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Applied Computational Intelligence and Soft Computing
Volume 2016, Article ID 1709827, 7 pages
http://dx.doi.org/10.1155/2016/1709827
Research Article

Computational Intelligence Approach for Estimating Superconducting Transition Temperature of Disordered MgB2 Superconductors Using Room Temperature Resistivity

1Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2Physics and Electronics Department, Adekunle Ajasin University, Akungba Akoko, Ondo State 342111, Nigeria
3Institute for Digital Communications, School of Engineering, University of Edinburgh, UK
4Computer Information Systems Department, University of Dammam, Dammam 31451, Saudi Arabia

Received 22 November 2015; Revised 15 April 2016; Accepted 24 April 2016

Academic Editor: Sebastian Ventura

Copyright © 2016 Taoreed O. Owolabi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Doping and fabrication conditions bring about disorder in MgB2 superconductor and further influence its room temperature resistivity as well as its superconducting transition temperature (). Existence of a model that directly estimates of any doped MgB2 superconductor from the room temperature resistivity would have immense significance since room temperature resistivity is easily measured using conventional resistivity measuring instrument and the experimental measurement of wastes valuable resources and is confined to low temperature regime. This work develops a model, superconducting transition temperature estimator (STTE), that directly estimates of disordered MgB2 superconductors using room temperature resistivity as input to the model. STTE was developed through training and testing support vector regression (SVR) with ten experimental values of room temperature resistivity and their corresponding using the best performance parameters obtained through test-set cross validation optimization technique. The developed STTE was used to estimate of different disordered MgB2 superconductors and the obtained results show excellent agreement with the reported experimental data. STTE can therefore be incorporated into resistivity measuring instruments for quick and direct estimation of of disordered MgB2 superconductors with high degree of accuracy.

1. Introduction

Superconductor is a material that allows perpetual flow of current as a result of disappearance of its electrical resistivity when it is cooled below a particular temperature called superconducting transition temperature (). Several practical applications of superconductors such as magnetic resonance imaging in hospitals, magnetic levitation train, particle accelerators, superconducting quantum interference devices, magnetoencephalography, and filters for mobile communication and their future applications (which include transmission of electricity, fast computing, and high temperature superconducting generator) depend mainly on the value of . Phenomenon of superconductivity was first observed in mercury in 1911 when electrical resistance of pure mercury went to zero around 4 K [1]. Understanding of this behavior was not clearly known until 1957 when three physicists propounded BCS theory which governs the emergence of superconductivity [2]. This theory attributes the emergence of superconductivity in materials to the ability of electrons to pair (formation of cooper pairs) which pave ways for the free movement of electrons in a coordinated manner. Collision of electrons with lattice, other electrons, and defects among others bring about resistivity in materials and this is circumvented in superconductors due to the formation of cooper pairs. Efforts have been made to realize room temperature superconductor and to raise the values of of known superconductors. However, MgB2 holds a promising future since its can be altered through doping and fabrication conditions [35].

The awareness of superconductivity in magnesium di-boride (MgB2) in 2001 marked a new advancement that diversifies the applications of superconductors [6]. MgB2 is a two-band superconductor with several practical applications as a result of its unique properties such as transparency of the grain boundaries which permits flow of current, lower anisotropy, and large coherence length. Its economic affordability as well as its unique properties offers it a special place in practical applications despite its low as compared with high temperature superconductors. The techniques that are usually employed for improving the superconducting properties of MgB2 include introduction of disorder through chemical doping and several thermomechanical processing techniques which alter its room temperature resistivity [3, 5, 724]. In order to improve the superconducting properties of this material (in particular ), we develop STTE for accurate, quick, and direct estimation of of disordered MgB2 superconductors as an alternative to conventional method which is experimentally intensive. This developed model allows the potential of several dopants to be assessed within short period of time so as to ascertain the specific dopant that improves the superconducting property of MgB2 superconductor.

The normal state strong electron-phonon scattering which gives rise to the large room temperature resistivity in intermetallic superconductors is associated with the observed high [25]. Low resistivity of MgB2 at room temperature and temperature just above its could be attributed to its two-band nature characterized with different strength of electron-phonon coupling [25]. Its superconductivity comes from the large coupling in 2D band and MgB2 demonstrates anisotropy in the normal state resistivity. The proposed SVR based model has been reported as a novel tool of estimating superconducting properties of doped MgB2 superconductor using few descriptive features [26]. The choice of SVR in building STTE is due to its many unique features which include sound mathematical foundation, nonconvergence to local minima, accurate generalization, and predictive ability when trained with few descriptive features [27].

SVR tackles real life problems using the principles of artificial intelligence in the field of machine learning. It acquires pattern or relation that exists between the target () and descriptor (room temperature resistivity) and adopts the acquired pattern for future estimation of the unknown target with the aid of the descriptors. Its excellent predictive ability is made which is used in tackling several problems in medical field [28], material science [26, 2932], and oil and gas industries [33, 34], to name but a few. The excellent predictive and generalization ability of SVR in solving numerous problems coupled with the need to have accurate, direct, and effective way of estimating the effects of disorder on of MgB2 superconductor serves as motivation for carrying out this research work.

2. Description of the Proposed Model

SVR follows a learning algorithm formulated from support vector machines which was originally proposed by Vapnik for classification purposes [35]. It uses loss function that controls the smoothness of the response of SVR and the number of support vectors which ultimately affect the generalization capacity as well as the complexity of the model. The loss function represents the maximum error (deviation of the target vectors tolerated by the model). Equation (1) represents a linear decision function in which indicates a dot product in space :where (input space) and .

Minimization of Euclidean norm is necessary since flatness of the defined decision equation is desired. The optimization problem for the regression is presented inThe assumption entailed in the optimization problem presented in (2) is the existence of a function that controls the error in all training pairs in such a way that the error is less than . In order to build up a robust system that caters for external constraints, slack variables ( and ) are introduced in the optimization problem and presented in [27]The regularization factor introduced in (3) trades off the frequency of error and the complexity of decision rule by altering the size of slack variables.

The optimization problem is solved by using Lagrangian multipliers (, , , and ) to transform the problem to dual space representation. The constraint equations are multiplied by the multipliers and the result is taken away from (objective function). The resulting Lagrangian is illustrated inIn order to obtain the solution of the optimization problem, the saddle points of the Lagrangian function are obtained by equating the partial derivative of Lagrangian with respect to , , , and to zero which gives rise to (5) as described in [29]:By putting (5) in (4), we have The estimation problem presented in (1) is finally represented by (8) after incorporating and obtained from (6):Kernel function represents a nonlinear mapping function that maps nonlinear regression problem to high dimensional feature space where linear regression is performed. The choice of kernel function depends on the nature of the problem and could be linear, polynomial, Gaussian, or sigmoid kernel function.

During the course of optimizing the SVR model, SVR variables such as the regularization factor (), epsilon (), hyperparameter (), and kernel option were defined and adjusted until the model attained optimum performance.

The training stage of the proposed model entails acquisition of pattern that exists between the descriptor and the target which could further be generalized by the model so as to accurately estimate unknown target with the aid of the descriptor. High correlation coefficient, low root mean square error, and low mean absolute error signify the accuracy and efficiency of the model. In the case of STTE developed in this research work, the SVR model was taken through training and testing stages and then used to estimate of several MgB2 superconductors.

2.1. Evaluation of the Performance of the Developed Model

The performance of the developed model was evaluated on the basis of the correlation coefficient (CC) between the experimental and the estimated , root mean square error (RMSE), and mean absolute error (MAE). The generalization performance was evaluated using where and represent error (difference between the experimental and estimated ) and number of data points, respectively. and , respectively, represent the experimental and the estimated , while and represent their mean value, respectively.

In the case of the developed model, high correlation coefficient of 100%, low root mean square error of 1.29, and low absolute error of 0.279 were obtained during the testing phase of the model.

3. Empirical Study

3.1. Description of the Dataset

The dataset that was employed in modeling STTE comprises a total number of ten experimental values of room temperature resistivity and the corresponding of disordered MgB2 superconductors. The adopted dataset was drawn from the literatures [3638] and presented in Table 1.

Table 1: The dataset used for modeling STTE.

Statistical analysis was carried out on the dataset and the results of the analysis are presented in Table 2.

Table 2: Statistical analysis of the dataset used in modeling STTE.

Deduction from the values of the mean, median, standard deviation, minimum, and maximum gives insight into the dataset as well as the level of its discrepancies. The correlation coefficient between and room temperature resistivity was calculated and found negatively correlated. Meanwhile, the obtained correlation suggests the level of relationship between the target and descriptor.

3.2. Computational Methodology

The modeling and simulations involved in this work were conducted within the computing environment of MATLAB. The dataset presented in Table 1 was randomized prior to the commencement of the modeling. The randomization was carried out purposely to enhance computational efficiency and to ensure consistency. The dataset was further divided into training and testing set in the ratio of 8 to 2. The best values of SVR hyperparameters (i.e., regularization factor, kernel option, lambda, and epsilon) were used for generating support vectors through training dataset and the testing set of data was used to validate the model. The developed model (STTE) was then used to estimate the effect of starch doping, nano-silicon carbide, and preparation condition on of MgB2 superconductors and the obtained values are compared with the experimental results.

3.3. Optimization Strategy

The optimization strategy employed during the development of STTE was the test-set cross validation technique. In this case, the values of correlation coefficient, root mean square error, and mean absolute error were monitored in every run of the training and testing dataset for a group of parameters (regularization factor, kernel option, epsilon, and hyperparameter). In the course of searching through all the possible values of the parameters in a given range, the best performance measures were identified with the corresponding values of parameters for the fixed set of features. The whole process involves searching for the initial kernel option from the pool of the available kernel options and the identification of the best values of the parameter and so as to identify the corresponding performance measures. The best values of the hyperparameters were further used to train SVR model. The values of hyperparameters that ensure optimum performance of the model are presented in Table 3.

Table 3: Optimum parameters for the developed model (STTE).

4. Results and Discussion

4.1. Development of the Model (STTE)

The STTE was developed using randomly selected ten experimental values of room temperature resistivity with their corresponding . Excellent correlation coefficient of 100% was obtained in the course of testing the developed model prior to its usage. Low values of 1.29 and 0.279 for the root means square and mean absolute errors were, respectively, recorded. The pairwise comparison of the experimentally reported and estimated values is depicted in Figure 1.

Figure 1: Correlation between the experimental and estimated value of during the testing phase of STTE.

The actual (experimental value) and estimated values of obtained during the development stage of STTE are presented in Table 4. The percentage errors were also calculated in order to observe the percentage deviation of the estimated values.

Table 4: Comparison between the experimental and estimated value of during the course of developing STTE.
4.2. Application of the Developed Model (STTE) in Estimating Disordered MgB2 Superconductors

The developed STTE was used to estimate of different disordered MgB2 superconductors and the obtained values are compared with the experimental data. The utilization of the developed model involves feeding the model with the value of the room temperature resistivity and the model employs this value to generate the corresponding .

4.2.1. Effect of Starch Doping on of MgB2 Superconductor Using the Developed Model

Addition of starch to MgB2 superconductor alters its transition temperature as reported in the literature [4]. In order to assess the estimation strength of the developed model, the model was used to estimate the effect of starch doping on of MgB2 superconductor and the results are compared with the experimental values. Figure 3 shows the reduction in the transition temperature of MgB2 superconductor when 1% concentration of starch was added. The transition temperature further reduced upon increase in the concentration of starch. The estimated results agree excellently with the experimentally reported values [4] (see Figure 2).

Figure 2: Effect of starch on of MgB2 superconductor using the developed model.
Figure 3: Effect of nano-silicon carbide and graphene on of MgB2 superconductor using the developed model.
4.2.2. Effect of Nano-Silicon Carbide and Graphene on of MgB2 Superconductor Using the Developed Model

The effect of nano-silicon carbide (nano-SiC) and graphene on of MgB2 superconductor is also investigated and presented in Figure 3. The results of the developed model also agree well with the experimental values [39]. Introduction of graphene to MgB2 already doped with 5% nano-SiC has little effect on as depicted in Figure 3.

4.2.3. Effect of Preparation Condition (Heating Rate) on of MgB2 Superconductor Using the Developed Model

In order to further justify the effectiveness of the developed model, the effect of preparation condition on of MgB2 superconductor was estimated and presented in Figure 4. The rate of heating has significant effect on reducing of MgB2 superconductor, especially, when the rate was increased to 30 K/min as depicted in the graph. The results of the developed model show good agreement with the reported experimental values [40].

Figure 4: Effect of heating rate on of MgB2 superconductor using the developed model.

5. Conclusion and Recommendation

SVR was used to develop STTE through training and testing using test-set cross validation optimization technique with the aid of ten experimental values of room temperature resistivity and their corresponding . The developed model (STTE) was then used to estimate the effect of starch doping, nano-silicon carbide, graphene, and preparation condition on of MgB2 superconductors and the obtained values are compared with the experimental results. High accuracy obtained in the estimated suggests that the developed model is capable of estimating of disordered MgB2 superconductors with slight deviation from the experimental values. The developed model is therefore recommended for quick estimation of of disordered MgB2 superconductors and can as well be incorporated into resistivity measuring instrument for direct estimation of of disordered MgB2 superconductors.

Competing Interests

The authors declare that they have no competing interests.

References

  1. D. Van Delft and P. Kes, “The discovery of superconductivity the discovery of superconductivity feature,” Physics Today, vol. 63, no. 9, pp. 38–42, 2010. View at Google Scholar
  2. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” APS Journals Archive, vol. 108, no. 5, p. 1175, 1957. View at Publisher · View at Google Scholar
  3. S. Zhou, A. V. Pan, and S. X. Dou, “An attempt to improve the superconducting properties of MgB2 by doping with Zn-containing organic compound,” Journal of Alloys and Compounds, vol. 487, no. 1-2, pp. 42–46, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. D. Tripathi, S. S. Moharana, and T. K. Dey, “The role of starch doping on the superconducting properties of MgB2,” Cryogenics, vol. 63, pp. 85–93, 2014. View at Publisher · View at Google Scholar · View at Scopus
  5. Z. Zhang, H. Suo, L. Ma, T. Zhang, M. Liu, and M. Zhou, “Critical current density in MgB2 bulk samples after co-doping with nano-SiC and poly zinc acrylate complexes,” Physica C: Superconductivity and its Applications, vol. 471, no. 21-22, pp. 908–911, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, “Superconductivity at 39 K in magnesium diboride,” Nature, vol. 410, no. 6824, pp. 63–64, 2001. View at Publisher · View at Google Scholar · View at Scopus
  7. K. Hušeková, I. Hušek, P. Kováč, M. Kulich, E. Dobročka, and V. Štrbík, “Properties of MgB2 superconductor chemically treated by acetic acid,” Physica C: Superconductivity and its Applications, vol. 470, no. 5-6, pp. 331–335, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. M. Paranthaman, C. Cantoni, H. Y. Zhai et al., “Superconducting MgB2 films via precursor postprocessing approach,” Applied Physics Letters, vol. 78, no. 23, pp. 3669–3671, 2001. View at Publisher · View at Google Scholar · View at Scopus
  9. M. Iavarone, R. Di Capua, A. E. Koshelev et al., “Effect of disorder in MgB2 thin films,” Physical Review B, vol. 71, no. 21, article 214502, 2005. View at Publisher · View at Google Scholar · View at Scopus
  10. C. B. Eom, M. K. Lee, J. H. Choi et al., “High critical current density and enhanced irreversibility field in superconducting MgB2 thin films,” Nature, vol. 411, no. 6837, pp. 558–560, 2001. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Buzea and T. Yamashita, “Review of the superconducting properties of MgB2,” Superconductor Science and Technology, vol. 14, no. 11, pp. R115–R146, 2001. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Eisterer, R. Müller, R. Schöppl, H. W. Weber, S. Soltanian, and S. X. Dou, “Universal influence of disorder on MgB2 wires,” Superconductor Science and Technology, vol. 20, no. 3, pp. 117–122, 2007. View at Publisher · View at Google Scholar · View at Scopus
  13. Sudesh, N. Kumar, S. Das, C. Bernhard, and G. D. Varma, “Effect of graphene oxide doping on superconducting properties of bulk MgB2,” Superconductor Science and Technology, vol. 26, no. 9, Article ID 095008, 2013. View at Publisher · View at Google Scholar · View at Scopus
  14. J. H. Lim, S. H. Jang, S. M. Hwang et al., “Effects of the sintering temperature and doping of C60and SiC on the critical properties of MgB2,” Physica C: Superconductivity, vol. 468, no. 15–20, pp. 1829–1832, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. V. Ferrando, P. Manfrinetti, D. Marré et al., “Effect of two bands on critical fields in MgB2 thin films with various resistivity values,” Physical Review B, vol. 68, no. 9, Article ID 094517, 2003. View at Google Scholar · View at Scopus
  16. M. Putti, V. Braccini, E. Galleani d'Agliano et al., “Thermal conductivity of MgB2 in the superconducting state,” Physical Review B, vol. 67, no. 6, Article ID 064505, 2003. View at Google Scholar
  17. D. Tripathi and T. K. Dey, “Effect of (Bi, Pb)-2223 addition on thermal transport of superconducting MgB2 pellets,” Journal of Alloys and Compounds, vol. 618, pp. 56–63, 2015. View at Publisher · View at Google Scholar · View at Scopus
  18. Y. Cui, Y. Chen, Y. Yang, Y. Zhang, C. Cheng, and Y. Zhao, “Hg substitution effect on superconductivity and crystal structure of MgB2,” Journal of Electronic Science and Technology, vol. 6, no. 2, pp. 152–156, 2008. View at Google Scholar
  19. P. Toulemonde, N. Musolino, H. L. Suo, and R. Flükiger, “Superconductivity in high-pressure synthesized pure and doped MgB2 compounds,” Journal of Superconductivity, vol. 15, no. 6, pp. 613–619, 2002. View at Publisher · View at Google Scholar
  20. N. Ojha, V. K. Malik, C. Bernhard, and G. D. Varma, “Enhanced superconducting properties of Eu2O3-doped MgB2,” Physica C: Superconductivity and its Applications, vol. 469, no. 14, pp. 846–851, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. Sudesh, S. Das, C. Bernhard, and G. D. Varma, “Enhanced superconducting properties of rare-earth oxides and graphene oxide added MgB2,” Physica C: Superconductivity, vol. 505, pp. 32–38, 2014. View at Publisher · View at Google Scholar · View at Scopus
  22. H. Ağıl, H. Yetis, M. Akdogan, C. Altug, S. Akturk, and A. Gencer, “Fabrication and characterization of C15H12O2 doped MgB2 bulk superconductors,” Cryogenics, vol. 63, pp. 138–142, 2014. View at Publisher · View at Google Scholar · View at Scopus
  23. S. Lee, T. Masui, A. Yamamoto, H. Uchiyama, and S. Tajima, “Carbon-substituted MgB2 single crystals,” Physica C: Superconductivity, vol. 397, no. 1-2, pp. 7–13, 2003. View at Publisher · View at Google Scholar · View at Scopus
  24. J. H. Kim, S. Zhou, M. S. A. Hossain, A. V. Pan, and S. X. Dou, “Carbohydrate doping to enhance electromagnetic properties of MgB 2 superconductors,” Applied Physics Letters, vol. 89, no. 14, Article ID 142505, 2006. View at Publisher · View at Google Scholar · View at Scopus
  25. P. C. Canfield and G. W. Crabtree, “Magnesium diboride: better late than never,” Physics Today, vol. 56, no. 3, pp. 34–41, 2003. View at Publisher · View at Google Scholar
  26. T. O. Owolabi, K. O. Akande, and S. O. Olatunji, “Estimation of superconducting transition temperature TC for superconductors of the doped MgB2 system from the crystal lattice parameters using support vector regression,” Journal of Superconductivity and Novel Magnetism, vol. 28, no. 1, pp. 75–81, 2014. View at Publisher · View at Google Scholar · View at Scopus
  27. T. O. Owolabi, K. O. Akande, and S. O. Olatunji, “Application of computational intelligence technique for estimating superconducting transition temperature of YBCO superconductors,” Applied Soft Computing, vol. 43, pp. 143–149, 2016. View at Publisher · View at Google Scholar
  28. M. A. Shini, S. Laufer, and B. Rubinsky, “SVM for prostate cancer using electrical impedance measurements,” Physiological Measurement, vol. 32, no. 9, pp. 1373–1387, 2011. View at Publisher · View at Google Scholar · View at Scopus
  29. T. O. Owolabi, K. O. Akande, and S. O. Olatunji, “Development and validation of surface energies estimator (SEE) using computational intelligence technique,” Computational Materials Science, vol. 101, pp. 143–151, 2015. View at Publisher · View at Google Scholar · View at Scopus
  30. T. O. Owolabi, K. O. Akande, and S. O. Olatunji, “Estimation of surface energies of transition metal carbides using machine learning approach,” International Journal of Materials Science and Engineering, vol. 3, no. 2, pp. 104–119, 2015. View at Google Scholar
  31. T. O. Owolabi, K. O. Akande, and S. O. Olatunji, “Estimation of surface energies of hexagonal close packed metals using computational intelligence technique,” Applied Soft Computing Journal, vol. 31, pp. 360–368, 2015. View at Publisher · View at Google Scholar · View at Scopus
  32. T. O. Owolabi, K. O. Akande, and O. O. Sunday, “Modeling of average surface energy estimator using computational intelligence technique,” Multidiscipline Modeling in Materials and Structures, vol. 11, no. 2, pp. 284–296, 2015. View at Publisher · View at Google Scholar · View at Scopus
  33. K. O. Akande, T. O. Owolabi, and S. O. Olatunji, “Investigating the effect of correlation-based feature selection on the performance of support vector machines in reservoir characterization,” Journal of Natural Gas Science and Engineering, vol. 22, pp. 515–522, 2015. View at Publisher · View at Google Scholar · View at Scopus
  34. S. O. Olatunji, A. Selamat, and A. A. A. Raheem, “Predicting correlations properties of crude oil systems using type-2 fuzzy logic systems,” Expert Systems with Applications, vol. 38, no. 9, pp. 10911–10922, 2011. View at Publisher · View at Google Scholar · View at Scopus
  35. V. N. Vapnik, The Nature of Statistical Learning Theory, Springer, New York, NY, USA, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  36. A. Serquis, X. Z. Liao, Y. T. Zhu et al., “Influence of microstructures and crystalline defects on the superconductivity of MgB2,” Journal of Applied Physics, vol. 92, no. 1, pp. 351–356, 2002. View at Publisher · View at Google Scholar · View at Scopus
  37. J. Jiang, B. J. Senkowicz, D. C. Larbalestier, and E. E. Hellstrom, “Influence of boron powder purification on the connectivity of bulk MgB2,” Superconductor Science and Technology, vol. 19, no. 8, pp. L33–L36, 2006. View at Publisher · View at Google Scholar
  38. L. Schultz, A. Handstein, D. Hinz et al., “Fully dense MgB2 superconductor textured by hot deformation,” Journal of Alloys and Compounds, vol. 329, no. 1-2, pp. 285–289, 2001. View at Publisher · View at Google Scholar · View at Scopus
  39. K. S. B. De Silva, X. Xu, X. L. Wang et al., “A significant improvement in the superconducting properties of MgB2 by co-doping with graphene and nano-SiC,” Scripta Materialia, vol. 67, no. 10, pp. 802–805, 2012. View at Publisher · View at Google Scholar · View at Scopus
  40. Q. Zhao, Y. Liu, Y. Han, Z. Ma, Q. Shi, and Z. Gao, “Effect of heating rates on microstructure and superconducting properties of pure MgB2,” Physica C: Superconductivity and its Applications, vol. 469, no. 14, pp. 857–861, 2009. View at Publisher · View at Google Scholar · View at Scopus