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![Page 1: Designing a Solenoid for Low Temperature Resistance Measurements of Nanostructures PHYS 4300 May 15, 2009 Jon Caddell Dr. Murphy.](https://reader036.fdocuments.us/reader036/viewer/2022062713/56649f585503460f94c7dbad/html5/thumbnails/1.jpg)
Designing a Solenoid for Low Temperature Resistance Measurements
of Nanostructures
PHYS 4300
May 15, 2009
Jon Caddell
Dr. Murphy
![Page 2: Designing a Solenoid for Low Temperature Resistance Measurements of Nanostructures PHYS 4300 May 15, 2009 Jon Caddell Dr. Murphy.](https://reader036.fdocuments.us/reader036/viewer/2022062713/56649f585503460f94c7dbad/html5/thumbnails/2.jpg)
OutlineOutline• Motivation
• Spintronics• Weak Localization• Minimizing Spatial Variation of Resistance
• Laboratory Details• Current Experimental Setup• New Setup (add Independent Perpendicular Field)
• Modeling B Field Spatial Variation• Non-infinite Solenoid B Field Non-uniformities
• Conclusion• Optimizing Solenoid Design to Minimize Spatial
Variation of Resistance
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SpintronicsSpintronics• Digital technology has two states corresponding to
logic True/False• If the parametersparameters associated with spinspin are included,
then you can double the number of logic states• Go from binary computing to four-level logic
computing(T/F T↑ /T↓ /F↑ /F↓ )
• This could boost computing power (more information stored per bit)
• InSb interesting for spintronics; need to know more about fundamental spin behavior
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Fundamental Spin Behavior: Fundamental Spin Behavior: Weak Localization (no spin-orbit)Weak Localization (no spin-orbit)• Infinite number of scattering trajectories starting from origin• Subset of these trajectories lead back to the origin• Each path around, there’s also a path in the opposite
direction (time reversal invariant)
Scattering site (defect)Origin
Clockwise Clockwise (cw)(cw)
Counter- Counter- clockwise clockwise (ccw)(ccw)
e- path in disordered material
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Weak Localization (cont.)Weak Localization (cont.)• Classical Probability for returning to
origin:
Pcw + Pccw = Pclasstotal
• Q.M. Probability:
Ψcw2 + Ψccw
2 + <Ψcw|Ψccw> = PQMtotal
• Probabilityclass. < ProbabilityQ.M.
Resistanceclass. < ResistanceQ.M.
• Add a Bperp field Aharonov-Bohm Effect
Ψcw picks up a phase change opposite in sign to Ψccw
<Ψcw|Ψccw> term now has less constructive interference
Constructive interference
B Field
R
Q.M.
Classical
B
(ccw)(ccw)
(cw)(cw)
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Weak Localization Weak Localization →→ Weak Anti-Localization Weak Anti-Localization• IncludeInclude spin-orbit coupling Weak
Anti-Localization
• For Bperp=0, Q.M. interference term <Ψcw|Ψccw> now destructive
Resistanceclass. > ResistanceQ.M.
• Phase change, from Bperp field, as before destroys the interference
• Result Graph is inverted for spin-orbit coupling
B
Q.M.
ClassicalR
Weak Anti-Localization
B
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Spin/OrbitSpin/Orbit• Looking at spin/orbit
• Orbit depends on Bperp. (Lorentz Force), F=q(v x B)
• Spin depends on Btotal (Zeeman Energy), E=g µB B
• Adding Parallel Magnetic Field
• Bperp stays the same
• But Btotal changes (Btotal=Bperp+Bparallel)
e-
v
B
F
Lorentz Force
B=0 B≠0
up and down spin at same energy level
up and down spin at different energy level
Zeeman Effect
gyromagnetic ratio Bohr magneton
So applying Bparallel separates spin from orbital motion
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Weak Anti-Localization and BWeak Anti-Localization and Bllll Magnetic Field Magnetic Field• One spin energetically favorable
So applying Bparallel separates spin from orbital motion• Goal: seeing how weak anti-localization changes with
magnetic field
B=0 B≠0
up and down spin at same energy level
up and down spin at different energy level
Zeeman Effect
Q.M.
ClassicalR
Weak Anti-Localization
![Page 9: Designing a Solenoid for Low Temperature Resistance Measurements of Nanostructures PHYS 4300 May 15, 2009 Jon Caddell Dr. Murphy.](https://reader036.fdocuments.us/reader036/viewer/2022062713/56649f585503460f94c7dbad/html5/thumbnails/9.jpg)
OutlineOutline• Motivation
• Spintronics• Weak Localization• Minimizing Resistance Spatial Variation
• Background• Current Experimental Setup• New Setup (add Independent Perpendicular Field)
• Modeling B Field Spatial Variation• Non-infinite Solenoid B Field Non-uniformities
• Conclusion
![Page 10: Designing a Solenoid for Low Temperature Resistance Measurements of Nanostructures PHYS 4300 May 15, 2009 Jon Caddell Dr. Murphy.](https://reader036.fdocuments.us/reader036/viewer/2022062713/56649f585503460f94c7dbad/html5/thumbnails/10.jpg)
Current SetupCurrent Setup
Large SolenoidLarge Solenoid
Sample HolderSample Holder
Cryostat Cryostat casingcasing
SampleSample
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Current Experimental SetupCurrent Experimental Setup• Need a Magnetic Field for experiment• Already got one• Want to change B parallel and B
perpendicular separately• Need field to be spatially UNIFORM
Bperp.
Bperp.Bparallel
New Method
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Future SetupFuture Setup• Low Temp.• NO power dissipation• Superconducting
I
I
I
I
Bz = Bperp.
new magnet
Bx = B// existing magnet
y
z
x
y
x
~1 Tesla
~10 mT
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Manufacturability, Economics, Environmental, SafetyManufacturability, Economics, Environmental, Safety
• Manufacturability• Materials: order off-the-shelf NbTi wire, machine the
coil form and wind coil ourselves, pot in standard epoxy
• Constraints: solenoid must fit inside 2” diameter larger solenoid (limits length and diameter), wire diameter from what is commercially available
• Economics• Coil design and construction in-house to avoid
outside custom work• Environmental
• No power dissipation since coil is superconducting; materials recyclable (except epoxy) and non-toxic
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Manufacturability, Economics, Environmental, SafetyManufacturability, Economics, Environmental, Safety
• Safety – Cryogenic Temp.• Quench Protection
P=VI=0 superconducting, I<Ic
P=VI≠0 non-superconducting, I>Ic
Power dissipation → boil He (liquid → gas)Expands x700• Quench valve, open to relieve over-pressure
• Air content
• Air 22% O2 if a quench, lots of He, O2 content drop
Evacuate room.
![Page 15: Designing a Solenoid for Low Temperature Resistance Measurements of Nanostructures PHYS 4300 May 15, 2009 Jon Caddell Dr. Murphy.](https://reader036.fdocuments.us/reader036/viewer/2022062713/56649f585503460f94c7dbad/html5/thumbnails/15.jpg)
OutlineOutline• Motivation
• Spintronics• Weak Localization• Minimizing Resistance Spatial Variation
• Background• Current Experimental Setup• New Setup (add Independent Perpendicular Field)
• Modeling B Field Spatial Variation• Non-infinite Solenoid B Field Non-uniformities
• Conclusion
![Page 16: Designing a Solenoid for Low Temperature Resistance Measurements of Nanostructures PHYS 4300 May 15, 2009 Jon Caddell Dr. Murphy.](https://reader036.fdocuments.us/reader036/viewer/2022062713/56649f585503460f94c7dbad/html5/thumbnails/16.jpg)
Modeling, Exploit SymmetryModeling, Exploit Symmetry• Biot-Savart Law for Current Loop• Stack rings, approximate Solenoid• For center plane of Solenoid,
Radial components of B cancel• Only have to consider spatial variation
of Axial B Field to evaluate, Bz
Biot-Savart Law
Sum vectors
radial comp. cancelB
Plane of sample
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0.2 0.4 0.6 0.8 1.0N ormalized R adius
8101214161820
B Fie ld T
Finding Spatial BFinding Spatial Bzz Distribution, Single Loop Distribution, Single Loop• Integrate to find B
• Symmetry, Integrate half, x 2• Involves Elliptical Integrals
• B Increases Radially!
Biot-Savart Law
rR
dl
θ
Integrate theta 0 → π
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Modeling SolenoidModeling Solenoid• Method A: Summing
stacked current loopsUses Elliptical Integrals
Elliptic Integral of the first kind
Elliptic Integral of the second kind
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Modeling SolenoidModeling Solenoid• Method B: Solenoid
ModelUses Legendre
Polynomials
Previous citation &
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0.2 0.4 0.6 0.8alpha
8.10
8.15
8.20
8.25
8.30
mTB z E llip tic S o l. B lu e, P o lyn o m ial S o l. R ed
Comparing Method A to Method BComparing Method A to Method B• Know answer for infinite solenoid B=µ I N / L• Method A correct• Method B required effort
Compare with many loopsClose agreement, w/factor 0.1
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0.2 0.4 0.6 0.8alpha
0.30
0.35
0.40
0.45
0.50
0.55
mTB z E llip tic S o l. B lu e, P o lyn o m ial S o l. R ed
Comparing Method A to Method BComparing Method A to Method B• Compare in single loop limit• Close agreement, w/factor 0.1
α = normalized radius
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Modeling SolenoidModeling Solenoid• Method B
• Spatial Variation for short and long coils
Radial Dependence
B
0.2 0.4 0.6 0.8alpha
8.10
8.15
8.20
8.25
8.30
mTB z Lo n g S o len o id
Normalized Radius
4% variation
0.2 0.4 0.6 0.8alpha
0.30
0.35
0.40
0.45
0.50
0.55
mTB z Lo n g S o len o id
Normalized Radius
140% variation
Bz – Short Solenoid
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Change of B over sample areaChange of B over sample area
B Field
Strong B
Weak B
• Want sample to have spatially uniform B Field
• Need to find tolerance for ΔB
B Field
Steps=0.013 T
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Transition From B to Conductivity• Now know B(r,I)• σ(B), need σ(r,I)
B Field
σ
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Finding Acceptable Finding Acceptable ΔΔB(r)B(r)• Recall Weak Anti-Localization Signal• Conductivity σ(B) → but B(r,I) → σ(r,I)
I, current
r, radius
σ, conductivity
I, current
r, radius
∂σ/ ∂r
B Field
σ
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Finding Acceptable Finding Acceptable ΔΔB(r)B(r)• 3% variation of B acceptable• Data fit uses up to B(I = 2 Amps)• Sample only in center of coil
I, current
r, radius
∂σ/ ∂r
I, current
r, radius
∂σ/ ∂r
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Finding Acceptable Finding Acceptable ΔΔB(r)B(r)• 3% variation of B acceptable• Coil is within tolerance
0.2 0.4 0.6 0.8alpha
8.10
8.15
8.20
8.25
8.30
mTB z Lo n g S o len o id
4% variation
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1 2 3 4 5coil diameter, 0 , 5cm
7.5
8.0
8.5
9.0
9.5B z mT
I 4 A mps, C enter, L ength 5cm , W ire dia . 0.0753cm
5 10 15 20 25 30length , 0 , 5cm
0.002
0.004
0.006
0.008B z TeslaI 4 A mps, C enter, W ire dia . 0.0753, C oil dia . 3cm
0.08 0.09 0.10 0.11 0.12w ire diameter, .065, .125cm
0.00600.00650.00700.00750.00800.0085
B z TeslaI 4 A mps, C enter, L ength 5cm , C oil dia .3cm
1 2 3 4 5 6coil diameterlength , 0 , 10cm
0.005
0.010
0.015
0.020B z TeslaI 4 A mps, C enter, W ire dia . 0.0753cm
Trends of Field on CenterTrends of Field on Center
B(coil dia.) B(length)
B(wire dia.) B(coil dia.&length coupled)
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Bz(coil dia.,length,wire dia.), Fixed IBz(coil dia.,length,wire dia.), Fixed Imax.max.
B field D=0.2cmL=4.996cm
D=0.3cmL=4.991cm
D=0.4cmL=4.984cm
D=0.5cmL=4.975cm
wire diameter0.033cm
51.2mT 55.7mT 51.8mT 44.1mT
0.043cm 34.9mT 39.5mT 39.0mT 35.0mT
0.054cm 25.0mT 28.9mT 29.8mT 28.0mT
0.0643cm 19.3mT 22.7mT 23.9mT 23.2mT
0.0753cm 15.3mT 18.1mT 19.5mT 19.3mTB
D = Coil Diameter
L = Coil Length
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ConclusionConclusion• Steps
• Mathematica Routine to model B Field• Optimize Field by minimizing B variations• Design superconducting coil
• Future Steps• Build superconducting coil• Test at cryogenic temperatures (4.2K)• Perform Measurements
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ReferencesReferencesSources for Formulas: Elliptical Integral, Legendre
Polynomial from:• SOME USEFUL INFORMATION FOR THE
DESIGN OF AIR-CORE SOLENOIDS by D.Bruce Montgomery and J. Terrell., published November, 1961, under Air Force Contract AF19(604)-7344.
• Dimensionless Prefactor from:• THE DESIGN OF POWERFUL
ELECTROMAGNETS: Part II. The Magnetizing Coil, by F. Bitter, published December 1936, R.S.I. Vol. 7
• Current Setup pictures courtesy of Ruwan Dedigama
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Acknowledgements• Dr. Murphy, Capstone Advisor• Ruwan Dedigama, Graduate Student• Dilhani Jayathilaka, Graduate Student