ABB Scenario in Superconducting Heterostructures

• The ABB scenario is a superconducting mechanism featuring exciton-mediated Cooper pairing at metal–semiconductor interfaces.
• It employs a detailed second-order pairing interaction using exciton propagators and momentum conservation to couple electrons across layers.
• Extensions with high-energy optical phonons and forward scattering better account for the high critical temperatures observed in FeSe/STO systems.

The Allender–Bray–Bardeen (ABB) scenario describes a mechanism for Cooper pairing in superconducting heterostructures, where conduction electrons in a metal layer ("M") are coupled to interband excitations (excitons) in an adjacent semiconductor layer ("S"). Originally introduced by Allender, Bray, and Bardeen in 1972 for metal–semiconductor interfaces, this framework models additional electron pairing mediated not by phonons but by virtual excitonic transitions in the adjacent material. The ABB mechanism has received renewed attention in the context of high-temperature superconductivity in FeSe monolayers on SrTiO₃ (STO), but the scenario faces significant quantitative challenges for explaining the observed high critical temperatures in these systems (Sadovskii, 2016).

1. Physical Structure and Interaction Mechanism

The ABB scenario involves a layered heterostructure where a metal (M) and a semiconductor (S) are in intimate contact. The physical picture is that two electrons in the metal undergo a virtual process: they excite an electron–hole pair (an exciton) in the adjacent semiconductor and subsequently re-absorb the virtual excitation. In lowest order, the Feynman process can be described as:

• M electron with momentum $k_1\uparrow$ transitions to $k_2\uparrow$ plus a virtual exciton in S.
• M electron with momentum $-k_1\downarrow$ re-absorbs the exciton, becoming $-k_2\downarrow$.

Momentum conservation across the interface is expressed as $$\mathbf{q} = \mathbf{k}_2 - \mathbf{k}_1 = \mathbf{k}_v - \mathbf{k}_c + \mathbf{G}$$,

where $\mathbf{k}_v$ and $\mathbf{k}_c$ are semiconductor valence- and conduction-band momenta, and $\mathbf{G}$ is a reciprocal lattice vector of S. The net effect is an effective retarded attractive interaction between electrons in the metal, analogous to phonon-mediated BCS pairing, but with an excitonic propagator.

2. Mathematical Framework and Parameters

The effective second-order pairing interaction for M electrons due to exciton exchange can be written:

$$V_{\rm ex}(\mathbf{k}, \mathbf{k}') = |g_{\rm ex}|^2\,D_{\rm ex}(\mathbf{k}-\mathbf{k}', \omega_n - \omega_{n'})$$

where the exciton propagator for nearly dispersionless S-excitons of energy $\omega_g$ is

$$D_{\rm ex}(\mathbf{q}, \Omega) = -\frac{1}{\Omega^2 + \omega_g^2}, \quad \omega_g = E_c - E_v,$$

with $E_c$ and $E_v$ as conduction and valence band edges.

Projecting onto the Fermi surface and summing over Matsubara frequencies yields an effective BCS-type gap equation with a dimensionless excitonic coupling constant:

$$\lambda_{\rm ex} = N_M(0)\,\langle\,|g_{\rm ex}|^2/\omega_g^2\,\rangle_{\rm FS},$$

where $N_M(0)$ is the Fermi-level density of states per spin in M, and the average is over Fermi-surface points connected by small $\mathbf{q}$. Neglecting the Coulomb pseudopotential, the critical temperature is:

$$T_c^{(\rm ex)} \simeq \omega_g\,\exp(-1/\lambda_{\rm ex}).$$

ABB estimated that for a good M–S interface:

$$\lambda_{\rm ex} \approx b\,a\,\mu\,\frac{\omega_p^2}{\omega_g^2},$$

with

$$\begin{array}{rl} \mu &\sim 0.1\text{–}0.2\ \text{(bare Coulomb repulsion in M)}\ \omega_p &\sim 10\,\mathrm{eV}\ \text{(S plasma frequency)}\ \omega_g &\sim 3\text{–}4\,\mathrm{eV}\ \text{(S gap)}\ b &\sim0.2,\, a\sim0.2\text{–}0.3\ \text{(geometric/screening factors)} \end{array}$$

The optimistic result is $\lambda_{\rm ex}\sim 0.1$.

3. Application to FeSe Monolayer/SrTiO₃ Systems

In the FeSe/STO system, FeSe constitutes the metallic layer (M) and STO acts as the wide-gap semiconductor (S). The relevant parameters are:

• $\omega_g \approx 3.25$ eV (STO indirect gap)
• $\omega_p \approx 10$ eV
• $\mu \approx 0.1$–$0.2$
• $b \approx 0.2$
• $a \approx 0.25$
• $N_M(0) \approx 2$ states/eV per Fe spin per cell

Inserting these values:

$$\lambda_{\rm ex} \sim (0.2)(0.25)(0.15)\frac{(10)^2}{(3.25)^2} \approx 0.05\text{–}0.13$$

The resulting $T_c^{(\rm ex)}$ for even the most optimistic $\lambda_{\rm ex}=0.13$ falls dramatically short of observed values:

$$T_c^{(\rm ex)} \simeq 3\,\text{eV}\,\exp(-7.7) \sim 10^{-3}\,\text{eV} \sim 10\,\text{K}$$

This is well below the $T_c \approx 65$–$75$ K observed by ARPES and transport probes in FeSe/STO (Sadovskii, 2016).

4. Quantitative Failure and Theoretical Limitations

Even when combining FeSe intrinsic phonon pairing ($\lambda_{\rm ph} \approx 0.44$, which alone gives $T_c \sim 9$ K) with the ABB excitonic term, and renormalizing by a Coulomb pseudopotential ($\mu^* \sim 0.1$), the total calculated $T_c$ does not exceed $T_c \lesssim 20$ K:

$$T_c \approx \frac{\omega_{\rm ph}}{1.45}\exp\left[-\frac{1}{\lambda_{\rm ph}^* + \lambda_{\rm ex}^* - \mu^*}\right]$$

with $\omega_{\rm ph}\sim350$ K. This suggests the ABB excitonic scenario in its classic formulation is quantitatively insufficient for FeSe/STO (Sadovskii, 2016).

5. Extensions: Optical Phonons and Forward Scattering

Replacing the excitonic exchange with coupling to high-energy optical phonons in STO ($\omega_{op}\sim 100$ meV) results in much larger coupling constants, $\lambda_{op} \sim 0.5$–$0.6$. In a two-boson model, the effective pairing constant becomes:

$$g_{\rm eff} \approx \lambda_{\rm FeSe}^* + \frac{\lambda_{op}^* - \mu^*}{1-(\lambda_{op}^*-\mu^*)\ln(\omega_{op}/\omega_{\rm ph})}$$

The critical temperature is then:

$$T_c = \frac{\omega_{\rm ph}}{1.45}\exp\left(-1/g_{\rm eff}\right)$$

With realistic parameters, this can push $T_c$ into the 60–80 K range [see Fig. 6b in (Sadovskii, 2016)].

Further enhancement arises from dominant forward-scattering ($q\approx 0$) electron–phonon interactions, which concentrate the coupling in the small-$\mathbf{q}$ channel. For $|g(\mathbf{q})|^2 = g_0^2 N \delta_{\mathbf{q},0}$, the linearized Eliashberg gap equation yields:

$T_c\simeq \frac{\lambda_m}{2+3\lambda_m}\,\Omega_0$

where $\lambda_m = g_0^2/\Omega_0^2$, $\Omega_0 \approx 100$ meV, and $\lambda_m \approx 0.15$–$0.2$. This yields $T_c \simeq 60$–$80$ K, consistent with experiment, despite modest overall coupling [see Fig. 7 in (Sadovskii, 2016)].

6. Implications and Context within Superconductivity Research

The ABB scenario originally aimed to explain enhanced superconducting critical temperature via interfacial excitonic coupling mechanisms in layered metallic–semiconductor systems. In FeSe/STO monolayers, the distinct two-dimensional electronic structure motivates reassessment of the ABB mechanism. However, the classic ABB model, with $\lambda_{\rm ex} \lesssim 0.1$ and resulting $T_c \lesssim 10$ K, fails to capture the observed high $T_c$. Replacement of excitonic mediation with high-energy optical phonons, especially under strongly forward-peaked electron–phonon coupling, provides a more plausible microscopic basis for the elevated $T_c$, yielding $\lambda \sim 0.2$–$0.6$ in the small-$\mathbf{q}$ channel and quantitatively matching experiments (Sadovskii, 2016). A plausible implication is that the interface's role is to provide access to bosonic excitations (here optical phonons) with appropriate coupling characteristics, not strictly limited to the classic exciton exchange envisioned by ABB.

7. Summary

The ABB scenario describes exciton-mediated superconducting pairing at M–S interfaces, generating new Cooper pairing channels distinct from conventional phonon exchange. While predictive for some parameters, its application to FeSe monolayer/SrTiO₃ systems underestimates the observed $T_c$ due to limited coupling strength. Extensions incorporating optical phonon modes with strong forward-scattering reproduce the magnitude of experimentally observed $T_c$, signifying a shift from the classic ABB excitonic mechanism to phonon-mediated interactions in contemporary interface superconductivity frameworks (Sadovskii, 2016).