Characterization of the intra-unit-cell magnetic order in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$

As in ${{\mathrm{YBa}}_2{\mathrm{Cu}}_3\mathrm{O}}_{6+x}$ and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{8+\delta }$, the pseudogap state in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ is characterized by the existence of an intra-unit-cell magnetic order revealed by polarized neutron scattering technique. We report here a supplementary set of polarized neutron scattering measurements for which the direction of the magnetic moment is determined and the magnetic intensity is calibrated in absolute units. These data allow a close comparison between bilayer systems ${{\mathrm{YBa}}_2{\mathrm{Cu}}_3\mathrm{O}}_{6+x}$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ and raise important questions concerning the range of the magnetic correlations and the role of disorder around optimal doping.

Figure 1

Hole doping ($p$) dependencies of the pseudogap (PG) temperature $T^{☆}$ measured by ARPES in Bi2212 system [26] and the characteristic temperature associated with the appearance of the IUC magnetic phase $T_{\text{mag}}$, as determined by polarized neutron scattering technique. In the PG state, following the ARPES measurements [26], one can distinguish two regimes, labeled I and II, as explained in the text.

Figure 2

(a) Characterization of the $T_c$ onset measured by neutron depolarization technique for three samples: UD-85 (black curve) [13, 33], OD-87 (red curve) [37], OD-70 (green curve) [38]. While the superconducting transition is rather steep for the overdoped samples, it becomes much smoother for the underdoped sample, a characteristic feature of the underdoping. (b) Rocking scans across the Bragg peak $\mathbf{Q}=(1,1,0)$. The measured intensity for sample UD-85 is multiplied by a factor of 2.5, for sake of comparison with the other samples. (c) Low-energy phonon measurements at the wave vector $\mathbf{Q}=(0,0,13)$ on samples UD-85 and OD-87. The shaded area corresponds to the incoherent elastic scattering on top of the nuclear background.

Figure 3

(a) Variation of the nuclear Bragg intensities as a function of $\mathbf{Q}=(1,0,L)$ in the UD-85 sample. The measurement is performed at room temperature in the NSF channel with the neutron polarization Hx. The nuclear Bragg reflections are located at odd integer $L$ values. Contaminations from Al powder lines also show up at noninteger $L$ values and large wave vectors. (b) Temperature dependencies of the raw neutron scattering intensities measured at the Bragg peak $\mathbf{Q}=(1,0,1)$ (full symbols) and at $\mathbf{Q}=(0.9,0,1)$ (crosses), the background position, in the NSF channel (light blue) and in the SF channel (black). (c) Temperature dependencies of intrinsic Bragg scattering (background subtracted) at $\mathbf{Q}=(1,0,1)$ in the SF channel (black) and in the NSF channel (light blue) divided by a $T$-independent bare flipping ratio ${\mathrm{FR}}_x^o=52.6$ (${1/\mathrm{FR}}_x^o=0.019$). The solid lines are discussed in the text.

Figure 4

(a)–(d) Temperature dependencies of ${1/\mathrm{FR}}_x$ at $\mathbf{Q}=(1,0,L)$ and (2,0,0) for the samples: UD-85 (left axis: full squares) and OD-70 (right axis: open circles). Solid lines are guides to the eye.

Figure 5

(Top panel) Orientation of the three orthogonal polarizations $\mathrm{H}\alpha$ ($\alpha =x,y,z$) with respect to the scattering plane $[1,0,0]$/$[0,0,1]$ and the transferred wave vector $\mathbf{Q}$. (a)–(c) Temperature dependencies of the inverse-flipping ratio $1/{\mathrm{FR}}_{\alpha }$ for the UD-85 sample (left axis, full symbols) and OD-70 (right axis, open symbols) for different polarizations: (a) $\mathrm{Hx}$ (black), (b) $\mathrm{Hy}$ (green), (c) $\mathrm{Hz}$ (red). Solid lines are guides to the eye.

Figure 6

Bi2212 OD-87 sample: $1/\mathrm{FR}x\left(T\right)$ measured at the wave vectors $\mathbf{Q}=(1,0,L)$ (left axis, full symbols) and $\mathbf{Q}=(2,0,0)$ (right axis, open symbols): (a) $L=1$, (b) $L=3$.

Figure 7

(a) Calibration of the nuclear Bragg intensities in absolute unit (barn) for Bi2212 samples. (b) Magnetic intensity at Bragg wave vectors, measured in the normal state at $T_c+25$ K, as a function of $Q_L=\frac{2\pi }{c}$ in Bi2212 (UD-85, OD-87, OD-70) and in Y123 (UD-54 [2, 3], UD-63 [4], UD-30%–2%Zn [6]). (c) $Q_L$ dependence of the normalized magnetic intensity in the monolayer cuprates Hg1201 (UD-61 [10]) and La214 (UD-22 [12]). Lines correspond to a unique function for which the intensity varies only (see text).

Figure 8

(a) Left axis: hole doping dependence of the pseudo-gap (PG) temperature $T^{☆}$ in Bi2212, measured by ARPES [26], $ab$ resistivity [44], and ERS [27]. Hole doping dependencies of the characteristic temperature associated with the appearance of the IUC magnetic phase $T_{\text{mag}}$ for the bilayer systems Bi2212 and for Y123 [2, 4, 6]. Right axis: variation of magnetic Bragg intensity measured at $\mathbf{Q}=(1,0,L)$ with $L=3$ for Bi2212 ($\left|Q\right|=1.75$ Å${}^{-1}$), $L=1$ for Y123 ($\left|Q\right|=1.71$ Å${}^{-1}$). (b) Evolution of the magnetic intensity as a function of $T_{\text{mag}}$.