Ferromagnetic superconductors

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Bruno Phelps
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1 Department of Physics, Norwegian University of Science and Technology Pisa, July

2 Outline 1 2 Analytical framework Results 3 Tunneling Hamiltonian Josephson current 4 Quadratic term Cubic term Quartic term 5

3 Outline 1 2 Analytical framework Results 3 Tunneling Hamiltonian Josephson current 4 Quadratic term Cubic term Quartic term 5

4 Conventional vs. Unconventional Conventional non-magnetic superconductors: Cooper pairs have zero spin, zero angular momentum. Unconventional superconductors: Cooper pairs have non-zero angular momentum and may carry a net spin. Bulk or surface effect? Uniform coexistence or phase-separated? Pairing symmetry of SC order parameter?

5 Conventional vs. Unconventional Conventional non-magnetic superconductors: Cooper pairs have zero spin, zero angular momentum. Unconventional superconductors: Cooper pairs have non-zero angular momentum and may carry a net spin. Bulk or surface effect? Uniform coexistence or phase-separated? Pairing symmetry of SC order parameter?

6 Conventional vs. Unconventional Conventional non-magnetic superconductors: Cooper pairs have zero spin, zero angular momentum. Unconventional superconductors: Cooper pairs have non-zero angular momentum and may carry a net spin. SC deep within the FM phase [see S. S. Saxena et al., Nature 406, 687 (2000), D. Aoki et al., Nature 413, 613 (2001)]. Bulk or surface effect? Uniform coexistence or phase-separated? Pairing symmetry of SC order parameter?

7 Conventional vs. Unconventional Conventional non-magnetic superconductors: Cooper pairs have zero spin, zero angular momentum. Unconventional superconductors: Cooper pairs have non-zero angular momentum and may carry a net spin. SC deep within the FM phase [see S. S. Saxena et al., Nature 406, 687 (2000), D. Aoki et al., Nature 413, 613 (2001)]. Bulk or surface effect? Uniform coexistence or phase-separated? Pairing symmetry of SC order parameter?

8 Conventional vs. Unconventional Conventional non-magnetic superconductors: Cooper pairs have zero spin, zero angular momentum. Unconventional superconductors: Cooper pairs have non-zero angular momentum and may carry a net spin. SC deep within the FM phase [see S. S. Saxena et al., Nature 406, 687 (2000), D. Aoki et al., Nature 413, 613 (2001)]. Bulk or surface effect? Uniform coexistence or phase-separated? Pairing symmetry of SC order parameter?

9 Conventional vs. Unconventional Conventional non-magnetic superconductors: Cooper pairs have zero spin, zero angular momentum. Unconventional superconductors: Cooper pairs have non-zero angular momentum and may carry a net spin. SC deep within the FM phase [see S. S. Saxena et al., Nature 406, 687 (2000), D. Aoki et al., Nature 413, 613 (2001)]. Bulk or surface effect? Uniform coexistence or phase-separated? Pairing symmetry of SC order parameter?

10 Model Our model of a ferromagnetic superconductor [M. Grønsleth, J. L., J.-M. Børven, A. Sudbø, Phys. Rev. Lett. 97, (2006)]: Thin film with in-plane magnetization no orbital pair-breaking effect and no diffraction pattern in tunneling currents. Uniform coexistence of both order parameters - no vortex flux lattice (although it has been suggested to exist, see Tewari et al., Phys. Rev. Lett. 93, ). No s-wave pairing (see Phys. Rev. B 67, ), but equal-spin triplet pairing anaus to superfluid 3 He.

11 Model Our model of a ferromagnetic superconductor [M. Grønsleth, J. L., J.-M. Børven, A. Sudbø, Phys. Rev. Lett. 97, (2006)]: Thin film with in-plane magnetization no orbital pair-breaking effect and no diffraction pattern in tunneling currents. Uniform coexistence of both order parameters - no vortex flux lattice (although it has been suggested to exist, see Tewari et al., Phys. Rev. Lett. 93, ). No s-wave pairing (see Phys. Rev. B 67, ), but equal-spin triplet pairing anaus to superfluid 3 He.

12 Model Our model of a ferromagnetic superconductor [M. Grønsleth, J. L., J.-M. Børven, A. Sudbø, Phys. Rev. Lett. 97, (2006)]: Thin film with in-plane magnetization no orbital pair-breaking effect and no diffraction pattern in tunneling currents. Uniform coexistence of both order parameters - no vortex flux lattice (although it has been suggested to exist, see Tewari et al., Phys. Rev. Lett. 93, ). No s-wave pairing (see Phys. Rev. B 67, ), but equal-spin triplet pairing anaus to superfluid 3 He.

13 Model Our model of a ferromagnetic superconductor [M. Grønsleth, J. L., J.-M. Børven, A. Sudbø, Phys. Rev. Lett. 97, (2006)]: Thin film with in-plane magnetization no orbital pair-breaking effect and no diffraction pattern in tunneling currents. Uniform coexistence of both order parameters - no vortex flux lattice (although it has been suggested to exist, see Tewari et al., Phys. Rev. Lett. 93, ). No s-wave pairing (see Phys. Rev. B 67, ), but equal-spin triplet pairing anaus to superfluid 3 He.

14 Model Ĥ c,c = k c k, c k, k, k,k,q V k,k c k+q/2, c k+q/2, c k +q/2, c k +q/2, q J q S q S q,

15 Model Z =Tr e Ĥ c,c = e F

16 Model S eff = k,q, d k, k, + k k, + k, k k, k+q/2, M q k q/2, k q/2 M q k+q/2 + k,q k+q/2, k+q/2, k,q + k,q k+q/2, k+q/2, q 1 J q M q M q k,q V 1 k,k k,q. k,k,q

17 Superconducting order parameter

18 Superconducting order parameter ˆ k = k k k k = = id k ˆ ˆ y, d x k + id y k d z k d z k d x k + id y k

19 Superconducting order parameter d k = k k 2, i k + k, 2 k

20 Superconducting order parameter S k = id k d k * Unitary states: S k = 0 Non-unitary states: S k 0 : Non-unitary

21 Superconducting order parameter S k = 1/2 k 2 k 2 ẑ

22 Superconducting order parameter k,σ,σ = k,σ,σ e i(θ k+θ R(L) σσ ), θ k = θ k + π Analog of He 3 A-phase k = k 0 Analog of He 3 A1-phase kσσ 0; k σ σ = 0 Analog of He 3 A2-phase k k 0 All A-phases feature k,, = 0

23 Model Hamiltonian H FMSC = H 0 + k Â k k, k H 0 = JN 0 m k + 2 k k k b k.

24 Model Hamiltonian k = k z, =, =±1

25 Model Hamiltonian k = c k c k c k c k T

26 Model Hamiltonian k k k Â k = 2 1 k k k k k k k k k

27 Model Hamiltonian Ĥ = k kσ ξ k + INM2 2 ( ĉ kσĉ kσ 1 Δ kσσ 2 b kσσ kσ ) ( ξ kσ Δ kσσ Δ kσσ ξ kσ )( ĉkσ ĉ kσ )

28 Self-consistent equations + solutions Δ kσσ = 1 N M = 1 N k kσ V kk σσ σξ kσ 2E kσ tanh(βe kσ /2) Δ k σσ tanh(βe k 2E σ/2) k σ k,σ,σ = σ,0 Yl=1 σ (θ, φ) 3/8π Y σ l=1 (θ, φ) = σ 3/8πe iσθ sin(φ)

29 Self-consistent equations + solutions Free energy difference x 10 3 x F FM F NU Ĩ F U F NU Coexistent phase energetically favored compared to the unitary state Ĩ

30 Self-consistent equations + solutions F U F NU F PM F FM Free energy difference T c, /T c, Tc,U /T c, T/T c,

31 Self-consistent equations + solutions x Magnetization enhances the majority spin gap 4 Δ,0 3 2 Zero DOS for minority spin at Fermi level 1 Δ, M

32 Self-consistent equations + solutions 7 x M 5 Δ,0 4 Δ,0 3 2 Spontaneous magnetization arises: coexistence of ferromagnetism and triplet superconductivity Ĩ

33 Self-consistent equations + solutions M A2-phase A1-phase Pure FM Ĩ

34 Self-consistent equations + solutions 5 x M Ĩ = Δ,0 3 2 A2-phase A1-phase Pure FM phase 1 Δ, T

35 Self-consistent equations + solutions M Tc,M Ĩ Ĩ =1.02 Ĩ = Ĩ =1.005 Ĩ = T

36 Self-consistent equations + solutions Ĩ =1.01 Ĩ =1.005 Double peak in C V as signature of A2-phase CV /N(0)Tc, 10 Jump at T = T c, 5 CV /N(0)Tc, 10 5 Jump at T = T c, T/T c, T/T c,

37 Self-consistent equations + solutions The ratio C V /C V T =Tc 1.3 BCS-result for s-wave superconductors C V = ,0 N (0) 2T c, 1 C V C V T =Tc = M 1+ M ) T=Tc, ( ΔCV /CV Numerical results Analytical solution M

38 Outline Analytical framework Results 1 2 Analytical framework Results 3 Tunneling Hamiltonian Josephson current 4 Quadratic term Cubic term Quartic term 5

39 Analytical framework Results Calculations: BdG-equations (BTK-formalism) FM/FMSC-interface in xy-plane. BdG-equations describe quasiparticle states in FMSC: ( ) ( ) ˆM k ˆ k ukσ ˆM k v kσ ˆ k ( ) ukσ = E kσ, v kσ FM ẑ Barrier Normally-reflected electron θ Incident electron with spin σ Andreev-reflected hole Re{Δ (θ)} Re{Δ (θ)} α β Transmitted electronlike quasiparticle: Δσ(θ+) θ σ s Transmitted holelike quasiparticle: Δσ(θ ) FMSC with ˆM k = ε kˆ1 ˆσz U R and ˆ k = ˆσ d k iˆσ y. Barrier described by dimensionless parameter Z. Spin-generalized BTK formalism [Phys. Rev. B 25, 4515 (1982)].

40 Analytical framework Results Calculations: BdG-equations (BTK-formalism) tot z = z z + z z

41 Analytical framework Results Calculations: BdG-equations (BTK-formalism) z = e ik sin y 1 0 e ik cos z + r e E, 1 0 e ik cos z + r h E, 0 1 e ik cos z, z = e iq sin y t u e E, s+ v s+ * s+ e iq cos s z + t h E, v s s u e iq cos s z, s

42 Analytical framework Results Calculations: BdG-equations (BTK-formalism) k = 2m E F + U L 1/2 q = 2m E F + U R 1/2 u s± = s± /E 2 1/2 v s± = s± /E 2 1/2

43 Analytical framework Results Calculations: BdG-equations (BTK-formalism) /2 G E d cos g = /2 E, P L P R, g E, =1+ r h E, 2 r e E, 2, /2 F = /2 d cos f P L P R, f =1 1 2k cos / + 2. P L(R) σ = (E F + σu L(R) )/2E F

44 Analytical framework Results Conductance spectrum: Analogue of A2-phase +0.05E F +0.50E F 0.50E F Normalized conductance Metallic contact (Z =0) R =2 R =3 1.5 R =4 R = Intermediate regime (Z = 3) Voltage ev/δ, Tunneling (Z ) FM exchange U L. R =,0 /,0. σ = σ σ,0 e i[θ+φσ α(β)].

45 Analytical framework Results Conductance spectrum: Analogue of A2-phase Normalized conductance Z =0 Z = h/e F =0.15 h/e F =0.30 h/e F =0.45 Z ev/δ,0

46 Analytical framework Results Conductance spectrum: Simple odd-parity model 1.5 {α, β} =0 1.5 {α, β} = π/2 G(E) G(E) R Δ =2 R Δ =3 R Δ =4 R Δ = E/Δ,0 E/Δ,0 =,0 cos(θ α), =,0 cos(θ β) (Z = 3).

47 Analytical framework Results Conductance spectrum: Simple odd-parity model Normalized conductance h/e F =0.15 h/e F =0.30 h/e F =0.45 {α, β} = {α, β} = π/2 CV /N(0)Tc, h/e F = ev/δ, T/T c, =,0 cos(θ α), =,0 cos(θ β) (Z = 3).

48 Outline Tunneling Hamiltonian 1 2 Analytical framework Results 3 Tunneling Hamiltonian Josephson current 4 Quadratic term Cubic term Quartic term 5

49 Tunneling Hamiltonian Tunneling in thin-film structure

50 Tunneling Hamiltonian Tunneling in thin-film structure Ṅ = i H T,N = i Dˆ 1/2 T kp c k d p Dˆ 1/2 T * kp d p c k kp

51 Tunneling Hamiltonian Tunneling in thin-film structure Charge currents, 1- and 2-particle channels I C (t) = I C 1p (t) + IC 2p (t) = e α Ṅαα Spin currents, 1- and 2-particle channels I S (t) = I S sp(t) + I S tp(t) = αβ σ αβ Ṅαβ(t)

52 Tunneling Hamiltonian Tunneling in thin-film structure DC Charge/Spin Josephson current, A1-phase I C tp = e cos 2 /2 X S I tp,z = cos 2 /2 X,, where we have defined the quantity X = kp T kp 2 k p F kp sin cos pk E k E p + cos sin pk,

53 Tunneling Hamiltonian Tunneling in thin-film structure DC Charge/Spin Josephson current, A2-phase I C(S) J(,z) = I C(S) ( θ p,k, θσσ L θαα) R k,k,σ,α I C (φ 1, φ 2 ) = e 2 [1 + σα cos(ϑ)] T k,p 2 kαα pσσ E kα E pσ cos(φ 1 ) sin(φ 2 )F kpασ I S (φ 1, φ 2 ) = 1 2 α[1 + σα cos(ϑ)] T k,p 2 kαα pσσ E kα E pσ cos(φ 1 )sin(φ 2 )F kpασ F kpασ = f (±E kα ) f (E pσ ) E ± kα E pσ

54 Outline Quadratic term Cubic term Quartic term 1 2 Analytical framework Results 3 Tunneling Hamiltonian Josephson current 4 Quadratic term Cubic term Quartic term 5

55 Preliminaries Quadratic term Cubic term Quartic term S eff = d k,k k G 1 k q tr k,q V 1 k,k k,q. k,k,q 1 J q M q M q

56 Preliminaries Quadratic term Cubic term Quartic term G 1 =G 0 1

57 Preliminaries Quadratic term Cubic term Quartic term = M D M D

58 Preliminaries Quadratic term Cubic term Quartic term D = d 0 1+d = i y

59 Preliminaries Quadratic term Cubic term Quartic term F GL = TrlnG d q + tr k,q V k,k k,q 1, k,k,q 1 J q M q M q

60 Quadratic term Cubic term Quartic term Magnetic and superconducting contributions E 2 = 1 2 TrG 0 G 0 = 1 2 k 1,k 2 G 0k1 k1,k 2 G 0k2 k2,k 1

61 Quadratic term Cubic term Quartic term Magnetic and superconducting contributions F 2 = k,q k,0 J q 1 g 0,k+q/2g 0,k q/2 + c.c. M q M q n 1 g * 0,k+q/2g 0,k q/2 + c.c. +2/V d k,q 2. 16

62 Quadratic term Cubic term Quartic term Magnetic and superconducting contributions c F 2,m = 2 q 4N 0 tanh m J M q M q.

63 Quadratic term Cubic term Quartic term Magnetic and superconducting contributions d F 2,m = 2 q N 0 v F 2 m J q 2 M q M q. tanh 2 m 1 tanh 2 m

64 Quadratic term Cubic term Quartic term Magnetic and superconducting contributions d k,q =A i q kˆi

65 Quadratic term Cubic term Quartic term Magnetic and superconducting contributions c F 2,S = 1 2 q 4N 0 3 T T c tr AA T c

66 Quadratic term Cubic term Quartic term Magnetic and superconducting contributions d F 2,S = 7v 2 FN q 2 traa + q i A i q j A j + q j A i q i A j

67 Quadratic term Cubic term Quartic term Cubic coupling of magnetic and superconducting contributions E 3 = 1 G 0k1 k1,k 3 2 G 0k2 k2,k 3 G 0k3 k3,k 1 k 1,k 2,k 3

68 Quadratic term Cubic term Quartic term Cubic coupling of magnetic and superconducting contributions E 3 =4 g 0,k g * * 0,k g 0,k + c.c. id k, q1 d k,q3 M q2 k, q i

69 Quadratic term Cubic term Quartic term Cubic coupling of magnetic and superconducting contributions F 3 = 3 q 1,q 2,q 3 3! i A i A * i M,

70 Quadratic term Cubic term Quartic term Cubic coupling of magnetic and superconducting contributions 3/3!=4Re n,k 1 n 2 + k 2 kˆikˆ j 1 = N i n + 0 ij k /3 n 0 n + 1/2.

71 Quadratic term Cubic term Quartic term Quartic coupling of magnetic and superconducting contributions E 4 = 1 4 k 1,k 2 k 3,k 4 G 0k1 k1,k 2 G 0k2 k2,k 3 G 0k3 k3,k 4 G 0k4 k4,k 1.

72 Quadratic term Cubic term Quartic term Quartic coupling of magnetic and superconducting contributions F 4,m = 4m M q1 M q2 M q3 M q4. 31 q i Here, the coefficient of the M q factors is given by the trace over the electron propagators m 4m =Re g 0,k 4 1 = N 0 Re d k, n n 0 i n + = 3 2 4! tanh m 2 1 tanh 2 m

73 Quadratic term Cubic term Quartic term Quartic coupling of magnetic and superconducting contributions F 4,S = 4S traa T 2 +2 traa 2 +2tr AA T AA T * 4! q i +2tr AA 2 2tr AA AA *, 37

74 Quadratic term Cubic term Quartic term Quartic coupling of magnetic and superconducting contributions F 4,Sm = 8g 0,k g 0,k g * 0,k g * * 0,k 2 M q1 d q2 M q3 d q4 k, q i, n * M q1 M q3 d q2 d q4 16g 0,k g 0,k g 0,k g * * 0,k d q3 d q4 M q1 M q2 = q i 7N M A j M A * j,

75 Quadratic term Cubic term Quartic term Quartic coupling of magnetic and superconducting contributions F 4 = 1/4! 4m M q1 M q2 M q3 M q4 + 4Sm M A j q i M A * j + 4S traa T 2 +2 traa 2 +2tr AA T AA T * +2tr AA 2 2tr AA AA *. 39

76 Quadratic term Cubic term Quartic term Complete Ginzburg-Landau model F GL = r d 3 S T 2 traa + S 2 D2 traa + D i A i D j A j + D j A i D i A j + m T M M + m M M + 3 3! i A * i A i M + 4S 4! 2 traa 2 traa 2 +2tr AA T AA T * +2tr AA 2 2tr AA AA * + 4m 4! M M 2 + 4ms 4! M A i M A * i, 40

77 Outline 1 2 Analytical framework Results 3 Tunneling Hamiltonian Josephson current 4 Quadratic term Cubic term Quartic term 5

78 Information obtained through spectroscopy The conductance spectrum of a FM/FMSC junction may reveal information about: The magnitude of the SC gap(s) [A1- or A2-phase]. The relative orientation of the gaps with respect to the crystalline axis.

79 Information obtained through spectroscopy The conductance spectrum of a FM/FMSC junction may reveal information about: The magnitude of the SC gap(s) [A1- or A2-phase]. The relative orientation of the gaps with respect to the crystalline axis.

80 Information obtained through spectroscopy The conductance spectrum of a FM/FMSC junction may reveal information about: The magnitude of the SC gap(s) [A1- or A2-phase]. The relative orientation of the gaps with respect to the crystalline axis.

81 Information obtained through spectroscopy The conductance spectrum of a FM/FMSC junction may reveal information about: The magnitude of the SC gap(s) [A1- or A2-phase]. The relative orientation of the gaps with respect to the crystalline axis. Very useful in terms of determining the correct pairing symmetry in the FMSC.

Quantum Processes in Josephson Junctions & Weak Links. J. A. Sauls

CMS Colloquium, Los Alamos National Laboratory, December 9, 2015 Quantum Processes in Josephson Junctions & Weak Links J. A. Sauls Northwestern University e +iφ 2 e +iφ 1 111000 00000000 111111110000000