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Explore the design of a solenoid for low-temperature resistance measurements of nanostructures in the context of spintronics and weak localization. The text covers motivation, laboratory details, experimental setups, and modeling of magnetic field spatial variations. It discusses the impact of adding an independent perpendicular field and the implications for resistance and interference patterns. The goal is to minimize spatial resistance variations and enhance understanding of fundamental spin behaviors. The article also delves into the potential of spintronics to revolutionize computing power and introduces the concept of weak anti-localization with and without magnetic fields. Additionally, safety considerations for superconducting coils and environmental impact are addressed.
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Designing a Solenoid for Low Temperature Resistance Measurements of Nanostructures PHYS 4300 May 15, 2009 Jon Caddell Dr. Murphy
Outline • 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
Spintronics • Digital technology has two states corresponding to logic True/False • If the parameters associated with spin 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: Weak Localization (no spin-orbit) Clockwise (cw) Counter- clockwise (ccw) Scattering site (defect) Origin e- path in disordered material • 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)
Weak Localization (cont.) Constructive interference B R Q.M. (ccw) Classical B Field (cw) • Classical Probability for returning to origin: Pcw + Pccw = Pclasstotal • Q.M. Probability: Ψcw2 + Ψccw2 + <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
Weak Localization → Weak Anti-Localization R B Classical Q.M. B Weak Anti-Localization • Include 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
Spin/Orbit Lorentz Force e- v Zeeman Effect B B≠0 B=0 up and down spin at different energy level F up and down spin at same energy level • 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) gyromagnetic ratio Bohr magneton So applying Bparallel separates spin from orbital motion
Weak Anti-Localization and Bll Magnetic Field R Classical Zeeman Effect Weak Anti-Localization B≠0 B=0 Q.M. up and down spin at same energy level up and down spin at different energy level • One spin energetically favorable So applying Bparallel separates spin from orbital motion • Goal: seeing how weak anti-localization changes with magnetic field
Outline • 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 Setup Sample Cryostat casing Sample Holder Large Solenoid
Current Experimental Setup Bperp. Bperp. Bparallel New Method • Need a Magnetic Field for experiment • Already got one • Want to change B parallel and B perpendicular separately • Need field to be spatially UNIFORM
Future Setup I x y x z I y • Low Temp. • NO power dissipation • Superconducting Bx = B// existing magnet ~1 Tesla I I Bz = Bperp. new magnet ~10 mT
Manufacturability, 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, 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.
Outline • 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 Symmetry B Biot-Savart Law • 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 Sum vectors radial comp. cancel Plane of sample
Finding Spatial Bz Distribution, Single Loop dl r R θ Biot-Savart Law • Integrate to find B • Symmetry, Integrate half, x 2 • Involves Elliptical Integrals • B Increases Radially! Integrate theta 0 → π
Modeling Solenoid • Method A: Summing stacked current loops Uses Elliptical Integrals Elliptic Integral of the first kind Elliptic Integral of the second kind
Modeling Solenoid Previous citation & • Method B: Solenoid Model Uses Legendre Polynomials
Comparing 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 loops Close agreement, w/factor 0.1
Comparing Method A to Method B • Compare in single loop limit • Close agreement, w/factor 0.1 α = normalized radius
Modeling Solenoid 4% variation 140% variation B Radial Dependence Normalized Radius Normalized Radius • Method B • Spatial Variation for short and long coils Bz – Short Solenoid
Change of B over sample area B Field Steps=0.013 T • Want sample to have spatially uniform B Field • Need to find tolerance for ΔB Strong B Weak B B Field
Transition From B to Conductivity σ B Field • Now know B(r,I) • σ(B), need σ(r,I)
Finding Acceptable ΔB(r) σ B Field • Recall Weak Anti-Localization Signal • Conductivity σ(B) → but B(r,I) → σ(r,I) r, radius σ, conductivity I, current ∂σ/ ∂r I, current r, radius
Finding Acceptable ΔB(r) I, current ∂σ/ ∂r r, radius • 3% variation of B acceptable • Data fit uses up to B(I = 2 Amps) • Sample only in center of coil ∂σ/ ∂r I, current r, radius
Finding Acceptable ΔB(r) • 3% variation of B acceptable • Coil is within tolerance 4% variation
Trends of Field on Center B(coil dia.) B(length) B(wire dia.) B(coil dia.&length coupled)
Bz(coil dia.,length,wire dia.), Fixed Imax. B D = Coil Diameter L = Coil Length
Conclusion • 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
References Sources 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