CHAPTER 3 Model and simulation method for the vortex dynamics in clean system
3.3 Simulation result and discussion
3.3.1 Parameters for YBCO
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= 8c2 2 = 5 103ohm 1cm 1= 1:15 103 1012sec 1 = 4:545 1015
! = 8c22 n= 8 (1:53140)2(4:545 1015)
(3 1010)2 = 1:06 s=cm2
tGL= 2=2 = 1:24 10 14s; (3.22)
and unit of electric …eld
EGL= Hc2
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 ah = 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
ah(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 = 4c2 2, the dimensionless nonlinear conductivity in the LLL approximation is proportional to the super‡uid density,
LLL= i
2vh LLL@y LLL LLL@y LLLi =D
j LLLj2E
= ah(v)
A(v)ev2: (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 ah(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 ah(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 ah become small.
The theory is inapplicable when ah(v) is large, so I consider two cases with large ah(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 ah, without thermal noise on the mesoscopic level). Electric …eld ranges from 0 to 0:2. The parameters ah(0) for this case are therefore, 0:3 and 0:15 respectively. See Fig.3-5. The red triangles are values for ah = 0:3, while the blue squares are values for ah = 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 ah(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 ah (…x b). The red triangles are values for ah(0) = 0:3.
The blue squares are values for ah(0) = 0:15.
Figure 3-6: J-E curve for di¤ erent ah (…x t). The red triangles are values for ah(0) = 0:3. The blue squares are values for ah(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 ra 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
+ Tc 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 = jn+ js, where
jn = GL (4.4)
js = 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 (rn ra)) = 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 Tc become smaller.
However, this work compared systems with the same critical temperature, a constant is added into the condensation potential, so that
1 N
XN n=1
(1 t u (rn ra) + ) = 1 N
XN n=1
(1 t) ; (4.7)
where
= XN n=1
u(rn ra)
N : (4.8)
The comparison of J E cures for di¤erent strength of pinning potentials as shown in Fig.4-8. The parameters for the simulation are: the electric …eld form 0 to 0:4. The pinning factor f = 1, and size of pinning center is 1. The strength of pinning potential are 0 (clean system), 1 and 10. The values of critical current are dependent on the strength of pinning potential. When pinning potential is large, the vortices are more di¢ cult to escape from the pinning centers. The resistance is zero when vortices are trapped. As shown in Fig4-8, the superconductors with pinning array can bear stronger electric …eld. The reason is the velocity of moving vortices is reduced by the pinning force, in other words, the energy dissipation is reduced. All in all, the pinning e¤ect increase both the critical current and critical electric …eld (critical velocity of moving lattice), while the critical temperature is decrease.
Figure 4-8: Comparison of J E curve for di¤ erent strength of pinning potential ( u = 0; u = 0:5; u = 1). The red triangles are values for clean superconductor, the blue circle are values for superconductors with weaker pinning array (u = 1), the purple diamond are values for superconductors with stronger pinning array (u = 10). The pinning e¤ects are obvious when electric …eld is small, and when electric …eld near critical electric …eld.
Chapter 5
Conclusion
The statics and dynamics of vortex lattices in highly anisotropic layered type II superconductors has been studied by using the 2D time-depentent Ginzburg-Landau equation. The U algorithm was used to simulate the TDGL equation numerically. Both the order parameters describing the vortex lattice and the non-linear J E characteristics were calculated. Two di¤erent cases were investigated in this thesis, one is the clean superconductors and another one is superconductors with arti…cial pinning array commensurate with Abrikosov lattice.
In clean superconductors, the static super‡uid density (square of the order parameter j j2) decreases as the magnetic …eld increases approaching the upper critical …eld Hc2(T ) and van-ishes when magnetic …eld is larger then Hc2(T ). The statics con…guration of vortex lattice has hexagonal symmetry. When the electric …eld enters the superconductors, the vortices would move due to the Lorentz force and the vortex con…guration is deformed. Furthermore, the super‡uid density also decreases with electric …eld increase. The relation between super‡uid density, magnetic …eld and electric …eld are well represented by the bifurcation expansion for-mula j j2= ah= A;where the parameter ah = (1 t b b22)=2 (the super‡uid density becoming zero when ah 0). Deviations from this formula at large currents (electric …elds) were found.
In small electric …led region the J E characteristic is linear, while in large electric …eld the J E characteristic is non-linear. The simulation result for J E characteristic …t in with analytical result in small ah. I also compared J E characteristics with di¤erent ah including di¤erent temperature t and di¤erent magnetic …eld b.
In superconductors with arti…cial pinning array, the vortices are trapped by the pinning
centers. Two di¤erent strength of pinning arrays are considered in this work. I compared vortex con…guration in statics and dynamics for di¤erent pinning sizes (form small to large), numbers and kinds (from a very strong pinning by a normal islands to weak Tc pinning). The order parameter and supercurrent con…guration, which is in clean material dominated by the repulsive inter - vortex forces is deformed by the pinned vortices. The J E characteristics are also strongly in‡uenced by the pinning array. When the electric …eld is smaller than the pinning force, the vortices are standing on the pinning centers and reduce the energy dissipation(Joule heat). When the electric …eld is larger than the pinning force, the velocity of moving vortex is reduced by the pinning array. Therefore the superconductors with pinning array can bear stronger electric …eld. All in all, the pinning e¤ect increase both the critical current and critical electric …eld (critical velocity of moving lattice), while the critical temperature is decrease.
The model in this thesis only consider the uniform electric …eld, therefore this model only can applied when normal current is larger than supercurrent. The full electrodynamics have to be considered when normal current smaller than supercurrent. In addition, the model does not contain the in‡uence of thermal ‡uctuation which are especially important for high Tc
superconductors. When temperature approach Tc, the thermal ‡uctuation would dominate the physical quantities such as vortex structure (structure factor), heat capacity and conductivity.
In the future, the full electrodynamics and thermal noise would add into the model.
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Appendix A
Dimensionless formulas
Start from physical unit Gibb’s free energy without pinning potential
F = gauge is H , unit of order parameter isp
2 0 =p
0Hc 2 = 1. Let dimensionless energy f = 8H22
c F , therefore the dimensionless Gibb’s free energy can rewrite as following f =
Next, I chose symmetric gauge, A = 21By;12Bx cEt , while the dimensionless gauge as
following Third, the TDGL equation
2 the dimensionless supercurrent density
j = i
2 r0 r0 + j j2a: (A.8)
Appendix B
Link variables
The link variable vector U de…ne as following
U = exp i n1;n2 ; (B.1)
where phase n1;n2 =Rn+ =s
n a dr is the Aharonov-Baohm(A-B) phase, with a is the dimen-sionless vector potential, n is the link’s origin, while it ends at n + (a =s) .he line integral is taken along the straight line. In order to simplify the calculation of the line integration, we de…ne a parameter 0 < < 1:
d
d r ( ) = a4
s ;
where = 1; 2; 3 and is the unit vector.
The general formula for n1;n2 is
n1;n2 = Z 1
0
d d
d x ( ) ax(r ( )) + d
d y ( ) ay(r ( )) (B.2)
= aM s
Z 1 0
d xax(r ( )) + yay(r ( )) ;
where ax and ay were de…ned in Eq.(A.4). For hexagonal grid, the unit vectors are
1= (1; 0) ; 2= 1 The A-B phase for 1 direction:
r1( ) = a Similarly, the A-B phase for 2 and 3 direction
r2( ) = a
For rectangular grid, the unit vectors are
1 = (1; 0) ; 2= 0;
p3 2
!
: (B.9)
The A-B phase for 1 and 2 directions
r1( ) = a
Appendix C
Contimuum limit
In this work, I introduced lattice gauge theory to get the discretized TDGL equation and free energy. This appendix proof that the limit of discretized formulas are the same as continuum formulas. About the free energy on hexagonal grid:
f =
with lattice distance as = d, where symbol U represent the link variable which de…ned in Eq.(B.1).
fgrad = 2
The Eq.(C.3) become as following
fgrad = 2
1
Put fgardtand fpot together, It’s clear that the formula for discretized free energy and continuum free energy are the same. Similarly, the discretized TDGL equation is the same as continuum
Put fgardtand fpot together, It’s clear that the formula for discretized free energy and continuum free energy are the same. Similarly, the discretized TDGL equation is the same as continuum