How the electron-phonon coupling mechanism work in metal superconductor Qiankai Yao1,2 1College of Science, Henan University of Technology, Zhengzhou450001, China 2School of physics and Engineering, Zhengzhou University, Zhengzhou450001, China Abstract Superconductivity in some metals at low temperature is known to arise from an electron-phonon coupling mechanism. Such the mechanism enables an effective attraction to bind two mobile electrons together, and even form a kind of pairing system(called Cooper pair) to be physically responsible for superconductivity. But, is it possible by an analogy with the electrodynamics to describe the electron-phonon coupling as a resistivity-dependent attraction? Actually so, it will help us to explore a more operational quantum model for the formation of Cooper pair. In particularly, by the calculation of quantum state of Cooper pair, the explored model can provide a more explicit explanation for the fundamental properties of metal superconductor, and answer: 1) How the transition temperature of metal superconductor is determined? 2) Which metals can realize the superconducting transition at low temperature? PACS numbers: 74.20.Fg; 74.20.-z; 74.25.-q; 74.20.De ne is the mobile electron density, η the damping coefficient 1. Introduction that is determined by the collision time τ . In the BCS theory[1], superconductivity is attributed to a In metal environment, mobile electrons are usually phonon-mediated attraction between mobile electrons near modeled to be a kind of classical particles like gas molecules, Fermi surface(called Fermi electrons). The attraction is each of which performs a Brown-like motion and satisfies the sometimes referred to as a residual Coulomb interaction[2] that Langevin equation can glue Cooper pair together to cause superconductivity. dυ m υ κ(r,t) (2) Historically, the electron-phonon coupling concept was e dt suggested by Fröhlich in 1950[3], and confirmed by the Where, κ denotes the random potential with an average equal discovery of isotope effect[4]. The BCS theory led to a to the thermal motion energy of mobile electrons(in natural fundamental understanding of superconducting phenomena in units) momentum space, but it needs to give a more heuristic π 2T 2 picture[5,6]. κ (3) 4E In this paper, we begin with a decision of Brown-like F motion with a particle-field Lagrangian for mobile electrons in EF is Fermi energy. Eq.(2) can be yielded by a particle-field metal environment. This Lagrangian takes a mathematical form Lagrangian similar to the electromagnetic one, and the similarity allows us 1 L(υ, r,t) m υ 2 aA (r,t) υ κ(r,t) (4) to adopt a unified framework to describe the electromagnetic 2 e a and acoustic phenomena. By the definition, it can yield an endowed with a differential relation effective attraction between Fermi electrons to be physically dA A responsible for electronic pairing. The effective Hamiltonian of a a υ A (5) dt t a paired electrons and its Schrödinger solution are presented, which will directly derive the transition temperature of metal for the damping vector Aa (υt r) 4πεa , a η / 4πεa . materials, including a physical criterion for their working as Although the presented Lagrangian only can lead to a superconductors. Brown-like motion, rather than a bound state, it still gives us a firm base to develop the electronic pairing model. 2. Acoustic interaction From Lagrangian (4) we see that, the damping interaction 1) From damping motion to acoustic interaction As well is mathematically similar to the electromagnetic interaction. The known that, the electric conduction of metal materials is similarity inspires us to treat Aa as an acoustic vector potential, achieved by means of the drift motion of large number mobile and a as an acoustic charge with mass density εa playing the electrons. When conducting in metal, mobile electrons will role of acoustic permittivity[7], analogous to the electric one εe . suffer a damping collision with other objects, and thus cause a This means, a mobile electron in metal is actually acting as a collision-dependent resistivity dyon(quoted from Schwinger’s invention[8]) of carrying dual m charge (e,a) , rather than a pure electrically charged particle. ρ ηn e2 , η e (1) e τ Importantly, the electromagnetic interaction essence of material elasticity and tension suggests, the acoustic equations should screening length, namely have the basic structure of electromagnetic equations. The π π suggestion allows us to formally write a joint electric-acoustic (ξ , ξ ) , e a 2 2 equation group[7] 4mee PF 4mea PF (12) 10 6 ρe ~ (10 m,10 m) E , B 0 εe B E P is Fermi momentum. The two characteristic lengths E 0, B μ ε μ j F t e e t e e (6) respectively determine two modified frequent dispersions ρ b a , e 0 ω2 k 2 1/ ξ 2 . It is tantamount to the photon and phonon ε e,a e,a e,a a obtaining their masses 1/ ξ . Eq.(11) provides us a physical e b e,a b 0, e μ ε μ j t a a t a a framework to study the electron-phonon coupling. For example, by the equation we can write a composite Yukawa potential for a ρa , ja denote the acoustic charge and current densities, μa mobile electron with dual charge (e,a) at the zero point the reciprocal of elastic modulus(analogous to magnetic e r/ξ a r/ξ permeability μe ). Correspondingly, the related fields are e a (e,a) (r) e e (13) defined by r r Ae It shows that, when the distance satisfies ξe r ξa , two E e , B Ae t (7) acoustically charged electrons will produce a net Coulomb-like A b a , e A attractive force a t a a 2 1 ρ 2 The vibrating and shearing fields b , e can be thought of as F (r) ~ (14) att r 2 4πε n 2e 4 r 2 the quantum of acoustic wave in term of phonon, by which the a e acoustic interaction is mediated(analogous to the electro- The force is explicitly dependent on resistivity ρ and mass magnetic interaction mediated by photon). When the medium density εa , that is, the bigger resistivity and smaller density, the stronger attraction. condition of 1/ μa εa 1/ μeεe is met, Eq.(6) can be amalgamated into a Heaviside-like form[9] Consequently, the electromagnetic analogy also allows us to express the acoustic force on an acoustically charged electron Bˆ Eˆ ρ , Eˆ μ ε j in a Maxwell-style form e e e t a (8) aA Eˆ a ˆ ˆ Fa a a (15) B ρa , B μeεe je t t Of which, the first term describes the damping effect, and the with second the acoustic attraction. In the follows, we will present Eˆ ε E 1/ μ e μ j j , 1/ε ρ ρ how the acoustic attraction can glue two Fermi electrons e a , e e e e e e (9) together to form a Cooper pair at low temperature. ˆ B 1/ μe B εa b μa ja ja , 1/εa ρa ρa 2) Cooper system According to the BCS theory, two paired electrons must possess opposite momentum and spin[1]. In Eq.(8), the acoustic current ( ja , ρa ) just occupies the position of the magnetic source. And hence, it is appropriate for However, in our model, opposite momentum implies opposite us to phenomenologically understand the acoustic charge a as damping, and only the opposite damping can determine the zero an acoustic “magnetic charge”, and the vibrating(shearing) field resistance characteristic of Cooper system. So, in this sense, b ( e ) as an acoustic “magnetic (electric) field”. superconductivity doesn’t exclude the existence of damping, but Not only that, with regard to Eq.(6), if introducing the requires a zero damping effect on the whole. following 2-component arrays In practice, once the attraction of acoustic charge is introduced, we can use Hamiltonian method to study electronic μ J 1/ε ρ (J, ρ) e e , e e pairing. To this end, let us consider two acoustically charged μa Ja 1/εa ρa Fermi electrons with opposite momentum and spin, which (10) separated by a distance 2r( ξa ) , only involve radial motion Ae / μe εee (A, ) , relative to their mass centre. Then, by canonical momentum Aa / μa εa a L P p a F Aa (16) we also can equivalently express it in the Proca form[10] υ 2 2 ξ 0 we have the following paired Hamiltonian 2 , e (11) 2 (A, ) (J , ρ) 2 t 0 ξa 2 2 2 Pj a H (P , r ) F κ (17) where, the acoustic interaction range(also called the coherent j j Fj j1 2me 4rj length) ξa achieves its definition by following the electric a F is the effective acoustic charge carried by Fermi electrons P j 0 s s js M s (shorter collision time determines a F a ), κF the average , (22) t s thermal motion energy of Fermi electrons. In operator form, the 0 s Ps t Hamiltonian yields a s-wave symmetrical Schrödinger solution with the effective polarization and magnetization defined by r1 r2 4 P n p , M n m (23) a0 s s s s s s Ψ (r1,r2 ) ~ e , a0 2 (18) mea F ps , ms denote the induced dipole and magnetic moments of a followed by a single-particle bound energy, i.e. disassembly super-atom. According to Cooper’s suggestion[11], the induced energy supercurrent can be used to replace the electromagnetic 4 potential to describe the electromagnetic properties of m a e F E(T ) E0 κF (T ) , E0 (19) superconductor.