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

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TU Braunschweig

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TU Braunschweig Yb-based heavy-fermion compounds: Fermi surface, quasiparticles, magnetization dynamics Gertrud Zwicknagl

Institut für Mathematische Physik Technische Universität Braunschweig

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TU Braunschweig 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 Investigate behaviour behaviour in the vicinity of Quantum in the vicinity of Quantum Critical Points Critical Points Stochiometric Stochiometric Ce Ce-

• HFS: (Often) SDW

HFS: (Often) SDW Yb Yb-

• HFS: Local quantum criticality

HFS: Local quantum criticality

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TU Braunschweig Introduction: Introduction: “ “Local Local” ” quantum criticality in quantum criticality in YbRh2Si2 => => Fermi surface changes ? Fermi surface changes ?

(after Gegenwart et al (2008))

0.01 0.1 1 10 0.01 0.1 1 10

• QCP

large FS large FS large FS small FS small FS small FS small FS

T* H0 T0 nFL FL AF

Temperature (K) In-plane magnetic field (T)

dHvA Experiments

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TU Braunschweig Introduction: Enigmatic magnetic properties Introduction: Enigmatic magnetic properties Hoc genus in rebus Hoc genus in rebus firmandumst firmandumst multa multa prius prius quam quam Ipsius Ipsius rei rei rationem rationem reddere reddere possis possis Et Et nimium nimium longis longis ambagibus ambagibus est est adeundum adeundum; ; Quo Quo magis magis attentae attentae auris auris animumque animumque reposco reposco. . T.

• T. Lucreti

Lucreti Cari Cari “ “De De rerum rerum natura natura” ”, Lib. VI, , Lib. VI, De lapide magnete 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

Translation Rolfe Humphries, (1968)

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TU Braunschweig

f itinerant f localized

Introduction: Fermi surface changes in Ce-HFS

Multi-sheeted FS

CeRu2Si2 CeRu2Ge2

Tautz et al., King et al.

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TU Braunschweig Introduction: Fermi surface in Ce-HFS

f states localized Fermi surface of LaRu2Si2 Fermi surface of CeRu2Si2 at intermediate T T ~ 6TK: Large Z hole pocket like in La Hypothesis confirmed: local f character at EF no (coherent) quasiparticles f states itinerant (low T) Fermi surface of CeRu2Si2

(Tautz et al) (Denlinger et al)

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TU Braunschweig

f itinerant f localized

Introduction: H-induced Fermi surface changes in CeRu2Si2

• H. Aoki et al., (1994)

metamagnetic transition

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TU Braunschweig Introduction: H-induced Fermi surface changes in CeCu2Si2

Bruls et al., (1994) Lifshitz transition due to Zeeman splitting?

• U. Pulst, GZ (1993)

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TU Braunschweig Introduction: Yb-based HFS

Heavy quasiparticles: Kondo mechanism Yb compounds as hole analogue of their Ce counterparts Ground state: Linear combination of f-valence: 13+x

13 1 14

, f h f h = =

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TU Braunschweig

Minimum in electrical resistivity of dilute magnetic alloys deHaas, deBoer, van den Berg (1934)

Introduction: Heavy fermions in 4f systems Introduction: Heavy fermions in 4f systems andKondo andKondo effect effect “ “Confinement Confinement“ “

Low temperatures: High temperatures Heavy Quasiparticles Free moments+conduction electrons

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TU Braunschweig Introduction: Procedure

• Calculate Fermi surface and quasiparticles

corresponding to small and large Fermi surface

• Test of the method: Compare predictions with

LuRh2Si2 => 4f localized YbIr2Si2 => 4f delocalized

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TU Braunschweig Outline

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

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TU Braunschweig Electronic structure: 4f localized

Tetragonal structure ThCr2Si2 Dirac-relativistic band structure calculation for conduction electrons Selfconsistent LDA potentials 13 4f electrons treated as part of ion core Comparison with Lu compounds

Ce M X

Yb Rh,Ir Si

2 2 2 2

2 2

5 / ( ) 4 / ( )

YbRh Si YbIr Si

mJ mole K mJ mole K γ γ ฀ ฀

2 2 2 2

2 2

8 / ( ) 4 / ( )

LuRh Si LuIr Si

mJ mole K mJ mole K γ γ ฀ ฀

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TU Braunschweig Electronic structure: 4f localized

Dispersion similar to LDA+U Ir-compound: Bands broader

YbRh2Si2 YbIr2Si2

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Fermi surface:Two major sheets Z-centered hole surface and multiply-connected surface Similar Fermi surface for Ir-compound

Electronic structure: 4f localized

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TU Braunschweig 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

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TU Braunschweig 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., 4f14 configuration States at Fermi energy have very little 4f-character LDA results (after Norman, 2005)

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TU Braunschweig 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 (Herbst, Wilkins, 1984)

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TU Braunschweig Renormalized band method

Quasiparticle bands Phase shift:

~ arctan ~ ~ η ε

f f f

E E

af=

− Γ

Condition: No re-distribution of charge ⇒ ~

~ ε f

f

Γ

d i

Single parameter : adjusted to specific heat

~ Γf

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TU Braunschweig Renormalised band method

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

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TU Braunschweig Renormalised band method:

Yb as hole-analogue of Ce

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TU Braunschweig Renormalised band method:

Yb as hole-analogue of Ce Center of gravity below Fermi energy Invert hierarchy of multiplets and CEF states

Yb Ce

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TU Braunschweig Local 4f dynamics in YbRh2Si2 and YbIr2Si2:

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

from simplified NCA (ZZF, 1992)

• Assumptions for Rh-system:

TKondo=25K, close to integer valence

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TU Braunschweig Local 4f dynamics in YbRh2Si2 and YbIr2Si2:

CEF states

b20 = 0.5246; b40 = 0.01195; b60 = - 0.0004725; b44 = 0.03538; b64 = - 0.01206;

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TU Braunschweig Local 4f dynamics in YbRh2Si2:

Compare INS data with calculated averaged spectrum Scaling: Adjust weight in energy range 10 meV - 60 meV

( ) ( ) ( )

( )

'' , 2 '' , '' , / 3 T T T χ ω χ ω χ ω

⊥

= = = + =

฀

Kondo

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TU Braunschweig Local 4f dynamics in YbRh2Si2:

Quadrupole moment: Non-monotonic T-dependence due to subtle interplay of Kondo effect and CEF excitations Suggestion: Measure qudrupole splitting in Moessbauer data

( )

2

( ) 3 1 ( )

z Q

Q T J J J E T = − + → Δ

min min

( ) (0) ( ) (0) ( ) (0) ( ) (0)

Q Q Q Q

E T E Q T Q Q T Q E T E Δ − Δ − = − Δ − Δ

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TU Braunschweig Local 4f dynamics in YbRh2Si2:

Quadrupole moment: YbRh2Si2 Non-monotonic T-dependence due to subtle interplay of Kondo effect and CEF excitations Input CEF states as determined from fit TKondo=25 K, 4f valence close to 1 Q(T) Qred(T)

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TU Braunschweig Local 4f dynamics in YbRh2Si2 and YbIr2Si2:

CEF ground state Weak hybridization with conduction states Anisotropic effective masses

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TU Braunschweig

Fermi surface:Two major sheets Z-centered hole surface and multiply-connected surface Similar Fermi surface for Ir-compound Multiply-connected FS qualitatively different from 4f localized result

Renormalised bands: Results

2 2

2

680 / ( )

YbRh Si

mJ mole K γ ฀

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TU Braunschweig

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

Renormalised bands:

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TU Braunschweig Comparison with experiment: Hall coefficient Comparison with experiment: Hall coefficient

Conductivity tensor; Boltzmann approximation Conductivity tensor; Boltzmann approximation with with where where

(0) 2 3 (1) 2 1

1 ( ) ( , ) ( , ) ( , ) ( , ) 1 ( ) ( , ) ( , ) ( , ) ( , ) ( , )

k k

f i e i k v i k v i k E i k e f i i k v i k v i k i k c E i k

αβ α β αβγ γδσ α σ βδ

σ τ σ ε τ

−

⎛ ⎞ ∂ = − ⎜ ⎟ Ω ∂ ⎝ ⎠ ⎛ ⎞ ∂ = − ⎜ ⎟ Ω ∂ ⎝ ⎠

∑ ∑

M

r r

r r r r r r r r r

( )

(0) (1)

( ) ( ) ( )

i

i i B

αβ αβγ

αβ γ

σ σ σ = + +

∑

K B

1 2

1 1 ( , ) ( , ) ; ( , ) ( , ) v i k E i k i k E i k k k k

α αβ α α β −

∂ ∂ = = ∂ ∂ ∂ M r r r r h h

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TU Braunschweig Comparison with experiment: Hall coefficient Comparison with experiment: Hall coefficient

Assumption Assumption Calculate correction to Calculate correction to Drude Drude result result

( )

( )

, i k i τ τ → r

( ) ( ) ( )

( )

( )

2 (0) (0) 3 2 (1) (1) 2 xx xx xyz xyz

e i n i m e i n i m c σ τ σ σ τ σ = =

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TU Braunschweig Comparison with experiment: Hall coefficient Comparison with experiment: Hall coefficient

Transport integrals Transport integrals

p System Method i ¯ n(i) ¯ σ(0)

xx (i)

¯ σ(1)

xyz(i) ¯

n(i)¯ σ(1)

xyz(i)

YbRh2Si2 4f core 1 1.76 0.197

• 0.289
• 0.50864

2 1.22 0.384

• 0.153
• 0.18666

YbRh2Si2 4f ren band 1 1.37 0.014

• 0.00275
• 0.0037675

2 0.63 0.075 +0.00652 +0.0041076 I-YbIr2Si2 4f ren band 1 1.42 0.051

• 0.00323
• 0.0045866

2 0.58 0.138 +0.01003 +0.0058174

Partial compensation

• f particle and hole

contributions

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TU Braunschweig

• S. Friedemann et al (2008)

System Method RH(calc.) YbRh2Si2 4f core 5.16.10−10m3/C YbRh2Si2 4f ren band −0.39 · 10−10m3/C I-YbIr2Si2 4f ren band −0.26 · 10−10m3/C

Comparison with experiment: Hall effect Comparison with experiment: Hall effect

Good agreement for Ir-compound => 4f-itinerant FS confirmed

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TU Braunschweig

3

40 30 20 10 1 2 3 4 5 6 7 8 10 20 30 40 50 60 d12 d11 D12 D11 d8 d10 d9 d7 d6 d5 D10 D9 D8 D7 D6 D5 d4 d3 D4 D3 d2 D2 d1 D1

dHvA frequency (kT) (100) (110) → (001) Magnetic field angle (degrees)

This study (14-16 T) Knebel et al. (12-28 T) LDA+SOC: small FS

(a)

40 30 20 10 1 2 3 4 5 6 7 8 10 20 30 40 50 60

dHvA frequency (kT) (100) (110) → (001) Magnetic field angle (degrees)

This study (14-16 T) Knebel et al. (12-28 T) LDA+SOC: large FS

(b)

Ren bands

YbRh2Si2 Fermi surface has two major sheets

Comparison with experiment: Comparison with experiment: deHaas deHaas-

• vanAlphen

vanAlphen

Exp.: Rourke et al (2008) Renormalized bands seem consistent with dHvA data

Problem: Magnetic field dependence of Fermi surface

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TU Braunschweig

YbRh2Si2 H || 110 F~13 kT

Comparison with experiment: Comparison with experiment: deHaas deHaas-

• vanAlphen

vanAlphen

• T. Westerkamp (2008)

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TU Braunschweig Summary and outlook Summary and outlook: :

• Fermi surface and quasiparticles calculated for localized

and delocalized 4f

• Local 4f dynamics: Subtle interplay between Kondo and

CEF excitations INS data qualitatively described Quadrupole moment Q(T)?

• Transport integrals (->Hall coefficients) calculated for

Yb-HFS 4f localized: good agreement with Lu reference compound Renormalized bands: Coherent 4f-derived quasiparticle bands => (Almost) compensation of contributions from two bands almost

• dHvA frequencies
• In progress: Comparison with ARPES