Simulations of the Current and Magnetic Field Distribution for

Superconducting Thin Films

Prepared by: Yunong Hu

PHYS 437A/B Research Project

Supervisor: Dr. Adrian Lupascu

Institute of Quantum Computing

Department of Physics of Astronomy

University of Waterloo

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CONTENTS

I. Purpose of the ProjectII. Theoretical BackgroundIII. Numerical Method for Calculating Current Density and

Magnetic Field Distribution of the Superconducting Thin-film

IV. Results of Simulation

Purpose of Project

• Simulate and calculate the current density and the magnetic field distribution for superconducting qubits

• Find a simulation method for current distribution which can be applied to a irregular shape superconducting thin film or superconducting qubits.

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An rf-SQUID consists of a Josephson junction within applied magnetic field.

22

0

cos 22 2

eJ

self

qH Em L

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THEORETICAL BACKGROUND

• Meissner EffectThe expulsion of magnetic field for a superconductor in superconducting state.• Starting from susceptibility

Source: www.chm.bris.ac.uk

0 1 0B H

1

0 ( ) 0B H M

M H

Numerical Method for Calculating Current Density of the Superconducting Thin-film

• Grouping each of the four neighboring cubes in set of four. Ref[1]

• Inside the nth set of four cubes, the direction of current is described by unit vectors ,

Source: Ref[1] 5

B

nth set

4 4

, ,1 1

ˆa bnm n m n a m b

a b

M M U U

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Numerical Method for Rectangular Superconducting Thin Film

• The calculated result

• This integral has been solved in Ref[1]

3 30, ,

, ,

22 3

0 , 2

1ˆ4a b

a b

a ba

n mn m n a m bn m

n m n a m b

nn m L n an

n

J JM d r d r

I I r r

Jd rI

20 0

0

20

1.88 , ( 0)4

ˆ 0.98 , ( )4

,4

a bn m

s ds

M s d s

s d sd

d=0, overlapped

d=s, neighbouring cubes

d>sSchematic of conditions of the three expressions

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Numerical Method for Calculating Current Density of the Superconducting Thin-film• The mutual inductance

between each sets were calculated and plotted as a matrix.

( , ) ( , )nm i j k lM M

• The current distribution minimized the flux in the interior.

• Need to numerically solve and find the current of nth set

1

0N

nm m n nm

M I S B

1,...,n N

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RESULTS OF SIMULATION• After the convergence

factor has reached

• k is the convergence factor controls the extent of the external flux is totally expelled.

• Or the desired iteration time has reached, whichever comes first, the loop stops and print the matrix of Matrix plot for the magnitude of current in

each set of cubes. (Resolution 11*11)

𝑆𝑛 ⋅𝐵𝑛−∑𝑚=1

𝑁

𝑀𝑛𝑚 𝐼𝑚=𝑘

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RESULTS OF SIMULATION• Sum up the in one cube,

applying the unit vectors and divide by surface area, the current density Jn can be calculated.

Up: vector density plot for 11*11 sheet of cubes. Down: vector density plot .

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Up: Simulation for current distribution in a 21*21 square thin filmDown: Simulation for current distribution in a 51*51 square thin film

Current magnitude Vector density plot Stream density plot

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Magnetic field distribution for a rectangular thin film

Superconducting thin film is placed in the middle of the plot.

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Comparison to the results of PHYS 437AUp: Current magnitude and current distribution simulated in PHYS 437A (Resolution 11*11)

Down: Current magnitude and current distribution simulated in PHYS 437b (Resolution 81*51)

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Circular disk

A circular disk on 50*50 grid paper

Vector plot for the circular diskThe magnitude of small current loops inside the circular disk.

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Current distribution inside a cross

Magnitude of Current loop inside a cross.

Vector stream plot for current inside a cross.

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Simulation for the irregular shaped thin-film

• A random irregular shape drew on AutoCAD.

• The current density vector plot of the irregular shaped film.

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Simulation for the irregular shaped thin-film

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Analysis• Simulation should based on enough discretization,

otherwise obvious error may occur.• This simulation method can apply to

superconducting thin film of any shapes. However, discretization is limited by computational speed.

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AcknowledgementsI am grateful to Dr. Adrian Lupascu, who guided me through this project, suggested me related literatures and helped me with coding problems in Mathematica.

I also thank Pol Forn-Diaz, Ali Yurtalan, Guiyang Han and all other members in Superconducting Quantum Device Group for their support and suggestions for this project.

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References[1] Cano, Daniel, Brian Kasch, H. Hattermann, Reinhold Kleiner, Claus Zimmermann, Dieter Koelle, and József Fortágh. "Meissner effect in superconducting microtraps." Physical review letters 101, no. 18 (2008): 183006.

[2] Kittel, Charles “ Chapter 11: Supreconductivity.” Introduction to Solid State Physics. 3rd ed. New York: Wiley, 1966, 334-369.

[3] Grover, Frederick Warren. Inductance calculations: working formulas and tables. Courier Corporation, 1946.

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