Spin and Charge Order around Vortices and Impurities in High-$T_c$ Superconductors

A comparative study is made for the spin and charge structure around superconducting vortices and unitary impurities, by solving self-consistently an effective Hamiltonian including interactions for both antiferromagnetic spin-density wave (SDW) and $d$-wave superconducting orderings. Around vortices, we show the induction of an SDW two-dimensionally modulated with a period of eight lattice constants ($8a_0$) and an associated charge-density wave (CDW) with a period of $4a_0$, which explains very well recent experimental observations. In the case of unitary impurities, an SDW modulation with identical periodicity, but without an associated CDW, is also predicted.

Figure 1

The spatial distribution of the $d$-wave SC order parameter $|{\Delta }_d|$ (a), the staggered magnetization $M_s$ (b), and the electron density $\delta n=\sum _{\sigma }{n}_{i\sigma }-n_f$ (c) around one vortex located at the center of an area of $24\times 24$ sites. The size of the whole magnetic cell is $48\times 24$. Parameter values: $t^{\prime }=-0.2$, $U=2.5$, $V=1$, and $n_f=0.83$.

Figure 2

The Fourier spectrum of the spatial modulation of the spin-density $M_s$ (a) and the charge-density $\delta n$ (b) around the vortex. Parameter values are the same as in Fig. 1.

Figure 3

The spatial distribution of the $d$-wave SC order parameter $|{\Delta }_d|$ (a), the staggered magnetization $M_s$ (b), and the electron density $\delta n=n_i-n_f$ (c) around a nonmagnetic unitary impurity (${ϵ}_0=100$) located at the center of the unit cell of $36\times 36$ sites. Parameter values are the same as in Fig. 1.

Figure 4

The Fourier spectrum of the spatial modulation of the spin-density $M_s$ (a) and the charge-density $\delta n$ (b), around a nonmagnetic unitary impurity. Parameter values are the same as in Fig. 3.