Vortex phase diagram of $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ single crystals

Irreversible magnetic properties of $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ single crystals were studied by using several techniques including transport, magnetization, and magneto-optics. The mixed-state response was found to be very similar to high-$T_c$ cuprates, such as Y-Ba-Cu-O and Nd-Ce-Cu-O. In particular, we find a nonmonotonic “fishtail” shape of the magnetization loops $M\left(T=\text{const},H\right)$ and a corresponding feature in $M\left(T,H=\text{const}\right)$ scans, as well as a very large magnetic relaxation rate. The supercurrent density, $j$, in the critical state was evaluated from magnetization and from direct transport measurements close to $T_c$. At 5 K we estimate a moderate $j\approx 2.6\pm 0.2\times {10}^5\text{A}/{\text{cm}}^2$. Analysis of the temperature and field dependencies of the magnetic relaxation is consistent with the collective pinning and creep (weak pinning and fast creep) model and suggests a crossover from the collective to the plastic creep regime in fields exceeding the field at which the fishtail magnetization is maximum. While the zero-field anisotropy of the magnetic properties in the superconducting state of $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ is small, the vortex behavior changes for different orientations of magnetic field with respect to the crystal axes, implying field-dependent anisotropy of pinning properties.

Figure 1

(Color online) The samples are cuboids with the shortest side along the crystallographic $c$ axis. For the convenience of calculations sample dimensions are denoted as $2c\le 2a\le 2b$ with $a$ and $b$. The notation for the currents flowing in the critical state are shown for (a) $H\parallel c$ and (b) $H\parallel a$.

Figure 2

(Color online) Magnetization loops measured in a $0.28\times 0.7\times 1.26{\text{mm}}^3$ single crystal of $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ in a magnetic field $H\parallel c$ at $T=5$, 10, 15, and 18 K. The left inset zooms at the higher temperature region for $T=15$, 16, 17, 18, and 19 K where the evolution of the fishtail magnetization is clearly seen. The right inset shows a superconducting transition measured after zero-field cooling at $H=10\text{Oe}$.

Figure 3

(Color online) Magnetization loops for $H\parallel a$ in the $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ single crystal. The crystal is shown in the right inset and has dimensions $0.28\times 0.7\times 1.26{\text{mm}}^3$. Left inset: comparison of $M\left(H\right)$ loops at $T=15\text{K}$ for two field orientations.

Figure 4

(Color online) Supercurrent density in same-batch single crystals $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$. “Magnetic” current density estimated from Eq. (1) at zero applied field by annealing of the remanent magnetization (filled circles) from $M\left(H\right)$ loops measured at different temperatures (squares), from magneto-optical profiles using Eq. (2) (stars), and from direct transport measurements (open circles). The lines show fits to $j\left(T\right)=j\left(0\right){\left[1-{\left(T/T_c\right)}^p\right]}^n$ of the data set shown in full circles with indicated exponents and adjusted ${\chi }^2$ values. Inset zooms at the region of $T_c$ for easier comparison with the transport data obtained from Fig. 8 (circles) and Fig. 9 (triangles).

Figure 5

(Color online) Supercurrents $j\left(T\right)$ measured in a single crystal $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ upon warming after 5.5-T field was reduced to the indicated values of the applied magnetic field at 5 K. Open circles mark the temperatures where $j\to 0$ (the irreversibility line above which the vortex lattice pinning vanishes). The squares mark the temperatures at which $j\left(T\right)$ curves exhibit a change in behavior (marking the fishtail feature as discussed below).

Figure 6

(Color online) Magneto-optical images of a crystal $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ $\left(0.053\times 0.24\times 0.55{\text{mm}}^3\right)$ in the remanent state. The field was applied along the $c$ axis (perpendicular to the page) above $T_c$ and removed after cooling to 5 K. Frames (a), (b), and (c) show images for $T=10$, 13, and 15 K, respectively. Frame (d): a three-dimensional profile of the induction $B_z\left(x,y\right)$ perpendicular to the sample surface obtained from the data shown in (a).

Figure 7

(Color online) Profiles of the magnetic induction for a few different temperatures along the line shown in Fig. 6b. Definition of the maximum variation in the magnetic induction, $\Delta B$, is shown for $T=8.5\text{K}$.

Figure 8

(Color online) Zero-field current-voltage characteristics at indicated temperatures in a bar-shaped, $3\times 0.1\times 0.02{\text{mm}}^3$, crystal of $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ with the current flowing in the $ab$ plane. Upper inset: A photograph of the sample with contacts. Lower inset: The derivative $\text{d}V/\text{d}j$ used to identify the current $j$ which drives the system normal.

Figure 9

(Color online) Resistance as a function of temperature for different dc transport currents in a $3\times 0.1\times 0.02{\text{mm}}^3$ crystal. The onset of nonzero resistance was used to determine the temperature dependence of the maximum transport supercurrent density. Inset: normalized resistance in a full-temperature range measured by two- and four-probe methods.

Figure 10

(Color online) Time dependence of magnetization during 22 h at $T=15\text{K}$ with $H=1\text{T}$ after reducing magnetic field to zero from 5 T. Lower inset: fit to Eq. (4) with parameters indicated. Upper inset: ln-ln plot of $M\left(t\right)$ for $H=1$ and 3 T.

Figure 11

(Color online) The field dependence of magnetization relaxation rate $S$ in a single crystal of $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ shown along with the $M\left(H\right)$ loop measured at the same temperature $T=15\text{K}$.

Figure 12

(Color online) Comparison of the relaxation rate $S\left(H\right)$ for two field orientations at $T=15\text{K}$.

Figure 13

(Color online) Logarithmic relaxation rate as a function of temperature measured in a single crystal of $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ in the remanent state, $H=0$, in three orientations of the trapped vortex direction with respect to the sample. For comparison, $S\left(T\right)$ at $H=3\text{T}$ is shown. (Note different origins and scales of the left and right axes).

Figure 14

(Color online) The $H\text{−}T$ phase diagram of $\text{Ba}{\left({\text{Fe}}_{0.93}{\text{Co}}_{0.07}\right)}_2{\text{As}}_2$ single crystals in the mixed state. The solid lines are fits to the commonly used form $H_{xx}\left(T\right)=H_{xx}\left(0\right){\left[1-{\left(T/T_c\right)}^p\right]}^n$. We obtained for the fishtail peak position $H_p\left(0\right)\approx 13.6\text{T}$, $n=3/2$, and $p=1$. For the irreversibility line: $H_{\text{irr}}\left(0\right)\approx 15.2\text{T}$, $n=3/2$, and $p=2$. For the upper critical field: $H_{c2}\left(0\right)\approx 66.7\text{T}$, $n=4/3$, and $p=1$.