Scaling of the physical properties in Ba(Fe,Ni)${}_2$As${}_2$ single crystals: Evidence for quantum fluctuations

We report on local magnetization, tunnel diode oscillators, and specific-heat measurements in a series of Ba(Ni${}_x$Fe${}_{1-x}$)${}_2$As${}_2$ single crystals ($0.26\le x\le 0.74$). We show that the London penetration depth $\lambda \left(T\right)=\lambda \left(0\right)+\Delta \lambda \left(T\right)$ scales as $\lambda \left(0\right)\propto 1/T_c^{0.85\pm 0.2}$, $\Delta \lambda \left(T\right)\propto T^{2.3\pm 0.3}$ (for $T<T_c/3$), and $\partial \Delta \lambda /\partial T^2\propto 1/T_c^{2.8\pm 0.3}$ in both underdoped and overdoped samples. Moreover, the slope of the upper critical field [$H_{c2}^{\prime }=-{(d{H}_{c2}/dT)}_{|T\to T_c}$] decreases with $T_c$ in overdoped samples but increases with decreasing $T_c$ in underdoped samples. The remarkable variation of $\lambda \left(0\right)$ with $T_c$ and the nonexponential temperature dependence of $\Delta \lambda$ clearly indicates that pair-breaking effects are important in this system. We show that the observed scalings strongly suggest that those pair-breaking effects could be associated with quantum fluctuations near three-dimensional superconducting critical points.

Figure 1

(a) $B_{\mathrm{rem}}^{\gamma }$ as a function of the applied field for the indicated temperature in Ba(Ni${}_x$Fe${}_{1-x}$)${}_2$As${}_2$ with $x=3.4$$\%$ [$B_{\mathrm{rem}}$ is the remanent field trapped in the sample after a field excursion up to $H_a$ (see text for details) and $\gamma$ is an exponent $\sim$0.4–0.6]. The first penetration field is deduced from the linear extrapolation of $B_{\mathrm{rem}}^{\gamma }$ to zero. The inset is the same as main panel for different $x$ values at $T=1.3$ K (the different curves are shifted vertically by $0.2$ for clarity). (b) Temperature dependence of the lower critical field $H_{c1}$ in Ba(Ni${}_x$Fe${}_{1-x}$)${}_2$As${}_2$ single crystals (for the indicated doping concentrations $x$). The solid lines are the values expected from Eq. (3), introducing only one free parameter (corresponding to the absolute value at zero $T$) for the whole set of data (see text for details). The shaded cones correspond to the values expected from the amplitude of the specific-heat jump at $T_c$ (see Fig. 4) and the slope of the upper critical field (see Fig. 5 and text for details).

Figure 2

(a) London penetration depth at $T\to 0$ [$\lambda \left(0\right)$] as a function of the critical temperature $T_c$ in Ba(Ni${}_x$Fe${}_{1-x}$)${}_2$As${}_2$ crystals deduced from $H_{c1}$ measurements (squares; see Fig. 1) and specific-heat measurements (crosses; see text for details). The thick line is a guide to the eye varying as $1/T_c^{0.85}$. (b) Slope of the $\Delta \lambda$ vs $T^2$ curve (see Fig. 3) as a function of $T_c$ in samples from the same batch deduced from tunnel diode oscillator measurements (squares). The crosses correspond to $\lambda \left(0\right)/2T_c^2$ (see text for details), introducing the $\lambda \left(0\right)$ values deduced from $C_p$ measurements [see (a)].

Figure 3

London penetration depth ${\lambda }_{\mathrm{mix}}={\lambda }_{ab}+(d/w){\lambda }_c$ ($\sim$${\lambda }_{ab}$ for $w\gg d$) as a function of ${(T/T_c)}^2$ in Ba(Ni${}_x$Fe${}_{1-x}$)${}_2$As${}_2$ single crystals for the indicated $x$ values. As in Fig. 2, solid, half-solid, and open symbols refer to underdoped, optimally doped, and overdoped samples, respectively. The inset shows the best fit values for the exponent $n$, assuming that $\Delta \lambda \propto T^n$.

Figure 4

Temperature dependence of the field-dependent part of the specific heat for the indicated field values in Ba(Ni${}_x$Fe${}_{1-x}$)${}_2$As${}_2$.

Figure 5

Temperature dependence of the upper critical field $H_{c2}$ deduced from the midpoint of the specific-heat anomaly (squares; see also Fig. 2 for symbol definition). The crosses correspond to the irreversibility field $H_{\mathrm{irr}}$ deduced from the onset of diamagnetic screening, and circles correspond to $H_{c2}$ values deduced from $H_{\mathrm{irr}}$ (see text for details).

Figure 6

(a) $1/\xi \lambda$ (solid symbols), $T_c^{1.5}$ (small open symbols), and $T_c{(\Delta C_p/T_c)}^{0.5}$ (large open symbols) as a function of doping in both Ba(Fe${}_x$,Ni${}_{1-x}$)${}_2$As${}_2$ (present work, black circles) and Ba(Fe${}_x$,Co${}_{1-x}$)${}_2$As${}_2$ (blue squares). The $x$ values in Co-doped samples have here been rescaled by a factor $\sim$0.6. (b) Left scale: slope of the critical field $H_{c2}^{\prime }=d{H}_{c2}/d{T}_{|T\to T_c}$ [solid lines in Fig. 5b] as a function of the composition $x$ in Ba(Ni${}_x$Fe${}_{1-x}$)${}_2$As${}_2$ single crystals where $H_{c2}$ has been deduced from specific-heat measurements (open squares) and from the irreversibility field (solid squares). Right scale: critical temperature $T_c$ (solid circles) as a function of $x$.