Chromium at high pressures: Weak coupling and strong fluctuations in an itinerant antiferromagnet

The spin- and charge-density-wave order parameters of the itinerant antiferromagnet chromium are directly measured with nonresonant x-ray diffraction as the system is driven toward its quantum critical point with high pressure using a diamond anvil cell. The exponential decrease of the spin and charge diffraction intensities with pressure confirms the harmonic scaling of spin and charge, while the evolution of the incommensurate ordering vector provides important insight into the difference between pressure and chemical doping as means of driving quantum phase transitions. Measurement of the charge density wave over more than two orders of magnitude of diffraction intensity provides the clearest demonstration to date of a weakly coupled BCS-type ground state. Evidence for the coexistence of this weakly coupled ground state with high-energy excitations and pseudogap formation above the ordering temperature in chromium, the charge-ordered perovskite manganites, and the blue bronzes, among other such systems, raises fundamental questions about the distinctions between weak and strong coupling.

Figure 1

(Color online) Experimental setup and representative diffraction scans. (a) Schematic of the first Brillouin zone in Cr. Magnetic electron and hole Fermi surfaces are connected by nesting wave vector $Q$; nonmagnetic bands are omitted for clarity. (b) Map of satellite CDW and SDW Bragg reflections. CDW peaks appear as a jack around the allowed bcc reflections. SDW peaks appear around the forbidden positions. Only SDW reflections corresponding to a $Q\parallel L$-type domain are shown for clarity. (c) Micrograph of the diamond anvil cell sample chamber showing typical arrangement of an oriented single-crystal Cr sample. Visible as well are Ag and ruby grains that allow pressure measurement in situ at low temperature and room temperature, respectively. (d) Scans of SDW, CDW, and lattice Bragg peaks at 4.3 GPa, which is our highest-pressure SDW measurement. All intensities have been normalized to the value for the nearby [(020) or (110)] lattice reflection. The lack of a measurable peak at the $\left(1+2\delta ,1,0\right)$ position corresponds to a $\mathbf{Q}\parallel \left(H00\right)$ volume occupation of below 0.5%. The difference in width of the two SDW scans is attributed to unequal crystal mosaicities in the different $S$-domain volumes.

Figure 2

(Color online) Lattice and CDW Bragg peaks at 5.0 GPa; sample oriented with the [111] direction along the diamond anvil cell compression axis. All intensities have been normalized to the value for the $\left(1\overline{2}1\right)$ lattice reflection.

Figure 3

(Color online) (a) Evolution of $Q$-domain distribution with increasing pressure for one Cr crystal at fixed $T=6.9\text{K}$. The changing domain distribution underscores the need to measure all domain types in order to accurately determine the true amplitude of the order parameters. (b) $\theta \text{−}2\theta$ scans of lattice Bragg peaks at a series of pressures; Bragg’s law is used to convert from $2\theta$ to Cr lattice constant so that (110) and (200) reflections may be included on the same plot. All scans (except for ambient pressure) are taken at $T<8\text{K}$. The scale bars associated with each scan give the instrument resolution, taking into account the detector slit resolution, energy resolution, and beam divergence (typical values: $70\mu \text{rad}$, $2\times {10}^{-4}$, and $4\mu \text{rad}$, respectively).

Figure 4

(Color online) Total CDW diffraction intensity $I_{\text{CDW}}=I\left(2Q,0,0\right)$ normalized by the lattice reflection $I\left(200\right)$ as a function of pressure at $T<8\text{K}$. Data represent measurements of ten different crystals using two different sample geometries and are adjusted to account for the measured $Q$-domain distributions. The intensity decreases exponentially by more than two decades as the lattice shrinks by 1.2% between 0 and 6.7 GPa.

Figure 5

(Color online) Suppression of order parameters and ordering temperatures with applied pressure $P$ (right) and vanadium doping $x$ (left) vs low-temperature lattice constant. Vertical dashed line marks the lattice constant for pure Cr $\left(2.8822\text{Å}\right)$ at ambient $P$ and base $T$. All quantities are normalized to their values for pure Cr at low $T$ and ambient $P$. $I_{\text{SDW}}^{1/2}$ and $I_{\text{CDW}}^{1/4}$ have been vertically displaced by $+0.2$ and $+0.4$, respectively. Diffraction intensities $I_{\text{SDW}}$ and $I_{\text{CDW}}$ were measured at $T<8\text{K}$ and account for the measured $Q$- and $S$-domain distributions. $T_N\left(P\right)$ is taken from Ref. 20, using our measured $\left(1/a_0\right)\left(da/dP\right)=-1.76\times {10}^{-3}{\text{GPa}}^{-1}$ and the pressure scale reduced by a factor of 1.3 to account for the difference in calibrations. $T_N\left(x\right)$ is taken from Refs. 21, 39 and the lattice constants $a\left(x,T\to 0\right)$ are linearly extrapolated between the measured values at $x=0\%$ and 3.2% $\left(2.8850\pm 0.0004\text{Å}\right)$. Dashed lines are exponential fits as follows: $I_{\text{CDW}}^{1/4}\propto \text{exp}\left(\frac{C_{\text{CDW}}}{4}\frac{\Delta a}{a}\right)$, $C_{\text{CDW}}=436\pm 28$ (68% confidence level); $I_{\text{SDW}}^{1/2}\propto \text{exp}\left(\frac{C_{\text{SDW}}}{2}\frac{\Delta a}{a}\right)$, $C_{\text{SDW}}=227\pm 10$ (68% c.l.); $T_N\left(\text{pressure}\right)\propto \text{exp}\left(C_N^P\frac{\Delta a}{a}\right)$, $C_N^P=110$; $T_N\left(\text{V}\text{doping},x<2.5\%\right)\propto \text{exp}\left(C_N^x\frac{\Delta a}{a}\right)$, $C_N^x=-1120$.

Figure 6

(Color online) Evolution of the SDW wave vector $Q$ with pressure, temperature (Refs. 19, 27), and V doping (Refs. 34, 39). The rapid variation of $Q$ with $T$ (shown here for $4.2\text{K}<T<300\text{K}$) reflects the temperature dependence of the Fermi surface as well as the influence of entropy on the magnetic ordering free energy (Ref. 2). The equally rapid variation of $Q$ with doping reflects changes to the Fermi surface resulting from a lowered valence electron count. By contrast, $Q$ in pure Cr varies slowly with $P$, leveling off above 4 GPa even as the order parameter continues its exponential decrease (Fig. 4). Inset: Evolution of $Q$ with pressure for ${\text{Cr}}_{1-x}{\text{V}}_x$, $x=3.2\%$, at $T<2\text{K}$ from neutron diffraction. $Q$ is nearly independent of pressure. The highest pressure (smallest lattice constant) plotted here is $4.6\pm 0.1\text{kbar}$, where $P_C=7.5\text{kbar}$ (Ref. 21).

Figure 7

(Color online) Radial scans of CDW satellite pair $\left(2-2\delta ,0,0\right)$ and $\left(2+2\delta ,0,0\right)$ and lattice (200) reflection at 295 K and ambient pressure; $x$ axis scaling is the same for all three scans. The asymmetry in the CDW scans is not seen in the (200) reflection, which is sharp and symmetric. The CDW line shapes are therefore intrinsic to the underlying magnetic order rather than resulting from the lattice constant distribution. The reflection symmetry of the CDW line shapes around the (200) position suggests an asymmetric distribution of $Q$ vectors due to deviations of the Fermi surface from idealized flat nesting sheets.

Figure 8

(Color online) Excess electrical resistivity at high temperature shows effects of incipient long-range order. (a) $\rho \left(T\right)$ for pure Cr and ${\text{Cr}}_{1-x}{\text{V}}_x$, $x=3.2\%$ $\left(T_N=52\text{K}\right)$. All samples are single crystals; data have been vertically offset to compensate for the very different metallic resistivities at 300 and 50 K. Cr samples are adjacent cuts from the same wafer, but only one was annealed. Annealing both sharpens the signature at $T_N$ and slightly enhances the excess $\rho$ above $T_N$; therefore, this excess $\rho$ cannot be due to lattice strains. Solid lines are cubic fits to the data in the range $T_N+35\text{K}<T<T_{\text{max}}$, where $T_{\text{max}}=400\text{K}$ for pure Cr and 115 K for ${\text{Cr}}_{1-x}{\text{V}}_x$, $x=3.2\%$. (b) Excess resistivity, $\rho \left(T\right)-{\rho }_{\text{FIT}}\left(T\right)$, for the data displayed in (a). The resistivity signature of magnetic fluctuations is seen at a higher normalized temperature in the nearly critically doped 3.2% ${\text{Cr}}_{1-x}{\text{V}}_x$ than in pure Cr. (c) $\rho \left(T\right)$ of a series of ${\text{La}}_{1-x}{\text{Ca}}_x{\text{MnO}}_3$ samples (Ref. 48). Curves are labeled by doping level $x=0.52,0.55,0.60,0.75,0.80$, for which $T_{\text{CO}}=188$, 222, 259, 217, and 176 K, respectively. Black bars mark the pseudogap temperatures $T^{\ast }\left(x\right)$ determined from optical spectroscopy (Ref. 48); the solid lines are linear fits to $\rho \left(T,x\right)$ over the range $T>T^{\ast }(x\phantom{)}$). (d) $\rho \left(T\right)$ for pure and doped blue bronze ${\text{Rb}}_{0.3-x}{K}_x{\text{MoO}}_3$ (Ref. 50); $T_{\text{CO}}=183$ and 179 K for $x=0$ and 0.15, respectively. The solid line is a linear fit to $\rho \left(T\right)$ for $T>T_{\text{CO}}+35\text{K}$.

Figure 9

Schematic dispersion relation for an itinerant electron system exhibiting an enhanced susceptibility at wave vector $Q$, leading to a low-temperature BCS-type ground state. The canonical spin-wave dispersion (solid line) is steep and these modes contribute little to the free energy in the ordered state. The enhanced susceptibility at $Q$ leads to collective slow modes (dashed line) around this point, an effect which is enhanced by coupling of the spin to the charge and lattice degrees of freedom. The population of these collective modes with energy below the single-particle excitation gap $2g_0$ drives the thermal phase transition; above $T_C$, pseudogaps remain and $2g_0/k_B{T}_C$ is larger than the canonical BCS value of 3.5.