Persistent spin current in mesoscopic spin ring MingChe

Persistent spin current in mesoscopic spin ring Ming-Che Chang Dept of Physics Taiwan Normal Univ Jing-Nuo Wu (NCTU) Min-Fong Yang (Tunghai U. )

A brief history • persistent current in a metal ring (Hund, Ann. Phys. 1934) • related papers on superconducting ring • Byers and Yang, PRL 1961 (flux quantization) • Bloch, PRL 1968 (AC Josephson effect) • persistent current in a metal ring charge • Imry, J. Phys. 1982 • diffusive regime (Buttiker, Imry, and Landauer, Phys. Lett. 1983) • inelastic scattering (Landauer and Buttiker, PRL 1985) • the effect of lead and reservoir (Buttiker, PRB 1985 … etc) • the effect of e-e interaction (Ambegaokar and Eckern, PRL 1990) • experimental observations (Levy et al, PRL 1990; Chandrasekhar et al, PRL 1991) • electron spin and spin current • textured magnetic field (Loss, Goldbart, and Balatsky, PRL 1990) spin • spin-orbit coupling (Meir et al, PRL 1989; Aronov et al, PRL 1993 … etc) • FM ring (Schutz, Kollar, and Kopietz, PRL 2003) • AFM ring (Schutz, Kollar, and Kopietz, PRB 2003) • this work: ferrimagnetic ring

Aharonov-Bohm (AB) effect (1959) magnetic Φ flux solenoid r 0 path 1 path 2 Φ Phase shift = = 2πΦ/Φ 0 flux quantum Φ 0 = h/e (0. 4× 10 -6 Gauss-cm 2)

AB resistance oscillation in a mesoscopic ring (Buttiker, Imry, and Landauer, Phys. Lett. 1983) Webb et al, PRL 1985

Persistent charge current in a normal metal ring Similar to a periodic system with a large lattice constant R Phase coherence length L=2 R … … Persistent current I -1/2 / 0 Smoothed by elastic scattering… etc

Berry phase (1984) Solid angle Ω = area on the sphere / R 2 Ω R Berry phase = spin×Ω

Metal ring in a textured B field (Loss et al, PRL 1990, PRB 1992) R After circling once, an electron acquires • an AB phase 2πΦ/Φ 0 (from the magnetic flux) • a Berry phase ± (1/2)Ω(C) (from the “texture”) C Ω(C) B Electron energy:

Persistent charge and spin current (Loss et al, PRL 1990, PRB 1992)

Ferromagnet (FM), antiferromagnet (AFM), and ferrimagnet (FIM) Spin wave in ferromagnet Spin-wave quantum is called “magnon”

Ferromagnetic Heisenberg ring in a non-uniform B field (Schütz, Kollar, and Kopietz, PRL 2003) Large spin limit, using Holstein-Primakoff bosons:

vector/triad on curved surface: rules of parallel transport mi Δ is the sum of the angles of the triangle defect angle (≡Δ－π) = solid angle Ω traced out by the path = rotation angle of a parallel-transported vector mi+2 mi+1 Twist angle of the triad = solid angle traced out by m Ref: Littlejohn, PRB 1988; Balakrishnan et al, PRL 1990, PRB 1993.

3 Local triad and parallel-transported triad 2 1 solid angle traced out by m +) A gauge-invariant expression

Longitudinal part to order S, Transverse part Choose the triads such that Then, mi mi+1

Hamiltonian for spin wave Persistent spin current (NN only, Ji. i+1 ≡－J) Magnetization current • Im vanishes if T=0 (no zero-point fluctuation!) Choose a gauge such that Ω spreads out evenly • Im vanishes if N>>1 ε(k) ka Schütz, Kollar, and Kopietz, PRL 2003

Experimental detection (from Kollar’s poster) • measure voltage difference ΔV at a distance L above and below the ring • magnetic field • temperature Estimate: L=100 nm N=100 J=100 K T=50 K B=0. 1 T → ΔV=0. 2 n. V

Antiferromagnetic Heisenberg ring in a textured B field (Schütz, Kollar, and Kopietz, PRB 2004) Large spin limit • half-integer-spin AFM ring has infrared divergence (low energy excitation is spinon, not spin wave) • consider only integer-spin AFM ring. need to add staggered field to stabilize the “classical” configuration (modified SW) for a field not too strong v

Ferrimagnetic Heisenberg chain, two separate branches of spin wave: (S. Yamamoto, PRB 2004) • Gapless FM excitation well described by linear spin wave analysis • Modified spin wave qualitatively good for the gapful excitation

Ferrimagnetic Heisenberg ring in a textured B field (Wu, Chang, and Yang, PRB 2005) • no infrared divergence, therefore no need to introduce the self-consistent staggered field • consider large spin limit, NN coupling only Using HP bosons, plus Bogolioubov transf. , one has where

Persistent spin current At T=0, the spin current remains non-zero Effective Haldane gap

System size, correlation length, and spin current (T=0) AFM limit Magnon current due to zero-point fluctuation Clear crossover between 2 regions FM limit no magnon current

Magnetization current assisted by temperature Assisted by quantum fluctuation (similar to AFM spin ring) • At low T, thermal energy < field-induced energy gap (activation behavior) • At higher T, Imax(T) is proportional to T (similar to FM spin ring)

Issues on the spin current • Charge is conserved locally, and charge current density operator J is defined through the continuity eq. • The form of J is not changed for Hamiltonians with interactions. • Spin current is defined in a similar way (if spin is locally conserved), However, • Even in the Heisenberg model, Js is not unique when there is a non-uniform B field. (Schütz, Kollar, and Kopietz, E. Phys. J. B 2004). • Also, spin current operator can be complicated when there are 3 spin interactions (P. Lou, W. C. Wu, and M. C. Chang, Phys. Rev. B 2004). • Beware of background (equilibrium) spin current. There is no real transport of magnetization. • Similar problems in spin-orbital coupled systems (such as the Rashba system).

Open issues: • spin ring with smaller spins • spin ring with anisotropic coupling • diffusive transport • leads and reservoir • itinerant electrons (Kondo lattice model. . etc) • connection with experiments • methods of measurement • any use for such a ring?