content_image/443151.jpg content_image/443152.jpg content_image/443155.jpg content_image/443156.jpg content_image/443167.jpg content_image/443168.jpg content_image/443179.jpg content_image/443147.jpg content_image/443169.jpg content_image/443170.jpg content_image/443146.jpg content_image/443177.jpg content_image/443178.jpg content_image/443159.jpg content_image/443160.jpg content_image/443166.jpg content_image/443176.jpg
The spinors ˆ a$_{↑}$ and ˆ a$_{↓}$ are eigenstates of excitations in the normal graphene. We also introduce vectors in the Nambu space, 

Each component here is a pseudo-spinor. We find for the upper sign in Eq. (11) 

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For k | | 

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For the lower sign in Eq. (11) we have 

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and 

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Here 

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The different wave functions are orthogonal, ˇ ψ + α ˇ ψ$_{β}$ = δ$_{αβ}$ . Equation (13) goes over into Eq. (15) under the transformation E → − E and µ → − µ . Using Eq. (10) one can check that 

where z$_{0}$ is the unit vector in the z direction perpendic- ular to the graphene layer plane. 

For the current-carrying state the solvability condition of the Bogoliubov–de Gennes equations (8), (9) takes the form 

<img src='content_image/443179.jpg'>

where p$_{±}$ = p ± k / 2. Equation (20) 

cannot be solved analytically for nonzero k , except for the zero doping limit µ = 0. In 

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## III. CURRENT-CARRYING STATE 

the latter case the solvability condition Eq. (20) becomes bi-quadratic and yields the energy spectrum 

<img src='content_image/443146.jpg'>

In this limit, the Bogoliubov–de Gennes equations can also be solved analytically (see Appendix B 2). One sees that the energy, Eq. (21), for µ = 0 does not have the usual Doppler term proportional to the vector k . This may lead to a confusion 15 when calculating the supercur- rent. 

Let us consider the linear correction to the energy and to the wave functions due to superconducting momentum k assuming v$_{F}$ | k | ≪ µ . We put E = E (0) + E ′ where E ′ ≪ E (0) and E (0) is the energy of one of the states with k = 0 determined by Eqs. (12), (14). Within the linear approximation in k we find from Eq. (20) for any finite µ ̸ = 0 

for the upper (lower) sign in Eq. (11). Therefore, correc- tions to the energies are 

The energy E$_{D}$ = ( dξ$_{p}$/dp )( k / 2) is the usual Doppler shift for the normal-state energy ξ$_{p}$ = v$_{F}$ p . Equation (22) coincides with the result of Ref. 10 obtained in the linear approximation in k . At the same time, it differs from the linear in k term obtained from Eq. (21) for µ = 0. This means that the undoped case µ = 0 requires a special consideration. This will be done later in Section III B (see also Appendix B). 

First-order corrections to the wave functions can be found by expanding the total functions in terms of the zero-order functions ˆ u (0) β , ˆ v (0) β given by Eqs. (13), (15): 

Inserting this into Eqs. (1), (2) we find 

One can check that B$_{βα}$ = − B ∗ $_{αβ}$. We find B$_{12}$ = B$_{21}$ = B$_{34}$ = B$_{43}$ = 0 while 

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## A. Linear response 

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