Andreas Kreisel. University of Florida, Gainesville, FL. Bosons: Spin-waves and BEC in thin-film ferromagnets spin-wave theory interactions and BEC

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1 Andreas Kreisel University of Florida, Gainesville, FL Bosons: Spin-waves and BEC in thin-film ferromagnets spin-wave theory interactions and BEC Fermions: Spin fluctuation pairing and symmetry of order parameter in K xfe2-yse2 Fe based superconductors spin-fluctuation theory spin-orbit coupling 1

2 Spin-waves and BEC in thin-film ferromagnets From textbook knowledge towards BEC of magnons Andreas Kreisel, Francesca Sauli, Johanes Hick, Andreas Rückriegel, Lorenz Bartosch, Peter Kopietz Institut für Theoretische Physik Goethe Universität Frankfurt Germany Christian Sandweg, Matthias Jungfleisch, Vitaliy Vasyucka, Alexander Serga, Peter Clausen, Helmut Schultheiss, Burkard Hillebrands Fachbereich Physik Technische Universität Kaiserslautern Germany SFB TRR 49 European Physical Journal B 71, 59 (2009) Rev. Sci. Instrum. 81, (2010) European Physical Journal B 78, 429 (2010) Phys. Rev. B 85, (2012) Phys. Rev. B 86, (2012) 2

3 1. Introduction: Spin-wave theory H= ~i S ~j Jij S Heisenberg model ij determine ordered classical groundstate ferromagnet classical groundstate=quantum groundstate anti-ferromagnet (2 sublattices, Néel groundstate) AK, Hasselmann, Kopietz, ('07) AK, Sauli, Hasselmann, Kopietz, ('08) triangular anti-ferromagnet (3 sublattices, frustration) Chernychev, Zhitomirsky ('09) Veillette et al. ('05) AK, et al. ('11) 3

4 1. Spin wave theory expand in terms of bosons (1/S expansion), Holstein-Primakoff transformation ~i S ~j H= Jij S ij h i ^b; ^by = 1 S^z = S n ^ n ^ = ^by^b r p n ^ ^ S^+ = 2S 1 b 2S r p n ^ S^ = 2S^by 1 2S r 1 n ^ n ^ 1 = 1 + O( 2 ) 2S 4S S Holstein, Primakoff, Phys. Rev. 58, 1098 (1940) H= determine properties of resulting interacting theory of bosons ~ k E~k b~yk b~k + ~ k1 ;~ k2 ;~ k3 3 (~k1 ; ~k2 ; ~k3 )b~yk b~k2 b~k (1; 2; 3; 4)by1by2 b3 b4 + : : : 1;2;3;4 4

5 1. Spin wave theory: General results ferromagnet quadratic excitation spectrum vanishing interaction vertices 4» (~k1 ~k2 + ~k3 ~k4 ) antiferromagnet linear spectrum (Goldstone mode) two modes in magnetic field (2 sublattices) divergent interaction vertices 4» s j~k1 jj~k2 j j~k3 jj~k4 j à ~k1 ~k2 1 j~k1 jj~k2 j! Hasselmann, Kopietz ('06) 5

6 2.1 Spin-wave theory for thin film ferromagnets Motivation: Experiments on YIG Crystal structure: space group: Ia3d Y: 24(c) white Fe: 24(d) green Fe: 16(a) brown O: 96(h) red Gilleo et al. 58 Magnetic system: 40 magnetic ions in elementary cell 40 magnetic bands Elastic system: 160 atoms in elementary cell 3x160 phonon bands low spin wave damping good experimental control Observation of the occupation number using microwave antennas or Brillouin Light Scattering (BLS) Sandweg, AK, et al., Rev. Sci. Instrum. 81, (2010) BEC of magnons at room temperature! Demokritov et al. Nature 443, 430 (2006) Parametric pumping of magnons at high k-vectors creates magnetic excitations Question: Time evolution of magnons: Nonequilibrium physics of interacting quasiparticles 6

7 2.1 Simplifications to relevant physical properties crystal structure of YIG microscopic Hamiltonian dipole-dipole quantum spin S Zeeman term interactions ferromagnet ^ mag = 1 H 2 AK, Sauli, Bartosch, Kopietz ('09) ij Jij S i S j h Siz i 1 ¹2 ^ij )(S j r ^ij ) S i S j ] [3(S i r 3 2 i jrij j 7 j6=i

8 2.2 Linear spin-wave theory Geometry (thin film) Numerical approach Ewald summation technique Diagonalization of 2N x 2N matrix H2 = Analytic approaches Approximation for lowest mode Bogoliubov transformation q E~k = Ã A ~k B ~k B ~k A T ~k structure factor [h + ½ex~k 2 + (1 f~k ) sin2 ~k ][h + ½ex~k 2 + f~k ] = 4¼¹MS No dipolar interaction: known result E~k = h + ½ex~k2! =0 8

9 2.3 Results for magnon spectra Parallel mode Perpendicular mode Minimum for BEC Demokritov et al. Nature ( 06) d = 400a ¼ 0:5¹m N = 400 He = 700 Oe p E0 = h(h + 4¼¹Ms ) Hybridization: surface mode 9

10 2.4 Comparison to experiments Excitation and detection of spinwaves using Brillouin light scattering spectroscopy (BLS) ~k = 90± Side questions excitation efficiency for parametric pumping Serga, AK, et al. ('12) Time-dependent spin-wave theory Rückriegel, AK, Kopietz ('12) intrinsic Damping Chernychev ('12) 10

11 2.5 BEC at finite momentum Hamiltonian (after removing time dependence) ²~k b~y b~k k H2 = ~ k y y b b + bb parallel pumping new features for YIG system condensate at finite wave-vectors Ák = ±k;kmin Á0 possible 2 condensates ²~k = ² ~k Ák = ±k;kmin Á+ + ± Á k; k min 0 0 explicitly symmetry breaking term Hick, Sauli, AK, Kopietz ('10) Napoleons hat potential 11

12 2.5 Gross-Pitaevskii equation two component BEC does not solve GPE Á¾k = ±k;q Ãn¾ + ±k; q Ãn¾ interactions provoke condensation at integer 1 p Á¾k = N ±k;nq Ãn¾ multiples of ~kmin n= 1 condensate density ½(r) = = jáa (r)j2 4 jãn j2 cos2 (nq r) k = kmin finite size effects n Interaction vertices from analytical approach H4» (2;2) by by bb + (3;1) by bbb + h.c. (2;2)» Jk 2 ferromagnetic magnons 12

13 2.6 Results BEC in YIG discrete Gross-Pitaevskii equation (²nq 1 + 3! n n n ¹)Ãn¾¹ n Ãn¾ = 1 ¾¾1 ¾2 ¾1 ¾2 ±n;n1 +n2 Vnn à Ãn2 n n n n ¾ ¾ ¾¾1 ¾2 ¾3 ¾1 ¾2 ¾3 ±n;n1 +n2 +n3 Unn à n1 Ãn2 Ãn3 : 1 n2 n3 ¾1 ¾2 ¾3 13

14 3 Summary description of magnetic insulators: Spin-wave theory development of interacting spin-wave theory with dipole dipole interactions interesting properties of the energy dispersion interactions: possible condensation of bosons at finite wave-vectors and integer multiples 14

15 Spin fluctuation pairing and symmetry of order parameter in KxFe2-ySe2 Andreas Kreisel, Yan Wang [ 王彦 ], Peter Hirschfeld Department of Physics, University of Florida, Gainesville, FL , USA Thomas Maier Center for Nanophase Materials Sciences and Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN , USA Douglas Scalapino Department of Physics, University of California, Santa Barbara, CA , USA Phys. Rev. B 88, (2013)

16 A) Fe based superconductors discovery: LaO1 xfxfeas Tc=26K Kamihara et al. JACS 2006

17 A.1 Fe based superconductors: Materials Materials Crystal structure Tc Fermi surface (DFT) 28 K (55 K for Sm) 38 K 18 K 8K

18 A.2 Open questions (many) here we concentrate on symmetry of the order parameter realistic models that capture main parameters of materials under investigation

19 B.1 KxFe2-ySe2 Experimentally Different phases 245 vacancy phase Ye et al. PRL (2011) Pure SC phases K0.6Fe2Se2, K0.3Fe2Se2? Ying et al. JACS (2013) Absence of hole pocket? Evidence for fully gapped SC state Specific heat Zeng et al. (2011) ARPES Mou et al. (2011) Spin-lattice relaxation in NMR Ma et al. (2011) Quian et al. PRL (2011)

20 B.2 Band structure: Tight binding model 10 orbital, I4/mmm for K-122 Wannier projection of DFT results 10 1 H0 = t`ij1 `2 cyi`1 ci`2 N ij `1 ;`2 =1 no hole pocket n=6.25 K0.8Fe1.7Se2 n=6.12 K0.85Fe1.8Se2

21 B.2 Interactions Hubbard-Hund Hamiltonian ¹ Hint = U ni`" ni`# + U i;` + J¹ ¹0 i;`0 <` ¾;¾0 extended Hubbard model ni` ni`0 i;`0 <` cyi`¾ cyi`0 ¾0 ci`¾0 ci`0 ¾ + J¹0 i;`0 6=` cyi`" cyi`# ci`0 # ci`0 " Kuroki et al. PRL (2008) exchange interaction pair hopping

22 B.3 Spin fluctuation mediated pair scattering Susceptibility in normal state (orbital resolved) Â0`1 `2 `3 `4 (q) periodic with 1 Fe zone 1 ¹º = M`1 `2 `3 `4 (k; q)g¹ (k + q)gº (k) 2 k;¹º Interactions: RPA approximation ÂRPA 0`1 `2 `3 `4 (q) Â0 = 1 U s Â0 ÂRPA 1`1 `2 `3 `4 (q) peaked at Q=(1.65,0.35) pi Â0 = 1 + U c Â0

23 B.4 Spin fluctuation mediated pair scattering Scattering vertex in singlet channel 3 ¹ s RPA ij ~ ¹s ij (k; k ) = Re M`1 `2 `3 `4 U Â1 (k k0 ; 0)U 2 `1 `2 `3 `4 band space 1 ¹ s 1 ¹ c RPA 1 ¹c 0 c ¹ + U U Â0 (k k ; 0)U + U Graser et al. NJP (2009) `1 `2 `3 `4 0

24 B.5 Gap equation Decompose gap function into magnitude and dimensionless symmetry function g(k) k = g(k) ij ij Pairing strength functional variation leads to eigenvalue equation g (k) = I i Cj dk ij (k; k )g (k ) 2 0 (2¼) vf (k )

25 B.5 Gapfunction leading instabilities (n=6.25) dx2-y2 wave state vertical nodes (accidental) dxy wave state horizontal nodes (symmetry enforced) s-wave state loop nodes

26 B.5 Gapfunction doping dependence underdoped case (n=6.12) dx2-y2 wave state horizontal nodes (accidental) overdoped case (n=6.25) dx2-y2 wave state vertical nodes (accidental)

27 C.1 Hybridization effects Transition from d to s-wave Khodas, Chubukov PRL (2012) ±» hybridization e ' =»k ellipticity h vf (k)kf (k) i quasi-nodeless (fully gapped)

28 C.2 Hybridization in KxFe2-ySe2 cut along nonsymmetry path Origin of hybridization : I4/mmm space group spin-orbit coupling HSO = 3d Fe i =x;y;z L i Si ; Friedel ('64) approximate with atomic wave functions for spin-orbit coupling

29 C.3 Gapfunction with spin-orbit coupling weakening of superconducting instability (all symmetries) no spinorbit coupling spin-orbit coupling (λ=0.05 ev)

30 D Summary KxFe2-ySe2 different from other Fe based SC missing hole-pocket makes s-wave instability less likely (spin-fluctutation theory), dominant dx2-y2 wave symmetry quasinodes (vertical or horizontal) small hybridization regime also with spin-orbit coupling differences to experimental results small effect of Z-centrered hole pockets quasinodal behavior makes detection difficult missing ingredients as correlations or deviations from normalstate properties due to doping / impurities

31 Acknowledgements External collaborators Hirschfeld group (UF) Kopietz group (UF)