Transverse-Field Measurements

In this chapter, transverse-field (TF) $\mu$SR measurements on the Pr$_{2-x}$Ce$_x$CuO$_4$ single crystals are presented. For all the measurements described here, the initial muon-spin polarization $\vec{P}(0)$ was directed parallel to the CuO$_2$ layers while the applied magnetic field $H$ was directed perpendicular to the CuO$_2$ layers.

Figure 5.1 shows the time evolution of the muon-spin polarization taken under field-cooled (FC) and zero-field cooled (ZFC) conditions at $T$ = $2.4$ K. In the FC procedure, the sample is cooled to temperatures below $T_c$ in an applied magnetic field. Generally speaking, for intermediate applied fields $H_{c_{1}}$ < $H$ < $H_{c_{2}}$, this results in a well-ordered vortex lattice in the superconducting state. Consequently, the time spectra show an oscillating signal with an amplitude damped out over time due to the inhomogeneous internal magnetic field distribution of the vortex lattice. In the ZFC procedure, the sample is cooled below $T_c$ in zero external magnetic field and then an external field is applied. In this case, pinning at the sample edges usually prevents the magnetic flux from entering the bulk of the sample at low external magnetic fields ( $H \leq 300$ Oe). Consequently the ZFC asymmetry spectra resemble those in the ZF measurements. At higher field ($H$ = $500$ Oe) under ZFC conditions, the magnetic flux fully penetrates the bulk of the sample, but in the absence of a strong repulsive interaction between the vortices, random defects in the sample exert a pinning force on the vortices, resulting in a highly disordered arrangement.

Fig. 5.1: Time evolution of the muon-spin polarization in Pr$_{2-x}$Ce$_x$CuO$_4$ taken under (a) FC and (b) ZFC conditions at $T$ = $2.4$ K. Note that the time spectrum at $H$ = 500 Oe is presented in the rotating reference frame.

Figure 5.2 shows the time evolution of the muon-spin polarization taken under FC and ZFC conditions at $H$ = $91$ Oe at a number of temperatures. In the FC procedure [Fig. 5.2(a)], the muon-spin depolarization rate decreases with increasing temperature due to the growth of the in-plane magnetic penetration depth $\lambda _{ab}$, which is the length scale over which magnetic field decays outside of the vortex core. On the other hand, under ZFC conditions [Fig. 5.2(b)], pinning at the sample surface prevents the magnetic flux from entering the bulk of the sample at low temperatures. At higher temperature ($10$ K), magnetic flux penetrates deeper into the sample due to thermal depinning and an increase of $\lambda _{ab}$.

Fig. 5.2: Time evolution of the muon-spin polarization in Pr$_{2-x}$Ce$_x$CuO$_4$ taken under (a) FC and (b) ZFC conditions at $H$ = $91$ Oe.

Figure 5.3 shows fast Fourier transforms (FFTs) of the muon-spin precession signals at $H$ = $91$ Oe recorded under FC conditions. To eliminate systematic errors, these measurements were taken at different temperatures in a random sequence. The FFTs give an approximate picture of the internal magnetic field distribution $n(B)$. Although the sharp features of $n(B)$ are smeared due to the effects described in section 2.4, the FFTs still provide useful insight into the nature of $n(B)$. Above $T_c$, the field distribution is broadened by both the electronic and nuclear magnetic moments. The $\mu$SR linewidth was observed to increase with increasing $H$. This behaviour suggests that there is an inhomogeneous distribution of local magnetic susceptibilities in the sample, which results in a distribution of muon frequency shifts. It is likely that the inhomogeneity arises from spatial variations of charge doping. Below $T_c$, the field distribution is further broadened and becomes asymmetric, due to the inhomogeneous field produced by a vortex lattice. To minimize the effects of the inhomogeneous local magnetic susceptibility, the magnetic penetration depth measurements were performed at low external magnetic fields.

Fig. 5.3: FFTs of the muon-spin precession signal in Pr$_{2-x}$Ce$_x$CuO$_4$ plotted as a function of the local magnetic field $B_{\mu}$ sensed by the muons. The value of the external magnetic field ($H$ = $91$ Oe) is represented by the vertical dashed line.

In Nd$_2$CuO$_4$, which is the parent compound of the related electron-doped cuprate
Nd$_{2-x}$Ce$_x$CuO$_4$, the muon stopping site was found to be near an O(2) oxygen midway between two CuO$_2$ planes [2] (Fig. 3.1). In particular, the implanted muon hydrogen bonds to the oxygen ion with a bonding radius of $\sim 1$ Å. This single muon stopping site assignment is consistent with Fig. 5.3, which shows only one well-resolved signal with an average frequency shift relative to the Larmor precession frequency of the positive muon in vacuum.

Figure 5.4 shows the temperature dependence of the difference between the average local magnetic field $B_{0}$ at the muon site and the external magnetic fields $H$ = $91$ Oe, $200$ Oe, $500$ Oe and $2000$ Oe. The open circles represent measurements at $H$ = $200$ Oe on a second Pr$_{2-x}$Ce$_x$CuO$_4$ single crystal. Above $T_c$, $B_{0}$ - $\mu_{0}H$ is a linear function of $H$ (see inset of Fig. 5.4) which corresponds to a $\mu ^{+}$ Knight shift,

$$K_{\mu}^{\perp} = \frac{B_{0} - \mu_{0}H}{\mu_{0}H}\, .$$ (6.1)

Note that ``$\perp$'' refers to $H$ directed perpendicular to the CuO$_2$ plane. This is the measured $\mu ^{+}$ Knight shift. The intrinsic $\mu ^{+}$ Knight shift is obtained after correcting for the sample geometry such that,

$$K^{\perp} = K_{\mu}^{\perp} + (\frac{1}{3} - N)\chi^{\perp}\, ,$$ (6.2)

where $N$ is a geometrical demagnetization factor that reduces the effective local field sensed by the muons. In particular, the demagnetization field is given by

$$\vec{B}_{dem} = -\mu_{0}N\vec{M}\, ,$$ (6.3)

where $\vec{M}$ = $\chi^{\perp}\vec{H}$. For a thin plate-like crystal $N \approx 1$. The $\frac{1}{3}$ factor in Eq. (5.2) is due to the Lorentz field [20], $\vec{B}_L$ = $\mu_{0}\vec{M}/3$.

Fig. 5.4: Temperature dependence of the difference between the average local magnetic field $B_{0}$ at the muon stopping site and the external magnetic field $H$ = $91$, $200$, $500$ and $2019$ Oe. The open circles represent measurements on a second Pr$_{2-x}$Ce$_x$CuO$_4$ single crystal at $H$ = $200$ Oe. Inset: Magnetic field dependence of $B_{0} - \mu _{0}H$ at $T$ = $5$ and $25$ K.

Figure 5.5 shows a plot of the intrinsic $\mu ^{+}$ Knight shift $K^{\perp }$ against the measured bulk magnetic susceptibility $\chi ^{\perp }$. The negative value of $K^{\perp }$ in the normal state results from the dipolar fields of electronic Pr magnetic moments, which are induced by the external magnetic field. Below $T_c$, the average internal magnetic field increases such that for low $H$, $B_{0}$ is actually greater than the value of the applied field. This is unusual because a fundamental property of a superconductor is it that it expels magnetic field. In contrast to the normal state, $B_{0}$ - $\mu_{0}H$ is not a linear function of $H$ and is reduced with increasing $H$.

Fig. 5.5: The intrinsic $\mu ^{+}$ Knight shift$K^{\perp }$ vs. the bulk susceptibility $\chi ^{\perp }$ at $H$ = $3$ kOe. Inset: Temperature dependence of the bulk magnetic susceptibility measured in an external magnetic field $H$ = $10$ kOe applied perpendicular ( $\chi ^{\perp }$) and parallel ($\chi^{\Vert}$) to the $\hat{a}$-$\hat{b}$ plane.

As shown in Fig. 5.6, the increase of $B_{0}$ with decreasing temperature nearly coincides with the diamagnetic shift observed below $T_c$ in bulk magnetic susceptibility measurements taken under FC conditions. Such behaviour could be associated with the so-called paramagnetic Meissner effect (PME) [27], which is a reported anomalous paramagnetic response in dc FC magnetization measurements. However, magnetization measurements taken under both FC and ZFC conditions show a diamagnetic response that is inconsistent with this interpretation. A second possible explanation is that the Pr electronic moments induce screening currents in the CuO$_{2}$ planes [28]. However, unlike the large rare-earth moments in the RBa$_2$Cu$_3$O$_7$ (R $\equiv$ Gd, Er) compound studied in Ref. [28], the Pr moments induced at low fields are too weak to induce a significant shift of the local magnetic field. In particular, at $H$ = $90$ Oe and $T$ = $25$ K (see Fig. 5.4 inset), $B_0$ - $\mu H \approx -0.3$ G, whereas $B_0$ - $\mu H \approx 10$ G at $T$ = $5$ K.

Fig. 5.6: Temperature dependence of the average local magnetic field at the $\mu ^{+}$ stopping site $B_{0}$ (solid circles) and the bulk magnetic susceptibility $\chi ^{\perp }$ (open circles) measured under FC conditions at $H$ = $91$ and $500$ Oe.

A third possible origin of the increased internal magnetic field below $T_c$ is that the vortices stabilize magnetic order of the Cu spins. This has recently been observed in a neutron scattering study of underdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$ ($x$ = $0.1$) single crystals [29]. Although the difference between the average local magnetic field in the superconducting state ($B_{0}^{S}$) and the normal state ($B_{0}^{N}$) of Pr$_{2-x}$Ce$_x$CuO$_4$ decreases as $H$ increases, it does not necessarily imply that the additional magnetism below $T_c$ weakens with increasing $H$. Figure 5.7 shows a plot of $B_{0}^{S}$ - $B_{0}^{N}$ as a function of $B_{0}^{N}$ at temperatures $T$ = $5$ K and $10$ K. The solid curves are fits to the relation

$$B_{0}(T) = \sqrt{(B_{0}^{N})^{2} + B_{\Vert}^{2}}\, ,$$ (6.4)

where $B_{\Vert}$ is an additional component of field in a direction perpendicular to $B_{0}^{N}$. These fits give $B_{\Vert}$ = $43$ G and $26$ G at $T$ = $5$ K and $10$ K, respectively. The quality of these fits indicate that the orientation of the additional field that appears at the muon site below $T_c$ is directed primarily parallel to the CuO$_2$ planes.

Fig. 5.7: Magnetic field dependence of $B_{0}^{S}$ - $B_{0}^{N}$ at $T$ = $5$ K (closed circles) and $10$ K (open circles). The solid curves are fits of the data to Eq. (5.4) with $B_{\Vert}$ as a free parameter.

Figure 5.8 shows two proposed Cu-spin structures for the parent compound Pr$_2$CuO$_4$ [30,31]. In both structures, the Cu spins are antiferromagnetically ordered in the CuO$_2$ planes. However, in the collinear structure, Cu-spins in adjacent CuO$_2$ planes are aligned either parallel or antiparallel to each other, whereas in the non-collinear structure, Cu-spins in adjacent CuO$_2$ planes are aligned perpendicular to one another. Also shown in Fig. 5.8 is a plot of the calculated magnitude of the magnetic field at the muon stopping site due to the dipolar fields of the Cu moments as a function of the canting angle $\theta$ between the Cu spins and the CuO$_2$ plane. The calculation assumes the ordered Cu moment value of 0.40 $\mu_B$ determined by neutron scattering in Pr$_2$CuO$_4$ [30]. The same field-dependent neutron scattering measurements suggest that the correct Cu-spin structure is the non-collinear one. Our calculation shows for both Cu-spin structures, the magnitude of the magnetic field at the muon stopping site due to the Cu moments is the same. It also shows that this magnetic field is directed parallel to the CuO$_2$ plane. This indicates that the onset of antiferromagnetic order is the source of the additional local magnetic field sensed by the muons below $T_c$.

Fig. 5.8: Top: Two proposed ordered Cu-spin structures for Pr$_{2}$CuO$_{4}$. Bottom: Magnetic field strength at the muon stopping site due to Cu dipole moments plotted as a function of the canting angle $\theta$ between the Cu spins and the CuO$_2$ plane. The arrows at the oxygen atoms indicate the direction of the net local magnetic field due to the Cu moments. The results are the same for both models.

To determine the in-plane magnetic penetration depth $\lambda _{ab}$, the individual time histograms from the four counters (U, D, L, R) were fit simultaneously to Eq. (2.4), assuming the spatial magnetic field profile given by Eq. (2.8). To model the effect of the internal magnetism due to electronic and nuclear moments on the muon-spin precession signals, several depolarization functions $G(t)$ were tested. For example, we first tried $G(t)$ = $e^{-\sigma^2t^2/2}$ and $G(t)$ = $e^{-\Delta t}$ which assumes these static magnetic moments are dense and dilute respectively. In the final analysis a power exponential function, $G(t)$ = $\exp{[-(\Delta t)^{\beta}]}$, with the value of $\beta$ fixed to $1.2$. This gave fits of similar quality over the whole temperature range below $T_c$. Figure 5.9 shows the FFTs of the muon-spin precession signal and the two different ``best-fit'' theory functions from the time domain at $T$ = $3.15$ K. The FFT of the fit function with $G(t)$ = $e^{-(\Delta t)^{1.2}}$ clearly gives a more accurate representation of the measured internal magnetic field distribution than that with $G(t)$ = $e^{-(\sigma t)^2/2}$.

Fig. 5.9: FFTs of the muon-spin precession signal (solid line) and the theoretical muon-spin polarization functions with $G(t)$ = $e^{-(\Delta t)^\beta }$ where $\beta$ = $1.2$, (dashed line) and $G(t)$ = $e^{-(\sigma ^2 t^2)/2}$ (dotted line).

At low external magnetic field, where the density of vortices is small, few muons stop in the vicinity of the vortex cores. Consequently, measurements at low $H$ here are not sensitive to the high-field cutoff of the measured field distribution, which corresponds to the vortex core region. This prevents an accurate determination of $\xi _{ab}$. Figure 5.10 shows the temperature dependence of the in-plane magnetic penetration depth determined with the value of $\xi _{ab}$ fixed in the fitting procedure. The plot shows results for two different values of $\xi _{ab}$. The value of $\lambda_{ab}(T)$ at any given temperature differs by less than 2% for the two values of $\xi _{ab}$. In the end, we fixed the value of $\xi _{ab}$ at $60$ Å, which is consistent with the value of the upper critical field $H_{c2}$ = $\Phi_{0}/2\pi\xi^{2}_{ab}$ measured in Ref. [33].

Fig. 5.10: Temperature dependence of $\lambda _{ab}$ for a Pr$_{2-x}$Ce$_x$CuO$_4$ single crystal with $\xi _{ab}$ = 30 Å (open triangles) and 60 Å (solid squares) at $H$ = $91$ Oe.

Figure 5.11 shows the temperature dependence of $\lambda _{ab}^{-2}(T)$ at $H$ = $91$ Oe. The solid curve is a fit to the relation

$$\lambda_{ab}^{-2}(T) = \lambda_{ab}^{-2}(0)[1-(T/T_c)^2]\, ,$$ (6.5)

where $\lambda_{ab}(0)$ = $3369 \pm 73$ Å and $T_c$ = $15.9 \pm 0.2$ K. We note that the value of $T_c$ is roughly equal to the temperature below which the dc bulk magnetic susceptibility flattens off under ZFC conditions (see Fig. 3.2).

Fig. 5.11: Temperature dependence of the magnetic penetration depth $\lambda _{ab}^{-2}(T)$ at $H$ = $91$ Oe. The solid curve is fit of the data to Eq. (5.5).

Figure 5.12 shows the temperature dependence of $\lambda_{ab}(T)$ in a second Pr$_{2-x}$Ce$_x$CuO$_4$ single crystal at $H$ = $200$ Oe. The large error bars indicate that $\lambda _{ab}$ is poorly determined due to the broadening effects with increased applied magnetic field described earlier.

Fig. 5.12: Temperature dependence of the magnetic penetration depth in a second Pr$_{2-x}$Ce$_x$CuO$_4$ single crystal at $H$ = $200$ Oe.

Fig. 5.13: Normalized magnetic penetration depth $\lambda ^{2}_{ab}(0)/\lambda ^{2}_{ab}(T)$ as a function of reduced temperature $T/T_{c}$ for Pr$_{2-x}$Ce$_x$CuO$_4$ single crystals at $H$ = $91$ Oe, La$_{2-x}$Sr$_x$CuO$_4$ at $H$ = $2$ kOe and YBa$_2$Cu$_3$O$_{6.95}$ at $H$ = $5$ kOe. The solid curve is a guide for the eye.

Figure 5.13 shows the normalized magnetic penetration depth $\lambda_{ab}^{-2}(T)/\lambda_{ab}^{-2}(0)$ plotted as a function of the reduced temperature $T/T_{c}$ at $H$ = $91$ Oe. Also shown are measurements in La$_{2-x}$Sr$_x$CuO$_4$ at $H$ = $2$ kOe [13] and YBa$_2$Cu$_3$O$_{6.95}$ at $H$ = $5$ kOe [34] from earlier $\mu$SR studies. The graph, which assumes a value of $\lambda_{ab}(0)$ = $3300$ Å for Pr$_{2-x}$Ce$_x$CuO$_4$ shows reasonable agreement with the hole-doped cuprates at temperatures down to $T \sim 0.2$ $T_c$. Unfortunately, the cryostat used for cooling the crystals was limited to $T \geq 2.3$ K. This prevented a precise determination of the low temperature limiting behaviour of $\lambda _{ab}^{-2}(T)$.