# Superconducting properties of eutectic high-entropy alloy superconductor
NbScTiZr

Jiro Kitagawa<EMAIL_ADDRESS>Haruto Ueta  Yuto Watanabe  Takeru Seki
Yoshikazu Mizuguchi  Terukazu Nishizaki

###### Abstract

The influence of annealing on the superconducting properties of eutectic high-
entropy alloy NbScTiZr was examined by measuring magnetization, electrical
resistivity, and zero-field specific heat. Additionally, the extent of lattice
strain was assessed via the analysis of lattice parameters and Vickers
microhardness. The greatest lattice strain was inferred in the sample annealed
at 400 ∘C. Field-dependent magnetization datasets indicate enhanced flux
pinning in the as-cast, 400 ∘C annealed, and 600 ∘C annealed samples. The
superconducting parameters such as upper critical field, Ginzburg-Landau
coherence length, and magnetic penetration depth do not strongly depend on the
annealing temperature. However, the Maki parameter shows a peak at 400 ∘C
annealing temperature and correlates with the magnitude of lattice strain.
This suggests that greater lattice strain may enhance the Maki parameter by
altering the orbital-limited field through the modification of flux pinning
strength. Although the influence of lattice strain is less discernible in
zero-field specific heat measurements, the superconducting parameters deduced
from specific heat data suggest a potential for strong-coupled
superconductivity in samples heat-treated above 600 ∘C. Our investigation
underscores the substantial impact of lattice strain on the Maki parameter of
eutectic HEA superconductors, primarily through the modulation of flux pinning
strength.

###### keywords:

High-entropy alloys , Superconducting properties , Magnetization , Specific
heat , Hardness

††journal: Materials Today Communications

[inst1]organization=Department of Electrical Engineering, Faculty of
Engineering, Fukuoka Institute of Technology,addressline=3-30-1 Wajiro-
higashi, Higashi-ku, city=Fukuoka, postcode=811-0295, country=Japan

[inst2]organization=Department of Electrical Engineering, Faculty of Science
and Engineering, Kyushu Sangyo University,addressline=2-3-1 Matsukadai,
Higashi-ku, city=Fukuoka, postcode=813-8503, country=Japan

[inst3]organization=Department of Physics, Tokyo Metropolitan
University,city=Hachioji, postcode=192-0397, country=Japan

##  1 Introduction

In traditional alloys, a minor quantity of additional elements is incorporated
into a predominant element. The concept of high-entropy alloy (HEA),
introduced in 2004, has disrupted the conventional paradigm of alloy design[1,
2]. HEAs typically consist of four or more principal elements with
approximately equal atomic proportions[3, 4]. These metallic alloys have
garnered considerable attention owing to their array of unconventional
properties, including high-temperature strength, heightened resistance to
radiation, exceptional thermoelectric figure of merit, improved magnetic
refrigeration characteristics, good soft ferromagnetism, and interesting
superconductivity[5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Superconducting HEAs
have emerged as a prominent research focus since the discovery of an HEA
superconductor in 2014[15]. Presently, HEA superconductivity is reported in
diverse crystal structures such as body-centered cubic (bcc)[16, 17, 18, 19,
20, 21, 22, 23, 24], hexagonal close-packed (hcp)[25, 26, 27], face-centered
cubic[28], CsCl-type[29], A15[30, 31], NaCl-type[32, 33], $\alpha$ (or
$\beta$)-Mn-type[34, 35], $\sigma$-phase type[36], CuAl2-type[37],
BiS2-based[38], and YBCO-based[39] structures. Noteworthy discoveries in the
domain of HEA superconductors include the robustness of superconductivity
against high pressures and irradiation, deviations from the Matthias rule, and
the attainment of high critical current densities[40, 41, 42, 43, 44, 45, 46,
47]. Additionally, the high hardness or shape memory effect exhibited by
certain superconducting HEAs holds promise for developing multifunctional
devices[23, 48].

The concept of eutectic HEA represents a recent innovation that has sparked a
revolution in traditional HEAs[49]. Eutectic high-entropy alloys combine the
benefits of both eutectic alloys and HEAs, showcasing excellent castability
and superior mechanical properties. Notably, high strength and good ductility
can be concurrently achieved by controlling the two phases in eutectic
HEAs[50]. However, the superconductivity in eutectic HEAs remains relatively
unexplored[51]. Our recent investigation on eutectic superconducting HEA
NbScTiZr has unveiled the modulation of the superconducting critical
temperature ($T_{\mathrm{c}}$) through thermal annealing[46]. The eutectic HEA
comprises the bcc and hcp phases, with the bcc phase undergoing the
superconducting transition. We observed a systematic enhancement of
$T_{\mathrm{c}}$ from 7.9 to 9 K with increasing thermal annealing
temperatures up to 800 ∘C. Thermal annealing above 400 ∘C results in grain
size coarsening. Moreover, analysis of lattice parameters suggests the
presence of lattice strain, particularly at lower annealing temperatures. We
have also demonstrated a remarkably high critical current density for the as-
cast NbScTiZr. Nevertheless, the impact of thermal annealing on
superconducting parameters, such as critical fields and coherence length in
NbScTiZr, remains unexplored. Investigating this aspect is essential for
comprehending the role of lattice strain associated with the eutectic
structure in HEA superconductivity.

In this study, we estimated the fundamental superconducting parameters through
magnetic, transport, and zero-field thermal measurements of NbScTiZr samples
prepared under various annealing temperatures. Magnetization measurements
provide the lower and upper critical fields, facilitating the extraction of
the Ginzburg-Landau coherence length and the magnetic penetration depth. We
discuss the annealing temperature dependences of these parameters and the
influence of lattice strain on flux pinning strength. The electrical
resistivity measurements as a function of magnetic field are also informative
for discussing upper critical field. Zero-field specific heat measurements
provide insights into the bulk nature of the transition associated with
chemical homogeneity. Utilizing specific heat data, we compare thermal
parameters with those of the BCS model in the weak coupling limit.
Furthermore, lattice strain is probed via Vickers microhardness analysis.

Table 1: Chemical compositions of bcc and hcp phases and lattice parameters of prepared samples. sample | composition | lattice parameter (Å)
---|---|---
as-cast | bcc:Nb26.6(4)Sc22.3(5)Ti25.5(5)Zr25.6(4) |  $a$=3.359(2)
| hcp:Nb21.3(4)Sc30(1)Ti23.0(7)Zr25.8(5) |  $a$=3.262(3), $c$=5.152(3)
400 ∘C | bcc:Nb27.9(3)Sc21.6(2)Ti25.3(3)Zr25.2(2) |  $a$=3.308(2)
| hcp:Nb22.1(5)Sc32.3(9)Ti21.3(5)Zr24.2(2) |  $a$=3.252(1), $c$=5.131(1)
600 ∘C | bcc:Nb29.1(6)Sc19.1(4)Ti29.8(9)Zr22(1) |  $a$=3.314(1)
| hcp:Nb18.0(5)Sc32(1)Ti19(1)Zr30.8(7) |  $a$=3.254(1), $c$=5.133(2)
800 ∘C | bcc:Nb33(1)Sc15(1)Ti32(1)Zr20(1) |  $a$=3.325(1)
| hcp:Nb13(1)Sc39.1(9)Ti15.1(4)Zr32.6(6) |  $a$=3.252(2), $c$=5.138(3)
1000 ∘C | bcc:Nb39.0(8)Sc4.2(6)Ti38.0(6)Zr18.9(8) |  $a$=3.360(1)
| hcp:Nb2.6(7)Sc56(1)Ti7.3(7)Zr34.1(5) |  $a$=3.260(1), $c$=5.141(2)

##  2 Materials and Methods

The as-cast sample was made employing a homemade arc furnace under an argon
atmosphere. The constituent materials comprised Nb wire (Nilaco, 99.9 %), Sc
chips (Kojundo Chemical Laboratory, 99 %), Ti wire (Nilaco, 99.9 %), and Zr
wire (Nilaco, 99.5 %). The mixture of elements with the atomic ratio of
Nb:Sc:Ti:Zr=1:1:1:1 underwent melting on a water-cooled Cu hearth. Subsequent
annealing at 400 ∘C, 600 ∘C, 800 ∘C, or 1000 ∘C for 4 days was conducted in an
electric furnace, with the sample enclosed within an evacuated quartz tube.

X-ray diffraction (XRD) patterns were acquired utilizing an X-ray
diffractometer (Shimadzu XRD-7000L) employing Cu-K$\alpha$ radiation in the
Bragg-Brentano configuration. Thin slabs cut from the samples were utilized
because of the difficulty in obtaining fine powders due to high ductility. The
temperature-dependent dc magnetization $M$($T$) and isothermal magnetization
curves were recorded employing a SQUID magnetometer (MPMS3, Quantum Design).
The temperature dependences of electrical resistivity $\rho$($T$) under
several external fields up to 9 T were measured by the four-probe method using
PPMS equipment (Quantum Design). Specific heat $C$ was determined via the
thermal-relaxation method employing PPMS equipment (Quantum Design). Vickers
microhardness was assessed on polished surfaces under a 2.94 N load applied
for 10 seconds using a Shimadzu HMV-2T microhardness tester.

Figure 1: (a) XRD patterns of NbScTiZr samples prepared under various
annealing conditions. The simulation patterns of the bcc ($a$=3.359 Å) and hcp
($a$=3.262 Å, $c$=5.152 Å) structures are also presented. The origin of each
XRD pattern is shifted by a value for the clarity. (b) Annealing temperature
dependences of lattice parameter (filled blue circles) and that calculated
using the hard-sphere model (filled blue squares). The annealing temperature
dependence of lattice deviation, defined as
$(a_{\mathrm{exp}}-a_{\mathrm{hard}})/a_{\mathrm{hard}}\times 100$, where
$a_{\mathrm{exp}}$ and $a_{\mathrm{hard}}$ are the lattice parameters of the
experimental data and the hard-sphere model, respectively, is also shown.
Figure 2:  Temperature-dependent ZFC magnetization measured under external
fields, as denoted in figure for (a) as-cast, (b) 400 ∘C annealed, (c) 600 ∘C
annealed, (d) 800 ∘C annealed, and (e) 1000 ∘C annealed samples. $M$ is
normalized by the absolute value of $M$ at 2.0 K.

##  3 Results and Discussion

Figure 1(a) illustrates the XRD patterns of both as-cast and thermally
annealed samples. The XRD pattern of the as-cast sample manifests peaks
corresponding to bcc and hcp phases, denoted by filled circles and triangles,
respectively (also refer to the simulated patterns of bcc and hcp structures).
The Bragg reflection peaks of the thermally annealed samples are likewise
assignable to the bcc or hcp structure. Notably, the positions of the Bragg
reflection peaks shift towards higher angles as the annealing temperature
progresses from the as-cast state to 400 ∘C annealing and subsequently towards
lower angles with further annealing above 400 ∘C (refer to the dotted curves
marked for the bcc phase in Fig. 1(a)). After assigning Miller indices, the
lattice parameter of each phase is determined employing the least square
method, as summarized in Table 1. The obtained parameters of the bcc phase are
plotted against the annealing temperature in Fig. 1(b), considering the as-
cast state as equivalent to 0 ∘C annealing. We check the influence of lattice
strain by employing the hard-sphere model with no consideration of lattice
strain. In this model, the ideal lattice parameter is estimated based on the
chemical composition derived from energy-dispersive X-ray spectrometer coupled
with field-emission scanning electron microscopy[46]. The chemical composition
of the bcc phase, as reported in a previous study[46], is provided in Table 1
for each sample. The hard-sphere model predicts the lattice parameter
expressed as 2.309$\bar{r}$, where $\bar{r}$ represents the composition-
weighted atomic radius. By utilizing atomic radii of 1.429 Å for Nb, 1.641 Å
for Sc, 1.4615 Å for Ti, and 1.6025 Å for Zr[52], the calculated lattice
parameters are also depicted in Fig. 1(b). Lattice strain is assessed by the
disparity between the experimental lattice parameter and that derived from the
hard-sphere model. This disparity is denoted as lattice deviation and
quantified by $(a_{\mathrm{exp}}-a_{\mathrm{hard}})/a_{\mathrm{hard}}\times
100$, where $a_{\mathrm{exp}}$ and $a_{\mathrm{hard}}$ are the lattice
parameters of the experimental data and the hard-sphere model, respectively.
Figure 1(b) further illustrates the annealing temperature dependence of the
lattice deviation. The deviation exhibits a shallow minimum at 400 ∘C,
reaching -6 %. This value is relatively higher compared to single-phase HEA
superconductors such as HfMoNbTiZr[23] (lattice deviation: -1.7 %) and
(Ti35Hf25)(Nb25Ta5)Re10[43] (lattice deviation: -2.3 %). This outcome supports
the manifestation of lattice strain, particularly in samples annealed at lower
temperatures. Additionally, we evaluated the crystallite size $D$ of the bcc
phase utilizing the Scherrer equation
$D=K\lambda/\beta_{\mathrm{XRD}}\mathrm{cos}\theta$, where
$\beta_{\mathrm{XRD}}$ represents the full width at half maximum of the
diffraction peak, $\lambda$ denotes the X-ray wavelength, and $K$ signifies a
numerical coefficient. Assuming $K$=0.9, the $D$ values computed from the
principal peaks of all samples are 14 nm (as-cast), 17 nm (400 ∘C), 20 nm (600
∘C), 18 nm (800 ∘C), and 15 nm (1000 ∘C), respectively.

Figure 3:  (a) Temperature dependences of upper critical field of NbScTiZr
samples. The figure is derived from Figs. 2(a)-(e). The solid curves show the
fitting results using eq. (1). (b) Annealing temperature dependence of
$T_{\mathrm{c}}$ extracted using eq. (1).

Figures 2(a)-(e) display the temperature dependences of magnetization measured
under the zero-field cooled (ZFC) condition, with the external field
$\mu_{0}H$ ranging from 0.4 mT to 7 T for all samples. Each $M$($T$) dataset
is normalized by the absolute value of $M$ at 2 K. The onset temperature of
diamagnetism at $\mu_{0}H$=0.4 mT mirrors $T_{\mathrm{c}}$ under zero magnetic
field (also indicated by arrows in Figs. 2(a)-(e)). As the annealing
temperature increases to 600 ∘C, $T_{\mathrm{c}}$ experiences enhancement.
However, further increments in the annealing temperature marginally diminish
$T_{\mathrm{c}}$. We determined the upper critical field
$\mu_{0}H_{\mathrm{c2}}$ using the $M$($T$) datasets in Figs. 2(a)-(e). The
onset of the diamagnetic signal is defined as $T_{\mathrm{c}}$ at a $\mu_{0}H$
value (see also Fig. S1 to S10 in Supplementary Material), enabling the
construction of a $\mu_{0}H_{\mathrm{c2}}$ vs. temperature plot as illustrated
in Fig. 3(a). The temperature dependences of $\mu_{0}H_{\mathrm{c2}}$ are
analyzed utilizing the Werthamer–Helfand–Hohenberg (WHH) model:

$\mathrm{ln}\frac{1}{t}=(\frac{1}{2}+\frac{i\lambda_{\mathrm{SO}}}{4\gamma})\psi\left(\frac{1}{2}+\frac{h+\lambda_{\mathrm{SO}}/2+i\gamma}{2t}\right)+(\frac{1}{2}-\frac{i\lambda_{\mathrm{SO}}}{4\gamma})\psi\left(\frac{1}{2}+\frac{h+\lambda_{\mathrm{SO}}/2-i\gamma}{2t}\right)-\psi\left(\frac{1}{2}\right)$
(1)

, where $t=T/T_{\mathrm{c}}$, $\lambda_{\mathrm{SO}}$ is the spin-orbital
scattering parameter,
$\gamma=[(\alpha_{\mathrm{M}}h)^{2}-(\lambda_{\mathrm{SO}}/2)^{2}]^{1/2}$
($\alpha_{\mathrm{M}}$: Maki parameter), and $\psi(x)$ is the digamma
function. In equation (1), $h$ is defined as

$h=\frac{4H_{\mathrm{c2}}(T)}{\pi^{2}T_{\mathrm{c}}(-dH_{\mathrm{c2}}(T)/dT)_{T=T\mathrm{c}}}.$
(2)

For reliable fitting, we refer to the literature reporting the WHH model
parameters of NbTi alloy[53]. The reported values of $\alpha_{\mathrm{M}}$ and
$\lambda_{\mathrm{SO}}$ are 1.3 and 4.5, respectively, which are suitable for
the 800 ∘C and 1000 ∘C annealed samples with weak lattice strain. The same
parameters can reproduce the temperature dependences of
$\mu_{0}H_{\mathrm{c2}}$ of these samples, as shown in Fig. 3(a). The obtained
$\mu_{0}H_{\mathrm{c2}}$(0) and $\alpha_{\mathrm{M}}$ values of all samples
are summarized in Table 2. The values of $\lambda_{\mathrm{SO}}$ are 1.0, 1.0,
2.0, 4.5, and 4.5 for the as-cast, 400 ∘C annealed, 600 ∘C annealed, 800 ∘C
annealed, and 1000 ∘C annealed samples, respectively. In Table 2, we use the
notation $\mu_{0}H_{\mathrm{c2}}^{M}$(0) and $\alpha_{\mathrm{M}}^{M}$ for
$\mu_{0}H_{\mathrm{c2}}$(0) and $\alpha_{\mathrm{M}}$, respectively. The
annealing temperature dependence of $\mu_{0}H_{\mathrm{c2}}$(0) or
$\alpha_{\mathrm{M}}$ is extensively discussed below. The $T_{\mathrm{c}}$
values extracted from the analysis of $\mu_{0}H_{\mathrm{c2}}$ are plotted as
a function of annealing temperature in Fig. 3(b), elucidating the impact of
annealing temperature on $T_{\mathrm{c}}$ as mentioned earlier.

We have also examined $\mu_{0}H_{\mathrm{c2}}$($T$) by measuring $\rho$($T$)
under external fields, as depicted in Figs. 4(a)-(e). The onset of the
superconductive transition is employed as $T_{\mathrm{c}}$, and
$\mu_{0}H_{\mathrm{c2}}$($T$) is again analyzed using the WHH model for each
sample (see Fig. 4(f)). Table 2 also displays $\mu_{0}H_{\mathrm{c2}}$(0) and
$\alpha_{\mathrm{M}}$ for each sample, denoted as
$\mu_{0}H_{\mathrm{c2}}^{\rho}$(0) and $\alpha_{\mathrm{M}}^{\rho}$,
respectively. The estimated values of $\lambda_{\mathrm{SO}}$ are 0.7, 0.3,
2.0, 4.5, and 4.5 for the as-cast, 400 ∘C annealed, 600 ∘C annealed, 800 ∘C
annealed, and 1000 ∘C annealed samples, respectively. The dataset of
$\mu_{0}H_{\mathrm{c2}}^{\rho}$(0) does not significantly differ from that of
$\mu_{0}H_{\mathrm{c2}}^{M}$(0). Furthermore, the annealing temperature
dependence of $\alpha_{\mathrm{M}}^{\rho}$ is similar to that of
$\alpha_{\mathrm{M}}^{M}$.

Table 2: Superconducting parameters and Vickers microhardness of NbScTiZr. The superscripts $M$ and $\rho$ of $\mu_{0}H_{\mathrm{c2}}$(0), $\alpha_{\mathrm{M}}$, and $\xi_{\mathrm{GL}}$(0) denote the parameters evaluated through the temperature-dependent measurements of $M$ and $\rho$, respectively. $\lambda_{\mathrm{GL}}^{M}$(0) is calculated using $\mu_{0}H_{\mathrm{c1}}$(0) and $\xi_{\mathrm{GL}}^{M}$(0). $\lambda_{\mathrm{GL}}^{\rho}$(0) is calculated using $\mu_{0}H_{\mathrm{c1}}$(0) and $\xi_{\mathrm{GL}}^{\rho}$(0). $\kappa_{\mathrm{GL}}^{M}$ ($\kappa_{\mathrm{GL}}^{\rho}$) is calculated using $\xi_{\mathrm{GL}}^{M}$(0) ($\xi_{\mathrm{GL}}^{\rho}$(0)) and $\lambda_{\mathrm{GL}}^{M}$(0) ($\lambda_{\mathrm{GL}}^{\rho}$(0)). sample | as-cast | 400 ∘C | 600 ∘C | 800 ∘C | 1000 ∘C
---|---|---|---|---|---
$T_{\mathrm{c}}$ (K) | 8.11 | 8.71 | 9.36 | 9.13 | 9.03
$\gamma_{\mathrm{el}}$ (mJ/mol$\cdot$K2) | 6.47 | 6.79 | 5.77 | 5.46 | 6.62
$\beta$ (mJ/mol$\cdot$K4) | 0.150 | 0.127 | 0.127 | 0.129 | 0.145
$\mu_{0}H_{\mathrm{c1}}$(0) (mT) | 29.1 | 32.9 | 22.9 | 22.2 | 4.46
$\mu_{0}H_{\mathrm{c2}}^{M}$(0) (T) | 13.1 | 14.1 | 14.6 | 13.9 | 14.3
$\alpha_{\mathrm{M}}^{M}$ | 1.7 | 1.8 | 1.4 | 1.3 | 1.3
$\mu_{0}H_{\mathrm{c2}}^{\rho}$(0) (T) | 12.8 | 12.7 | 14.5 | 15.0 | 14.5
$\alpha_{\mathrm{M}}^{\rho}$ | 1.7 | 1.9 | 1.4 | 1.3 | 1.3
$\xi_{\mathrm{GL}}^{M}$(0) (nm) | 5.0 | 4.8 | 4.8 | 4.9 | 4.8
$\lambda_{\mathrm{GL}}^{M}$(0) (nm) | 137 | 128 | 159 | 161 | 405
$\kappa_{\mathrm{GL}}^{M}$ | 27 | 27 | 33 | 33 | 84
$\xi_{\mathrm{GL}}^{\rho}$(0) (nm) | 5.1 | 5.1 | 4.8 | 4.7 | 4.8
$\lambda_{\mathrm{GL}}^{\rho}$(0) (nm) | 137 | 127 | 159 | 162 | 405
$\kappa_{\mathrm{GL}}^{\rho}$ | 27 | 25 | 33 | 34 | 84
Hardness (HV) | 336(1) | 345(4) | 264(6) | 239(2) | 230(2)
Figure 4:  Temperature-dependent electrical resistivity measured under
external fields, as denoted in figure for (a) as-cast, (b) 400 ∘C annealed,
(c) 600 ∘C annealed, (d) 800 ∘C annealed, and (e) 1000 ∘C annealed samples.
(f) Temperature dependences of upper critical field of NbScTiZr samples. The
figure is derived from (a)-(e). The solid curves show the fitting results
using eq. (1).

Figures 5(a)-(e) exhibit the isothermal $M$-$H$ curves measured at various
temperatures for all examined samples. Notably, all samples exhibit negative
broad peaks at $\mu_{0}H$ values, surpassing the anticipated lower critical
field $\mu_{0}H_{\mathrm{c1}}$ for transition metal alloys. This phenomenon
can generally be elucidated by strong flux pinning mechanisms[54]. In an ideal
type-II superconductor with no defects, a completely reversible $M$-$H$ curve
manifests, characterized by a sharp peak at $\mu_{0}H_{\mathrm{c1}}$.
Introducing defects acting as flux pinning sites causes a sharp peak to be
absent at the first flux quantum penetration, resulting in irreversibility in
the $M$-$H$ curve. Consequently, the initial magnetization process displays a
broad peak, as observed in Figs. 5(a)-(e). It is worth noting that NbScTiZr in
the as-cast state and heat-treated at 800 ∘C exhibit extensive $M$-$H$ loops,
as reported in the previous paper[46]. The area of the $M$-$H$ hysteresis loop
for the as-cast sample exceeds that of the sample annealed at 800 ∘C at the
same temperature, indicative of robust flux pinning in the as-cast sample.
Indeed, the external field demonstrating the negative peak
($\mu_{0}H_{\mathrm{p}}$) for the as-cast sample exceeds that of the 800 ∘C
annealed sample at the fixed temperature (refer to Figs. 5(a) and (d)). The
lattice deviation discussed in Fig. 1(b) reflects lattice strain, correlating
with flux pinning strength as the lattice strain yields pinning sites.
Consequently, we examined the correlation between $\mu_{0}H_{\mathrm{p}}$ at 2
K and lattice deviation for each sample, as depicted in Fig. 5(f). The figure
demonstrates that a more significant lattice deviation corresponds to greater
$\mu_{0}H_{\mathrm{p}}$ values. Therefore, flux pinning is most pronounced in
the sample annealed at 400 ∘C. In Figs. 5(a)-(e), $\mu_{0}H_{\mathrm{c1}}$
value in each figure is determined by identifying the point at which the
initial $M$-$H$ curve deviates from the linear line (see also solid line in
each figure). The evaluated $\mu_{0}H_{\mathrm{c1}}$ increases with decreasing
temperature, as demonstrated in Fig. 5(g), which is well reproduced by the
Ginzburg-Landau equation,

$H_{\mathrm{c1}}(T)=H_{\mathrm{c1}}(0)\left(1-\left(\frac{T}{T_{\mathrm{c}}}\right)^{2}\right)$
(3)

, where $\mu_{0}H_{\mathrm{c1}}$(0) represents the zero temperature
$\mu_{0}H_{\mathrm{c1}}$. Table 2 summarizes the values of
$\mu_{0}H_{\mathrm{c1}}$(0) for all samples.

Figure 5: Field-dependent magnetization of (a) as-cast, (b) 400 ∘C annealed,
(c) 600 ∘C annealed, (d) 800 ∘C annealed, and (e) 1000 ∘C annealed samples.
(f) Annealing temperature dependences of $\mu_{0}H_{\mathrm{p}}$ at 2 K and
lattice deviation. $\mu_{0}H_{\mathrm{p}}$ is defined as $\mu_{0}H$ showing
the negative peak in the $M$-$H$ curve. (g) Temperature dependences of lower
critical field of NbScTiZr samples. The solid curves show the fitting results
using eq. (3).

For each sample, the Ginzburg-Landau coherence length $\xi_{\mathrm{GL}}$(0)
is calculated using the equation
$\xi_{\mathrm{GL}}(0)=\sqrt{\frac{\Phi_{0}}{2\pi\mu_{0}H_{\mathrm{c2}}(0)}}$,
where $\Phi_{0}$ denotes the magnetic flux quantum of 2.07$\times$10-15 Wb.
The resultant values are listed in Table 2. It is noted that
$\xi_{\mathrm{GL}}$(0) is slightly smaller compared to typical HEAs
(NbReHfZrTi: 6.1 nm[17], Ta1/6Nb2/6Hf1/6Zr1/6Ti1/6: 5.2 nm[55], and
HfMoNbTiZr: 7.7 nm[23]). The magnetic penetration depth
$\lambda_{\mathrm{GL}}$(0) can be extracted using the relation
$\mu_{0}H_{\mathrm{c1}}(0)=\frac{\Phi_{0}}{4\pi\lambda_{\mathrm{GL}}(0)^{2}}\mathrm{ln}\frac{\lambda_{\mathrm{GL}}(0)}{\xi_{\mathrm{GL}}(0)}$.
The extracted $\lambda_{\mathrm{GL}}$(0) values are also displayed in Table 2.
Subsequently, the Ginzburg-Landau parameter
$\kappa_{\mathrm{GL}}$=$\lambda_{GL}(0)/\xi_{GL}(0)$ is obtained, yielding
values ranging from 25 to 84, as shown in Table 2. All values exceed
1/$\sqrt{2}$, indicating type-II superconductors. Figures 6(a) and (b)
encapsulate the annealing temperature dependences of fundamental
superconducting parameters. The annealing temperature dependences of
$\mu_{0}H_{\mathrm{c2}}$(0), $\lambda_{\mathrm{GL}}(0)$, and
$\xi_{\mathrm{GL}}(0)$ do not exhibit systematic behaviors, indicating that
flux pinning strength does not significantly alter the fundamental
superconducting parameters. However, as shown in Fig.6 (c), the overall
annealing temperature dependence of $\alpha_{\mathrm{M}}$ aligns well with
that of lattice deviation. The value of $\alpha_{\mathrm{M}}$ estimated in
each measurement reaches its maximum at an annealing temperature of 400 ∘C,
where the strongest flux pinning is observed. In type-II superconductors, the
Pauli paramagnetic field and the orbital limiting field are the principal
sources of superconductivity disruption under a magnetic field. The critical
magnetic field, which disrupts Cooper pairing due to the excess of Zeeman
splitting energy over the superconducting gap energy, is termed the Pauli
limit. In the orbital limiting field mechanism attributed to the Lorentz
force, the field-induced kinetic energy of a Cooper pair surpasses the
superconducting condensation energy. The Maki parameter $\alpha_{\mathrm{M}}$
is given by
$\frac{\sqrt{2}\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}}{\mu_{0}H_{\mathrm{c2}}^{\mathrm{Pauli}}}$,
where $\mu_{0}H_{\mathrm{c2}}^{\mathrm{Pauli}}$ is the Pauli limiting field
and $\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}$ is the orbital-limited upper
critical field. $\mu_{0}H_{\mathrm{c2}}^{\mathrm{Pauli}}$ is defined by
1.84$T_{\mathrm{c}}$. Employing refined $T_{\mathrm{c}}$ values listed in
Table 2, $\mu_{0}H_{\mathrm{c2}}^{\mathrm{Pauli}}$ values are obtained as 14.9
T (as-cast), 16.0 T (400 ∘C), 17.2 T (600 ∘C), 16.8 T (800 ∘C), and 16.6 T
(1000 ∘C). Substituting $\alpha_{\mathrm{M}}$ and
$\mu_{0}H_{\mathrm{c2}}^{\mathrm{Pauli}}$ values into
$\alpha_{\mathrm{M}}=\frac{\sqrt{2}\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}}{\mu_{0}H_{\mathrm{c2}}^{\mathrm{Pauli}}}$,
the datasets of $\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}$ (T) are estimated as
(17.9, 20.4, 17.0, 15.4, 15.3) obtained through $M$($T$) measurements and
(17.9, 21.5, 17.0, 15.4, 15.3) obtained through $\rho$($T$) measurements for
(as-cast, 400 ∘C, 600 ∘C, 800 ∘C, 1000 ∘C) samples. In each dataset,
$\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}$ is the highest at an annealing
temperature of 400 ∘C, and the annealing temperature dependence of
$\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}$ roughly correlates with that of
lattice deviation (also see Fig. 6(c)). The larger lattice deviation seems to
enhance $\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}$. As reported previously[46],
the lamellar structure exhibits fineness with a thickness of $\sim$ 70 nm in
the as-cast state. Although the 400 ∘C annealed sample maintains a fine
microstructure, systematic grain size coarsening occurs with increasing
annealing temperature above 400 ∘C. It is anticipated that lattice strain is
more pronounced in the fine eutectic structure due to the greater influence of
the mismatch of structural properties at the interface of bcc and hcp phases,
resulting in stronger flux pinning at lower annealing temperatures. Stronger
flux pinning can elevate the orbital limiting field, which is consistent with
our analysis of $\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}$.

Figure 6:  Annealing temperature dependences of (a)
$\mu_{0}H_{\mathrm{c1}}$(0) and $\mu_{0}H_{\mathrm{c2}}$(0), (b)
$\xi_{\mathrm{GL}}$(0) and $\lambda_{\mathrm{GL}}$(0), and (c)
$\alpha_{\mathrm{M}}$. In (c), the annealing temperature dependence of lattice
deviation is also shown for comparison. (d) VEC dependences of
$\mu_{0}H_{\mathrm{c2}}$(0) for NbScTiZr and Nb-Ti-Zr alloys.

In the preceding discussion, we neglect the alteration of chemical composition
upon annealing (refer to Table 1). The binary or ternary Nb-based bcc alloys
adhere to the Matthias rule, which suggests that $T_{\mathrm{c}}$ is highly
influenced by the density of states. Within transition metal bcc alloys, the
density of states is delineated by the valence electron concentration (VEC),
and $T_{\mathrm{c}}$ ascends with increasing VEC up to approximately 4.6[13].
The VEC values of bcc phases of NbScTiZr alloys are 4.04, 4.06, 4.10, 4.18,
and 4.35 for the as-cast, 400 ∘C annealed, 600 ∘C annealed, 800 ∘C annealed,
and 1000 ∘C annealed samples, respectively. Thus, as depicted in Fig. 3(b),
the trend of $T_{\mathrm{c}}$ versus annealing temperature conforms to the
Matthias rule below 600 ∘C. However, a departure from the Matthias rule is
discerned above 600 ∘C. To gain further insight into the composition effects
on superconducting properties, the VEC dependence of
$\mu_{0}H_{\mathrm{c2}}$(0) of NbScTiZr is juxtaposed with that of the Nb-Ti-
Zr alloy, which bears resemblance to NbScTiZr, as shown in Fig. 6(d). In the
Nb-Ti-Zr system, (VEC, $\mu_{0}H_{\mathrm{c2}}$(0) (T)) datasets manifest as
(4.12, 2) for Nb0.12Ti0.44Zr0.44, (4.17, 6) for Nb0.17Ti0.62Zr0.21, (4.3, 9)
for Nb0.30Ti0.52Zr0.18, and (4.3, 9) for Nb0.33Ti0.33Zr0.33[56]. Should the
chemical composition comprehensively expound the annealing temperature
dependence of $\mu_{0}H_{\mathrm{c2}}$(0), the VEC dependence of
$\mu_{0}H_{\mathrm{c2}}$(0) in NbScTiZr would mirror that of Nb-Ti-Zr.
However, a distinct behavior is observed. This suggests that the alteration of
chemical composition does not sufficiently explain the annealing temperature
dependence of the superconducting properties in NbScTiZr.

In contemplating the potential application as a superconducting wire, a
comparison of $T_{\mathrm{c}}$ and $\mu_{0}H_{\mathrm{c2}}$(0) between
NbScTiZr and NbTi alloy is imperative. The $T_{\mathrm{c}}$ and
$\mu_{0}H_{\mathrm{c2}}$(0) of the NbTi alloy are 10 K and 14.5 T,
respectively[54]. Although most of the $\mu_{0}H_{\mathrm{c2}}$(0) values in
NbScTiZr are comparable to that of the NbTi alloy, the $T_{\mathrm{c}}$ values
of NbScTiZr are 1 to 2 K lower compared to the NbTi alloy. Hence, the pursuit
of augmenting $T_{\mathrm{c}}$ without compromising
$\mu_{0}H_{\mathrm{c2}}$(0) emerges as the subsequent paramount research
endeavor.

Figure 7: (a) $C/T$ vs. $T^{2}$ plots of NbScTiZr samples investigated. The
solid lines represent the fitting results using
$\frac{C}{T}=\gamma_{\mathrm{el}}+\beta T^{2}$ relation. Temperature-dependent
$C_{\mathrm{el}}$ of (b) 400 ∘C annealed, (c) 600 ∘C annealed, (d) 800 ∘C
annealed, and (e) 1000 ∘C annealed samples. The dotted lines show the fitting
results using
$A\mathrm{exp}\left(-\frac{\Delta(0)}{k_{\mathrm{B}}T}\right)+\gamma_{\mathrm{el}}^{{}^{\prime}}T$.
The solid lines serve as guides for extracting values of discontinuity in
$C_{\mathrm{el}}$ at $T_{\mathrm{c}}$.

Figure 7(a) elucidates a comparison of the specific heat data for all samples,
presented as $C/T$ vs. $T^{2}$ plots. In each sample, the conspicuous specific
heat jump at $T_{\mathrm{c}}$ supports the bulk nature of the superconductive
transition. Notably, the transition of the as-cast sample appears relatively
broad, with an additional small anomaly detected at approximately 6.3 K. The
broad superconductive transition is observed in some bcc HEA superconductors.
For instance, the single-phase bcc (NbTa)0.67(MoHfW)0.33, a 4.3 K
$T_{\mathrm{c}}$ HEA superconductor, exhibits a broad specific heat jump,
although the underlying cause of this broadening remains unclear[57].
Additionally, phase segregation occasionally yields a broadening of the
transition, as seen in the case of bcc Ti-Hf-Nb-Ta-Re alloy represented by
(Ti20Hf10)(Nb35Ta25)Re10 and (Ti15Hf5)(Nb35Ta35)Re10 [43]. These alloys are
composed of two bcc phases with slightly different chemical compositions. The
specific heat jump of phase-segregated alloy tends to be broad due to an
inhomogeneity of chemical composition. In the present case, the additional
small anomaly below $T_{\mathrm{c}}$ may suggest the existence of an
inhomogeneous distribution of chemical composition in the as-cast sample.
Therefore, the origin of the broad superconductive transition in the as-cast
NbScTiZr alloy may be akin to that observed in the Ti-Hf-Nb-Ta-Re alloy. With
increasing annealing temperature up to 800 ∘C, the specific heat jump becomes
sharper, but in the case of annealing at 1000 ∘C, the specific heat jump
exhibits slight broadening. As previously mentioned, the degree of lattice
strain is most pronounced in the sample annealed at 400 ∘C. The non-systematic
behavior of the sharpness of the superconductive transition against the
annealing temperature indicates the weak influence of lattice strain on the
zero-field specific heat. Each data above $T_{\mathrm{c}}$ can be fitted by
the equation

$\frac{C}{T}=\gamma_{\mathrm{el}}+\beta T^{2}$ (4)

, where $\gamma_{\mathrm{el}}$ and $\beta$ are the Sommerfeld coefficient and
the lattice contribution, respectively. The resulting parameters are presented
in Table 2. It is noteworthy that both $\gamma_{\mathrm{el}}$ and $\beta$
encompass contributions attributed to the bcc and hcp phases.

The superconducting parameters regarding the specific heat, except for the as-
cast sample, are estimated through the analyses of Figs. 7(b)-(e), which
depict temperature dependences of the electronic specific heat
$C_{\mathrm{el}}$($T$), obtained by subtracting $\beta T^{3}$ from $C$($T$).
In the superconducting state, each $C_{\mathrm{el}}$($T$) can be modeled by
$A\mathrm{exp}\left(-\frac{\Delta(0)}{k_{\mathrm{B}}T}\right)+\gamma_{\mathrm{el}}^{{}^{\prime}}T$,
where $A$ is a proportional constant, $\Delta(0)$ is the superconducting gap,
$k_{\mathrm{B}}$ is the Boltzmann constant, and
$\gamma_{\mathrm{el}}^{{}^{\prime}}$ represents the electronic specific heat
coefficient associated with the hcp phase. The determined values of
$\Delta(0)$ (meV) and $\gamma_{\mathrm{el}}^{{}^{\prime}}$ (mJ/mol$\cdot$K2))
are (1.16, 2.8) for the 400 ∘C annealed sample, (1.73, 2.7) for the 600 ∘C
annealed sample, (1.84, 2.8) for the 800 ∘C annealed sample, and (1.66, 2.6)
for the 1000 ∘C annealed sample. The $\gamma_{\mathrm{el}}^{{}^{\prime}}$
values are necessary to achieve good fittings, as demonstrated by the dotted
curves in Figs. 7(b)-(e), indicating a non-superconducting behavior of the hcp
phase down to 2 K. The values of
$\frac{2\Delta(0)}{k_{\mathrm{B}}T_{\mathrm{c}}}$ are calculated as 3.22,
4.43, 4.85, and 4.56 for the 400 ∘C, 600 ∘C, 800 ∘C, and 1000 ∘C annealed
samples, respectively. The value of
$\frac{2\Delta(0)}{k_{\mathrm{B}}T_{\mathrm{c}}}$ for the 400 ∘C annealed
sample closely approximates 3.52, as expected for an s-wave BCS
superconductor. Conversely, the remaining three values exceed 3.52, suggesting
a possible strong-coupled superconductor. In the as-cast sample, the
broadening of the transition impedes the evaluation of
$\frac{2\Delta(0)}{k_{\mathrm{B}}T_{\mathrm{c}}}$. Next, we discuss the
normalized specific heat jump $\frac{\Delta
C_{\mathrm{el}}}{(\gamma_{\mathrm{el}}-\gamma_{\mathrm{el}}^{{}^{\prime}})T_{\mathrm{c}}}$,
expected to be 1.43 for the BCS model in the weak coupling limit. In our
analysis, we eliminate the contribution of
$\gamma_{\mathrm{el}}^{{}^{\prime}}$ from the $\gamma_{\mathrm{el}}$ obtained
in Fig. 7(a) because the hcp phase does not undergo a superconducting
transition. $T_{\mathrm{c}}$ is employed as the midpoint of the transition.
The discontinuities of $C_{\mathrm{el}}$ at $T_{\mathrm{c}}$ were estimated by
extrapolating the data below and above $T_{\mathrm{c}}$, as depicted by solid
lines in Figs. 7(b)-(e). The estimated values of $\frac{\Delta
C_{\mathrm{el}}}{(\gamma_{\mathrm{el}}-\gamma_{\mathrm{el}}^{{}^{\prime}})T_{\mathrm{c}}}$
are 1.63, 2.83, 3.97, and 2.68 for the 400 ∘C, 600 ∘C, 800 ∘C, and 1000 ∘C
annealed samples, respectively. Similar to the case of
$\frac{2\Delta(0)}{k_{\mathrm{B}}T_{\mathrm{c}}}$, the sample annealed above
600 ∘C appears to largely deviate from the BCS superconductor in the weak
coupling limit. It is worth noting that strong-coupled superconductivity has
recently been reported in several transition metal-based bcc alloys, such as
TiHfNbTa and Ta-Hf[24, 58]. The sharp jump in specific heat at
$T_{\mathrm{c}}$, particularly for the samples annealed at 600 ∘C or 800 ∘C,
is intriguing, notwithstanding the fact that the volume fraction of the bcc
phase approximates 50 %. In these specimens, the
$\gamma_{\mathrm{el}}^{{}^{\prime}}$ value of the hcp phase parallels
$\gamma_{\mathrm{el}}-\gamma_{\mathrm{el}}^{{}^{\prime}}$ associated with the
bcc phase. This implies that the contribution of specific heat from the hcp
phase does not substantially surpass that of the bcc phase, thereby preventing
the burying of the abrupt jump in specific heat of the bcc phase at
$T_{\mathrm{c}}$.

We now address the annealing temperature dependence of $T_{\mathrm{c}}$.
$T_{\mathrm{c}}$ generally depends on the density of states at the Fermi level
and the electron-phonon interaction. The density of states at the Fermi level
reflects the Sommerfeld coefficient of the bcc phase, represented by
($\gamma_{\mathrm{el}}-\gamma_{\mathrm{el}}^{{}^{\prime}}$). The annealing
temperature dependence of
($\gamma_{\mathrm{el}}-\gamma_{\mathrm{el}}^{{}^{\prime}}$) does not align
with that of $T_{\mathrm{c}}$ displayed in Fig. 3(b). Therefore, the electron-
phonon interaction would significantly influence $T_{\mathrm{c}}$. As depicted
in Fig. 1(b), thermal annealing has a notable impact on lattice strain, which
would also alter the electron-phonon interaction, thereby affecting
$T_{\mathrm{c}}$.

Figure 8: Annealing temperature dependence of Vickers microhardness for
NbScTiZr. The annealing temperature dependence of lattice deviation is also
shown for comparison.

Table 2 also presents the Vickers microhardness values of the examined samples
(see also Fig. 8). Although annealing at 400 ∘C leads to a slight enhancement
of hardness compared to the as-cast state, increasing the annealing
temperature above 400 ∘C systematically diminishes the hardness. Lamellar
structures are generally expected to coarsen at elevated annealing
temperatures[59]. The microstructure of the as-cast NbScTiZr exhibits a
significantly fine lamellar-like arrangement with a thickness of $\sim$ 70 nm.
While the microstructure size remains consistent at 400 ∘C annealing, grain
size coarsening is observed as the annealing temperature exceeds 400 ∘C[46]. A
spheroidization-like morphology emerges in the sample annealed at 1000 ∘C[46].
The microstructure-hardness relationship of NbScTiZr closely resembles that of
CoCrFeNiNbx eutectic HEA[60]. The microstructure of CoCrFeNiNbx maintains a
very fine lamellar arrangement up to 600 ∘C annealing, beyond which the
lamellar size systematically increases with rising annealing temperature.
Above 600 ∘C annealing, hardness shows a systematic decline. In the fine
lamellar structure, interface strengthening is suggested as the origin of high
hardness[61, 62]. The interface between two phases of the eutectic structure
acts as a resistance to dislocation motion due to the mismatch of mechanical
and structural properties at the interface. Consequently, the coarsened
microstructure with reduced interface area results in decreased hardness. The
microstructure-hardness relationship of NbScTiZr or CoCrFeNiNbx aligns well
with the scenario of interface strengthening. Figure 8 illustrates the
annealing temperature dependence of Vickers microhardness alongside that of
lattice deviation. As lattice deviation decreases, microhardness tends to
decline. Lattice deviation reflects the degree of lattice strain, which is
more pronounced in the fine eutectic structure due to the significant
influence of structural property mismatches at the interface. Hence, the
hardness-lattice deviation relation also supports the interface strengthening
mechanism. Moreover, our results suggest that Vickers microhardness serves as
an effective indicator of lattice strain in eutectic HEAs.

##  4 Summary

We investigated the superconducting properties of the as-cast and thermally
annealed NbScTiZr samples, characterized by a eutectic structure comprising
bcc and hcp phases, employing magnetic, transport, and zero-field thermal
measurements. The lattice deviation, defined as the deviation of the
experimental lattice parameter from that derived by the hard-sphere model with
no lattice strain, suggests the greatest lattice strain at the annealing
temperature of 400 ∘C. We determined $\mu_{0}H_{\mathrm{c1}}$(0),
$\mu_{0}H_{\mathrm{c2}}$(0), $\xi_{\mathrm{GL}}$(0), and
$\lambda_{\mathrm{GL}}$(0) for all samples through magnetization and
electrical resistivity measurements. The $M$-$H$ curves exhibiting negative
peaks at $\mu_{0}H_{\mathrm{p}}$ much higher than $\mu_{0}H_{\mathrm{c1}}$
reveal stronger flux pinning for the as-cast, 400 ∘C annealed, and 600 ∘C
annealed samples. Although $\mu_{0}H_{\mathrm{c2}}$(0),
$\xi_{\mathrm{GL}}$(0), and $\lambda_{\mathrm{GL}}$(0) do not systematically
depend on the annealing temperature, the annealing temperature dependence of
the Maki parameter strongly correlates with that of lattice deviation. Our
analysis suggests that larger lattice strain can enhance the Maki parameter by
elevating the orbital-limited field through the modification of flux pinning
strength. Specific heat measurements confirm the bulk nature of
superconducting transition in all samples. However, the broadness of the
superconductive transition in the as-cast state suggests the presence of
chemical inhomogeneity. In the case of zero-field specific heat, the effect of
lattice strain is less discernible compared to magnetization measurements. The
possibility of strong-coupled superconductivity is supported by
$\frac{2\Delta(0)}{k_{\mathrm{B}}T_{\mathrm{c}}}$ and $\frac{\Delta
C_{\mathrm{el}}}{(\gamma_{\mathrm{el}}-\gamma_{\mathrm{el}}^{{}^{\prime}})T_{\mathrm{c}}}$,
which exceed BCS values in the weak coupling limit. The annealing temperature
dependence of Vickers microhardness correlates well with that of the lattice
deviation. Our study indicates that the lattice parameter and Vickers
microhardness effectively reflect the degree of lattice strain in eutectic
HEAs. Additionally, lattice strain significantly influences the Maki parameter
of eutectic HEA superconductors by modifying flux pinning strength.

## Author Contributions

Jiro Kitagawa: Conceptualization; Supervision; Formal analysis; Investigation;
Writing – original draft; Writing – review & editing. Haruto Ueta: Formal
analysis; Investigation. Yuto Watanabe: Formal analysis; Investigation. Takeru
Seki: Formal analysis; Investigation. Yoshikazu Mizuguchi: Formal analysis;
Investigation; Writing - reviewing & editing. Terukazu Nishizaki: Formal
analysis; Investigation; Writing - reviewing & editing.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

J.K. acknowledges the support from a Grant-in-Aid for Scientific Research
(KAKENHI) (Grant No. 23K04570) and the Comprehensive Research Organization of
Fukuoka Institute of Technology. Y.M. acknowledges the support from a Grant-
in-Aid for Scientific Research (KAKENHI) (Grant No. 21H00151). T.N.
acknowledges the support from a Grant-in-Aid for Scientific Research (KAKENHI)
(Grant No. 20K03867) and the Takahashi Industrial and Economic Research
Foundation. We are grateful to Kousuke Ueda of Kyushu Sangyo University for
support of magnetization measurements.

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