Structure and equation of state of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+4+\delta }$ from x-ray diffraction to megabar pressures

Pressure is a unique tuning parameter for probing the properties of materials, and it has been particularly useful for studies of electronic materials such as high-temperature cuprate superconductors. Here we report the effects of quasihydrostatic compression produced by a neon pressure medium on the structures of bismuth-based high-${\mathit{T}}_{\mathit{c}}$ cuprate superconductors with the nominal composition ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+4+\delta }$ ($n=1,2,3$) up to 155 GPa. The structures of all three compositions obtained by synchrotron x-ray diffraction can be described as pseudotetragonal over the entire pressure range studied. We show that previously reported pressure-induced distortions and structural changes arise from the large strains that can be induced in these layered materials by nonhydrostatic stresses. The pressure-volume equations of state (EOS) measured under these quasihydrostatic conditions cannot be fit to single phenomenological formulation over the pressure ranges studied, starting below 20 GPa. This intrinsic anomalous compression as well as the sensitivity of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+4+\delta }$ to deviatoric stresses provide explanations for the numerous inconsistencies in reported EOS parameters for these materials. We conclude that the anomalous compressional behavior of all three compositions is a manifestation of the changes in electronic properties that are also responsible for the remarkable nonmonotonic dependence of ${\mathit{T}}_{\mathit{c}}$ with pressure, including the increase in ${\mathit{T}}_{\mathit{c}}$ at the highest pressures studied so far for each. Transport and spectroscopic measurements up to megabar pressures are needed to fully characterize these cuprates and explore higher possible critical temperatures in these materials.

Figure 1

Unit cells of Bi-2201, Bi-2212, and Bi-2223 presented in a tetragonal ($I4$/mmm) representation. Structural parameters for Bi-2201 from Matheis et al. [101] and for Bi-2212 and Bi-2223 from Wesche et al. [102].

Figure 2

Representative integrated x-ray diffraction patterns for BSCCO samples. All patterns were fit to a tetragonal $I4$/mmm symmetry unit cell with ambient pressure parameters given in the text. (a) Diffraction patterns for Bi-2201 taken at 0.9 GPa (black) and 46 GPa (red) ($\lambda =0.4959$ Å). Tick marks indicate predicted Bragg peaks for Bi-2201 at 0.9 GPa. Peaks from unreacted ${\text{Bi}}_2{\text{O}}_3$ from synthesis are identified in the pattern. The inset is a reflected light micrograph of the sample loaded in the DAC at 0.9 GPa with a $150\mu \mathrm{m}$ sample chamber diameter surrounded by the Ne pressure-transmitting medium. (b) Diffraction patterns of mixed-phase Bi-2212 $+$ Bi-2223 at 2.3 GPa (black, $\lambda =0.406626$ Å) and 119 GPa (red, $\lambda =0.4133$ Å). Both are indexed using tetragonal $I4$/mmm symmetry. Tick marks indicate predicted Bragg peaks of Bi-2223 (black) and Bi-2212 (red) at 2.3 GPa. Diffraction from the pressure-transmitting medium Ne (above freezing pressure) and W from the gasket are identified. The inset is a reflected light micrograph of mixed-phase Bi-2212 and Bi-2223 at 2.3 GPa in the DAC with a $150\mu \mathrm{m}$ sample chamber surrounded by the Ne medium.

Figure 3

Pressure dependence of the structural parameters of Bi-2201. (a) $a$ (red) and $c$ (blue) parameters of Bi-2201. (b) $c/a$ ratio of Bi-2212, showing a change in the observed trend between 8 and 10 GPa. Lines inserted as guides to the eye. (c) Unit-cell volume with Vinet EOS fit to the data over three pressure ranges, 0.1 MPa–48 GPa [$V_0=714.7$ Å${}^3, K_0=32.0\left(1.3\right)\mathrm{GPa}, K_0{}^{\prime }=10.7\left(3\right)$], 0.1 MPa–8 GPa [$V_0=714.7$ Å${}^3, K_0=54.6\left(1.3\right)\mathrm{GPa}, K_0{}^{\prime }=0.57\left(3\right)$], and 8–48 GPa [$V_0=696.6\left(4.0\right)$ Å${}^3, K_0=45.8\left(3.4\right)\mathrm{GPa}, K_0{}^{\prime }=9.6\left(4\right)$]. For pressure ranges including zero pressure, the fixed value of $V_0=714.7$ Å${}^3$ is that determined from ambient pressure x-ray diffraction. All error bars represent $1\sigma$ uncertainty.

Figure 4

Pressure dependence of the structural parameters of Bi-2212 measured under quasihydrostatic conditions (this work, green solid circles) and nonhydrostatic conditions (NHS; this work, open purple squares) and previous results, Olsen et al. [32], Gavarri et al. [33], Zhang et al. [34], and Zhou et al. [21]. (a) $a$ (triangles), $b$ (inverted triangles), and $c$ (circles) lattice parameters. Lines inserted as guides to the eye. (b) $c/a$ axial ratio, showing a change in compression mechanism at 23 GPa, coinciding with the stiffening of $c$. (c) Pressure dependence of the unit-cell volume and Vinet EOS fit to the data over three pressure ranges, 0.1 MPa–108 GPa [$V_0=898.0$ Å${}^3, K_0=53.9\left(1.6\right)\mathrm{GPa}, K_0{}^{\prime }=8.7\left(2\right)$] 0.1 MPa–23 GPa [$V_0=898.0$ Å${}^3, K_0=67.7\left(2.3\right)\mathrm{GPa}, K_0{}^{\prime }=4.9\left(3\right)$], and 23–108 GPa [$V_0=793.9\left(4.6\right)$ Å${}^3, K_0=262.8\left(22.4\right)\mathrm{GPa}, K_0{}^{\prime }=2.9\left(4\right)$]. For pressure ranges containing ambient pressure (0.1 MPa), $V_0$ was fixed to $898.0{\AA }^3$, a value obtained by ambient pressure XRD. Low-pressure and high-pressure fits produce $K_0$ and $K_0{}^{\prime }$ values in agreement with each other over similar pressure ranges. For all plots, error bars represent $1\sigma$ uncertainty.

Figure 5

Pressure dependence of the structural parameters of Bi-2223. (a) $a$ and $c$ parameters. Lines inserted as guides to the eye. (b) $c/a$ ratio for Bi-2223. (c) Unit-cell volume with Vinet EOS fit to the data over three pressure ranges, 0.1 MPa–155 GPa [$V_0=1085.0$ Å${}^3, K_0=77.7\left(9\right)\mathrm{GPa}, K_0{}^{\prime }=6.4\left(1\right)$], 0.1 MPa–30 GPa [$V_0=1085.0$ Å${}^3, K_0=84.9\left(2\right)\mathrm{GPa}, K_0{}^{\prime }=5.6\left(4\right)$], and 30–155 GPa [$V_0=1179\left(36\right)$ Å${}^3, K_0=35.5\left(9\right)\mathrm{GPa}, K_0{}^{\prime }=7.9\left(5\right)$]. For pressure ranges including ambient pressure (0.1 MPa), $V_0$ was fixed to 1085.0 Å${}^3$, a value obtained from ambient pressure x-ray diffraction. All error bars represent $1\sigma$ uncertainty.

Figure 6

Stress-strain relationships for the three compounds showing changes in compression mechanisms. The pressures corresponding to particular strains are indicated for each. (a) Bi-2201. (b) Bi-2212. (c) Bi-2223. Lines inserted as guides to the eye.

Figure 7

(a) Previously measured pressure dependence of ${\mathit{T}}_{\mathit{c}}$ for BSCCO compounds as reported by Chen et al. [19, 103] and Deng et al. [20]. (b) Strain $(1-\eta )$ dependence of previously reported ${\mathit{T}}_{\mathit{c}}$ values using the EOS measured in this work to convert pressure to strain. Dashed line inserted as a guide to the eye indicating strain where a second rise in ${\mathit{T}}_{\mathit{c}}$ occurs.

Figure 8

Sensitivity of Vinet EOS fit parameters for BSCCO compared to selected simple compounds. (a) Sensitivity of $K_0$ to pressure cutoff for BSCCO and for MgO, CaO, LiF, and NaCl. Data points with an X indicate values measured at ambient pressures using ultrasonic techniques. For simple compounds, the fit values of $K_0$ are relatively independent of cutoff pressure and agree with the ultrasonic zero-pressure values, whereas for BSCCO the fit values of $K_0$ disagree with those measured at zero pressure when a large cutoff is used in the fitting process. Reducing the upper pressure cutoff used in the fit causes the resultant $K_0$ to approach the zero-pressure values. (b) Sensitivity of $K_0{}^{\prime }$ to upper pressure cutoff, demonstrating that the fit $K_0{}^{\prime }$ for simple materials is cutoff-independent and typically between 2 and 8 (shaded region), whereas for BSCCO, $K_0{}^{\prime }$ rapidly increases up to approximately 150 as the cutoff is decreased. For all samples, the increase in uncertainties as the cutoff pressure is lowered is due to fewer data points being used in the fits. Data from Speziale et al. [87], Richet et al. [88], Dewaele [89], Fanggao et al. [39], Seetawan [41], Dominec et al. [35], Solunke et al. [38], Munro et al. [40], and Zha et al. [104].