The disordered and correlated insulator ${\mathrm{Bi}}_2{\mathrm{CrAl}}_3{\mathrm{O}}_9$

The $3d$ transition metal insulator ${\mathrm{Bi}}_2{\mathrm{CrAl}}_3{\mathrm{O}}_9$ forms with a quasi-one-dimensional structure characterized by linear chains of edge-sharing, Cr-and Al-centered, distorted octahedra. The UV/Vis spectrum of high-quality single crystals is marked by broad absorption edges corresponding to direct transitions across a 1.36-eV insulating gap. Measurements of dc magnetic susceptibility $\chi$ reveal a fluctuating moment of $2.60\pm 0.01{\mu }_B/\mathrm{Cr}$—reduced from the $3.87{\mu }_B/\mathrm{Cr}$ expected for ${\mathrm{Cr}}^{3+}$, while the Weiss temperature ${\mathrm{\Theta }}_W=-21\pm 1$ K implies that the prevailing local moment interactions are weakly antiferromagnetic in nature. Some 10% of the fluctuating moment is quenched, presumably due to the onset of an antiferromagnetic or spin glass phase at temperature $T^{★}=98\pm 3$ K, while measurements of magnetization versus field $H$ at $T\le 10$ K scale as $H/T^{0.68\left(4\right)}$, suggesting the presence of quantum fluctuations associated with a disordered phase. Density functional theory calculations carried out within the generalized gradient approximation are in excellent agreement with experimental results, asserting that short-range magnetic interactions remnant above $T^{★}$ stabilize the insulating state.

Figure 1

(a) The unit cell of ${\mathrm{Bi}}_2{\mathrm{CrAl}}_3{\mathrm{O}}_9$ with ${\mathrm{Cr}}_{0.5}{\mathrm{Al}}_{0.5}$-centered distorted octahedra colored in red, Al-centered tetrahedra colored in blue, Bi drawn in white, and O drawn in black, as indicated. The obtained space group is $Pbam$, and the lattice parameters are $a=8.1354\left(12\right)$ Å, $b=7.7532\left(10\right)$ Å, and $c=5.7518\left(9\right)$ Å. Refinement parameters are residual $R=2.58$, weighted residual $wR=2.45$, and goodness of fit $S=1.02$. (b) An optical microscope image of a typical crystal illustrating its plate-like habit. (c) A view of the chain of ${\mathrm{Cr}}_{0.5}{\mathrm{Al}}_{0.5}$-centered, distorted octahedra oriented along $\mathbf{c}$ with nearest-neighbor (${\mathrm{Cr}}_{0.5}{\mathrm{Al}}_{0.5}$)-(${\mathrm{Cr}}_{0.5}{\mathrm{Al}}_{0.5}$) distances $d_1$ and $d_2$ and O-(${\mathrm{Cr}}_{0.5}{\mathrm{Al}}_{0.5}$)-O angles ${\theta }_1$ and ${\theta }_2$ indicated. Colors are as in (a). (d) The ($0kl$) reciprocal space map showing well-indexed, clean, round reflections and no evidence of an occupancy-related superstructure.

Figure 2

(a) The UV-Vis diffuse reflectance $R$ spectrum (black) of a mosaic of ${\mathrm{Bi}}_2{\mathrm{CrAl}}_3{\mathrm{O}}_9$ single crystals plotted as a function of incident photon energy $h\nu$ from 1.38 to 4.00 eV, corresponding to wavelength $\lambda =900$ to 310 nm. The green dashed lines are guides for the eye to indicate the positions of likely ${}^4{\mathrm{A}}_2\left({\mathrm{t}}_2^3\right){\to }^4{\mathrm{T}}_2\left({\mathrm{t}}_2^2\mathrm{e}\right)$ and ${}^4{\mathrm{A}}_2\left({\mathrm{t}}_2^3\right){\to }^4{\mathrm{T}}_1\left({\mathrm{t}}_2^2\mathrm{e}\right)$ transitions, as indicated. (Inset) The fine structure of the R lines, resulting from ${}^4{\mathrm{A}}_2\left({\mathrm{t}}_2^3\right){\to }^2\mathrm{E}\left({\mathrm{t}}_2^3\right)$ transitions. The dashed blue lines are guides for the eye. (b) The square of the Kubelka-Munk function $f{\left(R\right)}^2$ plotted vs $h\nu$ (black), where the solid red lines are best fits of the Tauc relation to the linear regions (see text).

Figure 3

(a) The temperature $T$ dependence of the dc magnetic susceptibility $\chi =M/H$ with applied field $H=10000$ Oe of a collection of ${\mathrm{Bi}}_2{\mathrm{CrAl}}_3{\mathrm{O}}_9$ single crystals (black circles). Solid lines are fits as described in (b). (b) The $T$ dependence of $1/\chi$ (black circles). The red and blue solid lines are fits to the Curie-Weiss law across temperature ranges $T>80$ K and $T<75$ K, respectively, and correspond to fluctuating moments of $2.60\pm 0.01{\mu }_B/\mathrm{Cr}$ and $2.33\pm 0.01{\mu }_B/\mathrm{Cr}$, as indicated. (c) $\chi T$ plotted as a function of $T$ (black circles), clarifying the discontinuity in slope at $79\pm 3$ K. The solid violet line is a smoothed representation of the data, obtained by 5 point moving window average. (d) The $T$ dependence of Fisher's specific heat $d\left(\chi T\right)/dT$ computed from the smoothed curve in (d).

Figure 4

(a) Magnetic isotherms measured with $T$ incremented in 1 K steps as indicated. (b) Magnetization $M$ plotted as a function of $H/T$ to illustrate deviation from Brillouin functionlike behavior. Colors as in (a), as indicated. The upper and lower solid black lines are the $g=2, J=3/2$ Brillouin function with magnitude arbitrarily adjusted to the $T=10$ and 4 K data, respectively. (c) $M$ plotted as a function of $H/T^{\gamma }$ with the critical exponent $\gamma =0.68\left(4\right)$ indistinguishable from $\frac{2}{3}$, colors as in (a). (d) $\chi T^{\gamma }$ plotted as a function of $T$ for $\gamma =1$ (red), 0.68(4) (green), and 0.5 (blue), with near $T$ independence in the second case supporting the value of the critical exponent $\gamma =0.68\left(4\right)$. The solid green line serves to provide a horizontal guide for the eye.

Figure 5

(a) Density functional theory (DFT) calculation of the density of states $N\left(E\right)$ of ${\mathrm{Bi}}_2{\mathrm{CrAl}}_3{\mathrm{O}}_9$ calculated in the absence of spin polarization (black line). The partial Cr-$3d$ contribution to $N$ is shown in red. (Inset) The input crystal structure of ${\mathrm{Bi}}_2{\mathrm{CrAl}}_3{\mathrm{O}}_9$ that models the statistical occupancy of the ${\mathrm{Cr}}_{0.5}{\mathrm{Al}}_{0.5}$ sites for the calculations. Cr are colored in red, Al in blue, Bi in white, and O in black. (b) Spin polarized DFT calculations with an static antiferromagnetic moment corresponding to the experimental value, colors as in (a). (c) Nonspin polarized DFT calculations with Hubbard $U$ calculated from first principles, colors as in (a). (c) Nonspin polarized DFT calculations with Hubbard $U$ and Hund's $J_H$ calculated from first principles, colors as in (a).