Computational Models of Superconducting Quantum Effects

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1 JOURNAL ON POTONICS AND SPINTRONICS VOL. NO.4 NOVEMBER 014 ISSN (Print) ISSN (Online) Computational Models of Superonduting Quantum Effets Roío Cayetano, Member of GI-TESCA and Teaher of Informatis Engineering Division Abstrat Through of omputational and numerial models are reated geometrial models of superonduting of type II, that is to say, Abrikosov superonduting to obtaining of a Fermioni prohibit energy trenhes (r), whose transition from a BCS - theory and through the appliation to any Majorana state (r), it meets that ( r) ( r), where the BCS-theory haves as limit GLAG extensions of magneti field, when, having valid the theory GLAG, to all temperature until ritial temperature. Then by the Cooper pairs and under the spatial variation of the energy trenhes and their magneti images are developed these numerial models mentioned to their after study in superonduting of type II materials. Keywords Critial magneti field, energy trenhes, Fermioni superonduting region, fluxes, superonduting materials of type II. A I. INTRODUCTION N study of the mirosopi effets of the superondutivity (BCS-theory [1]) in superondutors of type II, and some superondutors of type I save very speial onditions, establish the existene of a differentiated energy regions (energy bands or ondution bands, and trenhes [ite]) studied in a metal whose Cooper pairs [1] an be formed by ouplings that ondue us to them. Other phenomenon that appears in the study of the superondutors of type II, are the Abrikosov vorties [] that are intimately linked with the behavior of the magneti field very near to the ritial magneti field. Finally to the respet of the vorties state, Abrikozov shows (and experimental is onfirmed) that the struture of the vorties (filaments) is not arbitrary; these vorties are formed in the superontuting material drawing a triangular net. To the beginning, when is had the first vorties (we are near to ) the separation between them is big, and if is followed the inrease of the value of the applied magneti field, eah time penetrate more fluxoids [4, 5] (are formed more vorties) until that are so together that the distane between them is so only of the order of the oherent length. Then we are in the value of the applied field equal to their seond ritial haraterizing inside of the lassifiation of the superondutors of type II, and that in this last point, the vorties ollapse and the material return to the resistive normal state (superonduting zone or normal zone). The determination and evaluation of superonduting materials near to the ritial temperaturet, establish that the major ontribution of the energy of superondutivity omes of the Fermioni behavior of the orresponding Fok spae [6] that an be determined around of the superonduting material provoking an energy trenhes whose variation of energy, that an be ontrolled spatially by magnetism; result of the onsideration of the trenhes (r), as a omplex funtion that is annulled of a ontinuous way, non-sharp to T [7]. The theory of Gor kov [8], using the tehnique of the Green funtions re-interpret the mean of the BSC, parameter (prohibit energy trenhes) doing possible the spatial variation mentioned. The following researh haves as target to obtain many geometrial models of these re-interpreting of the BSC-, and the formatting of their orbital spaes predited by the mirosopi theories having to limit the theory GLAG, to all temperature until ritial temperature to superondutors of type II. One of the phenomenons of quantum harater that onform the appearing of orbital spaes in the superonduting phenomena is the tunnel effet, whih is risked to differened onditions between superondutor states densities and expressed as differential ondutivity di / dv. In all it, the fermioni behavior is determinant to omposite a suffiient Fermi liquid to onform a Fermi surfae in the limit of the energy trenhes. *Roío Cayetano, distinguished teaher of the Informatis Engineering Division, TESCA, and is a member of Dr. Bulnes s Researh Group ( orr.author@univers.edu). 19 The existene of an energy trenhes in the superondutor state haves the tunnel effet in a formed struture for superondutor-insulator-metal in normal state, haves speial

2 JOURNAL ON POTONICS AND SPINTRONICS VOL. NO.4 NOVEMBER 014 ISSN (Print) ISSN (Online) harateristis [9]. It s lear that we need previously, to dispose of normal eletrons for over of the superonduting trenhes, that is to say, is neessary to break Cooper pairs as first option (that is to say, minor applied potential differenes than the trenhes not will produe tunnel effet). Then is had that potential differenes applied to the barrier not produe tunnel urrent I, save, that defeat a threshold value that is preisely the bandwidth of trenhes. II. MEASUREMENTS AND SUPERCONDUCTING PARAMETERS A problem that is required is the forming of strength oupling in the Cooper pairs, whih requires of a phonons type that an be distinguishable through an extension of the BCS-theory that does distinguish the different ontributions of the different phonons of the net of a superonduting material of type II. If we ould do a systemati study of the elements, alloys and superonduting omposites seeing as response to the BSC-theory preditions, we would see that if the eletron-phonon interation is more intense (for example, are very highest the values of the resistivity in the normal state), major is the deviation in the theory. The extension of the BCS-theory to the situations in the eletron-phonon oupling is strength, that are the majority of the superondutors, is one of the themes more ompliated in the solid state physis. This theory was realized by Eliashberg and ompleted by M Millan [10, 11]. Assuming that the phononi oupling (oupling ( ) te ) is onstant near the Fermi energy is defined an average eletron-phonon oupling as [11, 1]: 1 F( ) g k,k', ( k' ) ( k ) ( q N E kk' Theorem. (F. Bulnes). Applied geometrial models to the strength eletron-phonon interation in the BCS-theory an be obtained for bounded variation using the spetral densities os, or os, in a superondutor of type II, in a superonduting state that an be extended to the Eliashberg-MMillan theory. The Feynman diagram orresponding to the interation eletron-phonon-eletron haves spetra in the purely BCS-theory, the density F( ), where g k, k', a oeffiient that obeys to a Fermi energy of oupling between phonons and eletrons to eah mode in a superonduting metal of type II, in a superonduting state that an be extended to the Eliashberg-MMillan theory. ),, is If the variations of (), have a behavior as,, then these an be refleted in the interation as a phonon link-wave (whose variation obeys to the theorem of bounded variation [1] onsidering the derivatives of ( t) sin t 1 ) f 0 in the orresponding Feynman diagram (see the figure 1). Indeed, if we onsider as ordinary spetral eletron-phonon density the funtion, for example, F ) [ ( ) ( )], the ( 0 0 funtion F (), haves Fourier anti-transform os 0t, os, is the th f ( t) sin 0t. or and order derivative of Figure 1. Spetra of eletron-phonon-eletron interation. Observe that link-wave represented in the Feynman diagram (whih a phonon wave) an be represented as pulse that inrease at infinite in the two diretions. This is explained for the exitation of eletrons [7]. III. NUMERICAL MODELS AND COMPUTATIONAL WORK Confirming the existing relations between the parameters of temperature near to the ritial, the oherene parameter and the penetration longitude parameter in superondutors of type II, and their dependene with respet to the energy trenhes, we an onstrut ertain models that onfirm the BCS-theory and their limit in the GLAG extensions of magneti field - theory. In this analysis, we will use the numerial models exalting the harateristis of magnitude in the values of superonduting energy always existing to the reating of energy trenhes and their variation in the study and design of superonduting materials with speifi properties to different appliations. Considerations in photonis and of superonduting mirosopi effets establish that the step of modeling haves that go given shape to single objets that after will be used in the sene to reate the quantum effets required. Likewise, as first model we obtain the energy surfae z E k 1, k ), to prohibit energy trenhes that produes a ( planar onstant of momentary superondutivity [14]. To it we 1 The funtion ( t) t sin(1/ t), interval [ 0,/ ].. f is of bounded variation on the 0

3 JOURNAL ON POTONICS AND SPINTRONICS VOL. NO.4 NOVEMBER 014 ISSN (Print) ISSN (Online) onsider the amiltonian z z : sin(1 1) sin(1 1) sin( ) 10, sin(10 ) (1) z 10 sin(1 ), sin(1 ) () The temperature ondition establishes that inside a Cooper pair is had the intrinsi oherent longitude [4]: F , () k T B The proess of modeling an inlude some related ativities with the preparation of the D-model to their measurement of others parameters in superondutivity (see the figure ). In a range x, and y, in D. min 5 max 5 Then the intrinsi oherent longitude (whih is a intrinsi material parameter) depends only of the ritial temperature and whose physial mean is the size of the Cooper pairs, while that the oherent longitude due to Ginzburg-Landau (T ), [10, 11] depends of temperature T, that is to say, onforms the zone in that is had the spatial variation of the trenhes (r). The size of the Cooper pairs an be put in a limit ondition on the finite length that must omply of differene temperatures in the border before and after of the trenhes (onsidering the flow of eletrons and the establishment of phonons predited in the figure 1): T ( ) T( ), (4) Figure. Prohibit energy trenhes to superondutors. The energy surfae E( k 1, k ), that produe a planar onstant of momentary superondutivity [14]. The ase that we see is with major generality, onsidering two different onstants of superondutivity 1, and, (for example, to two metals to an alloy). If we onsider the ritial temperature and the magneti field orresponding to the temperature range T T, and the limit whih is good ondition to our measures as given in (). The omputational work in the figures and, require that model let designed in the spae applying mathematial ompute ombining D and D-models to reate enrihed ontrols of the fine measures doing vary the superondutivity, the temperature and Fermi surfae parameters. An expanded alulus to Cooper pairs in the Fermion interation an be interpreted as photon field where the trenhes is evidened (Figure 4). Figure. Representation of the energy trenhes phenomenon onsidering the ritial temperature of the magneti field orresponding to the range T T, with 1. T T, the free half walked of the eletrons in the normal state (distane that walk between two onseutive interations with the net) will be superior to the oherent length (T ), whih is known as lean superondutor (see the figure ). ere the form of amiltonian is: Figure 4. Fermioni Fok spae orresponding that an be determined around of the superonduting material provoking an energy trenhes whose variation of energy, that an be ontrolled spatially by magnetism. The D-model of the figure 4, an some ativities of the orbit spaes forming and related with the preparing of the D-model to their after animation. The following animation in the next paper will be in a simulating ANSYS or COMSOL [15]. The Eliashberg-MMillan theory is enough ompliated as to that one expression of the ritial superonduting 1

4 JOURNAL ON POTONICS AND SPINTRONICS VOL. NO.4 NOVEMBER 014 ISSN (Print) ISSN (Online) temperature F( ),, T, in funtion of the spetral parameter be very little manageable. Now, when is had that (ome from the relation between strength oupling and the orretions of the energy of the quasi-partiles due to purely Coulumbian interations (fermions showed in the figure 4) ). Then an be use an approximated expression known as MMillan equation: T 1 0.4(1 ) D exp, 1.45 *(1 0.6 ) (5) F ( ) os, or os, used to prove their neessity of bounded variation in ertain finite interval. B) Spetral density of eletron-phonon interation for the intensity of the eletron-phonon oupling obtained for tunnel effet in metal alloys superondutors [4]. IV. APPLICATIONS Through the relation between a magneti field near to, and the range of temperature tends to the ritial temperature we an have the behavior of an eletro-twistor generated by the magneti field-superonduting interation to levitation [7] (see the figure 6). where D, is the Debye temperature and the rest of the parameters have the signifiant sine aquaintane in the mirosopi theory of the superondutivity. Using the funtional derivative of the ritial temperature given by T C F( ),we ompute to the ritial temperature T, to a value of F ), and then is alulated a new ( 0 ritial temperature to the value of modified F( ), with ertain weight, to ertain near frequeny to the before. That is to say, are used the little differenes of the bounded variations suggested by the theorem and the existing statistial weight in the Feynman diagram orresponding to the figure 1, stay intimately related with the weight of said frequeny (see figure 5). A) B) Figure 6. Eletro-anti-gravitational effet having present the proess of superondutor magneti field that produes the Abrikozov effet. At extremely low temperatures an eletron an draw the positive ions in a superonduting material towards it. The movement of the ions reates a more positive region that attrats another eletron to the area (see the figure 7). Figure 5. A) Eletron-phonon spetral density models m* m Equation 1, where the parameter, arry on itself the idea of the lothing eletron (means that the eletron in their movement polarize the net and is aompanied for a phonons loud, for example, to the metal Pb, their value is ).

5 JOURNAL ON POTONICS AND SPINTRONICS VOL. NO.4 NOVEMBER 014 ISSN (Print) ISSN (Online) Figure 7. Cooper pairs make superondutors in their rystalline net. Finally, and doing use of the omputational modeling of the spatial variation of the energy trenhes we an obtain a final representation onsidering other effets due to the matter ondensation phenomena (see the figure 8) through of fog reated by the thermal onditions obtained around of the magneti field-superonduting interation (and other whose origin let be Fok spae reated [6]) in the eletro-anti-gravitational effet to magneti levitation (figure 6). magneti in their two phases (passing from the superondutors of type I to typeii), and the forming of the energy trenhes due to the fermioni interations establishing two orbital spaes good differentiated by the energy trenhes, whih are ahieved with relations between of the oherent length parameter (T ), in funtion of the temperature in the range T T, the penetrating length (T ), and their Ginzburg-Landau parameter k, not being less important the interation eletron-phonon whih in reality mark the fundamental proess to ahieve the fundamental property of the superonduting material, if this is designed in their rystalline net as a superondutor of type II, with a extended theory as GLAG extensions of magneti field, and whose variations in the ouplings eletron-phonon an be bounded and thus measured through the spetral density normed in F ( ). In this ase, the interval of definition must be extremely short, and this ould represent a range of ontrol via Fourier spetra of the ahievement of the ouplings and interations eletron-phonon in the searh of the adequate alloy [16] with a temperatures ambient range. VI. ACKNOWLEDGMENTS I am very grateful with the JPS, for their help and orientation. Figure 8. Energy trenhes with amiltonian sin(1 1) /sin( - ) sin(1 1) 10/sin(10 - The graphis for hardware use the Goround shaped, whih interpolates the olors to along of the surfae starting of the ompute of the illumining in the vertex of the primitive obtaining that the objet has a more realisti aspet (see the figure 8). V. CONCLUSIONS The reating of geometrial models obeys to one orret interpretation of the prohibit energy trenhes forming inside of the range of ritial temperature and the tendeny of the magneti field to a ritial magneti field in the ase of the superondutors of type II. But due the asymmetri inonsistenies in the oneptual map of the superonduting theory, arisen between their different mirosopi theories as the GLAG, BCS, and Eliashberg-MMillan theories, are needed models that permit establish the variations of some parameters and their different behavior under relations between them. One of these relations are, for example, the ritial ). Referenes [1] J. Bardeen, L. N. Cooper, J. R. Shrieffer (April 1957). Mirosopi Theory of Superondutivity, Physial Review 106 (1): [] L. Cooper (November 1956). "Bound Eletron Pairs in a Degenerate Fermi Gas".Physial Review 104 (4): [] A.A. Abrikosov, Zh. Eksp. Teor. Fiz., 144 (1957) (English translation: Sov. Phys. JETP (1957)].) Abrikosov's original paper on vortex struture of Type-II superondutors derived as a solution of G L equations for κ > 1/ [4] M. A. Alario, J. L.Vient, Superondutivity, Eudema Fortuny, Madrid, Spain, [5] F. Bulnes, E. ernandez, J. Maya, Design and Development of an Impeller Synergi System of Eletromagneti Type for Levitation Suspension and Movement of Symmetrial Bodies, ASME, Vol. 7: Fluid Flow, eat Transfer and Thermal Systems, Parts A and B, Vanouver, British Columbia, Canada, November 1 18Paper No. IMECE , pp. 1-; 1 pages. doi: /imece [6] I. Verkelov, R. Goborov, F. Bulnes, Fermioni Fok Spae in the Superonduting Phenomena and their Appliations, Journal of Photonis and Spintronis, Vol., no. 4, pp [7] F. Bulnes, J. Maya and I. Martínez, "Design and Development of Impeller Synergi Systems of Eletromagneti Type to Levitation/Suspension Flight of Symmetrial Bodies," Journal of Eletromagneti Analysis and Appliations, Vol. 4 No. 1, 01, pp doi: 10.46/jemaa [8] L.P. Gor'kov, Sov. Phys. JETP 6, 164 (1959). [9] Ivar Giaever - Nobel Leture. Nobelprize.org. Retrieved 16 De er-leture.html [10] D. Saint-James, G. Sarma and E. J. Thomas, Type II Superondutivity Pergamon (Oxford 1969) [11] V. L. Ginzburg and L.D. Landau (1950). "On the theory of superondutivity," Zhurnal Eksperimental noi I Teoretiheskoi Fisiki 0: 1064.

6 JOURNAL ON POTONICS AND SPINTRONICS VOL. NO.4 NOVEMBER 014 ISSN (Print) ISSN (Online) [1] J. Bardeen, L. N. Cooper and J. R. Shrieffer (1957). "Theory of Superondutivity". Physial Review 108 (5): [1] C. Adams, A. James (19), On definition of bounded variation for funtions of two variables, Transations of the Amerian Mathematial Soiety 5 (4): 84 84, doi: /s [14] J. Mahmoud, Spintronis in Devies: A Quantum Multi-Physis Simulation of the all Effet in Superondutors, Journal on Photonis and Spintronis, vol., no. 1, pp.-7. [15] K. E. Jones, Simulations: Team develops omputational model for prediting superondutivity, [16] De Gennes, P. G., Superondutivity of metals and alloys, Benjamin, New York, USA,