Enhancement of superconducting properties in the La–Ce–H system at moderate pressures

Ternary hydrides are regarded as an important platform for exploring high-temperature superconductivity at relatively low pressures. Here, we successfully synthesized the hcp-(La,Ce)H9-10 at 113 GPa with the initial La/Ce ratio close to 3:1. The high-temperature superconductivity was strikingly observed at 176 K and 100 GPa with the extrapolated upper critical field Hc2(0) reaching 235 T. We also studied the binary La-H system for comparison, which exhibited a Tc of 103 K at 78 GPa. The Tc and Hc2(0) of the La-Ce-H are respectively enhanced by over 80 K and 100 T with respect to the binary La-H and Ce-H components. The experimental results and theoretical calculations indicate that the formation of the solid solution contributes not only to enhanced stability but also to superior superconducting properties. These results show how better superconductors can be engineered in the new hydrides by large addition of alloy-forming elements.

needle to scratch the La-Ce layer on the glass slide to get particles, which left traces (Fig. S14a). La and Ce were both uniformly distributed except for the scratched area (Fig. S14b). Si and O mainly come from the glass slide, thus the scratched area has high intensity (Fig. S14d). On the La-Ce surface, there was a tiny amount of oxygen because of the oxides formed during the transfer. Moreover, Al, C and Ca exist all over the selected region and can be viewed as a background signal. Fig. S16 shows that the scratched particles have a uniform distribution of La and Ce. The sputtered La-Ce can form clusters or smooth solid solutions.           In this run, we tested the nano-polycrystalline diamond (NPD) 3 . NPD consists of randomly oriented fine diamond nanocrystals and has higher Knoop hardness than SCD 4,5 .

Analysis of the superconducting parameters
To get more information on the new superconducting La-Ce-H phase, we have estimated the superconducting parameters by carefully analyzing the temperature-dependent resistance R(T) data. The Debye temperature D of the synthesized La-Ce-H compound is calculated by fitting the experimental R(T) data using the Bloch-Grüneisen (BG) formula 17 , which has been used for some reported polyhydrides 18,19,20,21 : where A, D , and R0 were found using the least-squares method. These data also allow us to obtain the electron-phonon coupling parameter using the numerical reversal of the Allen-Dynes formula Tc = F(λ, ωlog) λ = G(Tc, ωlog), where ωlog is replaced by 0.827×θ D , and ω2 is 4/3 ωlog for simplicity. In other words, θ D and ωlog are in the same ratio as the original McMillan formula and its subsequent modification by Allen and Dynes. We summarized the deduced parameters in Table S2 and plotted the data in Fig. S39. The fitted D and ωlog remain around 700-900 K at 90-150 GPa, suggesting the need of relatively high λ = 2-3 to ensure that the observed Tc is above 160 K. Rather high λ corresponds to the weakly ordered, soft and highly defective structure of this superconducting La-Ce-H phase.  It is interesting that the dependence of the EPC parameter (λ) and the logarithmically averaged frequency (ωlog) are non-monotonic functions of pressure for the studied polyhydrides. It is generally assumed that the Debye temperature and ωlog will increase with increasing pressure. However, as Fig. S39b shows, this is not always the case. As the pressure increases, a phase transition to a more highly symmetric polyhydride modification may occur with a change in its stoichiometry. Such processes are usually accompanied by the appearance of "soft" phonon modes, a significant increase in the EPC coefficient, and a decrease in the average phonon energy.
In the cases studied, the phase transformations occur in the low-pressure interval of 90-100 GPa for Ce-La polyhydrides and in the high-pressure interval > 130 GPa for lanthanum hydrides. Simultaneous changes in all three parameters λ, TC, ωlog and high values of λ correspond to these regions. In particular, for (La,Ce)H9+x (Fig. S39a), an abnormally high λ > 5 is observed at 100 GPa, which rapidly decreases to ≈1. The resulting Eliashberg functions require some discussion. As can be seen from Fig. S40d, the large values of the electron-phonon interaction coefficient (λ > 4.7) for hexagonal LaH9 at 120 GPa indicate its dynamic instability 22 . This is one of the reasons why we do not observe the formation of this compound in the experiment. However, various dopants can stabilize a hexagonal P63/mmc-XH9 structure at this pressure. One of them is Ce, which forms a stable hexagonal CeH9 23,24,25 . Obviously, there is a certain critical concentration of cerium that stabilizes the hexagonal structure of XH9. Both 25 at.% (this work) and 50 at.% 6 of cerium will definitely stabilize such a structure in the experiment. However, as we can see from Fig. S40c, the polyhydride with 25 at.% cerium has α 2 F(ω) characteristic of amorphous alloys and films (see for example 26,27 ), which indicates the beginning of destruction of the ordered hexagonal structure. As the cerium concentration decreases to the critical value, the stability of hexagonal (La,Ce)H9 will decrease, while the electron-phonon interaction coefficient (λ) and the critical temperature of superconductivity (TC) will increase until (La,Ce)H9 decomposes. This explains why the critical temperature for La0.75Ce0.25H9 is greater than for La0.5Ce0.5H9. An additional factor is also the suppression of superconductivity due to spin-flipping scattering on the local magnetic moments of Ce atoms 28 .  unstable, but LaCeH18 has shorter imaginary "tail" which allows us to say that cerium somewhat stabilizes the hexagonal XH9 structure.
Many recent experimental studies of polyhydrides (e.g., LaH10 30,31 , LaYH20 32 , CeH9 25 , etc.) have shown that they are stable at much lower pressures than the dynamic stability criterion based on the harmonic approximation predicts. This serious discrepancy between the theoretical predictions and the experiment greatly devalues the harmonic calculations. We also investigated the applicability of Born's criteria 33,34 for hexagonal and cubic crystals under static isotropic pressure P to our polyhydrides. Table S5 shows that all compounds (LaH10, LaCeH18 and La3CeH36) are mechanically stable except P63/mmc-LaH9, for which two criteria are not met.  35 and the Allen-Dynes formula (A-D) 36 with μ* = 0.15-0. The calculation of the Debye temperature using elastic constants (Table S5)

Theoretical Methods
To calculate isotope coefficient α, the Allen-Dynes interpolation formulas were used: , , where the last two correction terms are usually small (~0.01).
The superconducting transition temperature TC was estimated by Allen-Dynes formula in the following form: where The EPC constant λ, logarithmic average frequency ωlog and mean square frequency ω2 were calculated as: where µ * is the Coulomb pseudopotential, for which we used widely accepted lower and upper bounds of 0.10 and 0.15.
The Sommerfeld parameter was found as where N(0) -is the total density of electronic states at the Fermi level per spin. It was used to estimate the upper critical magnetic field and superconducting gap using the known semiempirical equations of the BCS theory (see Ref. 38 , equations 4.1 and 5.11), working satisfactorily for TC/ωlog < 0.25: where γ' = 2γ. Regarding formula (A8), the following remark should be made. The upper critical field by nature depends on the amount of impurities and defects in the sample, so it changes significantly from sample to sample. Only the lower limit for μ0HC2, which corresponds to an ideal crystal, can be estimated.
However, we noticed that formula (S8), originally designed to calculate the thermodynamic critical field 2 log log 2 (0) 3.53 1 12.5 ln 2 μ0HC of superconductors, also gives a good estimate of the upper critical field of hydrides if the full DOS expressed per mole of a hydride is used in the Sommerfeld parameter (γ'). In this case, μ0HC2 is expressed in Tesla in formula (S8).
The lower critical magnetic field was calculated according to the Ginzburg-Landau theory 39 (11) where λL is the London penetration depth can be estimated by the formula (12) here c -is the speed of light, e -is the electron charge, me -is the mass of an electron, and ne -is an effective concentration of charge carriers expressed via the average Fermi velocity (VF) in the Fermi-gas model: The average Fermi velocity can be estimated as (14) where ξ is the coherence length calculated from the experimental upper critical magnetic field using the formula .