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The nuclear spin–lattice relaxation rate in the PuMGa5 materials
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 634–636 www.elsevier.com/locate/jmmm
The nuclear spin–lattice relaxation rate in the PuMGa5 materials Yunkyu Banga,, M.J. Grafb, N.J. Currob, A.V. Balatskyb a
Department of Physics, Chonnam National University, Kwangju 500-757, Republic of Korea b Los Alamos National Laboratory, Los Alamos, NM 87545, USA Available online 3 November 2006
Abstract We examine the nuclear spin–lattice relaxation rates 1=T 1 of PuRhGa5 and PuCoGa5 , in particular, in normal state. PuRhGa5 exhibits a gradual suppression of ðT 1 TÞ1 below 25 K far above T c 8:5 K, while measurements for PuCoGa5 reveal a monotonic increase down to T c . We propose that this behavior is consistently understood by the crossover from the two-dimensional quantum antiferromagnetic regime of the local 5f-electron spins of Pu to the concomitant formation of the fermion pseudogap based on the twocomponent spin-fermion model. r 2006 Elsevier B.V. All rights reserved. PACS: 74.70.Tx; 75.20.Hr; 76.60k Keywords: Pseudogap; Heavy fermion; NMR
1. Introduction The discovery of superconductivity (SC) in plutoniumbased compounds [1,2] such as PuCoGa5 and PuRhGa5 has stimulated the study of unconventional superconductivity in the f-electron systems. Recently, Curro et al. [3] have established, based on their measurements of the Knight shift and spin–lattice relaxation rates, that the pairing symmetry of PuCoGa5 is a d-wave type. With a record high superconducting transition temperature T c of the order of 20 K and its unconventional pairing symmetry in PuCoGa5 , this class of materials provides hope for a unifying pairing mechanism from heavy fermion superconductors to the high-T c cuprates [3,4]. More recently, Sakai et al. [5] have measured 1=T 1 in PuRhGa5 and found the universal T 3 law in 1=T 1 in superconducting phase indicating a d-wave type gap just as in PuCoGa5 but with a lower T c 8:5 K. However, its normal state 1=T 1 behaves qualitatively different than in PuCoGa5 . In this work we argue that there is a pseudogap Corresponding author. Asia Pacific Center for Theoretical Physics, Pohang 790-784, Republic of Korea. Tel.: +82 62 5303363. E-mail address: [email protected] (Y. Bang).
0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.277
(PG) formed in PuRhGa5 above T c in a range of 20 K, which is again an intermediate energy scale compared with the PG energies of HTSC and heavy fermion (HF) superconductors. 2. 1=T 1 in normal state In Fig. 1, we plot the measured 1=T 1 T for PuCoGa5 and PuRhGa5 versus temperature for normal state only. The succinct features are: (1) at high temperatures (T\25 KÞ both data show T 1 behavior; (2) 1=T 1 T in PuCoGa5 shows monotonic increase up to T c ; (3) the most interesting feature of the curve is the gradual round-off of 1=T 1 T in the PuRhGa5 data below 25 K far above T c ¼ 8:5 K. This roundoff in 1=T 1 T starting above T c has been frequently observed in underdoped HTSC and attributed to the suppression of low-energy spin fluctuations associated with the PG behavior. Recently, several heavy fermion compounds also showed such a PG behavior but only in a very narrow temperature range of a few Kelvin— however, the HF cases provide more convincing evidence for its magnetic origin. In this work, as an attempt to explain the PG in PuRhGa5 , we proposed the two-component spin-fermion
ARTICLE IN PRESS Y. Bang et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 634–636
1/T1T (sec K)-1
1/T1T (sec K)-1
10
5
3.6
3.2 0
20 T (K)
40
0 0
40
80
120
160
T (K) Fig. 1. Plot of 1=T 1 T versus T for the normal state for PuRhGa5 (open diamonds) and PuCoGa5 (open circles) and the theoretical fits. Parameters for the theoretical fits are DSG ¼ 20 K and DFPG ¼ 8 K for PuRhGa5 and DSG ¼ 0 K and DFPG ¼ 8 K, respectively. Inset: close-up view for PuRhGa5 at low temperatures. Theoretical fits with (dotted line) and without (solid line) the fermion pseudogap (FPG) correction are shown for comparison. (PuRhGa5 data are from Ref. [4].)
model [6], as follows: X X H¼ cya ðkÞeðkÞca ðkÞ þ JSðrÞ sðrÞ þ H S .
(1)
r
k;a
The last term H S represents an effective low-energy Hamiltonian for the local spins and the local spins SðrÞ interact with conduction electron spins ðsÞ through the second term. When the local spins have a short range AFM correlation, the spin correlation function has the general form wðq; oÞ ¼
wðQ; 0Þ , 1 þ x jq Qj2 o2 =D2SG io=o ¯ 2
(2)
where DSG is the spin gap, Q the AFM ordering vector, x the magnetic correlation length, and o ¯ the spin relaxation energy scale, which comes from Landau damping of the fermionic sector. With the above form of wðq; oÞ and assuming the 2D AFM correlation, it can be shown that 1=T 1 T1=o ¯ [7]. Further, it is known that ox ¯ 1 for the z ¼ 1 quantum-critical phase of the 2D quantum antiferromagnet (QAFM) and x1 displays the following behavior [8]: ( T for quantum criticality ðQCÞ ðT4T Þ; x1 const: for quantum disorder ðQDÞ ðToT Þ: (3)
635
Putting together all these and assuming that PuRhGa5 crossovers from the QC to the QD regime at around T DSG 20 K, we plot this theoretical result (solid line) with experimental data (open diamonds) in the inset of Fig. 1. While the result describes the QC regime at high temperatures and the crossover to the QD regime, it is still not sufficient to explain the additional fall-off of 1=T 1 T from 20 K to T c 8 K. We propose that this additional suppression of the spin-fluctuations is caused by the suppression of the fermion DOS. Since the term io=o ¯ in Eq. (2) originates from Landau damping of the fermionic sector, more correct form of it is ioNðE F ; TÞ=o. ¯ With a strong coupling between fermions and the spin-fluctuations, the fermion DOS NðE F ; TÞ develops a gap like structure (fermion pseudogap (FPG), DFPG ), and then it leads to the additional suppression in 1=T 1 T NðE F ; T; DFPG Þ=oðTÞ. ¯ Combining the FPG DFPG and the magnetic correlation of the 2D QAFM, we can write 1=T 1 T in the twocomponent spin-fermion model as " !# DFPG 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=T 1 T ¼ AN 0 1 tanh ð4Þ 2 T 2 þ G2 1 þ B. ½DSG þ T expð4DSG =TÞ The first factor N 0 ½1 is a phenomenological form of the fermion DOS NðE F ; T; DFPG Þ with DFPG and damping rate G. The second factor 1=½DSG þ is a smooth crossover function [8] for xðTÞ describing the QC to QD behavior of Eq. (3) with the spin gap DSG T . B is a constant describing a temperature independent contribution and A is an overall constant. Using this formula, we fit the experimental data of PuCoGa5 and PuRhGa5 in normal state in Fig. 1. Two key fitting parameters provide estimates for the important energy scales of this phenomenological model. For PuRhGa5 , we used DSG ¼ 20 K and DFPG ¼ 8 K; the damping rate G is not a very sensitive model parameter, so we use G ¼ 25 K in all cases. For PuCoGa5 the monotonically increasing 1=T 1 T at lower temperatures implies increasing xðTÞ and stronger magnetic correlations than in PuRhGa5 . Therefore, we use DFPG ¼ 8 K and DSG ¼ 0 K for PuCoGa5 . In summary, with the proposed two component spinfermion model, we found that the two gap energy scales (DFPG and DSG ) naturally appears in the PG region. We obtain a successful fitting of 1=T 1 T data both for PuCoGa5 and PuRhGa5 in normal phase. Acknowledgments We thank John Sarrao, Joe Thompson, David Pines and Eric Bauer for many stimulating discussions. Y.B. was supported by the KOSEF through the CSCMR and the Grant no. KRF-2005-070-C00044. This research was supported by the US Department of Energy at Los Alamos National Laboratory under contract No. W-7405-ENG-36.
ARTICLE IN PRESS Y. Bang et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 634–636
636
References [1] [2] [3] [4]
J.L. Sarrao, et al., Nature (London) 420 (2002) 297. F. Wastin, et al., J. Phys. Condens. Matter 15 (2003) S2279. N.J. Curro, et al., Nature (London) 434 (2005) 622. Y. Bang, A.V. Balatsky, F. Wastin, J.D. Thompson, Phys. Rev. B 70 (2004) 104512.
[5] H. Sakai, et al., J. Phys. Soc. Japan 74 (2005) 1710. [6] Y. Bang, I. Martin, A.V. Balatsky, Phys. Rev. B 66 (2002) 224501. [7] A. Sokol, D. Pines, Phys. Rev. Lett. 71 (1993) 2813. [8] S. Chakravarty, B.I. Halperin, D.R. Nelson, Phys. Rev. B 39 (1989) 2344.
Journal of Magnetism and Magnetic Materials 310 (2007) 634–636 www.elsevier.com/locate/jmmm
The nuclear spin–lattice relaxation rate in the PuMGa5 materials Yunkyu Banga,, M.J. Grafb, N.J. Currob, A.V. Balatskyb a
Department of Physics, Chonnam National University, Kwangju 500-757, Republic of Korea b Los Alamos National Laboratory, Los Alamos, NM 87545, USA Available online 3 November 2006
Abstract We examine the nuclear spin–lattice relaxation rates 1=T 1 of PuRhGa5 and PuCoGa5 , in particular, in normal state. PuRhGa5 exhibits a gradual suppression of ðT 1 TÞ1 below 25 K far above T c 8:5 K, while measurements for PuCoGa5 reveal a monotonic increase down to T c . We propose that this behavior is consistently understood by the crossover from the two-dimensional quantum antiferromagnetic regime of the local 5f-electron spins of Pu to the concomitant formation of the fermion pseudogap based on the twocomponent spin-fermion model. r 2006 Elsevier B.V. All rights reserved. PACS: 74.70.Tx; 75.20.Hr; 76.60k Keywords: Pseudogap; Heavy fermion; NMR
1. Introduction The discovery of superconductivity (SC) in plutoniumbased compounds [1,2] such as PuCoGa5 and PuRhGa5 has stimulated the study of unconventional superconductivity in the f-electron systems. Recently, Curro et al. [3] have established, based on their measurements of the Knight shift and spin–lattice relaxation rates, that the pairing symmetry of PuCoGa5 is a d-wave type. With a record high superconducting transition temperature T c of the order of 20 K and its unconventional pairing symmetry in PuCoGa5 , this class of materials provides hope for a unifying pairing mechanism from heavy fermion superconductors to the high-T c cuprates [3,4]. More recently, Sakai et al. [5] have measured 1=T 1 in PuRhGa5 and found the universal T 3 law in 1=T 1 in superconducting phase indicating a d-wave type gap just as in PuCoGa5 but with a lower T c 8:5 K. However, its normal state 1=T 1 behaves qualitatively different than in PuCoGa5 . In this work we argue that there is a pseudogap Corresponding author. Asia Pacific Center for Theoretical Physics, Pohang 790-784, Republic of Korea. Tel.: +82 62 5303363. E-mail address: [email protected] (Y. Bang).
0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.277
(PG) formed in PuRhGa5 above T c in a range of 20 K, which is again an intermediate energy scale compared with the PG energies of HTSC and heavy fermion (HF) superconductors. 2. 1=T 1 in normal state In Fig. 1, we plot the measured 1=T 1 T for PuCoGa5 and PuRhGa5 versus temperature for normal state only. The succinct features are: (1) at high temperatures (T\25 KÞ both data show T 1 behavior; (2) 1=T 1 T in PuCoGa5 shows monotonic increase up to T c ; (3) the most interesting feature of the curve is the gradual round-off of 1=T 1 T in the PuRhGa5 data below 25 K far above T c ¼ 8:5 K. This roundoff in 1=T 1 T starting above T c has been frequently observed in underdoped HTSC and attributed to the suppression of low-energy spin fluctuations associated with the PG behavior. Recently, several heavy fermion compounds also showed such a PG behavior but only in a very narrow temperature range of a few Kelvin— however, the HF cases provide more convincing evidence for its magnetic origin. In this work, as an attempt to explain the PG in PuRhGa5 , we proposed the two-component spin-fermion
ARTICLE IN PRESS Y. Bang et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 634–636
1/T1T (sec K)-1
1/T1T (sec K)-1
10
5
3.6
3.2 0
20 T (K)
40
0 0
40
80
120
160
T (K) Fig. 1. Plot of 1=T 1 T versus T for the normal state for PuRhGa5 (open diamonds) and PuCoGa5 (open circles) and the theoretical fits. Parameters for the theoretical fits are DSG ¼ 20 K and DFPG ¼ 8 K for PuRhGa5 and DSG ¼ 0 K and DFPG ¼ 8 K, respectively. Inset: close-up view for PuRhGa5 at low temperatures. Theoretical fits with (dotted line) and without (solid line) the fermion pseudogap (FPG) correction are shown for comparison. (PuRhGa5 data are from Ref. [4].)
model [6], as follows: X X H¼ cya ðkÞeðkÞca ðkÞ þ JSðrÞ sðrÞ þ H S .
(1)
r
k;a
The last term H S represents an effective low-energy Hamiltonian for the local spins and the local spins SðrÞ interact with conduction electron spins ðsÞ through the second term. When the local spins have a short range AFM correlation, the spin correlation function has the general form wðq; oÞ ¼
wðQ; 0Þ , 1 þ x jq Qj2 o2 =D2SG io=o ¯ 2
(2)
where DSG is the spin gap, Q the AFM ordering vector, x the magnetic correlation length, and o ¯ the spin relaxation energy scale, which comes from Landau damping of the fermionic sector. With the above form of wðq; oÞ and assuming the 2D AFM correlation, it can be shown that 1=T 1 T1=o ¯ [7]. Further, it is known that ox ¯ 1 for the z ¼ 1 quantum-critical phase of the 2D quantum antiferromagnet (QAFM) and x1 displays the following behavior [8]: ( T for quantum criticality ðQCÞ ðT4T Þ; x1 const: for quantum disorder ðQDÞ ðToT Þ: (3)
635
Putting together all these and assuming that PuRhGa5 crossovers from the QC to the QD regime at around T DSG 20 K, we plot this theoretical result (solid line) with experimental data (open diamonds) in the inset of Fig. 1. While the result describes the QC regime at high temperatures and the crossover to the QD regime, it is still not sufficient to explain the additional fall-off of 1=T 1 T from 20 K to T c 8 K. We propose that this additional suppression of the spin-fluctuations is caused by the suppression of the fermion DOS. Since the term io=o ¯ in Eq. (2) originates from Landau damping of the fermionic sector, more correct form of it is ioNðE F ; TÞ=o. ¯ With a strong coupling between fermions and the spin-fluctuations, the fermion DOS NðE F ; TÞ develops a gap like structure (fermion pseudogap (FPG), DFPG ), and then it leads to the additional suppression in 1=T 1 T NðE F ; T; DFPG Þ=oðTÞ. ¯ Combining the FPG DFPG and the magnetic correlation of the 2D QAFM, we can write 1=T 1 T in the twocomponent spin-fermion model as " !# DFPG 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=T 1 T ¼ AN 0 1 tanh ð4Þ 2 T 2 þ G2 1 þ B. ½DSG þ T expð4DSG =TÞ The first factor N 0 ½1 is a phenomenological form of the fermion DOS NðE F ; T; DFPG Þ with DFPG and damping rate G. The second factor 1=½DSG þ is a smooth crossover function [8] for xðTÞ describing the QC to QD behavior of Eq. (3) with the spin gap DSG T . B is a constant describing a temperature independent contribution and A is an overall constant. Using this formula, we fit the experimental data of PuCoGa5 and PuRhGa5 in normal state in Fig. 1. Two key fitting parameters provide estimates for the important energy scales of this phenomenological model. For PuRhGa5 , we used DSG ¼ 20 K and DFPG ¼ 8 K; the damping rate G is not a very sensitive model parameter, so we use G ¼ 25 K in all cases. For PuCoGa5 the monotonically increasing 1=T 1 T at lower temperatures implies increasing xðTÞ and stronger magnetic correlations than in PuRhGa5 . Therefore, we use DFPG ¼ 8 K and DSG ¼ 0 K for PuCoGa5 . In summary, with the proposed two component spinfermion model, we found that the two gap energy scales (DFPG and DSG ) naturally appears in the PG region. We obtain a successful fitting of 1=T 1 T data both for PuCoGa5 and PuRhGa5 in normal phase. Acknowledgments We thank John Sarrao, Joe Thompson, David Pines and Eric Bauer for many stimulating discussions. Y.B. was supported by the KOSEF through the CSCMR and the Grant no. KRF-2005-070-C00044. This research was supported by the US Department of Energy at Los Alamos National Laboratory under contract No. W-7405-ENG-36.
ARTICLE IN PRESS Y. Bang et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 634–636
636
References [1] [2] [3] [4]
J.L. Sarrao, et al., Nature (London) 420 (2002) 297. F. Wastin, et al., J. Phys. Condens. Matter 15 (2003) S2279. N.J. Curro, et al., Nature (London) 434 (2005) 622. Y. Bang, A.V. Balatsky, F. Wastin, J.D. Thompson, Phys. Rev. B 70 (2004) 104512.
[5] H. Sakai, et al., J. Phys. Soc. Japan 74 (2005) 1710. [6] Y. Bang, I. Martin, A.V. Balatsky, Phys. Rev. B 66 (2002) 224501. [7] A. Sokol, D. Pines, Phys. Rev. Lett. 71 (1993) 2813. [8] S. Chakravarty, B.I. Halperin, D.R. Nelson, Phys. Rev. B 39 (1989) 2344.