Structure and Pinning of the Moving Vortex Matterin Type II Superconductors

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Structure and pinning of the moving vortex matter in type II superconductors

View the table of contents for this issue, or go to the journal homepage for more 2009 J. Phys.: Conf. Ser. 150 052028

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Structure and Pinning of the Moving Vortex

Matter in Type II Superconductors

D.T. Bu and B. Rosenstein

Electrophysics Department, National Chiao Tung University,

Hsinchu, Taiwan, R.O.C.

July 30, 2008

Introduction and Summary. In type II superconductors for which the pen-etration depth exceeds the correlation length the magnetic …eld penetrates the sample in a form of Abrikosov vortices, which strongly interact thereby cre-ating an elastic "vortex matter". Impurities always present in a sample lead to inhomogeneities which greatly a¤ect the thermodynamic and especially dynamic properties of the vortex matter. In addition, thermal ‡uctuations also signi…-cantly in‡uence the vortex matter, either directly by melting the vortex lattice into a vortex liquid or by reducing the e¢ ciency of the disorder. Once electric current J is injected into a sample, one is faced with a problem of description of the dynamical phase diagram which should be drawn in the three dimensional space T H J . This makes the analysis essentially more complicated, since one has to go beyond linear response.

In this note we investigate dynamics of the vortex matter beyond linear response using the disordered Ginzburg - Landau model (GL). In statics the replica method of handling disorder in the framework of GL model was utilized [1] to obtain the irreversibility line along with other properties of the disordered vortex matter. Dynamics in the presence of thermal ‡uctuations and disorder is phenomenologically described using the time dependent Ginzburg - Landau (TDGL) model in which the coe¢ cients have random components. Such an attempt was made by Dorsey, Huang, and Fisher [2] in the homogeneous (liquid) phase using a dynamic Martin-Siggia-Rose (MSR) formalism. They obtained the irreversibility line and formulated the linear response theory of the vortex matter.

In strongly type II superconductors magnetic and electric …elds inside the su-perconductor are homogeneous over a wide range of parameters since the mixed state originates from superposition of many (~B=Hc1 1) vortices. The same

argument is valid for electric …eld which arises due to vortex motion. Indeed, the electric …eld is related by Lorentz transformation to the homogeneous magnetic …eld appearing in the frame moving with vortices. Finite electric …elds without disorder have been considered within framework of TDGL in [3].

We study the dynamic correlation function and the response functions of the

2 Abstract

The Martin - Siggia - Rose method is applied to the time dependent Ginzburg - Landau model in the presence of both disorder and thermal ‡uctuations beyond linear response. This allows calculation of correla-tors of the order parameter, the critical current as function of magnetic …eld and temperature and the I-V curves. The static or moving ‡ux line lattice in type II superconductors undergoes a transition into three dis-ordered phases: moving vortex liquid (not pinned), homogeneous vortex glass (pinned) and crystalline Bragg glass (pinned) due to both thermal ‡uctuations and random quenched disorder. The location of the glass transition line is determined and compared to experiments. The line is clearly di¤erent from both the melting line and the second peak line de-scribing the translational and rotational symmetry breaking at high and low temperatures respectively. Applications to thermal transport and Nernst e¤ect are considered.

25th International Conference on Low Temperature Physics (LT25) IOP Publishing Journal of Physics: Conference Series 150 (2009) 052028 doi:10.1088/1742-6596/150/5/052028

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D.T. Bu and B. Rosenstein

Electrophysics Department, National Chiao Tung University,

Hsinchu, Taiwan, R.O.C.

July 30, 2008

Introduction and Summary. In type II superconductors for which the pen-etration depth exceeds the correlation length the magnetic …eld penetrates the sample in a form of Abrikosov vortices, which strongly interact thereby cre-ating an elastic "vortex matter". Impurities always present in a sample lead to inhomogeneities which greatly a¤ect the thermodynamic and especially dynamic properties of the vortex matter. In addition, thermal ‡uctuations also signi…-cantly in‡uence the vortex matter, either directly by melting the vortex lattice into a vortex liquid or by reducing the e¢ ciency of the disorder. Once electric current J is injected into a sample, one is faced with a problem of description of the dynamical phase diagram which should be drawn in the three dimensional space T H J . This makes the analysis essentially more complicated, since one has to go beyond linear response.

In this note we investigate dynamics of the vortex matter beyond linear response using the disordered Ginzburg - Landau model (GL). In statics the replica method of handling disorder in the framework of GL model was utilized [1] to obtain the irreversibility line along with other properties of the disordered vortex matter. Dynamics in the presence of thermal ‡uctuations and disorder is phenomenologically described using the time dependent Ginzburg - Landau (TDGL) model in which the coe¢ cients have random components. Such an attempt was made by Dorsey, Huang, and Fisher [2] in the homogeneous (liquid) phase using a dynamic Martin-Siggia-Rose (MSR) formalism. They obtained the irreversibility line and formulated the linear response theory of the vortex matter.

In strongly type II superconductors magnetic and electric …elds inside the su-perconductor are homogeneous over a wide range of parameters since the mixed state originates from superposition of many (~B=Hc1 1) vortices. The same

argument is valid for electric …eld which arises due to vortex motion. Indeed, the electric …eld is related by Lorentz transformation to the homogeneous magnetic …eld appearing in the frame moving with vortices. Finite electric …elds without disorder have been considered within framework of TDGL in [3].

We study the dynamic correlation function and the response functions of the

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order parameter within an appropriately generalized gaussian approximation in both the ‡ux ‡ow and the pinned phases. We consider the stationary case only, namely when the correlation function depends on the time di¤erence and is therefore characterized by the spectrum C!. The critical surface Tg(H; J ) in the

three dimensional space T H J , separating the pinned and unpinned phases is obtained as a surface at which C!! 1 for ! ! 0. Above this surface the

real space correlator decays exponentially, while below it is a constant at large time scales. The constant is proportional to the Edwards - Anderson (EA) order parameter characterizing transition to a glassy state. Approaching criticality in the parameter space T ! Tg _{various quantities diverge power-wise in (T} _Tg₎

with critical exponents calculated in mean …eld. The static glass transition line, namely the line at zero electric …eld, coincides with the one obtained using the replica method [1].

Basic equations and methods. Our starting point is the TDGL equation in the presence of thermal ‡uctuations which on the mesoscopic scale are repre-sented by a white noise :

h2

4m D = F + : (1)

The covariant time derivative is D _@@ +ie_h ; where is the scalar elec-tric potential describing the driving force and the inverse di¤usion constant is =2:The variance of the thermal noise determines the temperature T = tTc.

The static GL free energy including the Tc disorder is:

F = Z d3r h 2 2m j(r+ ie hcA) j 2₊ h 2 2m_cj@z j 2 _T c(1 t) (1 + U (r)) j j2+ b0 2j j 4_:

The random component of Tc, U (r), is modeled by a white noise

character-ized by variance depending on the pinning, U (r) U (r0_{) =} _(r _r0₎ 2

znp: The

dimensionless pinning strength np is proportional to the density of pinning

cen-ters in units of the coherence volume 2 _z, where _z = _a with anisotropy parameter _a=qm

mc.

A Langevin type dynamics can be formulated as a functional problem with the dynamical "partition function", de…ned by the MSR functional integral over the order parameter and an additional "ghost" …eld . The ghost …eld allows exact integration over the white noise :

Z = Z

D D D _{D exp f A}M SR[ ; ]g : (2)

The explicit form is given elswhere[4]. The functional approach enable us to calculate both the dynamics correlators and response function of the system in close analogy to calculation of the static correlators in statistical physics. For example dynamical correlator is

C (r; ; r0; 0) = Z 1 Z

D D D D _r( ) _r0( 0) exp f AM SR[ ; ]g :

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The system has two dimensionless couplings n = np

4 p2Gi (1 t)2

t characterizing

relative strength of disorder compared to interactions and g = 8 btp2Gi charac-terizing interactions compared to thermal ‡uctuations withp2Gi = ₄ 2b_T0_c 2

z

and units of Hc2 and E0 = _e4h3 are used for magnetic b = _HB

c2 and electric

E = EE0 …elds.

Since the model is highly nontrivial even in the simplest cases, one has to use an approximation scheme. We utilize a method which evolved from the gaussian variational approach to quantum mechanics called here gaussian approximation. Since higher correlators are needed, a certain generalization of the gaussian approximation introduced in detail in [4] is used. An additional advantage of this approach over the resummation of diagram technique used in [2] (borrowed from the physics of weak localization) is that it is a systematic and an unambiguous without any reference to the "large number of components" limit.

Correlators near the glass line and critical exponents. It is convenient to rescale the correlation functions as c! = ₈g

2=3

C! and ! = 8g 2=3

R!:

An appropriate units of time is GL = 2 and we use the dynamical scaled

temperature aT = (8 =g)2=3ah, ah = 1₂ 1 t b E2=b2 . In the "liquid"

phase in which disorder does not alter the vacuum structure signi…cantly, the solution can be found assuming the validity of the dissipation - ‡uctuation theorem (DFT), which subsequently can be checked by substitution back into the gap equation. A closed form cubic equation for the frequency dependent response function _!is 4n 3=2_! 4 1=2₀ _! aT 1 i! 4ah ! + 1 = 0; ₀= aT+ a 2 Td 1=3+ d1=3 12 (1 n)4=3 (4) d = a3_T + 12 (1 n) q 324 (1 n)2 3a3 T 216 (1 n) 2 ; which determines the correlator c! due to DFT:

c!= 8 g 2=3 ! ! 1=2 ! + !1=2 1=2 ! + !1=2 4n ! ! : (5)

The critical surface in the "space" of dimensionless scaled parameters (t; b; E) is de…ned as a set of values of the parameters for which the correlator C! at

! = 0 diverges: Eg2=b2= 1 t b + ng 4 2=3 3 2 n : (6)

The critical electric …eld Egwhich destroys the "glassy" state forcing vortices to

move in a direction perpendicular to the …eld. The expansion of the correlation functions near the glass line in a small parameter T = aT agT = (8 =g)

2=3

and leads to the following critical behavior: c!= p rel T

2 (2n)1=3 1 + 1 + ( rel!)

2 1=2 1=2

, (7)

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where the characteristic time, rel = ₃82 T

2 n2_g

2=3

; determines a long scale decay of the correlator C ( ) / e = rel_{. On the critical surface, where}

T = 0

and rel diverges, the correlator and response function, g != 1 (2n)2=3 " 1 2e i =4 p 6 4 ng 1=3 !1=2 # ; cg_!= r 1 3 8 g 1=3 1 np!; (8) both have a fractional power dependence on !. One therefore observes criticality with exponent 1₂. In the static limit near transition, the response function

0 is continuous, while the correlator c0 = ₃p8₂ 8g 2=3

1 (2n)5=3

Tdiverges with

critical exponent 1.

In the glass phase, namely when aT < agT,

c!= 8 g 2=3 ! ! 1=2 ! + !1=2 1=2 ! + !1=2 4n ! ! + (!) ; = (2n)1=3 T: (9) where the constant is the EA order parameter which vanishes on the dynamical glass line and increases below it.

The expression for _! is given by a solution of 4n 3=2_! 4 (2n) 1=3 _! aT 1

i! 4ah !

+ 1 = 0: (10) Indeed, in addition to a regular part obeying the DFT, there is a singular (at zero !) contribution expressing the persistent correlation c ( 0_{) ! : Near}

the glass line it diverges as 1=2_{due to HLL part. Nernst coe¢ cient is derived}

in a similar way.

We are grateful to D.P. Li, B. Shapiro. Work was supported by NSC of R.O.C. 952112M009048 and MOE ATU program.

References

[1] D.P. Li, B. Rosenstein and V. Vinokur, J. of Superconductivity and Novel Magnetism 19, 369 (2006); H. Beidenkopf et al, Phys. Rev. Let. 95, 257004 (2005); 98, 167004 (2007).

[2] A.T. Dorsey, M. Huang, and M.P.A. Fisher, Phys. Rev. B 45, 523 (1992). [3] R.S. Thompson and C.R. Hu, Phys. Rev. Lett. 27, 1352 (1971); D. Li, A.M.

Malkin and B. Rosenstein, Phys. Rev. B70, 214529 (2004). [4] B. Rosenstein and V. Zhuravlev, Phys. Rev. B76, 014507 (2007)