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Local density of states of high-Tc superconductors with long-range Coulomb interaction
Physics Letters A 374 (2010) 1779–1783
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Local density of states of high-Tc superconductors with long-range Coulomb interaction Hong-Wei Zhao, Hao Meng, Ling-feng Zhang, Lin Wen, Liang-ma Shi, Guo-Qiao Zha, Shi-Ping Zhou ∗ Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, PR China
a r t i c l e
i n f o
Article history: Received 7 December 2009 Received in revised form 22 January 2010 Accepted 29 January 2010 Available online 2 February 2010 Communicated by R. Wu Keywords: High-Tc superconductors Local density of states Long-range Coulomb interaction
a b s t r a c t The local density of state (LDOS) has been studied by solving the mean-field Bogoliubov–de Gennes equations based on a model Hamiltonian with competing antiferromagnetic (AF) and d-wave superconducting orderings in the presence of a long-range Coulomb interaction. The vortex-core state with energies slightly above and below the Fermi level shows a four-fold symmetry with two-dimensional modulations under small on-site repulsion strength U , whereas it has a stripe structure for large U . The four-fold symmetry modulation with a periodicity of 4a0 is recovered as a long-range Coulomb interaction has been included. Recent scanning-tunneling-microscopy experiments may be understood in terms of the present results. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Vortex core states of high-temperature superconductors (HTSCs) have attracted a lot of attention in the recent years [1–16]. A fundamental phenomenon of the core states is the splitting of the zero bias conductance peak (ZBCP), as observed in the scanningtunneling-microscopy (STM) experiments in YBa2 Cu3 O7−x (YBCO) [2] and in Bi2 Sr2 CaCu2 O8+x (BSCCO) [3]. The conventional BCS theory that predicts a strong enhancement of the quasiparticle peak around the Fermi level for a superconductor of energy nodes (lines) seemed unlikely to provide a good account for such observations [4]. Several modified models have been proposed [5–10]. On the assumption of the coexistence of an antiferromagnetic (AF) order with the d-wave superconductivity (DSC), Zhu and Ting [10] obtained a local density of states (LDOS) spectrum of double peaks slightly below and above the Fermi level. The AF fluctuation around the vortex cores has been probed by nuclear magnetic resonance experiments [11], which support the coexistence of the AF and d-wave orderings. It has also been found that LDOS maps of HTSCs are closely related to the on-site repulsion induced AF order. For instance, the stripe modulations were found in underdoped copper oxide superconductors [12] where the AF order is strong due to a large U , whereas a square pattern was observed in the low-energy scanning-tunneling-microscopy experiments [13,14] for overdoped
*
Corresponding author. E-mail address: [email protected] (S.-P. Zhou).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.01.064
samples where the AF order is negligible. Since a large on-site repulsion is assumed to induce a stripe phase, the recent STM experiments for underdoped BSCCO [15] and Nax Ca2−x CuO2 Cl2 [16] samples that have probed a checkerboard pattern with a twodimensional (2D) modulation are very interesting. As we will show in this Letter, an inclusion of the long-range Coulomb interaction into the model Hamiltonian is expected to provide a good interpretation for above observations, regarding both the double peak LDOS spectrum and the four-fold symmetric shape of the lowlying quasiparticle states oriented with the nodal directions of the dx2 − y 2 wave superconductor. An inclusion of the long-range coulomb interaction in the model Hamiltonian lies mainly on the following two points. Firstly, vortex cores will intrinsically charged up in HTSCs of a small value of k F ξ due to Coulomb repulsion between conducting electrons, as commonly observed in a neutral superfluid system. This would induce a charge density wave (CDW) competing with the AF order associated spin-density wave (SDW) that may modulate the vortex structure and the LODS spectrum. Secondly, unlike what occurred to a metallic superconductor, the Thomas–Fermi screening effect was substantially reduced for HTSCs since the short wave length part of the Coulomb interaction between conducting electrons, which has been omitted in the TF approximation, becomes important [17–19]. In the present Letter, we investigate the LDOS for the underdoped HTSCs based on an effective mean-field t–t –U –V d –V c model [see below]. By solving the Bogoliubov–de Gennes equations in a 2D Lattice, we show that the splitting of quasiparticle excitations at the Fermi level and the core states maps are closely
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related to the model parameters. For instance, in the absence of the long-range Coulomb repulsion, spatial distribution of the core states shows a two-dimensional pattern to the one-dimensional stripe transition with the on-site repulsion strength U . Meanwhile, a well-defined double peak LDOS spectrum developed for a reasonable large U . The long-range coulomb interaction can provide an intrinsic electron depletion mechanism by inducing the so-called ferromagnetic polarons or the ‘ferros’ [20,21], which reside at the intersections of a crossed array of antiphase domain walls of the AF order parameter to accommodate holes. The ensuing CDW competes with the existing AF-like SWD that favors to lower-lying quasiparticle excitations. As a result, the two featured peaks of the core states would move more close to the Fermi level. The four-fold symmetric pattern is expected to recover in the underdoped case with a reasonable large long-range Coulomb repulsion strength. Interestingly, the LDOS maps shows a change in the periodicity from 5a0 to 4a0 with the long-range Coulomb repulsion strength V c varying from 0 to 0.15 [in units of the nearest-neighboring hopping integrals]. Recent scanning-tunneling-microscopy experiments [15,16] may be understood in terms of the present results. 2. Model
†
(t i j ,σ )c iσ c jσ +
ij,σ
+ +
†
U niσ¯ − μ c iσ c iσ
∗ij
j
† † ij c i↑ c j↓
+ H.c.
= εn
uniσ
v niσ¯
(2)
, √
† † where Hijσ = −t ij + V2c (c i+δ,σ c iσ +c √
i+ 2δ,σ
nl −¯n
c iσ / 2 )+[U niσ¯ −
]δij and ij = V2d c i↑ c j↓ − c i↓ c j↑ − V2c c i↓ c j↑ . √ The index i + δ and i + 2δ are the four nearest and next-nearest
μ+
Vc 2
l=i
r i ,l
neighbors of the i-th site, respectively. The self-consistent conditions are:
ni↑ =
2N n 2 u f ( E n ),
(3)
i
n =1
ni↓ =
2N n 2
v 1 − f ( E n ) , i
(4)
n =1
2N Vd n =1
4
uni vnj ∗ + vni ∗ unj tanh
V c n∗ n − vi u j 1 − f ( E n ) ,
En
2k B T (5)
2
where uni = (− v ni↑∗ , uni↑ ) and vni = (uni↓∗ , v ni↓ ) denote the 2Ndimensional wave function vectors, and f ( E ) = 1/(e β E + 1) is the Fermi–Dirac distribution function. The vortex charge density (in †
iD,i+e y − iD,i−e y )/4, where ijD = ij exp[i Φπ0 (1)
†
Here c iσ (c iσ ) is the electron creation (annihilation) operator of spin index σ on the i-th site. μ is the chemical potential determining the averaged electron density n¯ = iσ niσ / N (N = N x × N y is the linear dimension of the unit cell of the vor†
tex lattice). niσ = c iσ c iσ is the number operator. The on-site repulsion will generate an AF-like spin-density wave (SDW) order †
SDW = U c i↑ c i↑ − c i↓ c i↓ . The d-wave superconducting (DSC) ori V
der parameter is defined at site i as iD = (iD+e ,i + iD−e ,i − x x
l=i
†
−Hij∗ σ¯
jσ v njσ¯
†
V c nl − n¯ † c c iσ . 2 | rl − r i | iσ i,σ
un
ij
units of −|e |) is ni = c i↑ c i↑ + c i↓ c i↓ = ni↑ + ni↓ . The DSC or-
i,σ
ij
N Hijσ
ij =
We start with an effective mean-field t–t –U –V d –V c model defined on to a two-dimensional lattice of YBCO, where the onsite repulsion U is responsible for the AF order, the nearestneighbor attraction V d for the d-wave superconducting paring, and V c denotes the long-range Coulomb interaction strength. The effect Hamiltonian can be written as
H =−
The Hamiltonian [Eq. (1)] can be diagonalized by solving Bogoliubov–de Gennes (BdG) Eq. (2) self-consistently with the periodic boundary condition.
†
†
der is defined as i j = 2d c i↑ c j↓ − c i↓ c j↑ , where V d represents the strength of DSC order. The sum over i , j in the hopping term includes both the nearest- and next nearest-neighbors, whereas the pairing attraction involves only the nearest-neighbor. In the long-range Coulomb interaction term the sum over l runs over the whole magnetic unit cell. The long-range Coulomb interaction strength is V c = e 2 /(2εe a2 ) with εe the effective dielectric constant that has a general tendency of increase with hole concentration in HTSCs [22]. Consequently, one can reasonably use a large value of V c for underdoped sample, but a small one for overdoped case. In = ∇ × A, the hopping integral the presence of a magnetic field h can be expressed as t i j = t¯ exp[i Φπ 0
ri rj
· d r ]. Assume each magA
netic unit cell of dimension ( N x × N y )a2 = 48 × 24a2 can accommodate two superconducting vortices. The average field approximates to B 24 Tesla( H c1 ). Since the magnetic field induced by the supercurrent around the vortex core is so small compared with the external magnetic field, the screen current effect can be ne = B (− y , x, 0), with glected. We then choose a Landau gauge of A 2 x, y as the x, y component of the position vector r .
( ri + rj )/2 ri
· d r ]. Each A
time when the on-site repulsion or the long-range Coulomb interaction V c is varied, the chemical potential μ needs to be adjusted ¯ The iteration processes have been carried out until for a fixed n. the relative difference of order parameter between two consecutive iteration steps is less than 10−4 . In our calculations, the length is measured in units of the lattice constant a, and the energy is in the nearest-neighbor hoping integral parameter t¯. We set V d = 1 and the next-nearestneighbor hopping t = −0.2. The averaged charge density is chosen as n¯ = 0.875, corresponding to an underdoping level x = 0.125. We report on the results for V c ∼ (0 ∼ 0.35), which corresponds to a static dielectric constant from εe → ∞ to εe ∼ 15 for doped cuprate superconductors [22]. The calculation is performed in a very low temperature region. 3. Results and discussions First, we study the effect of the on-site repulsion U on the LDOS without including the long-range Coulomb interaction. The LDOS is defined by
ρi ( E ) = −
1 Mx M y
2N n,k 2 u f ( E n,k − E ) i
n,k
2
+ vni ,k f ( E n,k + E ) .
(6)
Here f ( E ) is the derivative of the Fermi distribution function. The summation is averaged over M x × M y wave vectors in first Brillouin zone. Fig. 1 plots the LDOS spectrum at the vortex-core center and at the middle point between two nearest neighbor vortices along the x direction for a small on-site repulsion (U = 1.6) case. The
H.-W. Zhao et al. / Physics Letters A 374 (2010) 1779–1783
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Fig. 1. (Color online.) The LDOS spectrum as a function of energy for the on-site repulsion of U = 1.6. The spin-up and spin-down LDOS at the core center are denoted by the blue thick line (a) and by the black thin line (b). The total LODS is presented in (c) with the black thick line for that at the vortex-core center, and red thin line at the midpoint between two nearest-neighbor vortices along the x-direction. The wave vectors in first Brillouin zone are M x × M y = 24 × 24.
Fig. 2. (Color online.) The LDOS spectrum as a function of energy for the on-site repulsion of U 1 = 2.2 [(a) and (c)], and U 2 = 2.4 [(b) and (d)]. The spin-up and spin-down LDOS at the core center are denoted by the blue thick line and by the black thin line in the left panel [(a) and (b)]. In the right panels ((c) and (d)), the black thick line is at the vortex-core center, and red thin line is at the midpoint between two nearest-neighbor vortices along the x-direction. The arrow points to the vortex core states (VCS). The wave vectors in first Brillouin zone are M x × M y = 24 × 24.
AF order amplitude is in the order of 10−6 (not shown), and the system behaves as a pure d-wave ordering. Figs. 1(a) and 1(b) show the spin-up and spin-down LDOS at the vortex-core center, they are overlapping with energy and have a single peak around the Fermi energy. The total LDOS at the core center in Fig. 1(c) (black thick line) shows a single peak at the Fermi energy, which refers to the ZBCP for superconductors of energy nodes (lines). On the contrary, at a sufficiently large on-site repulsion strength, for example, U 2.0, the AF-like SDW amplitude would become significant that will lift out the energy degeneracy of spinup and spin-down electrons, and a reduction or even a complete depression of the ZBCP. Figs. 2(a) and 2(b) depict the LDOS of spin-up and spin-down electrons at the core center for two different on-site repulsion strengths of U = 2.2 and U = 2.4. As shown in Figs. 2(c) and 2(d) for U 1 = 2.2 and for U 2 = 2.4, respectively, the total LDOS spectrum of the vortex core states splits into a double-peak structure around the Fermi level or the zero energy, which are qualitatively in agreement with the differential conductance spectrum of the STM experiments on YBCO [2] and BSCCO [3]. Furthermore, a lower LDOS in Fig. 2(d) than that in Fig. 2(c) around the Fermi level (E = 0), coupled with a shift in the core state peak energies from E 1 = −0.24 for U 1 to E 2 = −0.25 for U 2 , implies an enhancement in the spin-split of vortex core states with
the on-site repulsion strength. The LDOS far away from the center (refer to the red thin line in (c) and (d)) is almost invariant with the on-site repulsion strength, because of a substantially reduction in the AF order on the site. These results confirm that the splitting of the ZBCP is closely related to the AF order in the vortex cores. We present the LDOS maps under a fixed energy, for different U values in Fig. 3, which indicate how the vortex core state patterns vary with the on-site repulsion. The fixed energies we chose are close to the peak-energies for the VCS. For the on-site repulsion strength of U 1 = 2.2 (Figs. 3(a) and 3(b)), close to the core center the LDOS map shows a four-fold symmetry structure with 2D modulations. For a large repulsion of U 2 = 2.4, however, the LDOS map shows a strip structure with a periodicity of 5a0 that extends to the whole unit cell. This seems to be in agreement with the early common consensus that a large on-site repulsion will induce a SDW of stripe texture. However, it doesn’t sound well for the recent STM observations of the checkerboard pattern with the two-dimensional modulations in underdoped Nax Ca2−x CuO2 Cl2 [15] and BSCCO [16] samples of a strong AF order. We now discuss the effect of the long-range Coulomb interaction on LDOS spectrum and on the core states patterns. For straightforward, we investigate the large on-site repulsion case of U = 2.4, in which the SDW of the stripe texture existed when the
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H.-W. Zhao et al. / Physics Letters A 374 (2010) 1779–1783
Fig. 3. (Color online.) He LDOS maps at the energies close to those of the VCS peaks. The left and right panels are for U 1 = 2.2 and U 2 = 2.4, respectively. The bottom row shows the 2D density plots of the top row. The wave vectors in first Brillouin zone are M x × M y = 24 × 24.
Fig. 4. (Color online.) The LDOS as a function of energy with the long-range Coulomb interaction, (a) and (c) for V c = 0.06, and (b) and (d) for V c = 0.15. In the left panels are the spin-up LDOS (blue thick line) and spin-down LDOS (black thin line) at the core center. In the right panels are the LDOS at the vortex-core center (black thick line) and at the midpoint between two nearest-neighbor vortices along the x-direction (red thin line). The arrow points to the vortex core states (VCS). The on-site repulsion is fixed at U = 2.4.
long-range Coulomb interaction has not been included. We first check the LDOS spectrum. As shown in Figs. 4(c) and 4(d), the relative height of the vortex core states reduced [referred to those in Fig. 2], accompanied with a shift in the VCS peaks from E = −0.25 to E 1 = −0.20 for V c = 0.06, and to E 2 = −0.10 for V c = 0.15, respectively. Clearly, this is due to the competition between the long-range Coulomb repulsion with the on-site repulsion induced AF-like SDW ordering. As shown in the effective model Hamiltonian (2), opposite signs present for the long-range Coulomb interaction term and the on-site repulsion term when the local density of quasiparticles is larger than the averaged density. That is the case for an AF core where an extra electron accumulation has been tested [23]. The long-range coulomb interaction can provide an intrinsic electron depletion mechanism by inducing the so-called ferromagnetic polarons or the ‘ferros’ [20,21], which reside at the intersections of a crossed array of antiphase domain walls of the AF order parameter to accommodate holes. This generates a CDW ordering competition with the SDW ordering, leading in general to
a reduction in the net AF order parameter amplitude and favors to lower-lying quasiparticle excitations. We should emphases that the long-range Coulomb interaction may also split the spin degeneracy of core states once the associated CDW and the already existing SDW orderings are in phase for some wave vectors in the first Brillouin zone. As a result, the two featured peaks of the core states may move more close to the Fermi level, but remain distinctive, as shown in Figs. 4(a) and 4(b). Finally, we examine the effect of the long-range Coulomb interaction on the LDOS maps. The energies chosen here are those close to the VCS peaks. The left and right panels in Fig. 5 are the LDOS maps at the energy levels of E 1 = −0.20 and E 2 = −0.10 for V c = 0.06 and V c = 0.15, respectively. The other parameters are the same with those in Fig. 4. The left panel shows that the spatial modulation of the LDOS map remains the stripe structure for a small long-range Coulomb interaction strength (V c = 0.06), and so is the periodicity of the stripe modulation of 5a0 . When the strength of the long-range Coulomb interaction becomes large, for
H.-W. Zhao et al. / Physics Letters A 374 (2010) 1779–1783
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Fig. 5. (Color online.) The LDOS maps at the fixed energy close to the VCS peak-energy. The left and right panels are for V c = 0.06 and V c = 0.15. The bottom row shows the 2D density plots of the top row. The on-site repulsion is fixed at U = 2.4. The wave vectors in first Brillouin zone are M x × M y = 24 × 24.
example, V c = 0.15, the one-dimensional stripe structure was replaced by a two-dimensional modulation with four-fold symmetry. A close examination reveals that there are four unit cell with period of 4a0 localized near the vortex core, in agreement with the STM experiments [15,16]. 4. Conclusions In conclusion, we have studied the effect of the antiferromagnetism and the long-range Coulomb interaction on the LDOS spectrum in underdoped HTSCs by numerically solving the BdG equations based on an effective model Hamiltonian on a square lattice. We show that the splitting of the ZBCP is closely related to the on-site repulsion induced AF-like SDW ordering. The LDOS map shows a transition between two-dimensional modulation and onedimensional stripe structure with the effective AF order strength. The long-range Coulomb interaction can both induce a spin-flip which gives rise to a hopping between spin-up and spin-down electrons and can split the spin degeneracy of core states as well. As a result, the two featured LDOS peaks of the core states may move more close to the Fermi level, but remain distinctive. The four-fold symmetry modulation with a periodicity of 4a0 is recovered for the underdoped case of large on-site repulsion as a reasonable large long-range Coulomb interaction has been included. Recent scanning-tunneling-microscopy experiments [15,16] may be understood in terms of the present results. Acknowledgements This work was supported by National Natural Science Foundation of China under Grants Nos. 60671042 and 60971053, and by Science and Technology Committee of Shanghai Municipal for key research projects under Grant No. 09JC1406000, and by Shanghai Municipal Education Committee under Grant No. 10zz63. H.W.Z. acknowledges Innovation Funds for Graduates of Shanghai University, China (2010).
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Local density of states of high-Tc superconductors with long-range Coulomb interaction Hong-Wei Zhao, Hao Meng, Ling-feng Zhang, Lin Wen, Liang-ma Shi, Guo-Qiao Zha, Shi-Ping Zhou ∗ Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, PR China
a r t i c l e
i n f o
Article history: Received 7 December 2009 Received in revised form 22 January 2010 Accepted 29 January 2010 Available online 2 February 2010 Communicated by R. Wu Keywords: High-Tc superconductors Local density of states Long-range Coulomb interaction
a b s t r a c t The local density of state (LDOS) has been studied by solving the mean-field Bogoliubov–de Gennes equations based on a model Hamiltonian with competing antiferromagnetic (AF) and d-wave superconducting orderings in the presence of a long-range Coulomb interaction. The vortex-core state with energies slightly above and below the Fermi level shows a four-fold symmetry with two-dimensional modulations under small on-site repulsion strength U , whereas it has a stripe structure for large U . The four-fold symmetry modulation with a periodicity of 4a0 is recovered as a long-range Coulomb interaction has been included. Recent scanning-tunneling-microscopy experiments may be understood in terms of the present results. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Vortex core states of high-temperature superconductors (HTSCs) have attracted a lot of attention in the recent years [1–16]. A fundamental phenomenon of the core states is the splitting of the zero bias conductance peak (ZBCP), as observed in the scanningtunneling-microscopy (STM) experiments in YBa2 Cu3 O7−x (YBCO) [2] and in Bi2 Sr2 CaCu2 O8+x (BSCCO) [3]. The conventional BCS theory that predicts a strong enhancement of the quasiparticle peak around the Fermi level for a superconductor of energy nodes (lines) seemed unlikely to provide a good account for such observations [4]. Several modified models have been proposed [5–10]. On the assumption of the coexistence of an antiferromagnetic (AF) order with the d-wave superconductivity (DSC), Zhu and Ting [10] obtained a local density of states (LDOS) spectrum of double peaks slightly below and above the Fermi level. The AF fluctuation around the vortex cores has been probed by nuclear magnetic resonance experiments [11], which support the coexistence of the AF and d-wave orderings. It has also been found that LDOS maps of HTSCs are closely related to the on-site repulsion induced AF order. For instance, the stripe modulations were found in underdoped copper oxide superconductors [12] where the AF order is strong due to a large U , whereas a square pattern was observed in the low-energy scanning-tunneling-microscopy experiments [13,14] for overdoped
*
Corresponding author. E-mail address: [email protected] (S.-P. Zhou).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.01.064
samples where the AF order is negligible. Since a large on-site repulsion is assumed to induce a stripe phase, the recent STM experiments for underdoped BSCCO [15] and Nax Ca2−x CuO2 Cl2 [16] samples that have probed a checkerboard pattern with a twodimensional (2D) modulation are very interesting. As we will show in this Letter, an inclusion of the long-range Coulomb interaction into the model Hamiltonian is expected to provide a good interpretation for above observations, regarding both the double peak LDOS spectrum and the four-fold symmetric shape of the lowlying quasiparticle states oriented with the nodal directions of the dx2 − y 2 wave superconductor. An inclusion of the long-range coulomb interaction in the model Hamiltonian lies mainly on the following two points. Firstly, vortex cores will intrinsically charged up in HTSCs of a small value of k F ξ due to Coulomb repulsion between conducting electrons, as commonly observed in a neutral superfluid system. This would induce a charge density wave (CDW) competing with the AF order associated spin-density wave (SDW) that may modulate the vortex structure and the LODS spectrum. Secondly, unlike what occurred to a metallic superconductor, the Thomas–Fermi screening effect was substantially reduced for HTSCs since the short wave length part of the Coulomb interaction between conducting electrons, which has been omitted in the TF approximation, becomes important [17–19]. In the present Letter, we investigate the LDOS for the underdoped HTSCs based on an effective mean-field t–t –U –V d –V c model [see below]. By solving the Bogoliubov–de Gennes equations in a 2D Lattice, we show that the splitting of quasiparticle excitations at the Fermi level and the core states maps are closely
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H.-W. Zhao et al. / Physics Letters A 374 (2010) 1779–1783
related to the model parameters. For instance, in the absence of the long-range Coulomb repulsion, spatial distribution of the core states shows a two-dimensional pattern to the one-dimensional stripe transition with the on-site repulsion strength U . Meanwhile, a well-defined double peak LDOS spectrum developed for a reasonable large U . The long-range coulomb interaction can provide an intrinsic electron depletion mechanism by inducing the so-called ferromagnetic polarons or the ‘ferros’ [20,21], which reside at the intersections of a crossed array of antiphase domain walls of the AF order parameter to accommodate holes. The ensuing CDW competes with the existing AF-like SWD that favors to lower-lying quasiparticle excitations. As a result, the two featured peaks of the core states would move more close to the Fermi level. The four-fold symmetric pattern is expected to recover in the underdoped case with a reasonable large long-range Coulomb repulsion strength. Interestingly, the LDOS maps shows a change in the periodicity from 5a0 to 4a0 with the long-range Coulomb repulsion strength V c varying from 0 to 0.15 [in units of the nearest-neighboring hopping integrals]. Recent scanning-tunneling-microscopy experiments [15,16] may be understood in terms of the present results. 2. Model
†
(t i j ,σ )c iσ c jσ +
ij,σ
+ +
†
U niσ¯ − μ c iσ c iσ
∗ij
j
† † ij c i↑ c j↓
+ H.c.
= εn
uniσ
v niσ¯
(2)
, √
† † where Hijσ = −t ij + V2c (c i+δ,σ c iσ +c √
i+ 2δ,σ
nl −¯n
c iσ / 2 )+[U niσ¯ −
]δij and ij = V2d c i↑ c j↓ − c i↓ c j↑ − V2c c i↓ c j↑ . √ The index i + δ and i + 2δ are the four nearest and next-nearest
μ+
Vc 2
l=i
r i ,l
neighbors of the i-th site, respectively. The self-consistent conditions are:
ni↑ =
2N n 2 u f ( E n ),
(3)
i
n =1
ni↓ =
2N n 2
v 1 − f ( E n ) , i
(4)
n =1
2N Vd n =1
4
uni vnj ∗ + vni ∗ unj tanh
V c n∗ n − vi u j 1 − f ( E n ) ,
En
2k B T (5)
2
where uni = (− v ni↑∗ , uni↑ ) and vni = (uni↓∗ , v ni↓ ) denote the 2Ndimensional wave function vectors, and f ( E ) = 1/(e β E + 1) is the Fermi–Dirac distribution function. The vortex charge density (in †
iD,i+e y − iD,i−e y )/4, where ijD = ij exp[i Φπ0 (1)
†
Here c iσ (c iσ ) is the electron creation (annihilation) operator of spin index σ on the i-th site. μ is the chemical potential determining the averaged electron density n¯ = iσ niσ / N (N = N x × N y is the linear dimension of the unit cell of the vor†
tex lattice). niσ = c iσ c iσ is the number operator. The on-site repulsion will generate an AF-like spin-density wave (SDW) order †
SDW = U c i↑ c i↑ − c i↓ c i↓ . The d-wave superconducting (DSC) ori V
der parameter is defined at site i as iD = (iD+e ,i + iD−e ,i − x x
l=i
†
−Hij∗ σ¯
jσ v njσ¯
†
V c nl − n¯ † c c iσ . 2 | rl − r i | iσ i,σ
un
ij
units of −|e |) is ni = c i↑ c i↑ + c i↓ c i↓ = ni↑ + ni↓ . The DSC or-
i,σ
ij
N Hijσ
ij =
We start with an effective mean-field t–t –U –V d –V c model defined on to a two-dimensional lattice of YBCO, where the onsite repulsion U is responsible for the AF order, the nearestneighbor attraction V d for the d-wave superconducting paring, and V c denotes the long-range Coulomb interaction strength. The effect Hamiltonian can be written as
H =−
The Hamiltonian [Eq. (1)] can be diagonalized by solving Bogoliubov–de Gennes (BdG) Eq. (2) self-consistently with the periodic boundary condition.
†
†
der is defined as i j = 2d c i↑ c j↓ − c i↓ c j↑ , where V d represents the strength of DSC order. The sum over i , j in the hopping term includes both the nearest- and next nearest-neighbors, whereas the pairing attraction involves only the nearest-neighbor. In the long-range Coulomb interaction term the sum over l runs over the whole magnetic unit cell. The long-range Coulomb interaction strength is V c = e 2 /(2εe a2 ) with εe the effective dielectric constant that has a general tendency of increase with hole concentration in HTSCs [22]. Consequently, one can reasonably use a large value of V c for underdoped sample, but a small one for overdoped case. In = ∇ × A, the hopping integral the presence of a magnetic field h can be expressed as t i j = t¯ exp[i Φπ 0
ri rj
· d r ]. Assume each magA
netic unit cell of dimension ( N x × N y )a2 = 48 × 24a2 can accommodate two superconducting vortices. The average field approximates to B 24 Tesla( H c1 ). Since the magnetic field induced by the supercurrent around the vortex core is so small compared with the external magnetic field, the screen current effect can be ne = B (− y , x, 0), with glected. We then choose a Landau gauge of A 2 x, y as the x, y component of the position vector r .
( ri + rj )/2 ri
· d r ]. Each A
time when the on-site repulsion or the long-range Coulomb interaction V c is varied, the chemical potential μ needs to be adjusted ¯ The iteration processes have been carried out until for a fixed n. the relative difference of order parameter between two consecutive iteration steps is less than 10−4 . In our calculations, the length is measured in units of the lattice constant a, and the energy is in the nearest-neighbor hoping integral parameter t¯. We set V d = 1 and the next-nearestneighbor hopping t = −0.2. The averaged charge density is chosen as n¯ = 0.875, corresponding to an underdoping level x = 0.125. We report on the results for V c ∼ (0 ∼ 0.35), which corresponds to a static dielectric constant from εe → ∞ to εe ∼ 15 for doped cuprate superconductors [22]. The calculation is performed in a very low temperature region. 3. Results and discussions First, we study the effect of the on-site repulsion U on the LDOS without including the long-range Coulomb interaction. The LDOS is defined by
ρi ( E ) = −
1 Mx M y
2N n,k 2 u f ( E n,k − E ) i
n,k
2
+ vni ,k f ( E n,k + E ) .
(6)
Here f ( E ) is the derivative of the Fermi distribution function. The summation is averaged over M x × M y wave vectors in first Brillouin zone. Fig. 1 plots the LDOS spectrum at the vortex-core center and at the middle point between two nearest neighbor vortices along the x direction for a small on-site repulsion (U = 1.6) case. The
H.-W. Zhao et al. / Physics Letters A 374 (2010) 1779–1783
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Fig. 1. (Color online.) The LDOS spectrum as a function of energy for the on-site repulsion of U = 1.6. The spin-up and spin-down LDOS at the core center are denoted by the blue thick line (a) and by the black thin line (b). The total LODS is presented in (c) with the black thick line for that at the vortex-core center, and red thin line at the midpoint between two nearest-neighbor vortices along the x-direction. The wave vectors in first Brillouin zone are M x × M y = 24 × 24.
Fig. 2. (Color online.) The LDOS spectrum as a function of energy for the on-site repulsion of U 1 = 2.2 [(a) and (c)], and U 2 = 2.4 [(b) and (d)]. The spin-up and spin-down LDOS at the core center are denoted by the blue thick line and by the black thin line in the left panel [(a) and (b)]. In the right panels ((c) and (d)), the black thick line is at the vortex-core center, and red thin line is at the midpoint between two nearest-neighbor vortices along the x-direction. The arrow points to the vortex core states (VCS). The wave vectors in first Brillouin zone are M x × M y = 24 × 24.
AF order amplitude is in the order of 10−6 (not shown), and the system behaves as a pure d-wave ordering. Figs. 1(a) and 1(b) show the spin-up and spin-down LDOS at the vortex-core center, they are overlapping with energy and have a single peak around the Fermi energy. The total LDOS at the core center in Fig. 1(c) (black thick line) shows a single peak at the Fermi energy, which refers to the ZBCP for superconductors of energy nodes (lines). On the contrary, at a sufficiently large on-site repulsion strength, for example, U 2.0, the AF-like SDW amplitude would become significant that will lift out the energy degeneracy of spinup and spin-down electrons, and a reduction or even a complete depression of the ZBCP. Figs. 2(a) and 2(b) depict the LDOS of spin-up and spin-down electrons at the core center for two different on-site repulsion strengths of U = 2.2 and U = 2.4. As shown in Figs. 2(c) and 2(d) for U 1 = 2.2 and for U 2 = 2.4, respectively, the total LDOS spectrum of the vortex core states splits into a double-peak structure around the Fermi level or the zero energy, which are qualitatively in agreement with the differential conductance spectrum of the STM experiments on YBCO [2] and BSCCO [3]. Furthermore, a lower LDOS in Fig. 2(d) than that in Fig. 2(c) around the Fermi level (E = 0), coupled with a shift in the core state peak energies from E 1 = −0.24 for U 1 to E 2 = −0.25 for U 2 , implies an enhancement in the spin-split of vortex core states with
the on-site repulsion strength. The LDOS far away from the center (refer to the red thin line in (c) and (d)) is almost invariant with the on-site repulsion strength, because of a substantially reduction in the AF order on the site. These results confirm that the splitting of the ZBCP is closely related to the AF order in the vortex cores. We present the LDOS maps under a fixed energy, for different U values in Fig. 3, which indicate how the vortex core state patterns vary with the on-site repulsion. The fixed energies we chose are close to the peak-energies for the VCS. For the on-site repulsion strength of U 1 = 2.2 (Figs. 3(a) and 3(b)), close to the core center the LDOS map shows a four-fold symmetry structure with 2D modulations. For a large repulsion of U 2 = 2.4, however, the LDOS map shows a strip structure with a periodicity of 5a0 that extends to the whole unit cell. This seems to be in agreement with the early common consensus that a large on-site repulsion will induce a SDW of stripe texture. However, it doesn’t sound well for the recent STM observations of the checkerboard pattern with the two-dimensional modulations in underdoped Nax Ca2−x CuO2 Cl2 [15] and BSCCO [16] samples of a strong AF order. We now discuss the effect of the long-range Coulomb interaction on LDOS spectrum and on the core states patterns. For straightforward, we investigate the large on-site repulsion case of U = 2.4, in which the SDW of the stripe texture existed when the
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Fig. 3. (Color online.) He LDOS maps at the energies close to those of the VCS peaks. The left and right panels are for U 1 = 2.2 and U 2 = 2.4, respectively. The bottom row shows the 2D density plots of the top row. The wave vectors in first Brillouin zone are M x × M y = 24 × 24.
Fig. 4. (Color online.) The LDOS as a function of energy with the long-range Coulomb interaction, (a) and (c) for V c = 0.06, and (b) and (d) for V c = 0.15. In the left panels are the spin-up LDOS (blue thick line) and spin-down LDOS (black thin line) at the core center. In the right panels are the LDOS at the vortex-core center (black thick line) and at the midpoint between two nearest-neighbor vortices along the x-direction (red thin line). The arrow points to the vortex core states (VCS). The on-site repulsion is fixed at U = 2.4.
long-range Coulomb interaction has not been included. We first check the LDOS spectrum. As shown in Figs. 4(c) and 4(d), the relative height of the vortex core states reduced [referred to those in Fig. 2], accompanied with a shift in the VCS peaks from E = −0.25 to E 1 = −0.20 for V c = 0.06, and to E 2 = −0.10 for V c = 0.15, respectively. Clearly, this is due to the competition between the long-range Coulomb repulsion with the on-site repulsion induced AF-like SDW ordering. As shown in the effective model Hamiltonian (2), opposite signs present for the long-range Coulomb interaction term and the on-site repulsion term when the local density of quasiparticles is larger than the averaged density. That is the case for an AF core where an extra electron accumulation has been tested [23]. The long-range coulomb interaction can provide an intrinsic electron depletion mechanism by inducing the so-called ferromagnetic polarons or the ‘ferros’ [20,21], which reside at the intersections of a crossed array of antiphase domain walls of the AF order parameter to accommodate holes. This generates a CDW ordering competition with the SDW ordering, leading in general to
a reduction in the net AF order parameter amplitude and favors to lower-lying quasiparticle excitations. We should emphases that the long-range Coulomb interaction may also split the spin degeneracy of core states once the associated CDW and the already existing SDW orderings are in phase for some wave vectors in the first Brillouin zone. As a result, the two featured peaks of the core states may move more close to the Fermi level, but remain distinctive, as shown in Figs. 4(a) and 4(b). Finally, we examine the effect of the long-range Coulomb interaction on the LDOS maps. The energies chosen here are those close to the VCS peaks. The left and right panels in Fig. 5 are the LDOS maps at the energy levels of E 1 = −0.20 and E 2 = −0.10 for V c = 0.06 and V c = 0.15, respectively. The other parameters are the same with those in Fig. 4. The left panel shows that the spatial modulation of the LDOS map remains the stripe structure for a small long-range Coulomb interaction strength (V c = 0.06), and so is the periodicity of the stripe modulation of 5a0 . When the strength of the long-range Coulomb interaction becomes large, for
H.-W. Zhao et al. / Physics Letters A 374 (2010) 1779–1783
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Fig. 5. (Color online.) The LDOS maps at the fixed energy close to the VCS peak-energy. The left and right panels are for V c = 0.06 and V c = 0.15. The bottom row shows the 2D density plots of the top row. The on-site repulsion is fixed at U = 2.4. The wave vectors in first Brillouin zone are M x × M y = 24 × 24.
example, V c = 0.15, the one-dimensional stripe structure was replaced by a two-dimensional modulation with four-fold symmetry. A close examination reveals that there are four unit cell with period of 4a0 localized near the vortex core, in agreement with the STM experiments [15,16]. 4. Conclusions In conclusion, we have studied the effect of the antiferromagnetism and the long-range Coulomb interaction on the LDOS spectrum in underdoped HTSCs by numerically solving the BdG equations based on an effective model Hamiltonian on a square lattice. We show that the splitting of the ZBCP is closely related to the on-site repulsion induced AF-like SDW ordering. The LDOS map shows a transition between two-dimensional modulation and onedimensional stripe structure with the effective AF order strength. The long-range Coulomb interaction can both induce a spin-flip which gives rise to a hopping between spin-up and spin-down electrons and can split the spin degeneracy of core states as well. As a result, the two featured LDOS peaks of the core states may move more close to the Fermi level, but remain distinctive. The four-fold symmetry modulation with a periodicity of 4a0 is recovered for the underdoped case of large on-site repulsion as a reasonable large long-range Coulomb interaction has been included. Recent scanning-tunneling-microscopy experiments [15,16] may be understood in terms of the present results. Acknowledgements This work was supported by National Natural Science Foundation of China under Grants Nos. 60671042 and 60971053, and by Science and Technology Committee of Shanghai Municipal for key research projects under Grant No. 09JC1406000, and by Shanghai Municipal Education Committee under Grant No. 10zz63. H.W.Z. acknowledges Innovation Funds for Graduates of Shanghai University, China (2010).
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