Raman spectroscopy and pressure-induced structural phase transition in UTe₂

Abstract

Results of the Raman scattering experiments, heat capacity measurements, ab initio simulations of the Raman spectra and pressure-induced phase transition in UTe₂ single crystal are reported. Assignment of symmetries to particular Raman-active phonons follows directly from a comparative analysis of the measured and calculated Raman spectra. Theoretically determined lattice contribution to the specific heat of UTe₂ allows for better description of its heat capacity measured over the temperatures ranging from 30 to 400 K. The orthorhombic-to-tetragonal phase transition pressure of 3.8 GPa is predicted at room temperature in very good agreement with the recent experimental studies. The phase transition remains almost phonon-independent with the transition pressure weakly temperature-dependent below 500 K. The strong local Coulomb correlations between U-5f electrons and spin−orbit interaction are shown to be important for realistic theoretical description of phonons and pressure-induced phase transition in UTe₂.

Graphical abstract

Keywords

Raman spectroscopy / heat capacity / phase transition / ab initio simulations

1 Introduction

Intensive experimental and theoretical research of the heavy-fermion unconventional superconductor UTe₂ performed over last years [1–6] has established topological and odd-parity superconducting state of this compound below 1.6−2.0 K [7, 8]. Even though UTe₂ does not order magnetically down to 25 mK under ambient pressure [9], it is qualified to the family of unconventional uranium-based superconductors [10] that comprise such compounds as UGe₂ [11], URhGe [12], and UCoGe [13], in which spin fluctuations without an ordered magnetic moment play a significant role in Cooper pairing [14, 15]. Uranium ditelluride is known from its complex temperature-pressure-field phase diagram, interplay between magnetism and superconductivity [16], multiple magnetic and superconducting phases emerging and evolving upon external pressure [17–23], high-magnetic field [24–30] as well as combination of both magnetic field and pressure [31–34]. More recent experimental studies [18, 35] show that UTe₂ undergoes a structural phase transition from its body-centered orthorhombic $Immm$ structure (space group No. 71) to a body-centered tetragonal structure of $I4/mmm$ symmetry (space group No. 139) between 3 and 5 GPa. On the other hand, theoretical calculations suggested that the $Immm\to I4/mmm$ structural transformation takes place at much higher pressure of 9 GPa [36].

In general, a majority of experimental and theoretical efforts undertaken up to now addressed mainly the understanding and clarification of mechanism underlying unconventional superconductivity in UTe₂ along with exploration of its unique properties. Notably less attention has been paid, however, to the phonon dynamics in uranium diteluride and its role (if any) in the structural phase transition between low-pressure orthorhombic phase (O-phase) and high-pressure tetragonal phase (T-phase). The effect of external pressure on the phonon-dependent quantities of UTe₂ remain largely unexplored as well. Hence, the present work extends description of UTe₂ compound by investigating its lattice dynamics, which we gain from the Raman scattering experiments performed on single-crystalline samples. We employ state-of-the-art density functional theory (DFT) incorporating both strong local Coulomb correlations and spin-orbit interaction (DFT + U + SO method) to support and explain results of our Raman scattering experiments and heat capacity measurements. We also reconsider the $Immm\to I4/mmm$ phase transformation and demonstrate that by applying the DFT + U + SO scheme one can predict transition pressure between these phases in very close agreement with experimental observations.

2 Methodology

2.1 Experimental

Single crystals of UTe₂ were grown by the molten salt flux (MSF) method as described in detail by Sakai et al. [37]. This method has been proved to produce high-quality UTe₂ single crystals with limited content of uranium vacancies, and hence showing a critical temperature $T_{\mathrm{c}}=2.1$ K as compared to $T_{\mathrm{c}}\sim 1.6$ K of single-crystalline samples prepared by chemical vapor transport (CVT) method [7, 23, 38]. The Energy Dispersive X-ray analysis performed on the grown crystals confirmed their 1:2 stoichiometry. The structure of our single-crystalline samples was determined from the X-ray diffraction (XRD) experiments performed at 300 K and using Rigaku OD Supernova diffractometer with an Atlas S2 CCD detector and a mirror-collimated Mo-K$\alpha$ ($\lambda$ = 0.71073 Å) radiation from a micro-focused sealed X-ray tube. Integration, absorption correction and scaling were done using the CrysAlisPro 1.171.42.49 (Rigaku Oxford Diffraction, 2022) program package. The crystal structure was solved by charge flipping with the program SUPERFLIP [39] and refined with the Jana2020 program package [40] by a full-matrix least-squares technique on $F^2$. All atoms were refined anisotropically. The residual factor for the 287 significantly intense reflections with $I>3\sigma (I)$ was 0.0296 and goodness of the fit was 2.0001.

The room temperature (RT) Raman spectra were excited with the 514.5 nm line of an Ar laser at a power of about 20 mW ($\sim$ 2 mW on the sample) and recorded in a back-scattering geometry using a RM-1000 RENISHAW micro-Raman spectrometer, equipped with Bragg grating filters enabling good stray light rejection. The diameter of the laser spot on the sample surface amounted to 2−3 μm and the spectral resolution was better than 1.5 cm⁻¹ with a 2400 l/mm grating. Single crystals of UTe₂ for Raman spectroscopy measurement were cleaved in a glove box under a protective Ar atmosphere and subsequently sealed in a thin quartz capillary to prevent the formation of TeO₂ while exposed to air.

The heat capacity of UTe₂ single crystal was measured by standard thermal relaxation technique in the temperature range between 0.4 and 370 K using PPMS Quantum Design Inc. apparatus.

2.2 Theoretical

Calculations have been carried out within the DFT method implemented in Vienna ab initio Simulation Package (VASP) [41, 42]. The projector augmented-waves (PAW) pseudopotentials, generalized gradient approximation (GGA) exchange-correlation functional with the Perdew, Burke and Ernzerhof (PBE) parametrization [43, 44], and the plane-wave energy cutoff of 330 eV were used in the non-spin-polarized calculations. Relativistic effects associated with heavy uranium atoms were taken into account via introduction of the spin-orbit interaction (SO). The interplay between strong localization and itinerant character of the U-5f states were considered by employing a moderate value of the Coulomb repulsion described by the effective Hubbard potential $U_{\mathrm{e}\mathrm{f}\mathrm{f}}=1.0$ eV ($U_{\mathrm{e}\mathrm{f}\mathrm{f}}=U-J$, $J=0.5$ eV). This value of $U_{\mathrm{e}\mathrm{f}\mathrm{f}}$ has been suggested by the recent theoretical studies [2, 4, 5], as the most optimal for UTe₂. Nevertheless, the Coulomb repulsion $U_{\mathrm{e}\mathrm{f}\mathrm{f}}$ has been calculated for both orthorhombic ($Immm$, No. 71) and tetragonal ($I4/mmm$, No. 139) structures of UTe₂ along with its change upon external pressure. These computations were performed within the density functional perturbation theory (DFPT) implemented in the QUANTUM ESPRESSO (QE) package [45, 46]. Here, the PBE approximation and PAW pseudopotentials adopted from the QE database were employed. Results of these calculations are presented and discussed in Fig. A1 (see Appendix).

The Brillouin zones of the orthorhombic ($Immm$, No. 71) and tetragonal ($I4/mmm$, No. 139) structures of UTe₂ were sampled with the 126 and 196 irreducible k-points, respectively. Calculations were carried out as a function of external pressure ranging from 0 to 6 GPa. The structures were fully optimized (lattice parameters and atomic positions) at each pressure with convergence criteria for total energy and forces acting on each atom less than ${10}^{-7}$ eV and ${10}^{-5}$ eV/Å, respectively. The dynamical properties of the orthorhombic lattice at ambient pressure and tetragonal lattice at elevated pressure were determined within the approximation of harmonic phonons [47, 48]. The calculated density of phonon states was used to obtain the lattice contribution to the heat capacity [49]. To study the pressure-induced structural phase transition in UTe₂, we have determined Gibbs free energies of the orthorhombic and tetragonal structures at the given pressure [49]. Contributions to the Gibbs free energy ($G$) from a rigid lattice at 0 K ($E_0$), vibrational term ($F_v$) computed within the quasi-harmonic Debye−Grüneisen formalism [50, 51], the electronic term ($F_e$) obtained under assumption of the temperature independent density of electron states [52, 53], and the term $F_{pV}$ arising from the applied pressure were computed. Details of the applied methodology can be found elsewhere [54–56]. Peak intensities of the nonresonant Raman spectra were calculated according to the expression [57]: $I\sim |{\mathit{e}}_i\mathit{R}{\mathit{e}}_s{|}^2{\omega }^{-1}(n+1)$, where $(n+1)$ is the population factor for Stokes scattering with $n=[\exp (\hslash \omega /(k_{B}T))-1{]}^{-1}$ denoting the Bose-Einstein thermal factor, ${\mathit{e}}_i$(${\mathit{e}}_s$) is the polarization of incident (scattered) radiation, $\mathit{R}$ is the Raman susceptibility tensor. The components of $\mathit{R}$ tensor were determined from derivatives of the electric polarizability tensor over the atomic displacements [58, 59]. Electric polarizabilities were calculated within the linear-response method implemented [60] in the VASP code.

3 Results and discussion

3.1 Crystal and electronic structures of O-phase

The body-centered orthorhombic crystal structure with the space group $Immm$ (No. 71) is schematically shown in the inset of Fig.1. The U-atoms are located at ($4i$) Wyckoff positions with fractional coordinates (0, 0, $z_{\mathrm{U}}$). There are two symmetry non-equivalent Te-atoms residing at ($4j$) and ($4h$) sites with fractional coordinates of (0.5, 0, $z_{\mathrm{T}{\mathrm{e}}_1}$) and (0, $y_{\mathrm{T}{\mathrm{e}}_2}$, 0.5), respectively. The parameters of the orthorhombic UTe₂ structure determined from the present room-temperature single-crystal X-ray diffraction experiments are collected in Tab.1 and compared to the optimized structure resulting from of our ab initio calculations carried out within the DFT + U + SO method. Both experimental and theoretical structural parameters remain consistent with the values reported in the previous experimental studies [61–63], our experimental XRD and EDX details are summarized in Table A1. There is also an overall good agreement between the calculated and experimental values of the bulk modulus, which amount to 48 GPa and 46 GPa (no pressure-transmitting medium) [35] at RT, respectively. The atomic and orbital projected electronic densities of states, which are displayed in Fig.1(a), clearly indicate a dominant contribution from the U-$5f$ states with $J=5/2$ manifold in close vicinity of the Fermi energy (E_F) as well as small semiconducting gap of about 20 meV. The $J=7/2$ manifold is shifted to higher energies by about 1 eV. More pronounced contribution from the U-$6d$ and Te-$5f$ electron states is observed respectively above and below $\sim 2$ eV of E_F, where these states remain hybridized. Our calculations confirm experimental observations about metallicity of UTe₂ [8, 64], i.e., no energy gap is present when both spin−orbit and the Coulomb-type strong electron correlations are taken into account. This also remains in accord with the recent theoretical studies of UTe₂ suggesting importance of the inclusion of the on-site Coulomb repulsion in predicting metallic character of this compound, without which UTe₂ would become small-gap semiconductor [3–5].

Fig.1 (a) Atomic and orbital-projected electron densities of states and (b) band structure of nonmagnetic orthorhombic UTe2 calculated within the DFT + U + SO scheme. Inset: The $2\times 2\times 1$ supercell of the orthorhombic UTe2. The U($4i$), Te1($4j$), and Te2($4h$) atoms are represented by blue, dark gray, and light gray balls, respectively.

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Tab.1 Lattice parameters and fractional coordinates of U, Te1, and Te2 atoms of orthorhombic UTe2 measured at ambient pressure and T = 300 K compared to the optimized crystal structure within the DFT+U+SO scheme ($U_{\mathrm{e}\mathrm{f}\mathrm{f}}=$ 1 eV).

Lattice parameters
	a (Å)	b (Å)	c (Å)
Exp.	4.161(2)	6.133(2)	13.971(3)
Cal.	4.1630	6.2016	13.9844
Fractional coordinates
	$z_{\mathrm{U}}$	$z_{\mathrm{T}{\mathrm{e}}_1}$	$y_{\mathrm{T}{\mathrm{e}}_2}$
Exp.	0.13517(1)	0.29781(3)	0.25087(15)
Cal.	0.13369	0.29837	0.25275

The calculated band structure of the O-phase, Fig.1(b), allowed us to determine the effective electron band mass tensor $m^{\ast }$ according to the following relationship:

(1)${\left({\displaystyle \frac{1}{m^{\ast }}}\right)}_{ij}={\displaystyle \frac{1}{{\hslash }^2}}{\displaystyle \frac{{\mathrm{\partial }}^2{E}_n(\mathit{k})}{\mathrm{\partial }{k}_i\mathrm{\partial }{k}_j}},i,j=x,y,z,$

where $x,y,z$ denote directions in the $\mathit{k}$-space, while $E_n(\mathit{k})$ stands for the dispersion relation of the $n$-th electronic band. The derivatives were evaluated numerically at the bands extrema using the finite difference method. In the O-phase, the computed effective electron band mass at the $\mathrm{\Gamma }$-point amounts to ${⟨m^{\ast }⟩}_{\mathrm{\Gamma }}=7.1m_0$, where $⟨\cdots ⟩$ is the $\frac{1}{3}$ of the band effective mass tensor trace and ${\mathrm{m}}_0$ is the free electron mass.

3.2 Phonon dynamics in O-phase

The dispersion relations of phonons and densities of phonon states in orthorhombic UTe₂ at ambient pressure that are depicted in Fig.2 clearly show that phonons gather into two bands separated by a small gap ($\sim 5$ cm ⁻¹). The lowest frequency band, which extends up to about 80 cm⁻¹ corresponds to the acoustic and transverse optical (TO) phonon modes. Here, the acoustic phonons (mainly the longitudinal acoustic modes) mix with the low-lying TO phonons. In this part of the phonon spectrum contribution from the U-sublattice vibrations is slightly larger than that from the Te-sublattice. The higher-frequency phonon band, covering the range of 86−185 cm⁻¹, is also constituted by shared vibrations of the U and Te sublattices with dominating contributions from Te atoms. The top of this phonon band involves pure Te vibrations.

Fig.2 Phonon dispersion relations and phonon densities of states $G(\omega )$ in the low-pressure orthorhombic structure of UTe2. The high-symmetry points are labeled according to the Brillouin zone of the $Immm$ space group.

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3.3 Zone-center phonon modes and the Raman spectra of O-phase

The primitive unit cell of the orthorhombic UTe₂ with space group $Immm$ contains 6 atoms, i.e., 2 formula units, and this leads to 18 phonon modes in total. The site symmetries of the U, Te₁, and Te₂ atoms in $Immm$ ($D_{2h}^{25}$) space group are $C_{2v}$. Thus, the phonons at the Brillouin zone center ($\mathrm{\Gamma }$-point) can be classified by a factor group analysis as follows:

$\begin{array}{rl}\mathrm{U}(4i)& \mathrm{\Gamma }=A_g\oplus B_{2g}\oplus B_{3g}\oplus B_{1u}\oplus B_{2u}\oplus B_{3u},\\ {\mathrm{T}\mathrm{e}}_1(4j)& \mathrm{\Gamma }=A_g\oplus B_{2g}\oplus B_{3g}\oplus B_{1u}\oplus B_{2u}\oplus B_{3u},\\ {\mathrm{T}\mathrm{e}}_2(4h)& \mathrm{\Gamma }=A_g\oplus B_{1g}\oplus B_{3g}\oplus B_{1u}\oplus B_{2u}\oplus B_{3u},\end{array}$

where $A_g$, $B_{1g}$, $B_{2g}$, and $B_{3g}$ phonons are Raman active, whereas $B_{1u}$, $B_{2u}$, and $B_{3u}$ are infrared (IR) active. There are no silent (optically inactive) modes in UTe₂. After subtracting three acoustic modes ($B_{1u}\oplus B_{2u}\oplus B_{3u}$) one obtains irreducible representations of the optically active vibrational modes in UTe₂: ${\mathrm{\Gamma }}_{\mathrm{o}\mathrm{p}\mathrm{t}}=3A_g\oplus B_{1g}\oplus 2B_{2g}\oplus 3B_{3g}\oplus 2B_{1u}\oplus 2B_{2u}\oplus 2B_{3u}$, where the IR-active $B_{1u}$, $B_{2u}$, and $B_{3u}$ phonons are related to the dipole moment oscillations along $c$, $b$, and $a$ axes, respectively. The modes with $A_g$ symmetry correspond to vibrations of U and Te₁ atoms along $c$-axis and vibrations of Te₂ atoms along $b$-axis. Displacements of Te₂ atoms along $a$-axis give rise to the $B_{1g}$ modes, while the $B_{2g}$ phonons come from displacements of U and Te₁ atoms along the $a$-axis. The $B_{3g}$ modes are due to movements of U and Te₁ atoms along $b$-axis and Te₂ atoms along the $c$-axis.

According to the polarization selection rules [65] the modes with $A_g$ symmetry can be observed in the polarized backscattering Raman spectra when the polarizations of incident (${\mathit{e}}_i$) and scattered (${\mathit{e}}_s$) radiations are parallel (${\mathit{e}}_i\parallel {\mathit{e}}_s$). Hence, the Raman spectra measured at $x(yy)\overline{x}$, $x(zz)\overline{x}$, $y(xx)\overline{y}$, $y(zz)\overline{y}$, $z(xx)\overline{z}$, and $z(yy)\overline{z}$ reveal peaks arising from the $A_g$ phonons. At the crossed polarization configuration (${\mathit{e}}_i\perp {\mathit{e}}_s$), the scattering geometries of $z(xy)\overline{z}$, $y(xz)\overline{y}$, and $x(yz)\overline{x}$ enable us to detect the phonons with $B_{1g}$, $B_{2g}$, and $B_{3g}$ symmetries, respectively. In order to calculate intensities of the Raman active modes in orthorhombic UTe₂ one needs to consider the following polarizability Raman tensors:

$\begin{array}{rl}& {\mathit{R}}_{A_g}=\left(\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right),{\mathit{R}}_{B_{1g}}=\left(\begin{array}{ccc}0& d& 0\\ d& 0& 0\\ 0& 0& 0\end{array}\right),\\ & {\mathit{R}}_{B_{2g}}=\left(\begin{array}{ccc}0& 0& e\\ 0& 0& 0\\ e& 0& 0\end{array}\right),{\mathit{R}}_{B_{3g}}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& f\\ 0& f& 0\end{array}\right).\end{array}$

The simulated backscattering Raman spectra at $y(zz)\overline{y}$, $y(xz)\overline{y}$, and $x(yz)\overline{x}$ configurations are shown in Fig.3(a). The spectrum at $z(xy)\overline{z}$ geometry which contains a single Raman peak arising from the $B_{1g}$ phonon is not shown. Among the Raman peaks representing the $A_g$ modes, the peak appearing at $\sim 140$ cm⁻¹ is the most intense one. Remaining $A_g$ peaks have almost equal intensities. The intensity of the higher-energy $B_{2g}$ phonon is nearly twice of that calculated for the lower-energy $B_{2g}$ mode. Finally, the simulated polarized spectrum revealing modes with the $B_{3g}$ symmetry shows only a two-peak structure with the higher and lower intensities of the lower- and higher-energy $B_{3g}$ phonons, respectively. The middle-energy $B_{3g}$ mode at 98 cm⁻¹ is hardly visible due to its extremely low intensity.

Fig.3 (a) Polarized Raman spectra of UTe2 crystal calculated at $y(zz)\overline{y}$ ($A_g$ modes), $y(xz)\overline{y}$ ($B_{2g}$ modes), and $x(yz)\overline{x}$ ($B_{3g}$ modes) backscattering geometries. Spectra are simulated at 300 K and with laser excitation wavelength of 514 nm. Peaks are represented by Lorentzians with artificial FWHMs of 2 cm−1. (b) Room-temperature experimental Raman spectrum of the UTe2 single crystal measured in a backscattering geometry with the laser excitation $\lambda =514$ nm (solid symbols). Solid line connecting experimental points corresponds to the multi-Lorentzian fit. The shaded area below experimental spectrum stands for the simulated unpolarized Raman scattering spectrum of the UTe2 single crystal from the ac-plane ($\lambda =514$ nm, T = 300 K).

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The experimental Raman spectrum measured in backscattering geometry at RT and averaged over different excitation polarization angles to show majority of the Raman-active phonons in UTe₂ is displayed in Fig.3(b). It is compared to the backscattering Raman spectrum form the ac-plane simulated for the unpolarized beam. A close correspondence between experimental and calculated spectra allowed us to correlate positions of the measured peaks with the calculated frequencies of individual Raman phonons and assign them appropriate symmetry associated with particular irreducible representation. Results are also collected in Tab.2. In general, the experimental spectrum is composed of six intense peaks representing three $A_g$, two $B_{2g}$, and only one of three $B_{3g}$ modes. The remaining two $B_{3g}$ Raman modes have significantly smaller intensities. In order to observe the $B_{1g}$ phonon mode one should set up different scattering configuration then that applied in the present experiment, c.f. polarization selection rules discussed above.

Tab.2 Experimental and calculated frequencies of the Raman and IR-active phonon modes in orthorhombic UTe2. Frequencies are expressed in cm−1.

Mode symmetry	Raman Exp.	Active Calc.	IR-active Calc.
$B_{2g}$	55.3	55.2
$B_{3g}$	72.4	69.7
$A_g$	83.2	83.2
$B_{2u}$			87.7
$B_{3u}$			97.3
$B_{3g}$	92.2	98.0
$B_{1g}$		101.6
$B_{2u}$			110.2
$B_{2g}$	112.2	112.8
$B_{1u}$			113.0
$B_{3u}$			120.9
$B_{3g}$	122.1	123.9
$B_{1u}$			138.4
$A_g$	140.5	139.4
$A_g$	177.7	178.3

3.4 Heat capacity of O-phase

The electronic specific heat of UTe₂ below 12 K, which is depicted in the inset of Fig.4 was obtained by subtracting the DFT calculated lattice (phonon) contribution $C_{\mathrm{p}\mathrm{h}}$ from the measured specific heat. It reveals the emergence of bulk superconductivity below $T_{\mathrm{c}}=1.92$ K, a clear $\lambda$-anomaly in the vicinity of $T_{\mathrm{c}}$, and a large linear electronic contribution to the low-temperature heat capacity $\gamma =104\mathrm{m}\mathrm{J}\cdot {\mathrm{K}}^{-2}\cdot {\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$. We note that both $T_{\mathrm{c}}$ and $\gamma$ determined in the present investigations closely correspond to those reported in previous studies [1, 66]. A bare linear electronic contribution ${\gamma }_b=20.4\mathrm{m}\mathrm{J}\cdot {\mathrm{K}}^{-2}\cdot {\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$, which is estimated from our DFT calculated density of electron states at the Fermi energy $N(E_{\mathrm{F}})=$ 8.64 states/eV per primitive unit cell of UTe₂ and assuming a spherical Fermi surface ${\gamma }_b=\frac{1}{3}{\pi }^2{k}_B^{2}N(E_{\mathrm{F}})$ with $k_B$ denoting the Boltzmann constant, results in rather small conduction-band effective mass enhancement of 5.1 ($m^{\ast }/m_b\sim \gamma /{\gamma }_b=1+\lambda$). Such underestimation is mainly due to severe limitation and deficiency of the applied spherical Fermi surface model to describe adequately the Fermi surface topology in UTe₂ [3–6]. Nevertheless, the ratio $\mathrm{\Delta }{C}_{\mathrm{e}\mathrm{l}}=(C-C_{\mathrm{p}\mathrm{h}})/(\gamma T_{\mathrm{c}})=2.1$, where $\mathrm{\Delta }{C}_{\mathrm{e}\mathrm{l}}$ is the height of superconducting jump, exceeds the weak-coupling BCS value of $1.43$ [67, 68], and hence indicates the strong-coupling superconductivity consistent with the pairing of heavy-electrons in UTe₂. The mass enhancement sets only at low temperatures and its inclusion at $T>30$ K significantly overestimates the experimental specific heat. Therefore, substantially reduced electronic contribution to the specific heat of UTe₂ maintains above 30 K and the experimental specific heat can be very well described even by ${\gamma }_b+C_{\mathrm{p}\mathrm{h}}/T$ over the temperature range of 30−400 K. Certainly, the sole lattice term remains insufficient to adequately represent the specific heat of UTe₂ at elevated temperatures (c.f. solid and dashed curves in Fig.4).

Fig.4 Reduced specific heat $C/T$ of UTe2. Experimental data are indicated by solid circles. Solid and dashed lines represent calculated lattice contribution to the heat capacity $C_{\mathrm{p}\mathrm{h}}/T$ and ${\gamma }_b+C_{\mathrm{p}\mathrm{h}}/T$, respectively. Inset: Experimental low-temperature specific heat corrected for the calculated lattice (phonon) contribution $(C-C_{\mathrm{p}\mathrm{h}})/T$.

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3.5 Phase transition and characterization of T-phase

The recent theoretical [36] and experimental [18, 35] studies of the structural phase transition in UTe₂ indicated that its low-pressure body-centered orthorhombic structure of $Immm$ symmetry ($Z=4$) transforms into the body-centered tetragonal structure with $I4/mmm$ space group ($Z=2$) at elevated pressures. The transition pressure, $p_{tr}$, predicted by ab initio calculations using the DFT + U method with $U=7$ eV [36] amounts to 9 GPa, whereas significantly lower values are determined from experiments by Huston et al. [35] ($p_{tr}\sim 5$ GPa) and Honda et al. [18] ($p_{tr}=3.5-4$ GPa). Also, the experimental studies [18] show some difference in $p_{tr}$ between data obtained for single-crystalline and powder samples. The transition occurs just above 7 GPa in powder samples due to a coexistence of the low- and high-pressure phases between 5 and 7 GPa, while such a spread in $p_{tr}$ is not observed for single crystals. Additionally, the pressure-transmitting medium plays some role in determining the transition pressure in UTe₂, as evidenced by experimental results of Huston et al. [35]. It was found that transformation takes place between 5 and 8 GPa in the absence of a pressure-transmitting medium, whereas at 5 GPa in the presence of Ne as the pressure-transmitting medium. At the phase transition a relative decrease in unit cell volume ($\mathrm{\Delta }V/V_0$, where $V_0$ denotes the volume at ambient pressure) of 10%−11% is reported by experiments [18, 35], while a slightly higher value (12%) is given by the computational studies [36]. Indeed, a relatively low value of the UTe₂ bulk modulus, determined either experimentally (46 GPa [35], 59 GPa [18]) or theoretically (48 GPa), indicates that this material is rather soft and suggests that it may easily undergo structural phase transition induced by external pressure.

To explore the effect of both strong electron correlations of Coulomb-type in the $f$-electron shell and spin-orbit coupling on the pressure-mediated orthorhombic-to-tetragonal phase transformation in UTe₂, we have undertaken appropriate theoretical reinvestigation within the DFT + U + SO approach. Despite experimentally suggested transition from the low-pressure $Immm$ structure to the high-pressure structure of the $I4/mmm$ symmetry with Te atoms occupying the $4e(0,0,z)$ Wyckoff positions, we have additionally considered two modifications of the $I4/mmm$ structure: (i) with Te atoms residing in $4c(0,0.5,0)$ sites and (ii) Te atoms located at $4d(0,0.5,0.25)$ positions. Results of our calculations show that the high-pressure tetragonal structures with Te atoms residing either at $4c$ or $4d$ sites are less energetically favorable in the whole investigated pressure range as compared to the structure with Te atoms located at $4e$ positions. Therefore, we do not observe pressure-induced $Immm\to I4/mmm$ transformation when Te atoms occupy $4c$ or $4d$ sites. This, of course, confirms the recent experimental observations [18, 35].

The calculated pressure of the $Immm\to I4/mmm$ structural phase transformation amounts to 3.8 GPa at RT and 3.5 GPa at 0 K, and it closely corresponds to the experimentally determined $p_{tr}$. On one hand, the $p_{tr}$ obtained in the present theoretical research remains significantly lower than that predicted by Hu et al. [36] from their DFT+U computations using quite large $U=$ 7 eV, but on the other one finds a good correlation between results of both ab initio and experimental studies when the strong electron correlations between the U-$5f$ states are described by Hubbard potential of about 1 eV. Indeed, we do not observe strong dependence of $p_{tr}$ on the applied $U_{\mathrm{e}\mathrm{f}\mathrm{f}}$. The transition pressure in UTe₂ remains almost constant for the $U_{\mathrm{e}\mathrm{f}\mathrm{f}}$ ranging between 0 and 3 eV, as established by our theoretical investigations. Moreover, there is also a rather weak dependence of $p_{tr}$ on temperature below 500 K, as shown in the inset of Fig.5. The pressure dependence of the UTe₂ crystal molar volume ($V_m$) below and above $p_{tr}$, which is illustrated in Fig.5, gives the relative drop in the crystal volume at $p_{tr}=3.8$ GPa of about 10%, which matches experimental observations (10%−11%) [18, 35].

Fig.5 Pressure dependence of the UTe2 molar volume ($V_m$). Experimental data (solid symbols) for single crystals are adopted from Ref. [18]. Theoretical data are denoted by dashed lines. Inset: Temperature dependence of the calculated transition pressure ($p_{tr}$) in UTe2.

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The optimized $I4/mmm$ phase at 4 GPa is characterized by the lattice parameters $a_{\mathrm{c}\mathrm{a}\mathrm{l}}=3.9666$ Å, $c_{\mathrm{c}\mathrm{a}\mathrm{l}}=10.1041$ Å, the atomic fractional coordinate of Te($4e$) atoms $z_{\mathrm{T}\mathrm{e}}=0.34456$, and U atoms occupying $(2a)$ Wyckoff positions. We note that the structural parameters of the high-pressure tetragonal $I4/mmm$ phase resulting from our calculations agree quite well with the experimental data reported by Huston et al. [35] ($a=3.967$ Å, $c=10.123$ Å) and Honda et al. [18] ($a=3.98$ Å, $c=9.80$ Å). Furthermore, the resulting room temperature bulk modulus of the $I4/mmm$ structure, which equals to 79.2 GPa at 4 GPa corresponds almost exactly to that provided by the experimental studies performed at RT with no applied pressure-transmitting medium, i.e., 78.9 GPa [35]. The structure of UTe₂ formed above $p_{tr}$ remains denser than its low-pressure counterpart by about 9.4%, which additionally ensures its greater stability at elevated pressures.

Alike in orthorhombic UTe₂, the density of electrons states in its tetragonal phase at 4 GPa, see Fig.6(a), is dominated by the U-$5f$ states, but in contrast to the O-phase no semiconducting gap is observed. Also, the T-phase shows much more significant hybridization of the U-$6d$ and Te-$5p$ electrons below and above $\sim 2$ eV with respect to E_F as compared to the O-phase. This hybridization cover much extended energy range in the T-phase than in the O-phase. The differences in the electronic structures of the O- and T-phase arise mainly from their somehow distinct crystal structures having respectively 8-fold and 10-fold coordinated uranium atoms as well as from effects of exerted pressure on the tetragonal phase. Again, using the calculated band structure, Fig.6(b) and Eq. (1) one obtains the ${⟨m^{\ast }⟩}_{\mathrm{\Gamma }}=2.3m_0$ for the T-phase at 4 GPa, which remains about 3 times smaller in comparison with the ${⟨m^{\ast }⟩}_{\mathrm{\Gamma }}$ of the O-phase at 0 GPa.

Fig.6 (a) Atomic and orbital-projected electron densities of states and (b) band structure of nonmagnetic tetragonal UTe2 calculated within the DFT + U + SO scheme at 4 GPa. Inset: The $2\times 2\times 1$ supercell of the tetragonal UTe2. The U($2a$) and Te($4e$) atoms are represented by blue and gray balls, respectively.

Full size|PPT slide

It is worth exploring in more detail the behavior of particular contribution to the Gibbs free energies of the orthorhombic and tetragonal phases. The major contribution to the Gibbs free energies of both phases comes from the energies of their rigid lattices ($E_0$). The $E_0(Immm)<E_0(I4/mmm)$ holds both at 0 and 4 GPa, although with difference in energies $\mathrm{\Delta }{E}_0=E_0(Immm)-E_0(I4/mmm)$ decreasing with increased hydrostatic pressure. The phonon term $F_v$ arising from lattice vibrations is significantly smaller than the $E_0$ term and does not exceed 65 meV/f.u. even at 4 GPa. Both phases have almost the same $F_v$ at 0 GPa, whereas at 4 GPa $F_v(I4/mmm)>F_v(Immm)$ by only $\sim 6$ meV/f.u. Such small contribution from $F_v$ does not ensure the T-phase to have Gibbs free energy lower than the Gibbs energy of the O-phase and this applies in the whole investigated pressure range. Because the electronic contributions $F_e$ are negligible, the only term which allows the T-phase to be lower in energy than the O-phase is the pressure term $F_{pV}$. Indeed, the $F_{pV}$ contribution in tetragonal phase at 4 GPa is smaller by 215 meV as compared to $F_{pV}$ in the orthorhombic phase, and hence $G(I4/mmm$) becomes lower than $G(Immm)$ by 71 meV/f.u. We also note that above phase transition the $Immm$ structure becomes dynamically unstable, as indicated by the calculated dispersion of phonons depicted in Fig.7 that reveal imaginary frequencies (soft-phonon mode) in the vicinity of the $S$ high-symmetry point. The phonon band structures of the O- and T-phases are substantially different between one another. This difference is related to distinct crystal structures, different number of non-equivalent atoms in the respective primitive unit cells and external pressure which stabilizes the tetragonal phase of UTe₂.

Fig.7 Phonon dispersion relations and phonon densities of states $G(\omega )$ in (a) orthorhombic and (b) tetragonal structures of UTe2 at 4 GPa. Imaginary frequencies are denoted by negative values ${\omega }^2(\mathit{k},j)<0$. The high-symmetry points are labeled according to the respective Brillouin zones of the $Immm$ and $I4/mmm$ space groups.

Full size|PPT slide

Finally, we would like to pay attention to the uranium valence in the O- and T-phases. Although it is generally accepted that UTe₂ is a compound with intermediate valence, i.e., exhibiting a non-integer U-$5f$ occupancy [69, 70] which ranges between $5f^3$ (U³⁺) and $5f^2$ (U⁴⁺) at elevated pressures below O$\to$T transition, the $5f$ electron count is still being debated [2, 5, 33, 64]. Our electronic structure calculations result in the $5f$ electrons count of 2.90 and 2.75 for the O-phase at ambient pressure and T-phase at 4 GPa, respectively. This confirms results of the recent X-ray absorption and magnetic circular dichroism experiments [70], reporting the $5f$ electron count to be in-between 2.6 and 2.8 at ambient pressure and approaching 3 above 4 GPa in the tetragonal phase [18]. An electronic configuration close to U³⁺, which deviates slightly from a value characteristic for free uranium ion (3⁺), indicates moderate delocalization and hybridization effects of the U-$5f$ states in the ambient pressure O-phase and the high-pressure T-phase of UTe₂.

4 Summary and conclusions

This joint experimental and theoretical research on the phonon dynamics, heat capacity, and pressure-mediated structural phase transition in UTe₂ confirms its bulk superconductivity below critical temperature of 1.92 K, the heavy-electron-mass state at very low temperatures, and non-negligible residual electronic term contributing to its specific heat even far above 30 K. Detailed comparative analysis of the measured and simulated Raman spectra, that are probably the first of that kind, provides information on the atomic vibrations and symmetries of phonon modes at the Brillouin zone center, which could be useful for further studies on the off-stoichiometric as well as doped UTe₂. It is shown that vibrational properties, phonon-dependent thermodynamical functions, and pressure-induced structural transformation in UTe₂, that are described theoretically within the DFT + U + SO method, are predicted accurately and follow results of experiments, provided the effects of spin−orbit interactions are taken into account and the U-$5f$ electrons are considered as partly delocalized with strong local Coulomb correlations not exceeding 1−2 eV.

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Declarations

The authors declare no competing interests and no conflicts.

Data availability statement

All data that support the findings of this study are included within the article.

Acknowledgements

The Czech Science Foundation (GACR) project No. 22-22322S, the e-INFRA CZ (ID:90254) and QM4ST (CZ.02.01.01/00/22_008/0004572) projects supported by the Ministry of Education, Youth and Sports of the Czech Republic are acknowledged. The Interdisciplinary Center for Mathematical and Computational Modeling (ICM), Warsaw University, Poland are acknowledged for providing the computer facilities. Crystal growth, X-ray characterization and heat capacity measurements were performed in MGML (mgml.eu), which is supported within the program of Czech Research Infrastructures (Project No. LM2023065).

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