electrons or holes. $U_{\mathbf{q}}$ and $V_{\mathbf{q}}$ are, respectively, intra- and interspecies Coulomb interactions. The bilayer is located inside a planar cavity that confines the cavity photon mode ($C$). The (blue) shaded area represents the finite-size external laser pump spot. (b) Same setup in a single quantum-well geometry. Here, the majority $\sigma =1$ and minority $\sigma =2$ species belong to the same well.

Figure 2

Particle-hole excitation process via a photon without (a) and with (b) a Fermi sea—all photon-mediated transitions are approximately vertical in a cavity.

Figure 3

Photon energy shift in the presence of a Fermi gas ${\omega }_{\text{C}\mathbf{0}}^{\left(E_{\text{F}}\right)}-{\omega }_{\text{C}\mathbf{0}}$ as estimated from Eq. (23) (solid line) and from Eq. (25) (dashed line) for either fixed Fermi energy $E_{\text{F}}$ and varying Rabi splitting $\mathrm{\Omega }$ (a),(b) or conversely fixed $\mathrm{\Omega }$ and varying $E_{\text{F}}$ (c),(d). Parameters are for a GaAs single quantum well ($d=0$), mass ratio $m_2/m_1=0.25$, and screened interactions $N_s=1$.

Figure 4

Phase diagram of photon-exciton detuning $\delta$ and majority particle Fermi energy $E_{\text{F}}$ for a GaAs heterostructure with either a single quantum well ($d=0$) [panels (a),(b),(e),(f)] or a bilayer geometry at a distance $d=a_{\text{X}}$ [(c),(d),(g),(b)]. Panels (a)–(d) are for RPA screened Coulomb interactions ($N_s=1$), while (e)–(h) are for unscreened interactions ($N_s=0$). The mass ratio is fixed to either $m_2/m_1=0.25$, i.e., one electron in a Fermi sea of holes, or $m_2/m_1=4$, i.e., one hole in a Fermi sea of electrons. The Rabi splitting is fixed to $\mathrm{\Omega }=2R_{\text{X}}$ for the $d=0$ case and to $\mathrm{\Omega }=2|E_{\text{X}}^{\left(d\right)}|\simeq 0.64R_{\text{X}}$ for the bilayer at the $d=a_{\text{X}}$ case. Solid lines are first-order transitions (SF-FF and SF-N). The dashed almost vertical line is the second-order FF-N transition occurring for screened interactions. First- and second-order transitions meet at a critical end point. The diamond symbols indicate the value of the density, $E_{\text{F}0}$, at which the SF-FF transition occurs in the absence of the cavity field $\mathrm{\Omega }=0={\alpha }_{\mathbf{Q}}$. The color map represents the photon fraction $|{\alpha }_{\mathbf{Q}}{|}^2$.

Figure 5

Momentum $Q_{\text{min}}$ (filled [blue] circles) minimizing the many-body polaritonic energy $E_{\mathbf{Q}}$ as a function of the majority Fermi energy $E_{\text{F}}$ for a single quantum well $d=0$, mass ratio $m_2/m_1=0.25$, Rabi splitting $\mathrm{\Omega }=2R_{\text{X}}$ and detuning $\delta =8R_{\text{X}}$. Interactions are RPA screened ($N_s=1$) in panel (a), and unscreened ($N_s=0$) in panel (b). Solid (violet) lines represent the value of $Q_{\text{min}}$ in the absence of light-matter coupling ($\mathrm{\Omega }=0$), while the thick dashed (black) line is the Fermi momentum $k_{\text{F}}$. The corresponding photon fraction $|{\alpha }_{\mathbf{Q}}{|}^2$ is plotted with open (red) circles and the corresponding axes are on the right side of each panel.

Figure 6

In panel (a), the solid lines are SF-FF (or SF-N) phase boundaries for different values of the Rabi splitting $\mathrm{\Omega }$, for a single quantum well with hole doping, $d=0$ and $m_2/m_1=0.25$, and for screened interactions $N_s=1$. In particular, the region above a solid line is either FF (on the left of the dashed line) or N (on the right). The almost vertical dashed line is the approximately $\mathrm{\Omega }$-independent FF-N boundary (see Fig. 4). Below each solid line, the phase is SF. Each symbol represents the minimal detunings ${\delta }_{\text{min}}$ of the boundaries—special values are $\mathrm{\Omega }=0$ (filled diamond) and the $\mathrm{\Omega }$ at which ${\delta }_{\text{min}}={\delta }^{*}$ (filled circle). A special value common to all boundaries is $({\delta }^{*},E_{\text{F}}^{*})$ (filled circle). In panel (b), the solid line and symbols give the behavior of ${\delta }_{\text{min}}$ as a function of $\mathrm{\Omega }$ for screened $N_s=1$ interactions, while the dot-dashed line represents ${\delta }_{\text{min}}$ for unscreened $N_s=0$ interactions.

Figure 7

Different possible topologies of the phase diagram as a function of the Rabi splitting $\mathrm{\Omega }$ and the majority particle Fermi energy $E_{\text{F}}$ for a GaAs heterostructure with a single quantum well ($d=0$), $m_2/m_1=0.25$, and screened interactions $N_s=1$. The detuning has been fixed to $\delta =0<{\delta }_0$ [panel (a)], ${\delta }_0<\delta =1.5R_{\text{X}}<{\delta }^{*}$ [panel (b)], and $\delta =4R_{\text{X}}>{\delta }^{*}$ [panel (c)]. In all panels, the vertical dot-dashed line is the value of $E_{\text{F}}^{*}$ (see Fig. 6), while all other lines and labels are as in Fig. 4.

Figure 8

Phase diagram for a ${\mathrm{MoSe}}_2$ monolayer embedded into a planar cavity as a function of photon-exciton detuning $\delta$ and electron Fermi energy $E_{\text{F}}$. We take the Rabi splitting $\mathrm{\Omega }=40\text{meV}$. The solid line is the first-order SF-FF transition, while the diamond symbol indicates the value of $E_{\text{F}0}$ at which the SF-FF transition occurs in the absence of the cavity field ($\mathrm{\Omega }=0={\alpha }_{\mathbf{Q}}$) [12]. The color map represents the photon fraction $|{\alpha }_{\mathbf{Q}}{|}^2$.

Figure 9

Comparison between detuning $\delta$ (12) and Rabi splitting $\mathrm{\Omega }$ (13), as defined in the renormalization procedure of Sec. 2a, and the respective quantities ${\delta }_{\text{eff}}$ and ${\mathrm{\Omega }}_{\text{eff}}$ obtained by a least-squares fitting procedure described in Appendix pp1. Parameters are for a GaAs microcavity with a single quantum well ($d=0$), $m_2/m_1=0.25$, and $E_{\text{F}}=0$.

Figure 10

Rescaled shift of exciton energies $E_{\text{X}\mathbf{Q}}^{(d,E_{\text{F}})}-E_{\text{X}}^{\left(d\right)}$ at $\mathbf{Q}=\mathbf{0}$ (solid) and ${\mathbf{Q}}_{\text{min}}$ (dashed) as a function of density. Parameters are for a GaAs single quantum well ($d=0$, $E_{\text{X}}^{(d=0)}=-R_{\text{X}}$), two different mass ratios $m_2/m_1=0.25$ and 4, and for both screened ($N_s=1$) [panels (a),(b)] and unscreened ($N_s=0$) [panels (c),(d)] interactions. Dot-dashed and dotted lines show the low- and high-density fittings, respectively.

Figure 11

Rescaled shift of exciton energies $E_{\text{X}\mathbf{Q}}^{(d,E_{\text{F}})}-E_{\text{X}}^{\left(d\right)}$ at $\mathbf{Q}=\mathbf{0}$ (solid) and ${\mathbf{Q}}_{\text{min}}$ (dashed) as a function of the density. Parameters are for a GaAs single quantum well ($d=0$, $E_{\text{X}}^{(d=0)}=-R_{\text{X}}$), $m_2/m_1=0.25$, and screened interactions $N_s=1$. The diamond symbol represents the Fermi energy $E_{\text{F}0}$ at which the two energies split as a consequence of a second-order transition where ${\mathbf{Q}}_{\text{min}}$ moves away from $\mathbf{0}$. The color map is the photon fraction $|{\alpha }_{\mathbf{0}}{|}^2$ of the many-body polariton state at $\mathbf{Q}=\mathbf{0}$ for $\mathrm{\Omega }=0.2R_{\text{X}}$. The color map is plotted against $E_F/R_{\text{X}}$ ($x$-axis) and detuning $\delta /R_{\text{X}}$ ($y$-axis).

Figure 12

Momentum $Q_{\text{min}}$ minimizing the many-body exciton energy $E_{\mathbf{Q}}=E_{\text{X}\mathbf{Q}}^{(d,E_{\text{F}})}$ obtained by solving Eq. (18) as a function of the majority Fermi energy $E_{\text{F}}$ for a single quantum well ($d=0$), and for different values of the mass ratios $m_2/m_1$ as indicated. In panel (a) the interaction is screened, $N_s=1$, while in panel (b) it is unscreened, $N_s=0$.

Figure 13

Many-body polariton ground-state energy $E_{\mathbf{Q}}$ with respect to the normal-state energy $E_{\text{N}}$ (solid lines) vs momentum $Q$. Parameters are for a GaAs heterostructure with a single quantum well ($d=0$), mass ratio $m_2/m_1=0.25$, Rabi splitting $\mathrm{\Omega }=2R_{\text{X}}$, detuning $\delta =8R_{\text{X}}$, and screened interactions ($N_s=1$). Dashed colored lines are the many-body exciton energies $E_{\text{X}\mathbf{Q}}^{(d,E_{\text{F}})}$ evaluated in the absence of the light-matter coupling, $\mathrm{\Omega }=0$. The gray dotted line indicates where the minima at $Q=0$ and $Q\ne 0$ are equal. Panel (a) shows the first-order SF-FF transition when increasing the system density, while panel (b) shows the N-SF first-order transition.

Figure 14

Illustration of the dependence of energies on Rabi splitting close to $E_{\text{F}}^{*}$. The parameters are for a GaAs heterostructures with a single quantum well $d=0$, mass ratio $m_2/m_1=0.25$, and for screened [panels (a) and (c)] and unscreened [panels (b) and (d)] interactions. (a) and (b) Energy difference between the many-body polariton SF energy $E_{\mathbf{0}}$ and the FF energy $E_{{\mathbf{Q}}_{\text{min}}}$ [panel (b)] or between $E_{\mathbf{0}}$ and the normal-state energy $E_{\text{N}}$ [panel (a)]. For each Fermi energy, $E_{\text{F}}$, the detuning is fixed according to Eq. (D1), describing the SF-FF boundary at $\mathrm{\Omega }=0$. (c) and (d) Photon energy ${\omega }_{\text{C}\mathbf{0}}-E_g$ satisfying Eq. (D9) at $\mathbf{Q}=\mathbf{0}$ and $E_{{\mathbf{Q}}_{\text{min}}}$—for the values of $E_{\text{F}}$ considered in the plot and for screened interactions, $E_{{\mathbf{Q}}_{\text{min}}}$ coincides with the normal-state energy $E_{\text{N}}$, i.e., ${\mathbf{Q}}_{\text{min}}=k_{\text{F}}{\widehat{\mathbf{k}}}_{\text{F}}$.