Charge density wave fluctuations, heavy electrons, and superconductivity in KNi${}_2$S${}_2$

Understanding the complexities of electronic and magnetic ground states in solids is one of the main goals of solid-state physics. Materials with the canonical ThCr${}_2$Si${}_2$-type structure have proved particularly fruitful in this regard, as they exhibit a wide range of technologically advantageous physical properties described by “many-body physics,” including high-temperature superconductivity and heavy fermion behavior. Here, using high-resolution synchrotron x-ray diffraction and time-of-flight neutron scattering, we show that the isostructural mixed valence compound KNi${}_2$S${}_2$ displays a number of highly unusual structural transitions, most notably the presence of charge density wave fluctuations that disappear on cooling. This behavior occurs without magnetic or charge order, in contrast to expectations based on other known materials exhibiting related phenomena. Furthermore, the low-temperature electronic state of KNi${}_2$S${}_2$ is found to exhibit many characteristics of heavy-fermion behavior, including a heavy electron state ($m^{*}/m_e\sim$ 24), with a negative coefficient of thermal expansion, and superconductivity below $T_c=0.46\left(2\right)$ K. In the potassium nickel sulfide, these behaviors arise in the absence of localized magnetism, and instead appear to originate in proximity to charge order.

Figure 1

Representative Rietveld analyses of SXRD data collected at (a) $T=300$ K and (b) $T=6.4$ K.

Figure 2

(a) From analysis of temperature-dependent high-resolution synchrotron x-ray diffraction (11-BM, Advanced Photon Source), a clear transition in unit cell volume of KNi${}_2$S${}_2$ is observed at $T\sim 75$ K (dot-dashed line); no transition is observed at $T=250$ K (dotted line). (b) The temperature dependence of the unit cell volume of KNi${}_2$S${}_2$ shows two anomalies below $T<13$ K, with an overall negative coefficient of thermal expansion below $T<8.9$ K. (dashed lines guide the eye; inset: unit cell of KNi${}_2$S${}_2$). Error bars are contained within the size of the symbols.

Figure 3

(a) Illustration of the (100) crystal plane of KNi${}_2$Se${}_2$ and Fourier difference maps from data collected at $T=300$ K, viewed on the (001) plane reveal the displacement of Ni atoms from the (b) ideal $4d$ (0, 0.5, 0.25) position to the (c) displaced site at $8g$ (0, 0.5, $z$). The schematic unit cell on the right illustrates the location of the Ni atoms (blue spheres). (d) Coincident with the transition in unit cell volume are charge density wave (CDW) fluctuations that disappears on cooling as $T<75$ K; these CDW fluctuations manifest as a displacement of the Ni atom ($z$) from the ideal $4d$ (0, 0.5, 0.25) site to the $8g$ (0, 0.5, $z$) site. Through all measured temperatures, the space-group symmetry, $I4/mmm$ is retained, with the displaced $z$ position requiring a split-site occupancy ($4d$ to $8g$). The color denotes the normalized Rietveld goodness-of-fit (${\chi }^2/{\chi }_{\mathrm{best}}^2$; brighter $=$ better fit) to show the significance of the displacement shift at $T\sim 75$ K.

Figure 4

(a) Temperature dependence of the neutron pair distribution function derived from neutron total scattering data (HIPD, Lujan Center) of KNi${}_2$S${}_2$ shows the presence of multiple nearest-neighbor Ni-Ni distances observed at $T=300$ K that disappear on cooling (dashed line: ideal Ni-Ni separation). (b) Quantitative modeling of the PDF (NPDF, Lujan Center) measured at $T=300$ K reveals a poor least-squares fit of the $I4/mmm$ crystal structure determined by SXRD (LS) due to the presence of multiple Ni-Ni distances (arrows). Reverse Monte Carlo (RMC) simulations accurately describe the data. (c) In contrast, the $T=15$ K PDF (NPDF) is quantitatively described by a least-squares fit of the average crystallographic structure with a single Ni-Ni distance.

Figure 5

Least-squares fit of a linear combination of tetragonal $I4/mmm$ and orthorhombic $Fmmm$ models of KNi${}_2$S${}_2$ to the $T=300$ K PDF allows extraction of the local Ni-S and Ni-Ni distances; however, the structural model fails to describe the data beyond $r>3.5$ Å.

Figure 6

(a) Representative reverse Monte Carlo simulation of the pair distribution function. (b) Projection of all 24 000 atoms from the RMC supercell (small spheres) back onto the idealized crystallographic unit cell (large spheres) reveals that the ThCr${}_2$Si${}_2$-type connectivity of KNi${}_2$S${}_2$ is maintained after the simulation, as is the average $I4/mmm$ crystallographic symmetry. (c) A representative [Ni${}_2$S${}_2$] layer extracted from a 24 000 atom RMC supercell ($T=300$ K) illustrates the incoherent distribution of short (dark, $d_{\mathrm{Ni}-\mathrm{Ni}}<2.52$ Å) and long bonds (light, $d_{\mathrm{Ni}-\mathrm{Ni}}>2.86$ Å). Intermediate length bonds are omitted for clarity. (d) Scaled atomistic representations of short and long Ni-Ni bonds extracted from the $T=300$ K RMC supercell.

Figure 7

(a) Histogram of Ni-Ni distances extracted from the RMC supercell. The orange (light gray) line is the result from a simulation of the measured data. The solid black line corresponds to a histogram of Ni-Ni distances simulated from anisotropic and harmonic displacements. Below is their difference. (b) The ensemble bond valence sums (BVS) computed from the RMC supercell, which is highly sensitive to local charge disproportionation (Ref. 27), indicate a single population of charged Ni species centered around a $\mathrm{BVS}=2.0\left(2\right)$. The solid line is a fit to a single Gaussian expression.

Figure 8

Neutron powder diffraction (NPD) data and from the NPDF instrument (90${}^{\circ }$ bank) acquired at (a) $T=300$ K and (b) $T=15$ K and Rietveld analyses illustrating the absence of additional reflections, as would appear from long-range ordered magnetism and enlargement of the $I4/mmm$ nuclear unit cell (circles: data; line: fit; difference curve below). (c) Direct subtraction of NPD data (black) collected at $T=50$ K (orange/light gray) and $T=5$ K (blue/dark gray) on the HIPD instrument (90${}^{\circ }$ bank) to illustrate an absence of additional scattering from long-range ordered magnetism. The inset shows data collected on the forward scattering detector bank (14${}^{\circ }$), plotted on a logarithmic scale.

Figure 9

(a) Temperature dependence of electrical resistivity of KNi${}_2$S${}_2$, measured on a polycrystalline pellet, indicates metallic transport and a first-order transition near $T\sim 250$ K. (b) The isotropic atomic displacement parameters for K, Ni, and S show transitions concomitant with the K${}^{+}$ sublattice freezing at $T\sim 250$ K (dotted line). Colored lines are guides to the eye. (c) Rietveld analysis of the SXRD data reveals that the S position remains undisturbed, with only a minor transition observed at $T\sim 250$ K.

Figure 10

The linear magnetic susceptibility (closed squares; thin line as a guide to the eye), as determined by the high-field slope of isothermal magnetization, is nearly temperature-independent. The magnetic susceptibility determined by measurements at constant fields (open circles) reveals a gradual upturn at $T\sim 75$ K.

Figure 11

(a) Isothermal field-dependent magnetization curves at various temperatures (open circles) illustrate the linear contribution at high magnetic field (solid lines) and increased curvature at low temperatures. (b) Global fit (thin lines) of a Brillouin function and linear susceptibility term [Eq. (1)] to the $M-M_{300\mathrm{K}}$ data (open circles).

Figure 12

The electronic specific heat ($C_p-C_{\mathrm{latt}}$) reveals a bulk superconducting transition at $T_c=0.46\left(2\right)$ K (thick solid line: equal entropy construction). Magnetic fields suppress the transition, and by ${\mu }_{0}H=14$ T a sharp upturn at $T<0.2$ K is present and described by Schottky anomalies corresponding to impurity ($\sim {10}^{-6}$) and nuclear spins (thin solid line). Inset: applied magnetic fields have a weak influence on $\gamma$.

Figure 13

(a) The temperature dependences of the unit cell parameters of KNi${}_2$S${}_2$ ($a$ axis, circles on left; $c$ axis, squares on right) show a negative coefficient of thermal expansion of both axes below $T<8.9$ K (dashed lines guide the eye, vertical lines denote transitions). (b) The excess entropy below $T\sim 10$ K is only well described when an enhanced-mass $\gamma T$ electronic contribution is included. The Dulong-Petit limit ($3NR/T$; dotted line) is achieved near $T=300$ K. (c) The specific heat normalized by $T^3$ indicates additional thermodynamic degrees of freedom, well described by the Einstein expression for localized vibrational modes, as described in the text ($\gamma T$ contribution: dashed line; Debye lattice: black dot-dashed line; Einstein modes: shaded peaks).

Figure 14

The temperature/magnetic-field phase diagram displays the complex structural and electronic states observed in KNi${}_2$S${}_2$, including superconducting (SC), heavy mass, intermediate but noncharge density wave (CDW), Ni-Ni charge density wave, and high K${}^{+}$ mobility phases. The K${}^{+}$ sublattice melting transition is a first-order transition at $T=250$ K with an associated two-phase region. The other transitions appear to be second-order, but the data are ambiguous regarding their thermodynamic nature.