1. Determining superconducting vortices configurations with stochastic processes in a GPU-based code N. Molero Puerto1, R. Mayo García2 1 URJC 2 CIEMAT
March, 27th – 29th 2018

2. Index
The Physical Problem
Experiment
Simulations
Results
Conclusions

3. Many effects can be observed
The Physical Problem
Superconducting vortex lattice dynamics and vortex lattice pinning are strongly modified by arrays of nanodefects embedded in superconducting films
By using arrays of holes (antidotes), which thread the films or dots embedded in the sample, this effect can be studied
Many effects can be observed
reconfiguration of the vortex lattice
Effects induced by arrays made with different materials and different diameters of the pinning centers
Channeling effects
Etc. 3

4. Different interactions take part
The Physical Problem
Magnetoresistance measurements are a perfect tool to study these effects
Resistance vs. applied magnetic fields shows deep minima when the vortex lattice matches the unit cell of the array due to geometric matching occurs when the vortex density is an integer multiple of the pinning center density
Different interactions take part
Vortex–vortex
Vortex–artificially induced pinning center (array of nanodefects)
Vortex–intrinsic and random pinning centers
The magnetoresistance minima show up only when the temperature is close to Tc 4

5. The diameter of the Ni dots is 200 nm
Experiment
Samples are Nb film on top of array of submicrometric Ni dots which have been fabricated by electron beam lithography on Si substrates
Size 40 mm2
400×600 nm2 rectangular arrays of Ni dots have been selected as the artificially fabricated pinning arrays for the present work
The thickness of the Ni dots is 40 nm, while the thickness of the Nb film is 100 nm
The diameter of the Ni dots is 200 nm 5

6. Minima appear at applied magnetic fields
Experiment
Minima appear at applied magnetic fields
Hn = n· φ0 / (a· b)
a and b are the lattice parameters of the rectangular array
φ0 = 2.07· 10−15 Wb
Other parameters
l = 2.6 mm
f0 = 3.08· 10-6 T2nm
rp = 100 nm
The number of vortices n per array unit cell can be known by simple inspection of the magnetoresistance curves, in which the first minimum corresponds to one vortex per unit cell, the second minimum to two vortices per unit cell, and so on 6

7. The Physical Problem 7

8. DiVoS finds the minimum of the following equation
Simulations
The DiVoS code can simulate squares, rectangles, and triangles of any size
DiVoS finds the minimum of the following equation
DiVoS does not take advantages neither of matching conditions with respect to the vortices lattices nor computational cutoff approximations to place the vortices 8

9. Simulations: GA Genetic Algorithm
A population of candidate solutions (genes) to an optimization problem is evolved toward better solutions
Previous objective function to be minimized by the movement of the vortices
The evolution starts from a set of populations of randomly generated individuals
Each candidate solution has a set of properties which is a repetitive application of the mutation, crossover, inversion, and selection operators
The algorithm ends when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population 9

10. Simulations: SA Simulated Annealing
Minimize a function by identifying with the energy of an imaginary physical system undergoing an annealing process
Move from Fi to Fi+1 via a proposal. If the new state has:
Lower energy, accept Fi+1
Higher energy, accept with probability A=exp(-DF/KT)
Stochastic acceptance of higher energy states, allows the process to escape local minima
When T is high, the acceptance of these uphill moves is higher, and local minima are discouraged
As T is lowered, more concentrated search near current local minimum, since only few uphill moves will be allowed 10

11. Simulations: SA Simulated Annealing
Reannealing interval, or epoch length, is the number of points to accept before reannealing (change the temperature)
L = iterations at a particular temperature
Larger decreases in T require correspondingly longer L to re-equilibrate
Running long L at larger temperatures is not very useful  Decrease T rapidly at first
Vortices
Select 1 movement out of 8 randomly
1 by 1 (~Molecular Dynamics)
All at the same time (Multidimensional Gaussian) 11

12. Simulations: SA Simulated Annealing
After the epoch length (L steps), the comparison Fi+1 vs Fi is made
After the comparison, L and T are updated
Reannealing interval evolves with Lk+1=bLk with b>1
Thermostat
Linear: Tk+1=aTk, (with 1<a<0) or Tk–a (with a>0)
Exponential: Tk+1=Tk· 0.95a (with a≥1)
Logarithmic: Tk+1=Tk/log(a) (with a≥10) 12

13. Simulations: BH Basin Hopping
Similar to SA, though it represents an improvement
Random perturbation of coords (location on error surface)
Local minimization – find best (lowest) error locally
Accept/reject new position based on function value at that point
Acceptance test is usually Metropolis criterion from Monte Carlo (Metropolis-Hastings) methods
MH is more generally used to generate random samples from a prob. distribution from which direct sampling is difficult (maybe we don’t know what it looks like) 13

14. Computational outcomes
Results
Computational outcomes
Over 99% of the computing time gets into the evaluation of interactions
Original scalability is of O(N²), being N the number of vortices
Parallel scalability is of O(N²/2G), being G the number of GPU cores
By using a cache mechanism, a GPU core is faster than a CPU for this problem 14

15. Experiments have been executed 5 times each
Results GA
200 genes randomly created
1 gene per Nvdia core
500 generations
Mutation and crossover rate were constant
SA and BH
Initial value of T was 25,000 K
Temperature decreased as logarithmic function
Population in the simulated annealing is 1
Experiments have been executed 5 times each
Tesla K20 15

16. Results 16

17. Results 17

18. Results 18

19. Results 19

20. Results 20

21. Results 21

22. BH presents better results than SA and GA
Conclusions
From the computational point of view, the code executes and scales pretty well
BH presents better results than SA and GA
Further tests are needed to be carried out in order to properly match experiment and simulation
Current mathematical model does not correspond with experiments though
A plethora of results can be found in “Determinación de configuraciones de vórtices superconductores con procesos estocásticos con aceleradores gráficos en entornos de supercomputación”. BSc Thesis. N. Molero Puerto 22

23. THANK YOU. rafael. mayo@ciemat. es CIEMAT Avda
THANK YOU!!! CIEMAT Avda. Complutense, 40 – Madrid 23