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Mesoscopic Anisotropic Magnetoconductance Fluctuations in Ferromagnets Shaffique Adam Cornell University PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 For details: S. Adam, M. Kindermann, S. Rahav and P.W. Brouwer, Phys. Rev. B 73 212408 (2006) Mesoscopic Anisotropic Magnetoconductance Fluctuations in Ferromagnets PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 Mesoscopic Anisotropic Magnetoconductance Fluctuations in Ferromagnets PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 Quantum Magnetism Mesoscopic Anisotropic Magnetoconductance Fluctuations in Ferromagnets PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 Quantum Magnetism Electron Phase Coherence Mesoscopic Anisotropic Magnetoconductance Fluctuations in Ferromagnets PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 Quantum Magnetism Electron Phase Coherence Ferromagnets Mesoscopic Anisotropic Magnetoconductance Fluctuations in Ferromagnets PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 Quantum Magnetism Electron Phase Coherence Ferromagnets Phase Coherent Transport in Ferromagnets Mesoscopic Anisotropic Magnetoconductance Fluctuations in Ferromagnets PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 Quantum Magnetism Electron Phase Coherence Ferromagnets Phase Coherent Transport in Ferromagnets • Motivation (recent experiments) • Introduction to theory of disordered metals • Analog of Universal Conductance Fluctuations in nanomagnets Motivation: Picture taken from Davidovic group Cu-Co interface are good contacts ~ 5 nm L ~ 30 nm C ( ) G(m)G( cos( ) m m L , disordered regime 2 C ( ) G(m)G(m) G(m) Physical System we are Studying [2] C ( ) G(m)G(m) G(m) 2 cos( ) m m 10 [0, / 2] B(T) •Aharanov-Bohm contribution • Spin-Orbit Effect - 10 V ( mV) Data/Pictures: Y. Wei, X. Liu, L. Zhang and D. Davidovic, PRL (2006) Physical System we are Studying [3] 2 c is correlatio n angle C ( ) G(m)G(m) G(m) cos( ) m m Aharanov-Bohm Spin-Orbit Effect contribution c ~ 2 ~ m a c ~ 10 VSO ~ (k k ) S ~ (k k ) m Introduction to Phase Coherent Transport Smaller and colder! V I A Image Courtesy (L. Glazman) Sample dependent fluctuations are reproducible (not noise) Ensemble Averages Need a theory for the mean <G> and fluctuations <GG> [Mailly and Sanquer, 1992] Introduction to Phase Coherent Transport [2] Electron diffusing in a dirty metal Impurities path ~ path, flux Electron A C e i PA B | PA B A | A A | A | A A * 2 , 2 * Classical Contribution Quantum Interference Diffuson Cooperon Introduction to Phase Coherent Transport [3] Weak Localization in Pictures Introduction to Phase Coherent Transport [3] Weak Localization in Pictures Introduction to Phase Coherent Transport [3] Weak Localization in Pictures For no magnetic field, the phase depends only on the path. Every possible path has a twin that is exactly the same, but which goes around in the opposite direction. Because these paths have the same flux and picks up the same phase, they can interfere constructively. Therefore the probability to return to the starting point in enhanced (also called enhanced back scattering). In fact the quantum probability to return is exactly twice the classical probability Introduction to Phase Coherent Transport [4] Weak Localization in Equations A C e i PA B | 2 A | , PA B | 2 A | * A A A A * path Classical Contribution Quantum Interference Cooperon Diffuson A C e i ~ path, flux PA B | 2 | A , PA B | PA B reversepath A | | A | 2 * A A A A A A | A | A A * 2 2 , * , * Universal Conductance Fluctuations [Mailly and Sanquer (1992)] [C. Marcus] Theory: Lee and Stone (1985), Altshuler (1985) Review of Diagrammatic Perturbation Theory (Kubo Formula) Conductance G ~ Diffuson 1 ( Dq 2 i ) Cooperon 1 ( Dq~ 2 i ) Cooperon defined similar to Diffuson upto normalization Calculating Weak Localization and UCF G G Weak Localization GG c Universal Conductance Fluctuations Calculation of UCF Diagrams 2 e GG h 2 2 e h 2 3 Tr[ JDJD JCJC ] 2 2 Lz 3 1 4 4 n 2 C ,D 2 n Sum is over the Diffusion Equation Eigenvalues scaled by Thouless Energy L n nz2 nx2 z Lx 2 2 Lz ny L y 2 iE , ET nz 1,2, nx , n y 0,1,2, ET D Lz 2 Quasi 1D can be done analytically, and 3D can be done numerically: Var G = 0.272 1 4 n n 4 90 e GG h 2 2 1 2 C ,D 15 ~ 15 x 4 for spin Effect of Spin-Orbit (Half-Metal example) [1] Energy Ferromagnet Fermi Energy Spin Up DOS(E) Spin Down DOS(E) Half Metal Effect of Spin-Orbit (Half-Metal example) [2] Without S-O H Vk k ' With S-O V Vq Vq ' (q q' ) 2 V 1 ( Dq 2 i ) 1 ( Dq 2 i ) k k ' iV so k k ' Vqso Vqso' m (k ' k ) / k F2 (q q' ) 2 so V 1 1 m m' Dq 2 i so 1 2 1 m m' Dq i so NOTE: For m=m’, Spin-Orbit does not affect the Diffuson (classical motion) but large S-O kills the Copperon (interference) Calculation of C(m,m’) in Half-Metal Without S-O With S-O C ( ) G(m)G(m) G(m) 2 GG 1 m m m’ m ( ) ( ) ( Dq 2 i ) m’ = m’ = = 1 Dq 2 i 1 n n 4 90 4 1 cos so 1 4 n (n2 A)2 4 x 4 [2 x coth( x) x 2 sinh 2 ( x)] A 1 cos( ) ET so x 1 cos( ) ET so Results for Half-Metal D=1, Analytic Result 3 e2 C ( ) 2 h 2 1 cos F ET s 0 1 cos F ET s 0 2 x coth( x) x 2 sinh 2 ( x) F ( x) x4 D=3, Done Numerically , changes definition of , We can estimate correlation angle for parameters and find about five UCF oscillations for 90 degree change Full Ferromagnet Half Metal H V k k ' iV so k k ' Vqso Vqso' m (k ' k ) / k F2 (q q' ) 2 so V Ferromagnet z Vk k ' E z iV so k k ' (m z e1 x e2 y ) (k ' k ) / k F2 1 2 1 m m' Dq i so n n 2 A( ) are Eigenvalue s of Diffusion Equation A 1 cos ET so n 2 a ( ) are Eigenvalue s of 2 2 Diffusion Equation Results for C(m,m’) in Ferromagnet 2 3 e2 C ( ) F a ( ) F a ( ) 2 h 2 x coth( x) x 2 sinh 2 ( x) F ( x) x4 Limiting Cases for m = m’ m=m’ SO C D spin Total - 1/15 1/15 4 8/15 Half Metal No 1/15 1/15 1 2/15 Half Metal Strong 0 1/15 1 1/15 Ferromagnet Weak 1/15 1/15 2 4/15 Ferromagnet Strong 0 1/15 1 1/15 Normal Metal Conclusions: Showed how spin-orbit scattering causes Mesoscopic Anisotropic Magnetoconductance Fluctuations in half-metals (This is the analog of UCF for ferromagnets) This effect can be probed experimentally 10 VSO ~ (k k ) S ~ (k k ) m 5 c ~ 2 B(T) - 10 V ( mV) Backup Slides Magnetic Properties of Nanoscale Conductors Shaffique Adam Cornell University Backup Slide 2 c is correlatio n angle C ( ) G(m)G(m) G(m) cos( ) m m Aharanov-Bohm Spin-Orbit Effect contribution 0 c ~ ~ L 2 c ~ 2 so c ~ so ET ~ ~ L1 L c ~ 10 Backup Slide Density of States quantifies how closely packed are energy levels. DOS(E) dE = Number of allowed energy levels per volume in energy window E to E +dE DOS can be calculated theoretically or determined by tunneling experiments DOS(E) Energy Fermi Energy Fermi Energy is energy of adding one more electron to the system (Large energy because electrons are Fermions, two of which can not be in the same quantum state). Backup Slide Energy DOS(E) Fermi Energy Spin Up Spin Down Energy Fermi Energy DOS(E) DOS(E) Backup Slide • Magnetic Field shifts the spin up and spin down bands Energy Ferromagnet Fermi Energy Spin Up DOS(E) Spin DOS Spin Down DOS(E) Half Metal Backup Slide Weak localization (pictures) For no magnetic field, the phase depends only on the path. Every possible path has a twin that is exactly the same, but which goes around in the opposite direction. Because these paths have the same flux and picks up the same phase, they can interfere constructively. Therefore the probability to return to the starting point in enhanced (also called enhanced back scattering). In fact the quantum probability to return is exactly twice the classical probability Backup Slide Weak localization (equations) A C e i PA B | PA B path i * Quantum Interference Diffuson ~ path, flux A C e , 2 Classical Contribution reversepath Cooperon PA B | 2 | A , PA B | PA B A | A A | A | A A * 2 A | | A | 2 * A A A A A A | A | A A * 2 2 , * , * Backup Slide Weak Localization and UCF in Pictures <G> <G G> Backup Slide Backup Slide Backup Slide Backup Slide Introduction to Quantum Mechanics Energy is Quantized Wave Nature of Electrons (Schrödinger Equation) Wavefunctions of electrons in the Hydrogen Atom (Wikipedia) Scanning Probe Microscope Image of Electron Gas (Courtesy A. Bleszynski)