Effect of uranium deficiency on normal and superconducting properties in unconventional superconductor UTe₂

Table 1 summarizes the characterization of representative samples grown from different conditions. Among them, the sample NS was grown following the condition described in ref. [8]. The resistivity and specific heat measurements showed absence of superconductivity down to 0.45 K. On the other hand, SC1 samples grown following the Ran’s prescription [1], where tellurium deficient starting composition is employed, showed superconductivity around 1.6 K. In the SC2 samples, we kept the starting composition the same but applied lower temperature condition. The resulting crystals showed fairly high superconducting transition temperatures. This demonstrates that the starting composition affect $T_{\rm c}$ and residual resistivity ratio (RRR) rather than the growth temperature. Higher $T_{\rm c}$ samples with $T_{\rm c}$ up to 2 K were recently reported from much lower growth temperatures[24]. The RRR for the superconducting samples are always higher than that of non-superconducting sample, indicating that the non-superconducting sample shows more electron scattering due to impurities or defects.

Table 2 shows the crystallographic parameters determined by single-crystal X-ray diffraction. Fractional coordinates, anisotropic atomic displacement parameters and secondary extinction effect were taken into account during the fitting. The lattice parameters of sample NS are slightly smaller than SC1 and SC2, while there are no significant differences in the fractional coordinates. We note that the atomic displacement parameter for uranium site in NS is fairly large. It is even larger than that of tellurium sites. Such a large atomic displacement parameter is indicative of a large spread of spatial distribution or low atomic concentration at the corresponding site. To clarify this we attempt to fit the data by changing the uranium occupancy. As shown in Fig. 1 fractional coordinates are very insensitive to the uranium occupancy. On the other hand, the reliability factor $R_{1}$ shows a minimum when the occupancy is around 0.96. At this occupancy, the atomic displacement parameter of uranium is smaller than the two tellurium sites, as seen in most of conventional metallic compounds. The similar analysis for the samples SC1 and SC2 converges around full uranium occupancy. Uranium deficiency was found also in uranium monotelluride UTe. See Appendix for details.

The occupancy 0.962(2) of the uranium ($Z$ = 92) site is equivalent to the existence of an atom with $Z$ = 88.3. There are two possibilities to realize such a situation. One is that $\sim 4$% of uranium atoms are missing, and the other is that the extra tellurium ($Z$ = 52) partially substitute the uranium site. In the latter case, the resulting composition would be (U_0.91Te_0.09)Te₂ = U_0.91Te_2.09 (U_0.87Te₂).

To verify this, the chemical composition of the samples is examined with the electron-probe microanalysis. There is a clear difference in the intensity ratio between signals from uranium and tellurium. By using the intensity ratio for sample SC2 as an internal standard for stoichiometric UTe₂, the composition of the sample NS is estimated as U_0.96±0.01Te₂. It is therefore likely that uranium site is not substituted by Te but rather left vacant to satisfy average 96 % occupancy. The possibility of the excess atoms at interstitial positions is examined by constructing an electron density map from the X-ray diffraction data by maximum entropy method [26]. The resulting electron density did not show peaks higher than 1 e/$\rm\AA^{3}$ at interstitial positions. Therefore this possibility can be excluded.

In the previous study of U-Te binary phase diagram, the stoichiometric UTe₂ was reported to appear by peritectic reaction with Te-rich liquid and U₃Te₅ phases at 1180^∘C, and then non-stoichiometry of UTe_2-x was reported to range from $x=0$ to $x=0.13$ [27], although the tetragonal UTe_1.87 phase was questionable [28]. Indeed, the Te-flux method using Te-rich composition yields single crystals of UTe₂ when temperature was gradually lowered from 1180 ^∘C [29], so this appears to be consistent with the phase diagram. However, the crystals grown by the Te-flux method [29, 30] do not show superconductivity or show $T_{\rm c}$ less than $\sim$1.6 K, which may correspond to our NS sample. It may suggest that the composition on the peritectic line at 1180 ^∘C would slightly deviate from UTe₂, i.e., U_1-xTe₂. Of course, in the case of CVT method, it is not in the chemical phase equilibrium, the phase diagram cannot be directly applicable. So far, for the synthesis of superconducting samples by CVT, a necessary condition is considered to start from a U-rich composition for UTe₂, i.e., U:Te=1:1.5, or 1:1.8. This requirement is, at least, consistent with the off-stoichiometric tendency near UTe₂ and our experimental result that the U-deficient crystals do not show superconductivity.

The sample dependence on superconductivity and chemical composition is extensively studied in ref. [23] and a significant difference in composition ratio Te/U is reported between superconducting and non-superconducting samples, where the non-superconducting sample shows systematically larger Te/U ratio. Qualitatively, their observation is consistent with the present EPMA result. In addition, we showed crystallographically that Te/U ratio is not due to the presence of excess Te, but due rather to uranium deficiency.

The deviation from the stoichiometry should also be reflected in the crystallographic parameters. As summarized in Table II, lattice parameters of NS1 sample is slightly smaller than those of SC samples. The difference in the unit cell volume is about 1.2 ${\mathrm{\AA}}^{3}$. Assuming that uranium atoms occupy roughly 1/3 of unit cell volume, a 4 % deficiency of uranium would result in 1.3 % of volume reduction if the vacancy is filled by other atoms. The fairly small experimental volume reduction 0.5 % suggests that the uranium deficient site is left vacant rather than being substituted by other atoms.

The difference in the fractional coordinates is extremely small between SC and NS samples, except for the $z$-coordinate of the uranium site. U_z = 0.1354(1) for NS samples is slightly larger than that of SC samples 0.13520(4). The resulting U-U distance 3.78 is, however, almost sample independent. Overall crystal structure seems to be maintained primarily by the tellurium lattice.

Temperature dependences of electrical resistivity and specific heat measured on the same piece of single crystal SC2 are shown in Fig. 2. Both resistivity and specific heat show a sharp transition at 1.8 K. The onset of the resistive transition begins at slightly higher temperature 1.93 K than the onset of specific heat anomaly 1.81 K, indicating that superconductivity is not occurring in entire sample in this temperature interval. Below 1.81 K, specific heat drastically increases, shows a sharp peak at 1.76 K and then decreases. With decreasing temperature, specific heat monotonically decreases down to 0.5 K. Unlike previous works reporting multiple superconducting transition, only a single and sharp superconducting transition is detected in this temperature range. Application of magnetic field along the $a$-direction (not shown) monotonically shifts the peak to the lower temperature side without splitting.

Specific heat below 1.3 K is approximately reproduced by $C/T=\gamma_{\rm N}+\beta T^{2}$ with $\gamma_{\rm N}$ = 0.047 J/K²mol, where $\gamma_{\rm N}$ is considered as the residual electronic specific heat in the superconducting state. It is now known that the $\gamma_{\rm N}$ correlates with $T_{\rm sc}$. The most recent study on the sample with higher $T_{\rm sc}$ = 2 K shows less $\gamma_{\rm N}$. It is expected that the highest $T_{\rm sc}$ sample would have significantly small $\gamma_{\rm N}$ term. [24, 31]

Specific heat is sample dependent both in low temperature and high temperature regions. In order to minimize experimental uncertainty in obtaining the absolute values, samples with the similar mass 10 mg were measured, where the molar specific heat is calculated using the nominal UTe₂ composition. At high temperature around 60 K, the electronic contribution to specific heat is expected to be less significant. The difference of specific heat can be regarded primarily as the difference of the lattice contribution. The smaller values for NS sample is indicative of a higher Debye temperature, most likely in agreement with the smaller lattice parameters for NS sample. With decreasing temperature $C/T$ value shows a maximum around 50 K, then decreases with decreasing temperature. $C/T$ of SC sample is larger at high temperatures, but crosses at 8 K. Below this temperature $C/T$ of SC sample is smaller. The difference becomes more significant with lowering temperature down to $T_{\rm sc}$ of SC sample. Note that, by taking into account the uranium deficiency in NS sample, uranium contribution to $C/T$ is even larger.

Between 8 and 2 K, $C/T$ for NS sample approximately follows $\gamma+\beta T^{\delta}$ with $\gamma=158~{}$mJ/K²mol, $\beta$ = 0.0039 J/K^2.5mol and $\delta=1.5$. A gradual increase of $C/T$ is observed below 1 K. The origin of this anomaly is not clear at present. In SC sample, on the other hand, the normal state $C/T$ cannot be reproduced by the same manner; $\delta$ is strongly temperature dependent. The large electronic contribution around 14 K, which is also reported by Willa et al., is prominent only for SC2 but is absent in NS sample.[32]

In order to estimate electronic contribution to specific heat, phonon contribution is approximated by Debye model with $T_{\rm D}$ = 200 K as shown by black line. The resulting electronic specific heat $\Delta C/T$ was obtained by subtracting Debye model from the experimental data (Figure 3). Here, the specific heat for NS sample is corrected for uranium deficiency.

In the SC sample, $\Delta C/T$ just above $T_{\rm c}$ is characterized by a heavy fermion behavior with Sommerfeld coefficient 120 mJ/K²mol. Assuming that the heavy fermion state arises from Kondo-type interaction, characteristic energy scale would be of the order of 100 K. The corresponding specific heat associated with the heavy fermion formation would then follow Coqblin-Schrieffer model [33]; the heavy fermion contribution would gradually decrease with increasing temperature. In experiment, however, a convex feature around 14 K is seen with increasing temperature. Subtracting the heavy fermion contribution mentioned above, remaining $\Delta C/T$ would look like a Schottky-like broad peak as also observed by Willa et al. [32].

In the NS sample, on the other hand, this feature is broadened; NS sample shows a lower maximum around 14 K, and $\Delta C/T$ decreases slowly with temperature and shows a larger electronic specific heat coefficient at 2 K. $\Delta C_{\rm NS}$ and $\Delta C_{\rm SC}$ cross at 8.2 K. Interestingly, the area defined by the two curves are approximately the same below and above this temperature. It means that entropy difference between 2 K and about 30 K is almost the same for both samples. The broad peak at 14 K in SC sample suggests the existence of a kind of energy gap with an excitation gap about 40-50 K. It seems that this gap does not open in the NS sample. The excess Sommerfeld term in NS sample may be accounted for by failing to open a gap in the magnetic/electronic excitation.

Recent inelastic neutron scattering (INS) experiments suggest a development of one dimensional ladder-type magnetic correlation characterized by a ferromagnetic coupling between nearest neighbor uranium atoms [11]. The characteristic temperature scale for this correlation is estimated as $T^{*}_{1}\sim 14$ K in agreement with the present observation. The present excitation ’gap’ is not completely consistent with the INS observation which suggests rather quasielastic response. However it is not easy in this case to distinguish quasielastic to inelastic responses because UTe₂ has enhanced density of states arising from heavy electron and consequently a quasi elastic scattering is expected.

The absence of the Schottky anomaly in the uranium-deficient NS sample is naturally understood by the breaking of the magnetic correlation between nearest uranium atoms. Note that 4 % of deficiency corresponds to one broken ferromagnetic neighbor in 13 ladder rungs ($\sim 50~{}{\rm\AA}$). Such defects would easily hinders the development of one-dimensional magnetic correlation.