Disorder-Induced Freezing of Dynamical Spin Fluctuations in Underdoped Cuprate Superconductors

We study the dynamical spin susceptibility of a correlated $d$-wave superconductor (dSC) in the presence of nonmagnetic disorder, using an unrestricted Hartree-Fock approach. This model provides a concrete realization of the notion that disorder slows down spin fluctuations, which eventually “freeze out.” The evolution of disorder-induced spectral weight transfer agrees qualitatively with experimental observations on underdoped cuprate superconductors. For sufficiently large disorder concentrations, static spin density wave (SDW) order is created when droplets of magnetism nucleated by impurities overlap. We also study the disordered stripe state coexisting with a dSC and compare its magnetic fluctuation spectrum to that of the disorder-generated SDW phase.

Figure 1

Intensity plot of $\chi (q,\omega )$ along a cut $\overrightarrow{q}=(q,\pi )$ (a),(c) and magnetization (b),(d) in the clean system with (without) stripes shown in (a),(b) $[U=2.6t]$ [(c),(d) $[U=2.9t]$]. The susceptibility in (c) is the sum of $\chi (q,\pi ,\omega )$ and $\chi (\pi ,q,\omega )$ representing domain averaging.

Figure 2

(a),(c),(e) Same as in Fig. 1a with the addition of disorder. Panels (b),(d),(f) show the associated real-space static magnetization. Parameters are $U=2.6t$ and $n_{\mathrm{imp}}\simeq 1\%$ (a),(b), $n_{\mathrm{imp}}\simeq 2\%$ (c),(d), and $n_{\mathrm{imp}}\simeq 3\%$ (e),(f). The results in the left-hand panels are averaged over ten random disorder configurations.

Figure 3

(a)–(c) $\chi (q_{\mathrm{IC}},\omega )$ as a function of $\omega$ for different $n_{\mathrm{imp}}$ and $U=2.5t$ (a), $U=2.6t$ (b), and $U=2.7t$ (c). Panel (d) shows the real-space magnetization induced by a single nonmagnetic impurity at $R=(12,12)$ for $U=2.7t$. Panels (e),(f) display the susceptibility at the IC wave vector $q_{\mathrm{IC}}=(\frac{3}{4},1)\pi$ in the bare ($U=0$) case (blue solid curve) and the full local susceptibility at $R=(11,12)$ (red dashed line) with $U=2.5t<U_{c1}$ (e) and $U=2.7t>U_{c1}$ (f). The black solid lines in (e),(f) show the spatially averaged full RPA susceptibility $\chi (q_{\mathrm{IC}},\omega )$.

Figure 4

(a),(b) Real-space magnetization with 2% (a) and 4% (b) disorder added to a stripe ordered system similar to Fig. 1d. Panels (c)–(f) show the associated susceptibility transverse (c),(d) and along (e),(f) the stripe direction for the same impurity concentrations, i.e., 2% disorder in (c),(e) and 4% in (d),(f).