- Email: [email protected]
Spin density wave instability induced by a magnetic field in quasi-two-dimensional d-wave superconductors
Physica C 460–462 (2007) 778–780 www.elsevier.com/locate/physc
Spin density wave instability induced by a magnetic field in quasi-two-dimensional d-wave superconductors A. BenAli *, R. Bennaceur LPMC, De´partement de Physique, Faculte´ des Sciences de Tunis, Universite´ El Manar, 2092 Tunis, Tunisia Available online 11 April 2007
Abstract We show that an applied magnetic field (AMF) perpendicular to the CuO2 planes of the cuprate superconductors (SCs) can induce a time reversal symmetry breaking spin density wave instability (SDWI) coexisting with the superconducting (SC) order and can open a gap over the whole Fermi surface (FS). The orbital effect of the AMF makes the quasi-particles (QPs) motion strictly 1D. The transition is of first-order from a d-wave SC state to a fully gaped state. Our results are in sound agreement with recent experimental observations in high-Tc SCs. Ó 2007 Elsevier B.V. All rights reserved. PACS: 78.67.Hc; 71.35.Gg; 71.35.Pq
The cuprates are quasi-2D SCs, with FS exhibiting nesting properties. Below Tc, a d-wave SC order is established, with lines of zero gap on the FS. In the node regions which correspond to the best FS nesting, superconductivity is weak enough to leave room for a field induced spin density wave (FISDW) gap. Recent experiments in high-Tc SCs indicate that under AMF superconductivity is suppressed in the core of the vortex and antiferromagnetism (AF) develops there [1–5]. Scanning tunneling spectroscopy measurements in YBCO [6] suggest the existence of localized QP states in the vortex core, which can exist only if there is an energy gap over the entire FS. Equally intriguing have been reports of an unusual plateaulike feature in the field dependence of the thermal conductivity j(H) in BSCCO [7–12]. More recent experiments measuring the tunneling conductance in YBCO under high AMFs are interpreted as a reduction of the low energy density of states leading to a fully gaped state [13]. The interesting information one can draw from all these experiments is that the AMF induces QPs current decrease in the nodal direction and
the transition for a new state characterized by a new symmetry gap. The high-Tc SCs are among the most complex systems studied in condensed matter physics. As a good starting point for modeling the strong correlation effects in the oxides should be the Hubbard or the t–J models at low doping concentrations. In the present work, we study a nearly square 2D FS with good nesting properties, as a worthwhile model for high-Tc SCs. The role of fluctuations of any kind has not been considered, since temperatures are assumed to be low enough to permit the Landau–Fermi liquid approach. We propose to treat the electron–electron interaction in a mean field approximation, with two different order parameters. The first is an effective attractive electron–electron interaction term. The second, the origin of which is the exchange interaction, is an electron–hole coupling term, corresponding to the FISDW state. The hamiltonian is given by H¼
X
nk c þ k;r ck;r
k;r *
Corresponding author. E-mail address: [email protected] (A. BenAli).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.04.013
X k;k 0
X k;k
gm ðk; k
0
þ 0 0 gs ðk; k 0 Þcþ k;" ck;# ck ;# ck ;"
0
þ 0 Þcþ kþq;# ck;" ck 0 q;" ck ;# ;
ð1Þ
A. BenAli, R. Bennaceur / Physica C 460–462 (2007) 778–780
k0
Dm ðkÞ ¼
X k
geff m ðk
þ q; k
0
0 ; H Þhcþ H ;k 0 þq# cH ;k " i:
ð3Þ
0
The approximated SC and FISDW coupling potentials displaying the gap symmetry are given by gs ðk; k 0 Þ ¼ gos fs ðkÞfs ðk 0 Þ;
ð4Þ
0 geff m ðk; k ; H Þ
ð5Þ
¼
0 geff om ðH Þfm ðkÞfm ðk Þ;
where fs(k) = cos (kxa) cos (kyb) and fm(k) = sin (kxa) sin (kyb). The field dependence of the magnetic coupling geff om ðH Þ is due to the orbital effect of the AMF, responsible for quantum interference effects [15]. This effect induces better nesting properties and, therefore, an effective coupling constant geff om ðH Þ rapidly increasing with the AMF. The temperature and the AMF dependences of the SC and the magnetic order parameters are insured by Eqs. (2) and (3), written in a simpler form given by the coupled gap equations in the mixed state
Magnetic coupling constant gom (eV)
0.46 FISDW
0.44 0.42 0.4
SC + FISDW First order transition lines
0.38 0.36 0.34
SC
0.32 15
30
60 45 Temperature (K)
75
60 50 FISDW 40 30 20 10
First order transition SC + FISDW
0 0.36 0.39 0.42 0.45 0.48 0.51 Magnetic coupling constant gom (eV) Fig. 2. FISDW gap versus geff om ðHÞ (gos = 0.5 eV, T = 10 K).
25
ð6Þ E
X tanhð2kBkT Þ 1 2 ¼ f ðkÞ ; ð7Þ m 2Ek geff om ðH Þ k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where Ek ¼ n0 þ ½Dos fs ðkÞ2 þ ½Dom fm ðkÞ2 . The curves shown in the phase diagram (Fig. 1) indicate the first-order transition lines separating the three phases. Figs. 2 and 3 depict numerical solutions of the gap equations, in the mixed phase, for different values of geff om ðH Þ. Obviously, Dom(Dos) increases (decreases) with geff om ðH Þ and the transition from one phase to another is first-order. The most interesting conclusion one can make from this study is that an AMF perpendicular to the conducting planes of high-Tc SCs can induce a first-order phase transi-
SC order parameter (meV)
E
X tanhð2kBkT Þ 1 ¼ fs2 ðkÞ ; 2Ek gos k
90
Fig. 1. geff om ðHÞ T phase diagram of high-Tc SCs, given for gos = 0.5 eV.
FISDW order parameter (meV)
where nk is the noninteracting electron dispersion relation, ck,r is an electron annihilation operator. gs(gm) is the SC (magnetic) coupling constant. In the nearly square FS, we approximate the electron dispersion relation by a strictly 2D linearized expression around the Fermi level for two opposite flat sheets of the FS: n(kx,ky) = n0(kx) + t?(ky) l. Here, n0(kx) = vF(jkxj kF) is linear along kx and t?(ky) = 2tcos (kyb) 2t 0 cos (2kyb) is a periodic function which describes the warping of the FS. vF is the Fermi velocity, b is the lattice parameter along the y-direction represents the periodic dependence on ky. kx(ky) is the electron wave vector component perpendicular (parallel) to the FS sheet and l is the chemical potential. The second harmonic term t 0 introduces the deviation from perfect nesting of the FS. Experimental observations provide evidence for dx2 y 2 symmetry of the SC order parameter. On the other hand, it is required that the FISDW order parameter does not vanish on the nodes of the dx2 y 2 -wave SC one and corresponds to the dxy-wave symmetry [14]. The SC and FISDW gap equations are given by the self-consistent conditions: X Ds ðkÞ ¼ gs ðk; k 0 ÞhcH ;k0 # cH ;k0 " i; ð2Þ
779
24
SC
23 22 21 First order transition 20 19 18
SC + FISDW 0.30 0.33 0.36 0.39 0.42 Magnetic coupling constant gom (eV)
Fig. 3. SC gap versus geff om ðHÞ (gos = 0.5 eV, T = 10 K).
780
A. BenAli, R. Bennaceur / Physica C 460–462 (2007) 778–780
tion to a mixed state. Consequently, a gap is open over the whole FS. The QPs existing at low temperature in the pure d-wave SC phase should, therefore, disappear discontinuously at the critical field. This might be the explanation of the j(H) plateau observed in BSCCO above a threshold AMF [7–12]. The sharp break in slope in j(H) is a signature of the first-order transition and the flat plateau that extends to high AMFs is the signature of the mixed phase, in which the QPs current decreases and the transition to a fully gaped state occurs. The field induced anomalies observed in the in-plane magnetic penetration depth [16] and the tunneling conductance [13] in YBCO might be ascribed to a fully gaped state, when SC order coexists with a FISDW one, that takes place and freezes out the QP excitations. Summarizing, we show that the orbital effect of a moderate AMF can induce a time reversal symmetry breaking SDWI coexisting with the SC order and can open a gap over the whole FS. Recent experiments in cuprates measuring the superfluid density, the specific heat, the tunneling conductance and the magnetization might be ascribed to
the generation of antiferromagnetic moments in the SC phase by the AMF and the disappearance of the QPs due to the elimination of the nodes of the SC gap by the magnetic one. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
V.F. Mitrovic´ et al., Nature 413 (2001) 501. D.P. Arovas et al., Phys. Rev. Lett. 79 (1997) 2871. M. Takigawa et al., Phys. Rev. Lett. 83 (1999) 3057. R. Wortis et al., Phys. Rev. B 61 (2000) 12342. D.K. Morr, R. Wortis, Phys. Rev. B 61 (2000) R882. I. Maggio-Aprile et al., Phys. Rev. Lett. 75 (1995) 2754. K. Krishana et al., Science 277 (1997) 83. H. Aubin et al., Science 280 (1998) 9a. H. Aubin et al., Phys. Rev. Lett. 82 (1999) 624. Y. Ando et al., Phys. Rev. B 62 (2000) 626. B. Zeini et al., Phys. Rev. Lett. 82 (1999) 2175. A.N. Taldenkov et al., J. Low Temp. Phys. 115 (1999) 15. R. Beck et al., Phys. Rev. B 72 (2005) 104505. R.B. Laughlin, Phys. Rev. Lett. 80 (1998) 5188. M. He´ritier et al., J. Phys. Lett. 45 (1984) L-943. J.E. Sonier et al., Phys. Rev. Lett. 83 (1999) 4156.
Spin density wave instability induced by a magnetic field in quasi-two-dimensional d-wave superconductors A. BenAli *, R. Bennaceur LPMC, De´partement de Physique, Faculte´ des Sciences de Tunis, Universite´ El Manar, 2092 Tunis, Tunisia Available online 11 April 2007
Abstract We show that an applied magnetic field (AMF) perpendicular to the CuO2 planes of the cuprate superconductors (SCs) can induce a time reversal symmetry breaking spin density wave instability (SDWI) coexisting with the superconducting (SC) order and can open a gap over the whole Fermi surface (FS). The orbital effect of the AMF makes the quasi-particles (QPs) motion strictly 1D. The transition is of first-order from a d-wave SC state to a fully gaped state. Our results are in sound agreement with recent experimental observations in high-Tc SCs. Ó 2007 Elsevier B.V. All rights reserved. PACS: 78.67.Hc; 71.35.Gg; 71.35.Pq
The cuprates are quasi-2D SCs, with FS exhibiting nesting properties. Below Tc, a d-wave SC order is established, with lines of zero gap on the FS. In the node regions which correspond to the best FS nesting, superconductivity is weak enough to leave room for a field induced spin density wave (FISDW) gap. Recent experiments in high-Tc SCs indicate that under AMF superconductivity is suppressed in the core of the vortex and antiferromagnetism (AF) develops there [1–5]. Scanning tunneling spectroscopy measurements in YBCO [6] suggest the existence of localized QP states in the vortex core, which can exist only if there is an energy gap over the entire FS. Equally intriguing have been reports of an unusual plateaulike feature in the field dependence of the thermal conductivity j(H) in BSCCO [7–12]. More recent experiments measuring the tunneling conductance in YBCO under high AMFs are interpreted as a reduction of the low energy density of states leading to a fully gaped state [13]. The interesting information one can draw from all these experiments is that the AMF induces QPs current decrease in the nodal direction and
the transition for a new state characterized by a new symmetry gap. The high-Tc SCs are among the most complex systems studied in condensed matter physics. As a good starting point for modeling the strong correlation effects in the oxides should be the Hubbard or the t–J models at low doping concentrations. In the present work, we study a nearly square 2D FS with good nesting properties, as a worthwhile model for high-Tc SCs. The role of fluctuations of any kind has not been considered, since temperatures are assumed to be low enough to permit the Landau–Fermi liquid approach. We propose to treat the electron–electron interaction in a mean field approximation, with two different order parameters. The first is an effective attractive electron–electron interaction term. The second, the origin of which is the exchange interaction, is an electron–hole coupling term, corresponding to the FISDW state. The hamiltonian is given by H¼
X
nk c þ k;r ck;r
k;r *
Corresponding author. E-mail address: [email protected] (A. BenAli).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.04.013
X k;k 0
X k;k
gm ðk; k
0
þ 0 0 gs ðk; k 0 Þcþ k;" ck;# ck ;# ck ;"
0
þ 0 Þcþ kþq;# ck;" ck 0 q;" ck ;# ;
ð1Þ
A. BenAli, R. Bennaceur / Physica C 460–462 (2007) 778–780
k0
Dm ðkÞ ¼
X k
geff m ðk
þ q; k
0
0 ; H Þhcþ H ;k 0 þq# cH ;k " i:
ð3Þ
0
The approximated SC and FISDW coupling potentials displaying the gap symmetry are given by gs ðk; k 0 Þ ¼ gos fs ðkÞfs ðk 0 Þ;
ð4Þ
0 geff m ðk; k ; H Þ
ð5Þ
¼
0 geff om ðH Þfm ðkÞfm ðk Þ;
where fs(k) = cos (kxa) cos (kyb) and fm(k) = sin (kxa) sin (kyb). The field dependence of the magnetic coupling geff om ðH Þ is due to the orbital effect of the AMF, responsible for quantum interference effects [15]. This effect induces better nesting properties and, therefore, an effective coupling constant geff om ðH Þ rapidly increasing with the AMF. The temperature and the AMF dependences of the SC and the magnetic order parameters are insured by Eqs. (2) and (3), written in a simpler form given by the coupled gap equations in the mixed state
Magnetic coupling constant gom (eV)
0.46 FISDW
0.44 0.42 0.4
SC + FISDW First order transition lines
0.38 0.36 0.34
SC
0.32 15
30
60 45 Temperature (K)
75
60 50 FISDW 40 30 20 10
First order transition SC + FISDW
0 0.36 0.39 0.42 0.45 0.48 0.51 Magnetic coupling constant gom (eV) Fig. 2. FISDW gap versus geff om ðHÞ (gos = 0.5 eV, T = 10 K).
25
ð6Þ E
X tanhð2kBkT Þ 1 2 ¼ f ðkÞ ; ð7Þ m 2Ek geff om ðH Þ k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where Ek ¼ n0 þ ½Dos fs ðkÞ2 þ ½Dom fm ðkÞ2 . The curves shown in the phase diagram (Fig. 1) indicate the first-order transition lines separating the three phases. Figs. 2 and 3 depict numerical solutions of the gap equations, in the mixed phase, for different values of geff om ðH Þ. Obviously, Dom(Dos) increases (decreases) with geff om ðH Þ and the transition from one phase to another is first-order. The most interesting conclusion one can make from this study is that an AMF perpendicular to the conducting planes of high-Tc SCs can induce a first-order phase transi-
SC order parameter (meV)
E
X tanhð2kBkT Þ 1 ¼ fs2 ðkÞ ; 2Ek gos k
90
Fig. 1. geff om ðHÞ T phase diagram of high-Tc SCs, given for gos = 0.5 eV.
FISDW order parameter (meV)
where nk is the noninteracting electron dispersion relation, ck,r is an electron annihilation operator. gs(gm) is the SC (magnetic) coupling constant. In the nearly square FS, we approximate the electron dispersion relation by a strictly 2D linearized expression around the Fermi level for two opposite flat sheets of the FS: n(kx,ky) = n0(kx) + t?(ky) l. Here, n0(kx) = vF(jkxj kF) is linear along kx and t?(ky) = 2tcos (kyb) 2t 0 cos (2kyb) is a periodic function which describes the warping of the FS. vF is the Fermi velocity, b is the lattice parameter along the y-direction represents the periodic dependence on ky. kx(ky) is the electron wave vector component perpendicular (parallel) to the FS sheet and l is the chemical potential. The second harmonic term t 0 introduces the deviation from perfect nesting of the FS. Experimental observations provide evidence for dx2 y 2 symmetry of the SC order parameter. On the other hand, it is required that the FISDW order parameter does not vanish on the nodes of the dx2 y 2 -wave SC one and corresponds to the dxy-wave symmetry [14]. The SC and FISDW gap equations are given by the self-consistent conditions: X Ds ðkÞ ¼ gs ðk; k 0 ÞhcH ;k0 # cH ;k0 " i; ð2Þ
779
24
SC
23 22 21 First order transition 20 19 18
SC + FISDW 0.30 0.33 0.36 0.39 0.42 Magnetic coupling constant gom (eV)
Fig. 3. SC gap versus geff om ðHÞ (gos = 0.5 eV, T = 10 K).
780
A. BenAli, R. Bennaceur / Physica C 460–462 (2007) 778–780
tion to a mixed state. Consequently, a gap is open over the whole FS. The QPs existing at low temperature in the pure d-wave SC phase should, therefore, disappear discontinuously at the critical field. This might be the explanation of the j(H) plateau observed in BSCCO above a threshold AMF [7–12]. The sharp break in slope in j(H) is a signature of the first-order transition and the flat plateau that extends to high AMFs is the signature of the mixed phase, in which the QPs current decreases and the transition to a fully gaped state occurs. The field induced anomalies observed in the in-plane magnetic penetration depth [16] and the tunneling conductance [13] in YBCO might be ascribed to a fully gaped state, when SC order coexists with a FISDW one, that takes place and freezes out the QP excitations. Summarizing, we show that the orbital effect of a moderate AMF can induce a time reversal symmetry breaking SDWI coexisting with the SC order and can open a gap over the whole FS. Recent experiments in cuprates measuring the superfluid density, the specific heat, the tunneling conductance and the magnetization might be ascribed to
the generation of antiferromagnetic moments in the SC phase by the AMF and the disappearance of the QPs due to the elimination of the nodes of the SC gap by the magnetic one. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
V.F. Mitrovic´ et al., Nature 413 (2001) 501. D.P. Arovas et al., Phys. Rev. Lett. 79 (1997) 2871. M. Takigawa et al., Phys. Rev. Lett. 83 (1999) 3057. R. Wortis et al., Phys. Rev. B 61 (2000) 12342. D.K. Morr, R. Wortis, Phys. Rev. B 61 (2000) R882. I. Maggio-Aprile et al., Phys. Rev. Lett. 75 (1995) 2754. K. Krishana et al., Science 277 (1997) 83. H. Aubin et al., Science 280 (1998) 9a. H. Aubin et al., Phys. Rev. Lett. 82 (1999) 624. Y. Ando et al., Phys. Rev. B 62 (2000) 626. B. Zeini et al., Phys. Rev. Lett. 82 (1999) 2175. A.N. Taldenkov et al., J. Low Temp. Phys. 115 (1999) 15. R. Beck et al., Phys. Rev. B 72 (2005) 104505. R.B. Laughlin, Phys. Rev. Lett. 80 (1998) 5188. M. He´ritier et al., J. Phys. Lett. 45 (1984) L-943. J.E. Sonier et al., Phys. Rev. Lett. 83 (1999) 4156.