Neutron, electron, and x-ray scattering investigation of ${\mathrm{Cr}}_{1-x}{\mathrm{V}}_x$ near quantum criticality

The weakness of electron-electron correlations in the itinerant antiferromagnet Cr doped with V has long been considered the reason that neither new collective electronic states nor even non-Fermi-liquid behavior are observed when antiferromagnetism in ${\mathrm{Cr}}_{1-x}{\mathrm{V}}_x$ is suppressed to zero temperature. We present the results of neutron and electron diffraction measurements of several lightly doped single crystals of ${\mathrm{Cr}}_{1-x}{\mathrm{V}}_x$ in which the archetypal spin density wave instability is progressively suppressed as the V content increases, freeing the nesting-prone Fermi surface for a new striped charge instability that occurs at $x_c=0.037$. This novel nesting driven instability relieves the entropy accumulation associated with the suppression of the spin density wave and avoids the formation of a quantum critical point by stabilizing a new type of charge order at temperatures in excess of 400 K. Restructuring of the Fermi surface near quantum critical points is a feature found in materials as diverse as heavy fermions, high-temperature copper oxide superconductors and now even elemental metals such as Cr.

Figure 1

Magnetic and charge modulation in Cr. (a) Nesting of the $\Gamma$ (electron) and H (hole) FS sheets of Cr by vectors $Q_{\mathrm{SDW}}^{+}$ and $Q_{\mathrm{SDW}}^{-}$. Electron lenses located along the vertices of the electron octahedron and hole pockets at N do not contribute to static SDW order. (b) SDW satellites at $Q_{\mathrm{SDW}}^{+}$ and $Q_{\mathrm{SDW}}^{-}$ in a 300 K neutron diffraction scan in pure Cr along the [h00] direction near the bcc extinct 1,0,0 Bragg peak. The intensities of the peaks at $Q_{\mathrm{SDW}}=2\pi /a(1\pm {\delta }_{\mathrm{SDW}},0,0)$ obey the magnetic form factor, as expected for the Cr SDW [10]. (c) $(h,k,0)$ plane from the TEM experiment on Cr at 300 K, where the apparent elongation of the $\overline{1}$10 Bragg peak arises from incommensurate CDW satellites that are only present below $T_N$ at $Q_{\mathrm{CDW}}=2Q_{\mathrm{SDW}}$. (d) CDW satellite at $Q_{\mathrm{SDW}}^{+}$ in a 300 K x-ray diffraction scan along $\left[H00\right]$ near the 2,0,0 Bragg peak. The incommensurability of CDW ${\delta }_{\mathrm{CDW}}=0.090\pm 0.01$ r.l.u., which is close to $2{\delta }_{\mathrm{SDW}}=0.082\pm 0.024$ taken from the neutron diffraction measurements in (b). Solid lines in (b) and (d) are the best fits to Gaussian line shapes.

Figure 2

Suppression of the SDW in ${\mathrm{Cr}}_{1-x}{\mathrm{V}}_x$. (a) [h00] neutron elastic scan in ${\mathrm{Cr}}_{0.963}{\mathrm{V}}_{0.037}$ near the bcc extinct (1,0,0) Bragg peak at 5 K. Solid line is best fit to a Gaussian line shape. Inset shows the temperature dependence of the intensity $I_{\mathrm{SDW}}$ of the SDW satellite at Q = (0.922,0,0) r.l.u., which vanishes at $T_N=19$ K. Solid line is a power law fit. (b) Ordered SDW moments $\mu$ and wave vectors $Q_{\mathrm{SDW}}$ for x = 0, 0.02, and 0.037. (c) Neél temperatures $T_N$ for ${\mathrm{Cr}}_{1-x}{\mathrm{V}}_x$ determined from our measurements (red filled circles) and from [18] $(\circ )$, [10] $(•)$, [11] $\left(\blacksquare \right)$, [22] $(\square )$, [23] $\left(▵\right)$, [19] $\left(▴\right)$, [9] $(▿)$, [24] $\left(▾\right)$. Linear fit has $T_N\to 0$ for $x_C=0.040\pm 0.001$. (d) Neél temperature as a function of $\alpha$, the ratio of the total FS area nested in ${\mathrm{Cr}}_{1-x}{\mathrm{V}}_x$, and ${\mathrm{Cr}}_{1-x}{\mathrm{Mn}}_x$ alloys to the total FS area gapped in pure Cr. Values of $T_N$ taken from neutron diffraction measurements (this work: red filled circles), and from more lightly V-doped samples [10] (red squares) and from a Mn-doped sample [21] $\left(\blacksquare \right)$. Line is a guide for the eye. Error bars in (b)–(d) are the size of the symbols.

Figure 3

Charge modulation in ${\mathrm{Cr}}_{0.963}{\mathrm{V}}_{0.037}$. (a) $(h,k,0)$ reciprocal plane at 300 K, where the characteristic splitting of the bcc Bragg peaks corresponds to a new charge modulation. The dashed line is the direction of the line cut shown in (b). (b) Line cut in the $\left[h00\right]$ direction in the reciprocal plane, showing the $(2,0,0)$ and $(\overline{2},0,0)$ r.l.u. bcc Bragg peaks, each with superlattice peaks with $Q_{SL}=1.661\pm 0.001$ r.l.u., and their second harmonics with $2Q_{SL}$. Ths solid line is a simulation of the line scan, where the lattice Bragg peaks, as well as the first and second harmonics of the charge modulation, are modeled as Lorentzian-shaped peaks.

Figure 4

(a)–(c) Calculated [001] electron diffraction patterns with (a) thickness = 5 nm, (b) thickness = 100 nm, and (c) thickness = 134 nm. The embedded lines in (a)–(c) are the intensity line scans from $-200$ to 200. (d) Schematics of Ewald's sphere in the vicinity of 200 reflection, showing that the excitation error, $S_g$, of ${\mathrm{SLR}}_{200L}$ is smaller than that of ${\mathrm{SLR}}_{200R}$. (e) Intensities of ${\mathrm{SLR}}_{200L}$ and ${\mathrm{SLR}}_{200R}$ as a function of thickness.

Figure 5

Charge stripes in quantum critical ${\mathrm{Cr}}_{0.963}{\mathrm{V}}_{0.037}$. (a) Direct space image with atomic resolution showing long vertical fringes that extend over length scales of $\simeq$5 nm. The region inside the red circle has charge stripe modulation; the region inside the green circle represents the unmodulated matrix. (b) The diffraction patterns obtained from the fast Fourier transform (FFT) of the real space image are different for the stripe regions (red circle) and for the matrix in which they are embedded (green circle). The FFT of the striped area encircled in red shows two incommensurate superlattice satellites near (0,0,0). (c) The FFT of the green encircled area shows no satellites, only the bcc Bragg peaks. (d) 300 K dark field image where the bright regions indicate parts of the sample with enhanced intensity at $Q=Q_{SL}$.

Figure 6

(a) Contour plot of the spatial variation of V concentration obtained from microprobe measurements. The V concentration varies by no more than 4% over the selected area. (b) Histogram of electron energy loss spectroscopy measurements: number of observations for a given V concentration versus deviations from the average V concentration probed on length scales from 58 to 920 nm. The average deviation from the mean V concentration is less than 0.1$\%$ for length scales from 58 to 920 nm.