Incoherent pair tunneling in the pseudogap phase of cuprates

Motivated by a recent experiment by Bergeal et al., we reconsider incoherent pair tunneling in a cuprate junction formed from an optimally doped superconducting lead and an underdoped normal-metallic lead. We study the impact of the pseudogap on the pair tunneling by describing fermions in the underdoped lead with a model self-energy that has been developed to reproduce photoemission data. We find that the pseudogap causes an additional temperature-dependent suppression of the pair contribution to the tunneling current. We discuss consistency with available experimental data and propose future experimental directions.

Figure 1

Incoherent pair tunneling contribution, $I_{\mathrm{pair}}=2e\mathrm{tr}\left(F^A{V}^{AB}{G}^B{G}^B{t}^B{G}^B{G}^B{V}^{BA}{F}^A\right)$, to the tunneling current. Here the trace includes summations over momenta and frequencies; double and single lines correspond, respectively, to Gor'kov functions $F_{\mathbf{p}}^A\left(i{\epsilon }_n\right)$ of the optimally doped (OD) lead $A$ and single-particle Greens functions $G_{\mathbf{k}}^B\left(i{\epsilon }_n\right)$ of the underdoped (UD) pseudogap lead $B$; circles represent one-electron tunneling matrix elements $V_{\mathbf{p}\mathbf{k}}^{AB}$,[13] and the wavy line denotes the particle-particle $t$ matrix for the pseudogap lead $B$ (here for $\mathbf{q}=0$). To arrive at Eq. (2) in the text, we assumed a pairing instability in the $d$-wave channel, $t_{\mathbf{k},{\mathbf{k}}^{\prime },\mathbf{q}}^R\left(\omega \right)={\mathcal{L}}_{\mathbf{q}}^R\left(\omega \right)\cos \left(2{\phi }_{\mathbf{k}}\right)\cos \left(2{\phi }_{{\mathbf{k}}^{\prime }}\right)$, and kept only the relevant $d$-wave part $\propto V_1$ in the harmonic expansion of the tunneling matrix elements, $\langle |V_{\mathbf{p}\mathbf{k}}^{AB}{|}^2\rangle =|V_0{|}^2+{\left|V_1\right|}^2\cos \left(2{\phi }_{\mathbf{p}}\right)\cos \left(2{\phi }_{\mathbf{k}}\right)$; see Refs. 3 and 13.

Figure 2

$T$ dependence of the vertex ${\mathcal{C}}^2$ within the single-lifetime model for the pseudogap lead, where $\mathcal{C}\propto \mathcal{B}(T,\Gamma ,{\Delta }_A,{\Delta }_B)$. Here $T^{*}$ is the pseudogap temperature, ${\Delta }_A\simeq {\Delta }_B=\Delta$, and $\Gamma /\Delta =\sqrt{3}T/T^{*}$, so that the Fermi arcs connect at $T=T^{*}$. The solid line shows the exact result with $\mathcal{B}$ given in (15) and ${\Gamma }_0={\Gamma }_1=\Gamma$, and the dash-dotted line shows the approximation ${\mathcal{B}}_0$ given in (17).

Figure 3

$T$ dependence of the vertex ${\mathcal{C}}^2$ within the two-lifetime model for the pseudogap lead, where $\mathcal{C}\propto \mathcal{A}(T,{\Gamma }_0,{\Gamma }_1,{\Delta }_A,{\Delta }_B)$. Here, ${\Delta }_A\simeq {\Delta }_B=\Delta$ and parameters similar to Ref. 6, ${\Gamma }_1=200$ meV, $\Delta =50$ meV, and ${\Gamma }_0=(16/\pi )(T-T_c)$, twice ${\Gamma }_{\mathrm{GL}}$, where $T_c=80$ K. The solid line shows the exact result $\mathcal{A}$ given in (15) and the dash-dotted line shows the approximation ${\mathcal{A}}_1$ given in (16).