Extended Josephson Relation and Abrikosov lattice deformation
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Extended Josephson relation and Abrikosov lattice deformation
Peter Matlock ∗ Research Department, Universal Analytics Inc., Airdrie, AB, Canada
From the point of view of time-dependent Ginzburg Landau (TDGL) theory, a Josephson-like rela-tion is derived for an Abrikosov vortex lattice accelerated and deformed by applied fields. Beginningwith a review of the Josephson Relation derived from the two ingredients of a lattice-kinematicsassumption in TDGL theory and gauge invariance, we extend the construction to accommodate atime-dependent applied magnetic field, a floating-kernel formulation of normal current, and finallylattice deformation due to the electric field and inertial effects of vortex-lattice motion. The result-ing Josephson-like relation, which we call an Extended Josephson Relation, applies to a much widerset of experimental conditions than the original Josephson Relation, and is explicitly compatiblewith the considerations of TDGL theory.
I. INTRODUCTION
The well-known Josephson Relation (JR) relates ap-plied electric and magnetic fields to motion of theAbrikosov vortex lattice in a thin superconducting sam-ple. Taking a simple form, h E i = − c v L × h B i , (1)the JR is easily understood intuitively via arguments in-volving the Faraday Law, and motion of magnetic flux,and thought to represent an approximation of generalapplicability, therefore being widely employed by exper-imentalists. It is thus time-tested and, though not of afundamental theoretical nature, is an essential and use-ful tool. Here, E and B will denote the local electric andmagnetic fields, and h· · · i denotes an averaging in space.Relation (1) has been known for decades, and does alsostand up to more modern considerations in the contextof time-dependent Ginzburg-Landau (TDGL) theory. In-deed, the JR is derived by Kopnin , using some simpleassumptions about the rigidity of the vortex-lattice mo-tion. In this analysis, the premise is that the vortex lat-tice already represents some solution to TDGL theory,and its motion must remain compatible with that the-ory; therefore there is an implicit assumption that thevortex lattice is triangular.More recently, a derivation along the same lines but ac-commodating inertial effects of the condensate has beenexhibited , which reproduces from the point of view ofTDGL theory an Inertial Josephson Relation (IJR), andwhich had been obtained much earlier in a hydrodynamiccontext . The IJR contains a term reflecting the iner-tia of the condensate itself, and is thus an extension ofthe JR, applicable at high frequencies. We shall reviewthis result below in section III, obtaining the IJR usingthe requirement of gauge invariance, combined with sim-ple ingredients of rigid lattice motion and the currentobtained from TDGL theory.In the present work we shall discuss ways in whichthis analysis can naturally be taken further, producingwhat we shall call an Extended Josephson Relation. TheInertial Josephson relation mentioned above represents one extension of the Josephson relation to accommodatecondensate inertia. By Extended Josephson Relation,we shall mean a relation further extended to include theeffects discussed below.The starting point is chosen to be the Floating-Kernel (FK) version of the TDGL equation, whichcontains the normal current; so named because it can beunderstood by shifting to the reference frame ‘floating’with this normal current.Not only can a normal-current correction be included,but it is not necessary to work with the assumption ofa rigid Abrikosov vortex lattice. Of course, should the(average) magnetic field change with time, so must thedensity of unit-flux vortices.Finally, since the electric field and also the inertia ofthe vortex lattice itself provide anisotropic stimuli in theplane of the vortex lattice, one ought consider deforma-tions of the lattice which include not only the densityfluctuations mentioned above, but a group of geomet-rical deformations which represent a response to theseanisotropies.An Extended Josephson Relation is calculated in sec-tion V which accommodates all these novel ingredients.The result can be considered an extension of the Joseph-son Relation (and Inertial Josephson Relation) which candescribe density fluctuations of the vortex lattice whichallow for a time-varying magnetic field, and anisotropicdeformations of the vortex lattice from the electric fieldand vortex acceleration. II. NOTATION, CONVENTIONS AND BASICS
We consider the usual configuration of a planar samplein the x - y plane, with E ⊥ ˆ z and vortices produced by B k ˆ z . We will make use of the usual definitions E = − c ∂ t A − ∇ φ and B = ∇ × A .We write a gauge transformation as A → A + c ∇ θ (2) φ → φ − ∂ t θ (3) ψ → e ie ∗ θ/ ~ ψ, (4)where θ is a function of space and time. Our calculationwill be gauge-invariant; we will make no gauge choice.This allows us to use the principle of gauge covariance. III. IJR FROM TDGL
Within this section, we carefully extract the InertialJosephson Relation from the TDGL equation. We followclosely the derivation of the Josephson Relation given byKopnin or the IJR given in [3]; we pay special attentionto gauge invariance, in anticipation of extending the IJRlater.The TDGL equation is − ~ m ∗ D ψ + αψ + βψ | ψ | = − Γ p C | ψ | (cid:0) ∂ t + i ~ e ∗ φ + c ∂ t | ψ | (cid:1) ψ (5)where the gauge-covariant derivative is D ψ := ( ∇ − ie ∗ ~ c A ) ψ. (6) J s resulting from (5) is given by variation of A ; J s = i ~ e ∗ m ∗ ¯ ψ (cid:0) ←− D − D (cid:1) ψ. (7)Now we must formulate the assumption of rigid motionof the vortex lattice. We define by ψ the configurationat time zero, ψ ( r ) := ψ ( r , r L ( t ), we thus require | ψ ( r , t ) | = | ψ ( r − r L ( t )) | (8)so that for some pure-gauge function ω we have ψ ( r , t ) = e − iω ( r ,t ) ψ ( r − r L ( t )) . (9)This equation must be gauge-invariant; transformingboth sides using θ ( r , t ) (see (2) – (4)), we see that ω must have the gauge transformation law ω ( r , t ) → ω ( r , t ) + e ∗ ~ (cid:0) θ ( r − r L ( t ) , − θ ( r , t ) (cid:1) . (10)Whatever physical assumption we specify, in the presentcase equation (9), must be independent of any gaugechoice. Although we do not at this stage know the formof the function ω , we do know how it must transform,and we shall make use of the transformation law (10) inwhat follows.Writing v L = ∂ t r L , we use (9) to calculate ∂ t (cid:2) ¯ ψ (cid:0) ←− D − D (cid:1) ψ (cid:3) = − v L · ∇ (cid:2) ¯ ψ (cid:0) ←− D − D (cid:1) ψ (cid:3) (11)+ 2 i | ψ | ( v L · ∇ + ∂ t ) (cid:0) ∇ ω + e ∗ ~ c A (cid:1) . Equation (10) implies that ( v L · ∇ + ∂ t ) ω gauge-transforms as( v L · ∇ + ∂ t ) ω → ( v L · ∇ + ∂ t ) ω − e ∗ ~ ( v L · ∇ + ∂ t ) θ, (12)which determines( v L · ∇ + ∂ t ) ω = e ∗ ~ (cid:0) φ − c v L · A (cid:1) . (13)Using v s := J s e ∗ | ψ | (14)we then obtain ∂ t v s = − v L ·∇ v s + e ∗ m ∗ (cid:2) c ∇ ( v L · A ) − c v L ·∇ A + E (cid:3) . (15)Finally, use the identity ∇ ( v · V ) − v · ∇ V = v × ∇ × V for constant v to write ∂ t v s = − v L · ∇ v s + e ∗ m ∗ c v L × B + e ∗ m ∗ E . (16)Note that this tells us the field v s does not move withthe lattice; otherwise we would have ( ∂ t + v L · ∇ ) v s = 0.We may now take the unit-cell average. Since v s isperiodic, the first term does not contribute and ∂ t h v s i = e ∗ m ∗ h E i + e ∗ m ∗ c v L × h B i . (17)This is the IJR as presented in [3]. It can be thought ofas the consequence of describing a rigidly moving vortexlattice using TDGL theory, and includes the inertial term ∂ t h v s i , absent from the original Josephson Relation. IV. FLOATING-KERNEL TDGL
Our starting point will not be the TDGL equation aspresented in (5), but a version of TDGL supplementedby a floating kernel (FK) term, m ∗ (cid:0) − i ~ ∇ − e ∗ c A − m ∗ en J n (cid:1) ψ + αψ + βψ | ψ | = − Γ p C | ψ | (cid:0) ∂ t i ~ e ∗ φ + c ∂ t | ψ | (cid:1) ψ. (18)The supercurrent is defined though variation of A , J s = e ∗ m ∗ ¯ ψ (cid:0) i ~ ←− D − D ) − m ∗ en J n (cid:1) ψ. (19)In the spirit of a two-fluid model of superconductivity,we write J s = e ∗ | ψ | ( v s − v n ) (20)and write the total current as a sum J := J s + J n (21)= e ∗ n s v s + en n v n (22)where v n := J n /en , J n = σ n E and n = n n + 2 n s .In the following section, we do not attempt to solveTDGL theory to determine the density or deformationdynamics of the vortex lattice, just as in the previous sec-tion there was no attempt to solve TDGL to determinerigid motion of the vortex lattice. Instead, we allow thatit may happen and study the consequences. In fact, rigidlattice motion may have been too strict an assumptioneven in the case of no normal current; there is no a pri-ori reason why the magnetic field must always remainconstant, and the lattice rigid. V. EXTENDED JOSEPHSON RELATION
In this section we allow much greater freedom for themoving vortex lattice; in particular, we allow the magni-tude of B to change with time, so that the vortex latticedensity is time-dependent. Further, we allow the shapeof the vortex lattice to undergo a global time-dependentdeformation, in response to anisotropic stimuli.We shall generalise the rigid-motion requirement (9) to ψ ( r , t ) = e − i ˜ ω ( r ,t ) ψ ( λ ( t )Σ( t )( r − r L ( t )) , . (23)Here λ ( t ) is a dynamic scaling factor for the vortex lat-tice. We can expect such a simple dynamic scaling to bevalid for a B field which does not vary excessively or tooquickly with time. We require r L (0) = 0 and λ (0) = 1,and define B as the magnitude of the average magneticfield when the vortex lattice density is λ = 1, therefore B ( t ) ≡ h| B ( t ) |i = B λ ( t ).Σ( t ) is a two-dimensional dynamic deformation ma-trix for the vortex lattice. Since we have accommodatedscaling with λ , we shall require that Σ be an element of SL (2 , R ), and we will writeΣ( t ) = e S ( t ) . (24)We shall leave discussion of the matrix S to the followingsection.Beginning with the vortex lattice (23) and the TDGLequation with floating kernel (18), we proceed as in theprevious section. Using equation (23) we calculate ∂ t ψ = Υ · ∇ ψ + i Ω ψ (25)where Υ = λ − Σ − ∂ t (cid:2) λ ( t )Σ( t )( r − r L ( t )) (cid:3) , (26)Ω = Υ · ∇ ˜ ω − ∂ t ˜ ω, (27)so that ∂ t D ψ = ( Υ · ∇ + i Ω) Dψ + ∇ ( Υ · ∇ + i Ω) ψ + ie ∗ ~ c ( Υ · ∇ − ∂ t ) A ψ (28) and ∂ t ¯ ψ ( ←− D − D ) ψ | ψ | = [ Υ · ∇ + ∇ Υ · ] ¯ ψ ( ←− D − D ) ψ | ψ | + 2 ie ∗ ~ c [ ∂ t − Υ · ∇ − ∇ Υ · ] A − i ∇ Ω . (29)This allows us to use (20) and evaluate ∂ t ( v s − v n ) = ( Υ · ∇ + ∇ Υ · )( v s − v n )+ e ∗ m ∗ c (cid:2) ∇ ( Υ · A ) − Υ × B (cid:3) + e ∗ m ∗ (cid:2) E + ∇ φ + ~ e ∗ ∇ Ω (cid:3) − n ∂ t J n (30)where this time we have used the identity ∇ ( v · V ) − v · ∇ V = v × ∇ × V + ∇ v · V . (31)Now, although Υ is gauge-invariant, it may be seen from(27) or (25) that Ω must transform asΩ → Ω − e ∗ ~ [ Υ · ∇ θ − ∂ t θ ] (32)which determinesΩ = − e ∗ ~ [ 1 c Υ · A + φ ] . (33)Substituting Ω and Υ into (30), we find ∂ t ( v s − v n ) = (cid:2) ∂ t λλ ( r − r L ) · ∇ − v L · ∇ (cid:3) ( v s − v n )+ (cid:2) ( r − r L ) ∂ t S T ∇ − ∂ t λλ + ∂ t S T (cid:3) ( v s − v n )+ e ∗ m ∗ (1 − τ ∂ t ) E − e ∗ m ∗ c (cid:2) ∂ t λλ ( r − r L )+ ∂ t S ( r − r L ) − v L (cid:3) × B (34)where τ = m ∗ σ n / e n . Finally we take the spatial aver-age and find ∂ t h v s − v n i = e ∗ m ∗ (1 − τ ∂ t ) h E i + e ∗ m ∗ c v L × h B i + (cid:2) ∂ t S T − ∂ t B B (cid:3)(cid:0) h v s − v n i + 2 n σ n h E i (cid:1) + e ∗ m ∗ c (cid:2) ∂ t B B r L + ∂ t S r L (cid:3) × h B i (35)where we have substituted for λ . We have used that v s isperiodic and v n is uniform, and for the sake of simplicitywe have taken the origin r = 0 to be at the centroid ofthe sample. Equation (35) is our Extended JosephsonRelation (EJR), the main result of this paper, and wepause to comment. The first line of (35) is simply theInertial Josephson Relation of (17), with the addition ofthe τ term accounting for the floating-kernel normal cur-rent contribution. The remaining corrections consist ofparts which depend on the time derivative of the mag-netic field, and parts which depend on geometrical latticedeformation; when deformation is absent, S = 0.In the following section we complete the expression ofthe EJR (35) by showing the explicit parametrisation ofthe deformation matrix S , VI. LATTICE DEFORMATION
A two-dimensional space may be expanded by a scalefactor e σ in the x direction, and e − σ in the y directionby the matrix Σ(0 , σ ) = (cid:20) e σ e − σ (cid:21) (36)which has unit determinant. Rotating so that the ex-pansion is along a line at angle ϑ and the contractionperpendicular,Σ( ϑ, σ ) = R ( − ϑ )Σ(0 , σ ) R ( ϑ )= exp (cid:26) σ (cid:20) − ϑ ϑ cos ϑ ϑ cos ϑ − ϑ (cid:21)(cid:27) := exp S µ . (37)Here we have parametrised the deformation by µ :=( σ cos ϑ, σ sin ϑ ). This set of deformations is not closed;composition can produce rotations. When combined withrotations all elements of SL (2 , R ) can be constructed.Some identities are Σ( ϑ, σ ) − = Σ( ϑ + π/ , σ ) =Σ( ϑ, − σ ), Σ( ϑ,
0) = 1 and Σ( ϑ, σ )Σ( ϑ, σ ) = Σ( ϑ, σ + σ ).If we define convenient matrices which depend on thedeformation parameter µ , m µ := (cid:20) µ x µ y µ y − µ x (cid:21) and n µ := (cid:20) µ x µ y − µ y µ x (cid:21) , (38)we may write S µ = m µ n µ /µ . For a given time-dependentdeformation parameter µ ( t ), we may calculate ∂ t S µ ( t ) = m µ µ (cid:2) n µ ′ − µ · µ ′ µ n µ (cid:3) . (39)Now, let us recall the S matrix introduced in sectionV, equation (24). Given some deformations parametrisedby µ , ν , . . . , we would set S = S µ + S ν + · · · . In fact,it is possible immediately to write down several plausibleexamples of physical sources of deformation.Let us anticipate that the application of an electricfield (in the plane), or a time-derivative of this field maytend to deform the vortex lattice. We can accommodatethis with a deformation parameter ν := E E + E ∂ t E . (40) Another possibility is that the global inertia or accelera-tion of the vortex lattice would coincide with a deforma-tion; in this case, µ := v v L + v ∂ t v L . (41)Obviously, there is an approximation involved here inthat our consideration is limited to a global deformation.Adding the above two deformation contributions to-gether, S ( t ) = S ν + S µ . (42) ∂ t S ( t ) may be calculated for each of the two terms byusing (39); ∂ t S ( t ) = m µ µ (cid:2) n µ ′ − µ · µ ′ µ n µ (cid:3) + m ν ν (cid:2) n ν ′ − ν · ν ′ ν n ν (cid:3) . (43)Upon substitution of ∂ t S and ∂ t S T the Extended Joseph-son Relation (35) will contain the parameters E , E , v and v . In principle, these parameters could be fit toexperimental data, characterising response beyond theusual JR or IJR. VII. CONCLUSIONS
As we mentioned in the introduction, the JosephsonRelation, though simple in form, is often used by exper-imentalists to understand the motion of vortices in thepresence of an applied field. It has been shown in theliterature how to recover the Josephson Relation anda form valid at higher frequencies, the Inertial Joseph-son Relation, in the context of TDGL theory. We havetaken this technique further, using the assumption of avortex lattice solution to TDGL theory, to extend theJosephson Relation to a form covering a floating-kernelformulation of TDGL with a normal current in the spiritof a two-fluid model. Additionally, we have shown howto include effects of time-varying magnetic flux, and alsoglobal vortex-lattice deformations; our considerations areexpected to be valid for small changes in magnetic fieldand small lattice deformations. This is not due to anyapproximation in the calculation itself, rather it is due tothe implicit assumption of the characteristics of the vor-tex lattice dynamics, equation (23), which neverthelessrepresents a far weaker assumption than that of perfectlyrigid global motion (8). Acknowledgements
The author is grateful to P.-J. Lin and P. Lipavsk´y forfruitful discussions regarding Josephson Relations. ∗ Electronic address: [email protected] B. D. Josephson, Phys. Lett. 16, 242 (1965) N. B. Kopnin,
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