Checkerboard Pattern and its Relation with Antinodal Electronic States

To address the relation between the anomalous behaviors observed in ARPES spectra and the charge ordering in the STM measurements of the HTSCs, a group of the US scientists in collaboration with Japanese scientists have been investigated this situation in Ca_2-xNa_xCuO₂Cl₂ (Na-CCOC). They have measured the ARPES spectra of Na-CCOC with x=0.05, 0.10, 0.12 (T_c=0, 13, and 22K, respectively), which also shows a two-dimensional charge order (2DCO) with periodicity of 4a, i.e. checkerboard.

Experimental results:

• Raw data (Fig. 1A-C) indicates strong anisotropy such that the maximum intensity in the nodal direction (π,π) drops off rapidly toward antinodal direction (π,0).
• The antinodal states with the smallest intensity in the momentum distribution curve, are well nested and separated by nesting wavevector |q|~2π/4a. q can be compared with the checkerboard modulation in this system (Fig. 2A&B).
• In addition to the correspondence in wavevector, there is a similar behavior in the bias voltage independence of the STM data and energy independence of the antinodal states below 50 meV in this study (Fig. 2C).  
• The antinodal and nodal k_F are nearly constant with doping (Fig. 2E).
• The anisotropy between the nodal and antinodal can also be observed in the energy distribution curves (EDCs) along Fermi surface (FS, Fig. 3A). For x=0.10 and 0.12, a sharp peak is observed only near the nodal direction and its spectral weight depletions over a wide range (~200 meV) by going to antinodal direction, in similar with the observation in underdoped cuprates  BSCCO and LSCO.   

Discussion:

• The relatively weak doping and ω dependence of the antinodal k_F is in stark contrast to the expected behavior of a near-E_F van Hove singularity.
• The intensity anisotropy in Fig. 1A-C is not due simply to the opening of a d-wave superconducting (d-SC) gap, because we expect that quasiparticles should remain well defined over the entire Brillouin zone.
• The seemingly incoherent antinodal features cannot be produced from FS nesting or charge ordering alone, because in charge density wave instability coherent quasiparticles still exist after formation of CDW.
• The similarity between antinodal spectra of Na-CCOC and La_1.2Sr_1.8Mn₂O₇ (in addition to the similarity in pseudogap state and CDW order with a nesting wavevector) indicates that cuprates and manganites may share similar general phenomenology in spite of differences in detailed of microscopic interactions.     
• The similarities of sharp antinodal excitations, the doping dependent growth of low-energy weight and the k anisotropy of FS in the Na-CCOC and LSCO systems, suggest an intrinsic commonality between the low-lying excitations for different HTSCs. This may imply a generic microscopic origin for these essential nodal states irrespective of other ordering tendencies.
• Concerning the same origin (typically electron-phonon interaction) of 2DCO and d-SC and their competing in many materials, both 2DCO and d-SC compete for the antinodes, such that the strength of one order parameter should come at the expense of the other.  So, we expect to see strong 2DCD in Na-CCOC which is a rather weak SC, and less pronounced 2DCO in BSCCO which is better HTSC, in agreement with experiment. It is possible that critical fluctuations between the 2DCO and another ordered state could result in the antinodal states, although it is not evident whether the nodal quasiparticles would still remain well defined. 
• Another explanation for broad antinodal states is based on Franck-Condon models, which explains the high energy pseudogap behavior in both the cuprates and manganites as a suppressed tail of incoherent spectral weight.

As a summary, this study reveals that the 2DCO in Na-CCOC is associated with strongly suppressed antinodal electronic states with nesting wavevector |q|~2π/4a.

Ref.: Kyle M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, Z.-X. Shen; Science 307 (2005) 901.   

Electronic Structure in the Vortex Cores of HTSCs

A group of scientists from the USA have measured the nuclear magnetic resonance (NMR) spectrum of optimally doped YBCO with ~60% ¹⁷O-enrichment (T_c=92.5 K) in the present of high magnetic fields (H₀=13-37 T). This study gives some information about the electronic structure of HTSCs near the vortex cores, and helps to understand the unexpected developing of antiferromagnetism order in the superconducting state of HTSCs by applying magnetic field. Applying high H₀ helps to increase the regions of samples occupied by vortex cores and finally the vortex core part of NMR spectrum (~15% in 37 T), and also suppresses spin diffusion and vortex vibrations which complicate the interpretation of the data at low H₀.  

The quasiparticles in the nodes of superconducting gap could be probed with NMR, with the rate given by:     

T₁^-1≈<|ε-Z+D_i||ε+Z+D_f|>           (1)

 

Where ε~k_BT, Z=-(1/2)γ_ehH₀ is the Zeeman energy and D_α=V_F,α.P_s is the Doppler shift from the supercurrent momentum P_s (α=i,f, V_F,α is the Fermi velocity for initial and final states). The temperature and filed dependence of T₁^-1 for different ranges of ε, Z and D (related to T, H₀ and position relative to the vortex core, respectively) for different processes (F₁, F₂, and F₃ in Fig. 3) give information about node excitations.   

Results:

• At H_int~0 where the D is very small due to the cancellation of supercurrents (Fig. 1), the increasing of T₁^-1 could be related to the Zeeman Effect because T₁^-1~<|Z²-ε²|> from Eq. 1 (Fig. 2).

• With going to higher H_int the T₁^-1 increases with a factor of 4-5. In this region the Doppler term dominates both ε and Z, so we expect T₁^-1~ D²~P_s²~r^-2 (r is distance from the core center) which has been observed experimentally in this study (Fig. 2).

• In the intermediate H_int (outside the vortex core) T₁^-1 is simply displaced in the vertical axis by H₀. This evidence could be related to the processes that Doppler term changes sign for initial and final states, i.e., F₁ and F₂ which T₁^-1~D²+Z², while T₁^-1~D²-Z² for F₁ process. The latter does not have a simple shift with H₀ especially when D>Z.

• Calculation using the imaginary part of electronic susceptibility (extracted from the data in this study) indicates that to derive experimental results of T₁^-1(H₀) in the intermediate H_int, the susceptibility must be strongly peaked near a wavevector corresponding to F₂ and F₃ processes relative to zero wavevector. This indicates that strong antiferromagnetic correlations coexist with superconductivity at low temperature. 

• Near the vortex core, the NMR rate increases considerably with H₀ and is most pronounced at the lowest temperatures (Fig. 4). For 13 T the rate decreases after reaching a maximum reflecting a drop-off in p_s. This substantial increasing could be related to vortex core states.

• The latter result is not compatible with the zero-bias anomaly peak in tunneling experiments of superconductors induced by localized states near surface or vortex cores. Instead there seems to be a mini-gap ~ ±5 meV much sharper than the variation of the density of states outside the vortex core.   

• An alternative explanation for the results in the vortex core regions is existence of antiferromagnetism order in the vortex cores, because filed induced antiferromagnetic excitations could strongly enhance the NMR rate in agreement with the result of this study.

Ref.: V. F. Mitrovic, E. E. Sigmund, M. Eschrig, H. N. Bachman, W. P. Halperin, A. P. Reyes, P. Kuhns & W. G. Moulton; Nature 413 (2001) 501.  

LDOS Modulations: Quasiparticle Interference or New Electronic State?

Observed local density of states (LDOS) modulations in the tunneling spectroscopy of HTSCs has started extensive research for determining its origin and consequences: two important possibilities are creation of new electronic order in the pseudogap, or interference of electronic states. To make clearer this subject, a group of scientist from the USA, Japan an Canada have investigated the differential tunneling conductance (G(V)≡ dI/dV α LDOS(E=eV)) for Bi2212 crystals between underdoped and slightly overdoped (T_c=78-85 K) at 4.2 K. They have analyzed Fourier transformation of real-space LDOS which gives valuable information for electronic states at k-space and scattering processes of quasiparticle, and enable to compare the STM and ARPES results. 

Their most important results:

• Periodic LDOS modulations have been observed in all samples with different voltage bias, although they have different spatial and wavelengths. The data shows checkerboard-like modulation both along and in 45˚ relative to CuO bond direction (see Fig. 2 and its caption).
• The longest coherence length for any modulation is l~80 Å (correspond to smallest ∆q~0.1 π/a in Fig.3), which may influence by several phenomena such as gap disorder, impurity resonance or oxygen disorder. 
• The q-space displaying LDOS modulation intensity is changing rapidly with energy (Fig. 3). Especially, the peaks in (±π,0) (and (0,±π)) directions appear at finite q at very low energy and goes to (0,0) with increasing of energy. The peaks in (±π,±π) directions, appear and move steadily to large q with increasing energy. The same phenomena have been observed in all six samples in this study, but the exact dispersion of the peaks systematically changes with doping.
• The dispersions of these two peaks are opposite, also it is slower in the (±π,±π). They are in excellent quantitative agreement with the dispersions of the connected Fermi surface segments in both directions, extracted from ARPES measurements (Fig. 4).

The agreement with ARPES confirms the general view that the peaks could be originated from interference of quasiparticles at the same energy and especial q of k-space which connected large joint DOS regions. The authors have discussed that these results could be interpreted as quasiparticles interference effects rather than existence of another electronic ordered state because: (1) all modulations have appreciable dispersion, (2) the dispersion are consistent with scattering between the identified k-space regions of high joint-DOS, (3) the evolution of the dispersions with doping is consistent with expected changes in the Fermi surfaces. New experiments will be required to distinguish between quasiparticle scattering effects or creation electronic ordered state or combination of them as a reason for the LDOS modulations surrounding the vortex core, which its intensity is very stronger than this study. In addition the findings of this study are experimental supports for the mechanism has been presented for incommensurate, dispersive, and small modulation of the superconducting electronic structure and magnetic phenomena detected by neutron scattering. 

Ref.: J. E. Hoffman, K. McElroy, D.-H. Lee, K. M Lang, H. Eisaki, S. Uchida, J. C. Davis; Science 297 (2002) 1148.

Electronic Liquid Crystals in HTSCs

Quantum fluctuations of electrons depending to its strength could create different kind of electronic order: solid, liquid, or liquid crystal such as nematic and smectic phases. A group of scientist from the United States has theoretically investigated these orders in the doped two dimensional (2D) antiferromagnet lattices, which is especially important to understand the various electronic phases in HTSCs. The isolated metallic stripe in 2D Mott insulator shows quantum critical behaviors such that its correlation length diverges and correlation functions fall off as a power of the distance in T=0. Due to existence of a large spin gap in this system the only low energy degrees of freedom are charge density wave (CDW) and superconductivity which their susceptibilities diverge in T→ 0 as

χ_CDW≈∆_sT^-(2-Kc) ; χ_SC~∆_sT^-(2-1/Kc)      (1)

Where K_c is a non-universal critical exponent depends on the magnitude and sign of the interaction (0<K_c<1 for repulsive). Although the CDW susceptibility is more divergent than the CS susceptibility (the system prefers to be CDW), the zero point energy (hω) of the transverse stripe fluctuations could have a significant effect on the competition between CDW and superconductivity. This study shows that by increasing the ratio of hω/V (V is Coulomb interaction) the system has transitions from a solid, to smectic, nematic, superconductivity, and finally isotropic phase (Fig. 2 and 3).

The Hamiltonian of this system could be written as:

H_c=∑_j∫dxV(∆_jY)cos[(√2π(∆_jφ)-2k_F(∆_jL)]       (2)

plus higher harmonics and a term related to the Josephson tunneling of superconducting pairs between adjacent stripes. Here j index the stripes, x is the coordinate along the stripes, Y indicates the magnitude of transverse oscillation, L_j is the arc-length of stripe j, ∆_jL=L_j+1-L_j (and for Y), φ defines the phase of the CDW, and V[∆_jY] includes the degree of coupling of the adjacent stripes. The effect of Josephson coupling could be considered in the same way, which the effective action to first order of J, is

<J>≈J₀exp{(α²/2)<[∆_jY]²>}     (3)

Here we present the most important results and the related experimental evidences:

• All terms (all orders in perturbation theory in powers of V in Eq. 2) that are not invariant under the transformation of φ_j→φ_j+δ_j (for arbitrary δ_j) are non-vanishing only near the surface. So, in the thermodynamic limit the phase φ_j of the CDW fluctuations on neighboring stripes are not locked together and there is no CDW order. For example in the first order, V⁽¹⁾(x;∆φ)~exp{-2k_F²<[∆L(x)]²>} fall-off as a exponential or power of x for T>0 and T=0, respectively (<[∆L(x)]²>~T|x| and ~hωlog|x|, respectively), and the same result for higher order.
• For K_c>1/2 the pair function on an individual stripe diverges as T→0 (Eq. 1) and hence for non-zero J, the smectic phase is always globally superconducting below a finite ordering temperature, T_c~(<J>∆_s)^Kc/(2Kc-1). Because of the broken rotational symmetry, the superconducting state is singlet and its symmetry is necessarily a mixture of ‘s-wave’ and ‘d-wave’.
• Starting from smectic with decreasing hω, the system becomes progressively more classical and has a first order phase transition to a crystalline state. In this phase: 1) the CDWs on neighboring stripes are locked, 2) the transverse stripe fluctuations become the phonons of a fully ordered crystal, 3) superconducting order is destroyed, and 4) the system becomes globally insulating.
• By increasing hω the system becomes more quantum and when the r.m.s. magnitude of the stripe transverse fluctuations become comparable to their spacing, there is a T=0 continuous transition to a quantum nematic phase with having the superconducting order. This phase could be enough narrow such that superconducting state could survive until some larger hω (Fig. 2) due to existence of significant local stripe correlation into the isotropic phase. This phase has a pure symmetry of s or d-wave.   
• At still larger hω there is another continuous transition to an isotropic phase with an anisotropic Fermi liquid ground state, or a low temperature superconductor if there remain sufficient residual interactions.
• The superconducting T_c increases with hω through the smectic and nematic phases (because of the enhancement of J) and decreases at larger hω when the system is going to the isotropic phase.
• For K_c<1/2 and a sufficiently small J, the system remains a (quantum critical) non-Fermi liquid at T=0, which are common in 1D and called ‘Luttinger liquids’.

There are some experimental signatures for the liquid crystal phases. Static peaks corresponding in incommensurate spin and charge stripe order have been observed in the neutron scattering of La_1.6-xNd_0.4Sr_xCuO₄. The stripes are along the CuO direction and the system is simultaneously bulk superconductor. Similar incommensurate peaks have been observed in magnetic neutron scattering factor of La_2-xSr_xCuO₄. Also, in YBa₂Cu₃O_7-δ dynamical incommensurate peaks corresponding to low energy dynamical stripe fluctuations have been found.

Ref.: S. A. Kivelson, E. Fradkin & V. J. Emery; Nature 393 (199 550.  

  

What Happens for Electrons Inside the HTSC Vortex?

The experimental observations of magnetic induced electronic structure are completely different from the pure d-wave BCS models predictions, for example: the absence of zero bias conductance peak or four-fold symmetric of LDOS (local density of states), low energy quasiparticle states about some meV in a radius of 75 Å, etc. Theories considering strong electronic correlation and antiferromagnetic spin fluctuations have presented more reliable results, e.g., phase transition into a magnetic order (either spin or orbital) in coexistence with superconductivity around the core, and resulted special modulations in the quasi-particle DOS. These predictions are in somehow agreement with the experimental results of INS, ENS, and NMR (Inelastic and elastic neutron scattering, and nuclear magneto resonance). To make clearer this subject, a group of scientists from Berkeley and Tokyo, have investigated the magnetic induced electronic structure for slightly overdoped Bi2212 crystals (T_c=89 K, and contain 0.5% of Ni) by STM measurements. To see better these effects two functions, S and power spectra of Fourier transform of S, PS[S]={FT[S]}², rather than the pure differential conductance conducting maps (~dI/dV α  LDOS) are considered.

Some electronic orders surrounding the vortex cores have been found for multiple samples and at 2-7 T magnetic range with the following characteristics:

• There are some peaks in PS pattern labeled by A, B1, and B2 (due to crystal structure and supermodulation) and C (due to electronic states). While the appearance of crystal peaks are related to some small spatial miss-matched of the subtracted LDOS patterns (Eq. 1), the peak C which is absent in the zero-filed LDOS maps is related to the vortex-induced quasiparticle states (Fig. 2B).   

• The C peaks are occurred at k-space radius 0.062 Å^-1 with width σ=0.011±0.002 Å^-1. Equivalently, this indicates a checkerboard pattern surrounding the vortex cores with periodicity of 4a oriented along Cu-O, and a correlation length of L=(1/πσ)≈30±5 Å (7.8±1.3 a).

• Vortex-induced LDOS is detected at least 50 Å away from the vortex core, i.e., at 5 T about 25% of the sample is under this influence while vortex cores only cover 2% of the sample surface. 

• The pattern is asymmetric such that the amplitude of PS in one Cu-O direction is stronger by a factor of 3 relative to its perpendicular direction. 

The authors have considered these observations as a sign of strong electronic correlation in the underlying lattice, and discussed them against the models based on formation of spin structure surrounding the vortex cores. Formation of antiferromagnetic insulating state inside the core cannot be directly tested in this study. Other proposals based on formation of magnetic order due to the coexistence spin density wave + superconductivity or charge order in the nematic stripe phase (due to 2-dimentionality of checkerboard) could also be considered. These models predict a magnetic ordered along Cu-O with 8a periodicity. The 8a periodicity (different from 4a in LDOS) can be understood because almost all microscopic theories suggest a ½ periodicity of the magnetic order for the associated charge ordered state. Also, good agreement between the autocorrelation patterns of S function and the models patterns, confirms the prediction of coupled spin-charge models (Fig. 3).

In conclusion, by combining the results of INS, ENS, NMR, and STM (with assuming the same phenomena for different HTSCs) a consistent picture of electronic and magnetic structure could be understood in the HTSC vortex cores.     

 Ref.: J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, H. Eisaki, S. Uchida, J. C. Davis; Science 295 (2002) 66.

           

Local Ordering in the Pseudogap State of HTSCs

Collaboration of the scientists from the United States and Japan has resulted in discovering an incommensurate local ordering in the pseudogap of HTSCs. They have investigated the scanning tunneling microscopy (STM) of the slightly underdoped single crystal of 0.6% Zn-doped Bi2212 (T_c=80 K, pseudogap energy ∆_ps~35-40 meV at 100 K). The Fourier analysis of the STM data shows some peaks arising from different origins of crystal and electronic structure. Although the crystal structure peaks have been already known from their energy dependence and wave vector magnitudes, another peak (vector Q in Fig. 2B) which is the most important finding of this study, has the following characteristic:

• This peak appears only in the tunneling energy range below the energy approximately equal to ∆_ps

• The intensity of this peak increases by decreasing energy in apposite trend with the crystal structure peaks

• This peak indicates an electronic order along the Cu-O with incommensuration periodicity of 4.7±0.2 a (Cu-Cu distance)

• Against the strong energy dependence of its intensity, its wavelength remains independent of energy in ±0.2 a resolution

• The width of the peak shows a correlation length of about 90 Å, four to five times of the electronic modulation, a behavior consistent with the locally disordered or glassy nature of the modulations

The authors have discussed that the peak could be related to the pseudogap state due to unexpected energy dependence of its intensity and its absence in the tunneling energy larger than ∆_ps. Also, this modulation in pseudogap seems to be fundamentally different from the similar bond-oriented modulations which have been observed in the superconducting state, because all of them indicate energy dependence period.

 

The main Question: What is the origin of this peak?

The opposite behavior of the energy dependence of its intensity with the crystal peaks (Fig. 2C) rules out lattice effects as an origin. Another possibility is the formation of standing waves caused by the quantum interference of quasiparticles elastically scattered from defects. Calculated interference patterns using various electron Green functions in this study (Fig. 4) indicates dispersion of the wavelengths (0.4 to 1.0 a over 40 meV) which is qualitatively inconsistent with the experimental STM data. Considering two electronic order such as charge and spin order: 1) the stripes can not explain the observed two-dimensional modulation because of its one-dimensional nature, and 2) there is no consensus that whether the pining of spin fluctuation by imperfections which can give rise to modulation in the local density of states (LDOS), alone could explain the opening of pseudogap. Also, within the models that explain pseudogap state as an incoherence electronic pair state, pair-liquid with phase fluctuation can show the energy-dependent modulated pattern in the LDOS, which is inconsistent with this study. However, if the performed pairs in this scenario localize and form a disordered static lattice the pair lattice modulation could only be detected below the pseudogap energy. This needs more study to clarify.

So, this study suggests that there is an ordered electronic state in the pseudogap which, competes or inconsistent with superconductivity, although its order parameter can not be identified well from this study.

Ref.: Michael Vershinin, Shashank Misra, S. Ono, Y. Abe, Yoichi Ando, Ali Yazdani; Science 303 (2004) 1995.

 

Does Checkerboard Indicate a Hidden Electronic Order in Pseudogap State?

Previous studies indicate the appearance of the checkerboard pattern in the pseudogap state of Bi2212 surrounding vortex cores or in strongly underdoped region. Scientists from Japan in collaboration with the United States have made a study to make clearer this situation and its relation with a hidden electronic order in the pseudogap. Existing of some hidden electronic ordered state could be a proposal to explain the pseudogap in HTSCs.

They have measured the spatial and energy-resolved differential tunneling conductance, g(r,E), for Ca_2-xNa_xCuO₂Cl₂ (CNCOC) crystals with x=0.08, 0.10, and 0.12 (T_c=0, 15, and 20 K, respectively) by scanning tunneling microscopy. This system which is Mott insulator in parent compound (x=0), goes to nodal metal in the zero temperature pseudogap regime by hole doping (Na) and then becomes superconductors for x≥0.10. The smaller measurement uncertainties due to the existence of less disorder at lower doping and temperatures makes CNCOC as a good candidate relative to Bi2212. The properties of g(r,E) should be determined by states in the CuO₂ plane because the CaCl layers are strongly insulating.

Important results:

• There is a V-shaped energy gap spanning approximately ±100 meV with minimum at E=0 (Fig. 1c). Similarity of these data with the data for Bi2212 indicates that this gap could represent the same phenomena at the pseudogap state of Bi2212. The strong bias asymmetry is kwon in lightly hole-doped Mott insulator.

• A clear checkerboard pattern of intense conductance modulations, with primary periodicity 4a (called as ‘tile’ below) and complex internal structure at the atomic scale has been observed.

• The Fourier transform, g(q,E), has a symmetry under 90° rotations (Fig. 2d-f). There important contribution of q-vector in the subgap (<100 meV) g(q,E) are in (±1,0), (±1/4,0), and (±3/4,0) in 2π/a unit (the similar is for (0,±1), (0,±1/4), and (0,±3/4) due to 90° symmetry). These vectors related to lattice periodicity, commensurate 4a×4a modulation, and an unanticipated modulation, respectively. They do not disperse with increasing energy and consistent with a crystalline electronic state.           

• Autocorrelation function, A(R,E)=<g(r,E)g(r+R,E)>, clearly exhibit 4a×4a modulation with a correlation length of ~10a and a 3a×3a intense incommensurate conductance maxima (q=(±3/4,0)) in the 4a×4a modulation structure (Fig. 3a).

• Interesting complex internal structure at the atomic scale has been observed: While, the lowest conductance coincides with the perimeter atoms of the tile, the local conductance maxima are not associated with atomic locations inside the tile (Fig 3d). Also, the average pseudogap in high conductance is smaller by a factor of 3 relative to the average pseudogap in low conductance in a tile (Fig. 3e).

• By increasing the doping the 4a×4a checkerboard state with its 4/3a×4/3a modulation and pseudogap variations change such that A(R,E) appear very similar between doping, i.e., checkerboard electronic state appears for all doping (0.08, 0.10, and 0.12).

The authors have discussed that the data of this study is consistent with the conjecture of two distinguish electronic states: 1) low energy states near nodal points which support metallic and superconducting states, 2) high energy states (pseudogap) in the antinodal directions which are incoherent at low doping and are unperturbed by doping from metal to superconductivity. The ARPES data confirm this conjecture: a large pseudogap (~200 meV) around (π,0) and nodal metallic or superconducting states (<10 meV). This study is inconsistent with the models such as orbital-current ordered state, one dimensional stripes (lacking of 90° symmetry), Fermi surface nesting, valence-bond solids, pair density waves, hole-pair or single hole Wigner crystals, and supersolids, but may still be consistent with the defect-dominated stripe orders or nematic liquid crystal strips.    

Finally the checkerboard state exemplifies a (no longer hidden) electronic order associated with the copper-oxide pseudogap, and the translationally invariant of the state indicates some form of electronic crystal (not liquid).  

Ref.: T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi & J. C. Davis; Nature 430 (2004) 1001.

What is the Origin of d-wave Gap in Nonsuperconducting Systems?

A group of scientists from United States have investigated the ground state of pesudogap by ARPES measurements of La_2-xSr_xCuO₄, in x=1/8 concentration, where the superconductivity suppressed strongly and the system has static spin and charge orders. The main idea of this study is to clear the origin of pseudogap in HTSCs, which have been previously related to the incoherence of superconducting pairs or existence of another ordered state which competes with superconductivity.

Here their results are summarized as:

• Fermi surface has been extracted for LBCO and LSCO, as a line in momentum space that connects the maxima of each of the measured momentum distribution curve (MDC) at energy ω=0, are in good agreement with previous data.

• The k-dependence of the gap is very similar to the superconducting gap in HTSCs: d-wave asymmetry and gapless at nodal points (Figs. 1B & 2A), even though superconductivity is essentially nonexistent in the 1/8 concentration.

• The gap in LBSO is larger at x≈1/8 than at x=0.095, and than in LSCO with x=0.07. Gathering with the pervious results, this gap indicates unexpected doping dependence with a maximum at x≈1/8, where the stripes are fully developed between to adjacent superconducting domes (Fig. 2C).This result is against the common belief that the gap monotonically increases as the antiferromagnetic phase is approached (underdoped region).

• The differential conductance spectra derived at many points in a wide range of the scanning tunneling microscopy data (for x=1/8 sample) confirms the existence of d-wave gap with amplitude ∆₀≈20 meV in agreement with ARPES result.

Is the observed gap at 1/8 concentration (or at least a portion of it) originated from the stripes order, in analogy with conventional two-dimensional charge density wave (CDW) systems?

Two scenarios could be considered to answer: 1) Segregation of carrier doped into one-dimensional stripes separated by charge-poor regions in the antiferromagnetism ground state, which is questionable with two-dimensional nature of Fermi surface and d-wave gap in HTSCs. 2) In a more conventional view the charge-spin-ordered state may formed by nesting of the Fermi surface segments, producing a divergent electronic susceptibility and a Peierls-like instability.    

The authors have discussed that due to the different value of CDW vector (q_CDW≈4k_F, k_F is the antinodal wave vector) with the usual one (2k_F), and its opposite doping dependence with magnetic incommensurability, the data are inconsistent with the second scenario. More fundamental problem of the data with nesting scenario is the fact that only four gapless points exist in the ground state (nodal points), not on whole part of the nested segments as we expect from the scenario.

These evidences (especially isolated nodal gapless points) reveal the pairing origin of the pseudogap, and imply that the strongly bound Cooper pairs are most susceptible to phase disorder and spatial ordering in this system. Also, the anticorrelation of the low energy pairing gap with T_c suggests that the phase coherence is strongly suppressed by quantum phase fluctuations. 

Ref.: T. Valla, A. V. Fedorov, Jinho Lee, J. C. Davis, G. D. Gu; Science 314 (2006) 1914. 

 

Is Phonon Softening Related to Stripes?

A group of scientists from Germany, France, Japan, and United States has recently reported some experimental evidences that relate the phonon softening of oxygen bond-stretching vibration mode in HTSCs to the stripe order. The basic idea of this study is that charge inhomogeneity with a periodic modulation (for example stripes) should affect interatomic Coulomb forces and soften lattice vibration at the same modulation wave vector. This vibration mode is the bond-stretching mode (oxygen oscillate along the Cu-O bond) with wave vector q=q_co=(0.25,0,0), perpendicular to the stripes. Phonon softening of this mode has been observed recently in some of HTSCs such as YBCO, LSCO, and HBCO.

This group has measured the dispersion of bond-stretching mode from inelastic neutron scattering for La_1.48Nd_0.4Sr_0.12CuO₄, La_1.875Ba_0.125CuO₄, and La_2-xSr_xCuO₄ (x=0.07, 0.15, 0.30), which some of them shows the static stripe order.

Their most important results could be summarized as:

• A single sharp peak of bond-stretching mode in q=(0.15,0,0) (different from q_co) converts to a two displaced peaks at q_co with different energy width but comparable integrated intensity. The main peak has a monotonic, cosine-like downward dispersion while the second peak anomalously softens by more than 10 meV near q_co (Fig. 2). It also has an anomalously large energy width. These evidences indicate the strong electron-phonon interaction for this mode near q_co.

• This phonon mode along the [110] direction has not such a softening. This could be explained because the stripes aligned along the Cu-O bonds.

• The anomaly is strongest at the lowest temperature, 10 K (below the stripe temperature, T_s≈ 60K), while at 330 K the low energy tail is strongly reduced and the main peak is enhanced (Fig. 3). This is against our expectation from conventional anharmonic behavior of phonons, which broadens and softens with increasing temperature.

• In La_2-xSr_xCuO₄, the peaks are considerably broader for x=0.07 and 0.15 than for undoped and overdoped samples. A similar feature has been observed in YBCO and HgBCO, indicates a correlation between the strong electron-phonon coupling of this phonon mode with a hole concentration in the range compatible with superconductivity.

The authors have discussed that the results can no be understood within the conventional models with assuming charge homogeneity. So, this study in somehow indicates the correlation of the anomalous phonon softening with stripes in HTSCs. Also, suggest that electron-phonon interaction may be important in the mechanism of superconductivity, although the contribution may be indirect.

Ref.: D. Reznik, L. Pintschovius, M. Ito, S. Iikubo, M. Sato, H. Goka, M. Fujita, K. Yamada, G. D. Gu & J. M. Tranquada, Nature 440 (2006) 1170. 

Two Distinct Fermi Momentum-Dependent Energy Gaps

Some scientists from United States, Japan, and Canada in collaboration with each other have discovered two different Fermi momentum-dependent gaps related to the superconducting state and pseudogap. They have measured the ARPES spectra of deeply underdoped Bi₂Sr₂Ca_1-xY_xCu₂O₈ crystals, with T_c of 30-50 K. Analyzing the data shows two distinct energy gaps with different characteristics:

1) One gap manifests itself as the spectral weight suppression near the antinodal region; becomes larger with decreasing doping,

2) The other gap is created near the nodal region, where quasiparticle peak can be observed, decreases with decreased doping. This gap has a d-wave momentum dependent with an amplitude that remains relatively constant for a range of doping below the optimal doping and then decreases in severely underdoped region.

This experiment also indicates that the length of Fermi arc, defined as the region where the coherence peak can be seen from the spectra, decrease with the decreasing of doping (Fig. 1-B to 1-G and 1-H inset). The opposite behavior of the magnitude of gap amplitude with doping (Fig. 2) suggests that the antinodal gap (No.1) is related to the pseudogap and not superconductivity, and the nodal gap (No. 2) more likely represents the real superconducting gap because of the existence of the coherence peak in the spectrum. So, the nodal gap seems to be superconducting gap, while the antinodal gap (pseudogap) may arise from other mechanism such as Umklapp scattering by antiferromagnetic correlation or from competing states such as stripes, polaronic, or charge-density wave.

This two-gap scenario is consistent with the results of thermodynamics and Raman studies. Also this scenario can explain the opposite doping dependence of superconducting gap deduced from Andreev reflection, penetration depth, and tunneling spectroscopy. While Andreev reflection and the penetration depth are sensitive to the superconducting state (related to gap No. 2), the scanning tunneling microscopy spectrum is sensitive to the antinodal gap (No. 1) because of larger phase space.

This scenario also has two important implications:

1) The pseudogap near the antinodal region is unlikely to be a precursor state of the superconducting state, 2) weekend superconductivity in the deeply underdoped regime arises not only from the loss of phase coherence but also a weakening of the pair amplitude.

These implications could be very important for developing a mechanism for high-T_c superconductivity.

 Ref.:  K. Tanaka, W. S. Lee, D. H. Lu, A. Fujimori, T. Fujii, Risdiana, I. Terasaki, D. J. Scalapino, T. P. Devereaux, Z. Hussain, Z.-X. Shen; Science 314 (2006) 1910.