Observations of the configuration of the high-field vortex lattice in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_7$: Dependence upon temperature and angle of applied field

We have mapped out the high-field phase diagram of the vortex lattice in lightly twinned ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_7$ by employing small-angle neutron scattering. This extends previous measurements [S. P. Brown et al., Phys. Rev. Lett. 92, 067004 (2004)] in which we reported a transition in local flux-line coordination from distorted triangular to square as a function of field applied parallel to the crystal $c$ axis. At high fields, we observe a transition back to triangular coordination with increasing angle between the applied field and $c$ axis or with increasing temperature. At lower fields, our results may be interpreted in terms of a field-induced reduction of the basal-plane $a\text{−}b$ anisotropy. Tilting the flux lines away from the $c$ axis reduces the pinning by twin planes, changes the orientation of the vortex lattice nearest-neighbor directions, and the transition between square and hexagonal vortex lattices becomes first order.

Figure 1

(Color online) (a) Raw data showing both VL signal and background, taken at 5 K, with 2 T, applied parallel to the crystal $c$ axis. The $a$ and $b$ axes are oriented as shown in Fig. 2f. The sample and magnet have been rocked in $\omega$ to satisfy the Bragg condition for VL diffraction spots on the right-hand side of the detector. (b) The same data with background subtracted. (c) The data of (b) with masking of Poisson noise near the beam stop, Gaussian smoothing of 3 pixels FWHM, and contours applied. The color scale used has the same range in these three panels and is linear in intensity.

Figure 2

(Color online) (a)–(e) Diffraction data represented as a sum over rocking curves at 5 K and various fields from 2–11 T applied parallel to the crystal $c$ axis. The orientation of the twinned crystal $a$ and $b$ axes for all these data are shown in (f). In (a) and (b), the sum is over rocking curves in both $\omega$ and $\varphi$. The rocks at higher field were carried out with $\left|\omega \right|=\left|\varphi \right|$, thus rocking directly through the weak peaks seen in the corners of the diffraction pattern. These and the weak peaks in the wings of the vertical and horizontal strong peaks are made clearer by the use of a logarithmic intensity scale. The intensity near the horizontal and vertical directions is visible by virtue of the relaxed resolution of our SANS setup and the form of the SANS resolution function (Refs. 46, 47); for instance, during a rock through the “top-right” region, the resolution will also encompass the peaks at the top and at the right of the diffraction pattern. Backgrounds taken at $T=90\text{K}>T_c$ have been subtracted. These data result from four VL domains (two in each crystal domain, each pinned to a different twin-plane orientation). In (f) we illustrate how the VL patterns, seen most clearly in (a) and (b), are formed from the two orientations of crystal anisotropy and two orientations of twin planes present in our sample. The reciprocal-lattice vectors for each VL domain lie on an ellipse which has axial ratio given by the mass anisotropy ${\gamma }_{ab}$ (Refs. 45, 48, 49) in the low-field region. The orientation of one of the vectors is known, since one of the VL planes is pinned to a set of twin planes. The magnitude of this reciprocal-lattice vector and the magnitude and direction of the others are then fixed by the shape of the ellipse and flux quantization. Defined in (f) are two angles, $\alpha$ and $\beta$, which characterize the configuration of the VL and its diffraction pattern. Due to the two dimensionality of the VL, the shape and distortion of the real-space VL is exactly the same as that of the reciprocal lattice, but rotated by $90°$ about the direction of the field; hence the patterns for each domain may also be viewed as real-space pictures of the vortex arrangement—after addition of a spot at the center of the pattern, corresponding to the position of the masked incident beam.

Figure 3

(Color online) Diffraction patterns taken with fields of 7–10.8 T applied at an angle of $5°$ to the crystal $c$ axis, yielding only two VL domains therefore providing easier resolution of the horizontal peaks. These data are the sums of rocks in $\omega$ and $\varphi$ through angles which satisfy the Bragg condition for peaks to the left/right and top/bottom of the pattern.

Figure 4

Variation of the angles $\alpha$ and $\beta$, defined in Fig. 2f, as a function of field applied at $\le 5°$ to $c$. The solid lines depict the VL structure predicted by anisotropic London theory using ${\gamma }_{ab}=1.28$. Inset: an illustration of the orthorhombic distortion of the two (degenerate) VL patterns expected in our sample.

Figure 5

(Color online) Variation of the estimated ratio of form factors $B_{11}/B_{10}$ versus field (see text for details of the calculation). Also shown is the form-factor ratio expected by anisotropic London theory, using the low-field value of the $a\text{−}b$ anisotropy ratio, but with the $\mathit{q}$ vectors of the observed VL structure. The horizontal line shows the form-factor ratio of 0.5 given by the isotropic London theory applied to a square vortex lattice.

Figure 6

(Color online) The diffraction peaks on the right-hand side of the detector at an induction of 10 T applied at $5°$ to the crystal $c$ axis, at selected temperatures in the range 10–70 K. Increasing temperature leads to a splitting of the spots, which is a signature of a return to a distorted-triangular lattice at high temperatures.

Figure 7

(Color online) Schematic indication of the VL coordination in our YBCO sample. The VL melting line from Ref. 44 is also shown. Since the transition is continuous, the dashed line separating the square and triangular phases has been drawn where the angle $\left(83°\right)$ between the two reciprocal-lattice vectors is halfway between the square and distorted-triangular limits. The temperature errors in the points used to draw this line are approximately equal to the size of the symbols.

Figure 8

Schematics showing (a) the direction of rotation, $\nu$, of the $c$ axis away from the field for our data presented in Sec. 5 and in (b) the effect of this rotation on the effective mass in the plane perpendicular to the vortex direction, where shorter lines represent larger effective masses. Since $m_c⪢m_a\approx m_b$, rotation as depicted in (a) increases the effective mass in the horizontal direction. Therefore crystal domains with $b$ vertical will become more anisotropic, while those with $a$ vertical will become less anisotropic.

Figure 9

(Color online) Field dependence of the VL domains present with the field applied at $\nu =10°$ to the $c$ axis in the rotation plane illustrated in Fig. 8a. Thus the $a/b$ axes are vertical in the figures and the field direction was at $\left(10/\sqrt{2}\right)°$ to the twin planes. All data were taken at 5 K. The pattern in (a) is due to at least two distinguishable VL domains. The diffraction pattern arising from one of these domains, indicated by the dashed blue lines, includes two horizontal spots and the VL is referred to as the “HTD” in the text. The other, indicated by the solid black lines, includes two vertical spots, and the VL is referred to as the “VTD.” The arrows, which lie parallel to the projection on the detector of the sample {110} directions, indicate the positions at which strong scattering would be observed with the field applied parallel to the sample $c$ axis (cf. Fig. 2); in the present arrangement, only weak scattering is observed at these points.

Figure 10

(Color online) Diffraction pattern taken at 5 K with the crystal $c$ axis rotated by $\gamma =30°$ away from the direction of the applied field of (a) 5 T and (b) 9 T. In (a) the HTD observed in the $10°$ data [Fig. 9a] is clearly visible, but the VTD is absent. In its place are two distorted-triangular VLs whose diffraction patterns are indicated by dashed blue and solid black lines. In (b) the HTD is still present but in addition there is an approximately square VL domain.