An intrinsic model for strain tensor effects on the density of states in A15 Nb3Sn

Cryogenics 97 (2019) 50–54

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Research paper

An intrinsic model for strain tensor effects on the density of states in A15 Nb3Sn

T

Li Qiaoa, , Xin Zhanga, He Dinga, Gesheng Xiaoa, Zhiqiang Lia, Lin Yangb ⁎

a

Institute of Applied Mechanics and Biomedical Engineering, College of Mechanics, Taiyuan University of Technology, Shanxi Key Laboratory of Material Strength and Structural Impact, Shanxi 030024, People’s Republic of China b College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, People’s Republic of China

ARTICLE INFO

ABSTRACT

Keywords: A15 Nb3Sn Density of states at the Fermi surface Strain tensor effects

The superconducting critical properties of the intermetallic compound Nb3Sn are negatively affected by strain applied to the material, which is important for scientific and technological applications. A recent hydrostatic pressure experiment (Phys. Rev. B 95, 184503 (2017)) emphasized the importance of the density of states (DOS) at the Fermi surface in understanding the inherent electromechanical properties of superconducting Nb3Sn. The Nb3Sn material is subject to various strain states due to fabrication, thermal mismatch, and operation. Hence, a detailed description of the comprehensive study of the relationship between the DOS at the Fermi surface and different strain components must be established. In this paper, a model of the strain tensor effects on the DOS in an Nb3Sn compound with an A15 lattice structure is proposed on the basis of Bhatt’s model in combination with

the k · p perturbation theory results. The magnitudes of the principle strain components, together with the differences between the three quantities, intrinsically account for the diverse characteristics of the DOS at the Fermi level in Nb3Sn material undergoing different deformation patterns. The model is helpful for identifying the origin of the strain sensitivity in Nb3Sn, and developing microphysics-based strain scaling laws in practical Nb3Sn superconductors in multiple strain states.

1. Introduction The A15 superconductor Nb3Sn has promising application prospects in international thermonuclear experimental reactors (ITER), nuclear magnetic resonance (NMR) spectroscopy, and high-energy physics (HEP) [1–3]. In these high-field magnet systems, strain from the fabrication process, thermal mismatch, and operation conditions causes severe degradation in the superconducting performance [4–12]. The DOS at the Fermi surface N (EF ) is an important physical quantity to understand the strain sensitivity of the critical superconducting properties of A15 Nb3Sn. When an Nb3Sn superconductor is subject to strain, the critical current density Jc , the critical upper field Hc2 , and the critical temperature Tc reduce nonlinearly and reversibly with the applied strain. These three physical parameters are interrelated. Identifying the inherent strain sensitivity of the critical superconducting properties in Nb3Sn compounds generally starts with studying the strain dependence of the critical temperature Tc . An A15 lattice distortion induced by strain can change its vibration modes (and therefore the electronphonon interaction spectrum, the phonon density of states, and the

⁎

interaction constant) and modify the electronic structure (N (EF )). An electromechanical coupling model is presented by Markiewicz [13], which is derived on the basis of the calculation of the phonon spectrum as a function of strain and the coupling of that to a change in the critical temperature Tc through the McMillan relation [14]. An improved scaling law for strain’s effects on Nb3Sn was subsequently developed by Arbelaez et al. [15] constructed on the basis of Markiewicz's analysis. The structural and superconducting properties of Nb3Sn in the GPa range were studied by Loria et al. [16] with the aid of angular dispersive synchrotron X-ray diffraction and ab initio calculations. The critical temperature behavior of Nb3Sn under hydrostatic pressure is dictated mostly by the electronic contribution, but phonons contribute to the evident anomalies up to 6 GPa. The recent experiment by Ren et al. [17] demonstrated that under high hydrostatic pressure, the normal state resistivity above Tc exhibits a square of the temperature dependence whose prefactor A decreases with increasing pressure and Tc depends linearly on A . Since the coefficient A [N (EF )]2 , the experiment suggested that the DOS at the Fermi level plays a significant role in governing the hydrostatic pressure dependence of the critical temperature in Nb3Sn. The Markiewicz strain analysis (as well as the

Corresponding author. E-mail address: [email protected] (L. Qiao).

https://doi.org/10.1016/j.cryogenics.2018.11.002 Received 14 June 2018; Received in revised form 7 November 2018; Accepted 9 November 2018 Available online 10 November 2018 0011-2275/ © 2018 Elsevier Ltd. All rights reserved.

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following developed model) provides a phonon properties-based explanation for the strain dependence of Tc . However, the strain-modified electronic DOS at the Fermi surface is implicit and not directly present in the analysis. Explaining the strain effects on Tc and developing a microphysics-based strain scaling law represents an important connection to the microscopic understanding of strain effects on N (EF ). However, the strain dependence of the critical current density Jc originates from the field-temperature phase boundary shifts, that is, Jc changes through a variation in the temperature-dependent upper critical field Hc2 (T ) [5]. The Werthamer, Helfand, and Hohenberg (WHH) theory [18] provides a satisfactory description of the temperature dependence Hc2 (T ) . A key parameter in the analysis is the slope of the Hc2 (T ) curve at T = Tc , which relates to the DOS at the Fermi surface N (EF ) by ( µ 0 Hc 2 (T )/ T )T = Tc = 4/ kB c N (EF ) . Here, µ 0 is the magnetic permeability of a vacuum, kB is the Boltzmann constant, c is the speed of light, and is the normal state electrical resistivity that is also related to N (EF ) [17]. In the framework of the WHH theory, the anomalies in the axis strain-dependent upper critical field Hc2 of Nb3Sn wires jacketed with AISI 316 stainless steel [12] were interpreted by Qiao and Zheng [19] with the aid of a degenerate one-dimensional model accounting for the lattice distortion-induced variations in the DOS at the Fermi surface, in which the shear deformation effects were not considered for simplicity. Understanding the strain effects on N (EF ) will help contribute to further studies on the microscopic-based strain characterization of the temperature-dependent upper critical field Hc2 (T ) and therefore the dependency of the critical current Jc of Nb3Sn on temperature, strain, and field (a microphysics-based strain scaling law for Jc ). A full understanding of strain’s effects on superconducting Nb3Sn requires obtaining the DOS at the Fermi surface N (EF ) as a function of strain. The Nb3Sn cable-in-conduit conductors (CICCs) are fabricated based on a cabling scheme that corresponds to a multistage design, and the material in the CICC layer is exposed to a combination of an extremely low temperature (4.2 K), a high current density (2400 A/mm2), and a strong magnetic field (12 T) [20,21]. It produces a complex strain state with a cross-effect, potentially making a general unified model more practically applicable. Hence, a model considering the effect of different strain components on N (EF ) is indispensable in scientific and engineering research. In this paper, an intrinsic model for characterizing the strain tensor effects on the density of states at the Fermi surface is established, the accuracy of which is verified by comparing its predictions to those resulting from the first-principle simulations. This model can describe the diverse variations of the DOS at the Fermi surface in A15 Nb3Sn that undergoes various external loading modes.

On the basis of Bhatt’s model [24], the expression of the density of states function of a single Nb3Sn crystal takes the form

N ( ) = Ntot /3

3 n( i=1

Ei ), (1)

n ( ) d = 1,

where Ntot is the total number of electrons and the energies Ei are the tight binding parameters involved in the model. Each of the three terms in the N ( ) function originates from a pair of saddle points of the opposite nature at the energies Ei (in the tight-binding model of the bands, they are the saddle points at M and along X for the three symmetry directions in the Brillouin zone). Of note, Bhatt’s model is a modified idealized Jahn-Teller model. Different from the earlier one-dimensional 1/2 density of states model (N ( ) [25]) and the Cohen-Cody-Hal( ) [26]), the three-diloran step-function density of states (N ( ) mensional densities of states do not show a singular variation; the sharpest variation is the type of near an extremum or a saddle point in the band. Concomitantly, for the reproduction of a large peak in the density of states demonstrated by the specific heat data and the magnetic susceptibility data, two close-lying saddle points are required in the density of states model. An idealized model exhibiting a secondorder Jahn-Teller effect (which is one of the features of the tightbinding model) is proposed by Bhatt. The simplified Bhatt’s model further puts the two saddle points at the same energy and supposes that each pair moves together, dragging the entire density function with it, which is not true in the band-structure model [24]. In this proposed model, to exactly describe the density of states of Nb3Sn, the saddle points are set at different energies. Bhatt’s model based on a tight-binding analysis of the electronic structure of the A15 compound is formulated from the 1 orbitals on the six transition metal atom sites. It includes both nearest-neighbor (intrachain) and next-nearest-neighbor (interchain) interactions and maintains only the two relevant bands. For interchain coupling of less than approximately 15% of the nearest-neighbor interaction, the density of states shows a sharp peak arising from two saddle points in the bands, one along X for the upper band and one at M for the lower band. Regarding the lower band, a Peierls gap at the X point and a Jahn-Teller splitting at the M point can be depicted by the model. In reference to the upper band, a Jahn-Teller splitting at the X saddle points in the tetragonal phase with sublattice pairing can also be captured and tracked by the model. The contribution of the bands to the total N ( ) curve for the Nb3Sn is considered in the given model to obtain a realistic estimate of the band structure DOS near the Fermi surface. The strain in response to an external stress or force applied to the Nb3Sn induces lattice distortions, accompanied by variations in the three energies of Ei (i = 1,2,3) . Each of the energies degenerates under a zero-stress state (the crystal structure of Nb3Sn in the tetragonal phase) and splits due to the applied stress. The shifts of the energies are strongly dependent on the strain state of the lattice. To reduce the complexity and retain a good approximation of the physics involved, the density of states function N ( ) is expanded in Taylor series in powers of energies. By assuming that the variation of the three energies Ei (i = 1,2,3) is mainly responsible for the behavior of N ( ) , it provides

2. The intrinsic model Strain can cause a change in the density of states in A15 Nb3Sn, which was evidenced by Lim et al.'s findings in a hydrostatic experiment conducted in 1983 [22]. In a recent experiment, an important role played by the hydrostatic pressure-modified density of states in the Tc degradation was identified [17]. The tensor characteristics of mechanical effects make it difficult to employ a unified model calculation for the evaluation of N (EF ) in strained Nb3Sn. The density of states at the Fermi level under the application of strain is denoted by N (EF ) , the quantity of which is scalar. The tensor function representation theory [23] shows that a scalar-valued tensor function can be expressed in terms of the principle invariants of its argument. Thus, the strainmodified DOS at the Fermi surface N (EF ) can be further represented as a function of the three principle invariants of strain tensor, that is, J1 , J2 , and J3 . The principle invariants of the deformation tensor in terms of the three principle strain components 1 , 2 , and 3 intrinsically define the strain tensor effects on the density of states in Nb3Sn. The establishment of the explicit expression for N (EF ) calls for a detailed analysis of the dependence of electron DOS at the Fermi surface on the three principle strain components.

N ( ) = N0( ) + +

1 Ntot 2 3

3 i=1

3 i=1

Ntot 3 d2n dEi2

Ei0

(Ei

( )

dn (E dEi E 0 i i

Ei0 )

Ei0) 2 + ......

(2)

Note that Eq. (2) is expanded in energies rather than in strain in consideration of the physics involved and the simplified mathematical expression. The aforementioned assumption is made on the basis that the variations in the band energy dispersions with the applied strain changes the DOS. It is a good approximation of the response of the DOS to a strain perturbation. Another important consideration is the simplicity of the expression. The choice of the strain scaling relation for Nb3Sn material engineering applications is a trade-off between the 51

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L. Qiao et al.

demand for better physics-based formalism and practical applicability to experimental measurements (there are few material and fitting parameters involved in the model function, and the values of the parameters can be easily deduced from the separate strain and temperature measurements). The quantities of the three energy bands are also scalars that are invariant under the rotation of coordinate axes. They can be further represented as a function of the principle strain components. Each energy band is an isotropic tensor function of the strain. The representation of the scalar function does not change with the rotation of the coordinate system [23]. The most simple way to determine the function forms is to obtain the energy band values in the multi-axial strain state in which the principle axes align with Cartesian coordinates as well as the loading directions (namely xx = 1, yy = 2 , and zz = 3 ).

f( )=1+ + +

E10 = a3 ( 1 +

+ [b12 + a1 (

E20 = a3 ( 1 +

E2 2 {{b21

2 2 1/2 {b11 + b12 } }

2

+ 3) + a2

E30 = a3 ( 1 +

2

+ 3) + a2

+

2

3

(3b)

(3c)

the electronic structure of A15 compounds in the generalized k · p model. The parameters involved account for the reduced symmetry of the three-dimensional distorted crystal structure, the band degeneracy, and the form of energy-band dispersion resulting from the strain. Using small strain approximations leads to the expression of N ( ) , which analytically describes the effect of the strain on the DOS in A15 Nb3Sn. Another important effect contributing to the strain sensitivity of the DOS at the Fermi surface is the Fermi level shift due to the strain. It is characterized by performing a second Taylor expansion. The expression of the Fermi level in terms of the principal strain components can thus be given as

EF 1

1

EF

+

2

=0

2

+

=0

EF 3

3 =0

+ ...

(4)

Substituting = EF into the formula N ( ) in Eq. (2) gives the expression of N (EF ) :

N (EF ) = N 0 (EF ) + = N 0 (EF0 ) + +

Ntot 3

3 i=1

3 i=1

Ntot 3

( ) dN 0 dEF

( )

EF

(EF

dn (Ei dEi E i

( )

dn (Ei dEi E i

Ei0)

EF0 ) + ... , Ei0) + ...

(5)

in which another Taylor expansion is utilized to estimate the expression of the function N 0 (EF ) (it characterizes the variation in the DOS at the Fermi surface arising from the strain-induced shift in the Fermi level). The combination of Eqs. (3), (4), and (5) thus yields

N (EF ) = N 0 (EF0 ) f ( )

+[ +[

22 32

+(

2 11

2 3)]

2

+

+(3

2 1)]

2 21

+

2 22 }

+(1

)]2

2 31

+

2 32 }

2

2 12 }

(7)

1 N 0 (EF0 )

i

=

i

= a1 Ntot

EF i

( )

=0

dN 0 dEF

EF0

dn /3N 0 (EF0 ) dEi E 0 i

+ a2

Ntot 3

( )

dn dEi E 0 i

+ a3

3),

mn .

(i = 1

3, n = 1

Ntot 3

3 n= 1

( )

dn dEn E 0 n

(i = 1

3) ,

2)

mensionless momentum. In the original k · p model proposed by Lee, Birman, and Williamson [27], the involved parameters are determined by comparing them to the Nb3Sn experiments under some simplifications. The deformation-potential constant a1 = 2.11 eV and the constants a2 and a3 are neglected for simplicity. The three band parameters involved in calculating the parameters bmn (m = 1 3, n = 1 2) are, respectively, given as T = 7.33 × 10 2 eV, = 0.7 , and = 0.427. Because of the lack of the exact value of the wave vector, the parameters bmn (m = 1 3, n = 1 2) cannot be directly obtained. The value of the Fermi level EF0 calculated by the first principle method is 10.5 eV [28]. The parameters EF / i (i = 1 3) arise from the first-order approximation of the strain-induced shift in the Fermi level; they are the linear proportional coefficients describing the variations in the Fermi level value with respect to the principle strain components and are utilized to predict i (i = 1 3) ; there are no reports on the values in the literature. Discrepancies exist between the reported values of the DOS at the Fermi level N 0 (EF0 ) of Nb3Sn under no applied strain. The experiment carried out by Lim et al. [22] demonstrated that the value is approximately 5.85 eV−1, while the first-principle calculations respectively conducted by Zhang et al. [29] and De Marzi et al. [30] provided values of 10.67 eV−1 and 8.4 eV−1. The total number of electrons Ntot is 3.93 × 1022 eV−1 cm−3 [27]. The involved parameters dn/ dEi (i = 1 3) describe the variations in the DOS on the energies Ei (i = 1 3) and contribute to the determinations of i (i = 1 3) and 3) ; due to the lack of the coefficient measuring the width of i (i = 1 the peak density function, the parameters cannot be directly given. Although some of the original physical quantities implicitly included in Eq. (7) are provided by previous studies [22,27–30], the exact determination of the ultimate parameters cannot be achieved because of the assumptions made in the modeling process. Hence, the coefficients in the proposed model are determined by comparing them to the firstprinciple calculations, which is an indirect way. The validity of the parameter estimates is checked by comparing the model predictions to the results of the first-principle calculations.

in which ai (i = 1 3) represents the deformation-potential constants and bmn (m = 1 3, n = 1 2) are the parameters used to characterize

EF = EF0 +

3{

2 31

12

function [24] and the three-dimensional k · p model for the electronic structure of A-15 compounds [27]. i (i = 1 3) and i (i = 1 3) characterize the strain-modified energy bands; they relate to the deformation-potential constants. mn (m = 1 3, n = 1 2) provides the energy bands of Nb3Sn under zero strain; they correlate with the di-

+

2 2 1/2 {b31 + b32 } }

2 1/2 2)] }

+ [b32 + a1 ( 1

(3a)

2 2 1/2 {b21 + b22 } }

2{

+[

Under the application of strain, the atomic system's point-group symmetries are lowered by the components of the strain tensor. It induces the evolution of the electronic bands and variations in the electron DOS at the Fermi surface. The characteristic parameters in the model determine the variation tendency of N (EF ) for Nb3Sn in different strained states. The analysis shows that the three principle strain components ( 1, 2 , and 3 ) and their differences intrinsically define the strain tensor effects on the density of states in Nb3Sn. It should be noted that in the model, the shear strains are implicit. The proposed model can provide a general description of strain effects on the DOS at the Fermi surface in A15 Nb3Sn, which is not limited to the case of a certain loading mode. The coefficients in Eq. (7) originate from Bhatt’s density of states

+ 3) + a2 1+

2 1/2 1)] }

+ [b22 + a1 ( 3 E3

2 {{b31

2

2 1/2 3)] }

2

+

2 11

1{ 2 21

= bmn / a1 (m = 1

from the k · p theory, the dispersions of the three-dimensional energy bands induced by small strains can thus be given by

E1

3 3

+

+

The values of the involved parameters describing the strain sensitivity are

In Ref. [27], a k · p Hamiltonian around the Brillouin zone corner or R point was constructed and the variations in the energies under the application of multi-axial strain were characterized. By drawing lessons

2 {{b11

2 2

1 1

(6)

with 52

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L. Qiao et al.

3. Results and discussion

magnitudes of the principle strain components, as well as the differences between the three quantities, account for the variation characteristics of the DOS at the Fermi level in the strained Nb3Sn material. The strain tensor effects on the DOS at the Fermi surface can be described by the analytical model. To further understand the differences between the curve patterns, the degenerate expressions characterizing the various strain states are respectively derived in terms of the model. For the hydrostatic loading mode, the principle strains are given by 1 2 P (in which P is the hydrostatic pressure and E and 1 = 2 = 3 = E are respectively the elastic modulus and Poisson's ratio). The expression of the model degenerates into

f( )=1+(

f( )=1+(

+

5

10

(a) 15

3{

20

The normalized DOS at the Fermi surface f( )

The normalized DOS at the Fermi surface f( )

0.9 0.8 0.7 the first-principle data[31] the model fitting curve

(c) 0.5

1.0

1.5

2.0

2.5

1

3)

2

P.

E

(8a)

2

3

)

a

+

2{

2 21

+[

a (1

22

+ )]2

2 22 },

+ 2 31

+[

+

32

a (1

2 31

+ )]2

+

2 32 }

(8b)

the first-principle data[30] the model fitting curve

0.9

-1.0

-0.5

0.0

(b)

0.5

1.0

The applied uniaxial strain (%)

1.0

0.0

+

1.0

The hydrostatic pressure (GPa)

0.6

2

Eq. (8b) shows that the variations in the DOS at the Fermi surface are

0.9

0

+

1 2 21

1.0

the first-principle data[29] the model fitting curve

1

Substituting the parameter values into the simplified model gives a linear decrease in the DOS at the Fermi surface with the increase in the hydrostatic pressure. It is just the law revealed by the experiments [17,22] and the first-principle calculations. The product of the sum of the parameter values 1 + 2 + 3 and the mechanical parameter 1 2 is E the proportional constant of the change. As the Nb3Sn material is subjected to the uniaxial loading mode, the principle strains are 1 = a (in which a is the applied uniaxial strain) and 2 = 3 = a . They lead to the following expression for the tendency of asymmetry:

The normalized DOS at the Fermi surface f( )

The normalized DOS at the Fermi surface f( )

To examine the model, a comparison of its predictions with the firstprinciple data [29–31] is presented. The strain-induced modifications of the DOS at the Fermi surface as the Nb3Sn single crystal undergoes diverse external loading modes: (a) the hydrostatic mode, (b) the uniaxial mode, (c) the shear loading mode, and (d) the torsion loading mode are shown in Fig. 1. Eq. (7) is utilized to fit the first-principle results, and the fitting curves are also displayed in Fig. 1. The optimized curve fitting yields sets of values given in Table 1 (the elastic modulus and Possion's ration of single crystal Nb3Sn is, respectively, 194.56 GPa and 0.4 [29]). The figure indicates that the decrease in the DOS at the Fermi level can be caused by the mechanical deformation, and the tendency of the curves depends on the applied loading mode. For the hydrostatic mode, the increase in the pressure causes the linear decrease in the DOS at the Fermi energy, while the other loading modes contribute to the obviously nonlinear response. When the Nb3Sn material is subject to uniaxial loading, the variation in N (EF ) changing with the axis strain demonstrates an asymmetrical profile (in a compressive zone and in a stretching zone). This is in qualitative accordance with the Tc , Hc , and Jc behaviors of uniaxial strained Nb3Sn material [5]. The variation curves show different characteristics for the shear loading mode and the torsion loading mode. With the increase in the applied strain, the N (EF ) value of the shear-deformed Nb3Sn decreases more rapidly than Nb3Sn under the torsion loading case as the critical strain point (approximately 1%) is approached, while the opposite trend is identified after the critical point. This is because the principal strain components exert different influences on the electronic structure variations and the Fermi surface topology changing in strained Nb3Sn. The

3.0

The shear strain (%)

1.0 0.9 0.8 0.7 0.6

the first-principle data[31] the model fitting curve

0.5 0.4

(d) 0.0

0.5

1.0

1.5

2.0

2.5

3.0

The torsion strain (%)

Fig. 1. The strain-induced variations in the DOS at the Fermi surface as a Nb3Sn single crystal experiencing diverse external loading modes: (a) the hydrostatic mode, (b) the uniaxial mode, (c) the shear loading mode, and (d) the torsion loading mode. 53

Cryogenics 97 (2019) 50–54

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Table 1 The strain sensitivity parameters in the developed model. 12.57

−9.225

12

11

0.00475

−35.5

21

0.002

2

1

3

2

1

0.975

−7.49 31

22

0.01528

0.002

0.0128

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3

−5.65 32

0.001

asymmetrical and nonlinear in the tensile and compressive regions and show obviously different patterns compared to the hydrostatic case. Meanwhile, because 2 equals 3 ( 2 3 = 0 ), the simplified expression shows no dependence on the parameter values of 1, 11, and 12 , and the variation in the nonlinear feature is mainly determined by the sets of parameter values for ( 2 , 21, and 22 ) and ( 3 , 31, and 32 ). The parameters i (i = 1 3) dominate in the first linear term, and this is the same as the result of the hydrostatic pressure mode. The degenerate expressions for the Nb3Sn material that undergoes external shear load or torsion load can be respectively written as: ( 1

f( )=1+ +

2 21

2{ 2 21

+

in which written as

1

+[

2 11

1{

+[

12

2 11

+ 2 ]2

+

2 12 }

]2

22 2 31

+[

32

2 31

+ 2 ]2

2 32 }

+

(8c)

is the applied shear strain, and the principle strains are = 2 , 2 = 0 , and 3 = 2 , and 2)

2

2{ 2 21

+

2 22 }+ 3 {

( 1

f( )=1+ +

3)

2

+

2 21

+

+[

22

2 22 }+ 3 {

2 11

1{

2 2 31

+[

12

2

]2

2 11

+

2 12 }

]2

+[

32

+

]2

2 31

+

2 32 }

(8d)

in which is the applied torsional strain and the principle strains are , and 3 = 0 . Eqs. (8c) and (8d) indicate that some dif1 = 2, 2 = 2 ferences in quantity have emerged in the two nonlinear curves. These originate from the magnitudes of the principle strain components as well as the differences between the three quantities. 4. Conclusions In conclusion, we propose an intrinsic model for strain tensor effects on the DOS in A15 Nb3Sn. A qualitative agreement between the theoretical predictions and the first-principle calculations is achieved. It offers a possibility of analytical study of the inherent electromechanical properties of superconducting Nb3Sn. The proposed model is helpful for understanding the strain sensitivity in Nb3Sn and establishing the microphysics-based strain scaling law in practical Nb3Sn superconductors in various strain states. Acknowledgments The authors thank the anonymous reviewers for improving the quality of this manuscript. This work was supported by the National Natural Science Foundation of China (grant no. 11772212 and grant no. 11402159). References [1] Zhou YH, Wang XZ. Review on some key issues related to design and fabrication of superconducting magnets in ITER. Sci Sin-Phys Mech Astron 2013;43:1558–69. [2] Devred A, Backbier I, Bessette D, Bevillard G, Gardner M, Jong C, et al. Challenges

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