Designing a Solenoid for Low Temperature Resistance Measurements

of Nanostructures

PHYS 4300

May 15, 2009

Jon Caddell

Dr. Murphy

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

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

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

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)

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

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

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

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

Current SetupCurrent Setup

Large SolenoidLarge Solenoid

Sample HolderSample Holder

Cryostat Cryostat casingcasing

SampleSample

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

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

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

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.

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

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

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

Modeling SolenoidModeling Solenoid• Method A: Summing

stacked current loopsUses Elliptical Integrals

Elliptic Integral of the first kind

Elliptic Integral of the second kind

Modeling SolenoidModeling Solenoid• Method B: Solenoid

ModelUses Legendre

Polynomials

Previous citation &

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

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

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

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

Transition From B to Conductivity• Now know B(r,I)• σ(B), need σ(r,I)

B Field

σ

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

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

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

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)

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

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

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

Acknowledgements• Dr. Murphy, Capstone Advisor• Ruwan Dedigama, Graduate Student• Dilhani Jayathilaka, Graduate Student