Angular-dependent upper critical field of overdoped Ba(Fe${}_{1-x}$Ni${}_x$)${}_2$As${}_2$

In-plane resistivity measurements as a function of temperature, magnetic field, and its orientation with respect to the crystallographic $ab$ plane were used to study the upper critical field, $H_{c2}$, of two overdoped compositions of the iron-based superconductor Ba(Fe${}_{1-x}$Ni${}_x$)${}_2$As${}_2$, $x=0.054$ and $x=0.072$. Measurements were performed using precise alignment (with accuracy less than 0.1${}^{\circ }$) of magnetic field with respect to the Fe-As plane. The dependence of the $H_{c2}$ on angle $\theta$ between the field and the $ab$ plane was measured in isothermal conditions in a broad temperature range. We found that the shape of $H_{c2}\left(\theta \right)$ clearly deviates from the Ginzburg-Landau functional form.

Figure 1

Temperature-dependent resistivity of two samples of BaFe${}_{1-x}$Ni${}_x$As${}_2$ used in this study, with $x=0.054$ (slightly overdoped) and $x=0.072$ (strongly overdoped), with doping level indicated with arrows with respect to the temperature-doping phase diagram of BaNi122 after Refs. 47 and 48 shown in the inset. Note pronounced curvature of the $\rho \left(T\right)$ for $T>T_c$, typical of overdoped compositions.[62] Sample resistivity value is defined with accuracy of about 20$\%$ due to uncertainty of geometric factors; see Refs. 46 and 63 for details.

Figure 2

During experiments in a single axis rotation system of a 35 T magnet, the direction of magnetic field was aligned parallel to the conducting plane by resistivity measurements in field $H$ slightly lower than $H_{c2\parallel }$, in which sample resistance shows strong angular dependence (black line in the top panel). The curve was measured in one-sided motion of the rotator to avoid backlash, with deep minimum corresponding to a $H\parallel ab$ or $\theta =0$ condition. The red open symbols show alignment measurements, taken in a second angular sweep of the same rotation direction, and stopped at $\theta =0$. $H$ and $T$ sweeps were used to determine the phase diagrams in the $H\parallel ab$ condition, and then magnetic field angle $\theta$ with respect to the plane was changed by continuing rotation of the sample in the same direction as during alignment. Because the orientation of the sample in the third direction, perpendicular to the rotation plane, was set by eye there may exist a nonzero angle $\phi$ between the field-rotation plane and the plane of the normal to the sample. In most cases, this angle should be less than 5${}^{\circ }$.

Figure 3

In-plane resistivity ${\rho }_a$ vs temperature for slightly overdoped Ba(Fe${}_{1-x}$Ni${}_x$)${}_2$As${}_2$, $x=0.054$ in magnetic fields (a) parallel to the conducting $ab$ plane (right to left, 0 T, 1 T, 2.5 T, 5 T, 7.5 T, 10 T, 15 T, 17.5 T, 20 T, 25 T, 30 T, and 35 T) and (b) parallel to the $c$ axis (right to left, 0 T, 1 T, 2 T, 4 T, 8 T, 10 T, 12 T, 15 T, 20 T, 30 T, and 35 T). Lines indicate 20$\%$, 50$\%$, and 80$\%$ of the resistivity value immediately above the superconducting transition. Bottom panel (c) shows $H_{c2}\left(T\right)$ phase diagrams for both directions of magnetic field.

Figure 4

In-plane resistivity ${\rho }_a$ vs temperature for heavily overdoped Ba(Fe${}_{1-x}$Ni${}_x$)${}_2$As${}_2$, $x=0.072$ in magnetic fields (a) parallel to the conducting $ab$ plane (right to left, 0 T, 1 T, 2.5 T, 5 T, 7.5 T, 10 T, 15 T, 17.5 T, 20 T, and 22.5 T) and (b) parallel to the $c$ axis (right to left, 0 T, 1 T, 2 T, 4 T, and 6 T). Lines indicate 20$\%$, 50$\%$, and 80$\%$ of the resistivity value immediately above the superconducting transition. Bottom panel (c) shows $H_{c2}\left(T\right)$ phase diagrams for both directions of magnetic field.

Figure 5

Field dependence of in-plane resistivity $\rho \left(H\right)$ of a slightly overdoped Ba(Fe${}_{1-x}$Ni${}_x$)${}_2$As${}_2$, $x=0.054$ sample at $T=13$ K [panel (a)] and $T=9$ K [panel (b)] with the magnetic field inclination angle $\theta$ as a parameter. (c) Isotherms $H_{c2}\left(\theta \right)$, obtained at 9 K and 13 K, using 80$\%$, 50$\%$, and 20$\%$ criteria. Solid line shows fit to Eq. (1).

Figure 6

Field dependence of in-plane resistivity $\rho \left(H\right)$ of strongly overdoped Ba(Fe${}_{1-x}$Ni${}_x$)${}_2$As${}_2$, $x=0.072$ sample at $T=1.5$ K [panel (a)] and $T=4$ K [panel (b)] with magnetic field inclination angle $\theta$ as a parameter. (c) Isotherms $H_{c2}\left(\theta \right)$, obtained 1.5 K and 4 K, using 80$\%$, 50$\%$, and 20$\%$ criteria. Solid line shows fit to Eq. (1).

Figure 7

Angular dependence $H_{c2}\left(\theta \right)$, determined from fixed temperature $\rho \left(H\right)$ of Fig. 5 using 20$\%$, 50$\%$, and 80$\%$ criteria (top to bottom), for slightly overdoped Ba(Fe${}_{1-x}$Ni${}_x$)${}_2$As${}_2$, where $x=0.054$ at 9 K (top panel) and 13 K (bottom panel). The lines are guides to the eyes. The data are plotted as $H_{c2}^{-2}\left({\sin }^2\theta \right)$, which according to Eq. (1) should be a straight line. The lines are guides to the eyes.

Figure 8

Angular dependence $H_{c2}\left(\theta \right)$, determined from fixed temperature $\rho \left(H\right)$ of Fig. 6 using 20$\%$, 50$\%$, and 80$\%$ criteria (top to bottom), for strongly overdoped Ba(Fe${}_{1-x}$Ni${}_x$)${}_2$As${}_2$, $x=0.072$ at 1.5 K (top panel) and 4 K (bottom panel). The lines are guides to the eyes. The data are plotted as $H_{c2}^{-2}\left({\sin }^2\theta \right)$, which according to Eq. (1) should be a straight line.

Figure 9

Analysis of the isothermal angular dependence of $H_{c2}$ on inclination angle to the highly conducting plane $\theta$, using linearization plot $H_{c2}^{-2}\left({\sin }^2\theta \right)$. Left panels show digitized $H_{c2}\left(\theta \right)$; right panels plot the same data as $H_{c2}^{-2}\left({\sin }^2\theta \right)$: (a) Graphite intercalation compounds[70] C${}_4$RbHg ($T_c=0.99$ K; measurements taken at $T_h=0.44$ K, open circles) and C${}_4$KHg ($T_c=0.73$ K, $T_h=0.40$ K, solid squares); (b) Sr${}_2$RuO${}_4$ ($T_c=1.43$ K, $T_h=0.10$ K, Ref. 15); (c) Mg(B${}_{1-x}$Al${}_x$)${}_2$, Ref. 26, $x=0.12$ ($T_c=30.8$ K, $T_h=14$ K, black solid squares, and $T_h=23$ K, red solid circles) and $x=0.21$ ($T_c=25.5$ K, $T_h=10$ K, blue open circles); (d) NbSe${}_2$, Ref. 71 ($T_c=7.2$ K, $T_h=4.2$ K); (e) (Ba${}_{1-x}$K${}_x$)Fe${}_2$As${}_2$, Ref. 60 ($T_c=28$ K, $T_h=20$ K, using different criteria for resistive transition: zero resistance, black triangles; midpoint, red circles; onset, green squares); (f) KFe${}_2$As${}_2$, Ref. 66 ($T_c=3.8$ K, $T_h=0.5$ K).