Hybridization effects in 4f systems as observed by photoemission spectroscopy

Transcription

1 Hybridization eects in 4 systems as observed by photoemission spectroscopy Yu. Kucherenko Institute or Metal Physics, Academy o Sciences o Ukraine, Kiev C. Laubschat, S.L. Molodtsov, S. Danzenbächer et et al. (TU Dresden) Experiment

2 Outline 1. Introduction 2. Single-impurity Anderson model -- Gunnarsson-Schönhammer approach -- Spectral density or emission 3. Resonant photoemission: intererence o excitation channels 4. Simple model: analytic solution 5. PES o rare-earth transition-metal compounds 6. Angle-resolved PES and simpliied periodic Anderson model Rare-earth atoms: strongly localized 4 levels + extended VB (5d,6s) states

3 4 n conigurations in Ce Main parameters: ε, U Energy E = + 2ε U 4 2 VB 4 E = E = ε Number o electrons

4 Single-impurity Anderson model H int. H H H VB + + = Hamiltonian: ) (, 2,. int,, ε ε k k k k k k k k d d V H U H d d H m m m m m m m m m m m m m VB = + + = = Localized degenerated 4 state in the sea o valence electrons P.W.Anderson, Phys.Rev. 124, 41 (1961)

5 -- the valence-band states are transormed to the same symmetry representation as the 4 states: 1 εm = V mδ( ε εk) k V ( ε ) m k k These states are orthogonal to each other i 2 Vk mvkm δ ( ε εk) = Vm( ε) δmm k -- VB states are discretized in N degenerated energy states (labeled by the index i) -- assumption: V ( ε) V( ε ) V m i i N N N ε i iν iv ε + ν ν i = 1 ν = 1 ν = 1 H = d d + + U + + V ( d + d ) + H% 2 ν ν N N ν ν ν ν i iν ν ν iν ν, ν i = 1 ν = 1 ν = ( m, ) -- contains the VB states which do not couple to the localized 4 states

6 Gunnarsson-Schönhammer approach The variational set o basis unctions: Coupling: V V N The eigenstates o H with energies E (m) are expressed as N N ( m) ( m) ( m) 0 i ij i= 1 i, j m = α 0, N + α 1, N 1; i + α 2, N 2; ij... The state with a minimal energy E g ground state g O.Gunnarsson, K.Schönhammer, Phys.Rev.B 28, 4315 (1983); 31, 4815 (1985)

7 N, inite U : minimal version o GS approach = V N N The hybridization parameters V i have to be scaled to i i N m N( N 1) = 1+ 2 N + basis unctions 2 Hybridization = i ρ( εi) N δe i ρ(ε) is the local valence-band DOS around 4 site allowed or hybridization R.Hayn, Yu.Kucherenko et al., Phys.Rev.B 64, (2001)

8 Electron removal states: 4 photoexcitation E = E ε, m, i = α φ, i ( m) ( m) ( m) i0 i0 0 φ 0 φ The spectral density in the PE process (sudden approximation): 1 I( ω, E ) = Im Nm N h mit, ( ω) kin. ( m) π m= 1 i= 1 ω kin. εi g E ( E E ) + iγ with the transition operator T( ω) = t ( ω) + t ( ω) d g ν d iν 2 I( ω, E ) = I ( ω, E ) + I ( ω, E ) + I ( ω, E ) kin. kin. d kin. int. kin. For VB photoemission a trivial result: I ( ω, E ) = t ( ω) % ρ( E hω) 2 d kin. d kin.

9 Resonant photoemission spectra hω e e 4 Direct channel Resonant channel hω Intensity Fano proile: 3d,4d Excitation energy Transition operator: r r T = t + v c t c + + ν A ν ν ν ( r) c ν ν r ν hω ( Eint. Eg ) + iγint.

10 Intensity (arb.units) CePd 3 on-res. o-res. 4 2 E F =0 Intensity (arb.units) α-ce γ -Ce E F =0 Binding energy (ev) Binding energy (ev) on-res. 4 emission, o-res. Pd d DOS Eect o the resonant channel: τ = vt A c t Parameters estimated or bulk α-ce: τ ε n Yu.Kucherenko et al., Phys.Rev.B 66, (2002)

11 H E E = ε ε + U 1 ( ) 2 2 = ε ε + 4 = 2 1 ( ) 2 2 = ε + ε Simple(st) model E g state interacts with only one degenerated VB state at E F I I Photoemission: ε E B E F =0 E hω E E ε ( ) kin = ( m g ) ( E E ) = 0 ( ) 1 g ( E E ) = ε + 4 ( ) g Ψ = A + B I 0 Ψ = B A A = B= 1 A ε 2 2 n = B E J.-M.Imer, E.Wuilloud, Z.Phys. B: Cond.Matter, 66, 153 (1987) ε I 1

12 ε = -1.5 U = 5.0 = 1.7 Eect o the VB DOS on the 4 emission Intensity (arb.units) Imer: Energy (ev) ε = 1.16 ± 0.02 = 0.18 _ to _ 0.57 Intensity (arb.units) Imer model is oversimpliied and can not be used or estimation o parameters or real systems! E F =0 Energy (ev) E F =0 Energy (ev)

13 GS approach to Pr, Nd... Energy o (bare) levels: Coniguration Ce (n=1) Pr (n=2) Nd (n=3) n-1 0 ε 2ε +U n ε 2ε +U 3ε +3U n+1 2ε +U 3ε +3U 4ε +6U - introducing an energy shit or diagonal matrix elements E( n-1 )=0 : Coniguration Ce (n=1) Pr (n=2) Nd (n=3) n n ε ε +U ε +2U n+1 2ε +U 2(ε +U )+U 2(ε +2U )+U -- new deinition or ε : ε = ε + ( n 1) % the same orm o H matrix U % ε = E E n n 1 ( ) ( )

14 Pr-TM compounds ε = 3.0 ev = 1.3 ev PrPd 3 Pr 5 Rh 4 ε = 2.9 ev = 1.32 ev Intensity (arb.units) on-res. Intensity (arb.units) PrPd PrAg ε = = 3.0 ev 0.95 ev o-res Energy (ev) ε = = 3.7 ev 0.90 ev 4d-4 resonance 3d-4 resonance E F =0 Energy (ev) Yu.Kucherenko et al., Phys.Rev.B 65, ; 66, (2002)

15 Tb-TM compounds Intensity (arb. units) Res. PE 155 ev TbRh TbPd (TbPd data shited by de=+0.17ev) 7 F 6 --> 8 S 7/2 Intensity (arb.units) TbRh TbPd TbRh TbPd Kinetic energy (ev) DyPd DyPd Energy (ev) S.L.Molodtsov et al., Phys.Rev.B 68, (2003)

16 W.A.Harrison, G.K.Straub, Phys.Rev.B 36, 2695 (1987): Hybridization matrix element: V 2 h = η ( l l r ) 1 l rl ll m ll m l+ l + 1 me d 0,24 CeIr 2 Hybridization parameter 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 RE-Pd RE-Rh 0, Ce Pr Nd Tb Dy V d0 (ev) 0,22 0,20 0,18 0,16 0,14 0,12 0,10 NdIr 2 NdRh 2 NdCo 2 PrCo 2 PrIr 2 PrRh 2 CeCo 2 CeRh 2 RE atomic number 1,1 1,2 1,3 1,4 1,5 1,6 (ev) Yu.Kucherenko et al., Phys.Rev.B 70, (2004)

17 Angle-resolved photoemission spectra

18 Periodic Anderson model U H = ε( k) d d + ε ( k) + n n + + k k k k i, i, k k 2 i, V ( ε)( d + d ) k k k k k k -- leads to a mixing o states with dierent k values = h ( k) + k 0 u U large contribution o the 4 2 coniguration small U Non-hybridized states create a band with no dispersion ε ( k) = ε V ( ε ) = c ( ε, k) k Diagonalizing h 0 (k); two parameters: ε and

19 ARPES: CePd3

20 ARPES: YbIr2Si2

21 LDA+U: YbIr 2 Si 2 9-layer slab calculations ( relativistic LMTO ) 4 bands or surace Yb layer

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26 Yb: considering holes instead o electrons Conigurations h 0 (4 14 ), h 1 (4 13 ), h 2 (4 12 ) -- or divalent atoms the h 2 coniguration may be neglected -- photoemission: h 0 h 1 transitions -- main peak (h 1 inal state) and satellite (h 0 inal state due to hybridization) -- energy spacing about 2

27 ARPES simulation: YbIr2Si2

28 Conclusions: - The 4 photoemission lineshapes in Ce, Pr, Nd, and even (under some assumptions) in Tb, Dy systems are properly described within the ramework o SIAM - Large energy splittings o the 4 ionization peak are obtained, i the energy position o the expected electron-removal state is close to the maximum o the valence-band DOS - The hybridization o the 4 states with the VB decreases (as expected) in the series o rare earth. However, even or heavier 4 elements it is not negligible - A simpliied version o PAM explains the main eatures o the angle-resolved Ce 4 photoemission using a realistic band structure - k dependence o hybridization strength is caused by the dispersive properties o VB states and their local character at the RE site.