TU Braunschweig Yb-based heavy-fermion compounds: Fermi surface, - PDF document

1 TU Braunschweig

Yb-based heavy-fermion compounds: Fermi surface, quasiparticles, magnetization dynamics TU Braunschweig Gertrud Zwicknagl Institut für Mathematische Physik Technische Universität Braunschweig 2

Motivation: Motivation: 4f systems as candidates for high figure of merit? 4f systems as candidates for high figure of merit? Enhanced figure of merit in the vicinity of Quantum Enhanced figure of merit in the vicinity of Quantum Critical Points? ? Critical Points Investigate behaviour behaviour in the vicinity of Quantum in the vicinity of Quantum Investigate Critical Points Critical Points TU Braunschweig Stochiometric Ce Ce- -HFS: (Often) SDW HFS: (Often) SDW Stochiometric Yb- -HFS: Local quantum criticality HFS: Local quantum criticality Yb 3

Introduction: “ “Local Local” ” quantum criticality in quantum criticality in Introduction: YbRh 2 Si 2 => => Fermi surface changes ? Fermi surface changes ? T 0 small FS 10 T* small FS Temperature (K) 1 nFL large FS TU Braunschweig 0.1 FL H 0 AF large FS small FS small FS large FS • 0.01 0.01 0.1 1 10 QCP dHvA Experiments In-plane magnetic field (T) (after Gegenwart et al (2008)) 4

Introduction: Enigmatic magnetic properties Introduction: Enigmatic magnetic properties Hoc genus in rebus firmandumst firmandumst multa multa prius prius quam quam Hoc genus in rebus Ipsius rei rei rationem rationem reddere reddere possis possis Ipsius Et nimium nimium longis longis ambagibus ambagibus est est adeundum adeundum; ; Et Quo magis magis attentae attentae auris auris animumque animumque reposco reposco. . Quo T. Lucreti Lucreti Cari Cari “ “De De rerum rerum natura natura” ”, Lib. VI, , Lib. VI, T. De lapide magnete TU Braunschweig Such things as this require a basic course In fundamentals, and a long approach By various devious ways, so, all the more, I need your full attention 5 Translation Rolfe Humphries, (1968)

Introduction: Fermi surface changes in Ce-HFS Multi-sheeted FS CeRu 2 Si 2 CeRu 2 Ge 2 f itinerant f localized TU Braunschweig Tautz et al., King et al. 6

Introduction: Fermi surface in Ce-HFS f states localized f states itinerant (low T) Fermi surface of LaRu 2 Si 2 Fermi surface of CeRu 2 Si 2 (Denlinger et al) (Tautz et al) TU Braunschweig Fermi surface of CeRu 2 Si 2 at intermediate T Hypothesis confirmed: T ~ 6T K : local f character at E F Large Z hole pocket no (coherent) quasiparticles like in La 7

Introduction: H-induced Fermi surface changes in H. Aoki et al., (1994) 8 metamagnetic transition f localized f itinerant CeRu 2 Si 2 TU Braunschweig

Introduction: H-induced Fermi surface changes in CeCu 2 Si 2 Lifshitz transition due to Zeeman splitting? TU Braunschweig Bruls et al., (1994) U. Pulst, GZ (1993) 9

Introduction: Yb-based HFS Heavy quasiparticles: Kondo mechanism Yb compounds as hole analogue of their Ce counterparts Ground state : Linear combination of TU Braunschweig = = 13 1 14 0 f h , f h f-valence: 13+x 10

Introduction: Heavy fermions in 4f systems Introduction: Heavy fermions in 4f systems andKondo effect effect andKondo “ Confinement Confinement“ “ “ Minimum in electrical resistivity of dilute magnetic alloys deHaas, deBoer, van den Berg (1934) TU Braunschweig Low temperatures: High temperatures Heavy Quasiparticles Free moments+conduction electrons 11

Introduction: Procedure • Calculate Fermi surface and quasiparticles corresponding to small and large Fermi surface • Test of the method: Compare predictions with LuRh 2 Si 2 => 4f localized TU Braunschweig YbIr 2 Si 2 => 4f delocalized 12

Outline 1. Introduction 2. Electronic structure: 4f localized 3. 4f delocalized: Difficulties of LDA 4. Renormalized band method TU Braunschweig 5. Local 4f dynamics 6. Renormalized bands 7. Comparison with experiment 8. Summary and outlook 13

Electronic structure: 4f localized Dirac-relativistic band structure calculation for conduction electrons Selfconsistent LDA potentials 13 4f electrons treated as part of ion core γ 2 ฀ 5 mJ / ( mole K ) YbRh Si 2 2 γ 2 ฀ 4 mJ / ( mole K ) YbIr Si 2 2 TU Braunschweig Comparison with Lu compounds Yb Ce Rh,Ir γ 2 M ฀ 8 / ( ) mJ mole K LuRh Si X 2 2 Si γ 2 ฀ 4 mJ / ( mole K ) LuIr Si 2 2 Tetragonal structure ThCr 2 Si 2 14

Electronic structure: 4f localized Dispersion similar to LDA+U Ir-compound: Bands broader YbRh 2 Si 2 YbIr 2 Si 2 TU Braunschweig 15

Electronic structure: 4f localized Fermi surface:Two major sheets Z-centered hole surface and multiply-connected surface TU Braunschweig Similar Fermi surface for Ir-compound 16

4f delocalized: Difficulties of LDA in Yb-based HFS Characteristic results for Ce-based HFS compounds f bands shifted to Fermi energy band widths too broad However: Topology of FS often correct in Ce-systems TU Braunschweig 17

4f delocalized: Difficulties of LDA in Yb-based HFS Few published LDA calculations Problem: Position of 4f-states Calculations tend to converge to ground state with filled 4f shell, i. e., 4f 14 configuration States at Fermi energy have very little 4f-character TU Braunschweig LDA results (after Norman, 2005) 18

4f delocalized: Difficulties of LDA in Yb-based HFS Inadequate treatment of 4f correlations => Compressibility too high, “overbinding“ LDA lattice constants in equilibrium too small Comparison Ce vs Yb TU Braunschweig (Herbst, Wilkins, 1984) 19

Renormalized band method Quasiparticle bands Phase shift : ~ Γ af = ~ η f E arctan ~ f ε − E f TU Braunschweig Condition: No re-distribution of d i ~ charge ⇒ ~ ε f Γ f ~ Γ f Single parameter : adjusted to specific heat 20

Renormalised band method Calculational scheme: Selfconsistent LDA band structure calculation starting from atomic potentials and lattice structure TU Braunschweig Selfconsistent potentials Heavy Masses Dispersion of conduction states Renormalized Bands 21

Renormalised band method: Yb as hole-analogue of Ce TU Braunschweig 22

Renormalised band method: Yb as hole-analogue of Ce Yb Ce TU Braunschweig Center of gravity below Fermi energy Invert hierarchy of multiplets and CEF states 23

Local 4f dynamics in YbRh 2 Si 2 and YbIr 2 Si 2 : Interplay between Kondo and CEF effects Determine CEF parameters from • CEF levels from INS (Stockert et al, 2006, Hiess et al (2006) easy plane anisotropy in χ • • Test: Calculate T-dependence of experimental quantities TU Braunschweig from simplified NCA (ZZF, 1992) • Assumptions for Rh-system: T Kondo =25K, close to integer valence 24

Local 4f dynamics in YbRh 2 Si 2 and YbIr 2 Si 2 : CEF states b20 = 0.5246; b40 = 0.01195; b60 = - 0.0004725; b44 = 0.03538; b64 = - 0.01206; TU Braunschweig 25

Local 4f dynamics in YbRh 2 Si 2 : Compare INS data with calculated averaged spectrum ( ) ( ) ( ) ( ) χ ω = = χ ω = + χ ω = '' , T 0 2 '' , T 0 '' , T 0 / 3 ⊥ ฀ Kondo TU Braunschweig 26 Scaling: Adjust weight in energy range 10 meV - 60 meV

Local 4f dynamics in YbRh 2 Si 2 : Quadrupole moment: Non-monotonic T-dependence due to subtle interplay of Kondo effect and CEF excitations ( ) = − + → Δ 2 Q T ( ) 3 J J J 1 E ( ) T z Q Suggestion: Measure qudrupole splitting in Moessbauer data TU Braunschweig Δ − Δ − E ( ) T E (0) Q T ( ) Q (0) = Q Q − Δ − Δ Q T ( ) Q (0) E ( T ) E (0) min Q min Q 27

Local 4f dynamics in YbRh 2 Si 2 : Quadrupole moment: YbRh 2 Si 2 Non-monotonic T-dependence due to subtle interplay of Kondo effect and CEF excitations Input CEF states as determined from fit T Kondo =25 K, 4f valence close to 1 TU Braunschweig Q(T) Q red (T) 28

Local 4f dynamics in YbRh 2 Si 2 and YbIr 2 Si 2 : CEF ground state Weak hybridization with conduction states Anisotropic effective masses TU Braunschweig 29

Renormalised bands: Results Fermi surface:Two major sheets Z-centered hole surface and multiply-connected surface TU Braunschweig γ 2 ฀ 680 mJ / ( mole K ) YbRh Si 2 2 Similar Fermi surface for Ir-compound Multiply-connected FS qualitatively different from 4f 30 localized result

Renormalised bands: Comparison with previous LDA results Z-centered hole surface and multiply-connected surface TU Braunschweig Similar Fermi surface for Ir-compound Multiply-connected FS qualitatively different for both schemes 31