Magnetic field dependent of the local density of states in cuprate superconductors

The d-wave superconductivity ($d$SC) with competing antiferromagnetic (AF) order as a function of magnetic fields $B$ has been studied by using a phenomenological $t-t'-U-V$ model and the Bogoliubov-de Gennes$'$ equations. Our calculations are performed in a magnetic field $B$ ranging from about $9T$ to $54T$ which corresponding to the magnetic unit cell with size from $40\times 80$ to $16\times 32$. Without the AF order $(U=0)$, the system is in the pure $d$SC state. We show that the local density of states (LDOS) far from the vortex core at the Fermi energy is proportional to $\sqrt{B}$ for $B<18T$, in agreement with the Volovik’s result. For $U=2.42$, we show that the field induced spin-density-wave (SDW) appears and it has the stripe-like structure with a periodicity of $8a$. The amplitude of the induced SDW stripes decreases as the strength of $B$ is reduced. We also point out that the LDOS at the Fermi energy far from the vortex core does not follow the Volovik’s prediction when the induced SDW is in presence.