2D bilayer electron-hole superfluidity with unequal and anisotropic masses

TL;DR

The paper addresses how mass imbalance and in-plane mass anisotropy affect interlayer electron-hole superfluidity in 2D bilayers. It uses a zero-temperature self-consistent Hartree-Fock approach to map the order parameter and the BEC-BCS crossover across layer separations and mass ratios for two scenarios: (i) isotropic but unequal masses and (ii) equal-average masses with orthogonal anisotropies, showing that both deviations weaken pairing by breaking Fermi-surface nesting and reduce the inferred $T_c$, while preserving a ground-state superfluid in the absence of screening. A key finding is that there is no zero-temperature transition to an unpaired plasma within the studied parameter range, indicating the robustness of the bilayer superfluid state and highlighting that nesting is not essential for condensation. The results establish a baseline for realistic bilayer platforms, including van der Waals heterostructures and anisotropic 2D semiconductors, and clarify how parameter details influence the strong-to-weak coupling crossover and the observability of superfluidity.

Abstract

We investigate the stability of electron-hole superfluidity in two-dimensional bilayers with unequal and anisotropic effective masses. Using a zero-temperature, self-consistent Hartree-Fock approach, we study two experimentally relevant deviations from the ideal equal-mass isotropic case: (i) isotropic but unequal conduction and valence band masses ($m_c^* \neq m_v^*$), and (ii) equal average masses with orthogonal in-plane anisotropies $(m_{c,x}^*, m^*_{c,y}) = (m_1^*, m_2^*)$ and $(m^*_{v,x}, m^*_{v,y}) = (m_2^*, m_1^*)$. For both scenarios, we compute the order parameter and analyze the BEC-BCS crossover as a function of layer separation and mass ratio. We find that both mass imbalance and mass anisotropy reduce the pairing strength and suppress the inferred critical temperature $T_c$ by breaking perfect Fermi surface nesting, and shift the BEC-BCS crossover. Despite these effects, superfluidity remains robust across the full range of densities and interlayer separations considered, with no transition to an unpaired plasma state in the absence of screening. Our results provide a baseline for understanding the interplay of mass mismatch and anisotropy in current and emerging bilayer platforms, including van der Waals heterostructures and anisotropic two-dimensional semiconductors. Our work also establishes that Fermi surface nesting is not a key ingredient for the bilayer superfluidity, which is always the ground state for all electron-hole bilayers although the resultant $T_c$ depends on the parameter details and may very well be unmeasurably low for large interlayer separations.

Figures (6)

• Figure 1: Self-consistent HF at $T=0$ for unequal and isotropic masses, $m_c^* \neq m_v^*$, with $m_c^* = 0.07 m_e$ fixed. Color maps show the maximum order parameter $\Delta^{\rm max} \equiv \max\{ \Delta_{\mathbf{k}}\}$ (in unit of Ry$^*$) as a function of interlayer spacing $d/a^*$ and valence band mass $m_v^*/m_e$ for electron densities $n_e \in [1, 8] \times 10^{10}$ cm$^{-2}$. The black dashed line in each figure marks the equal-mass case $m_v^* = m_c^* = 0.07m_e$. Here, Ry$^*$ and $a^*$ are defined using $m^* = 0.07 m_e$.
• Figure 2: Self-consistent HF at $T=0$ for unequal and isotropic masses, $m_c^* \neq m_v^*$, with $m_c^* = 0.07 m_e$ fixed. Line cuts from Fig. \ref{['fig1_diffmv_2d']} are shown for three representative valence band masses: $m_v^*/m_e = 0.02, 0.07$ and $0.21$. In all cases, the maximum order parameter $\Delta^{\rm max}$ decreases continuously and exponentially with increasing $d/a^*$. For small $m_v^*$ and large $d$, $\Delta^{\rm max}$ becomes extremely small, requiring very dense $k$-grid to resolve. Here, Ry$^*$ and $a^*$ in this figure are defined using $m^* = 0.07 m_e$.
• Figure 3: $\partial \Delta^{\rm max}/\partial \tilde{d}$ for unequal and isotropic masses, $m_c^* \neq m_v^*$, with $m_c^*=0.07m_e$ fixed. Here, $\Delta^{\rm max}$ is the $T=0$ self-consistent HF order parameter shown in Fig. \ref{['fig1_diffmv_2d']}. The blue dashed line in each figure marks the locus where $\partial \Delta^{\rm max}/\partial \tilde{d}$ equals the value for the equal-mass case $m_v^* = m_c^* = 0.07m_e$ at $\tilde{d} = d/a^* = 1$.
• Figure 4: Self-consistent HF at $T=0$ for average equal but anisotropic masses, $(m_{c,x}^*,m_{c,y}^*) = (m_1^*, m_2^*)$ and $(m_{v,x}^*,m_{v,y}^*) = (m_2^*, m_1^*)$, with $m_2^* = 0.07 m_e$ fixed. Color maps show the maximum order parameter $\Delta^{\rm max} \equiv \max\{ \Delta_{\mathbf{k}}\}$ (in unit of Ry$^*$) as a function of interlayer spacing $d/a^*$ and $m_1^*/m_e$ for electron densities $n_e \in [1, 8] \times 10^{10}$ cm$^{-2}$. The black dashed line in each figure marks the equal and isotropic mass case $m_1^* = m_2^* = 0.07m_e$. Here, Ry$^*$ and $a^*$ in this figure are defined using $m^* = 0.07 m_e$.
• Figure 5: Self-consistent HF at $T=0$ for average equal but anisotropic masses, $(m_{c,x}^*,m_{c,y}^*) = (m_1^*, m_2^*)$ and $(m_{v,x}^*,m_{v,y}^*) = (m_2^*, m_1^*)$, with $m_2^* = 0.07 m_e$ fixed. Line cuts from Fig. \ref{['fig4_anisom_2d']} are shown for three representative $m_1^*$ values, $m_1^*/m_e = 0.02, 0.07$ and $0.21$. In all cases, the maximum order parameter $\Delta^{\rm max}$ decreases continuously and exponentially with increasing $d/a^*$. The features are similar to unequal and isotropic masses case in Fig. \ref{['fig2_diffmv_line']}. Here, Ry$^*$ and $a^*$ in this figure are defined using $m^* = 0.07 m_e$.