Multigap superconductivity and Shubnikov–de Haas oscillations in single crystals of the layered boride ${\text{OsB}}_2$

Single crystals of superconducting ${\text{OsB}}_2$ $\left[T_{\text{c}}=2.10\left(5\right)\text{K}\right]$ have been grown using a Cu-B eutectic flux. We confirm that ${\text{OsB}}_2$ crystallizes in the reported orthorhombic structure (space group $Pmmn$) at room temperature. Both the normal and superconducting state properties of the crystals are studied using various techniques. Heat capacity versus temperature $C\left(T\right)$ measurements yield the normal state electronic specific heat coefficient $\gamma =1.95\left(1\right)\text{mJ}/\text{mol}{\text{K}}^2$ and the Debye temperature ${\Theta }_{\text{D}}=539\left(2\right)\text{K}$. The measured frequencies of Shubnikov–de Haas oscillations are in good agreement with those predicted by band structure calculations. Magnetic susceptibility $\chi \left(T,H\right)$, electrical resistivity $\rho \left(T\right)$, and $C\left(T,H\right)$ measurements ($H$ is the magnetic field) demonstrate that ${\text{OsB}}_2$ is a bulk low-$\kappa$ $\left[\kappa \left(T_{\text{c}}\right)=2\left(1\right)\right]$ type-II superconductor that is intermediate between the clean and dirty limits $\left[\phantom{(}\xi \left(T=0\right)/\ell =0.97)\right]$ with a small upper critical magnetic field $H_{\text{c}2}\left(T=0\right)=186\left(4\right)\text{Oe}$. The penetration depth is $\lambda \left(T=0\right)=0.300\mu \text{m}$. An anomalous (not single-gap BCS) $T$ dependence of $\lambda$ was fitted by a two-gap model with ${\Delta }_1\left(T=0\right)/k_{\text{B}}{T}_{\text{c}}=1.9$ and ${\Delta }_2\left(T=0\right)/k_{\text{B}}{T}_{\text{c}}=1.25$, respectively. The discontinuity in the heat capacity at $T_{\text{c}}$, $\Delta C/\gamma T_{\text{c}}=1.32$, is smaller than the weak-coupling BCS value of 1.43, consistent with the two-gap nature of the superconductivity in ${\text{OsB}}_2$. An anomalous increase in $\Delta C$ at $T_{\text{c}}$ of unknown origin is found in finite $H$; e.g., $\Delta C/\gamma T_{\text{c}}\approx 2.5$ for $H\approx 25\text{Oe}$.

Figure 1

(Color online) The crystal structure of ${\text{OsB}}_2$ viewed at a slight angle from the $b$ axis. The Os atoms are shown as large (red) spheres while the B atoms are shown as the small (blue) spheres. A single unit cell (shown) contains two formula units. (b) Projection of the ${\text{OsB}}_2$ structure onto the $ab$ plane.

Figure 2

(Color online) (a) Scanning electron microscope image of a typical ${\text{OsB}}_2$ crystal. (b) Rietveld refinement of the powder x-ray diffraction data of crushed ${\text{OsB}}_2$ crystals. The open black circles represent the observed x-ray pattern, the solid red line represents the fitted pattern, the dotted blue line represents the difference between the observed and calculated intensities, and the vertical black bars represent the peak positions.

Figure 3

(Color online) Electrical resistivity $\rho$ for a single crystal of ${\text{OsB}}_2$ versus temperature $T$ with current $I=5\text{mA}$ along the $b$ axis and with applied magnetic field $H=0\text{Oe}$. The inset shows the $\rho \left(T\right)$ data between $T=1.7$ and 2.6 K measured in various applied magnetic fields $H$ as indicated.

Figure 4

(Color online) (a) Isothermal magnetization $M$ versus magnetic field $H$ at various temperatures $T$ with $H$ applied along the $b$ axis. (b) Saturation magnetization $M_{\text{S}}$ versus $T$ obtained by fitting the $M\left(H\right)$ data above $H=2\text{T}$ at various $T$ by the expression $M\left(H,T\right)=M_{\text{S}}\left(T\right)+\chi \left(T\right)H$.

Figure 5

(Color online) Magnetic susceptibility $\chi$ versus temperature $T$ for a single crystal of ${\text{OsB}}_2$ with a magnetic field $H=3\text{T}$ applied along the $a$, $b$, and $c$ axes. The powder average $\chi \left(T\right)$ and the $\chi \left(T\right)$ for a polycrystalline sample (from Ref. 14) are also shown. The solid curve through the powder average $\chi \left(T\right)$ data is a fit by the expression $\chi \left(T\right)={\chi }_{\text{int}}+\frac{C}{T-\theta }$. The intrinsic susceptibility obtained by correcting the powder-averaged $\chi \left(T\right)$ for the ferromagnetic impurity contribution $M_{\text{S}}\left(T\right)$, ${\chi }_{\text{int}}\left(T\right)=\chi \left(T\right)-M_{\text{S}}\left(T\right)/H$ is shown as open diamond symbols.

Figure 6

(Color online) Temperature $T$ dependence of the zero-field-cooled (ZFC) and field-cooled (FC) dimensionless volume susceptibility ${\chi }_{\text{v}}$ in terms of the superconducting volume fraction $\left(4\pi {\chi }_{\text{v}}\right)$ of a single crystal of ${\text{OsB}}_2$ with a magnetic field $H=5\text{Oe}$ applied along the (a) $a$ axis, (b) $b$ axis, and (c) $c$ axis. (d) The $T$ dependence of the ZFC superconducting volume fraction $4\pi {\chi }_{\text{v}}$ of a single crystal of ${\text{OsB}}_2$ measured in various magnetic fields $H$ applied along the $b$ axis. The construction used to determine $T_{\text{c}}\left(H\right)$ is illustrated by the red straight line for $H=5\text{Oe}$. At low $T$, the $4\pi {\chi }_{\text{v}}$ values are all more negative than $-1$ due to demagnetization effects.

Figure 7

(Color online) Hysteresis loops at 1.7 K of the volume magnetization $M_{\text{v}}\left(H\right)$ normalized by $1/4\pi$, versus applied magnetic field $H$ applied along the (a) $a$ axis, (b) $b$ axis, and (c) $c$ axis. The arrows next to the data indicate the direction of field ramping during the measurement. (d) Normalized magnetization $M_{\text{v}}\left(H\right)$ at various temperatures $T$ versus $H$ along the $b$ axis. The construction used to determine $H_{\text{c}2}\left(T\right)$ is shown by the solid red line for $T=1.75\text{K}$.

Figure 8

(Color online) Dynamic susceptibility $\chi$ normalized by $1/4\pi$, versus temperature $T$ at a frequency of 10 MHz with various applied magnetic fields $H$ in units of Oe. The $4\pi \chi$ data have been normalized to a minimum value of $-1$ at the lowest $T$. The construction used to determine $T_{\text{c}}\left(H\right)$ is shown by the red line for $H=0$.

Figure 9

(Color online) Heat capacity $C$ versus temperature $T$ of a single crystal of ${\text{OsB}}_2$ between 1.75 and 48 K measured in zero magnetic field $H$. Inset (a) shows the $C\left(T\right)$ data between 1.75 and 3 K measured with $H=0$ and $H=1\text{kOe}$ applied along the $c$ axis. Inset (b) shows the data for $H=0$ plotted as $C\left(T\right)/T$ versus $T$ between $T=1.75$ and 2.5 K. The solid curve in inset (b) is a construction to estimate the heat capacity jump $\Delta C$ at $T_{\text{c}}$.

Figure 10

(Color online) (a) Heat capacity $C$ divided by temperature $T$ versus $T$ of a single crystal of ${\text{OsB}}_2$ in various applied magnetic fields $H_a$. A remanent magnetic field $H\approx 25\text{Oe}$ was present over the nominal value of the applied field $H_a$ listed in the figure legend. The true $H=0\text{Oe}$ data from Fig. 9 inset (b) are also included for comparison. (b) $C/T$ versus $T^2$ measured in various true magnetic fields $H$ for a different ${\text{OsB}}_2$ crystal.

Figure 11

(Color online) Upper critical magnetic field $H_{\text{c}2}$ versus temperature $T$ extracted from different types of measurements, as indicated. The straight line is a linear fit to the data near $T_{\text{c}}$. The dashed curve is a fit by the expression $H_{\text{c}2}\left(T\right)=H_{\text{c}2}\left(0\right)\left[1-{\left(\frac{T}{T_{\text{c}}}\right)}^{\alpha }\right]$. The circle at $T=0\text{K}$ labeled “WHH” is the estimate of $H_{\text{c}2}\left(T=0\right)$ using the WHH formula in the clean limit (see text).

Figure 12

(Color online) The normalized superfluid density ${\rho }_{\text{s}}\left(T\right)/{\rho }_{\text{s}}\left(0\right)$ versus reduced temperature $T/T_{\text{c}}$. The solid curve through the data is a fit by the two-gap $\gamma$ model for superconductivity. The curves ${\rho }_1$ and ${\rho }_2$ are the individual contributions from the two gaps.

Figure 13

(Color online) Temperature $T$ dependence of the two gaps ${\Delta }_1$ (top black solid curve) and ${\Delta }_2$ (bottom red solid curve). Also shown by the blue dashed curve is the BCS prediction for a single gap.

Figure 14

(Color online) The change $\Delta f=f\left(H\right)-f\left(0\right)$ in the tunnel diode oscillator (TDO) frequency $f$ versus magnetic field $H$ measured at $T=180\text{mK}$ with $H$ applied along the $c$ axis. The inset shows an image of the crystal used for the measurement. The $c$ axis is out of the plane of the image.

Figure 15

(Color online) The oscillating part of the TDO frequency $df=\Delta f$–smooth background versus the reciprocal of the magnetic field $1/H$ for $H$ applied along the $c$ axis.

Figure 16

(Color online) Power spectra of Shubnikov–de Haas oscillations obtained at the indicated temperatures from Fourier transformation of the quantum SdH oscillation data such as in Fig. 15 at the indicated temperatures. The spectra are shifted vertically by arbitrary amounts for clarity of presentation.

Figure 17

(Color online) (a) and (b) Fits to the amplitude versus $T$ data for the peaks $F_1$ and $F_2$, respectively, by Eq. (28). The values of the respective effective masses $m^{\ast }$ for the two Fermi-surface extrema are indicated in the figures.

Figure 18

(Color online) Power spectra of Shubnikov–de Haas oscillations obtained at temperatures $T=180\text{mK}$ for $H$ about $15°$ away from the $b$ axis in the $ab$ plane (see inset), obtained from Fourier transformation of quantum oscillation data (not shown) similar to those in Fig. 15. The indices $i$ for the quantum SdH oscillation frequencies $F_i$ are as indicated. These frequencies are listed in Table .