Hexagonal grid

Figure 2-4: Average super‡uid density D j j²E

as function of magnetic …eld(rectangular grid).

The red line is values for the analytic expression Eq.(2.40).The blue points are simulation values for rectangular grid.

where _A= h

4i

hj ²ji² = 1:16 for hexagonal structure[32].When the magnetic …eld is high enough(a_h 1), the order parameters belong in LLL. Fig2-4 show relation between ² and b in t = 0,

The red line is theory value for Eq.(2.40), and the blue points are simulation values. When a_h become smaller, the

D j j²

E

for theory and simulation are closer. Superconductivity would totally break down when a_h < 0, since there is no condensation potential. The vortex structures in di¤erent magnetic …elds are shown in following (Fig.2-5~Fig.2-7).

It’s clearly that the distribution of vortex structure is hexagonal(vortex solid). The vortex density become larger when magnetic …eld become larger. Moreover, the values for ² become smaller. The Cooper pairs are broken by the magnetic …eld.

2.3.2 Hexagonal grid

Hexagonal gird is similar as Arbikosove lattice, the relation between average order parameters

2 and magnetic …eld show in Fig.2-8

Figure 2-5: Vortex structure( b = 0:1). The picture (a) is vortex lattice in real space, the vortex structure have the hexagonal symmetry. The picture (b) is the vortex spectrum in qusi-monentum space, the peaks are sharp and the peak distribution has hexagonal symmetry, therefore, this tell that the vortices in the solid phase.

Figure 2-6: Vortex structure( b = 0:3). Similar as Fig.2-5, the vortex lattice in real space(picture(a)) has hexagonal symmetry, and the peak distribution for the vortex spec-trum(picture(b)) still has hexagonal symmetry. The order parameters(j j²) in large magnetic

…le(b = 0:3) are smaller than small magnetic …eld(b = 0:7), because of the Cooper pairs are broke by the magnetic …eld

Figure 2-7: Vortex structure( b = 0:7). Similar as Fig.2-5, the vortex lattice in real space(picture(a)) has hexagonal symmetry, and the peak distribution for the vortex spec-trum(picture(b)) still has hexagonal symmetry. The order parameters(j j²) in large magnetic

…le(b = 0:7) are samller than small magnetic …eld(b = 0:3).

Figure 2-8: Average super‡uid density D j j²E

as function of magnetic …eld(hexagonal grid).

The red line is values for the analytic expression Eq.(2.40). The purper points are simulation values in hexagonal grid, and almost the same as values for rectangular grid.

The red line is theory value where come form Eq.(2.40), the purple points are simulation value. The value for ² rectangular grid and for hexagonal grid are almost the same. The vortex structure in di¤erent magnetic …eld as shown in the following, see Fig.2-8~Fig.2-10.

Similar as vortex structure on rectangular grid, the vortex structure are also has hexagonal symmetry. The hexagonal grid is more symmetric than rectangular grid, however, the vortex lattice structure on the rectangular grid is more beautiful than on the hexagonal grid, since the relaxation time for hexagonal grid is larger than rectangular grid.

Figure 2-9: Vortex structure( b = 0:1). The vortex lattice has hexagonal symmetry.

Figure 2-10: Vortex structure( b = 0:3). The vortex lattice also has hexagonal symmetry.

Figure 2-11: Vortex structure( b = 0:7). The vortex lattice also has hexagonal symmetry, and the values of order parameters become samller, the Cooper pairs are broken by the magnetic

…eld.

Chapter 3

Model and simulation method for

the vortex dynamics in clean system

3.1 Time Dependent Ginzburg-Landau theory in continuum

3.1.1 Electric …eld in a mixed state superconductor and the ‡ux ‡ow

In previous chapter I discussed the static vortex system. The system would eventually relax to a lowest free energy state. The relaxation dynamics was not realistic in a sense that e¤ects of the electric …eld were neglected (which is not essential for the static properties of the system since the …nal state is the same). In this chapter I start to consider dynamics of a type II superconductor in external magnetic and electric …elds. When the electric …eld applied to the system, it acts on vortices as an external force. Consequently the TDGL equation should be modi…ed as following

2

~² 2m

@

@t +ie

~ = F [ ; A]

; (3.1)

where the free energy was de…ned in Eq.(chapter II). Here, is an electric potential and is the inverse di¤usion constant. The electric …eld is E = r ¹_c^@A_@t, while the magnetic …eld is given by B = r A. The TDGL equation is invariant under the gauge transformation

= exp ie

where is an arbitrary function of space and time. Choosing the zero scalar potential gauge is convenient for the simulations which follow:

= 0 (3.3)

A = 1

2 Bxbi + 1

2By cEt bj:

Here both magnetic and electric …elds are assumed to be constant. Dynamics of electromagnetic

…eld should in principle also taken into account by the Maxwell quietens. However as explained in chapter II, for strongly type II superconductors magnetization is of order 1= ² and hence negligible. Vortices overlap and their magnetic …elds become homogeneous. Electric …elds in the ‡ux ‡ow state are also homogeneous (except at very low values in the presence of pinning, see chapter IV) and their dynamics can be neglected as well except for the normal conductivity

J = _nE: (3.4)

Substituting the free energy F [ ; A] of Eq.(chapter II) into equation, the TDGL equation becomes The formula is formally identical to the static TDGL equation, however the vector potential now contains a time dependent part. The time derivative term is purely dissipative (relaxational) and carries an implicit the information about the vortex velocity. When vortices move faster the frequency of phase change becomes larger, and …nally, the vortex structure would break down and the order parameter goes to zero. Because of the energy dissipation, resistance of

type ll superconductors is not zero. The formula for supercurrent density is the same as in the static case.

J_s = ie ~

2m ( r r ) + e ²

mscj j²A: (3.6)

Note that the supercurrent density is also invariant under gauge transformation.

3.1.2 Dimensionless electric …eld and conductivity

Using the same units as in static case, ; t_GL, the derived units of electric …eld is E_GL =

ctGLHc2= ^c₂ Hc2. In Y BCO this …eld is very large. Therefore the dimensionless electric …eld will be de…ned as

= E=EGL: (3.7)

Consequently the dimensionless vector potential is

a= 1

2 by;1

2bx t ; (3.8)

here x and y in unit of , and t in unit of t_GL. The velocity of vortices, which in physical units is V = cE=B, in dimensionless units reads:

v = Vt_GL The dimensionless TDGL equation takes a form

@

@t = bH a_h + ( ) ; (3.9)

where bH = ¹₂D² ₂^b,

a_h = 1 t⁰ b

2 ; (3.10)

and the dimensionless supercurrent density

j_s= i

2( r r ) + j j²a; (3.11)

are the same as in the static case, with the only di¤erence being the electric …eld term in vector potential, Eq.(3.8).

3.2 Discrete TDGL equation on rectangular grid

3.2.1 Rectangular grid

The discretized version of the dynamic TDGL equation is similar to the statistic one, the only di¤erence being the link variables for the gauge …eld. The link variable

U_n1;n2 = exp _n1;n2 ; (3.12) For the nonequlibrum system, the link variables change with time, the link variables in ₁ and

2 directions are

where the distance between vortex a_M =q

p4

The discretized dynamic TDGL equation is similar as static case, the only di¤erence is the link variables. The formula show as following

d 3.2.2 Discretized supercurrent density

The the de…nition of supercurrent density

j = fgrad

a ; (3.17)

where = 1; 2, f is the dimensionless free energy. The supercurrent density in x and y direction as following

Next, I expand discretized supercurrent density formula to get continuum formula. Use Taylor expansion to expand link variables U_n1;n2 and order parameters /_n

1;n2,

where = 1; 2, both ¹_n1;n2 and ²_n1;n2 were de…ned in Eq.(3.14). The discretized supercurrent density in x direction can be write as

jx = i Similarly, The discretized supercurrent density in y direction

jy = i

Both the continuum limit for discretized supercurrent density in x and y direction are the some as continuum formulas Eq.(3.11).

3.3 Simulation result and discussion

3.3.1 Parameters for Y BCO

To the following parameters of Y BCO sample as used. Tc = 100K, Hc2 = 140T , m = 2me. The coherence length and the penetration depth:

= ~

The Lattice spacing is magnetic …eld b = 0:1 in this simulation a ' 13nm. The unit of time is:

n= ₈^c2 2 = 5 10³ohm ¹cm ¹= _1:1⁵ 10³ 10¹²sec ¹ = 4:545 10¹⁵

! = ⁸_c2² n= ⁸ (_1:53¹⁴⁰)^{2(4:545 1015)}

(3 10¹⁰)² = 1:06 s=cm²

t_GL= ²=2 = 1:24 10 ¹⁴s; (3.22)

and unit of electric …eld

E_GL= H_c2

3.3.2 Super‡uid density at ‡ux ‡ow

In contrast to the static case, under the in‡uence of the Lorentz force the vortices move.

At low electric …elds, the vortex structure approximately remains hexagonal for an isotropic material (assumed in the present study) in directions perpendicular to magnetic …eld. See Fig.

3.1(a) compared to perfect hexagonal lattice in statics at the same …eld b = 0:4. The moving lattice is not exactly hexagonal since electric …eld breaks explicitly the hexagonal symmetry,

Figure 3-1: Super‡uid density with small electric …eld. The picture(a) show the vortex lattice without electric …eld. As show in previous chapter, the vortex lattice has hexagonal symmetry.

The picture(b) show the vortex lattice with small electric …eld. The electric …eld apply in y direction, and the vortices move in x direction. The moving vortex lattice is not exactly hexagonal.

see theoretical symmetry considerations in[7]. In the large electric …elds, the vortex matter

‡ows in my simulation like a liquid(In fact, the theoretical results for moving lattice still has hexagonal). For example, when I consider the vortex ‡ow for a_h = 0:45 and electric …eld applied in the y direction, the vortex structures in di¤erent electric …eld show in Fig.3-1.

3.3.3 De…nition of nonlinear conductivity and comparison of simulation with the analytic results

Using expansion in small parameter

a_h(v) = 1

2 1 t b v² = 1

2 1 t b

2

b² : (3.27)

D. Li, B. Rosenstein and M. Malkin [15] calculated the nonlinear conductivity for clean type-ll superconductors. In the ‡ux ‡ow regime, in addition to the normal state conductivity, there is a large contribution form the Cooper pairs represented by order parameter …eld. The nonlinear

Figure 3-2: Moving lattices with pinning. Two snapshots at times t=2000 (a) and t=2100 (b).

The vortices move in the x direction

conductivity will be de…ned as follows

s = Js

E = i~e

2mE r + ie

~cA + cc: (3.28)

Using the unit 0 = ⁴_c2 ², the dimensionless nonlinear conductivity in the LLL approximation is proportional to the super‡uid density,

LLL= i

2vh LLL@_{y LLL LLL}@_{y LLL}i =D

j LLLj²E

= a_h(v)

A(v)e^v2: (3.29) Numerical simulation allows to check the analytic theory and extend it beyond perturbation theory applicability range.

I simulated the case b = 0:9, t = 0, for which the parameter a_h(v) = 0:05 is small, the order parameters are within the LLL approximation applicability range. For velocity is low (corresponding to low electric …eld), so that the Abrikosov beta does not change from its static value, _A(v) ' 1:16. The comparison of the analytic theory and the simulation is shown in Fig.3-3. For _A(v) 6= 1:16 at large electric …eld , the value for the theoretical line have to be modi…ed. As expected, the di¤erence between the theory and the simulation becomes smaller when a_h(v) become smaller. That is, when number of Cooper pairs in becomes smaller, most of them stay in LLL.

0.05 0.10 0.15 0.20 0.25 0.30 0.01

0.02 0.03 0.04 0.05

Figure 3-3: Comparison of simulated ‡ux ‡ow nonlinear conductivity with the analytic results.

The blue points are simulation values for j= . The red points are simulation value for ² =

LLL which provides the …rst approximation and the green line is the full analytic expression Eq.(3.29).

0.05 0.10 0.15 0.001

0.002 0.003 0.004 0.005 j

Figure 3-4: Comparison of simulated nonlinear j curve with analytic result. The green line is theory value Eq.(3.29). The red points are simulation values for _LLL .The blue points are simulation for supercurrent density values.

Next, the j characteristic is shown in Fig.3.4. The blue points are simulation result for supercurrent density, and the red points are simulation result for super‡uid density multiplied by electric …eld, ² . Note that ² is the LLL conductivity according to Eq.(3.29). The green line …t well with red points when a_h become small.

The theory is inapplicable when a_h(v) is large, so I consider two cases with large a_h(v). In the …rst the magnetic …eld b is …xed at b = 0:4, while temperature t⁰ has two values 0:0 and 0:3 (just change of a_h, without thermal noise on the mesoscopic level). Electric …eld ranges from 0 to 0:2. The parameters a_h(0) for this case are therefore, 0:3 and 0:15 respectively. See Fig.3-5. The red triangles are values for a_h = 0:3, while the blue squares are values for a_h = 0:15.

The supercurrent current density at higher temperature is smaller since the temperature breaks Cooper pairs.

In the case II, the magnetic …eld b has two values 0:4 and 0:7, while the temperature t⁰ is

…x, t⁰ = 0:0. Electric …eld from 0 to 0:2. The parameters a_h(0) for case Il are, therefore, 0:3 and 0:15 respectively. See Fig3-6. The supercurrent density in large magnetic …eld(b = 0:7) is

Figure 3-5: J-E curve for di¤ erent a_h (…x b). The red triangles are values for a_h(0) = 0:3.

The blue squares are values for a_h(0) = 0:15.

Figure 3-6: J-E curve for di¤ erent a_h (…x t). The red triangles are values for a_h(0) = 0:3. The blue squares are values for a_h(0) = 0:15:

smaller the small magnetic …eld(b = 0:3) under the same electric …eld, for Cooper pairs density are less in large magnetic. In addition, the j curve in b = 0:7 is more linear then in b = 0:3, the reason is that when vortex move faster, the Cooper pairs are more di¢ cult to transport.

Chapter 4

Vortex statics and dynamics in superconductor with periodic pinning.

4.1 Time dependent Ginzburg-Landau theory with periodic pin-ning

The di¤erence between the clean superconductor’s GL free energy and the energy in the presence of the arti…cial pinning centers located r_a is the pinning potential:

V (r) = Tc

X

a

U (r ra) : (4.1)

The TDGL equation should be modi…ed as follows

2

~² 2m

@

@t = ~

2m r ie

~cA

2

+ T_c 1 t⁰+ U (r ra) b⁰j j² : (4.2)

Using the same unitsas in the clean case, the dimensionless TDGL equation becomes

@

@t = 1

2(r ia)²+ 1

2 1 t⁰+ u (r ra) ( ) ; (4.3)

where u(r ra) is the dimensionless pinning potential. The dimensionless current density as de…ned before j = j_n+ j_s, where

j_n = _GL (4.4)

j_s = i

2( r r ) + j j²a:

As discussed in previous, the dissipation mainly comes from the vortex ‡ow. The electric resistance comes also from normal electrons, when the electric …eld is "allowed" to enter the sample due to the ‡ux ‡ow. Pinning slows and eventually stops vortices from moving. This is the reason why there exists the critical current.

4.2 Pinning distributions

Two kinds of pinning array are discussed in this thesis. The …rst one assumes that small parts of the sample are normal or even insulating. An example is the columnar defect or simply a hole. Obviously the order parameter inside such a center (neglecting the proximity e¤ects) is always zero. The weaker Tc pinning is just a local modi…cation of the critical temperature at the center’s site. It in‡uences the condensation potential, Eq.(4.6). In other words, the vortices will not be always con…ned to the pinning centers. The simulation results are shown in following subsections.

The shape of pinning sites I simulated are rectangles. The pining distribution is hexagonal, as shown in Fig.4-1. I assume that vortices outnumber the pinning sites, namely, the magnetic

…eld is above the so called matching …eld. Therefore the …lling factor parameter de…ned as,

f = pinning sites⁰ number

vortices⁰ number ; (4.5)

. For example the f = 1=2 case is shown in Fig.4-1

Figure 4-1: Pinning distribution. The black rectangles are pinning sites. The distribution of their centers is hexagonal, commensurate with the Abrikosov lattice.

4.3 Strong pinning array

The magnetic …eld b = 0:4 and the temperature t = 0 in the following simulation.

4.3.1 Super‡uid density

The vortex distribution is in‡uenced by pinning. In this simulation all the pinning distributions are hexagonal, however the size of the pinning sites varies (pinning size 4; 9; 16). The pinning factor f = 0:5. The results are shown in Fig.4-2~Fig.4-4. In Fig.4-2 for the size of 4, the vortex structure still has hexagonal symmetry, while for larger sited in Fig.4-3 and Fig.4-4, the vortex structure does not remain hexagonal.

4.3.2 Dynamics in the presence of pinning.

a. Interstitial vortices

The vortices in the superconductors with arti…cial pinning array can be separated into two di¤erent sets: the pinned vortices which are trapped on the pinning centers, and the interstitial vortices which can be considered as "free vortices". If the number of vortices is smaller than the number of pinning sites, all of them are likely to be pinned. If magnetic …eld is equal to the matching …eld, there are no interstitial vortices, however when vortices outnumber the

Figure 4-2: Vortex structure with periodic pinning (pinning size = 4 ). The vortex lattice still has hexagonal symmetry, while the vortex core in the pinning centers are larger then that of the intersticial vortices.

Figure 4-3: Vortex structure with periodic pinning(pinning size = 9 ). The vortex structure is di¤erent compared to the clean system. Pinning leads to expansion of the vortex cores.

Figure 4-4: Vortex Structure with periodic pinning(pinning size = 16 ). Similar to Fig.4-3, the pinning vortices lead the vortex distortiontion increasing the unit cell

sites, some of them will be "liberated" due to repulsion from the vortices already pinned. On Fig.4-5 the dynamics of the interstitial vortices is shown. The electric …eld is applied in the y direction, thus the vortices move along the x direction. The parameters for the simulation are:

the electric …eld = 0:0001. Pinning centers size is 4. Two snapshots at times t = 2000 (a) and t = 2100 (b) for the case when for each rectangular pinning center there are two vortices.

Two di¤erent kinds of vortices are seen: the interstitial vortices have smaller cores and move, and the pinned vortices which have larger cores (like in the static case) and don’t move.

For the pinning force is larger that the driving force, the pinned vortices are not moving, while the interstitial are pushed by driving force. The ‡ux ‡ow therefore is due to interstitial vortices.

b. I-V curves

In this subsection the dependence of the ‡ux ‡ow current on the electric …eld is presented. One should be warned that the simulation is reliable only when the supercurrent is smaller than the normal current. Otherwise the electric …eld in the superconductors is not uniform and a much more complicated system of equations including the Maxwell equations for the electric

…eld should be solved. Thus the simulation results for currents approaching the critical current

Figure 4-5: Moving lattices with pinning. Two snapshots at times t=2000 (a) and t=2100 (b) for the case when for each rectangular pinning center there are two vortices. Two di¤erent kinds of vortices are seen: the interstitial vortices have smaller cores and move, and the pinned vortices which have larger cores and don’t move.

are not correct.

In Fig. 4-6, the J E curves for the clean superconductors and the superconductors with pinning array are compared. The parameters in the simulation are: the electric …eld from 0 to 0:2, pinning factor f = 0 (clean) and 0:5. The size of pinning centers is 9. When the electric

…eld is large the behavior of J E curve for the clean superconductors and superconductors with pinning array are similar. However when electric …eld is small but larger than pinning force, the current for superconductors with pinning array is larger than the clan superconductors. The reason is that the pinning array reduces the velocity of moving vortices. If the driving force is smaller than the pinning, the resistance is zero inside the superconductors.

4.4 Weaker pinning array

The magnetic …eld b = 0:4 and the temperature t = 0 in the following simulation.

Figure 4-6: Comparison J E curve for clean superconductors and superconductors with pinning array. The red triangle represent clear system. The blue rectangle represent system with pinning in size of 9.

Figure 4-7: Comparison of vortex lattices for clean superconductors and superconductors with pinning array. The picture in the left hand side show the vortex lattice in clean superconductors, and the picture in the right hand side show the vortex lattice in superconductors with pinning array. The di¤erent between two pictures is that the positions of vortices are shifted by the pinning array.

4.4.1 Super‡uid density

The comparison of vortex lattice for the clean superconductors and superconductors with pin-ning array is shown in Fig.4-7. The pinpin-ning factor f = 0 (clean) and 1, and the pinpin-ning size is 9. As Fig.4-7 shown, the vortices are trapped on the pinning centers. The pinned vortices in strong pinning array and in weaker pinning array are di¤erent. For the strong pinning array the size of vortex cores dependent on the pinning size (see Fig.4-3), while for the weaker pinning array the size of vortex cores is almost the same as vortex in clean superconductors.

4.4.2 I-V curves

The condensation potential is in‡uenced by the pinning array, it can be seem as the critical temperature is changed. The average condensation potential is modi…ed as following

1 N

XN n=1

(1 t u (r_n r_a)) = 1 N

XN n=1

1 t⁰ ; (4.6)

where rarepresent the pinning positions, t is the original critical temperature and t⁰ is the new critical temperature. Note that t⁰ > t, which means the critical temperature T_c become smaller.

However, this work compared systems with the same critical temperature, a constant is added into the condensation potential, so that

However, this work compared systems with the same critical temperature, a constant is added into the condensation potential, so that

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