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Non-Fermi liquid behavior for U ions in cubic crystals
Physica B 281&282 (2000) 402}403
Non-Fermi liquid behavior for U ions in cubic crystals Mikito Koga!,*, Gergely ZaraH nd!,", Daniel L. Cox! !Department of Physics, University of California Davis, CA 95616, USA "Research Group of the Hungarian Academy of Sciences, Institute of Physics, TU Budapest, H-1521, Hungary
Abstract Various non-Fermi liquid (NFL) scenarios are studied using renormalization group methods for a nominally tetravalent (5f2) U ion in a cubic environment. Using an Anderson model, we "nd a small energy scale characterizing a crossover from an ionic spectrum to a NFL excitation spectrum, even in the mixed-valence regime. This scale is connected to the competition of two NFL "xed points (FPs): A two-channel Kondo FP dominated by a quadrupolar f2 doublet, and a novel FP dominated by an f2 triplet. For the latter, which may be relevant for uranium alloys like U Y Pd and UCu Pd , a quadrupolar exchange interaction gives rise to an intermediate temperature power law x 1~x 3 5~x x dependence (&¹~0.4) in the magnetic susceptibility. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Novel Kondo e!ect; Non-Fermi liquid; Numerical renormalization group
We study the Anderson model H"+ e cs c #+ E DMTSMD k km km M km M # < + (C cs DaTSMD#h.c.), (1) a,m_M km kmaM where the f2 states DMT represent the ! quadrupolar 3 doublet and ! triplet states coupled to the f1 ! excited 4 7 state DaT (E ,0) via the hybridization < with the a ! conduction electrons. The operator cs (c ) creates 8 km km (annihilates) a ! conduction electron with wave number 8 k, and kinetic energy e , and the C 's denote k a,m_M Clebsch}Gordan coe$cients. Using the numerical renormalization group method we identify two stable non-Fermi liquid (NFL) "xed points (FPs): The "rst corresponds to the two-channel Kondo model (2CKFP), and the low-lying excitations can be described by assuming coupling of only the ! 3 local moment to the conduction electrons. The second FP is dominated by the ! local moment (see below), and 4 it may also emerge, whenever the f2 ! triplet is the most 5 stable. The stability of the FPs depends on whether the
* Corresponding author. Fax: #1-530-752-4717. E-mail address: [email protected] (M. Koga)
bare crystal-"eld splitting D"E 4 !E 3 is above or ! ! below a critical value D . For D+D the two FPs com# # pete with each other and a crossover temperature ¹H appears [1]. For the crossover to the ! NFL state 4 3 we "nd ¹H&(DH)7@2 (see Fig. 1), where the e!ective 4 crystal-"eld splitting DH is associated with the occupancy weights of f-electrons nH and nH in the ! doublet and 3 4 3 ! triplet at the FP, respectively, DH"D ln (3nH/2nH) 4 0 3 4 (with D a non-critical factor), and it vanishes at the 0 critical point D . Fig. 2 shows the scaling of the non# linear magnetic susceptibility at the 2CKFP (EI & M E /D, CI &<2/D, with 2D the band width). A one paramM eter scaling is also obtained for the Zeeman part of the magnetic susceptibility dominated by the ! magnetic 7 excited doublet even if f2 is more stable than f1 (0.5(nH#nH[0.8). The trend of negative s(3) is in good 3 4 *.1 agreement with the non-linear susceptibility observed in UBe with the small energy scale of around 10 K. 13 The other NFL state's low-energy physics is described by the Kondo model H"+ e cs c k km km km # + cs c [J (S ) ) S#J (Q ) ) Q]. (2) k{m{ km D # m{m Q # m{m kk{mm{ Here S "3 (S"1) is the spin of the conduction elec# 2 trons (the local triplet), m, m@"$1,$3, and a potential 2 2
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 8 3 7 - 6
M. Koga et al. / Physica B 281&282 (2000) 402}403
Fig. 1. Relation between the small energy scale ¹H and the 4 e!ective crystal-"eld splitting DH for CI "0.10 ("), 0.15 (h) and 0.20 (e), for "xed EI 3 at ! 0.200. !
Fig. 2. Non-linear magnetic susceptibility of the two-channel Kondo FP. The curves are for CI "0.10, EI 3 "! 0.200; and ! EI 4 "! 0.160 (e), ! 0.165 (#), ! 0.170 (h), ! 0.175 (]). ! ¹ +10¹H. 4 4
scattering term is neglected. The quadrupolar operators are given by MQi, i"1,2,5N"MS S #S S , S S #S S , y z z y z x x z S S #S S , S2 !S2, (2S2!S2!S2)/J3N. The relx y y x x y z x y evance of the quadrupolar coupling J in Eq. (2) can be Q immediately seen from a multiplicative RG procedure [2]. In the absence of quadrupolar exchange the model scales to the dipolar FP `Da (see Fig. 3). This FP has been shown to be characterized by a critical exponent D"1 associated with its spin sector [3}5]. Obviously, 6 this dipolar FP is unstable to quadrupolar perturbations and for any non-zero J it #ows to a new FP `Sa with Q a pseudo S;(3) symmetry [2]. As shown in Fig. 4, for small to intermediate Kondo couplings, all impurity susceptibilities s(¹) collapse to a single-universal curve: At high temperatures s behaves according to the Curie} Weiss law ¹~a (a"1). For lower temperatures an intermediate region * speci"c to the presence of the quadrupolar coupling * appears with sK¹~0.4. As the temperature decreases further, s(¹) turns up and scales as &¹~2@3 near the novel "xed point. For larger couplings deviations from the universal scaling occur: the
403
Fig. 3. Scaling trajectories of the triplet model.
Fig. 4. Rescaled magnetic susceptibility of the triplet FP.
Curie}Weiss part is absent, s starts as &¹~1@3 at high ¹, and a gradually increases to 2 at low ¹. Photoemisson 3 data suggest that UCu Pd is in the extreme mixed 5~x x valence regime, compatible with a strong coupling limit of the Kondo model. We propose that the intermediate temperature range susceptibility may correspond to the `impuritya range identi"ed for this material [6]. The model may prove relevant to the alloy Y U Pd as 1~x x 3 well [2].
References [1] M. Koga, D.L. Cox, Phys. Rev. Lett. 82 (1999) 2575. [2] M. Koga, G. ZaraH nd, D.L. Cox, Phys. Rev. Lett. 83 (1999) 2421, and references therein. [3] T.-S. Kim, L.N. Oliveira, D.L. Cox, Phys. Rev. B 55 (1997) 12460. [4] M. Fabrizio, G. ZaraH nd, Phys. Rev. B 54 (1996) 10008. [5] A. Sengupta, Y.B. Kim, Phys. Rev. B 54 (1997) 14918. [6] M.C. Aronson et al., Europhys. Lett. 40 (1997) 245.
Non-Fermi liquid behavior for U ions in cubic crystals Mikito Koga!,*, Gergely ZaraH nd!,", Daniel L. Cox! !Department of Physics, University of California Davis, CA 95616, USA "Research Group of the Hungarian Academy of Sciences, Institute of Physics, TU Budapest, H-1521, Hungary
Abstract Various non-Fermi liquid (NFL) scenarios are studied using renormalization group methods for a nominally tetravalent (5f2) U ion in a cubic environment. Using an Anderson model, we "nd a small energy scale characterizing a crossover from an ionic spectrum to a NFL excitation spectrum, even in the mixed-valence regime. This scale is connected to the competition of two NFL "xed points (FPs): A two-channel Kondo FP dominated by a quadrupolar f2 doublet, and a novel FP dominated by an f2 triplet. For the latter, which may be relevant for uranium alloys like U Y Pd and UCu Pd , a quadrupolar exchange interaction gives rise to an intermediate temperature power law x 1~x 3 5~x x dependence (&¹~0.4) in the magnetic susceptibility. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Novel Kondo e!ect; Non-Fermi liquid; Numerical renormalization group
We study the Anderson model H"+ e cs c #+ E DMTSMD k km km M km M # < + (C cs DaTSMD#h.c.), (1) a,m_M km kmaM where the f2 states DMT represent the ! quadrupolar 3 doublet and ! triplet states coupled to the f1 ! excited 4 7 state DaT (E ,0) via the hybridization < with the a ! conduction electrons. The operator cs (c ) creates 8 km km (annihilates) a ! conduction electron with wave number 8 k, and kinetic energy e , and the C 's denote k a,m_M Clebsch}Gordan coe$cients. Using the numerical renormalization group method we identify two stable non-Fermi liquid (NFL) "xed points (FPs): The "rst corresponds to the two-channel Kondo model (2CKFP), and the low-lying excitations can be described by assuming coupling of only the ! 3 local moment to the conduction electrons. The second FP is dominated by the ! local moment (see below), and 4 it may also emerge, whenever the f2 ! triplet is the most 5 stable. The stability of the FPs depends on whether the
* Corresponding author. Fax: #1-530-752-4717. E-mail address: [email protected] (M. Koga)
bare crystal-"eld splitting D"E 4 !E 3 is above or ! ! below a critical value D . For D+D the two FPs com# # pete with each other and a crossover temperature ¹H appears [1]. For the crossover to the ! NFL state 4 3 we "nd ¹H&(DH)7@2 (see Fig. 1), where the e!ective 4 crystal-"eld splitting DH is associated with the occupancy weights of f-electrons nH and nH in the ! doublet and 3 4 3 ! triplet at the FP, respectively, DH"D ln (3nH/2nH) 4 0 3 4 (with D a non-critical factor), and it vanishes at the 0 critical point D . Fig. 2 shows the scaling of the non# linear magnetic susceptibility at the 2CKFP (EI & M E /D, CI &<2/D, with 2D the band width). A one paramM eter scaling is also obtained for the Zeeman part of the magnetic susceptibility dominated by the ! magnetic 7 excited doublet even if f2 is more stable than f1 (0.5(nH#nH[0.8). The trend of negative s(3) is in good 3 4 *.1 agreement with the non-linear susceptibility observed in UBe with the small energy scale of around 10 K. 13 The other NFL state's low-energy physics is described by the Kondo model H"+ e cs c k km km km # + cs c [J (S ) ) S#J (Q ) ) Q]. (2) k{m{ km D # m{m Q # m{m kk{mm{ Here S "3 (S"1) is the spin of the conduction elec# 2 trons (the local triplet), m, m@"$1,$3, and a potential 2 2
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 8 3 7 - 6
M. Koga et al. / Physica B 281&282 (2000) 402}403
Fig. 1. Relation between the small energy scale ¹H and the 4 e!ective crystal-"eld splitting DH for CI "0.10 ("), 0.15 (h) and 0.20 (e), for "xed EI 3 at ! 0.200. !
Fig. 2. Non-linear magnetic susceptibility of the two-channel Kondo FP. The curves are for CI "0.10, EI 3 "! 0.200; and ! EI 4 "! 0.160 (e), ! 0.165 (#), ! 0.170 (h), ! 0.175 (]). ! ¹ +10¹H. 4 4
scattering term is neglected. The quadrupolar operators are given by MQi, i"1,2,5N"MS S #S S , S S #S S , y z z y z x x z S S #S S , S2 !S2, (2S2!S2!S2)/J3N. The relx y y x x y z x y evance of the quadrupolar coupling J in Eq. (2) can be Q immediately seen from a multiplicative RG procedure [2]. In the absence of quadrupolar exchange the model scales to the dipolar FP `Da (see Fig. 3). This FP has been shown to be characterized by a critical exponent D"1 associated with its spin sector [3}5]. Obviously, 6 this dipolar FP is unstable to quadrupolar perturbations and for any non-zero J it #ows to a new FP `Sa with Q a pseudo S;(3) symmetry [2]. As shown in Fig. 4, for small to intermediate Kondo couplings, all impurity susceptibilities s(¹) collapse to a single-universal curve: At high temperatures s behaves according to the Curie} Weiss law ¹~a (a"1). For lower temperatures an intermediate region * speci"c to the presence of the quadrupolar coupling * appears with sK¹~0.4. As the temperature decreases further, s(¹) turns up and scales as &¹~2@3 near the novel "xed point. For larger couplings deviations from the universal scaling occur: the
403
Fig. 3. Scaling trajectories of the triplet model.
Fig. 4. Rescaled magnetic susceptibility of the triplet FP.
Curie}Weiss part is absent, s starts as &¹~1@3 at high ¹, and a gradually increases to 2 at low ¹. Photoemisson 3 data suggest that UCu Pd is in the extreme mixed 5~x x valence regime, compatible with a strong coupling limit of the Kondo model. We propose that the intermediate temperature range susceptibility may correspond to the `impuritya range identi"ed for this material [6]. The model may prove relevant to the alloy Y U Pd as 1~x x 3 well [2].
References [1] M. Koga, D.L. Cox, Phys. Rev. Lett. 82 (1999) 2575. [2] M. Koga, G. ZaraH nd, D.L. Cox, Phys. Rev. Lett. 83 (1999) 2421, and references therein. [3] T.-S. Kim, L.N. Oliveira, D.L. Cox, Phys. Rev. B 55 (1997) 12460. [4] M. Fabrizio, G. ZaraH nd, Phys. Rev. B 54 (1996) 10008. [5] A. Sengupta, Y.B. Kim, Phys. Rev. B 54 (1997) 14918. [6] M.C. Aronson et al., Europhys. Lett. 40 (1997) 245.