# Response to Tilted Magnetic Fields in Bi 2 Sr 2 CaCu 2 O 8 with Columnar Defects: Evidence for Transverse Meissner Effect

V. Ta Phuoc, Enrick Olive, R. de Sousa, A. Ruyter, L. Ammor, J. Soret

► To cite this version:

V. Ta Phuoc, Enrick Olive, R. de Sousa, A. Ruyter, L. Ammor, et al.. Response to Tilted Magnetic Fields in Bi 2 Sr 2 CaCu 2 O 8 with Columnar Defects: Evidence for Transverse Meissner Effect. Physical Review Letters, 2002, 88 (18), 10.1103/PhysRevLett.88.187002. hal-01794684

HAL Id: hal-01794684

https://univ-tours.hal.science/hal-01794684

Submitted on 25 Mar 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Response to Tilted Magnetic Fields in Bi₂Sr₂CaCu₂O₈ with Columnar Defects: Evidence for Transverse Meissner Effect

V. Ta Phuoc, E. Olive, R. De Sousa, A. Ruyter, L. Ammor, and J. C. Soret

LEMA, CNRS-FRE 2077, Université François Rabelais, 37200 Tours, France

(Received 19 January 2001; published 18 April 2002)

The transverse Meissner effect (TME) in the highly layered superconductor Bi₂Sr₂CaCu₂O₈+y with columnar defects is investigated by transport measurements. We present evidence for the persistence of the Bose glass phase for $H_⊥ < H_{⊥c}$: (i) the variable-range vortex hopping process crosses over to the half-loop regime; (ii) near $H_{⊥c}$ the energy barriers vanish linearly with $H_⊥$; (iii) the transition temperature is governed by $T_{BG}(H_‖, 0) - T_{BG}(H_‖, H_⊥) \sim |H_⊥|^{1/ν_⊥}$ with $ν_⊥ = 1.0 \pm 0.1$. Furthermore, above the transition as $H_⊥ \rightarrow H_{⊥c}^+$, moving kink chains consistent with a commensurate-incommensurate transition scenario are observed. These results show the existence of the TME for $H_⊥ < H_{⊥c}$.

DOI: 10.1103/PhysRevLett.88.187002

PACS numbers: 74.60.Ge, 74.72.Hs

The interplay between elasticity, interactions, thermal fluctuations, and dimensionality of magnetic flux lines subject to pinning yields a large variety of vortex phases in high-$T_c$ superconductors [1]. The nature of these phases and how they depend on the pinning is far from being completely elucidated. The theory of pinning by correlated disorder, such as twin planes or amorphous columnar tracks created by heavy ions, has been considered by Nelson and Vinokur (NV) [2] and Hwa et al. [3]. In the case of parallel tracks, NV have shown that, if the applied magnetic field **H** is aligned with the columnar defects, the low temperature physics of vortices is similar to that of the Bose glass (BG) [4], with the flux lines strongly localized in the tracks leading to zero creep resistivity. When **H** is tilted at an angle $θ$ away from the column direction, the BG phase with perfect alignment of the internal flux density **B** parallel to the columns is predicted to be stable up to a critical transverse field $H_{⊥c}$ producing the so-called transverse Meissner effect (TME) [2,5]. For $θ > θ_c \equiv \arctan(\mu_0 H_{⊥c} / B)$, the linear resistivity is finite resulting from the appearance of kink chains along the transverse direction as discussed in Ref. [6]. Finally, above a still larger angle, the kinked vortex structures disappear and **B** becomes collinear with **H**. Similar scenarios apply to vortex pinning by the twin boundaries [2] and by the layered structure of the compound itself [7].

Recently, the TME in untwinned single crystals of YBa₂Cu₃O₇ (YBCO) with columnar defects has been observed using Hall sensors [8]. Previous magnetic measurements in anisotropic 3D superconductors, such as YBCO, gave support to the presence of vortex lock-in phenomena, due to pinning by the twin boundaries [9] or by the interlayers between the Cu-O planes [10]. In the case of highly layered superconductors, such as Bi₂Sr₂CaCu₂O₈+y (BSCCO), the lock-in transition was observed for **H** tilted away from the layers [11], but the existence of the TME due to the pinning by columnar defects remained an open question.

In this Letter, we present measurements of the electrical properties near the glass-liquid transition in BSCCO single crystals with parallel columnar defects, as a function of $T$ and $θ$ for filling factors $f \equiv \mu_0 H_‖ / B_Φ < 1$. Below a critical angle $θ_c(T)$, we observe a vanishing resistivity $ρ(J)$ for low currents. A detailed quantitative analysis of $ρ(J)$ indicates that the creep proceeds via variable-range vortex hopping (VRH) at low currents due to some disorder [12], crossing over to the half-loop (HL) regime at high currents. For $θ > θ_c(T)$, the very signature of a kinked vortex structure, consistent with a commensurate-incommensurate (CIC) transition scenario in 1 + 1 dimensions [6], is deduced from the critical behavior of the linear resistivity. All these results clearly demonstrate that, when **H** is tilted at $θ < θ_c$ away from the defects, the flux lines remain localized on columnar defects, and hence the BG exhibits a TME.

The BSSCO single crystal was grown by a self-flux technique, as described elsewhere [13]. The crystal of 1 × 1 × 0.030 mm³ size with the *c* axis along the shortest dimension has a $T_c$ of 89 K, a transition width of ~1 K, and was irradiated along its *c* axis with 5.8 GeV Pb ions to a dose corresponding to $B_Φ = 1.5$ T at the GANIL (France). Isothermal I-V curves were recorded using a dc four-probe method with a sensitivity of ~$10^{-10}$ V and a temperature stability better than 5 mK. **H** was aligned with the tracks using the well-known dip feature occurring in the dissipation process for $θ = 0°$, and was tilted with an angular resolution better than 0.1° away from the column direction for *f* fixed.

Figure 1 shows a typical log-log plot of $V/I-I$ curves obtained by varying $θ$ for $f = 1/3$ and $T$ fixed below the BG transition temperature determined at $θ = 0°$ [12]. We observe a well-defined angular crossover at an angle $θ_c$. For $θ < θ_c$, the data suggest that the BG phase persists. On the contrary, the existence of an Ohmic regime for $θ > θ_c$ indicates a liquid vortexlike state.

We first focus on the angular range $θ < θ_c$, and we consider the scenario where the BG phase is stable. Thus,
<figure><img src="image_2.png" /><figcaption>FIG. 1. V/I-I curves for tilted magnetic fields. From the right to the left $\theta = 0^\circ, 5^\circ, 10^\circ, 15^\circ, 20^\circ, 25^\circ, 30^\circ, 35^\circ$, and $40^\circ$. The curved lines are a fit of the Bose glass theory to the data.</figcaption></figure>

one expects that excitations of some localized vortex lines lead to a nonlinear resistivity given by [2]

$$\rho(J) = \rho_0 \exp[-\tilde{E}_K(J_c/J)^\mu/k_B T], \quad (1)$$

where $\rho_0$ is a characteristic flux-flow resistivity and $\tilde{E}_K(J_c/J)^\mu$ represents the barriers against vortex motion. This expression is predicted to hold for various regimes of different behavior as the current probes different length scales in the BG. $\tilde{E}_K$ acts as a scaling parameter for the pinning energy and $J_c$ is the characteristic current scale of the creep process. When the current is large enough that the growth of vortex-loop excitations of a line from its pinning track does not reach out to the neighboring tracks, the HL excitations are relevant and yield an exponent $\mu = 1$ and $J_c = J_1 = U/(\Phi_0 d)$, where U is the mean pinning potential and $d = \sqrt{\Phi_0/B_\Phi}$ is the mean spacing between pins. With decreasing current, the size of half-loops increases with the result that some disorder in the pinning potential becomes relevant. This situation yields the VRH process characterized by $\mu = 1/3$ and by another important current scale $J_c = J_0 = 1/[\Phi_0 g(\tilde{\mu}) d^3]$, where $g(\tilde{\mu})$ denotes the density of pinning energies at the chemical potential of the vortex system. One expects that both $J_1$ and $J_0$ are insensitive to $H_\perp$. We therefore consider that the only important effect of $H_\perp$ is to lower $\tilde{E}_K$, according to the formula

$$\tilde{E}_K = E_K - \epsilon^2 \Phi_0 d H_\perp, \quad (2)$$

where $E_K$ is the mean kink energy, and the energy gain due to the tilt is obtained from the isotropic result $\Phi_0 d H_\perp$ by applying the scaling rule [Eq. (3.12)] of Ref. [1].

We now present the following two-step method of fitting Eqs. (1) and (2) to the data, which allows us to obtain a good understanding of the physics in our experiment. First, we plot in Fig. 2 the natural logarithm of $V/I$ versus $\tan\theta$ for $I$ fixed. Note that $\tan\theta$ is here directly proportional to $H_\perp$ since $f$ is kept constant. A linear variation

<figure><img src="image_3.png" /><figcaption>FIG. 2. Angular dependence of $\ln(V/I)$ for different $I$. The solid lines are a fit of $\ln(V/I) = M + N \tan\theta$ to the data; the dotted lines are a guide for the eye. Inset: $N/I_0^{1/3}$ vs $I$ (left axis) and $N/I_1$ vs $I$ (right axis) with $I_0$ and $I_1$ determined from theory. Inset (a): $f = 1/3$ and $T = 68.553$ K (filled symbols), $f = 2/15$ and $T = 69.136$ K (open symbols). Inset (b): $f = 1/3$ and $T = 59.960$ K in another crystal with $B_\Phi = 0.75$ T. The straight lines are fits whence we obtain $\mu = 0.33 \pm 0.05$ (solid line) for $I < I_2$ and $\mu = 1.0 \pm 0.1$ (dashed line) for $I > I_2$.</figcaption></figure>

$\ln(V/I) = M + N \tan\theta$ (expression designated below as E1) with $M$ and $N$ being current-dependent parameters is found for $\theta$ varying up to $\theta_c$ (solid lines), whereas above $\theta_c$ the data deviate from this behavior (dotted lines). This finding is another argument supporting a true angular transition at $\theta_c$, as suggested in Fig. 1. It should be noted that we draw such a conclusion from the observation of two independent regimes of different behavior. One leads to a vanishing linear resistivity as $\theta \to \theta_c^+$, and the other is evidenced by probing the regions of nonlinear resistivity above and below $\theta_c$. Another result is that $N \sim I^{-\mu}$ with $\mu$ undergoing a jump from $1/3$ to $1$ at $I_2 \approx 60$ mA. Such a current crossover is clearly visible in inset a of Fig. 2 (filled symbols), where we plot together $N$ normalized by $I_0^{1/3}$ and $N$ normalized by $I_1$ versus $I$ depending upon whether $I$ is $<I_2$ or $>I_2$, respectively. Here, $I_0$ and $I_1$ are two currents evaluated on the basis of the BG model, as will be seen below. It therefore follows that $N_{1/3}(I) = K(I_0/I)^{1/3}$ and $N_1(I) = KI_1/I$, where the labels $1/3$ and $1$ differentiate between two fits below and above $I_2$, respectively. In both these equations, $K \approx 0.6$ is a dimensionless constant. It should be noted that such a crossover at $I_2$ is clearly observed from V/I-I curves as well (see Fig. 1), where a sudden increase of $V/I$, suggesting an increasing vortex motion, occurs at $I_2$. Second, we plot in Fig. 3 and its inset the natural logarithm of $V/I$ versus $I^{-1/3}$ for $I < I_2$ and $I^{-1}$ for $I > I_2$, respectively. We verify that the expression $\ln(V/I) = P - Q_\mu I^{-\mu}$ (expression designated below as E2) fits very well our experimental data (solid lines) with an exponent $\mu = 1/3$ for $I < I_2$ and with $\mu = 1$ for $I > I_2$. In E2, $Q_\mu$ is labeled using the above convention. The result is that $P$ is a constant (within the experimental errors), while $Q_\mu$ depends
<figure><img src="image_4.png" /><figcaption>FIG. 3. $\ln(V/I)$ vs $I^{-1/3}$ for $I < I_2$ and $\theta < \theta_c$. Inset: $\ln(V/I)$ vs $I^{-1}$ for $I > I_2$ and $\theta < \theta_c$. In both plots the lines are a fit to the data using the form $\ln(V/I) = P - Q_\mu I^{-\mu}$ with $\mu = 1/3$ or 1 according to whether $I$ is <$I_2$ or >$I_2$.</figcaption></figure>

on $\theta$. In Fig. 4 and its lower inset, we show $Q_{1/3}$ and $Q_1$ versus $\tan\theta$, respectively. A linear variation in $\tan\theta$ is observed consistently with expression E1. In the upper inset of Fig. 4, we plot together $Q_{1/3}/I_0^{1/3}$ and $Q_1/I_1$ versus $\tan\theta$. The result is that the data are superimposable onto a single straight line with slope $\approx -0.6$ consistent with the value of $-K$, as may be verified by identifying the expression E1 with the expression E2.

In conclusion, $V/I = R_0 \exp[-(K' - K \tan\theta) \times (I_0/I)^{1/3}]$ is clearly observed in our experiment for $I < I_2$ (solid lines in Fig. 1) while $V/I = R_0 \times \exp[-(K' - K \tan\theta)(I_1/I)]$ is valid for data above $I_2$ (dashed lines in Fig. 1). In both fitting expressions, $K$ and $K'$ depend on $f$ and $T$. In the case of data presented in Fig. 1, $K \approx 0.6$ and $K' \approx 0.4$; furthermore $I_0 = 15$ Å

<figure><img src="image_5.png" /><figcaption>FIG. 4. Angular dependence of $Q_{1/3}$ defined in the regime $I < I_2$ (see Fig. 3). Lower inset: angular dependence of $Q_1$ defined in the regime $I > I_2$. Upper inset: angular dependence of $Q_{1/3}/I_0^{1/3}$ and $Q_1/I_1$. In each plot the straight line is a least-squares fit.</figcaption></figure>

and $I_1 = 0.4$ Å are evaluated from the BG model (see below), and $R_0 \approx 1.1 \ \mu\Omega$ is 5 orders of magnitude lower than the normal resistance. Therefore, for $\theta < \theta_c$ we have rather strong evidence of two separate (albeit related) vortex creep processes peculiar to a BG: the HL expansion with $\mu = 1$ which is cut off by the crossover at $I_2$ into the VRH process with $\mu = 1/3$. Moreover, we note that, for each filling factor investigated in our experiment ($f = 2/15, 1/3, 2/3$), we observe a *continuation* of the VRH process evidenced at $\theta = 0^\circ$ [12]. We therefore argue that the BG phase remains stable up to a critical tilting angle $\theta_c$, as predicted by NV [2].

To test the accuracy of the above view, we first estimate from the theory [1,2] the characteristic current scales, second the current crossover, and third the energy barriers in Eq. (1). $J_0$ depends only on $g(\tilde{\mu})$, the density of pinning energies evaluated at the chemical potential of the vortex system. Although a form of $g(x)$ is not yet available, an estimate of $g(\tilde{\mu})$ can be done in terms of the bandwidth of pinning energies $\gamma$, due to the disorder, and hence $g(\tilde{\mu}) \approx 1/(d^2\gamma)$ so that $J_0 \approx \gamma/\phi_0 d$. Following Blatter et al. [1], $\gamma = t_d + \gamma_i$, where $t_d \approx U/\sqrt{E_K/k_B T} \exp(-\sqrt{2} E_K/k_B T)$ estimates the dispersion in the pinning energies resulting from disorder in the position of tracks [2] and $\gamma_i$ arises from some on-site randomness. We shall assume random defect radii as the main source of on-site disorder. A reasonable estimate of $\gamma_i$ is then given by the width of the distribution of pinning energies $\tilde{P}(U_K) = P(c_K) dc_K/dU_K$, where $U_K$ is the binding energy of a defect with radius $c_K$ and $P(c_K)$ is the probability distribution of the defect radii. A realistic pdf is a normal law centered at $c_0 = 45$ Å with a standard deviation of 6 Å, as shown in Fig. 1 of Ref. [12]. Thus, by using the formula $U_K = (\phi_0^2/8\pi\mu_0\lambda_{ab}^2) \ln[1 + (c_K/\sqrt{2}\xi_{ab})^2]$ [1,2] where $\lambda_{ab}$ and $\xi_{ab}$ are, respectively, the planar penetration depth and the planar coherence length, we determine $U_0 = \langle U_K \rangle$ representing the energy scale for the vortex pinning. Another important energy scale corresponding to vortex positional fluctuations is $T^* = (k_B c_0 / 4 \xi_0 G_i) \sqrt{\ln(a_0 / \xi_{ab}) (T_c - T)}$, where $G_i$ is the 2D Ginzburg number and $a_0 = \sqrt{\phi_0 / B}$ is the vortex-lattice constant. $U_0$ and $T^*$ completely characterize the system of vortices and columnar defects. Thus, it can be inferred that $E_K = (d/\sqrt{2}\xi_{ab}) T^* \sqrt{f(T/T^*)}$, where $f(x) = x^2/2 \exp(-2x^2)$ accounts for entropic effects, and $U = U_0 f(T/T^*)$. Using appropriate parameters for BSCCO ($\lambda_0 \approx 1850$ Å, $\xi_0 \approx 20$ Å, and $G_i \approx 0.2$) we estimate $J_0 \approx 10^9$ A/m$^2$ and $J_1 \approx 2.6 \times 10^7$ A/m$^2$ for $f = 1/3$ and $T = 68$ K, respectively. The balance between the barrier energy for the VRH process $U_{VRH} = \tilde{E}_K(J_0/J)^{1/3}$ and the barrier for the HL regime $U_{HL} = \tilde{E}_K J_1 / J$ determines the crossover current $J_2 = (J_1/J_0)^{1/2} J_1 = U^{3/2} / (\phi_0 d \sqrt{\gamma})$. Thus, we have $J_2 \ll J_1 \ll J_0$, so that the crossover between the VRH process and the HL regime takes place at $J_2$. Assuming uniform currents in the sample, we find $J_2 \approx 60$ mA, in excellent agreement with
the experiment (Fig. 1). A more complete quantitative
evaluation of the consistency of the data with the model
is obtained for different values of *f* and *T*, as well as
for different values of *d* (inset of Fig. 2). In particular,
we find that the effect of *d* on *I*₂ is in good agreement
with the prediction *I*₂ ~ 1/*d*. Finally, evaluating *U*<sub>VRH</sub>
and *U*<sub>HL</sub> with *E*~K~ from Eq. (2), we find, using again
the above usual parameters of BSCCO with *ϵ* ~ 1/200,
that *E*<sub>K</sub>/<i>*k*<sub>B</sub>*T* ~ 0.3 and *ε*²*φ*₀*dH<sub>⊥</sub>/<i>*k*<sub>B</sub>*T* ~ 0.7 tan*θ* for
*f* = 1/3 and *T* = 68 K, which correspond to the values
of our fitting parameters *K*' and *K* (see upper inset of
Fig. 4). Thus, the experiment is also in good agreement
with the barriers *U*<sub>VRH</sub> and *U*<sub>HL</sub> theoretically predicted
below and above *I*₂, respectively.

Finally, we proceed to the linear regime observed above
θ_c (see Fig. 1). For θ > θ_c, Hwa et al. [6] predict, on the
basis of the CIC transition, the appearance of free moving
chains of kinks oriented in the H_⊥ direction leading to
a critical behavior of the linear resistivity ρ ~ (tanθ -
tanθ_c)ν characterized by an exponent ν = 1/2 (ν = 3/2)
in 1 + 1 (2 + 1) dimensions. This type of behavior with
ν = 1/2 is evident in Fig. 5 which shows a plot of R/R_c
versus tanθ - tanθ_c for different values of f (or equiva-
lently H_‖) and T < T_BG(H_‖, 0). Here, R_c is a scaling
parameter comparable to R_0, and, for f fixed, θ_c(T) is
determined by fitting R = R_c(tanθ - tanθ_c)¹/² to the
data. In Fig. 2, the arrow indicates θ_c obtained in this way.
The inset of Fig. 5 shows t(H_‖, H_⊥) ≡ [T_BG(H_‖, 0) -
T_BG(H_‖, H_⊥)]/T_BG(H_‖, 0) as a function of H_⊥, where
T_BG(H_‖, H_⊥) is obtained through the inversion of θ_c and
T. It turns out that t(H_‖, H_⊥) is independent of H_‖. The
solid line shown in the inset is a fit to the data following
t(H_‖, H_⊥) ~ |H_⊥|¹/ν_⊥ with ν_⊥ = 1.0 ± 0.1, as recently
suggested from numerical simulations [14] and observed
in (K, Ba)BiO₃ [15]. Moreover, such a value of ν_⊥ is ex-
cellently consistent with the result previously found using
the scaling theory of the BG transition at H_⊥ = 0 [12].

In summary, we demonstrate from transport measure-
ments the stability of the BG phase in a BSCCO single
crystal with columnar defects, when **H** is tilted θ ≤ θ_c
away from the column direction. We explain the critical
behavior of the linear resistivity on the basis of the CIC
transition of kink chains right above θ_c, as predicted in
1 + 1 dimensions for one-dimensional correlated disorder.
This implies that the appearance of the TME is concomi-
tant with the vanishing of the linear resistivity at θ_c. As a
consequence, our results support the scenario for a usual
BG-to-liquid transition in irradiated BSCCO in contrast to

<figure><img src="image_6.png" /></figure>

FIG. 5. Critical behavior of the linear resistance for different values of *f* and *T*. The curved line is a fit to the data using $R = R_c(\tan\theta - \tan\theta_c)^{1/2}$ in accordance with a CIC transition scenario. Inset: angular dependence of the Bose glass transition. The solid line is a fit of $T_{BG}(H_\parallel, 0) - T_{BG}(H_\parallel, H_\perp) \sim |H_\perp|^{1/\nu_\perp}$ with $\nu_\perp = 1.0 \pm 0.1$ to the data; the point (0,0) is included.

a puzzling two-stage BG-to-liquid transition recently ob-
served in untwinned YBCO with columnar defects [8].

[1] G. Blatter *et al.*, Rev. Mod. Phys. **66**, 1125 (1994).

[2] D. R. Nelson and V. M. Vinokur, Phys. Rev. B **48**, 13060 (1993).

[3] T. Hwa *et al.*, Phys. Rev. Lett. **71**, 3545 (1993).

[4] M. P. A. Fisher *et al.*, Phys. Rev. B **40**, 546 (1989).

[5] N. Hatano and D. R. Nelson, Phys. Rev. B **56**, 8651 (1997).

[6] T. Hwa, D. R. Nelson, and V. M. Vinokur, Phys. Rev. B **48**, 1167 (1993).

[7] D. Feinberg and C. Villard, Phys. Rev. Lett. **65**, 919 (1990);
L. Balents and D. R. Nelson, Phys. Rev. B **52**, 12951
(1995).

[8] A. W. Smith et al., Phys. Rev. B **63**, 064514 (2001).

[9] M. Oussena et al., Phys. Rev. Lett. **76**, 2559 (1996); A. A. Zhukov et al., Phys. Rev. B **56**, 3481 (1997); I. M. Obaidat et al., Phys. Rev. B **56**, R5774 (1997).

[10] A. A. Zhukov et al., Phys. Rev. Lett. **83**, 5110 (1999); Y. V. Bugoslavsky et al., Phys. Rev. B **56**, 5610 (1997).

[11] F. Steinmeyer et al., Europhys. Lett. **25**, 459 (1994);
S. Koleśnik et al., Phys. Rev. B **54**, 13319 (1996).

[12] J. C. Soret et al., Phys. Rev. B **61**, 9800 (2000).

[13] A. Ruyter et al., Physica (Amsterdam) **225C**, 235 (1994).

[14] J. Lidmar and M. Wallin, Europhys. Lett. **47**, 494 (1999).

[15] T. Klein et al., Phys. Rev. B **61**, R3830 (2000).