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Interfering antiferromagnetism and superconductivity in quasi-one-dimensional organic conductors
ARTICLE IN PRESS Physica B 405 (2010) S89–S91
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Interfering antiferromagnetism and superconductivity in quasi-one-dimensional organic conductors A. Sedeki a, D. Bergeron a, C. Bourbonnais a,b, a b
´seau Que´be ´cois sur les Mate´riaux de Pointe, De´partement de Physique, Universite´ de Sherbrooke, Sherbrooke, QC, Canada J1K-2R1 Re Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8
a r t i c l e in fo
abstract
Keywords: Antiferromagnetism Organic superconductivity Nuclear relaxation Electron–electron scattering rate
In this short note we summarize recent results obtained by the renormalization group approach to quasi-one-dimensional electron gas model. The approach is applied to the Bechgaard salts series ðTMTSFÞ2 X and the results are shown to give a satisfactory account of the interdependence between antiferromagnetism and superconductivity featured by their phase diagram, the anomalous enhancement of the nuclear relaxation and the electron–electron scattering rate under pressure. & 2009 Elsevier B.V. All rights reserved.
The Bechgaard salts have been among the first electron systems to show superconductivity in the close vicinity of itinerant antiferromagnetism or a spin-density-wave (SDW) state [1]. The attempt to understand on theoretical grounds the physics that governs the phase diagram of the Bechgaard salts soon raised the central issue of a non-conventional mechanism that would directly involve antiferromagnetic spin fluctuations in the origin of superconductivity [2–5]. The theoretical progress achieved since then in the framework of the quasi-one-dimensional electron gas model [6–10], has emphasized that in order to assure internal consistency, the model of Cooper pairing must conversely account for the behavior of spin fluctuations whose presence is ubiquitous in the phase diagram of these materials [11]. The experimental evidence for a mutual influence of Cooper pairing and spin correlations coming from the known nuclear magnetic resonance data and the latest electrical transport measurements under pressure corroborates this view [12,13]. The theoretical prediction of the one-loop weak coupling renormalization group (RG) approach to the phase diagram of quasi-one-dimensional electron gas model is shown in Fig. 1 [9,10]. The model is defined by an electron system with an open Fermi surface having a longitudinal Fermi energy EF an order of magnitude larger than the transverse hopping integral t? , in its turn much larger than the next to nearest-neighbor interchain 0 . The latter parametrizes nesting hopping integral denoted by t? deviations, which stands as the net effect of pressure in the model. Electrons interact through weak repulsive couplings and small Umklapp scattering [6,9,10,14]. Corresponding author at: Re´seau Que´be´cois sur les Mate´riaux de Pointe, De´partement de Physique, Universite´ de Sherbrooke, Sherbrooke, QC, Canada J1K-2R1. E-mail address: [email protected] (C. Bourbonnais).
0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.11.056
For band parameters and weak repulsive interactions compatible with experiments in the Bechgaard salts, the singularities and behaviors of various response functions wm allow the determination of the characteristic scales of the phase diagram 0 , nesting deviations are shown in Fig. 1. Thus at low ‘pressure’ or t? weak and wSDW is singular at TSDW signaling an instability of the 0 is raised, metal against a spin-density-wave (SDW) state. As t? TSDW is monotonically decreasing to ultimately dip sharply at the 0 . TSDW then merges with an approach of some critical value t? other instability line of the superconducting d-wave channel at Tc as a consequence of finite positive interference between density0 0 is increased above t? , the wave and Cooper pairings. When t? amplitude of spin fluctuations is further decreased and reduces the Cooper pairing interaction and then Tc , which goes down ‘under pressure’. The resulting transition lines capture the key features of the phase diagram of the Bechgaard salts under pressure [12,15]. Owing to the mutual reinforcement of the two pairing channels, it has been found that despite the existence of a singlet 0 0 4t? , SDW correlations superconducting ground state at a fixed t? continue to grow when the temperature is lowered. This is shown by the SDW response that varies according to a Curie–Weiss law, wSDW ðTÞp1=ðT þ YÞ [10]. The CW behavior takes place below the characteristic temperature TCW 10 K, which is essentially 0 , whereas the CW scale Y, which starts from independent of t? 0 0 (Fig. 1). zero at t? , increases rapidly with t? Spin fluctuations have a well-defined influence on the NMR spin lattice relaxation rate T11 for which a fair amount of experimental evidence is available [16–19]. According to the Moriya expression T11 ¼ T
Z
jAq j2
w00 ðq; oÞ 3 d q; o
ARTICLE IN PRESS S90
A. Sedeki et al. / Physica B 405 (2010) S89–S91
400
T(K)
SDW
TCW
200
C-W
0
∂ℓ
Σ
G
=
Σ
100
``
Θ Metal
102
`
600
-200 SCd 15
20
t'⊥(K)
25
-400
30
-600
Fig. 1. RG phase diagram of the quasi-1D electron gas model. The dashed line stands as the C-W scale Y in the superconducting sector. The dotted line defines the temperature domain of the C–W behavior. After Ref. [10].
-1
T1 (s-1)
150
50 20
40
-1
T1 (s-1)
T(K) 0
50
0
10
0 ω(K)
5000
10000
100
0
0
-5000
Fig. 3. Real and Imaginary parts of the retarded one-particle self-energy as a 0 ¼ 27:7 K ðTc ¼ 0:7 K). Inset: function of the frequency o obtained at T ¼ 2 K for t? two-loop RG flow equation in the diagrammatic form for the one-particle selfenergy.
150
100
-10000
20
30
40
50
T(K) Fig. 2. Calculated temperature dependence of the nuclear relaxation rate for 0 0 0 various ‘pressures’ t? , from the SDW (t? o t? , top curves) to the superconducting 0 0 4 t? , bottom curves). Inset: experimental results of Creuzet et al. After region (t? [10,16].
T11 is coupled by means of the hyperfine coupling to the imaginary part of the electronic spin response w00 ðq; oÞ for the low frequency spectrum of spin correlations at the wave vector q. As shown in Refs. [10,20], the relaxation rate superimposes a ‘Fermi liquid’ like component T11 C0 T, which dominates at high temperature and an antiferromagnetic piece, T11 C1 T wSDW ðTÞ, which is linked to the SDW response calculated by the RG method. The sum of the two contributions is shown in Fig. 2. As one moves along the ‘pressure’ axis of Fig. 1, the divergence of T11 , which is a signature of a SDW state, transformed into an anomalous enhancement in the superconducting sector of the 0 0 4t? . The amplitude of the anomaly is phase diagram when t? strongly ‘pressure’ dependent and decays as Tc decreases. A large part of the anomaly below TCW fits pretty well the Curie–Weiss form T1 T T þ Y [10], as first pointed out on experimental grounds by Brown et al. [18,19]. The overall calculated temperature profile of T11 reproduces the key features shown by experiments (inset of Fig. 2) [16,19]. We consider now the one-particle spectral properties of the model in the normal state of the superconducting sector of the phase diagram. For this purpose, RG calculations have to include two-loop diagrams for the flow equation of the inverse oneparticle Green function G1 ¼ G1 0 S (inset Fig. 3). The oneparticle self-energy SðkF ; on Þ can be computed as a function of
discrete Matsubara frequencies on . This quantity is analytically continued on the real frequency axis using a N-points Pade´ approximants procedure that finally leads to the real (S0 ðoÞ) and the imaginary (S00 ðoÞ) parts of the retarded one-particle selfenergy. In the superconducting sector of the phase diagram where 0 0 4t? , typical low temperature results of the analytical cont? 0 ¼ 27:7 K tinuation are shown in Fig. 3 at T ¼ 2 K for t? (Tc ¼ 0:65 K). Apart from the characteristic structure seen at high frequency for both quantities, clear features emerge in the relevant low frequency domain jojo t? , corresponding to energy region of coherent transverse single particle motion. The real part S0 ðoÞ shows a typical linear zero crossing with a negative slope as o-0, indicative of a finite quasi-particle weight zðkF Þ at the Fermi point—calculated here at kF ¼ ðkF ; k? ¼ 0ÞFfor the finite temperature considered. The imaginary part SðoÞ develops accordingly a sharp peak. The expression for minus the imaginary part of the self-energy defines the frequency dependent electron–electron scattering rate 1=tðoÞ ¼ S00 ðoÞ evaluated at the Fermi wave vector kF . This quantity is particularly pertinent at low energy. A close-up view of the T ¼ 2 K trace of S00 ðoÞ in Fig. 4 at jojo t? reveals indeed an unusual linear behavior of the form S00 ðoÞ a0 ajoj, at variance with the o2 variation expected for a Fermi liquid. The finite, albeit reduced and temperature dependent quasi-particle weight zðkF Þ shown by the real part, is consistent with the form S0 ðo-0ÞpolnL=T, reminiscent of that of a marginal Fermi liquid [21]. As we increase the temperature, however, the linear behavior of the imaginary part is progressively replaced by a o2 variation in a range of frequency that increases in size with T. This is exemplified in Fig. 4, where S00 ðoÞ b0 bo2 at T ¼ 15 K 4TCW . Such a crossover in the frequency behavior is present over the 0 0 4t? , characterized by a slope a of the linear whole range of t? 0 is increased. contribution that shrinks in size as t? This remarkable feature of S00 ðoÞ will be also present in the temperature dependence of quantities like resistivity. Adopting a simple Boltzman picture for the electrical conductivity in which the scattering rate as a function of energy is 1=tðeÞ ¼ S00 ðo ¼ eÞ, the conductivity will scale as ds 1=T in the low temperature region where T 5TCW and 1=tðeÞ is linear in energy. This leads to a contribution in resistivity drðTÞ AT that is linear in temperature. As the temperature increases well above TCW , namely where
ARTICLE IN PRESS A. Sedeki et al. / Physica B 405 (2010) S89–S91
pairing, are responsible for the development of marginal Fermi liquid at low temperature.
0
-50
C.B. acknowledges fruitful and continuous collaboration with D. Je´rome, P. Auban-Senzier, N. Doiron-Leyraud, S. Rene´ de Cotret and L. Taillefer. This work has received the financial support from the National Science and Engineering Research Council of Canada (NSERC), Re´seau Que´bcois des Mate´riaux de Pointe (RQMP) and the Quantum materials program of Canadian Institute of Advanced Research (CIFAR). The authors are also thankful to the Re´seau Que´be´cois de Calcul Haute Performance (RQCHP) for supercomputer facilities at the Universite´ de Sherbrooke.
Σ
T = 15K
-100 T = 2K -150 −500
S91
References
0 ω(K)
500
Fig. 4. Imaginary parts of the self-energy (Fig. 3) in the low frequency region well below (2 K) and above (15 K) the Curie–Weiss temperature scale TCW .
1=tðeÞ picks up a quadratic low energy dependence and the resistivity acquires a Fermi liquid component dr BT 2 that varies quadratically with temperature. This crossover between the linear and quadratic temperature dependence captures the key characteristics that were shown to take place in the electrical resistivity of the normal state of the Bechgaard salts [12,13]. Furthermore in these experiments, it was pointed out that the amplitude of the linear term in resistivity is correlated to the size of Tc over the whole range of pressure. A similar correlation between the linear coefficient A and Tc follows from the RG calculations. By increasing the ‘pressure’, as one 0 axis in Fig. 1, an order of magnitude reduction moves along the t? of Tc is concomitant with a similar reduction in the linear coefficient A for resistivity. The suppression of A under pressure is thus closely related to the decrease of antiferromagnetic fluctuations, namely those connected to the enhancement of the nuclear relaxation rate seen by NMR [10]. From the spectral properties, these same fluctuations, which interfere positively with superconducting d-wave
[1] D. Je´rome, A. Mazaud, M. Ribault, K. Bechgaard, J. Phys. (Paris) Lett. 41 (1980) L95. [2] V.J. Emery, Synth. Met. 13 (1986) 21. [3] L.G. Caron, C. Bourbonnais, Physica 143B (1986) 453. [4] M.T. Be´al-Monod, C. Bourbonnais, V.J. Emery, Phys. Rev. B 34 (1986) 7716. [5] D.J. Scalapino, E. Loh, J.E. Hirsch, Phys. Rev. B 34 (1986) R8190. [6] R. Duprat, C. Bourbonnais, Eur. Phys. J. B 21 (2001) 219. [7] Y. Fuseya, Y. Suzumura, J. Phys. Soc. Japan 74 (2005) 1263. [8] J.C. Nickel, R. Duprat, C. Bourbonnais, N. Dupuis, Phys. Rev. Lett. 95 (2005) 247001. [9] J.C. Nickel, R. Duprat, C. Bourbonnais, N. Dupuis, Phys. Rev. B 73 (2006) 165126. [10] C. Bourbonnais, A. Sedeki, Phys. Rev. B 80 (2009) 085105. [11] C. Bourbonnais, D. Je´rome, in: A. Lebed (Ed.), The Physics of Organic Superconductors and Conductors. Springer Series in Materials Science, vol. 110, Springer, Heidelberg, 2008, p. 357 arXiv:cond-mat/0904.0617. [12] N. Doiron-Leyraud, P. Auban-Senzier, S. Rene´ de Cotret, R. Homier, A. Sedeki, C. Bourbonnais, D. Je´rome, K. Bechgaard, L. Taillefer, arXiv:cond-mat/ 0905.0964; Phys. Rev. B, 2009, in press. [13] N. Doiron-Leyraud, P. Auban-Senzier, S. Rene´ de Cotret, K. Bechgaard, D. Je´rome, L. Taillefer, this issue. [14] V.J. Emery, R. Bruinsma, S. Barisic, Phys. Rev. Lett. 48 (1982) 1039. [15] D. Je´rome, H. Schulz, Adv. Phys. 31 (1982) 299. [16] F. Creuzet, et al., Synth. Met. 19 (1987) 277. [17] P. Wzietek, et al., J. Phys. I 3 (1993) 171. [18] W. Wu, et al., Phys. Rev. Lett. 94 (2005) 097004. [19] S.E. Brown, P.M. Chaikin, M.J. Naughton, in: A. Lebed (Ed.), The Physics of Organic Superconductors and Conductors. Springer Series in Materials Science, vol. 110, Springer, Heidelberg, 2008, p. 49. [20] C. Bourbonnais, et al., Phys. Rev. Lett. 62 (1989) 1532. [21] C.M. Varma, et al., Phys. Rev. Lett. 63 (1989) 1996.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Interfering antiferromagnetism and superconductivity in quasi-one-dimensional organic conductors A. Sedeki a, D. Bergeron a, C. Bourbonnais a,b, a b
´seau Que´be ´cois sur les Mate´riaux de Pointe, De´partement de Physique, Universite´ de Sherbrooke, Sherbrooke, QC, Canada J1K-2R1 Re Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8
a r t i c l e in fo
abstract
Keywords: Antiferromagnetism Organic superconductivity Nuclear relaxation Electron–electron scattering rate
In this short note we summarize recent results obtained by the renormalization group approach to quasi-one-dimensional electron gas model. The approach is applied to the Bechgaard salts series ðTMTSFÞ2 X and the results are shown to give a satisfactory account of the interdependence between antiferromagnetism and superconductivity featured by their phase diagram, the anomalous enhancement of the nuclear relaxation and the electron–electron scattering rate under pressure. & 2009 Elsevier B.V. All rights reserved.
The Bechgaard salts have been among the first electron systems to show superconductivity in the close vicinity of itinerant antiferromagnetism or a spin-density-wave (SDW) state [1]. The attempt to understand on theoretical grounds the physics that governs the phase diagram of the Bechgaard salts soon raised the central issue of a non-conventional mechanism that would directly involve antiferromagnetic spin fluctuations in the origin of superconductivity [2–5]. The theoretical progress achieved since then in the framework of the quasi-one-dimensional electron gas model [6–10], has emphasized that in order to assure internal consistency, the model of Cooper pairing must conversely account for the behavior of spin fluctuations whose presence is ubiquitous in the phase diagram of these materials [11]. The experimental evidence for a mutual influence of Cooper pairing and spin correlations coming from the known nuclear magnetic resonance data and the latest electrical transport measurements under pressure corroborates this view [12,13]. The theoretical prediction of the one-loop weak coupling renormalization group (RG) approach to the phase diagram of quasi-one-dimensional electron gas model is shown in Fig. 1 [9,10]. The model is defined by an electron system with an open Fermi surface having a longitudinal Fermi energy EF an order of magnitude larger than the transverse hopping integral t? , in its turn much larger than the next to nearest-neighbor interchain 0 . The latter parametrizes nesting hopping integral denoted by t? deviations, which stands as the net effect of pressure in the model. Electrons interact through weak repulsive couplings and small Umklapp scattering [6,9,10,14]. Corresponding author at: Re´seau Que´be´cois sur les Mate´riaux de Pointe, De´partement de Physique, Universite´ de Sherbrooke, Sherbrooke, QC, Canada J1K-2R1. E-mail address: [email protected] (C. Bourbonnais).
0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.11.056
For band parameters and weak repulsive interactions compatible with experiments in the Bechgaard salts, the singularities and behaviors of various response functions wm allow the determination of the characteristic scales of the phase diagram 0 , nesting deviations are shown in Fig. 1. Thus at low ‘pressure’ or t? weak and wSDW is singular at TSDW signaling an instability of the 0 is raised, metal against a spin-density-wave (SDW) state. As t? TSDW is monotonically decreasing to ultimately dip sharply at the 0 . TSDW then merges with an approach of some critical value t? other instability line of the superconducting d-wave channel at Tc as a consequence of finite positive interference between density0 0 is increased above t? , the wave and Cooper pairings. When t? amplitude of spin fluctuations is further decreased and reduces the Cooper pairing interaction and then Tc , which goes down ‘under pressure’. The resulting transition lines capture the key features of the phase diagram of the Bechgaard salts under pressure [12,15]. Owing to the mutual reinforcement of the two pairing channels, it has been found that despite the existence of a singlet 0 0 4t? , SDW correlations superconducting ground state at a fixed t? continue to grow when the temperature is lowered. This is shown by the SDW response that varies according to a Curie–Weiss law, wSDW ðTÞp1=ðT þ YÞ [10]. The CW behavior takes place below the characteristic temperature TCW 10 K, which is essentially 0 , whereas the CW scale Y, which starts from independent of t? 0 0 (Fig. 1). zero at t? , increases rapidly with t? Spin fluctuations have a well-defined influence on the NMR spin lattice relaxation rate T11 for which a fair amount of experimental evidence is available [16–19]. According to the Moriya expression T11 ¼ T
Z
jAq j2
w00 ðq; oÞ 3 d q; o
ARTICLE IN PRESS S90
A. Sedeki et al. / Physica B 405 (2010) S89–S91
400
T(K)
SDW
TCW
200
C-W
0
∂ℓ
Σ
G
=
Σ
100
``
Θ Metal
102
`
600
-200 SCd 15
20
t'⊥(K)
25
-400
30
-600
Fig. 1. RG phase diagram of the quasi-1D electron gas model. The dashed line stands as the C-W scale Y in the superconducting sector. The dotted line defines the temperature domain of the C–W behavior. After Ref. [10].
-1
T1 (s-1)
150
50 20
40
-1
T1 (s-1)
T(K) 0
50
0
10
0 ω(K)
5000
10000
100
0
0
-5000
Fig. 3. Real and Imaginary parts of the retarded one-particle self-energy as a 0 ¼ 27:7 K ðTc ¼ 0:7 K). Inset: function of the frequency o obtained at T ¼ 2 K for t? two-loop RG flow equation in the diagrammatic form for the one-particle selfenergy.
150
100
-10000
20
30
40
50
T(K) Fig. 2. Calculated temperature dependence of the nuclear relaxation rate for 0 0 0 various ‘pressures’ t? , from the SDW (t? o t? , top curves) to the superconducting 0 0 4 t? , bottom curves). Inset: experimental results of Creuzet et al. After region (t? [10,16].
T11 is coupled by means of the hyperfine coupling to the imaginary part of the electronic spin response w00 ðq; oÞ for the low frequency spectrum of spin correlations at the wave vector q. As shown in Refs. [10,20], the relaxation rate superimposes a ‘Fermi liquid’ like component T11 C0 T, which dominates at high temperature and an antiferromagnetic piece, T11 C1 T wSDW ðTÞ, which is linked to the SDW response calculated by the RG method. The sum of the two contributions is shown in Fig. 2. As one moves along the ‘pressure’ axis of Fig. 1, the divergence of T11 , which is a signature of a SDW state, transformed into an anomalous enhancement in the superconducting sector of the 0 0 4t? . The amplitude of the anomaly is phase diagram when t? strongly ‘pressure’ dependent and decays as Tc decreases. A large part of the anomaly below TCW fits pretty well the Curie–Weiss form T1 T T þ Y [10], as first pointed out on experimental grounds by Brown et al. [18,19]. The overall calculated temperature profile of T11 reproduces the key features shown by experiments (inset of Fig. 2) [16,19]. We consider now the one-particle spectral properties of the model in the normal state of the superconducting sector of the phase diagram. For this purpose, RG calculations have to include two-loop diagrams for the flow equation of the inverse oneparticle Green function G1 ¼ G1 0 S (inset Fig. 3). The oneparticle self-energy SðkF ; on Þ can be computed as a function of
discrete Matsubara frequencies on . This quantity is analytically continued on the real frequency axis using a N-points Pade´ approximants procedure that finally leads to the real (S0 ðoÞ) and the imaginary (S00 ðoÞ) parts of the retarded one-particle selfenergy. In the superconducting sector of the phase diagram where 0 0 4t? , typical low temperature results of the analytical cont? 0 ¼ 27:7 K tinuation are shown in Fig. 3 at T ¼ 2 K for t? (Tc ¼ 0:65 K). Apart from the characteristic structure seen at high frequency for both quantities, clear features emerge in the relevant low frequency domain jojo t? , corresponding to energy region of coherent transverse single particle motion. The real part S0 ðoÞ shows a typical linear zero crossing with a negative slope as o-0, indicative of a finite quasi-particle weight zðkF Þ at the Fermi point—calculated here at kF ¼ ðkF ; k? ¼ 0ÞFfor the finite temperature considered. The imaginary part SðoÞ develops accordingly a sharp peak. The expression for minus the imaginary part of the self-energy defines the frequency dependent electron–electron scattering rate 1=tðoÞ ¼ S00 ðoÞ evaluated at the Fermi wave vector kF . This quantity is particularly pertinent at low energy. A close-up view of the T ¼ 2 K trace of S00 ðoÞ in Fig. 4 at jojo t? reveals indeed an unusual linear behavior of the form S00 ðoÞ a0 ajoj, at variance with the o2 variation expected for a Fermi liquid. The finite, albeit reduced and temperature dependent quasi-particle weight zðkF Þ shown by the real part, is consistent with the form S0 ðo-0ÞpolnL=T, reminiscent of that of a marginal Fermi liquid [21]. As we increase the temperature, however, the linear behavior of the imaginary part is progressively replaced by a o2 variation in a range of frequency that increases in size with T. This is exemplified in Fig. 4, where S00 ðoÞ b0 bo2 at T ¼ 15 K 4TCW . Such a crossover in the frequency behavior is present over the 0 0 4t? , characterized by a slope a of the linear whole range of t? 0 is increased. contribution that shrinks in size as t? This remarkable feature of S00 ðoÞ will be also present in the temperature dependence of quantities like resistivity. Adopting a simple Boltzman picture for the electrical conductivity in which the scattering rate as a function of energy is 1=tðeÞ ¼ S00 ðo ¼ eÞ, the conductivity will scale as ds 1=T in the low temperature region where T 5TCW and 1=tðeÞ is linear in energy. This leads to a contribution in resistivity drðTÞ AT that is linear in temperature. As the temperature increases well above TCW , namely where
ARTICLE IN PRESS A. Sedeki et al. / Physica B 405 (2010) S89–S91
pairing, are responsible for the development of marginal Fermi liquid at low temperature.
0
-50
C.B. acknowledges fruitful and continuous collaboration with D. Je´rome, P. Auban-Senzier, N. Doiron-Leyraud, S. Rene´ de Cotret and L. Taillefer. This work has received the financial support from the National Science and Engineering Research Council of Canada (NSERC), Re´seau Que´bcois des Mate´riaux de Pointe (RQMP) and the Quantum materials program of Canadian Institute of Advanced Research (CIFAR). The authors are also thankful to the Re´seau Que´be´cois de Calcul Haute Performance (RQCHP) for supercomputer facilities at the Universite´ de Sherbrooke.
Σ
T = 15K
-100 T = 2K -150 −500
S91
References
0 ω(K)
500
Fig. 4. Imaginary parts of the self-energy (Fig. 3) in the low frequency region well below (2 K) and above (15 K) the Curie–Weiss temperature scale TCW .
1=tðeÞ picks up a quadratic low energy dependence and the resistivity acquires a Fermi liquid component dr BT 2 that varies quadratically with temperature. This crossover between the linear and quadratic temperature dependence captures the key characteristics that were shown to take place in the electrical resistivity of the normal state of the Bechgaard salts [12,13]. Furthermore in these experiments, it was pointed out that the amplitude of the linear term in resistivity is correlated to the size of Tc over the whole range of pressure. A similar correlation between the linear coefficient A and Tc follows from the RG calculations. By increasing the ‘pressure’, as one 0 axis in Fig. 1, an order of magnitude reduction moves along the t? of Tc is concomitant with a similar reduction in the linear coefficient A for resistivity. The suppression of A under pressure is thus closely related to the decrease of antiferromagnetic fluctuations, namely those connected to the enhancement of the nuclear relaxation rate seen by NMR [10]. From the spectral properties, these same fluctuations, which interfere positively with superconducting d-wave
[1] D. Je´rome, A. Mazaud, M. Ribault, K. Bechgaard, J. Phys. (Paris) Lett. 41 (1980) L95. [2] V.J. Emery, Synth. Met. 13 (1986) 21. [3] L.G. Caron, C. Bourbonnais, Physica 143B (1986) 453. [4] M.T. Be´al-Monod, C. Bourbonnais, V.J. Emery, Phys. Rev. B 34 (1986) 7716. [5] D.J. Scalapino, E. Loh, J.E. Hirsch, Phys. Rev. B 34 (1986) R8190. [6] R. Duprat, C. Bourbonnais, Eur. Phys. J. B 21 (2001) 219. [7] Y. Fuseya, Y. Suzumura, J. Phys. Soc. Japan 74 (2005) 1263. [8] J.C. Nickel, R. Duprat, C. Bourbonnais, N. Dupuis, Phys. Rev. Lett. 95 (2005) 247001. [9] J.C. Nickel, R. Duprat, C. Bourbonnais, N. Dupuis, Phys. Rev. B 73 (2006) 165126. [10] C. Bourbonnais, A. Sedeki, Phys. Rev. B 80 (2009) 085105. [11] C. Bourbonnais, D. Je´rome, in: A. Lebed (Ed.), The Physics of Organic Superconductors and Conductors. Springer Series in Materials Science, vol. 110, Springer, Heidelberg, 2008, p. 357 arXiv:cond-mat/0904.0617. [12] N. Doiron-Leyraud, P. Auban-Senzier, S. Rene´ de Cotret, R. Homier, A. Sedeki, C. Bourbonnais, D. Je´rome, K. Bechgaard, L. Taillefer, arXiv:cond-mat/ 0905.0964; Phys. Rev. B, 2009, in press. [13] N. Doiron-Leyraud, P. Auban-Senzier, S. Rene´ de Cotret, K. Bechgaard, D. Je´rome, L. Taillefer, this issue. [14] V.J. Emery, R. Bruinsma, S. Barisic, Phys. Rev. Lett. 48 (1982) 1039. [15] D. Je´rome, H. Schulz, Adv. Phys. 31 (1982) 299. [16] F. Creuzet, et al., Synth. Met. 19 (1987) 277. [17] P. Wzietek, et al., J. Phys. I 3 (1993) 171. [18] W. Wu, et al., Phys. Rev. Lett. 94 (2005) 097004. [19] S.E. Brown, P.M. Chaikin, M.J. Naughton, in: A. Lebed (Ed.), The Physics of Organic Superconductors and Conductors. Springer Series in Materials Science, vol. 110, Springer, Heidelberg, 2008, p. 49. [20] C. Bourbonnais, et al., Phys. Rev. Lett. 62 (1989) 1532. [21] C.M. Varma, et al., Phys. Rev. Lett. 63 (1989) 1996.