Magnetic and structural properties near the Lifshitz point in Fe${}_{1+x}$Te

We construct a phase diagram of the parent compound Fe${}_{1+x}$Te as a function of interstitial iron $x$ in terms of the electronic, structural, and magnetic properties. For a concentration of $x<10\%$, Fe${}_{1+x}$Te undergoes a “semimetal” to metal transition at approximately 70 K that is also first-order and coincident with a structural transition from a tetragonal to a monoclinic unit cell. For $x\approx 14\%$, Fe${}_{1+x}$Te undergoes a second-order phase transition at approximately 58 K corresponding to a semimetal to semimetal transition along with a structural orthorhombic distortion. At a critical concentration of $x\approx 11\%$, Fe${}_{1+x}$Te undergoes two transitions: the higher-temperature one is a second-order transition to an orthorhombic phase with incommensurate magnetic ordering and temperature-dependent propagation vector, while the lower-temperature one corresponds to nucleation of a monoclinic phase with a nearly commensurate magnetic wave vector. While both structural and magnetic transitions display similar critical behavior for $x<10\%$ and near the critical concentration of $x\approx 11\%$, samples with large interstitial iron concentrations show a marked deviation between the critical response indicating a decoupling of the order parameters. Analysis of temperature dependent inelastic neutron data reveals incommensurate magnetic fluctuations throughout the Fe${}_{1+x}$Te phase diagram are directly connected to the “semiconductor”-like resistivity above $T_N$ and implicates scattering from spin fluctuations as the primary reason for the semiconducting or poor metallic properties. The results suggest that doping driven Fermi surface nesting maybe the origin of the gapless and incommensurate spin response at large interstitial concentrations.

Figure 1

The magnetic and structural phase transitions in Fe${}_{1+x}$Te as a function of interstitial iron $x$. The first-order transition from paramagnetic to commensurate antiferromagnetic ordering corresponds to a tetragonal $P4/nmm$ to monoclinic $P{2}_1/m$ transition. The second-order transition to an incommensurate helimagnet corresponds to an orthorhombic $Pmmn$ distortion. The tricritical point is hence close to $x\approx 11\%–12\%$.

Figure 2

(a) Temperature evolution of the phase fraction in Fe${}_{1.057\left(7\right)}$Te from neutron powder diffraction data. The mixed tetragonal and monoclinic phases for the $T$ range enclosed by the dashed lines is clear evidence of the first-order nature of this transition. (b) The size of the magnetic moment per iron cation as a function of temperature. Comparison of the moment size with the monoclinic phase fraction demonstrates how $T_N$ and $T_S$ are simultaneous and sudden in this composition. Inset shows the antiferromagnetic unit cell within the $ab$ plane and the moment direction. In this figure and all others in this paper, when not indicated otherwise, error bars are the size of the data points and represent $\pm 1\sigma$.

Figure 3

Temperature evolution of the lattice parameters for Fe${}_{1.057\left(7\right)}$Te from neutron powder diffraction data. In (a), the $a$ and $b$ lattice parameters corresponding to the in plane Fe-Fe distance, in (b), the $c$ parameter, and in (c), the discontinuous volume change indicative of a first-order phase transition.

Figure 4

(a) The in-plane resistivity and (b) heat capacity as a function of temperature for Fe${}_{1.057\left(7\right)}$Te. The drop in resistivity marks a metal-to-semiconductor transition upon warming, which corresponds to a large increase in the specific heat. Note that the slight jump at $\sim$100 K is an artifact in the data collection. The solid filled circles display the estimated magnetic contribution to the specific heat, namely, $C_p-C_{\text{lattice}}$. The sawtooth structure at high temperature in the estimated magnetic contribution to the specific heat is a result of using nearest neighbor interpolation routine. The entropy is plotted in (b) in units of $R\ln 4$ and shows that the fully entropy for a $S=3/2$ moment is acquired above $T_N$.

Figure 5

Constant-$Q$ scans through the magnetic correlations in Fe${}_{1.057\left(7\right)}$Te. (a)–(c) show constant $Q$ slices taken over ($H,0,L\pm 0.15$). (d)–(f) illustrate constant $E=2.0$ meV cuts integrating around L $\pm$ 0.15. Note that the color images show $\chi$${}^{\prime \prime }$ as discussed in the text.

Figure 6

Constant $E=2.0$ meV scans through the magnetic correlations in Fe${}_{1.057\left(7\right)}$Te. (a)–(d) Constant energy slices at a series of temperatures. (e) The fitted position along the ($H,0,0.5$) direction based upon cuts through similar scans displayed in (a)–(d). (f) The peak intensity ($I_0$) as a function of temperature and the parameter $\alpha$ (described in the text), which parameterizes the correlations along $L$. (g) The dynamic correlation length taken at $E=2.0$ meV. The intensity around $\vec{Q}=(1,0,1)$ is due to a phonon.

Figure 7

Constant-$Q$ scans taken on the PUMA thermal triple-axis spectrometer. (a) and (b) Energy scans taken at $\vec{Q}=(0.45,0,0.5)$ at 100 and 60 K. (c) The half-width ($\Gamma$) as a function of temperature. (d) The temperature dependent parameter ${\chi }_0$.

Figure 8

(a) Temperature evolution of the orthorhombic to tetragonal distortion in Fe${}_{1.141\left(5\right)}$Te as measured by the (040) reflection form single crystal neutron diffraction experiments. (b) Comparison of the structural order parameter, ${\delta }^2$, and the magnetic order parameter as determined by measuring the (0.38 0 0.5) magnetic reflection. The critical exponent was fit only over the temperature range shown by allowing the transition temperature and exponent to vary.

Figure 9

A power-law fit to the intensity of the magnetic Bragg reflection in Fe${}_{1.141\left(5\right)}$Te yielding an exponent of $\beta =0.15\left(1\right)$, consistent with 2D-Ising behavior.

Figure 10

Temperature dependent in-plane electrical resistivity (a) and specific heat (b) for Fe${}_{1.141\left(5\right)}$Te. The entropy in units of $R\ln 4$ and, in contrast to the commensurate sample, shows only a gradual increase above $T_N$ and never reaches the full saturation value for $S=3/2$. The estimated magnetic contribution to the heat capacity is shown by the filled circles ($C_p-C_{\text{lattice}}$). The sawtooth structure to the magnetic portion at high temperatures is a result of using nearest neighbor interpolation from the data.

Figure 11

Constant-$Q$ scans through the magnetic correlations in Fe${}_{1.141\left(5\right)}$Te. (a)–(c) show constant $Q$ slices taken over (H,0,L$\pm$ 0.05). (d)–(f) Constant $E=2.0$ meV cuts integrating around $L\pm 0.05$. The magnetic intensity at 80 K is peaked at the incommensurate value of $H=0.375\pm 0.007$ $rlu$.

Figure 12

The calculated contribution to the resistivity from scattering off spin fluctuations in commensurate and incommensurate samples. The calculation is described in the main text.

Figure 13

Observed and calculated neutron powder patterns for Fe${}_{1.11}$Te at 48 K from the highest-resolution bank of the time-of-flight instrument HRPD (ISIS). The Rietveld refinement converged with an $R_{wp}$ of 10.7% and ${\chi }^2$ of 3.234. Upper tick marks indicate the Bragg reflections of the orthorhombic $Pmmn$ phase and lower tick marks the monoclinic $P{2}_{1}m$ phase. Inset shows a zoom-in of part of the powder pattern, demonstrating the high-resolution quality of the data to distinguish between the monoclinic and orthorhombic phases even close to the temperature of the nucleation of the monoclinic phase.

Figure 14

(a) Temperature evolution of the phase fraction in Fe${}_{1.11}$Te from neutron powder diffraction data. The amount of mixed orthorhombic and monoclinic phases is temperature dependent until a percentage of phase fraction locks in at $\approx$40 K. (b) The temperature dependence for the incommensurate wave vector, which is strongly temperature dependent for the orthorhombic phase up until the lock-in temperature of 40 K. (c) The size of the magnetic moment per iron cation as a function of temperature for each of the phases.

Figure 15

The lattice constants as a function of temperature for Fe${}_{1.11}$Te with (a) showing the in-plane $a$ and $b$ values for the two phases. (c) The $c$ lattice constant and the unit cell volume is shown in (c).

Figure 16

The magnetic order parameter plotted for both phases for Fe${}_{1.11}$Te for the two different phases. The data are from the HRPD spectrometer at ISIS. (a) The commensurate component and the hysteresis illustrating the first order nature of this transition. (b) The magnetic order parameter for the incommensurate phase that displays a second-order phase transition. There is no measurable difference in $T_N$ or the slope on both warming and cooling. The intensity difference reflects the low-temperature first-order transition. The exponent $\beta =0.380\left(5\right)$ belongs to the 3D Heisenberg universality class.

Figure 17

A map of the spin fluctuations in the high-temperature paramagnetic phase for Fe${}_{1.124\left(5\right)}$Te. (a) A constant-$Q$ slice along $\vec{Q}=(H,0,0.5\pm 0.15)$. (b) and (c) Constant $E=1.5$ meV slices at 100 and 200 K, respectively. (d) and (e) Constant energy cuts along $\vec{Q}=(H,0,0.5\pm 0.15)$ and $(0.425\pm 0.075,0,L),$ respectively, at 100 K. The scattering is peaked at the incommensurate wave vector of $H=0.373\pm 0.008$ $rlu$. The value is within error to that derived for the iron-rich incommensurate sample discussed above.