# Presentation: 818467

## Intermediate valence in Yb Intermetallic compounds 

Jon Lawrence	   
University of California, Irvine

This talk concerns rare earth intermediate valence (IV) metals, with a primary focus on certain Yb-based intermetallics.  The ground state is that of a heavy mass Fermi liquid.  The d.c. and optical conductivity reflect the existence of a Fermi surface with large effective masses.  On the other hand, properties such as the susceptibility, specific heat, valence and spin dynamics that are dominated by the spin fluctuations, which are highly localized, can be understood qualitatively (and sometimes quantitatively) as those of a collection of non-interacting Anderson/Kondo impurities.  We will attempt to explain and justify these statements.  

- Jon Lawrence
- University of California, Irvine
- This talk concerns rare earth intermediate valence (IV) metals, with a primary focus on certain Yb-based intermetallics.  The ground state is that of a heavy mass Fermi liquid.  The d.c. and optical conductivity reflect the existence of a Fermi surface with large effective masses.  On the other hand, properties such as the susceptibility, specific heat, valence and spin dynamics that are dominated by the spin fluctuations, which are highly localized, can be understood qualitatively (and sometimes quantitatively) as those of a collection of non-interacting Anderson/Kondo impurities.  We will attempt to explain and justify these statements.

- Talk at LBL, 12 April 2006

## Intermediate Valence Compounds

- CeSn3	     Fermi liquid (FL)	              	YbAgCu4, YbTlCu4
- CePd3	   FL with anomalies         		YbMgCu4, YbAl3

- High symmetry (cubic) (e.g. fcc, Cu3Au, C15B)
- Hence, isotropic (3D)
- Crystal fields unimportant:  spin decay rate kBTK/ħ > kBTcf/ħ
- Hence 4f moment degeneracy (NJ = 2J + 1 ) is
- NJ = 6 (Ce); 8 (Yb)
- Ground States:
- Moderately heavy Fermi Liquids
- (like a Fermi gas but with large effective mass)
***Hybridization gap*** important, but
- no antiferromagnetic (AF) correlations

## a  [4f13(5d6s)3> + b [4f14(5d6s)2>

**Basic physics:**
- Highly localized 4f orbital
- Hybridization  of 4f with conduction electrons
- Strong on-site Coulomb interaction U between 4f electrons
- preventing multiple occupancy of orbital
**Oversimplified single ion model:**
- Two nearly-degenerate localized configurations
-  form hybridized w.f.:
- a  [4f13(5d6s)3**>** + b [4f14(5d6s)2**>**
- where a = √nf   and b = √(1 – nf)
- Other states [4f12(5d6s)4**>, **etc.**, **are at very high energy.

**Intermediate Valence (IV) = Nonintegral valence**** **
**	****	****= **Partial occupation of the 4f shell
- Yb:		(5d6s)3 4f13	 nf  = 1      trivalent
-                            	(5d6s)2 4f14	 nf = 0       divalent
- YbAl3		(5d6s)2.75 4f13.25	 nf = 0.75      IV

##          High temperature limit: LOCAL MOMENT PARAMAGNET

- Integral valence:     nf  1     z = 2+nf = 3    Yb 4f13(5d6s)3
- Curie Law:    CJ/T    where  CJ = N g2 B2 J(J+1)/ 3 kB     J = 7/2 (Yb)
- Full moment entropy:    S  R *ln*(2J+1)
-                               
-               CROSSOVER
- at Characteristic temperature TK
-  		 
**  ****Low temperature limit:     FERMI  LIQUID**
**    **
- Nonintegral valence (nf < 1)  Yb 4f14-nf (5d6s)2+nf
- Pauli paramagnet:  (0) ~ B2 ρ(F)
- Linear specific heat:  Cv ~  T
-         = (1/3) 2 ρ(F) kB2
- Note:  Low T anomalies,
- relative the AIM,
- in χ(T) and C(T)/T.

**Basic behavior of IV metals**

- γ→

- χ(0) →

- nf →1

-  ← χ ~ CJ/T

- Cornelius et al, PRL 88 (2002) 117201

## FERMI LIQUID BEHAVIOR

- C/T vs. T2 for YbAl3

- Ebihara et al
- Physica B 281&282
- (2000) 754

**FERMI LIQUID BEHAVIOR**
- A Fermi liquid is a metal where, despite the strong electron-electron interactions,
- the statistics at low T are those of a free (noninteracting) Fermi gas,
- but with the replacement m  m* (the *effective mass*).
- The **specific heat** is linear in temperature C =  T
-           = {2 kB2 NA Z/(3 h3 2 N/V)2/3)} m*
- For simple metals (e.g. K):     = 2 mJ/mol-K2
-                                               m* = 1.25 me
- For YbAl3:                 = 45 mJ/mol-K2
-                                               m* ~ 25 me
- 	     “Moderately HEAVY FERMION” compound

-   = 45 mJ/mol-K2

- The Fermi liquid also exhibits** **
**Pauli paramagnetism:**
- YbAl3: (0) = 0.005 emu/mol

## Spin fluctuation spectra YbInCu4

- Q-independent, broad Lorentzian response 
**Primary excitation is a local, highly damped spin fluctuation (oscillation) at characteristic energy E****0 ****= k****B****T****K**

- Lawrence, Osborn et al
- PRB 59 (1999) 1134

-    YbInCu4
- Magnetic scattering
- Smag vs. E at two incident energies Ei

- E

**Spin fluctuation spectra YbInCu****4**

**Lorentzian power spectrum    **
- S(E) ~ ’ E P(E) = ’ E (/2)/{(E - E0)2 + 2}

## Anderson Impurity Model (AIM)

- Highly localized 4f13 impurity orbital surrounded by a sea of conduction electrons
- Nearly degenerate with 4f14 orbital
- 	Energy separation: Ef
- Strong on-site Coulomb interaction U between 4f electrons; 4f12 orbital at energy Ef + U where U >> V, Ef
- Hybridization V between configurations: conduction electrons hop on and off the 4f impurity orbital. Hybridization strength
- 	Γ = V2ρ where ρ is the density of final (conduction) states.
- Correlated hopping:
- 	when Γ ~ Ef << U then
- 	hopping from 4f14 to 4f13 is allowed
- 	but hopping from 4f13 to 4f12 is inhibited by the large value of U.

- e-h pairs
- ↓

## Characteristic features:

***Crossover***:  from ***Low temperature local Fermi liquid ***
- (nonintegral valence, Pauli paramagnetism, linear specific heat)
- to ***local moment*** behavior for T > TK
- (integral valence, Curie law magnetism, R*ln*(2J+1) entropy)
***Universality***:  Properties scale as T/TK,  E/kBTK,  BH/kBTK  [Wilson ratio: (2R/3CJ) (0)/   1 + (1/2J)]
- VERY MUCH LIKE ACTUAL BEHAVIOR OF IV COMPOUNDS

**Characteristic features:**
**Mixed valence due to hybridization (n****f**** **< 1)
**Energy lowering due to hybridization:**
- 	kBTK ~F exp{-Ef/[NJ V2 ρ(F)]}
- 		~ (1 – nf) NJ V2 ρ(F)
- The factor (1 – nf) reflects the correlations due to large U
***Spin/valence fluctuation:***  ***localized***, ***damped oscillator*** with ***characteristic energy*** E0 = kBTK :
- 		’’~  (T) E /((E-E0)2 +  2)

- Although intended for dilute alloys,
- 	(e.g. Lu1-xYbxAl3),  because the spin fluctuations are local, the AIM describes much of the physics of periodic IV compounds.

## Comparison to AIM

**Comparison to AIM**
**YbAl****3****:  Susceptibilty, Specific Heat, 4f occupation**

**AIM parameters**
- (Chosen to fit (0), nf(0) and (LuAl3))
- W = 4.33eV
- Ef = -0.58264eV
- V = 0.3425eV
- TK = 670K

- Cornelius et al
- PRL 88 (2002) 117201

## Low temperature spin dynamics

- For the low-T Lorentzian power function seen in YbAl3, experiment gives
- 	 E0 = 40meV and  = 25meV
- while the AIM calculation gives
- 	E0 = 40meV and  = 22meV
- The experiment also exhibits a crossover to quasielastic behavior that is expected in the AIM

**Comparison to AIM (continued)**

**Low temperature spin dynamics**
**Neutron scattering:**

- Murani, PRB 50 (1994) 9882

- At low temperature the neutron scattering exhibits an inelastic (IE)
- q-independent Kondo peak:
- ’’(E) =  E / ((E- E0)2 + 2)
- representing the strongly damped local  excitation.

## Fits to the AIM for YbAgCu4:

**Fits to the AIM for YbAgCu****4****:**

- O     data
- ___  AIM

- Excellent quantitative fits!

- Lawrence, Cornelius, Booth, et al

## Overall agreement with AIM (NCA):

Two parameters (Ef, V) chosen to fit (0) and nf(0) 
(plus one parameter for background bandwidth, chosen to agree with linear coefficient of nonmagnetic analogues LuAl3 and LuAgCu4)

YbAgCu4:  Good quantitative fits to the T dependence of  and nf 
		     and to the low T neutron spectrum

YbAl3:  	Fits neutron spectral parameters at T = 0
             	Fits specific heat coefficient to 20%
		It predicts the temperatures Tmax of the maxima in susceptibility  and 	specific heat to within 20%.

But the AIM predictions evolve more slowly with temperature than the data and there are low temperature anomalies in the susceptibility, specific heat and the neutron spectrum.

- Two parameters (Ef, V) chosen to fit (0) and nf(0)
- (plus one parameter for background bandwidth, chosen to agree with linear coefficient of nonmagnetic analogues LuAl3 and LuAgCu4)
- YbAgCu4:  Good quantitative fits to the T dependence of  and nf
- 		     and to the low T neutron spectrum
- YbAl3:  	Fits neutron spectral parameters at T = 0
-              	Fits specific heat coefficient to 20%
- 		It predicts the temperatures Tmax of the maxima in susceptibility  and 	specific heat to within 20%.
- But the AIM predictions evolve more slowly with temperature than the data and there are low temperature anomalies in the susceptibility, specific heat and the neutron spectrum.

## TRANSPORT BEHAVIOR OF IV COMPOUNDS

- The AIM predicts  a finite resistivity
- at T = 0 due to unitary scattering
- from the 4f impurity.
- In an IV compound, where the
-  4f atoms form a periodic array,
-  the resistivity must vanish.
-        (**Bloch’s law**)
- Typically in IV compounds
-         ~ A (T/T0)2
- This is a sign of Fermi Liquid
- “coherence” among the spin
- fluctuations.

-  YbAl3   vs T2
-          

- Ebihara et al
- Physica B 281&282 (2000) 754

##  Coherent Fermi Liquid behavior  Theory must treat 4f lattice

- for  (T), Cv(T), nf(T) and ’’(;T)
- to get correct transport behavior and to determine the Fermi surface
** ****Coherent Fermi Liquid behavior ******** Theory must treat 4f lattice**

**Two theoretical approaches to the Fermi Liquid State**
**Band theory: Itinerant 4f electrons:** Calculate band structure in the LDA.
- One-electron band theory (LDA) treats  4f electrons as itinerant;
- it does a good job of treating the 4f-conduction electron hybridization.
- It correctly predicts the topology of the Fermi surface.
- But:  **	****Band theory strongly underestimates the effective masses!**
- 			LDA:  m* ~ me        dHvA:  m*~ 15-25 me
- 			And, it can’t calculate the temperature dependence.
**Anderson Lattice Model:  Localized 4f electrons**
- Put 4f  electrons, with AIM interactions (Ef, V, U), on each site of a periodic lattice.
- This loses the details of the Fermi surface
- but gets the effective masses and the T-dependence correctly.
- Bloch’s law is satisfied for both cases.

## ANDERSON LATTICEIf we extend the Anderson impurity model by having f-electron sites on a lattice, then the model involves level crossing: a narrow 4f band at an energy Ef below the Fermi level εF degenerate with and hybridizing (with matrix element V) with a wide conduction band whose density of states is ρ.  In the absence of Coulomb correlations (U = 0) the hybridization and resulting level repulsion leads to a band structure with a hybridization gap of order Δ = NJV2 ρ.  The Coulomb interaction U inhibits the hopping when the site has the wrong occupation.  This leads to an effective hybridization Hamiltonian with renormalized parameters, that are smaller than the bare parameters: Veff = (1-nf)1/2 V and Ueff = 0.  This gives rise to a coherent band structure with renormalized hybridized bands near the Fermi energy.  The bands exhibit a hybridization gap Δeff; the indirect gap is of order TK. The Fermi level lies in the high DOS region due to the large admixture of 4f states.  The large DOS is responsible for the large m*. 

- The structure renormalizes back to the bare energies with increasing temperature:
- For very low T << TK, fully hybridized bands.
- For T >> TK, local moments uncoupled from band electrons.

## Optical conductivity BEST EVIDENCE FOR THE HYBRIDIZATION GAP AND ITS RENORMALIZATION WITH TEMPERATURE

**High temperature:**
- Normal Drude behavior:
-    ’() = (ne2/mb){ / (1 + 22)}
- mb  is the bare band mass,  is the relaxation time. Scattering from local moments.
-                        
-            CROSSOVER
-                       
**Low temperature:**
-  1)  IR absorption peak from vertical (Q = 0) transitions across hybridization gap
*Very* narrow Drude peak.  Both m and  renormalized:
-          mb   mb = m*               = *
-         ’() = ne2 [* /m*] {1/ (1 + *22)}

- YbAl3

## Optical conductivity YbAl3 Okamura et al, J Phys Soc Japan 73 (2004) 2045

## Neutron Scattering Both interband (across the gap) and intraband (Drude-like scattering near the Fermi energy) are expected in the neutron scattering, but in this case excitations at energy transfer ΔE can have finite momentum, transfer Q.

- The intergap excitations, whose intensities are proportional to the joint (initial and final) density of states (DOS), should be biggest for zone boundary ***Q*** which connects regions of large 4f DOS. The energy for this case is the indirect gap.  For smaller Q, the spectrum should be more like the optical conductivity (Q = 0), i.e. on the scale of the direct gap.

## Neutron scattering YbAl3The low temperature magnetic scattering shows two features: 1) A broad feature near E1 = 50 meV, which energy is essentially kBTK..  This ismost intense for zone boundary Q.2) A narrow feature near E2 = 30 meV, the energy of the deep minimum in the optical conductivity.  This is independent of Q.

- Christianson et al, PRL 96 (2006) 117206

## Neutron scattering YbAl3: Q dependence

**Neutron scattering YbAl****3****: Q dependence**
- This plot integrates over the E1 = 50 meV intergap excitation at various positions in the QK, QL scattering plane.  Peak intensity occurs near
- 		(QK,QL)  = (1/2, 1/2)
-       i.e. at the zone boundary,
- This Q-dependence is as expected for intergap transitions in the Anderson lattice

- This plot integrates over Q at QL = 0.  The band of constant color near
- 	E2 = 32 meV means that the excitation is independent of Q along the QK direction.
- Such an excitation does not occur in the theory of the Anderson lattice.  It corresponds to a localized excitation in the middle of the hybridization gap – like a magnetic exciton.

- Christianson et al, PRL 96 (2006) 117206

## ConclusionsProperties of IV compounds such as the susceptibility, specific heat, temperature-dependent valence and Q-integrated neutron scattering line shape, which are dominated by highly localized spin fluctuations, fit qualitatively and sometimes quantitatively to the Anderson impurity model.Properties that are highly sensitive to lattice order – d.c. transport (resistivity), optical conductivity – require treatment of the lattice periodicity.  Band theory gets the shape of the Fermi surface correctly, but can’t get the large mass enhancements or the temperature dependence.  Anderson lattice theory gets the mass enhancements and the temperature dependence but forsakes the Fermi surface geometry.  It predicts key features of the optical conductivity and the neutron scattering, in particular that there will be a hybridization gap, with intergap transitions strong for momentum transfer Q at the zone boundary.However, there are many anomalies: the susceptibility and specific heat are enhanced at low T relative the models, and evolve more slowly with temperature than expected based on the AIM. In addition, there appears to be a localized excitation in the hybridization gap that is also not predicted by the models.