HUMAN ENVIRONMENTS Environ 3
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Date Created: 09/12/15
Two energy scales and slow crossover in YbAl3 Jon Lawrence University of California Irvine httpwwwphysicsuciedujmlawrenResearchhtml YbAl3 is an intermediate valence IV compound with a large Kondo temperature scale TK 670K for the crossover from local moment behavior to nonmagnetic behavior Above 3050K the behavior is that of uncorrelated Kondo impurities and can be understood qualitatively in terms of the Anderson impurity model AIM Below a coherence temperature Tcoh 3050K the do and optical conductivity indicate that the system enters a Fermi liquid FL ground state in which the effective masses as determined by deA are large m 1525me Anomalies in the susceptibility speci c heat magnetotransport and spin uctuation spectra occur for T lt Tcoh The deA masses and the susceptibility anomaly are suppressed by application of a magnetic eld H gt 4OT kBTcohLLB In addition the crossover from the nonmagnetic to the local moment regime is slower than predicted by the AIM We discuss these results in terms of the Anderson lattice model with consideration given to the role of low electron density Collaborators Los Alamos Joe Thompson XT John Sarrao Crystal growth Mike Hundley RH pH NHlVIFLLANL Alex Lacerda pH LBL Corwin Booth nfT L3 XRA Shizuoka U Takao Ebihara Crystal growth dHVA U Nevada Andrew Cornelius MH deA Las Vegas Temple U Peter Riseborough Theory N CA IPNS ANL Ray Osborn x 0 neutron scattering HFBR BNL Steve Shapiro m m m Intermediate Valence IV Compounds Intermediate valence compounds CePd3 OLCe YbAgCu4 YbA13 etc Archetypal class of systems subject to electronelectron correlations Basic Physics Highly localized 4f electron degenerate with and hybridizing with conduction electrons with strong onsite Coulomb interactions between 4f electrons More complex than TM s eg Pd or 3D oneband Hubbard Model Less complex than TM oxides e g highTc which have Multiple bands 2D character possible hidden order quantum critical point QCP Comparison to Heavy Fermions HF HF IV CF doublet ground state CF unimportant TK lt ch TK gt ch NJ2J12 NJ6Ce8Yb Proximity to QCP Proximity to Kondo Insulator AF correlations Hybridization gap No AF correlations Low symmetry tet hex High symmetry cubic Anisotropy sometimes 2D effects Isotropic 3D Basic low temperature properties of IV compounds Intermediate Valence IV Nonintegral valence Partial occupation of the 4f shell Yb 5d6s3 4f13 nf 1 trivalent 5d6s2 4f14 nf O divalent YbA13 5d6s28 4f132 nf 08 1v Fermi Liquid with enhanced effective mass Tlinear specific heat for Fermi liquid 7 72 sz NA Z3 h3 72 NV23 m For simple metals eg K y 2mJmolK2 so m 125me For IV YbAl3 y 4OmJmolK2 so m 20me gt Moderately Heavy Fermion Pauli paramagnetism YbAl3 XO 004emumol Spin Fluctuation Spectra Neutron Scattering It is known from studies of single crystals of YbInCu4 Lawrence Shapiro et a1 PRESS 1997 14467 that the spin uctuations in IV compounds show very little Q dependence and the magnetic scattering exhibits a Lorentzian power spectrum X QE 11E1 f2Q X Q E PE PE 172 E02 F2391 E E02 1 91 PQ 4f form factor Qindependent broad Lorentzian response gt Primary excitation is a local highly damped spin uctuation oscillation at characteristic energy E0 kBT0 10 i i a l b d W For YbAl3 the parameters of the aka 6 I f t low temperature Lorentz1an are l res 4 t 39 i E0 40meV and l 25meV 2 i at Solid line 0 L 1 1 391 H a um i 39HNVH m Ile HN I39Y 39139 392 0 20 4390 6390 so 100 739 Energy Transfer meV Murani PRB 50 1994 9882 m 39m 39 a c At high temperature the scattering jug 39 K 6 a a n becomes quas1elast1c dashed line 3 W 4 A yquot e woe x E F E E2 F2 was a Crossover to 2 tquot A e e e l l relaxational spin dynamics o 20 so so so no X N e rt Enugy Transfer moV Murani PRB 50 1994 9882 Anderson Impurity Model AIM HA1 Ho Hi Vz H0 25ka Ic HM Ef39niU nninui Va ZVM c fi cc k N 53 H l Ef 6F E l R Although intended for dilute alloys eg La1XCeX because the spin uctuations are local the AIM describes much of the physics of periodic IV compounds It includes the basic physics of Highly localized 4f orbital at energy Ef Hybridization with conduction electrons With strength V Strong onsite Coulomb interaction U preventing other 4f occupancies Characteristic features K ondo Resonance a low energy peak in the renormalized densityof states DOS at kBTK 81 6Xp39EfNJV2 N8F Spinvalence actaation localized With characteristic energy kBTK Crossover to local moment behavior for T gt TK Universality Properties scale as TTK EkBTK uBHkBTK Basic predictions of the AIM High temperature limit Local Moment Paramagnet Integral valence nf a 1 z 2nf 3 Yb 4f135d6s3 Curie Law TxCJ 1 C N g2 LL32 JJl 3 kB Full moment entropy S a R ln2J1 J 72 Yb Quasielastic spin uctuations X X T E FE2 W U CROSSOVER at Characteristic temperature TK ll Low temperature limit 1 39 r TK TF3 Fermi Liquid n 2 Info l r Nonintegral valence nf lt 1 Yb 4f1439nf 5d6s2 f 7C Pauli paramagnet 960 N nfCJTK Fermi liquid entropy S CV 7 T Y thkB3 2J2Jl n K Iquot Damped inelastic spin uctuations S x X X T E 17EEo2 F2 E0 kBTK YbAl3 Susceptibilty Speci c Heat 4f occupation Data vs AIM 0 100 200 300 oOOo 00 0 oO O O A O 6 0005 E 5 g YbAI V 3 X omo NA 6 005 g V 39 E o 000 10 u 09 OOOoO OO C Oo 0000 O 08 00 o O o WO I I I 0 200 400 600 7 1140 lius et a1 Comparison to AIM continued Neutron scattering For the lowT Lorentzian power function experiment gives E0 40meV and F 25meV while the AHVI calculation gives E0 40meV and F 22meV The experiment also exhibits a crossover to quasielastic behavior that is expected in the AIM Wilson ratio The AIM predicts that the normalized ratio of susceptibility to speci c heat should be TEZR3CJXOy a 1 121 87 114 The experiment gives 1314 Overall agreement The AHVI with parameters chosen to t x0 and nf0 does an excellent job of tting the neutron spectral parameters and ts the speci c heat coef cient to within 20 It predicts the temperatures Tmax of the maxima in susceptibility and speci c heat to within 20 But the AIM predictions evolve more slowly with temperature than the data and there are low temperature anomalies TRANSPORT BEHAVIOR OF IV COMPOUNDS The AIM predicts a nite resistivity 5 f0 ml 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 3033 m2 1 Bloch s law quot Typically in IV compounds 0 p N A TO2 Fig 1 Temperature dependence of the electrical resistivity of YbA13 and LuAl3 The inset shows the Tzdependence of the This is a sign of Fermi Liquid resistivity coherence among the spin Ebihara et al uctuations Physica B281amp282 2000 754 In YbAl3 the T2 behavior of the resistivity is observed below 30K While the AIM is qualitatively good and sometimes quantitatively eg YbAgCu4 for x T CV0 new and moon to get correct transport behavior and to determine the Fermi surface Coherent Fermi Liquid behavior gt Theory must treat 4f lattice Two theoretical approaches to the Fermi Liquid State Band theory Itinerant 4f electrons Calculate band structure in the LDA Then either a Add correlations through LDA U or b Add Kondo physics through Renormalized Band Method Anderson Lattice Model Localized 4f electrons Various approximations for example a Ignore intersite contributions Yoshimori and Kasai Journ Mag Mag Mat 3134 1983 475 b Treat intersite contributions in NCA Georges et a1 PRL 85 2000 1048 c Treat intersite contributions in Dynamic Mean Field Appx Jarrell et a1 PRB 55 1997 R3332 Bloch s law is satisfied for all approximations Approximations such as a maintain single site thermodynamics b and 0 give anomalies relative to single site thermodynamics De Haas van Alphen and the Fermi surface Figures from Ebihara et al J Phys Soc Japan 69 2000 895 Yble3 H I lt1 11gt ZSmK a I I I i 130 150 170 H kOe n b B l A a k I A l 0 1 2 deA Frequency x1030e 39i3 1 a DHvA oscillation and b its FFT spectrum for the apan eld along 111 in YbAlg The frequency of the oscillations is determined by the areas S of the extremal cross sections of the Fermi surface in the direction perpendicular to the applied eld M A cos27tFH F hc2 rte S g l 39 I 2 E or 135 mo 5 Equot i W Hit 111 E E l I 005 010 Temperanne K Fig 3 Mass plot for the eld along 111 in YbAls The de Haas van Alphen experiment measures oscillations in the magnetization as a function of inverse magnetic eld YbAl3 band 13hole band 12hole band 13electron band l4electron Fig 7 Modi ed Fermi surfaces in YbAla The band 13hole and thef band 13electron Fermi surface represent the same Fermi sur ace The temperature dependence of the amplitude determines the effective mass m A 1sinhQmTH Where Q is a constant 9 I I deA Frequency x108 0e l 0 8 3 dIIvA Frequency x103 0e lt1 1 I I 7 107 l I 39 1 30 30 so 90 1 39 I lt110gt 5100gt ltlllgt lt110gt lt110gt30 lt100gt 30 1 lt1 Field Angle Degrees Field Angle Degrees Fig 2 Angular dependence of the deA frequency in min THEORY EXPT For IV compounds LDA gives the correct extremal areas Oneelectron band theory LDA treats 4f electrons as itinerant Does a good job of treating the 4fconduction electron hybridization Correctly predicts the topology of the Fermi surface seen by dHVA But LDA strongly underestimates the effective masses LDA smooths the very local Coulomb correlations This is a bad approximation for 4f electrons LDA badly overestimates the 4f band Widths Consequently it strongly underestimates the effective masses LDA m me dHVA m 1324 me LDA alone makes YbAl3 be divalent The correct valence and the Fermi surface shown above was obtained by forcing the 4f level energy Large masses and intermediate valence also can be obtained by LDA U ANDERSON LATTICE For a periodic IV compound there is one 4f on every site so the appropriate model is H 21lt 1lt111ltJr 21Ef n U at t 21ltka ckfi 00 This leads to a coherent band structure with renormalized hybridized bands near the Fermi energy The bands exhibit a hybridization gap 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 we Qatar 165 I 1 Ev AnnWren k Nia The structure renormalizes away with increasing temperature For very low T ltlt TK 39 Fully hybridized bands 39 i i For T TK 39 39 39 T No gap Incoherent KOHdO TCSOI IElI lCCS For T gtgt TK Local moments uncoupled from band electrons Optical conductivity BEST EVIDENCE FOR THE HYBRIDIZATION GAP AND ITS RENORMALIZATION WITH TEMPERATURE High temperature O wa Normal Drude behavior Tgt TK om nezmb c 1 1209 A mb bare band mass c relaxation time ll CROSSOVER T T U Low temperature 1 Infrared absorption peak from transitions across hybridization gap 2 Very narrow Drude peak Both m and c renormalized mbakmbm Takrrci 60 ne2 1 m 1 1 12m2 w rgt Q Okamura Ebihara and Namba unpublished YbAl3 f 30000 1 V cm a 20000 39 camMama 10000 0 01 02 03 04 Photon Energy eV Note IR conductivity invariant with T below 40K I Extended Drude analyses 3917 OCQ 3K 0 I 0 005 010 015 0 005 010 015 Photon Energy eV Photon Energy eV I Fermi liquid behavior 1 032 I Mass enhancement mmb 2530 9 Heavymass Fermi liquid Assuming frequencydependent scattering C00 nezmb MOD i001391 then the mass enhancement mquotlt kmb Moo Imyoo 00 is both frequency and temperature dependent For YbAl3 this procedure gives m2530 comparable to the deA masses Two energy scales and slow crossover in the Anderson Lattice While the transport behavior and the Fermi surface deA are affected by Fermi liquid coherence we have seen that experimental quantities such as the specific heat susceptibility valence and spin dynamics are qualitatively in good accord with the predictions of the AIM over a broad range of temperature This re ects highly localized spinvalence uctuations Nevertheless recent theory for the Anderson Lattice suggests that the behavior of these quantities can differ in two ways from the predictions of the AIM 1 Nonuniversal low temperature scale for coherence Low temperature anomalies Antoine Georges et al T T c39 quot PRL 85 2000 1048 2 Slow crossover from Fermi Liquid to Local Moment Mark Jarrell et al PRBSS 1997 R3332 Theory predicts that these differences become magnified when the conduction electron density is low Slow crossover in YbAl3 Slow crossover has been reported for xT and nfT for YbXCu4 Lawrence et a1 PRB 63 2001 054427 and correlated to electron density determined from Hall coefficient 116 lRHe YbAgCu4 YleCu4 YngCu4 YbZnCu4 ne lt leatom 0 100 200 300 A x 15 I O E 10 3 5 E D a 0 10 9 g 3 0005 05 3 a b x 0000 00 10 YbAI3 W433eV 0 9 E 058264eV C 39 V03425eV I O 08 c 116 gt leatom No slow crossover Slow crossover For YbAl3 116 lRHe 05 eatom and SLOW CROSSOVER OBSERVED for TX C J entropy susceptibility and 4f occupation number Symbols eXpt Data Lines AIM Low temperature anomalies in YbAl3 0006 g 0004 2 0 E E 00052 9 g 0002 E V x X M n A N x I E 3 39 E O M TK Cornelius et a1 PRL 88 2002 117201 A 00 0 5 0 T 15K n 003 5 l f 3 004 o 02 04 06 08 sine A A Hiess et al J Phys Cond Mat 12 2000 829 Above 40K the susceptibility and specific heat correspond qualitatively to the predictions of the AIM Below 3050K anomalies are observed No form factor anomaly The neutron form factor measures the spatial distribution of magnetization around the Yb site At most temperatures the form factor has the same Q or r dependence f24fQ as the 4f radial function In CePd3 and CeSn3 at low T a more diffuse 5d component f25dQ occurs f2Q a2 f24f 1a2 f25d This 5d contribution gives rise to an anomaly in the low temperature susceptibility similar to that of YbA13 However in YbA13 there is no form factor anomaly the magnetization density is that of the 4f orbital at all temperatures solid lines ARR RH10391 m3C New peak in low temperature spin dynamics We have seen that for most temperatures and energies the magnetic neutron scattering in YbAl3 follows the predictions of the AIM with a Lorentzian power spectrum with E0 40meV and F 25meV At low T there is an additional narrow peak with l b t Mrani PRB 50 1994 9882 data 5K t res o l A I Luklvl 391quot Irrl H H39Y 39139 H 4O 60 80 Energy Transfer meV 001 9000 E0 30meV and F SmeV This peak vanishes above 50K and hence appears to be a property of the fully coherent ground state 100 quot Magnetotransport anomalies Anomalies in the Hall coefficient and magnetoresistance are observed in this same temperature range Since RH lne this suggests a change in carrier density The onset of coherent Fermi liquid behavior appears to involve a change in the Fermi surface Cornelius et al PRL 88 2002 117201 Field dependent masses in YbAl3 Ebihara Cornelius Lawrence Uji and Harrison condmat0209303 20 I300 3 I xemumol Blt100gt 39Y 0 o 8ltgtltgt 0000 0006 0004 0002 20 xemumo L soc YbAIB H1T O Hlt4OT O Hgt4OT O 100 TK High eld deA shows that the effective masses for Hlt111gt decrease substantially for H gt 4OT This eld is much smaller than the Kondo eld BK kTKgJuB required to polarize the f electrons but is of order kBTcohHB 39 A eld of this magnitude also suppresses the low temperature susceptibility anomaly It is as though that the system exhibits a crossover from a anomalous high mass Fermi Liquid state to a nonanomalous moderately enhanced Fermi liquid state for uBH gt kBT coh39 EFFECT OF DISORDER ON THE LOW TEMPERATURE ANOMALIES I 39 I I 000000 AOOO55 YbHLuXAI3 o 0 O 39Q 0 quot0 O 00050 o39 00000000000000960251 O 39 GOOOAAAAAAMMA22 mg 00045 MOOOOAAAA 005 1 E 39 AAA VVVvWVWWVVVVV 800040 WA VVVvVv 010 V X 39 VVV 00035 WW 3 I I I I O 50 100 150 200 I I I 39 x 0 V V 0 15 O o 025 v v E 0 05 v A A A A O o O 010 0 o 01 C T JmoIeYbK 2 I I 439 8 a 39339 39 m O 25 50 75 100 The low temperature anomalies in the susceptibility and specific heat are very sensitive to alloy disorder and disappear for alloy concentrations as small as X 005 in Yb1XLuXAl3 Apparently the enhanced masses observed below Tcoh are very sensitive to lattice coherence LECTURE 3 2 Neighborhoods 10 min Let r denote a positive number and p a point on a coordinate line De nition The open interval of length Zr and with midpoint p ie the interval p 7 rp r is called an r7 neighborhood ofp or simply a neighborhood ofp An interval of the form vapr is called a rightesided reneighborhood ofp or simply a rightsided neighborhood of 10 An interval of the form p 7 rp is called a leftesided reneighborhood ofp or a leftesided neighborhood ofp Note By the above de nition a onesided neighborhood of a point does not include that point This terminology is not standard use but it simpli es our work in this course Note An open interval is a o rightsided neighborhood of its left endpoint o leftsided neighborhood of its right endpoint o neighborhood of its midpoint Example For the point p 239 write down the one and twosided neighborhoods of radius 12 Answer the leftsided neighborhood l227239 the rightsided neighborhood 239251 the neighborhood 227251 3 OneSided Limits 31 The RightSided Limit of a Function at a Point 15 min Let R be a positive number Suppose f is a function and a is a point such that the domain of f includes the open interval la a R The domain of f may or may not include the point 1 De nition A real number L is called a rightesided limit off at a if into every neighborhood of L the function f maps some rightsided neighborhood of a Note The intuition behind the concept of a limit is the problem of nding points 6 near and to the right of a such that is close to a given value L In other words we seek to approximate L by the values of fz and require the approximation to be accurate within tolerance e gt 0 Note A rightsided limit of f at a may not exist Examples 0 domain of f all z gt 0 R 111 0fz 11 0 domain of f all z 2 0 R 111 0 1 for z gt 0 and fa 0 Notation and Terminology lf L is a rightsided limit of f at a one writes limJr L 3 The notation 3 is often read fz tends to L as I approaches tends to a from the right77 Uniqueness of RightSided Limits If both L1 and L2 are rightsided limits of f at a then L1 L2 In short rightesided side limits are unique How to Verify That Right Sided Limits Are Unique To see this suppose the contrary ie that both L1 and L2 are right7sided limits of f at a but L1 7e L2 Then we could choose an 6 so small that the intervals L1 7 eL1e and L2 7 eL2 e do not overlap But then a point x in an interval aa 6 would have to map simultaneously into two non7overlapping intervals That is f would have to assume two di erent values at x This would violate the vertical line test for the function f The obtained contradiction shows that 1 How to Remember Why RightSided Limits Are Unique A function with two different rightsided limits at the same point would violate the vertical line test
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