Superconducting properties of pseudobinary telluride Chevrel phase Mo₄Re₂Te₈

The temperature dependence of resistivity, $\rho(T)$, for Mo₄Re₂Te₈ under a zero magnetic field is shown in figure 2(a). The onset of the sharp drop in resistivity is recorded at a temperature, $T_{C,\mathrm {onset}}$ = 3.54(2) K, with a transition width of $\Delta T$ = 0.3 K (figure 2(b)). The observed T_C matches well with the reported value [23, 24]. The increasing behavior of low temperature (normal -state) $\rho (T)$ demonstrates the metallic nature of Mo₄Re₂Te₈. Hence, the $\rho (T)$ temperature dependence is analyzed using the equation

$$\begin{equation} \rho(T) = \rho_0 + C\left(\frac{T}{\theta_\mathrm R}\right)^n \int_{0}^{\theta_R/T}\frac{x^n}{(e^x-1)(1-e^{-x})}dx\\ \end{equation} \tag{ 1 }$$

where the second term is the Bloch–Grüneisen (BG) expression accounting for the electron–phonon scattering [35]. ρ₀ is the residual resistivity due to defect scattering, $\theta_\mathrm R$ is the Debye temperature from the resistivity measurement, C is a material-dependent property, and n depends on the nature of the interaction. Below 100 K, the best fitting of $\rho (T)$ is for n = 3, providing $\theta_\mathrm R$ = 143(2) K, C = 0.51(1) m$\Omega{\,}$cm and ρ₀ = 2.95(1) m$\Omega{\,}$cm. Moreover, the resistivity trend in the high-temperature region for Mo₄Re₂Te₈consists of a negative curvature ($d\rho/dT$) with the nearly saturated resistivity, indicating an additional contribution in the system with a metallic nature. This behavior is reminiscent of that of other CPs, namely skutterudite and clathrate compounds, consisting of similar types of cluster structures [3, 36–39]. The same trend is also observed in the La₇X₃ (X = Rh, Ir) family and YCo₂, where the electronic scattering related to the excitations with a broad spectrum of energy is stated as a possible reason [40, 41]. At this point, the exact reason behind the observed trend in resistivity for our sample Mo₄Re₂Te₈ is unknown.

Zoom In Zoom Out Reset image size

Furthermore, the carrier density, n, is estimated from the Hall measurement. The transverse resistivity, ρ_xy , measured at 20 K under field variation, is shown in figure 2(c). The observed positive slope indicates the dominant hole carriers and provides the Hall resistivity, $R_\mathrm H = 5.2(1) \times 10^{-10}$ m⁻³ C. Considering only the single-band contribution, the estimated number density n is $1.21(3) \times 10^{28}$ m⁻³, which is again in good agreement with the known values for CPs and their selenide substitutes [42, 43].

The magnetic moment under temperature variation of Mo₄Re₂Te₈ was measured at 1 mT in both zero-field-cooled warming (ZFCW) and field-cooled cooling (FCC) mode, as shown in figure 3(a). The onset of the diamagnetic signal of the compound is noted at $T_{C, \mathrm{onset}}$ = 3.26(3) K, where the deviation of the FCC curve from the ZFCW depicts the flux pinning in the sample. The inset of figure 3(a) shows the magnetization loop under a magnetic field variation of ±4 T at 1.8 K for Mo₄Re₂Te₈. The full magnetization loop confirms the type-II superconductivity with bulk pinning in the sample, and represents an irreversible nature of magnetization, $H_\mathrm{irr}$ = 1.54 T, where further increase in the applied magnetic field de-pins the vortices.

Zoom In Zoom Out Reset image size

The lower critical field value, $H_{C1}$(0), is extracted from the magnetization dependence on the magnetic field measured at different temperatures up to T_C , as shown in the inset of figure 3(b). The linear deviation from the Meissner effect (entering the vortex region) is considered as the $H_{C1}$ for the respective isotherm. The temperature dependence of $H_{C1}$ for Mo₄Re₂Te₈ is shown in figure 3(b) and fitted using the Ginzburg–Landau (GL) equation, given as

$$\begin{equation} H_{C1}(T) = H_{C1}(0)\left[1-\left(\frac{T}{T_{C}}\right)^{2}\right]. \end{equation} \tag{ 2 }$$

The estimated value of $H_{C1}(0)$ from the fitting is 0.49(1) mT. Furthermore, the value of the upper critical field, $H_{C2}(0)$, is computed via variation of the transition temperature under - applied magnetic fields. The applied increasing magnetic field shifts the superconducting transition temperature to a lower value as observed in the resistivity, magnetization and specific heat measurements. The inset of figure 3(c) represents the same via the resistivity measurement, where T_C is recorded at the 90% drop in normal-state resistivity value. The variation of the upper critical field value, $H_{C2}$, with the reduced temperature, $t = T/T_{C}$, is fitted with the Ginzburg–Landau equation,

$$\begin{equation} H_{C2}(T) = H_{C2}(0)\left[\frac{(1-t^2)}{(1+t^2)}\right]. \end{equation} \tag{ 3 }$$

The well-fitted data are shown in figure 3(c), providing an upper critical field of $H_{C2}(0)$ = 5.08(7) T, 5.62(6) T and 5.78(4) T from the magnetization, specific heat and resistivity data, respectively.

The effect of an applied external magnetic field in superconductors is described by two mechanisms: (a) orbital decoupling and (b) the Pauli limiting effect. The orbital limiting field is where the increased kinetic energy of one electron breaks the Cooper pair, given by the Wartherm–Helfand–Hohenberg expression [44, 45],

$$\begin{equation} H_{C2}^{\,\mathrm{orbital}}(0) = -\alpha T_{C}\left.\frac{dH_{C2}(T)}{dT}\right|_{T = T_{C}} \end{equation} \tag{ 4 }$$

Here, α is considered to be 0.69 for the dirty limit superconductor. The initial slope $\frac{-dH_{C2}(T)}{dT}$ in the vicinity of T_C is 1.58(1) T for Mo₄Re₂Te₈, which evaluates the orbital limiting field, $H_{C2}^\mathrm{orbital}(0)$ = 3.55(4) T. From the Bardeen–Cooper–Schrieffer (BCS) theory of superconductors, another effect Pauli limiting field is represented as $H_{C2}^P$(0) = 1.86 T_C [46, 47], with T_C = 3.26(3) K, and $H_{C2}^P$(0) becomes 6.00(5) T. Moreover, the relative strength of the two magnetic effects, the orbital and Pauli limiting field, is measured by the Maki parameter [48], $\alpha_{M} = \sqrt{2}H_{C2}^\mathrm{orb}(0)/H_{C2}^{P}(0)$ = 0.84(2). The close value of α_M to one demonstrates the non-negligible effect of the Pauli limiting field in the Cooper pair breaking. Along with this, the proximity of the upper critical field value ($H_{C2,\rho}(0) = 5.78(4)$ T) to the Pauli limiting field is also compelling and similar to other Re-based superconductors, which shows unconventional superconductivity [25, 49, 50].

The superconducting characteristic length parameters were calculated from the upper and lower critical field values. The relation between $H_{C2}(0)$ and the Ginzburg–Landau coherence length, $\xi_\mathrm{GL}(0)$ [51], $H_{C2}(0) = \frac{\Phi_0}{2\pi\xi_\mathrm{GL}(0)^2}$, where $\Phi_0 = 2.07 \times 10^{-15}$ Tm² is the quantum flux, gives $\xi_\mathrm{GL}(0) = 80(1)$ Å. The lower critical field, $H_{C1}(0)$, and coherence length, $\xi_\mathrm{GL}(0)$, are related to the penetration depth, $\lambda_\mathrm{GL}(0)$, as follows [52]:

$$\begin{equation} H_{C1}(0) = \frac{\Phi_{0}}{4\pi\lambda_\mathrm{GL}^2(0)}\left(\mathrm{ln}\frac{\lambda_\mathrm{GL}(0)}{\xi_\mathrm{GL}(0)}+0.12\right) \end{equation} \tag{ 5 }$$

providing $\lambda_\mathrm{GL}(0)$ = 1325(40) nm. Moreover, the value of the Ginzburg–Landau parameter, $\kappa_\mathrm{GL} = \frac{\lambda_\mathrm{GL}(0)}{\xi_\mathrm{GL}(0)}$, is calculated to be 165(7), indicating a strong type-II superconductivity in Mo₄Re₂Te₈, consistent with the previously studied CPs. The thermodynamic critical field is also evaluated using the relation $H_C = \sqrt{\frac{H_{C1}(0)H_{C2}(0)}{{\ln} \kappa_\mathrm{GL}}}$ [52] and obtains the value H_C = 22(1) mT for Mo₄Re₂Te₈.

The bulk superconductivity is further probed by specific heat versus temperature measurement at zero applied magnetic field for Mo₄Re₂Te₈. Figure 4(a) shows the jump in specific heat data at $T_{C,\mathrm{mid}}$ = 3.23(6) K estimated from the midpoint. Above T_C , the normal-state region of specific heat is fitted using the Debye relation, $C/T = \gamma_n + \beta_3 T^2$, where γ_n is the Sommerfeld coefficient and β₃ is the lattice constant. Fitting of the data provides the parameters as γ_n = 24.7(8) mJ mol⁻¹ K⁻² and β₃ = 5.95(4) mJ mol⁻¹ K⁻⁴. Furthermore, the change in superconducting transition temperature with the applied magnetic field is shown in figure 4(b).

Zoom In Zoom Out Reset image size

The specific heat measurement also allows us to calculate several other parameters. The lattice constant parameter, β₃, is used to extract information about the phonons and provides the Debye temperature, $\theta_\mathrm D$. Using the relation $\theta_\mathrm D = (\frac{12\pi^4R N}{5\beta_3})^{1/3}$, where R is the universal gas constant and N = 14 the number of atoms per formula unit, the estimated value of θ_D is 166(1) K. The obtained value of $\theta_\mathrm D$ from the specific heat measurement matches the value from the resistivity measurement. The Sommerfeld coefficient, γ_n, is related to the single-particle DOS at the Fermi level by the expression $\gamma_n = (\frac{\pi^2 k_\mathrm B^2}{3}) D_c (E_{\mathrm{F}})$, and the operation yields $D_c(E_{\mathrm{F}})$ = 10.5(3) states eV⁻¹ f.u.⁻¹. The information on the electron–phonon coupling strength can also be extracted from McMillian theory, where the dimensionless quantity $\lambda_\mathrm{e-ph}$ is stated by the relation [53]

$$\begin{equation} \lambda_\mathrm{e-ph} = \frac{1.04+\mu^{*}\mathrm{ln}(\theta_\mathrm{D}/1.45T_{C})}{(1-0.62\mu^{*})\mathrm{ln}(\theta_\mathrm{D}/1.45T_{C})-1.04 } \end{equation} \tag{ 6 }$$

Here, $\mu^*$ is typically assumed to be 0.13 for intermetallic superconductors. The values θ_D = 166(1) K and T_C = 3.23(6) K, provide the value $\lambda_\mathrm{e-ph}$ = 0.67(2), suggesting that the telluride CP Mo₄Re₂Te₈ is a moderately coupled superconductor. Even though the high DOS at the Fermi level, large contribution to the lattice-specific heat, β₃, and parameter $\gamma_n/T_C$ = 7.6 mJ mol⁻¹ K⁻³ of Mo₄Re₂Te₈ are comparable to SnMo₆S₈, PbMo₆S₈ and Ag$_{1-x}$Mo₆S₈, possessing a strong electron–phonon coupling, Mo₄Re₂Te₈ has moderate coupling. The ratio $\gamma_n/T_C$ indicates the scale of the transition temperature with the DOS at $E_{\mathrm{F}}$ [3, 4].

The symmetry associated with the superconducting gap can be understood by analyzing the electronic specific heat temperature dependence in the superconducting region. The electronic contribution to the specific heat is calculated by subtracting the phononic contribution from the total specific heat at zero field. Theoretically, it can be computed from $C_\mathrm{el} = t \frac{dS}{dt}$, with S being the entropy in the superconducting region and within the BCS approximation has the form

$$\begin{equation} S = -\frac{6\gamma_n}{\pi^2}\left(\frac{\Delta(0)}{k_\mathrm{B}}\right)\int_{0}^{\infty}[\, \textit{f}\,\ln(f\,)+(1-f\,)\ln(1-f\,)]dy \end{equation} \tag{ 7 }$$

where $f(\xi) = [ \exp (E(\xi))/k_\mathrm BT)+1]^{-1}$ is the Fermi function. $E(\xi) = \sqrt{\xi^2 + \Delta(t)^2}$ is the excitation energy of the quasiparticle measured relative to the Fermi level, with $y = \xi/\Delta(0)$ and $\Delta(t)$ being the temperature-dependent gap function. In the isotropic s-wave BCS approximation, the gap function can be written as $\Delta(t)$ = tanh$\{1.82(1.018[(\mathit{1/t}) - 1])\}^{0.51}$ with $t = T/T_C$. Figure 4(c) displays the variation of electronic specific heat, $C_\mathrm{el}$, with temperature, fitted with the BCS s-wave model. The well-fitted data in the superconducting region yield $\Delta(0)/k_\mathrm B T_C$ = 1.81(4), while the specific heat jump value $\Delta C_\mathrm{el}/\gamma_n T_C$ = 1.44(8). Both of these values, $\Delta(0)/k_\mathrm B T_C$ and $\Delta C_\mathrm{el}/\gamma_n T_C$, are close to the BCS predicted values (1.76 and 1.43) in the weak coupling limit. Hence, this indicates that Mo₄Re₂Te₈ is a weakly coupled superconductor. However, to understand the exact structure of the superconducting gap and the nature of the pairing state, low-temperature specific heat below 1.9 K is required.

To gain an insight into the electronic properties, we have performed band structure calculations using the full potential linear augmented plane wave method as implemented in Wien2k [54, 55]. We used a k-point mesh of $10 \times 10 \times 10$ within the first Brillouin zone, and the generalized gradient approximation exchange-correlation functional of Perdew, Burke and Ernzerhof [56]. SOC was included for all the elements in the calculations. The energy and charge convergence criteria were set to 0.01 meV and 10⁻⁴ electronic charge per f.u., respectively. The Mo₄Re₂Te₈ structure was simulated by substituting Re in the place of two Mo atoms in Mo₆Te₈. The electronic band structure between the high-symmetry points [57] in the Brillouin zone and the DOS of Mo₄Re₂Te₈ is shown in figure 5. The inclusion of SOC for Re significantly modifies the band dispersion, while SOC in Mo and Te does not have much influence, indicating that the high SOC of Re plays an important role in Mo₄Re₂Te₈. Two doubly degenerate bands, B₁ and B₂, shown in red and blue color respectively, are dispersing in a large energy window in the vicinity of the Fermi level ($E_{\mathrm{F}}$) and cross $E_{\mathrm{F}}$ at different k points of the Brillouin zone. The Fermi level lies almost in the middle of band B₁, while band B₂ remains almost empty. The system is close to cubic (rhombohedral angle = 92.12${^{\circ}}$), and the high-symmetry points P, Z, Q, P₁ and Q₁ are almost equivalent where a small electron pocket (band B₂) is seen in the band structure. At the same time, a large hole pocket is centered around the Γ point (band B₁), as also observed from the Hall measurement, suggesting dominant hole carriers.

Zoom In Zoom Out Reset image size

Strong hybridization between the Mo/Re d states and Te p states is observed in the DOS plot (figure 5). The calculated value of total DOS at $E_{\mathrm{F}}$ is 9.87 states eV⁻¹ f.u.⁻¹, in qualitative agreement with the values obtained from the specific heat experiment.

Furthermore, in order to quantify the London penetration depth, λ_L , electronic mean free path, l_e , and to verify the dirty-limit superconductivity for Mo₄Re₂Te₈, a set of equations are implemented. The Fermi wave vector, $k_{\mathrm{F}}$, is estimated from the quasiparticle number density by the relation, $k_{\mathrm{F}} = (3\pi^2 n)^{1/3}$; here, $n = 1.21(3) \times 10^{28}$ m⁻³ from the normal Hall measurement, hence $k_{\mathrm{F}} = 0.71(2)$ Å⁻¹. The Sommerfeld coefficient, γ_n , and $k_{\mathrm{F}}$ are used for effective mass value estimation by the relation $m^{*} = (\hbar k_{\mathrm{F}})^2 \gamma_n/\pi^2 n k_\mathrm B^2$, with $k_\mathrm B$ being the Boltzmann constant, and $m^{*}$ becomes 3.2(3) m_e . Moreover, in consideration of Drude's model, the mean free path is defined as l_e = $v_{\mathrm{F}} \tau$, where τ is the scattering time given by $\tau^{-1} = n e^2 \rho_0/m^*$. $v_{\mathrm{F}}$ is the Fermi velocity, stated as $v_{\mathrm{F}} = \hbar k_{\mathrm{F}}/m^*$. Including the respective values of $m^{*}$, $k_{\mathrm{F}}$, n, and residual resistivity, ρ₀ = 2.95(1) mΩ cm, the scattering time τ and the Fermi velocity $v_{\mathrm{F}}$ are evaluated, which provides the mean free path, $l_\mathrm e = 0.81(9)$ Å. From BCS theory [51], the coherence length ξ₀ is approximated as $0.18 \hbar v_{\mathrm{F}}/k_\mathrm B T_\mathrm C$, hence becoming 1074(202) Å for $T_C = 3.26$ K. The huge difference between the coherence length and mean free path $\xi_0\gg$ $l_\mathrm e$ puts Mo₄Re₂Te₈ in the dirty-limit superconductor. Furthermore, the London penetration depth $\lambda_\mathrm{L}$ is expressed as $\lambda_\mathrm{L} = \left(\frac{m^{*}}{\mu_{0}n e^{2}}\right)^{1/2}$ and calculated to be 86(1) nm. The characteristic nature of poor metal with the short mean free path of the known CPs is also observed in our telluride CP compound Mo₄Re₂Te₈.

The CP has been classified as an unconventional superconductor based on their $\frac{T_\mathrm{C}}{T_{\mathrm{F}}}$ ratio provided by Uemura et al [1]. The ratio for the CP lies in the range $0.01 \leqslant \frac{T_{C}}{T_{\mathrm{F}}} \leqslant 0.1$ with high-T_C , organic, heavy-fermion, and other unconventional superconductors. For a 3D system, assuming a spherical Fermi surface, the Fermi temperature, $T_{\mathrm{F}}$, is given by the relation [58]

$$\begin{equation} k_\mathrm{B}T_{{\mathrm{F}}} = \frac{\hbar^{2}}{2}(3\pi^{2})^{2/3}\frac{n^{2/3}}{m^{*}}, \end{equation} \tag{ 8 }$$

where n is the quasiparticle number density per unit volume and $m^{*}$ is the effective mass of the quasiparticles. Considering the values of n and $m^{*}$as listed in table 2, from equation (8), we get $T_{{\mathrm{F}}}$ = 6980(845) K for Mo₄Re₂Te₈. The ratio $\frac{T_{C}}{T_{{\mathrm{F}}}}$ = 0.0005 places Mo₄Re₂Te₈ in the region of Re-based compounds, having Re₂₄Ti₅ [59] on its left side and other compounds having same composition on its right side [25, 26, 31, 49], with the elemental Re [31], as shown in figure 6.

Zoom In Zoom Out Reset image size

Table 2. Parameters in the superconducting and normal state of Mo₄Re₂Te₈.

Parameters	Unit	Mo4Re2Te8
TC	K	3.26(3)
$H_{C1}(0)$	mT	0.49(1)
$H_{C2}^{\rho}(0)$	T	5.78(4)
$H_{C2}^{P}(0)$	T	6.00(5)
$H_{C2}^{Orb}(0)$	T	3.55(4)
$\xi_\mathrm{GL}(0)$	Å	80(1)
$\lambda_\mathrm{GL}(0)$	nm	1325(40)
$k_\mathrm{GL}$		165(7)
γn	mJ mol−1 K−2	24.7(8)
$\theta_\mathrm{D}$	K	166(1)
$\Delta C_\mathrm{el}/\gamma_{n}T_{C}$		1.44(8)
$\Delta(0)/k_\mathrm{B}T_{C}$		1.81(4)
$v_{{\mathrm{F}}}$	105 m s−1	2.6(2)
n	1028 m−3	1.21(3)
$T_{{\mathrm{F}}}$	K	6980(845)
$T_{C}/T_{{\mathrm{F}}}$		0.0005(1)
$m^{*}$/$m_\mathrm{e}$		3.2(3)

Irrespective of the short mean free path, large contribution of lattice-specific heat, high DOS at the Fermi level, and moderate electron–phonon coupling in the pseudobinary compound Mo₄Re₂Te₈, the exotic character of the extremely high upper critical field value of CPs is absent. As -above mention parameters but with the strong electron–phonon coupling are accounted to be the possible reason for high upper critical field value in well-known CPs [15, 16]. Hence, strong electron–phonon coupling might be a possible requirement to enhance the upper critical field beyond a limit in CPs. However, the comparable value of the upper critical field to the Pauli paramagnetic limit suggests a possible unconventional nature of its superconducting ground state as observed in many Re-based superconductors [25, 49]. Notably, such a large value of the upper critical field has not been observed in any other Te superconductors, with the exception of the iron-based chalcogenide superconductors [62]. Moreover, the Uemura classification of Mo₄Re₂Te₈ places it in an interesting position between the two different Re concentration systems. In addition, a recent first-principles study of CP PbMo₆S₈ questions its position in the Uemura plot and suggests the phonon-driven superconductivity with a significant contribution of strong Coulombic repulsion in the compound [63]. Hence, to assure the unconventional nature and superconducting gap symmetry of Mo₄Re₂Te₈, further microscopic investigations are required.