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A model for the compositions of non-stoichiometric intermediate phases formed by diffusion reactions, and its application to Nb3Sn superconductors

Scientific Reports volume 6, Article number: 19096 (2016) | Download Citation


In this work we explore the compositions of non-stoichiometric intermediate phases formed by diffusion reactions: a mathematical framework is developed and tested against the specific case of Nb3Sn superconductors. In the first part, the governing equations for the bulk diffusion and inter-phase interface reactions during the growth of a compound are derived, numerical solutions to which give both the composition profile and growth rate of the compound layer. The analytic solutions are obtained with certain approximations made. In the second part, we explain an effect that the composition characteristics of compounds can be quite different depending on whether it is the bulk diffusion or grain boundary diffusion that dominates in the compounds, and that “frozen” bulk diffusion leads to unique composition characteristics that the bulk composition of a compound layer remains unchanged after its initial formation instead of varying with the diffusion reaction system; here the model is modified for the case of grain boundary diffusion. Finally, we apply this model to the Nb3Sn superconductors and propose approaches to control their compositions.


Intermediate phases with finite composition ranges represent a large class of materials, and their compositions may influence their performance in application, as demonstrated in a variety of materials, such as electrical conductivity of oxides (e.g., TiO2−y1), electromagnetic properties of superconductors (e.g., Nb3Sn and YBa2Cu3O7−y2), and mechanical properties of some intermetallics (e.g., Ni–Al0.4–0.553), etc. For instance, the superconducting Nb3Sn phase, which finds significant applications in the construction of 12–20 T magnets4,5,6, has a composition range of ~17–26 Sn at.%7,8, and its superconducting transition temperature and magnetic field decrease dramatically as Sn content drops4,8,9. The Nb3Sn phase, which is generally fabricated from Cu-Sn and Nb precursors through reactive diffusion processes, is always found to be Sn-poor (e.g., 21–24 at.%4,9,10), making composition control one of the primary concerns in Nb3Sn development since 1970s10. Although a large number of previous experiments (e.g.,4,9,10,11,12) have uncovered some factors that influence the Sn content, it is still a puzzle what fundamentally determines the Nb3Sn composition. This work aims to fill that gap. Here it is worth mentioning that the composition interval of a compound layer does not necessarily coincide with its equilibrium phase field ranges – the former can be narrower (e.g., the Nb3Sn example above) if the inter-phase interface reaction rates are slow relative to the diffusion rate across the compound, which results in discontinuities in chemical potentials at the interfaces.

There have been numerous studies regarding diffusion reaction processes, most of which have focused on layer growth kinetics (e.g.,13,14,15,16,17), compound formation and instability (e.g.,15,16,17), phase diagram determination (e.g.,18), and interdiffusion coefficient measurement (e.g.,19), while a systematic model exploring how to control compound composition is still lacking. We find it indeed possible to modify the model developed by Gosele and Tu14 for deriving the layer growth kinetics of compounds to calculate their compositions; however, certain assumptions (e.g., steady-state diffusion and first-order interface reaction rates) that the model was based on may limit the accuracy of the composition results. In this work, we aim to develop a rigorous, systematic mathematical framework for the compositions of intermediate phases.

The model

Let us consider that a non-stoichiometric AnB compound is formed in a system of M–B/A, where M is a third element that does not dissolve in AnB lattice20. The use of the third element M is to decrease the chemical potential of B, so that unwanted high-B A–B compounds (e.g., NbSn2 and Nb6Sn5 in the Nb-Sn system7) that would form in the B/A binary system can be avoided. With the M–B, AnB, and A-rich phases denoted as α, β, and γ, respectively, a schematic of the α/β/γ system for a planar geometry is shown in Fig. 1. Let us denote the α/β and β/γ inter-phase interfaces as I and II, respectively, and the mole fractions, chemical potentials, activities, and diffusion fluxes of B in the β phase at interfaces I and II as XIβ, μIβ, aIβ, JIβ, and XIIβ, μIIβ, aIIβ, JIIβ, respectively. The maximum and minimum mole fractions of B in β phase (i.e., AnB compound) from the phase diagram are set as XIβ,eq and XIIβ,eq, respectively. Let us also denote the μBs and aBs of α and γ as μBα, aBα, and μBγ, aBγ, respectively. Let us assume that the solubility of B in γ phase is negligible7. An isothermal cross section of such an M–A–B phase diagram at a certain temperature is shown in Fig. 2. This is the case we see for the Nb3Sn example above (for which A stands for Nb, B for Sn, and M for Cu), but the model below can be modified for other cases. Similar to the Cu-Nb-Sn system, let us assume B is the primary diffusing species in the β phase21 and that the α phase can act as an intensive sink for B vacancies in order for it to be an efficient source of B atoms for β layer growth, and that the diffusivity of B in α is high so that the α phase remains homogeneous during the growth of β layer11.

Figure 1: Schematics of (a) the α/β/γ diffusion reaction system in the planar geometry, and (b) XB profiles of the system.
Figure 1
Figure 2: Schematic of an isothermal cross section of the M–A–B ternary phase diagram.
Figure 2

The shaded region shows the equilibria among M-X1 B, A-XIβ,eq B, and γ phases, and the dashed line shows the equilibrium between α and β phases.

Here we assume that diffusion occurs by vacancy mechanism and the total atomic flux is balanced by the vacancy flux. As discussed in the papers by Svoboda and Fischer et al.22,23,24, the presence of various types of sinks or sources for vacancies may lead to quite different diffusional and conservation laws and equations. For this model, we assume that B vacancies are generated by the reaction at interface II (as will be discussed in detail later), and then diffuse across the β layer to interface I, where they are annihilated by B atoms from α phase (the B source). For the simplicity of the model, we assume that there are no sinks or sources for vacancies in the bulk or grain boundaries of β phase, while the only sink in the system for B vacancies is the α phase. The following model can be modified for cases with other types of sinks or sources for vacancies using the models by Svoboda and Fischer et al.22,23,24.

In this work let us assume the diffusivity of B in β phase, D, and the molar volume of β phase, Vmβ, do not vary with XB, in which case the continuity equation in the AnB layer is given by:

According to mass conservation, in a unit time the amount of B transferring across the interface I should equal to that diffusing into the β layer from the interface I, and the amount arriving at the interface II should equal to that transferring across it, i.e., dn/dt|I  = JIβAI, and dn/dt|II = JIIβAII, where AI and AII are the areas of the interfaces I and II, respectively. The molar transport rate dn/dt across an interface equals to rAint∙exp(−Q/RT)∙[1−exp(−Δμ/RT)], where r is the transfer rate constant for this interface with the unit of mol/(m2∙s), Aint is the interface area, Q is the energy barrier, R is the gas constant, T is the temperature in K, and Δμ is the driving force for atom transfer. For the interface I, Δμ|I = μBαμIβ. For the interface II, Δμ|II = μIIβμBγ, and μBγ = μB(A−XIIβ,eq B). With JB = −(D/Vm)∙(∂XB/∂x), we have:

Eqs. (2) and (3) are the boundary conditions for Eq. (1). Note that XB in α phase, XBα, drops with annealing time as B in α is used for β layer growth, so μBα drops with t:

where nM and nB0 are the moles of M and B in the M-B precursor. For those systems without the third element, μBα is constant, and Eq. (4) is not needed. In addition, since the B atoms diffusing to the interface II are used to form new β layers, we have:

Eqs. (1)–(5) are the governing equations for the system set up above, solutions to which give both the XB(x, t) and the l(t) of a growing AnB layer. It should be noted that for the systems with large volume expansion associated with transformation from γ to β, stress effects need to be considered25.

To simplify Eqs. (2) and (3), we notice that 1−exp[−(μBαμIβ)/RT] = 1−aIβ/aBα, since μBαμIβ = RTln(aBα/aIβ); similarly, 1−exp[−(μIIβμBγ)/RT] = 1−aBγ/aIIβ. Let us also denote D/[VmrI∙exp(−QI/RT)] as φI, and D/[VmrII∙exp(−QII/RT)] as φII: clearly φI and φII represent the ratios of diffusion rate over interface reaction rates, and have a unit of meter. Then Eqs. (2) and (3) can be respectively written as:

Analytic and Numerical Solutions

Let us first consider two extreme cases. First, for the case that the interface reaction rates are much higher than the diffusion rate across the β layer (i.e., diffusion-rate limited), φI and φII are near zero; according to Eqs. (2)–(3), , μBs are continuous at both interfaces, so XIIβ = XIIβ,eq. Suppose μBα and the position of interface I, xI, are both constant with time, then XIβ is also constant, and the solutions to Eqs. (1) and (5) are respectively XB(x, t) = XIβ−(XIβXIIβ,eq)∙erf{(xxI)/[2√(Dt)]}/erf(k/2) and l = k√(Dt) for the β layer, where k can be numerically solved from k∙exp(k2/4)∙erf(k/2) = 2/√π∙(XIβXIIβ,eq)/XIIβ,eq. For instance, for XIβ = 0.26 and XIIβ,eq = 0.17, k =0.953. On the other hand, if the interface reaction rates are much lower than the diffusion rate across β (e.g., as the β layer is thin), φI and φII are large; according to Eqs. (2) and (3), XB and JB are nearly constant in the entire β layer. Thus, (1−aB/aBα)/φI = (1−aBγ/aB)/φII, from which aB can be calculated. Integration of Eq. (5) gives: l t, and the pre-factor depends on the interface reaction rates.

For a general case between these two extremes, the equations can only be solved with the μ(X) relations of α and β provided. Next, let us consider a compound with a narrow composition range, so that as a Taylor series expansion is performed around XIIβ,eq for its a(XB) curve, high-rank terms can be neglected because |XXIIβ,eq| ≤ (XIβ,eqXIIβ,eq) is small; that is, aXaBγ + κ(XXIIβ,eq), where κ is the linear coefficient of the a(X) curve. Given the complex boundary conditions for Eq. (1), to obtain the analytic solutions we introduce a second approximation if the β composition range is narrow: the X(x) profile of the β layer is linear so that at a certain time J is constant with x, such that −(∂XB/∂x)|I ≈ −(∂XB/∂x)|II ≈(XIβXIIβ)/l. With these two approximations, we can solve Eqs. (6) and (7) and obtain that:

where η = φIIaBα/(φIaBα+κl). Then aIβ can be calculated from aIIβ, and XIβ and XIIβ can be calculated from aIβ and aIIβ using X = XIIβ,eq +(aXaBγ)/κ.

To verify the results, the equations are solved for a hypothetical system analytically and numerically, with and without the assumption that X(x) is linear, respectively. The obtained composition profiles are shown in Fig. 3(a). For simplicity, μBα of the system is set as μB(A−XIβ,eq B) and is constant (for Nb3Sn systems, this means that Nb6Sn5 serves as Sn source), and the other parameters are specified in the figure. The difference between the analytic and numerical solutions is <0.1%, showing that the approximation of linear X(x) is good if the composition range is small (2 at.% in this case). The l(t) result (where t is the annealing time after the incubation period) from the numerical calculations is shown in Fig. 3(b). While the analytic l(t) solution is complicated, some l(t) relations with simple forms can be used as approximations. The most widely used l(t) relation for the case of constant μBα is l = btm, in which m =1 for reaction-rate limited and m =0.5 for diffusion-rate limited; however, a defect with this relation is that as l increases from zero, it may shift from reaction-rate limited to diffusion-rate limited, so m may vary with t. Here a new relation l = q[√(t+τ)−√τ] (where q is a growth constant and τ is a characteristic time) is proposed. Such a relation is consistent with l2/v1+l/v2= t (where v1 and v2 are constants related to diffusion rate and interface reaction rates, respectively) proposed by previous studies14,15. This relation overcomes the above problem because as t τ, l  =[q/(2τ)]∙t and as t τ, l = qt. As can be seen from Fig. 3(b), a better fit to the numerical l(t) curve in the whole range is achieved by l = q(√(t+τ)−√τ).

Figure 3
Figure 3

(a) The calculated XB(x) profiles of the hypothetical system for the analytic and numerical solutions, with and without the assumption that XB(x) is linear, respectively. (b) The l(t) results from the numerical calculations, with the fits of l = q[√(t+τ)−√τ] and l = btm.

Results and Discussion

Before discussing the application of this model to a specific material system, it must be pointed out that all of the analysis and calculations above are for the case that B diffuses through β bulk. In such a case, for an α/β/γ system, as μBα drops with the growth of β layer, XB(x) of the entire β layer should decrease with μBα, because μBαμIβμIIβμBγ. Finally, one of two cases will occur: either μBα drops to μBγ (if A is in excess) so the system ends up with the equilibrium among γ, A–XIIβ,eq B, and M–X1 B (as shown by the shaded region in the isothermal M–A–B phase diagram in Fig. 2), or γ is consumed up and β gets homogenized with time and finally μB(β)= μB(α) (as shown by the dashed line in Fig. 2). In either case, β layer eventually reaches homogeneity.

However, we find that the composition could be different for a compound in which the bulk diffusion is low while grain boundary diffusion dominates. One such example is Nb3Sn, the composition of which displays some extraordinary features. As an illustration, the XSns of a Cu-Sn/Nb3Sn/Nb diffusion reaction couple after various annealing times are shown in Fig. 4. Clearly, as the XSn (and μSn) of Cu-Sn drop with time, the XSns of Nb3Sn do not drop accordingly; instead, they more or less remain constant with time. In addition, from 320 hours to 600 hours, although Nb has been fully consumed, the XSn of Nb3Sn does not homogenize (i.e., the XSn gradient does not decrease) with time. In many other studies on Cu-Sn/Nb systems with Nb in excess (e.g.,4,9,10), even after extended annealing times after the Nb3Sn layers have finished growing (which indicates that the Sn sources have been depleted, i.e., μSns have dropped to μBγ), XSns of Nb3Sn remain high above XIIβ,eq, without dropping with annealing time.

Figure 4: The measured XSns of a Cu-Sn/Nb3Sn/Nb system after various annealing times at 650 °C.
Figure 4

The standard deviation in the Sn content measurements is around ±0.5 at.%.

The reason for these peculiarities is that grain boundary diffusion in Nb3Sn dominates due to extremely low bulk diffusivity (e.g., lower than 10−23 m2/s at 650 °C)21,26,27 and small Nb3Sn grain size (~100 nm). In this case, our model and equilibrium-state analysis apply only to the diffusion zones (i.e., the grain boundaries and the inter-phase interfaces) instead of the bulk. To clarify this point more clearly, a schematic of the diffusion reaction process is shown in Fig. 5. At time t1, at the β/γ interface, high-B AnB (L2 layer) reacts with γ (L3 layer) to form some new AnB cells, leaving B vacancies (noted as VBs) in L2 layer (time t2). If bulk diffusivity is high, VBs simply diffuse through bulk (e.g., from L2 to L1, as shown by grey dotted arrows) to the B source. If bulk diffusion is frozen, the VBs diffuse first along β/γ inter-phase interface (as shown by green solid arrows), and then along β grain boundaries to the B source. This process continues until this L3 layer entirely becomes AnB (time t3), so the reaction frontier moves ahead to L3/L4, while the L2/L3 interface now becomes an inter-plane inside AnB lattice. If bulk diffusion is frozen, the VBs in the L2 layer that have not diffused to B source will be frozen in this layer forever, and will perhaps transform to A-on-B anti-site defects later (e.g., for Nb3Sn, Nb-on-Sn anti-sites are more stable than Sn vacancies28). Since these point defects determine the AnB composition, the XB in this L2 layer cannot change anymore regardless of μB variations in grain boundaries. That is to say, XB of any point is just the XIIβ of the moment when the reaction frontier sweeps across this point, i.e., the XB(x) of the β layer is simply an accumulation of XIIβs with l increase. Returning to Fig. 3(a), the dashed lines display the evolution of XB(x) with l increase for bulk diffusion, while that for grain boundary diffusion is shown by the solid lines. Since the energy dispersive spectroscopy (EDS) attached to scanning electron microscopes (SEM) that is used to measure the compositions typically has a spatial resolution of 0.5–2 μm, and thus mainly reflects the bulk composition, the composition characteristics of Nb3Sn layers as described above can be explained. It should be noted that knowledge of the difference between bulk diffusion and grain boundary diffusion is important in controlling the final composition of a compound. For instance, if bulk diffusivity is high, one method to form high-B AnB is increasing the starting B/A ratio so that after long annealing time for homogenization subsequent to the full consumption of A, μB(α)= μB(A−XIβ,eq B). However, our experiments demonstrate that for compounds with low bulk diffusivity (e.g., Nb3Sn), such an approach does not work; instead, controlling the XIIβs while the compounds are growing is the only way. For those compounds with low but non-negligible bulk diffusivities, their compositions would be between these two extremes.

Figure 5: A schematic of the diffusion reaction process for grain boundary diffusion.
Figure 5

Then what determines the bulk composition as grain boundary diffusion dominates? From Fig. 5, it can be clearly seen that there is a competition deciding the VB fraction in the frontier AnB layer: at t2 the reaction across the β/γ interface produces VBs in L2 layer, while the diffusion of B along β grain boundaries and α/β interface fills these VBs. Thus, if the diffusion rate is slow relative to the reaction rate at interface II (i.e., φII is low), a high fraction of VBs would be left behind as the interface II moves ahead, causing low B content; if, on the other hand, the diffusion rate is high relative to the reaction rate at interface II, the AnB layer has enough time to get homogenized with the B source, causing low XB gradient. In this case, the μB of B source and the reaction rate at interface I together set a upper limit for μB of β.

Next, we will modify the above model for the case of grain boundary diffusion for quantitative analysis. As pointed out earlier, the chemical potentials of grain boundaries can change with μBα and l, while those of the bulk cannot. In such a case, μIβ and μIIβ (suppose the diffusivities along the inter-phase interfaces are large) can still be calculated using our model, provided that the μ(X) relation and D of the β grain boundary (instead of the bulk) are used in all of the equations, and that φI and φII are multiplied by a factor of ∑AGB/Aint (where ∑AGB is the sum of the cross section areas of the grain boundaries projected to the inter-phase interfaces), because B diffuses only along β grain boundaries while reactions occur at the entire interfaces. Approximately, ∑AGB/Aint ≈ [1−d2/(d + w)2] ≈ 2w/d (where w is the β grain boundary width, and d is the grain size). Apparently, grain growth with annealing time reduces the diffusion rate. According to Eq. (8), aIIβ is determined by η and aBα, and increases with them, as shown by Fig. 6. Since η = φIIaBα/(φIaBα + κl)=1/[φI/φII + κl/(φIIaBα)], clearly η decreases as φI/φII and l increase, and the influence of l (which reflects the XIIβ-x gradient) is mitigated as φIIaBα increases. Thus, to improve XIIβ of AnB at l =0, one should increase μBα and the reaction rate at interface I, and reduce the reaction rate at interface II; meanwhile, to reduce XIIβ(x) gradient, one should increase φII (which means improving the diffusion rate or reducing the reaction rate at interface II) and aBα. Apparently, these quantitative conclusions are consistent with the above qualitative analysis.

Figure 6: The variation of aIIβ with η and aBα, according to Eq. (8).
Figure 6

Next let us compare this model with the example of Nb3Sn. It has been well established from experimental work that there are mainly two factors that can significantly influence the Sn content of Nb3Sn in a Cu-Sn/Nb3Sn/Nb diffusion reaction couple: heat treatment temperature and Cu-Sn source. The heat treatment temperature can simultaneously influence multiple factors of Eq. (8), such as aBα, D, and reaction rates at both interfaces, etc. Thus, the explanation of the influence of temperature on Sn contents using this theory requires knowledge of the quantitative variations of these factors with temperature. For the other factor, Cu-Sn source, the diffusion reaction couples can be classified into two types based on the Cu-Sn source: the type I uses bronze (with Sn content in Cu-Sn typically below 9 at.%) as Sn source, and the type II uses high-Sn Cu-Sn (e.g., Cu-25 at.% Sn). Previous measurements4,9,10,11,29 show that both types of samples have Sn contents above 24 at.% for the Nb3Sn layer next to the Cu-Sn source; however, they have distinct Sn content gradients as the Nb3Sn layers grow thicker: the type I generally has Sn content gradients above 3 at.%/μm29, while those of the type II are below 0.5 at.%/μm4,9,10. Such a difference in the Sn content gradients leads to distinct grain morphologies and superconducting properties. The different XSn gradients in the two types of samples with different Cu-Sn sources can be easily explained by our theory above: according to Eq. (8), increased μBα can decrease XSn gradients. It may also need further investigation regarding whether Cu-Sn source can also influence diffusion rates in Nb3Sn layer (e.g., via thermodynamic factor), because greater D leads to greater φII, which helps decreasing XSn gradients. As to the phenomenon that different wires have similar XSn in the Nb3Sn layer next to the Cu-Sn source, the relation between μSn(Cu-Sn) and μSn(Nb-XSn Sn) is required. The Cu-Sn system has been well studied, and the phase diagram calculated by the CALPHAD technique using the thermodynamic parameters given by ref. 30 is well consistent with the experimentally measured diagram31. Thus, the parameters from ref. 30 are used to calculate μSn of Cu-Sn, which is shown in Fig. 7. On the other hand, although thermodynamic data of Nb-Sn system were proposed by refs. 30 and 32, in these studies Nb3Sn was treated as a line compound. However, some information about μSn of Nb3Sn can be inferred from its relation with μSn of Cu-Sn: since Cu-7 at.% Sn leads to the formation of Nb-24 at.% Sn near the Cu-Sn source29, we have μSn(Cu-7 at.% Sn) ≥ μSn(Nb-24 at.% Sn). Thus, the expected approximate μSn(Nb-XSn Sn) curve in a power function is shown in Fig. 7. Furthermore, we can also infer that the Sn transfer rate at the Cu-Sn/Nb3Sn interface must be much faster than that at the Nb3Sn/Nb interface, so μSn discontinuity across the interface I is small. These explain why low-Sn Cu-Sn can lead to the formation of high-Sn Nb3Sn. It is worth mentioning that from Fig. 7, it is clear that the Taylor series for the true a(X) relation of Nb3Sn have more high-rank terms than a(X)≈aBγ + κ(XXIIβ,eq); however, our numerical calculations show that adding high-rank terms to the a(X) relation does not lead to different conclusions regarding the influences of aBα, φI, φII, and l on XIIβ. Thus, the above qualitative and quantitative analysis still applies.

Figure 7: The variation of μSn with XSn for Cu-Sn calculated based on thermodynamic data given in ref. 30, and a rough, speculative μSn(XSn) relation for Nb3Sn sketched according to the phase formation relation between Cu-Sn and Nb3Sn.
Figure 7

In summary, a mathematical framework is developed to describe the compositions and layer growth rates of non-stoichiometric intermediate phases formed by diffusion reactions. The governing equations are derived and analytic solutions are given for compounds with narrow composition ranges under certain approximations. We also modify our model for compounds in which bulk diffusion is frozen, of which the bulk is not in equilibrium with the rest of the system. Based on this model, the factors that fundamentally determine the compositions of non-stoichiometric compounds formed by diffusion reactions are found and approaches to control the compositions are proposed.


For the Cu-Sn/Nb3Sn/Nb diffusion reaction couples that were used for Sn content measurements (the results of which are shown in Fig. 4), the initial composition of the precursor Cu-Sn alloy was Cu-12 at.% Sn. The samples were reacted at 650 °C for 65 h, 130 h, 320 h, and 600 h. Then the surfaces of the samples were polished to 0.05 μm and the compositions were measured using an EDS system attached to an SEM. An accelerating voltage of 15 kV was used for the quantitative line scans. A standard Nb-25 at.% Sn bulk sample provided by Dr. Goldacker from Karlsruhe Institute of Technology was used for calibrating the Sn content of the samples. The standard deviation in the measurements was found to be about ± 0.5 at.%.

Additional Information

How to cite this article: Xu, X. and Sumption, M. D. A model for the compositions of non-stoichiometric intermediate phases formed by diffusion reactions, and its application to Nb3Sn superconductors. Sci. Rep. 6, 19096; doi: 10.1038/srep19096 (2016).


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The authors thank S. Dregia and J. Morral for useful discussions, and X. Peng and Hyper Tech Research Inc. for providing Nb3Sn samples for analysis. The work is funded by the US Department of Energy, Division of High Energy Physics, under an SBIR program.

Author information


  1. Department of Materials Science and Engineering, the Ohio State University, Columbus, OH 43210 USA

    • X. Xu
    •  & M. D. Sumption


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X.X. initiated this study, developed the model, and wrote the manuscript. M.D.S. supported this work, discussed the results, and reviewed the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to X. Xu.

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