Size Effects in the Ginzburg-Landau Theory
SSize Effects in the Ginzburg-Landau Theory
Miguel C N Fiolhais ∗ Department of Physics, City College of the City University of New York,160 Convent Avenue, New York 10031, NY, USA andLIP, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal
Joseph L Birman † Department of Physics, City College of the City University of New York,160 Convent Avenue, New York 10031, NY, USA (Dated: September 8, 2018)
Abstract
The Ginzburg-Landau theory is analyzed in the case of small dimension superconductors, a couple of orders of magnitudeabove the coherence length, where the theory is still valid but quantum fluctuations become significant. In this regime,the potential around the expectation value is approximated to a quadratic behavior, and the ground-state derived from theKlein-Gordon solutions of the Higgs-like field. The ground-state energy is directly compared to the condensation energy, andused to extract new limits on the size of superconductors at zero Kelvin and near the critical temperature.
INTRODUCTION
The Ginzburg-Landau (GL) theory [1], proposed in1950, is perhaps the most successful macroscopic descrip-tion of superconductivity, proceeding the famous phe-nomenological description of the electromagnetic fieldin a superconductor by the London brothers [2]. Theproposed macroscopic quantum theory by Ginzburg andLandau makes use of a quartic potential and the Higgsmechanism of spontaneous symmetry breaking [3–6] togenerate a local mass term of the vector potential. Assuch, it successfully describes the Meissner-Ochsenfeldeffect [7–10], for the magnetic field flux expulsion, thephenomenology associated with the phase transition [11–14], and also predicts the existence of a coherence lengthin superconductors, resulting from the local scalar or-der field fluctuations. Furthermore, in 1957, Abrikosov[15] predicted the penetration of strong magnetic fields intype-II superconductors through quantum vortices, giv-ing farther credibility to the GL model. The GL-theorycan be regarded nowadays as the three-dimensional ver-sion of the 3+1-dimensional scalar quantum electrody-namics, studied in detail by Coleman and Weinberg[16, 17], in the early seventies, as an attempt to generatethe scalar field mass through radiative corrections.The size limitations and effects in superconductiv-ity have been under study for several decades [18–22].In this letter, a new method is developed to derivethe approximate size limit of type-II superconductorsin the Ginzburg-Landau theory. The application ofthe Ginzburg-Landau theory is, first and foremost, con-strained by the magnitude of the coherence length, be-low which such a macroscopic theory is no longer valid.However, for type-II superconductors, with small coher-ence lengths, the quantum fluctuations may become sig-nificant enough to impose a new physical limit to su- perconductivity, above the coherence length. In addi-tion to this, the Ginzburg-Landau theory, in the Lon-don approximation, shall only be applied to type-II su-perconductors with large GL parameters, also known as“clean” superconductors, as opposed to type-I supercon-ductors, yielding non-local effects, where the Pippard’smodel must be taken into account [24]. The presenceof a macroscopic massive scalar Higgs-like field pervad-ing the superconducting region in the Ginzburg-Landautheory, corresponding to the collective excitation of theCooper pairs in the lattice [25, 26], implies the existenceof a ground-state energy resulting from the quantum fluc-tuations of the scalar field between the superconductorwalls. While for macroscopic superconductors, these fluc-tuations are usually negligible as the scalar field behavescontinuously in the classical approximation, they becomerelevant for small sized superconductors, leading to a dis-cretization of energy levels, and therefore, of the super-currents as well. In particular, for small superconductor,the ground state energy is expected to increase, even-tually to the point where it surpasses the condensationenergy and restores the vacuum symmetry of the GL po-tential. These quantum fluctuations can be parameter-ized by a quadratic approximation around the minimumof the potential, leading to the Klein-Gordon equationfor the scalar field. Therefore, the ground-state and theallowed energy levels can be directly extracted from theKlein-Gordon solutions for the scalar field in a box. Suchapproximation is, of course, limited to small fluctuations,but may also be extrapolated to large fluctuations, in or-der to provide an interesting lower limit on the size ofsuperconductors.The Ginzburg-Landau ground-state energies, derivedfrom the Klein-Gordon solutions, and the correspondingsize limits predictions at absolute zero and near the crit-ical temperature, are presented in this letter. In particu-1 a r X i v : . [ c ond - m a t . s up r- c on ] A p r ar, as the condensation energy vanishes near the second-order phase transition, the lower size limit increases, pos-sibly reaching macroscopic dimensions. THE GINZBURG-LANDAU THEORY AND THEHIGGS MECHANISM
At zero kelvin, the hamiltonian of the Ginzburg-Landau theory of superconductivity can be written as: H ( ψ, ∇ ψ, A , ∇ A ) = 12 m e | ( − i ¯ h ∇ − e A ) ψ | + α | ψ | + β | ψ | + 12 µ ( ∇ × A ) , (1)with the order parameter ψ ( x ) = ρ ( x )e iθ ( x ) , where ρ ( x )and θ ( x ) are real fields, 2 e is the electric charge of theCooper pairs, and the real constants α and β give thestrength of the quadratic and quartic terms, respectively.The vector field is represented by A . The ground stateof the potential, V ( ψ ) = α | ψ | + β | ψ | , is particularlyinteresting for a negative mass parameter α , (cid:104) ψ (cid:105) = ρ = − αβ , (2)corresponding to an infinite number of degenerate states.After the spontaneous symmetry breaking, i.e. fixing thegauge to θ ( x ) = 0, the energy density becomes H ( ψ, ∇ ψ, A , ∇ A ) = ¯ h m e ( ∇ ρ ) + V ( ρ ) + 2 e ρ m e A + 12 µ ( ∇ × A ) . (3)The mass term of the vector field, resulting from thespontaneous symmetry breaking, and also known as theMeissner-Higgs mass term [14], suppresses the magneticfield inside, with a corresponding London penetrationlength, λ L = (cid:114) m e µ e ρ . (4)On the other hand, the scalar field mass term can beassociated with the coherence length, ξ = (cid:115) ¯ h m e | α | . (5)Finally, the thermodynamic critical magnetic field canalso be extracted from the condensation energy, i.e. thenecessary energy to restore the vacuum symmetry, whichleads to B c = 14 ¯ he λ L ξ . (6) At finite temperature near T c , the quadratic parametervaries linearly with the temperature, α ( T ) ≈ α (cid:18) − TT c (cid:19) . (7) THE QUADRATIC APPROXIMATION AND THEKLEIN-GORDON EQUATION SOLUTIONS
As mentioned before, the quartic potential of theGinzburg-Landau theory can be approximated to aparabola around the expectation value, by expanding itto the second order of Taylor series, V ( ρ ) ≈ − α β + 2 α ( ρ − ρ ) , (8)where the fluctuations around the expectation valuecan be expressed in terms of a Higgs-like field, h ( x ) = ρ ( x ) − ρ . The quadratic approximation aroundthe expectation value is represented in Figure 1. ρ V( ρ ρ Higgs potentialQuadratic approx.
FIG. 1: Quadratic approximation (dashed line) of the Higgspotential (full line) around the expectation value.
In the particular case of small dimensions supercon-ductors, where significant quantum fluctuations arise, thefour-dimensional version of the Ginzburg-Landau theory,the Coleman-Weinberg model, comprising time depen-dencies, becomes a more accurate description. Assum-ing there is no external magnetic field, and the internalcontributions to the electromagnetic field are too small ,the dynamics of the scalar Higgs field in the quadraticapproximation of the Coleman-Weinberg model results This is only valid if the superconducting region is consideredto have a local net zero charge density, corresponding to themacroscopic limit, i.e. the Ginzburg-Landau regime.
2n the following time-dependent hamiltonian, H ( h, ∇ h ) = ¯ h m e (cid:18) ( ∇ h ) − c ( ∂ t h ) (cid:19) − α β +2 αh . (9)As such, in this approximation, the dynamic equation forthe scalar field simplifies to a Klein-Gordon equation,1 c ∂ h ( x ) ∂t − ∇ h ( x ) + 2 m e | α | ¯ h h ( x ) = 0 . (10)It should be stressed, however, that this approximationis more favorable for temperatures near criticality, ratherthan at zero Kelvin, as the expectation value of the Higgspotential approaches to zero giving rise to a second orderphase transition. In any case, it provides a good estimateon the magnitude of the ground-state energy one shallexpect in such potential, even at low temperature.The plane wave solutions for the Higgs-like field are,therefore, of the type, h ( r , t ) ∼ e i ( k · r − ωt ) , (11)where −| k | + ω = µ . In the particular case the scalarfield is constrained to a small sized three-dimensionalbox, like in a superconductor with small dimensions, thedifferent energy levels are given by the solutions for thefree particle of the infinite square potential [27], E n x ,n y ,n z = (cid:115) m e c | α | + (cid:80) i n i π ¯ h c L . (12)In the particular case the box dimensions are much largerthan the coherence length ( L >> ξ ), the second term inequation (12) becomes less relevant. Note that this con-dition is necessary to keep the validity of the Ginzburg-Landau theory intact in this regime. As a result, thekinetic component of the ground-state energy is approx-imately given by the non-relativist limit, E k1 ≈ √ π ¯ h c (cid:112) m e | α | L = 3 √ π ¯ hξcL . (13) THE GROUND STATE ENERGY AS THE SIZELIMIT OF SUPERCONDUCTIVITY
In small superconductors, but still larger than the co-herence length, as the ground-state energy arises, it maysurpass the condensation energy and restore the vacuumsymmetry. As such, one can establish a limit on thesize of the superconductor by imposing that the energydensity of its ground-state is equal to the condensationenergy, E L = B µ , (14) which leads to, L = 48 √ e µ c ¯ h ξ λ . (15)or, L = 1 . ξ / λ / . (16)At zero Kelvin, the size limit of a superconductor in theGinzburg-Landau theory depends on both the coherenceand penetration lengths, prominently on the first. Forthat reason, this result is only valid for clean supercon-ductors with a Ginzburg-Landau parameter [28] muchlarger than one, k GL >>
1, so that the minimum al-lowed scale can be at least one order of magnitude largerthan the coherence length, enabling the validity of theGinzburg-Landau theory.On the other hand, as the quadratic term vanishes nearcriticality, the ground-state energy of the scalar field atfinite temperature in this regime is no longer in the non-relativistic limit, E ≈ √ π ¯ hcL . (17)As a result, the minimal allowed scale in the Ginzburg-Landau theory near the critical temperature is, L ( T ) = 32 µ e √ π ¯ h λ L ( T ) ξ ( T ) , (18)or, L ( T ) ≈ . (cid:112) λ L ( T ) ξ ( T ) . (19)It should be noted, as well, that since the minimum al-lowed scale near criticality also depends on both the co-herence length and the London penetration depth, itstemperature dependence near the critical temperature isexpected to increase at a smaller rate than the coher-ence length. As such, the validity of this model is quitequestionable for temperatures extremely close to the crit-icality, even in clean superconductors. CONCLUSIONS
The lower size limits of clean high-temperature su-perconductors were derived in the framework of theGinzburg-Landau theory at zero Kelvin and near thecritical temperature. The existence of a ground-statefor the scalar Higgs-like field was predicted and de-rived, and compared with the condensation energy forextremely small superconductors. As result, the mini-mum allowed scale was found to depend on both the co-herence length and the London penetration depth in bothregimes, within the range of applicability of this model,limiting it to clean high-temperature superconductors.3s the experimental results on the size limits of cleanhigh-temperature superconductors are, to the best of ourknowledge, currently quite limited, we would like to chal-lenge experimental groups to test the predicted resultsobtained with this model. A concurrent result wouldprovide additional to the already successful Ginzburg-Landau theory, while the opposite would oblige furtherscrutiny on its range of validity. ∗ Electronic address: miguel.fi[email protected] † Electronic address: [email protected][1] V.L. Ginzburg and L.D. Landau,
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