- Email: [email protected]
Ordered stripe phases in the cuprates
PHYSlCA ELSEVIER
Physica C 341-348 (2000) 1789-1790 www.elsevier.nl/locate/physc
Ordered Stripe Phases in the Cuprates R.S. Markiewicz Physics Department and Barnett Institute, Northeastern U., Boston MA 02115 The photoemission spectrum of an ordered stripe array is calculated, including effects of long-range Coulomb repulsion. For all dopings, the dispersion is a superposition of two features, a magnetic stripe band with Mott gap and a charged stripe band with prominent superlattice minigaps, particularly one near the Fermi level EF. These two gaps resemble the experimental photoemission spectra of La2-~SrxCuO4 and Bi2Sr2CaCu2Os+~, which are modified by increasing stripe fluctuations.
Several recent calculations have modelled the stripe phases in the cuprates as a form of charge-density wave (CDW)[1]. However, evidence for long-range magnetic order in some La2-~SrxCuO4 (LSCO) compounds[2] suggests that the density modulation is more step-like than sinusoidal - closer to the discommensuration limit of CDW's. Here, a model of the extreme step-like limit is presented, based on a picture of nanoscale phase separation[3]. The free energy is assumed to have two minima[4], at x = 0 (magnetic stripes) and x0 = 0.25 (charged stripes), with a corresponding chemical potential. Coulomb effects are included via the electrochemical potential (dielectric constant e), and the local hole density on each row of atoms is self-consistently adjusted. The doping dependence of the stripe dispersion is analyzed, assuming the stripes are always an even number of cells (or Cu atoms) wide[5]. Two qualitatively new features are found. First, the dispersion is a superposition of two distinct components, associated with the magnetic and charged stripes, which can be identified with features seen in the experimentally observed photoemission dispersions. Secondly, these dispersions are modulated with superimposed doping dependent minigaps due to the stripe order. While most of these minigaps will be washed out by strong stripe fluctuations, there is one which is characteristically pinned at the Fermi level, and hence should survive fluctuation averaging. Its doping dependence resembles that of the normal-
state pseudogap. The magnetic stripes are described by a spindensity wave model[6], with Coulomb U and t and t' hopping parameters chosen to fit the dispersion in Sr2CuO2C12 (SCOC)[7]. The only doping dependence comes from the assumption that the equilibrium antiferromagnetic magnetization MQ vanishes on the charged stripes, MQ(x) = MQ(0)(1 - x/xo). Figure 1 shows the doping dependence of the dispersion from F -+ X - (% 0) --+ S - (~r, 7r), in the presence of Coulomb repulsion (e = 15). While the fine structures (minigaps) are strongly doping dependent, the overall dispersion is not, and is essentially a superposition of two contributions, one for the insulating magnetic stripes, one for the charged stripes. With increasing doping Coulomb effects push more holes onto the magnetic stripes, thereby shifting the Mott gap closer to EF. The overall doping dependence is quite similar to the experimental results of Ino, et al.[8] in LSCO (see their Fig. 3). Photoemission experiments in LSCO[8] and Bi2Sr2CaCu2Os+6 (BSCCO)[9] find two dispersion features near (Tr,0), Fig. 2. The feature nearer the Fermi level EF (open circles and squares) - the 'peaks' in BSCCO - correlate well with the calculated charge stripe miniband at EF (suns). The feature further from E~ ( x ' s and +'s) - the 'hump' in BSCCO - has the dispersion of the lower Hubbard band feature of the magnetic stripes (diamonds) but the shift from EF varies strongly with materials, in the sequence
0921-4534/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII S0921-4534(00)01061-3
R.S, M a r k i e w i c z / P h y s i c a C 3 4 1 - 3 4 8 (2000) 1 7 8 9 - 1 7 9 0
1790
(a),l
:>
ilpb/yV
0.60
0.50
~
i
--i
).04 //,a, A
030 !
r
= ).02
/
C,
/,.I
0.10
~d'
c
/
/
•
0.05
0.10
•
~'
pO
0.00 0.00
).03
4
m
o
).05
0.40
J
lj ,j
).06 III
0.15
[]
).01
<:'
0.20
).00 0.25
PAl
X
8
Figure 1. Stripe dispersion for dopings: x = 0 (solid line in a), 0.0625 (a), 0.125 (b), 0.167 (c), 0.1875 (d), and 0.25 (solid line in d). The +'s (circles) indicate >__80% of the wave function is on the magnetic (charged) stripes; diamonds indicate a mixture of both.
SCOC > LSCO > BSCCO. (The figure shows the calculated gap values divided by two, since the band parameters were chosen to agree with the larger gap in SCOC.) This material dependence is proposed to be due to fluctuations shifting holes into the magnetic stripes. Thus LSCO has well defined but fluctuating stripes, while fluctuations are stronger in BSCCO where direct evidence for stripes is hard to find. Moreover, above the superconducting To, fluctuations in BSCCO are even stronger and the peak and hump features collapse into a single broad peak. While the charge gaps grow with increased underdoping, spin gaps[10] grow with increased doping (triangles, Fig. 2). There is a duality crossover at xo/2 = 0.125, vertical line in Figure, where the system changes from a minority charge stripe ('rivers of charge') to a minority magnetic stripe phase (isolated spin ladders). These computations were carried out using the
Figure 2. Pseudogaps at (7r,0) in LSCO[8] (circles) and BSCCO[9] (diamonds) compared to calculated Mott gap (filled squares) and minigap (open squares); the calculated values are reduced by a factor of two. Triangles and fitted curve = spin gap A~ in YBa2Cu307-~.
facilities of Northesatern's Advanced Scientific Computation Center (NU-ASCC). REFERENCES
1. M.I. Salkola, et al. Phys. Rev. Lett. 77, 155 (1996); G. Seibold, et al. cond-mat/9906108. 2. T. Suzuki, et al. Phys. Rev. B57, 3229 (1998); H. Kimura, et al., Phys. Rev. B59, 6517 (1999). 3. R.S. Markiewicz, cond-mat/9911108. 4. R.S. Markiewicz, Phys. Rev. B56, 9091 (1997). 5. H. Tsunetsugu, et al., Phys. Rev. B51, 16456 (1995); S.R. White and D.J. Scalapino, Phys. Rev. Lett. 80, 1272 (1998). 6. J.R. Schrieffer, et al., Phys. Rev. B39, 11663 (1989). 7. B.O. Wells, et al., Phys. Rev. Lett. 74, 964 (1995). 8. A. Ino, et al., cond-mat/9902048. 9. J.C. Campuzano, et al., Phys. Rev. Lett 83, 3_709 (1999). 10. P. Bourges, et al. Physica B215, 30 (1995).
Physica C 341-348 (2000) 1789-1790 www.elsevier.nl/locate/physc
Ordered Stripe Phases in the Cuprates R.S. Markiewicz Physics Department and Barnett Institute, Northeastern U., Boston MA 02115 The photoemission spectrum of an ordered stripe array is calculated, including effects of long-range Coulomb repulsion. For all dopings, the dispersion is a superposition of two features, a magnetic stripe band with Mott gap and a charged stripe band with prominent superlattice minigaps, particularly one near the Fermi level EF. These two gaps resemble the experimental photoemission spectra of La2-~SrxCuO4 and Bi2Sr2CaCu2Os+~, which are modified by increasing stripe fluctuations.
Several recent calculations have modelled the stripe phases in the cuprates as a form of charge-density wave (CDW)[1]. However, evidence for long-range magnetic order in some La2-~SrxCuO4 (LSCO) compounds[2] suggests that the density modulation is more step-like than sinusoidal - closer to the discommensuration limit of CDW's. Here, a model of the extreme step-like limit is presented, based on a picture of nanoscale phase separation[3]. The free energy is assumed to have two minima[4], at x = 0 (magnetic stripes) and x0 = 0.25 (charged stripes), with a corresponding chemical potential. Coulomb effects are included via the electrochemical potential (dielectric constant e), and the local hole density on each row of atoms is self-consistently adjusted. The doping dependence of the stripe dispersion is analyzed, assuming the stripes are always an even number of cells (or Cu atoms) wide[5]. Two qualitatively new features are found. First, the dispersion is a superposition of two distinct components, associated with the magnetic and charged stripes, which can be identified with features seen in the experimentally observed photoemission dispersions. Secondly, these dispersions are modulated with superimposed doping dependent minigaps due to the stripe order. While most of these minigaps will be washed out by strong stripe fluctuations, there is one which is characteristically pinned at the Fermi level, and hence should survive fluctuation averaging. Its doping dependence resembles that of the normal-
state pseudogap. The magnetic stripes are described by a spindensity wave model[6], with Coulomb U and t and t' hopping parameters chosen to fit the dispersion in Sr2CuO2C12 (SCOC)[7]. The only doping dependence comes from the assumption that the equilibrium antiferromagnetic magnetization MQ vanishes on the charged stripes, MQ(x) = MQ(0)(1 - x/xo). Figure 1 shows the doping dependence of the dispersion from F -+ X - (% 0) --+ S - (~r, 7r), in the presence of Coulomb repulsion (e = 15). While the fine structures (minigaps) are strongly doping dependent, the overall dispersion is not, and is essentially a superposition of two contributions, one for the insulating magnetic stripes, one for the charged stripes. With increasing doping Coulomb effects push more holes onto the magnetic stripes, thereby shifting the Mott gap closer to EF. The overall doping dependence is quite similar to the experimental results of Ino, et al.[8] in LSCO (see their Fig. 3). Photoemission experiments in LSCO[8] and Bi2Sr2CaCu2Os+6 (BSCCO)[9] find two dispersion features near (Tr,0), Fig. 2. The feature nearer the Fermi level EF (open circles and squares) - the 'peaks' in BSCCO - correlate well with the calculated charge stripe miniband at EF (suns). The feature further from E~ ( x ' s and +'s) - the 'hump' in BSCCO - has the dispersion of the lower Hubbard band feature of the magnetic stripes (diamonds) but the shift from EF varies strongly with materials, in the sequence
0921-4534/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII S0921-4534(00)01061-3
R.S, M a r k i e w i c z / P h y s i c a C 3 4 1 - 3 4 8 (2000) 1 7 8 9 - 1 7 9 0
1790
(a),l
:>
ilpb/yV
0.60
0.50
~
i
--i
).04 //,a, A
030 !
r
= ).02
/
C,
/,.I
0.10
~d'
c
/
/
•
0.05
0.10
•
~'
pO
0.00 0.00
).03
4
m
o
).05
0.40
J
lj ,j
).06 III
0.15
[]
).01
<:'
0.20
).00 0.25
PAl
X
8
Figure 1. Stripe dispersion for dopings: x = 0 (solid line in a), 0.0625 (a), 0.125 (b), 0.167 (c), 0.1875 (d), and 0.25 (solid line in d). The +'s (circles) indicate >__80% of the wave function is on the magnetic (charged) stripes; diamonds indicate a mixture of both.
SCOC > LSCO > BSCCO. (The figure shows the calculated gap values divided by two, since the band parameters were chosen to agree with the larger gap in SCOC.) This material dependence is proposed to be due to fluctuations shifting holes into the magnetic stripes. Thus LSCO has well defined but fluctuating stripes, while fluctuations are stronger in BSCCO where direct evidence for stripes is hard to find. Moreover, above the superconducting To, fluctuations in BSCCO are even stronger and the peak and hump features collapse into a single broad peak. While the charge gaps grow with increased underdoping, spin gaps[10] grow with increased doping (triangles, Fig. 2). There is a duality crossover at xo/2 = 0.125, vertical line in Figure, where the system changes from a minority charge stripe ('rivers of charge') to a minority magnetic stripe phase (isolated spin ladders). These computations were carried out using the
Figure 2. Pseudogaps at (7r,0) in LSCO[8] (circles) and BSCCO[9] (diamonds) compared to calculated Mott gap (filled squares) and minigap (open squares); the calculated values are reduced by a factor of two. Triangles and fitted curve = spin gap A~ in YBa2Cu307-~.
facilities of Northesatern's Advanced Scientific Computation Center (NU-ASCC). REFERENCES
1. M.I. Salkola, et al. Phys. Rev. Lett. 77, 155 (1996); G. Seibold, et al. cond-mat/9906108. 2. T. Suzuki, et al. Phys. Rev. B57, 3229 (1998); H. Kimura, et al., Phys. Rev. B59, 6517 (1999). 3. R.S. Markiewicz, cond-mat/9911108. 4. R.S. Markiewicz, Phys. Rev. B56, 9091 (1997). 5. H. Tsunetsugu, et al., Phys. Rev. B51, 16456 (1995); S.R. White and D.J. Scalapino, Phys. Rev. Lett. 80, 1272 (1998). 6. J.R. Schrieffer, et al., Phys. Rev. B39, 11663 (1989). 7. B.O. Wells, et al., Phys. Rev. Lett. 74, 964 (1995). 8. A. Ino, et al., cond-mat/9902048. 9. J.C. Campuzano, et al., Phys. Rev. Lett 83, 3_709 (1999). 10. P. Bourges, et al. Physica B215, 30 (1995).