Competing magnetic and superconducting order in the rare-earth borocarbides $R{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$ $(R=\mathrm{Tm},\mathrm{Er},\mathrm{Ho},\mathrm{Dy})$

The competition between superconductivity and magnetic ordering in the rare-earth borocarbides is analyzed in terms of an estimate of the different contributions to the free energy. The two most important effects are the Anderson-Suhl screening [Phys. Rev. 116, 898 (1959)] of the indirect exchange interaction and the reduction of the number of Cooper-pair states by the superzone gaps created by the antiferromagnetic ordering. In the case of $\mathrm{Tm}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$, the theory accounts for the anisotropy of the upper critical field, the field dependence of the radius of the vortices, and the jump in the derivative of the bulk magnetization at the superconducting transition. The theory also gives a fair account of the magnetic effects on the upper critical fields in the Er, Ho, and Dy borocarbides.

Figure 1

The temperature dependence of the electronic susceptibility in the superconducting phase relative to the normal-phase value.

Figure 2

(Color online) Upper critical field in $\mathrm{Tm}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$. The experimental results are from Ref. 22. The dashed lines show the assumed values of $H_{c2}^0\left(T\right)$ [the temperature factor is the one of $H_{c2}$ in Eq. (18) with $\gamma =1$], when the field is parallel or perpendicular to $c$. The solid lines are the calculated results using $D_c=0.64$.

Figure 3

The upper figure shows the ratio $B_i∕B$ as a function of the field $B$ applied along the $c$ axis of $\mathrm{Tm}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$ at $1.9\mathrm{K}$. The points are the experimental results of Eskildsen (Ref. 20) and the solid curve shows the calculated variation when the demagnetization factor $D_c=0.72$. The lower figure displays the field dependence of the logarithm of the flux-line-lattice form factor under the same conditions as in the upper figure. The neutron-diffraction results are from Ref. 20. The solid line is the calculated result and the thin line indicates the upper critical field.

Figure 4

(Color online) Upper critical field in $\mathrm{Er}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$. The experimental results are from Ref. 27. The dashed line denotes the assumed value of $H_{c2}^0\left(T\right)$ when the field is along [001]. The solid lines are the calculated results using $D_c=0.6$.

Figure 5

(Color online) Upper critical field in $\mathrm{Ho}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$. The experimental results are from Ref. 32. The thin dashed line denotes the assumed $c$-axis value of $H_{c2}^0\left(T\right)$. The solid lines, and the dashed extrapolations, are the calculated results using $D_c=0.64$. Below about $3\mathrm{K}$, when the field is along [100] or [110], the calculated upper critical fields coincide with the metamagnetic transition fields of about $5\mathrm{kOe}$, as indicated by the thick dashed lines.

Figure 6

(Color online) Upper critical field in $\mathrm{Dy}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$. The experimental [100] results are from Ref. 33 and are those obtained for increasing fields. The $\mathit{H}\Vert c$ and $\mathit{H}\perp c$ data are from Ref. 34. The solid lines are the calculated results.