American Journal of Condensed Matter Physics 2014, 4(2): 32-35 DOI: 10.5923/j.ajcmp.20140402.02

A New Approach to Size Dependent Breakdown of Superconductivity in Ultranarrow Nanowires

I. Bogachan Tahirbegi

University of Barcelona Faculty of Physics, Av Diagonal 647-08028 Barcelona, Spain

Abstract Bardeen–Cooper–Schrieffer (BCS) came up with a groundbreaking theory in the field that has helped in the understanding of the superconductivity phenomena. In this paper, a new model is presented to explain localization of electrons using the basis of BCS theory. In order to achieve this, the necessary parameters for cooper pair formation were studied and an attempt has been made to explain cooper pair formation. The wave properties of the electrons and their diffraction patterns have been considered as waves similar to those observed in the double slit experiment. Using the subwaves, which are produced when waves penetrate the lattice structure of the material, a model has been proposed to explain how the cooper pairs move through a superconducting material. According to the relation between the lattice constant and wavelength of electrons at fermi level, cooper pair formation is analyzed. Moreover, in order to probe this model, a simulation has been used. This simulation has been studied for aluminum atoms because of its plane wave kind of behavior and size dependent breakdown of superconductive properties. According to the relation between the lattice constant and the atomic diameter, size dependent localization and delocalization of electrons were observed depending on the number of aluminium atoms, showing a strong correlation between the simulation and the experimental results. Keywords BCS theory, Condensed matter physics, Size dependent breakdown of superconductivity, Cooper pairs

Schrieffer. In this theory, electrons close to the Fermi level 1. Introduction are attracting each other because of phonon interactions. In summary, the proposed mechanism of the Cooper pair is as Since the discovery of superconductivity by Heike follows; when an electron moves through the lattice with a Kamerlingh Onnes, various theories have been developed to particular spin, it distorts the lattice and the distortion of a explain the phenomena. [1] There were difficulties in positive charge attracts a second electron to form a Cooper delineating a satisfactory microscopic theory for pair. [6] These electrons couple to the lattice resulting in superconductivity. In fact, the difficulty lies in focusing on pairs, called Cooper pairs. The pairs of electrons behave the part of the interaction that causes the transition from like bosons, with different properties to separate electrons; normal to superconducting state. A theory based on long one being that they manifest an energy gap above the Fermi wavelength components of the Coulomb interaction, and the level. In this paper, we have taken a theoretical standpoint fact the fluctuations in electron density cause electrons to be to explain localization of electrons and size dependent localized, was proposed by Heisenberg and Koppe. The breakdown of superconductivity in ultranarrow nanowires theory focused on perfect conduction rather than diamagne using the basis of BSC theory. The presented model aims to tism. [2, 3] Perfect diamagnetism in superconductors was shed light on the behavior of the solid material according to discovered by Meissner and Ochsenfeld in 1933. Magnetic the interference of the electron waves near the Fermi level. field lines were expulsed when the material changed from We considered that the wavefunctions are assumed as plane normal to superconducting state. [4] Moreover, the waves. The relationship between plane waves and the discovery of the isotope effect was a breakthrough for ordered atoms of a lattice explains the localization of superconductivity, because it proved that electron phonon electrons for superconductors. Our theory functions well in interactions are responsible for superconductivity. [5] A its application of explaining part of the interaction, which is more comprehensive theory of superconductivity was responsible for the transition from normal to modeled by John Bardeen, Leon Cooper, and Robert superconductive state and the localization of electrons. According to the relation between the lattice constant and * Corresponding author: [email protected] (I. Bogachan Tahirbegi) the atomic diameter depending on the number of atoms, size Published online at http://journal.sapub.org/ajcmp dependent localization and delocalization of electrons were Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved observed for aluminum atoms.

American Journal of Condensed Matter Physics 2014, 4(2): 32-35 33

2. A Cooper Pair Formation Model In Eq. (4), λ is the wavelength of an electron, h is the Planck constant (6.626068 × 10−34 m2 kg/s) and p is the Various analytical studies into conduction suggested that momentum of an electron. In Eq. (5), m is the effective mass the localization of electrons is a fundamental aspect of and v is the speed of the electron. Equation (6) shows the superconductivity. [8] A sine wave is often used to model the −31 relationship between the rest mass m0 (9.11 × 10 kg) and wavefunction of a particle, but since sine waves extend to the effective mass m of an electron, where “a” is a constant infinity and cannot represent a particle (1). In this analysis that changes according to the material and the temperature localized particles are represented by the superposition of [11]. waves with different wavelengths. We have proven that for the electrons near to Fermi level Ψ (x,t)= Ae i(kx-w t ) (1) the wavelength of them can be comparable with lattice In this equation, k is the wave number and w is the angular constants. frequency (angular speed). In order to describe a wave, these λ = h/p (4) wave functions must extend to zero in either direction and p = mv (5) also, as a requirement for the Schrodinger equation; the plane waves added together must be a solution to the Schrödinger m = m0a (6) equation itself, as is shown in the equation below; Combining the previous Eqs. (4)–(6), the next equation is iħdΨ(x, y, z, t)/dt=(-ħ2/2m)*∆2Ψ(x, y, z, t) obtained. +V(x, y, z, t)*Ψ(x,y,z,t) (2) λ =h/m0av (7) The electron velocity at Fermi level according to the free is a linear equation, if Ψ1(x, y, z, t) and Ψ2(x, y, z, t) are both 6 −1 solutions of the equation, thus, electron model is equal to Vf = 1.4×10 ms . [12] Taking into account this value and a, h, m0 cited previously, the Ψ(x, y, z, t)= a1* Ψ1(x, y, z, t)+a2* Ψ2(x,y,z,t) (3) wavelength value obtained is equal to 0.519/a nm. Taking where a1 and a2 are arbitrary constants. into account that for aluminum a =1,18 [13] and given that This linearity can be explained by interference and the distance between the atoms for aluminum 0,405 nm [14. diffraction of waves. For example, in the double slit 15], the aperture (lattice constant) is comparable to the experiment, if the wave emerging from one slit satisfies the wavelength. Thus, interference of plane waves can create wave equation, then addition of other waves will create a localized electrons near to the Fermi level (Figure 1.). new wave and this also will satisfy the equation. For localizing electrons the electron waves interact with the slits causing the diffraction of the waves and interference may result in the resulting subwaves. Using the results from this experiment, we propose that in this model the electrons act as waves penetrating through the interstices of neighboring atoms. This is analogous to the electron wave property theory that was proven in 1927 by Davisson and Germer, [9] in which electrons were accelerated through a slit and the diffraction pattern of these electrons was detected. Figure 1. Simplified representation of the model. The green circles The atoms in our model have been considered as spheres represent the atoms in the lattice structure and light blue waves represent the and the aperture between the atoms is considered as the slit, electron waves. The interferences of the subwaves (bright dots) represent localized electrons Figure 1 demonstrates a simplified scheme of this model, where the green circles represent the atoms in the lattice This localization of electrons has been studied by using structure. There is one slit between each pair of atoms in the the wave properties of the electrons from the external lattice. The penetrating waves from neighboring atoms are orbitals of the atoms and its diffraction patterns through the represented in light blue color. These waves interact with lattice structure of the materials. This hypothesis was probed each other and these “interferences” bring about the by using a simulation [16] and according to the relation destructive and constructive pattern of the electron waves. between the lattice constant (0,405 nm) and the atomic The distance between atoms is related to the size of the slit in diameter (0,286 nm) for the aluminum atoms [14, 15]. In this the Davisson and Germer experiment. Davisson and Germer model the atoms have been considered as spheres and the demonstrated that the wavelength of the electron and the slit aperture (lattice constant) between each pair of neighboring size must be comparable. This was achieved by accelerating atoms is considered as a slit. The atoms number (slit number) the electrons. [10] In this paper, we suggest that distance started to decrease from 12 to 2 causing to the delocalization between atoms and the wavelength of an electron can be of electrons (Figure 2 A-C). These results support comparable without accelerating the electrons for electrons experimental observations of the superconductive near to fermi level for some metals near to absolute zero. breakdown for ultranarrow sized aluminum nanowires [17].

34 I. Bogachan Tahirbegi: A New Approach to Size Dependent Breakdown of Superconductivity in Ultranarrow Nanowires

Figure 2. Irradiance distributions for single slit (green line) and 12, 6 and 2 slits (red line). These plots represent the electrons localization (A) and delocalization (B,C) by decreasing the number of aluminum atoms Localized electrons have been assumed as sine waves with So that, close frequencies and the equation (10) can be achieved by ∆k→0. the addition of two sine waves with trigonometric formula (8, This result supports the idea of infinite conductivity in a 9). In the following equations, k represents the wave number, perfect infinite crystal, since finite conductivity is caused by w represents the angular frequency (angular speed). Also, ∆k the motion of thermal ions. [18] However, according to and ∆w symbolize the difference in wavenumber and angular Cooper theory, superconductivity is caused by perfect frequency of electronic subwaves respectively. y1 and y2 in diamagnetism, not infinite conductivity. To account for this equations (8 and 9) represent subwaves, which account for discrepancy the behavior of electrons is considered the waves after the interference with the lattice. The analogous to those of waves in a body of water. The interference of subwaves is provided for in equation (10). In dispersion relation for water waves depends on the this instance, the system has been given for double slits, but wavelength of waves, different wavelength moves with can be extended for many slits. different speed according to the formula;

y1= ASin((k-∆k)x-(w-∆w)t) (8) w = C(k)k [19] (14)

y2= Asin((k+∆k)x-(w+∆w)t) (9) C(k) is a dispersion relation constant, which is a function of wavelength. For electrons, this dispersion relation y1+y2=2Asin(kx-wt)cos((∆k)x-(∆w)t) (10) constant is a function of the aperture between the atoms (d) In equation (10), the second term cos ((∆k)x-(∆w)t) alters and the equation will change to (15). the modulation of the first term, sin (kx-wt), corresponding to the oscillation at the average of the two frequencies. If ∆k w= C(d)k (15) and ∆w converge to zero, the difference between the The angular frequency (angular speed) of each subwave wavelengths is negligible. That situation can be provided, if will be affected according to the aperture of the atoms. Thus, the equation (10) turns to (11). angular frequency of each subwave will be related to the dispersion relation and the difference of angular frequencies y1+y2=2Asin(kx-wt) (11) of subwaves will be a function of the phonon frequency (f). cos((∆k)x-(∆w)t)→1 (12) ∆w= C(f) ∆k (16) In case the atoms are not vibrating, the creation of sine waves with close frequencies can be explained as follows; Thus, in a lattice with high phonon frequency, the all of the subwaves (λ1, λ2… λN ) of an electron are created behavior of each subwave will converge to each other from the same plane wave, so that all of them have the same (∆w→0). Convergences of ∆k and ∆w to zero give validity wavelengths. (λ1=λ2=…=λN). Even at very low temperatures, to these equations. (11, 12) These results imply that the the atoms continue to vibrate, albeit with a very low amplitude of lattice vibrations are converging to zero, which amplitude. The amplitude of the vibration determines the is possible at low temperatures and also phonon frequency limits of the distances between the atoms and therefore the must be as high as possible (∆w→0) for superconductivity limits of the wavelengths of the subwaves (k-∆k, k+ ∆k), to occur according to BCS theory predicts [6]. since different apertures between atoms will cause This theoretical approach also accounts for the formation variations in subwaves. For temperatures near to absolute of Cooper pairs. If we consider an electron moving in a zero, there are still lattice vibrations, but the amplitude of particular direction, there is also another electron moving in the vibrations is very small and the wavelengths of the the opposite direction (17, 18), due to the distortion of the subwaves converge to each other (λ1~λ2). lattice (Figure 1). Electrons travelling in opposite directions For 2 slit → λ1~λ2 can be represented by the equations below: For N slit → λ1~λ2 ~λ3~ ….~λN y1=2Asin(kx-wt) (17)

k=2π/ λ (13) y2=2Asin(kx+wt) (18)

k1~k2~ k3 ……~kN y= 2Asin(kx-wt) + 2Asin(kx+wt) (19)

American Journal of Condensed Matter Physics 2014, 4(2): 32-35 35

With trigonometric formula: gratitude for the funding of this study during his stay in Spain y=4Acos(wt)sinkx (20) by Fundacio Bosch i Gimpera. Equation (20) describes a wave oscillating, with a stationary spatial dependency, as a function of time. A Cooper pair of electrons couple and this manifests a new lower energy state, essentially forming a standing wave. REFERENCES This situation is analogous to the behavior of a wave inside [1] Onnes H K 1911 Proceedings of the Koninklijke Akademie a square well; for a one-dimensional infinite square well, Van Wetenschappen. 14 113. the energy levels behave like standing waves. The travelling waves are reflected by the wall of the infinite square [2] Heisenberg W 1948 Two lectures Cambidge University Press potential and a travelling wave is created that travels in the Cambridge. opposite direction. The interference of these waves creates a [3] Koppe H 1950 Ergeb. Exakt. Naturw. 23 283. standing wave [20]. All the electrons near to Fermi level will condense and [4] Meissner W and Ochsenfeld R 1933 Naturwissenschaften 21 787. create a band gap above the Fermi level. According to this theoretical approach, Cooper pair [5] Maxwell E 1950 Phys. Rev. 78 477. formation can be considered as behaving as follows. An [6] Bardeen J, Cooper L N and Schrieffer J R 1957 Phys. Rev. increase in temperature causes the amplitudes in the atomic 108 1175. vibrations to increase, resulting in an increase in the wavelengths of the electronic subwaves (∆k). At a critical [7] Leggett A J 2006 Nature physics 2 134. temperature Tc, equation (11) no longer provides for [8] Chaban I A 2006 Journal of Superconductivity and Novel standing waves and concurrently leads to the creation of Magnetism 19 1. energy states with different energies at the critical [9] Davidson C and Germer L 1927 Phys. Rev. 30 705. temperature (Tc). Also, the same situation can be created as explained above by decreasing the number of aluminum [10] Kartalopoulos S V 2004 Optical Bit Error Rate: An atoms. This situation causes to delocalization of electrons Estimation Methodology (John Wiley & Sons, New Jersey) p and proves the reason of size dependent breakdown of 41. superconductivity in ultranarrow nanowires. [11] Stradling R A and Wood R A 1970 Journal of Physics C 3 94. [12] Harker A H 2002 Free electron model lecture, Physics and Astronomy UCL, Vol. 14 http://www.cmmp.ucl.ac.uk/_ahh/ 3. Conclusions teaching/3C25/Lecture14p.pdf. Localization of electrons is a fundamental of [13] Levinson H J, Greuter F and Plummer E W 1983 Phys. Rev. B superconductivity. In agreement, it has been determined that, 27 727. for localizing electrons, the electron waves can interact with [14] H. Ibach and H. Luth, Solid-State Physics: An Introduction to slits. In this model, the atoms have been considered as Principles of Materials Science, (Springer, Berlin, Heidelberg, spheres and the aperture between each pair of neighboring 2009), p. 420. atoms is considered as a slit. It has been proven that the wavelengths of electrons near the Fermi level can be [15] C. Kittel, Introduction to Solid State Physics, ed. J. Wiley (Wiley, New York, 1996), 27 p. 129. comparable with the aperture between the atoms. That causes to the localization of the electrons. By decreasing the [16] Fraunhofer Diffraction simulationhttp://wyant.optics.arizona number of atoms, the electrons start to delocalize. That .edu/multipleSlits/multipleSlits.htm. causes the lost of the superconductivity in ultranarrow [17] Zgirski M, Riikonen K, Touboltsev V, and Arutyunov K 2005 nanowires. These theoretical tenets may provide the Nanoletters 5 1029. interaction which is responsible for the transition from normal to superconductive state and the reason of [18] Hermann C 2005 Statistical physics: including applications to condensed matter (Springer, New York) p 184. superconductive breakdown at ultra narrow nanowires. [19] Dean R G and Dalrymple R A 1991 Water wave mechanics for engineers and scientists (World Scientific, Singapore), p ACKNOWLEDGEMENTS 64. [20] Bes D R 2007 Quantum Mechanics: A Modern and Concise The author thanks to Mark Fields for the English Introductory Course (Springer, Berlin, Heidelberg), p 43. corrections. Also, the author wants to express his deepest