Spontaneous currents in ferromagnet-superconductor heterostructures

Transcription

1 Institute of Physics and Nanotechnology Center UMCS Spontaneous currents in ferromagnet-superconductor heterostructures Mariusz Krawiec Collaboration: B. L. Györffy & J. F. Annett Bristol Kazimierz 2005

2 Outline Proximity effect - basic facts Some remarkable experiments SC electrons in an exchange field - FFLO state Andreev bound states Self-consistent theory: negative U Hubbard model Current carrying ground state Effect of the normal metal slab Conclusions - % % % -

3 Proximity effect - basic facts = Uχ χ N SC U = 0 U < 0 ξ N-S F-S Holm et al, Z. Phys. (1932): pair breaking short superconducting vanishing of the resistance of the proximity effect S-N-S system (Josephson effect) Cooper, PRL (1961): de Jong & Beenakker, PRL (1995): first microscopic theory of the N-S system suppression of the Andreev reflections Andreev, JETP (1964): Clogston, PRL (1962): Andreev reflections E ex > / 2 no superconductivity

4 Some remarkable experiments superconducting transition temperature Lazar et al., PRB (2000) Fe/Pb/Fe resistivity + AC suceptibility 9. Dependence of the superconducting transition tem

5 differential conductance Kontos et al., PRL (2001) Nb/PdNi/Al 2 O 3 /Al planar tunneling spectroscopy.

6 density of states Cretinon et al., PRB (2005) Nb/CuNi scanning tunneling microscopy (STM)

7 Superconducting electrons in an exchange field - FFLO state e iqr Fulde & Ferrel, Phys. Rev. (1964) cos(qr) Larkin & Ovchinnikov, Sov. Phys. JETP (1965) F-S Demler et al., PRB (1997) Proshin & Khusainov, JETP Lett. (1997)

8 phase diagram T Izyumov et al., Phys. Usp. (2002) NM (T t,e t) BCS FFLO E ex properties of the FFLO state spatially dependent order parameter ( r) non-zero pairing momentum in the BCS theory spin polarization almost normal Sommerfeld specific heat almost normal single-electron tunneling characteristics unusual anisotropic electrodynamic behavior spontaneously generated current sensitivity to disorder strong dependence on the shape of the Fermi surface

9 Andreev bound states I-N-S Bohr-Sommerfeld: (α 1 + α 2 ) δϕ + β(ω) = 2nπ ( 0 : = ω/ = ±cos 2ωL ξcos(γ 2 ) de Gennes & Saint-James, PL (1963) ( π : = ω/ = ±sin Hu, PRL (1994) 2ωL ξcos(γ 2 ) ) ) I-F-S ω nσ (ϕ) = σcos((γ(ϕ) + σl n /ξ F ) /2) Kuplevakhskii & Fal ko, JETP Lett. (1990) cos(γ(ϕ)) = 1 2cos(ϕ/2) σ = ±1 ξ F = hv F /E ex

10 splitting of the zero-energy states self-induced Doppler shift: ω ω ± δ = ω ± ev F A below T (ξ/λ) T c

11 linear current response total current: J = J dia + J para 0 : ρ(ε F ) = 0 J para = 0 at T = 0 diamagnetic: J dia = e2 n mc A π : sharp peak at E F paramagnetic: J para = e2 n mc A dω ( ) df N(ω) dω N 0 overcompensation of the diamagnetic response instability: δf = JδA < 0 spontaneous current splitting of ZES spontaneous current magnetic field

12 Self-consistent theory: negative U Hubbard model M. K., B. L. Györffy & J. F. Annett, PRB (2002); EPJB (2003); Physica C (2003); PRB (2004); Physica C (2005) model H = Σ ijσ [ tij + (ε iσ µ)δ ij ] c + iσ c jσ + Σ iσ U i 2 ˆn iσˆn i σ rj - hopping integral: t ij = te ie A( r) d r r i - Coulomb interaction: U i = 0 (F M) and U i < 0 (SC) - site energies: ε iσ = 1 2 E exσ (F M) and ε iσ = 0 (SC) - magnetic field: B = (0, 0, Bz (x)) A = (0, A y (x), 0)

13 SPHFG equations Σ m k y ( ωˆτ0 δ nm Ĥ nm (k y ) ) Ĝ m m (ω, k y) = δ nm Ĥ nm (k y ) = 1 2 E exδ nm T n δ nm n δ nm 2 E exδ nm + T E exδ nm T n δ nm 0 0 n δ nm 1 2 E exδ nm + T + T ± = (tcos(k y ± ea y (n)) + µ)δ nm + tδ n,n+1 - principal layer technique Turek et al., Electronic structure..., Boston (1997) - finite temperature method Litak et al., Physica C (1995) ω = σ Ô = 2 π 2N 1 ν=0 ( ) e (2ν+1)πi/2N 1 ReG(ω ν )e (2ν+1)πi/2N ; N = βσ/2 ; σ > W/2

14 self-consistency and the Ampere s law electron concentration: n n = n n + n n n n = 2 β Σ k y νre { Tr(Ĝ nn (ω ν, k y ))e (2ν+1)πi/2N} spin polarization: m n = n n n n m n = 2 β Σ k y νre { Tr(Ĝ nn (ω ν, k y )ˆτ 3 )e (2ν+1)πi/2N} SC order parameter: n = U n Σ ky c n (k y )c n (k y ) n = 2U n β Σ k y νre { G 12 nn (ω ν, k y )e (2ν+1)πi/2N} current in the y direction: Jy tot(n) = J y (n) + J y (n) Jy tot 4et (n) = β Σ k y νsin(k y ea y (n))re { Tr(Ĝ nn (ω ν, k y ))e (2ν+1)πi/2N} polarization of the current: J y (n) = J y (n) J y (n) J y (n) = 4et β Σ k y νsin(k y ea y (n))re { Tr(Ĝ nn (ω ν, k y )ˆτ 3 )e (2ν+1)πi/2N} Ampere s law on a lattice: A( r) = µ0 j( r) A y (n + 1) 2A y (n) + A y (n 1) = µ 0 J y (n)

15 Current carrying ground state pairing amplitude χ n sin(n/ξ F ) (n/ξ F )

16 Andreev bound states zero-energy bound states (ZES): - splitting of the ZES δ 2etĀ y

17 spontaneous current FM SC B Cp (2e) e h (2e) J

18 spontaneous magnetic field Φ(n) = A y (n + 1) A y (n) A A n n+1 Φn FM SC H 10 2 H bulk c2

19 density of states vs temperature T < T c spontaneous current J = 0 J ξ S ρ tot (ε F ) 9 ξ S ξ S T/T c

20 ground state energy E = E J E 0 < 0 true ground state E kin /t E tot /t E ex /t E ex /t E /t E ex /t

21 Effect of the normal metal slab (F-N-S) M. K., J. F. Annett & B. L. Györffy, cond-mat (2005) model H = Σ ijσ [ tij + (ε iσ µ)δ ij ] c + iσ c jσ + Σ iσ U i 2 ˆn iσˆn i σ rj - hopping integral: t ij = te ie A( r) d r r i - Coulomb interaction: U i = 0 (F M and NM) and U i < 0 (SC) - site energies: ε iσ = 1 2 E exσ (F M) and ε iσ = 0 (NM and SC) - magnetic field: B = (0, 0, Bz (x)) A = (0, A y (x), 0)

22 pairing amplitude χ n sin(n/ξ F ) (n/ξ F ) d N = χ n n

23 spontaneous current J n d N = n ground state energy E tot d N

24 Conclusions F-S oscillatory behavior of the pairing amplitude zero-energy Andreev bound states in F M spontaneous current and magnetic field true ground state F-N-S pairing amplitude in F M: χ F NS χ F S difference in the GS energy: E F NS > E F S N M transparency of the interface