Superconductor/ferromagnet systems out of equilibrium

Transcription

1 Review: arxiv: Superconductor/ferromagnet systems out of equilibrium Tero T. Heikkilä Nanoscience Center/Department of Physics, University of Jyväskylä, Finland Collaborators: Sebastián Bergeret and Asier Ozaeta - San Sebastian F. Giazotto and Pauli Virtanen - SNS Pisa M. Silaev, I. Maasilta, Faluke Aikebaier, Risto Ojajärvi, Jyväskylä Funding by

2 Superconductivity and ferromagnetism Superconductor h " # i Ferromagnet ~ M

3 Coupling between ferromagnetism and superconductivity Superconductor Ferromagnet Equilibrium: proximity e ect length scales: S: nm Out of equilibrium: modified charge/spin/heat transport, effect on order parameter length scales: F: `m few nm `in `e ph several µm at low T `sf hundreds of nm

4 Long-range spin accumulation Hübler, Wolf, Beckmann, von Löhneysen, PRL 109, (2012) g NL = di det dv inj Measured spin di usion length 3 µm 10 `Nsf Also: Quay, Chevallier, Bena & Aprili, Nat. Phys. 9, 84 (2013) Yang, Yang, Takahashi, Maekawa & Parkin, Nature Mat. 9, 586 (2010)

5 Why was this strange? Normal state: `spin 300 nm E ect of superconductivity (Morten & Belzig, 2004): depends on the type of relaxation (spin-orbit or spin-flip) Spin-orbit: `Sspin `Nspin Spin-flip: `Sspin `Nspin In any case one should not get the observed length `Sspin 10`Nspin

6 Similar e ects on magnetic proximity from a ferromagnetic insulator: Wolf, Sürgers, Fischer & Beckmann, PRB 90, (2014) Ferromagnetic superconductor

7 Our task: 1. Describe such ferromagnetic superconductors S 2. Describe the F-S interface (tunnelling model): boundary conditions 3. Find the kinetic equations inside S (spin injection) Review: arxiv: Wolf, Sürgers, Fischer & Beckmann, PRB 90, (2014)

8 Theory: diffusive limit DNNFU spectrum ĝ R/K/A :4 4 matrices (spin Nambu) Di usive limit: Usadel equation exchange field pair potential ˇ = 1 Z D = d Tr[ 1 ǧ K ( )] 16 D ǧ 2 = ˇ1 self-energy for extra scattering ˇ = ˇ el + ˇ inel ˇ el = ˇ so + ˇ sf + ˇ orb ˇ inel = ˇ qp qp + ˇ qp ph

9 Theory: diffusive limit Equilibrium, no gradients: Dr (ĝ R rg R )+[i 3 ih 3 ˆ ˆ, ĝ R ]=0 DNNFU exchange field pair potential ˇ = 1 self-energy for extra scattering ˇ = ˇ el + ˇ inel ˇ el = ˇ so + ˇ sf + ˇ orb ˇ inel = ˇ qp qp + ˇ qp ph = 16 Z D D d Tr[ 1 (ĝ R ĝ A )] tanh 2T

10 Theory: diffusive limit DNNFU

11 Ferromagnetic superconductors S-F bilayer (FI=F insulator) Spin-splitting field h in a superconductor: kills singlet superconductivity 1 / Increasing T 0.6 / 0 Similar effect from Zeeman field h/tc h/(k B T C ) Exp: Meservey, Tedrow (1971) Moodera, et al. (1990, 2013) Beckmann, et al. (2014) and many others Chandrasekhar (1962), Clogston (1962), Fulde & Ferrell (1964), Larkin & Ovchinnikov (1965), Maki & Tsuneto (1965) Alexander, et al. (1985), Catelani, et al., (2008), and many others

12 Spin-split density of states BCS density of states With exchange field h: µ S Meservey & Tedrow, PRL 27, 919 (1971) (ALSO IN J. MOODERA S TALK)

13 Superconductor and an exchange field BCS density of states With exchange field h and spin-orbit scattering µ S Tedrow & Meservey, PRL 27, 919 (1971)

14 Our task: 1. Describe such ferromagnetic superconductors S 2. Describe the F-S interface (tunnelling model): boundary conditions 3. Find the kinetic equations inside S (spin injection) CONCENTRATE ON THE NONEQUILIBRIUM PART! Review: arxiv: Wolf, Sürgers, Fischer & Beckmann, PRB 90, (2014)

15 DNNFU Theory: Keldysh part and ǧ 2 = ˇ1 Satisfied by Nambu ( ) -spin( ) space: Interpretation: energy/charge/spin/spin energy (later in the talk)

16 Theory: boundary DNNFU conditions ere: tunnel barrier with (normal state) spin polarisation P = G " G # G " +G # Tunneling matrix ˆ = t 3 + u 3 (interface with ~ M û z ) B.c.: equate the matrix currents on both sides with ǰ k Left Right Bergeret, Verso, Volkov, PRB 86, (2012) N-state interface resistance/area N-state wire conductivity Completely general interface: M. Eschrig, A. Cottet, W. Belzig & J. Linder, NJP 17, (2015)

17 DNNFU Keldysh part ǰ k = ĵr k ĵ K k 0 ĵ A k Still a Nambu x spin matrix! For case of collinear magnetizations: N or F - insulator - spin-split superconductor: Spectral Charge current Energy current Spin current Spin energy current N ( ) = 1 8 Tr[ 3 3(ĝ R ĝ A )] Energy integral: in a few minutes

18 Huge thermoelectric effects in FIS junctions Details: Ozaeta, Virtanen, Bergeret & TTH, PRL 112, (2014) see also Machon, Eschrig, Belzig, PRL 110, (2013), NJP 16, (2014); Kalenkov & Zaikin, PRB 90, (2014), PRB 91, (2015) F S

19 Thermoelectric effects Linear response charge and heat currents across an interface: İ Q = G G th T V T/T Max. e ciency of thermoel. conversion Thermoelectric figure of merit ZT = 2 G th GT 2

20 Linear response Current density, linearised Boltzmann: Z j = e = e 2 Z 0 = e Z de (E)D(E)rf(E) = de (E)D(E)@ µ f(e) =e 2 Z de (E)D(E)@ T f(e) =e Z de de rµ/e + 0 rt (E)D(E) 4k B T cosh 2 E 2k B T E (E)D(E) 4k B T cosh 2 E 2k B T Symmetric Antisymmetric: requires E-dependence and e-h asymmetry! Sommerfeld expansion: Drude conductivity and Mott relation = e 2 D(E F ) (E F ) = elt 0, 0 = e 2 (D 0 (E F ) (E F )+D(E F ) 0 (E F ))

21 Thermoelectricity in metals Estimate: simple model with quadratic dispersion + Mott relation 0 N = 2 6 T 1 E F E F The size of thermoelectric coefficient in bulk superconductors: G(x) Small Small Gal perin, Gurevich & Kozub Sov. Phys. JETP (1974) x (Therefore, strongest thermoelectrics are semiconductors, with record-high ZT~3)

22 Interfaces Linear response charge and heat currents across an interface: İ G V = Q G th T T/T Tunneling limit: I = P R 1 1 2eR 1 den L (E)N R (E)[f 0 (E; µ L,T L ) f 0 (E; µ R,T R )] Q L = P R 1 1 2e 2 R 1 de(e µ L)N L (E)N R (E)[f 0 (E; µ L,T L ) f 0 (E; µ R,T R )]

23 Thermoelectric effects Linear response charge and heat currents across an interface: İ Q = G V G th T T/T antisymmetric Linear response thermoelectric coe cient: = P R 1 1 2eR 1 de EN L (E)N R (E) Requirement for large thermoelectric e ects: Strong energy dependence (density of states, scattering time) Electron-hole asymmetry 4k B T cosh 2 E 2k B T

24 Super-ferro: ingredients Ferromagnet: spin-dependent Fermi level spin-dependent contact resistance Polarization: P = R " R # R " +R # 2 [ 1, 1] superconductor with spin splitting h

25 Large thermoelectric effect: 1. Large e-h asymmetry per spin 2. For P 6= 0, di erent spin contributions weighed di erently

26 Spin-splitting field + polarization Linear response: Polarization! = 1 2eR T Z 1 1 de E[N "(E) N # (E)] 4k B T cosh 2 E 2k B T N/F P S Requirement for large thermoelectric e ects: Strong energy dependence (density of states, scattering time) Electron-hole asymmetry

27 DNNFU Keldysh part ǰ k = ĵr k ĵ K k 0 ĵ A k Still a Nambu x spin matrix! For case of collinear magnetizations: N or F - insulator - spin-split superconductor: Spectral Charge current Energy current Spin current Spin energy current N ( ) = 1 8 Tr[ 3 3(ĝ R ĝ A )] Energy integral: in a few minutes

28 Theory: General DNNFU structure

29 Thermally activated transport Behavior for k B T. h p2 G G T cosh( h)e, G th k r BG T e 2 G T e p2 e h 2 e h e h( h) 2 + e h( + h) 2i, sinh( h) i h cosh( h) Thermopower S = V T I=0 = P GT h = h k B T ; = kb T

30 FI(FS) Thermopower Thermopower S = V T I=0 = P GT T =0.3 S max k B e P k B T T =0.2 T =0.1 Figure of merit (ideal case): k B T ZT max h P 2 1 P 2 P =0.9 E ciency (max power): = CA ZT/(ZT + 2), CA =1 p Tcold /T hot

31 Only fit parameter: P 0.08

32 What to do with it? İ Q = Power conversion G V G th T T/T Cooling

33 Q N = G T e Z 1 Z 1 Electronic cooling Cooling even when h =0 1 N de(e µ N )N S (E)[f N (E) f S (E)] Heat balance: Q(V ; T I,T bath )=P e ph (T I,T ph ) S Pekola, TTH, et al., PRL (2004)

34 With spin splitting e 2 _Q=(GT " 2 ) h=0 h=0.1 h=0.2 h=0.3 h=0.4 h=0.5 h=0.6 kbtn=" h=0 h=0.1 h=0.2 h=0.3 h=0.4 h=0.5 h= ev=" ev=" Cooling power Electron temperature S -FI-N-FI-S P = 1, antiparallel magnetizations

35 Cooling of the superconductor Cooling power Electron temperature N-FI-S -FI-N P = 1, antiparallel magnetizations

36 S-F heat engine: efficiency FNF island (antiparallel magnetizations) S island (F electrodes) h =0.5! = 10!6 "; P = 0:8! = 10!6 "; P = 0:9! = 10!6 "; P = 0:95! = 10!4 "; P = 0:8! = 10!4 "; P = 0:9! = 10!4 "; P = 0:95 ZT k B T=" g relative strength of spurious heat conduction Dynes parameter, quality of the BCS gap

37 Thermoelectric detector TTH, Ojajärvi, Bergeret, Giazotto, Maasilta, submitted today MORE DETAILS: R. OJAJÄRVI S POSTER

38 Our task: 1. Describe such ferromagnetic superconductors S 2. Describe the F-S interface (tunnelling model) 3. Find the kinetic equations inside S (spin injection) Review: arxiv: Wolf, Sürgers, Fischer & Beckmann, PRB 90, (2014)

39 Spin injection into a spin-split superconductor Details: Silaev, Virtanen, Bergeret & TTH, PRL 114, (2015) see also Bobkov & Bobkova, Pis ma Zh. Eksp. Teor. Fiz. 101, 124 (2015), arxiv: , Krishtop, Houzet, Meyer, Phys. Rev. B 91, (R) (2015)

40 DNNFU Theory: Keldysh part and ǧ 2 = ˇ1 Satisfied by Nambu ( ) -spin( ) space: ˆf = f Lˆ1+f T 3 + f TS + f LS 3 Energy Charge Spin Spin energy

41 Modes of nonequilibrium f T f L3 Charge µ qp 6= µ S Spin energy T " 6= T # Spin Energy µ " 6= µ # T 6= T M. Silaev, P. Virtanen, F.S. Bergeret, and TTH, PRL 114, (2015)

42 Modes of nonequilibrium in superconductors E k E k E k 0 k 0 k 0 k Equilibrium Even mode T >T Odd mode Branch imbalance Charge imbalance Q > 0 µ QP >µ S (= 0) A. Schmid and G. Schön, J. Low. Temp. Phys. 20, 207 (1975)

43 DNNFU Kinetics We start from Usadel equation Spin-orbit Spin flips Orbital e ect of a field Charge and spin energy current: Coupling to condensate Energy and spin current: Spin heat relaxation M. Silaev, P. Virtanen, F.S. Bergeret, and TTH, PRL 114, (2015) Spin relaxation

44 Modes of nonequilibrium Charge Spin energy Interface with nonzero spin polarization Spin Energy Ferromagnetic superconductor N " 6= N # EXTENSION: F. AIKEBAIER S POSTER

45 Overall picture of the long-range effect i) Injection heats the electrons ii) Detector detects thermoelectrically I det / T e (V inj ), T e (V inj ) T e ( V inj ) ) g NL (V inj )= di det dv inj g NL ( V inj )

46 Fit to the experiments: (theory=red, exp=blue) Essential length scale: inelastic scattering length 5 10 µm `sf M. Silaev, P. Virtanen, F.S. Bergeret, and TTH, PRL 114, (2015)

47 Modes of nonequilibrium j Conservation law Charge current Potential Charge Energy current Temperature Energy Spin current Magnetization Angular momentum Spin energy current? Energy + angular t = r j

48 Conclusions Superconductivity + magnetism: rich set of noneq phenomena where some properties come only from the combination ZT ! = 10!6 "; P = 0:8! = 10!6 "; P = 0:9! = 10!6 "; P = 0:95! = 10!4 "; P = 0:8! = 10!4 "; P = 0:9! = 10!4 "; P = 0: k B T=" Other related effects: giant spin Seebeck effect, modified spin Hanle effect, rectification of ac signals,