Calculation and measurement of critical temperature in thin superconducting multilayers
Songyuan Zhao, David J. Goldie, Chris N. Thomas, Stafford Withington
CCalculation and measurement of critical temperature in thin superconductingmultilayers
Songyuan Zhao, ∗ D. J. Goldie, C. N. Thomas, and S. Withington Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 OHE, United Kingdom. (Dated: October 18, 2018)We have applied the Usadel equations to thin-film multilayer superconductors, and have calculatedthe critical temperature for general thin-film S − S (cid:48) bilayer. We extended the bilayer calculation togeneral thin-film multilayers. The model demonstrates excellent fit with experimental data obtainedfrom Ti-Al bilayers of varying thicknesses. Keywords: proximity effect, critical temperature, the Usadel equations, multilayers
I. INTRODUCTION
There is increasing interest in proximity-coupled super-conducting multilayers. This interest is fuelled by theirvarious potential applications in the fields of transition-edge sensors (TESs) [1], kinetic inductance detectors(KIDs) [2], superconducting electronics [3–5], supercon-ducting transmission line devices [6], Josephson junctions[7, 8], and SIS mixers [9]. In the field of TESs, multilay-ers have been studied to incorporate high conductivitynormal metals, and to reproducibly control the transi-tion temperature ( T c ) [10]. In the field of KIDs, multi-layers have been studied to achieve tuneable detectionfrequency thresholds, control over acoustic impedancematching, and protection of vulnerable materials throughusage of self-passivating outer layers [11–14].The superconducting properties of multilayers are gov-erned by their individual layer material properties, ge-ometries, and interface characteristics [15]. Calculationsof the multilayer T c from these factors are of consider-able value in reducing the time and effort spent on trial-and-error fabrications. In many of the above-mentionedapplications, it is important to reliably control the resul-tant multilayer T c so as to accommodate experimentalneeds (for example, bath temperature [12]), and to repro-ducibly fabricate sensors [10]. In addition, such calcula-tions allow the interface characteristics to be determinedfrom the measured T c [16, 17].The Usadel equations are a set of diffusive-limit equa-tions based on the Bardeen-Cooper-Schrieffer theoryof superconductivity. In thick superconducting layers,diffusive-limit equations are applicable in the presenceof impurities [18]. In thin, clean superconducting films,layer boundaries result in scattering and ensure the ap-plicability of diffusive-limit equations [15]. The Usadelequations have been widely used to analyse the T c ofmultilayers [19–22]. In particular, the work by Marti-nis et al. [10] provides an analytic analysis of thin-filmsuperconductor-normal conductor ( S − N ) bilayers. Theresults have been integrated in the analysis and designroutines of various multilayer devices, and demonstrategood predictive capabilities [23–26]. More recently, theframework of analysis has been extended to N − S − N trilayers [27]. The Usadel equations have also been numericallysolved by Brammertz et al. [28] for superconductor-superconductor ( S − S (cid:48) ) bilayers. The results demon-strate good agreement with measured T c for Ta-Al andNb-Al bilayers of various thickness combinations. Whilsta full numerical solution is the most accurate approach tosolving the Usadel equations, it is computationally inten-sive and requires users to be able to implement efficientUsadel equations solvers.Although the user-friendly results in [10] have provento be extremely useful in the design of bilayer S − N devices, they cannot be applied to S − S (cid:48) bilayers studiedin [29–31] and general higher-order multilayers studied in[14, 27, 32, 33]. In view of this, the aim of this paper is toextend the analysis framework of [10] to general thin-filmmultilayer systems.In section II of this paper, we extend the analysisframework of [10] to general S − S (cid:48) bilayer systems. Af-ter this, we describe the extension of the bilayer S − S (cid:48) solution to general trilayer systems and multilayer sys-tems. In section III, we present predictions and mea-surements of the T c of Ti-Al bilayer devices of variousthickness combinations. The measured T c data demon-strate good agreement with theoretical predictions, thusgiving assurance to the validity of our analysis scheme.We summarize this work in section IV. II. THEORYA. Usadel equations and boundary conditions
The Usadel equations in one dimension are [15, 34–36] (cid:126) D S ∂ θ∂x + iE sin θ + ∆( x ) cos θ = 0 , (1)and∆( x ) = N S V ,S (cid:90) (cid:126) ω D,S dE tanh (cid:18) E k B T (cid:19) Im (sin θ ) , (2)where θ ( x, E ) is a complex variable dependent on posi-tion x and energy E parametrising the superconductingproperties, N S is the electron single spin density of states, a r X i v : . [ c ond - m a t . s up r- c on ] A ug V ,S is the superconductor interaction potential, ∆ is thesuperconductor order parameter, (cid:126) ω D,S is the Debye en-ergy, T is the temperature of the multilayer, D S is thediffusivity constant, given by D S = σ S / ( N S e ) [10], e is the elementary charge, and finally σ S is the normalstate conductivity, at T just above T c . Equation (2) isknown as the self-consistency equation for order param-eter ∆( x ).The boundary conditions (BCs) relevant to the Usadelequations are presented in [12, 37]. At the open interfaceof layer S , the BC is given by ∂θ S ∂x = 0 . (3)At the S (cid:48) − S inner interface, the BCs are given by σ S ∂θ S ∂x = σ S (cid:48) ∂θ S (cid:48) ∂x , (4) R B σ S (cid:48) ∂θ S (cid:48) ∂x = sin( θ S − θ S (cid:48) ) , (5)where R B is the product of the boundary resistance be-tween the S (cid:48) − S layers and its area. B. Simplifications
Here we apply the same simplification scheme used in[10]:1. At T just above T c , superconductivity is weak, andas a result | θ | (cid:28) θ varies slowly,and can be accounted by a second order polynomialexpansion.With these simplifications, the Usadel equations de-scribing a thin multilayer at T just above T c become (cid:126) D S θ (cid:48)(cid:48) + iEθ + ∆( x ) = 0 , (6)∆( x ) = N S V ,S (cid:90) (cid:126) ω D,S dE tanh (cid:18) E k B T c (cid:19) Im ( θ ) , (7)where θ (cid:48)(cid:48) = ∂ θ/∂x . Equations (3,4) are unchanged.Equation (5) becomes R B σ S (cid:48) θ (cid:48) S (cid:48) = θ S − θ S (cid:48) , (8)where θ (cid:48) = ∂θ/∂x .Using the Usadel equations and the boundary condi-tions, one can straightforwardly show that there is a sin-gle resulting T c across a thin-film multilayer: AppendixA. 𝑥 −𝑑 𝑆 ′ 𝑑 𝑆 𝑆′ 𝑆 𝜃 𝑆 𝜃 𝑆 ′ 𝜃 𝑆 ′ ′ 𝜃 𝑆 ′ FIG. 1. Plot of bilayer superconductor parametrization func-tion θ again position x for a S (cid:48) − S bilayer. C. Bilayer Tc calculations
We refer to the geometry shown in figure 1. The inter-face gradients are given by θ (cid:48) S (cid:48) = d S (cid:48) θ (cid:48)(cid:48) S (cid:48) = − d S (cid:48) (cid:126) D S (cid:48) (∆ S (cid:48) + iEθ S (cid:48) ) (9) θ (cid:48) S = − d S θ (cid:48)(cid:48) S = 2 d S (cid:126) D S (∆ S + iEθ S ) (10)where the gradient is zero at the open boundaries dueto equation (3). Formulae for θ (cid:48)(cid:48) S (cid:48) and θ (cid:48)(cid:48) S are given byrearrangement of equation (6).The BCs, equations (4, 5), give − σ S (cid:48) d S (cid:48) (cid:126) D S (cid:48) (∆ S (cid:48) + iEθ S (cid:48) ) = σ S d S (cid:126) D S (∆ S + iEθ S )= 1 R B ( θ S − θ S (cid:48) ) . (11)Here we introduce a convenient physical constant C S = 2 d S R B σ S (cid:126) D S . (12)Equation (11) is used to express θ S , θ S (cid:48) in terms of ∆ S ,∆ S (cid:48) θ S = − C S ∆ S + C S (cid:48) ∆ S (cid:48) − iEC S (cid:48) C S ∆ S iEC S + iEC S (cid:48) + E C S C S (cid:48) (13) θ S (cid:48) = − C S ∆ S + C S (cid:48) ∆ S (cid:48) − iEC S (cid:48) C S ∆ S (cid:48) iEC S + iEC S (cid:48) + E C S C S (cid:48) . (14)Taking the imaginary partsIm( θ S ) = ( f C S + f )∆ S + f C S (cid:48) ∆ S (cid:48) (15)Im( θ S (cid:48) ) = ( f C S (cid:48) + f )∆ S (cid:48) + f C S ∆ S , (16)where f = E ( C S + C S (cid:48) )( E C S C S (cid:48) ) + E ( C S + C S (cid:48) ) (17) f = E C S C S (cid:48) ( E C S C S (cid:48) ) + E ( C S + C S (cid:48) ) . (18)Substituting equations (15,16) into equation (7), we haveresults of the form ∆ S = [ A ∆ S + B ∆ S (cid:48) ] (19)∆ S (cid:48) = [ A (cid:48) ∆ S (cid:48) + B (cid:48) ∆ S ] , (20)which yields a single equation for T c that is readily solvednumerically 1 = [ A + A (cid:48) − AA (cid:48) + BB (cid:48) ] . (21)Here A, B, A (cid:48) , B (cid:48) are functions of T c , and are given by A = N S V ,S (cid:90) (cid:126) ω D,S dE tanh( E k B T c ) ( f C S + f )(22) B = N S V ,S (cid:90) (cid:126) ω D,S dE tanh( E k B T c ) ( f C S (cid:48) ) (23) A (cid:48) = N S (cid:48) V ,S (cid:48) (cid:90) (cid:126) ω D,S (cid:48) dE tanh( E k B T c ) ( f C S (cid:48) + f )(24) B (cid:48) = N S (cid:48) V ,S (cid:48) (cid:90) (cid:126) ω D,S (cid:48) dE tanh( E k B T c ) ( f C S ) . (25)Practically, N S,S (cid:48) and ω D,S,S (cid:48) are obtained from collatedmaterial properties, and V ,S,S (cid:48) is obtained from the BCSequation [38] k B T c,S,S (cid:48) = 1 . (cid:126) ω D,S,S (cid:48) exp( − N S,S (cid:48) V ,S,S (cid:48) ) (26)where T c,S,S (cid:48) are the measured homogeneous critical tem-peratures of S and S (cid:48) layer materials.Using the above results for S − S (cid:48) bilayers, and set-ting the superconductor interaction potential of S (cid:48) layer V ,S (cid:48) = 0, one can recover the results previously derivedby Martinis et al. [10] for S − N bilayers. This confirmsour scheme of analysis for general S − S (cid:48) bilayers in the S − N limit. Analytical results for S − S (cid:48) bilayers canbe obtained in the clean limit where the interface is per-fectly transmissive, i.e. R B = 0. This result is presentedin Appendix B. D. Extension to trilayer and multilayer systems
In general, there are two types of trilayer systems: 1.The T c of middle layer is higher / lower than that of bothside layers; 2. The T c of middle layer is between that ofthe side layers. 𝑥𝑑 𝑑 𝑆 ′ 𝑑 𝑆 ′ + 𝑑 𝑆 ′′ −𝑑 𝑆 𝜃 𝑆 𝑆′ 𝑆′′
FIG. 2. Plot of bilayer superconductor parametrization func-tion θ again position x for a trilayer, where the middle layerhas higher T c compared to both side layers. For the first case, there exist a θ maximum/minimum,where θ (cid:48) = 0. Recognizing that θ (cid:48) = 0 is also theopen interface boundary condition, we can divide thetrilayer into 2 bilayers. As shown in figure 2, the maxi-mum/minimum location of a S − S (cid:48) − S (cid:48)(cid:48) trilayer is de-noted as d . Thus the left S − S (cid:48) equivalent bilayer hasthicknesses d S and d respectively, and the right S (cid:48) − S (cid:48)(cid:48) equivalent bilayer has thicknesses d S (cid:48) − d , and d S (cid:48)(cid:48) re-spectively. Equation (21) in the previous section attainsthe form F L ( d , T c ) = 1 , and (27) F R ( d , T c ) = 1 , (28)where F L ( d , T c ) = 1 is the analogue of equation (21) inthe left S − S (cid:48) equivalent bilayer, F R ( d , T c ) = 1 is theanalogue of equation (21) in the right S (cid:48) − S (cid:48)(cid:48) equivalent bilayer, and each has solution that traces out a curve of T c as a function of d . The intersection between the two T c ( d ) curves gives the solution for the overall device T c .In practice, the coupled pair of equations (27,28) can besolved using a standard numerical solver.The virtue of the above analysis is that it can bestraightforwardly extended to a general N-layer systemwhere the maxima/minima of θ can be found in all mid-dle layers. For a general N-layer system, there are N-2middle layers and as a result N-1 equivalent bilayers. De-note d i as the extremal point of the i-th middle bilayer.There are thus N-1 unknowns (N-2 values of d i and T c ),and N-1 analogues of equation (21). This system can alsobe solved using a standard numerical solver.For the second case of a S − S (cid:48) − S (cid:48)(cid:48) trilayer with-out maximum/minimum in the middle layer, as shownin figure 3, we need to connect the left and right innerboundaries via θ (cid:48) S (cid:48) ,R = θ (cid:48) S (cid:48) ,L + d S (cid:48) θ (cid:48)(cid:48) S (cid:48) (29)= θ (cid:48) S (cid:48) ,L − d S (cid:48) (cid:126) D S (cid:48) (∆ S (cid:48) + iEθ S (cid:48) ) . (30)Using the same analysis method in section II C, θ (cid:48)(cid:48) aboveis substituted away using equation (6). Again noting that 𝑥 𝑑 𝑆 𝑑 𝑆 + 𝑑 𝑆 ′ + 𝑑 𝑆 ′′ 𝑆 𝑆′′ 𝜃 𝑆′ 𝜃 𝑆 𝑑 𝑆 + 𝑑 𝑆 ′ 𝑆′ 𝜃 𝑆 ′ ,𝐿′ 𝜃 𝑆 ′ 𝜃 𝑆 ′ ,𝑅 ′ 𝜃 𝑆 ′′ ′ 𝜃 𝑆 ′′ FIG. 3. Plot of bilayer superconductor parametrization func-tion θ again position x for a trilayer, where the middle layerhas T c that is in between both side layers. at the open boundaries, θ (cid:48) = 0, it can then be shown that θ (cid:48) S (cid:48) ,L = − σ S σ S (cid:48) d S (cid:126) D S (∆ S + iEθ S ) (31)= 1 σ S (cid:48) R B ( θ S (cid:48) − θ S ) (32) θ (cid:48) S (cid:48) ,R = σ S (cid:48)(cid:48) σ S (cid:48) d S (cid:48)(cid:48) (cid:126) D S (cid:48)(cid:48) (∆ S (cid:48)(cid:48) + iEθ S (cid:48)(cid:48) ) (33)= 1 σ S (cid:48) R B ( θ S (cid:48)(cid:48) − θ S (cid:48) ) . (34)From the above set of equation, θ S,S (cid:48) ,S (cid:48)(cid:48) can be ex-pressed in terms of ∆ S,S (cid:48) ,S (cid:48)(cid:48) through simple rearrange-ments. These can then be substituted into equation (7),linking ∆ S,S (cid:48) ,S (cid:48)(cid:48) with each other. After these manipula-tions, we now have three unknowns (∆ S , ∆ S (cid:48) , and ∆ S (cid:48)(cid:48) )and three equations. Through some algebraic cancelling,a final equation of the form F tri ( T c ) = 1, akin to equa-tion (21), can again be obtained.The same procedure can be employed for higher-ordermultilayer systems. For each additional middle layer S (cid:48) ,five new unknowns are introduced ( θ S (cid:48) , θ (cid:48) S (cid:48) ,L , θ (cid:48) S (cid:48) ,R , θ (cid:48)(cid:48) S (cid:48) and ∆ S (cid:48) ). These five unknowns are compensated by onemore analogue of equation (6), one more geometry equa-tion linking θ (cid:48) S (cid:48) ,L and θ (cid:48) S (cid:48) ,R , one more self-consistencyequation linking θ S (cid:48) and ∆ S (cid:48) , and two more boundaryconditions. III. EXPERIMENTAL RESULTSA. Fabrication Details
Films were deposited onto 50 mm diameter Si wafersby DC magnetron sputtering at a base pressure of 2 × − Torr or below. Ti films were deposited at ambienttemperature and Al films were deposited after substratecooling to liquid nitrogen temperatures. Bilayer Ti-Alfilms were deposited without breaking vacuum. Titanium is deposited first for all bilayers studied. The wafers werediced into 13 . × . µ A. B. T c Measurements
TABLE I. Table of material properties.Aluminium Titanium T c (K) 1.20 a b , 0.588 c σ N (/ µ Ω m) d a a RRR e a a N (10 /J m ) 1.45 f f D (m s − ) 35 g g ξ (nm) 189 h h Θ D (K) 423 i ia Measured. b Measured for the first set of measurements in December 2017. c Measured for the second set of measurements in May 2018. d σ N is the normal state conductivity. e RRR is the residual resistivity ratio. f N is the normal state electron density of states, and is calcu-lated from the free electron model [39]. g Diffusivity constant D is calculated using D s = σ N,s / ( N ,s e )[10]. h Coherence length ξ is calculated using ξ s = [ (cid:126) D s / (2 πk B T c )] / [15], where k B is the Boltzmann constant. i Θ D is the Debye temperature, and is given by k B Θ D = (cid:126) ω D .Values are taken from [40]. Two sets of measurements were taken. The first setof measurements was performed in late 2017 for 7 Ti-Albilayers with d Al ranging from 0 nm −
125 nm. The secondset of measurements was performed 6 months later for 4Ti-Al bilayers with d Al ranging from 0 nm −
400 nm. d Ti is fixed at 100 nm for both sets of measurements. Toaccount for slight variations in the sputtering system, wemeasure T c, Ti for each set of measurements. The value of T c, Ti from the second set of measurements is 7% higherthan that from the first.The results of bilayer T c measurements are shown infigure 4, plotted against d Al . Diamonds indicate Ti-Al bi-layers measured in December 2017. Crosses indicate Ti-Al bilayers measured in May 2018. The dashed, blue lineis the theory plot for diamonds, and is calculated with R B = (0 +1 − ) × − Ωm , T c, Ti = 0 .
55 K. The best-fitvalue of R B is obtained by minimizing the squared-errorbetween the model and the data, and the uncertainty in-dicates the value of R B where the squared-error increasesby a factor of 1 /e , where e is Euler’s number. The dashed,red line is the theory plot for crosses, and is calculatedwith R B = (0 +4 − ) × − Ωm , T c, Ti = 0 .
588 K. Thetheory plots are generated using the physical parame-ters listed in Table I. The values of R B are small for d Al (nm) T c ( K ) T (K) R / R N a b c d FIG. 4. Plot of superconducting transition temperature T c versus Al thickness d Al , for fixed Ti thickness d Ti = 100 nm.Diamonds indicate Ti-Al bilayers measured in December2017. Crosses indicate Ti-Al bilayers measured in May 2018.The dashed, blue line is the theory plot for diamonds, and iscalculated with R B = (0 +1 − ) × − Ωm , T c, Ti = 0 .
55 K. Thedashed, red line is the theory plot for crosses, and is calculatedwith R B = (0 +4 − ) × − Ωm , T c, Ti = 0 .
588 K. Inset: nor-malized resistance
R/R N versus temperature for solid lines(a) d Al = 0 nm Ti-Al bilayer, (b) d Al = 25 nm Ti-Al bilayer,(c) d Al = 50 nm Ti-Al bilayer, and (d) d Al = 125 nm Ti-Albilayer. Here R is measured resistance, and R N is measuredthe normal state resistance. both theory plots, and indicate that our deposition tech-nique achieves very clean layer interfaces consistently.Given that the deposited bilayer interfaces are consis-tently very clean ( R B ≈ ), and that we have mea-sured both T c, Ti and T c, Al for each set of measurements,our theory plots are effectively generated without freeparameter. The measured T c values of Ti-Al bilayersdemonstrate good agreement with the theoretical plot,with χ = 1 × − for the December 2017 dataset, and χ = 4 × − for the May 2018 dataset. This close agree-ment between theory and measurements lends confidenceto the validity of our analysis scheme. Multiple mea-surements of three d Al = 0 nm bilayers (diamond datapoints) overlay closely with each other, demonstratingthe reproducibility of our depositions. Similar closenessof measured T c is observed in the two d Al = 50 nm bi-layers. The inset shows the normalized resistance R/R N versus temperature for solid lines (a) d Al = 0 nm Ti-Albilayer, (b) d Al = 25 nm Ti-Al bilayer, (c) d Al = 50 nmTi-Al bilayer, and (d) d Al = 125 nm Ti-Al bilayer. R is the measured resistance, and R N is the normal stateresistance just above T c . These bilayers display sharpsuperconducting state transitions, typically having tran-sition widths ∆ T ∼ IV. CONCLUSIONS
In this paper, we have described a general analysis ofthin-film multilayer T c , using the diffusive-limit Usadelequations. We performed a derivation of T c for a generalthin-film S − S (cid:48) bilayer. We have described methods ofextending this calculation to general thin-film trilayersand multilayers. Our model for S − S (cid:48) bilayer reduces toprevious results in [10] when the T c of S (cid:48) layer is set tozero. Our model is easy to implement and computation-ally fast. We find a five order-of-magnitude reduction in T c calculation time compared to solving the full Usadelequations as presented in [12]. Our experimental mea-surements of Ti-Al bilayer T c demonstrate good agree-ment with predictions from our model, thereby enablingour analysis method to be incorporated in the design ofsuperconducting multilayer devices. Work is currentlybeing conducted to extract R B of multilayers with lesstransmissive boundaries using our model, and is provingsuccessful. Appendix A: Critical temperature across layers
We consider a S − S (cid:48) bilayer whilst relaxing the as-sumption of a single resulting T c . We define T c, and T c, to be the resulting critical temperature of layer S and S (cid:48) respectively. Without loss of generality, we canorder the layer notation such that T c, < T c, .In layer S , at T > T c, , by definition of the criticaltemperature, ∆ S = 0 (A1) θ S = 0 . (A2)Using equation (6), we deduce θ (cid:48)(cid:48) S = 0. In the case ofthin films, we have θ (cid:48) S = d S θ (cid:48)(cid:48) S = 0. In layer S (cid:48) , BC equa-tion (4) ensures that θ (cid:48) S (cid:48) = 0, and equation (8) ensuresthat θ S (cid:48) = 0.Thus if layer S is in the normal state, layer S (cid:48) is alsoin the normal state, i.e. if T > T c, , then T > T c, . Fromthe ordering of the layers, T c, < T c, , we thus concludethat T c, = T c, = T c , i.e. there is a single T c acrossthe entire device. The above argument can be extendedto a general thin-film multilayer to show that there is asingle T c across the entire multilayer. This can be doneby repeatedly applying equation (4, 8) across each layerinterface.In this case of metamaterials made up of thin periodicmultilayers, this phenomenon of a single resulting T c hasbeen observed experimentally in [41]. Appendix B: Analytical bilayer solution in the cleanlimit
The clean interface limit is when R B = 0. Using equa-tion (8), the clean limit implies that θ S = θ S (cid:48) . Here weintroduce another convenient constant G S = 2 d S σ S (cid:126) D S . (B1)Equation (11) is used to express θ S , θ S (cid:48) in terms of ∆ S ,∆ S (cid:48) θ S = i Im( θ S ) = θ S (cid:48) = i Im( θ S (cid:48) ) (B2)= i E G S ∆ S + G S (cid:48) ∆ S (cid:48) G S + G S (cid:48) . (B3)Substituting the above equations into equation (7), weagain have results of the form∆ S = [ A ∆ S + B ∆ S (cid:48) ] (B4)∆ S (cid:48) = [ A (cid:48) ∆ S (cid:48) + B (cid:48) ∆ S ] . (B5)We can simplify coefficients A, B, A (cid:48) , B (cid:48) immensely dueto the simple 1 /E dependence of Im( θ S ) and Im( θ S (cid:48) ).We note that (cid:90) (cid:126) ω D dE E tanh( E k B T c ) = (cid:90) Θ D Tc dx x tanh( x ) (B6) ≈ ln (cid:18) e γ π Θ D T c (cid:19) , (B7)where γ ≈ . D / (2 T c ) (cid:29) K = e γ π . Using the BCS result1 N S V ,S = (cid:90) (cid:126) ω D,S dE E tanh( E k B T c,S ) (B8)= ln (cid:18) K Θ D,S T c,S (cid:19) , (B9)we can explicitly find the coefficients A = G S G S + G S (cid:48) ln (cid:18) K Θ D,S T c (cid:19) / ln (cid:18) K Θ D,S T c,S (cid:19) (B10) B = G S (cid:48) G S + G S (cid:48) ln (cid:18) K Θ D,S T c (cid:19) / ln (cid:18) K Θ D,S T c,S (cid:19) (B11) A (cid:48) = G S (cid:48) G S + G S (cid:48) ln (cid:18) K Θ D,S (cid:48) T c (cid:19) / ln (cid:18) K Θ D,S (cid:48) T c,S (cid:48) (cid:19) (B12) B (cid:48) = G S G S + G S (cid:48) ln (cid:18) K Θ D,S (cid:48) T c (cid:19) / ln (cid:18) K Θ D,S (cid:48) T c,S (cid:48) (cid:19) . (B13)By substituting the above coefficients into equa-tions (B4,B5), the following formula for T c can be readilyobtained T c = exp (cid:26) H S ln ( K Θ D,S ) + H S (cid:48) ln ( K Θ D,S (cid:48) ) − H S + H S (cid:48) (cid:27) , (B14)where H S = G S G S + G S (cid:48) / ln (cid:18) K Θ D,S T c,S (cid:19) (B15) H S (cid:48) = G S (cid:48) G S + G S (cid:48) / ln (cid:18) K Θ D,S (cid:48) T c,S (cid:48) (cid:19) . (B16) ∗ [email protected] Irwin K D 1995
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