Intermediate valence metals Jon Lawrence, UCI Colloquium, 12 October - PowerPoint PPT Presentation

Intermediate valence metals Jon Lawrence, UCI Colloquium, 12 October 2006 www.physics.uci.edu/~jmlawren 1. Introduction to concept of intermediate valence (IV) 2. Anderson impurity model Fits of χ (T), n f (T), Cp(T), χ ’’( ω ) to AIM (dominated by local spin fluctuations) PES(w) showing large vs. small energy scales 3. Coherent Fermi liquid ground state: Transport, specific heat and Pauli susceptibility De Haas van Alphen and LDA band theory Optical conductivity, neutron scattering and the Anderson Lattice 4. Anomalies Slow crossover from Fermi liquid to local moment Low temperature susceptibility and specific heat anomalies Low temperature anomaly in the spin dynamics Collaborators: LANL CMP: Zach Fisk, Joe Thompson, John Sarrao, Mike Hundley, Al Arko, George Kwei, Eric Bauer NHMFL/LANL: Alex Lacerda, Neil Harrison Argonne National Lab: Ray Osborn, Eugene Goremychkin Brookhaven National Lab: Steve Shapiro Shizuoka University: Takao Ebihara UCI Postdocs: Andy Cornelius (UNLV), Corwin Booth (LBL), Andy Christianson (ORNL) UCI students: Jerry Tang, Victor Fanelli 1 Temple U.: Peter Riseborough

Intermediate Valence Compounds CeBe 13 Fermi liquid (FL) YbAgCu 4 CePd 3 FL with anomalies YbAl 3 Ce 3 Bi 4 Pt 3 Kondo Insulator YbB 12 γ - α Ce Valence transition YbInCu 4 Intermediate Valence (IV) = Nonintegral valence Yb: (5d6s) 3 4f 13 n f = 1 trivalent (5d6s) 2 4f 14 n f = 0 divalent (5d6s) 2.75 4f 13.25 YbAl 3 n f = 0.75 IV Ce: (5d6s) 3 4f 1 n f = 1 trivalent (5d6s) 4 4f 0 n f = 0 tetravalent CePd 3 (5d6s) 3.2 4f 0.8 n f = 0.8 IV Oversimplified single ion model: Two nearly-degenerate localized configurations form hybridized w.f.: a [4f 13 (5d6s) 3 > + b [4f 14( 5d6s) 2 > where a = √ n f and b = √ (1 – n f ) 2

These IV properties are captured by Basic underlying physics: Anderson Impurity Model (AIM) Highly localized 4f 13 orbital surrounded by a sea of conduction electrons Nearly degenerate with 4f 14 orbital Energy separation: E f Strong on-site Coulomb interaction U e-h pairs between 4f electrons; 4f 12 orbital at ↓ energy E f + U where U >> V, E f Hybridization V between configurations: conduction electrons hop on and off the 4f impurity orbital. Hybridization strength Γ = V 2 ρ where ρ is the density of final (conduction) states. Correlated hopping: Although intended for dilute alloys, when Γ ~ E f << U then (e.g. Lu 1-x Yb x Al 3 ), because the spin hopping from 4f 14 to 4f 13 is allowed fluctuations are local, the AIM describes but hopping from 4f 13 to 4f 12 is much of the physics of periodic IV inhibited by the large value of U. compounds. 3 Classic correlated electron problem!

Key predictions of Anderson Impurity model: All approximately valid for periodic Energy lowering due to hybridization: k B T K ~ ε F exp{-E f /[N J V 2 ρ ( ε F )]} ~ (1 – n f ) N J V 2 ρ ( ε F ) IV metals. Kondo Resonance : Narrow 4f resonance at the Fermi level ε F . (Virtual charge fluctuations yield low energy spin fluctuations.) Contributes to the density-of-states (DOS) as ρ f ( ε F) ~ 1/T K . Mixed valence due to hybridization (n f < 1) Spin/valence fluctuation: localized, damped oscillator with characteristic energy E 0 = k B T K : χ ’’~ χ (T) E Γ /((E-E 0 ) 2 + Γ 2 ) Universality : Properties scale as T/T K , E/k B T K , μ B H/k B T K High temperature limit: LOCAL MOMENT PARAMAGNET Integral valence: n f → 1 z = 2+n f = 3 Yb 4f 13 (5d6s) 3 Curie Law: χ → C J /T where C J = N g 2 μ B 2 J(J+1)/ 3 k B J = 7/2 (Yb) Full moment entropy: S → R ln (2J+1) CROSSOVER at Characteristic temperature T K Low temperature limit: FERMI LIQUID Nonintegral valence (n f < 1) Yb 4f 14-nf (5d6s) 2+nf 2 ρ f ( ε F ) Pauli paramagnet: χ (0) ~ μ B 4 Linear specific heat: C v ~ γ T γ = (1/3) π 2 ρ f ( ε F ) k B 2

Spin fluctuation spectra YbInCu 4 YbInCu 4 Magnetic scattering S mag vs. E at two incident energies E i Lawrence, Osborn et al PRB 59 (1999) 1134 Lorentzian power spectrum S(E) ~ χ ’ E P(E) = χ ’ E ( Γ /2)/{(E - E 0 ) 2 + Γ 2 } E Q-dependence: In YbInCu 4 (Lawrence, Shapiro et al, PRB 55 (1997) 14467) no dependence of Γ or E 0 on Q and only a weak (15%) dependence of χ ’ on Q . Q-independent, broad Lorentzian response ⇒ Primary excitation is a local, highly damped spin fluctuation (oscillation) at characteristic energy E 0 = k B T K 5

0 100 200 300 T(K) YbAgCu 4 Fits to the AIM Data 0.02 Theory: W = 0.865 χ (emu/mol) YbAgCu 4 : Good quantitative fits E f = -0.4485 to the T dependence of χ , n f , γ V = 0.148 T K (K) = 95K 0.01 and to the low T neutron spectrum 0.00 1.0 O data ___ AIM n f 0.9 Two parameters (E f , V) chosen to fit χ (0) and n f (0) 0.8 20 (plus one parameter for background S(mb/sr-meV) bandwidth W, chosen to agree with specific heat of nonmagnetic 10 analogue LuAgCu 4 ) 0 0 5 10 15 20 Δ E(meV) Lawrence, Cornelius, Booth, et al 6

Comparison to AIM YbAl 3 : Semiquantitative agreement AIM parameters (Chosen to fit χ (0), n f (0) and γ (LuAl 3 )) W = 4.33eV E f = -0.58264eV V = 0.3425eV T K = 670K The AIM predictions evolve more slowly with temperature than the data ( slow crossover between the Fermi liquid and the local moment regime) and there are low temperature anomalies in the susceptibility, specific heat and the neutron spectrum. Cornelius et al 7 PRL 88 (2002) 117201

Comparison to AIM (continued) YbAl 3 Spin dynamics: Neutron scattering (Q- averaged) At T = 100K the neutron scattering exhibits an inelastic (IE) Kondo peak: χ ’’(E) = Γ E / ((E- E 1 ) 2 + Γ 2 ) representing the strongly damped local excitation. For YbAl 3 , E 1 /k B = 550 K which is of order T K . This Lorentzian is still present at 6 K where experiment gives E 1 = 50 meV and Γ = 18 meV while the AIM calculation gives E 1 = 40meV and Γ = 22meV ( Semiquantitative agreement ) In addition, there is a new peak ( low Lawrence, Christianson et al, unpublished data 8 temperature anomaly ) at 32 meV.

The AIM prediction for photoemission (Gives the relationship between large and small energy scales) Primary 4f 1 → 4f 0 emission at –E f ~ (-2.7 eV in CeBe 13 ) Hybridization width 1 eV = N J V 2 ρ ( ε F ) {implies exp[-E f / (N J V 2 ρ ( ε F ))] = 0.066} Kondo Resonance near Fermi energy ε F w/ width proportional to T K . Qualitative agreement, but there is a long- standing argument about the details: E.g. the relative weight in the KR is larger than expected, suggesting the 4f electrons are forming narrow bands near ε F . 5 4 3 2 1 0 The temperature dependence is also not as Binding Energy (eV) predicted, (perhaps “slow crossover” ) Many problems arise from the high surface Lawrence, Arko et al PRB 47 (1993) 15460 sensitivity of the measurement. 9

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 YbAl 3 ρ vs T 2 4f atoms form a periodic array, ↓ the resistivity must vanish. ( Bloch’s law ) Typically in IV compounds ρ ~ A (T/T 0 ) 2 This is a sign of Fermi Liquid “coherence” among the spin fluctuations. Ebihara et al 10 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 = γ T 2 N A Z/(3 h 3 π 2 N/V) 2/3 )} m* γ = { π 2 k B The Fermi liquid also exhibits For simple metals (e.g. K): γ = 2 mJ/mol-K 2 Pauli paramagnetism: m* = 1.25 m e YbAl 3 : χ (0) = 0.005 emu/mol For YbAl 3 : γ = 45 mJ/mol-K 2 m* ~ 25 m e → “Moderately HEAVY FERMION” compound C/T vs. T 2 for YbAl 3 γ = 45 mJ/mol-K 2 Ebihara et al Physica B 281&282 (2000) 754 11

The AIM is qualitatively good (and sometimes quantitatively, e.g. YbAgCu 4 ) for χ (T), C v (T), n f (T) and χ ’’( ω ;T) essentially because these quantities are dominated by spin fluctuations, which are highly local. BUT: to get the correct transport behavior and the coherent Fermi Liquid behavior ⇒ Theory must treat the 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* ~ m e dHvA: m*~ 15-25 m e And, it can’t calculate the temperature dependence. Anderson Lattice Model: Localized 4f electrons Put 4f electrons, with AIM interactions (E f , 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. 12 Bloch’s law is satisfied for both cases.

De Haas van Alphen and the Fermi surface Figures from Ebihara et al, J Phys Soc Japan 69 (2000) 895 The de Haas van Alphen The frequency of the oscillations is The temperature dependence of experiment measures determined by the areas S of the the amplitude determines the oscillations in the extremal cross sections of the Fermi effective mass m* magnetization as a surface in the direction A = 1/sinh(Qm*T/H) function of inverse perpendicular to the applied field. where Q is a constant M = A cos(2 π F/H) magnetic field. F = (hc/2 π e) S 13