Triplet proximity effect and odd-frequency pairing in graphene

We study the interplay between proximity-induced superconductivity and ferromagnetism in graphene by self-consistently solving the Bogoliubov de Gennes equations on the honeycomb lattice. We find that a strong triplet proximity effect is generated in graphene, leading to odd-frequency pairing correlations. These odd-frequency correlations are clearly manifested in the local density of states of the graphene sheet, which can be probed via scanning tunnel microscope measurements. Motivated by recent experiments on $\text{S}\left|\text{N}\right|\text{S}$ graphene Josephson junctions, we also study the spectrum of Andreev bound states formed in the normal region due to the proximity effect. Our results may be useful for interpreting spectroscopic data and can also serve as a guideline for future experiments.

Figure 1

(Color online) Local DOS on a large energy scale (left panel) and near the gapped region (right panels) in an $\text{S}\left|\text{N}\right|\text{S}$ graphene junction for a variety of lengths and CPMs. Large values of the DOS are indicated by a bright color while small values are given by a dark color. We have fixed $L_{\text{S}}=50$ sites and ${\mu }_{\text{S}}/t=0.6$, as well as set the superconducting phase difference to zero, $\varphi =0$. (a) ${\mu }_{\text{N}}={\mu }_{\text{S}}$ and $L_{\text{N}}=8$ sites ($\xi \sim 25$ sites). (b) ${\mu }_{\text{N}}={\mu }_{\text{S}}$ and $L_{\text{N}}=24$ sites. (c) ${\mu }_{\text{N}}/t=0.2$ and $L_{\text{N}}=8$ sites. (d) ${\mu }_{\text{N}}/t=0.2$ and $L_{\text{N}}=24$ sites. As seen, in-gap bound states, below the gap edge, are formed and persist even in the long-junction regime. The DOS is in general not symmetric around $E=0$ since there is a finite chemical potential in each region. Note that the value of the LDOS is determined by the height, and not the colors, in the middle three-dimensional surface plot of (a)–(d).

Figure 2

(Color online) Local DOS in an $\text{S}\left|\text{F}\right|\text{S}$ graphene junction with $L_{\text{S}}=50$ sites, $L_{\text{F}}=8$ sites, $h/t=0.05$, and ${\mu }_{\text{S}}/t={\mu }_{\text{F}}/t=0.6$. (a) Spatial profile for the superconducting order parameter. The position of the interfaces is marked with black vertical lines. The right inset shows the imaginary part of the induced odd-frequency correlations, which from bottom to top correspond to times $t\tau =1,2,3,4,5$. The left inset shows the same thing for $h/t=0.06$ and ${\mu }_{\text{F}}/t=0.2$. (b) Local DOS on a large energy scale (left panel) and near the gapped region (right panels). Large values of the DOS are indicated by a bright color while small values are given by a dark color. In (b), the value of the LDOS is given by the height of the curve similarly to Fig. 1. As seen, odd-frequency correlations give rise to a zero-energy peak in the DOS.

Figure 3

(Color online) Local DOS in an $\text{S}\left|\text{F}\right|\text{S}$ graphene junction with $L_{\text{S}}=50$ sites, $L_{\text{F}}=24$ sites, ${\mu }_{\text{S}}/t=0.6$, and $h/t=0.06$. (a) 0 phase, ${\mu }_{\text{F}}/t=0.6$. (b) $\pi$ phase, ${\mu }_{\text{F}}/t=0.6$. (c) 0 phase, ${\mu }_{\text{F}}/t=0.2$. (d) $\pi$ phase, ${\mu }_{\text{F}}/t=0.2$. Large values of the DOS are indicated by a bright color while small values are given by a dark color. As seen, the odd-frequency pairing amplitude is most pronounced in the 0 phase, both with and without a CPM. This is illustrated directly in (e), where the zero-energy peak value is plotted against ${\mu }_{\text{F}}/t$.