Relation of Quantum Magnetic Excitations with Stripes?!

Stripe formation is a good way to interpret the coexistence of antiferromagnetism order and superconductivity in HTSCs. However, the spectrum of magnetic excitation of YBCO is not consistent with the theoretical prediction for a material including stripes. A group of scientists from the USA, UK, and Japan have investigated this contraction by measuring inelastic neutron scattering of the La_1.875Ba_0.125CuO₄ (not superconductor with charge and stripe order). They have explained that the experimental data could be understood by stripe models by taking account of quantum excitations.  

Fig. 1 schematically shows the stripe order, and how the antiferromagnetism peak in the undoped system at Q_AF=(1/2,1/2) moved to the incommensurate peaks at (1/2±δ,1/2) and (1/2,1/2±δ) (δ=1/2p and p is the stripe spacing) when the system goes to the stripe order by hole doping. The dispersion of these peaks will be as a cone-like for the spin-wave behavior, ħω=c|Q-Q_AF| for ħω<J ~140 meV, (Fig. 1 d), or dip-hump-like for the tow-leg spin ladder theory (Fig. 1e). This difference makes hints to select more convenient theory for explaining the experimental data.  

Results:

• Experimental data shows four incommensurate peaks at 6 meV, while by increasing the energy the peaks have dispersed inward toward the Q_AF till 55 meV. At higher energy the excitations have started to disperse outwards and the shape change to the square-like with the corners rotated by 45 degree relative to incommensurate peaks. The square will be larger with increasing energy till 160 meV which its size is comparable with the antiferromagnetism Brillouin zone (Fig. 1).
• Calculations based on the spin-wave and two-leg spin ladder models indicate that the experimental data with square-shape dispersion are in agreement with the spin ladder interactions with quantum fluctuations (Fig. 3). In the spin-wave models the excitation should have a four-circle (projection of the cones at a constant energy) shape (Fig. 3a) which is completely different with the experimental observations.
• Magnetic scattering function of data, S(ω) (integrated on Q), indicates initially decreasing, then rising to a broad peak near 50-60 meV and gradually decreasing at higher energies. The computational S(ω) and dispersion along (1+q,q,0) based on a simple model of spin ladder theory are in consistent with the experimental data.

The authors have discussed that the ladder model, within the stripe picture, provides a more compelling explanation of the results. With considering the similar results for YBCO system, this study supports the concept that charge inhomogeneity, possibly dynamic in nature, is essential to achieve high T_c in HTSCs. 

Ref.: J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita & K. Yamada; Nature 429 (2004) 534.  

Coupling of Charge Carriers to the Spin Fluctuation

In an electron-phonon system the electron-phonon interaction spectral density, α²F(ω), is approximately equal to W(ω) defined as:

                           α²F(ω)≈W(ω)=(1/2π)d²[ωRe(σ^-1(ω))]/dω²   (1)

Where, Re(σ(ω)) is the real part of the optical conductivity. For the electron-spin interaction systems this relation could be converted to I²χ(ω)≈W(ω), where I is the coupling constant of electron-spin interaction and χ(ω) is the imaginary part of spin susceptibility. So, comparison of the W(ω) extracted from the experimental or computational study with the phonon or spin spectral density, gives suitable information about the type and coupling strength of the interaction with the carriers. Recently, the ~41 meV resonance magnetic mode has been observed in the spin spectral of the optimal doped YBCO and Bi2212 in the superconducting state. A group of scientist from Canada, Austria, and the USA has applied this method for both YBCO and Bi2212 systems to indicate the importance of the spin fluctuations in the superconducting state of HTSCs systems.  

Results:

• The calculated W(ω+Δ₀) from Eq. 1 for optimal doped YBCO (with the gap amplitude Δ₀~27 meV) traces the resonance pattern of the magnetic susceptibility and is in good agreement with the experimental I²χ(ω) data (Fig. 2a). The χ(ω) have been taken from the spin-polarized inelastic neutron experiment (integrated over momentum space), and W(ω+Δ₀) function has been derived (by Eq. 1) from the calculated σ(ω) in superconducting state considering d-wave gap.
• The W(ω+Δ₀) function has a peak in 68 meV (=ω_res+Δ₀=41+27 meV) and the coupling constant λ~2.6 (twice the first inverse moment of I²χ(ω)) is high enough to get T_c~100 K. The negative wiggles in W(ω) above the resonance peak is not consistent with the I²χ(ω), because the I²χ(ω) is not exactly equal to W(ω), i.e. the correspondence is expectable just near the resonance peak.  
• The experimental W(ω) for YBCO_x with x=6.95 and 6.6 have been derived from the experimental a-axis σ(ω) by Eq. 1 (Fig. 2b). The lower energy peak in x=6.6 sample could be related to the smaller ω_res in x=0.6 sample relative to x=6.95, and the lower amplitude of the peak (~smaller λ) could be calculated to lower T_c in x=0.6 sample.  
• The similarity of the experimental W(ω) pattern with the calculated W(ω+Δ₀) and experimental I²χ(ω) indicates that the σ(ω) has a trace of spin fluctuation in superconducting state rather than phonon.The similar behavior has been obtained in Bi-2212 system, which shows a magnetic resonance peak at ~43 meV.   

Finally, this study indicates the important role of the spin fluctuation in the superconducting state of HTSCs, and the strength of the electron-spin interaction is high enough to account for high T_c (~100 K) in these systems.

Ref.: J.P. Carbotte, E. Schachinger, and D.N. Basov, Nature 401 (1999) 354.  

Unusual or Usual Isotope Effect in HTSCs?!?

The isotope effect (IE) on T_c of optimally doped HTSCs (α=dlnT_c/dlnM <0.1) is small in comparison with the conventional BCS superconductors (α~0.5), an evidence that usually used against the phonon mediated pairing in HTSCs. However, the ARPES kink is one of the strongest evidence that there is strong electron-phonon coupling in HTSCs. To confirm this, a group of the US in collaboration with Japanese Scientists has measured the oxygen IE effect of the ARPES pattern for the optimally-doped Bi2212 system by substitution ¹⁶Oà¹⁸Oà¹⁶O_R (re-substituted again to confirm the pure IE). They have found a very strong and energy dependent (unusual) IE which is in contrast with the IE on T_c of this system.    

 Results:

• Both energy and momentum dispersive curves (EDCs & MDCs) show strong phonon renormalization of the electron dynamics in the 100–300 meV range, while the renormalization is very small for the low energy (<100 meV) and coherent peak (Fig. 1). This energy dependence IE and the dominant effect in energy higher than the kink energy (~ phonon frequencies of this system) go well beyond the Migdal–Eliashberg model.
• Isotope shift of the low energy is strongly anisotropic (k dependence) such that it changes from ~15 to -30 meV (stronger near the antinodal-cuts) with a subtle sign change through cuts 2-6 in Fig. 2d, while the kink shift is small ~(5-10)±5 meV for all cuts. Data shows a linear correlation of the isotope shift with the magnitude of superconducting gap along the Fermi surface (inset of Fig. 2a).
• The changes of the superconducting gap are small and random in both magnitude and sign. It varies by ~5 meV from one sample to another (for measuring different samples) regardless of the isotope mass.
• Both EDCs an MDCs data shows strong temperature dependence of the isotope shift below (T=20 K) and above (100 K) of T_c (Fig.3).

Discussion:

At first the author have discussed that the experimental error such as unintentional doping change induced by the substitution process or sample misalignment could be ruled out, because the temperature dependence, the reversibility and the reproducibility of the observed effects upon repeated measurements. The large IE not only confirms the phonon-originated of the ARPES kink, but also indicates strong electron-phonon interaction in this system. The strong E, k and T dependence of isotope shift are well beyond the usual electron-phonon coupling theory such as Migdal-Eliashberg theory for the strong coupling superconductivity.

The much stronger IE below T_c suggests a picture where pairing of electrons enhances their coupling to the lattice and vice versa, as in spin-Peierls physics. In this picture, the motion of electron pairs modifies the lattice distortion locally and vice versa. If the coupling is too strong: the pair will be localized, the lattice distortion becomes static, and finally the system becomes an insulator. Where the coupling is not strong enough, the dynamic spin-Peierls distortion follows the coherent motion of electron pairs in a superconducting state. The significant isotope dependences of pseudogap temperature T*, J and various low-temperature spin properties support this scenario.

Ref.: G.-H. Gweon, T. Sasagawa, S.Y. Zhou, J. Graf, H. Takagi, D.-H. Lee & A. Lanzara, Nature 430 (2004) 187. 

• Comment on this paper:

Based on the inconsistency of the IE on ARPES data and on T_c of optimally doped Bi-2212 of the above study, Douglas et al. have re-studied this effect in the almost identical material and experimental conditions. They are unable to detect the unusual IE on their new data and only a small average isotope shift of ~2 ± 3 meV has been found for 10 cuts (inset of Fig. 1b), inconsistent with the previous results of ~40 meV (by Gweon et al.). This small IE has been confirmed by using three different facilities, samples with multiple doping levels, including the optimal doping, and with various photon energies. The errors related to the sample misalignments, which is crucial for creating apparent energy shifts (probably the origin of the observed unusual IE by Gweon et al.), has been removed by collecting data on both sides of the (0,0) point (with ~0.1 degrees accuracy). Although, this study does not confirm the unusual large-scale IE, this result does not invalidate electron–phonon coupling as a potential pairing mechanism for HTSCs. 

Ref.: J. F. Douglas, H. Iwasawa, Z. Sun, A. V. Fedorov, M. Ishikado, T. Saitoh, H. Eisaki,H. Bando, T. Iwase, A. Ino, M. Arita, K. Shimada, H. Namatame, M. Taniguchi, T. Masui, S. Tajima, K. Fujita, S. I Uchida, Y. Aiura, D. S. Dessau; Nature 446 (2007) E5.   
  

Magnetic Resonance Mode and Its Relation with Thermodynamics of HTSCs

The most prominent feature of spin fluctuation in YBCO_6+x system is the 41 meV resonance peak which appears below the T_c of the optimal sample. This feature (and strong spin fluctuation for all x values in this system) has been considered as an important experimental evidence for the magnetic interaction pairing of HTSCs although its role on the pairing interaction and the details of the mechanism of superconductivity are not still clear. In this study scientists from the USA and the UK have studied this resonance peak in YBCO system and presented an interesting relation of this peak and the specific heat in this system. Inelastic neutron scattering is used to study the frequency and wave vector dependent magnetic structure of YBCO_6+x with x=0.6, 0.7, 0.8, 0.93 (T_c=62.7-92.5 K).

Results and discussion:

• In x=0.6 sample a resonance peak at 34 meV appears in the acoustic mode (in-phase magnetic fluctuation of the coupled CuO₂ planes) of the local [∫dq_xdq_yχ"(ω,q)] magnetic susceptibility by reducing temperature which is the most important feature of this measurement (Fig. 1).

• Well below the resonance peak the magnetic scattering is suppressed and a true spin-gap is developed in the superconducting state, while above it there is a broad feature extending at least to 220 meV with a maximum around 75 meV (Fig. 1).

• The resonance peak also exist in the temperature above T_c at 75-150 K for x=0.6 (Fig. 2). By increasing the temperature it broadens and its intensity decreases and finally vanishes at 200 K.

• The onset of the resonance occurs at T* which almost coincides with the T_c in optimal doped and is larger of T_c for underdoped samples as 115±15 K for x=0.8 and 150±20 K for x=0.6. This temperature is comparable with the temperatures that the dρ(T)/dT (electrical resistivity) and the Cu nuclear 1/(T₁T) relaxation rate reaches a broad maxima in this system, due to the formation of the pseudogap in this system.

• It is very interesting that the behavior of d<m²_res>/dT versus the doping and temperature is very similar with the behavior of electronic part of the specific heat (C^el(x,T); Fig. 3D-E). <m²_res>=(3h/4π²)∫dω χ”_res(ω)/[1-exp(hω/k_BT)] is the mean-squared (fluctuating) moment associated with the resonant part.

The authors have noticed this evidence as a track of the magnetic resonance fluctuation in the thermodynamics of the HTSCs. They have estimated the effect of the magnetic fluctuation on the thermodynamic properties of the HTSCs by calculating the dE_J/dT, where E_J is the exchange part of the simple t-J Hamiltonian. By some simple analytic calculation and considering the resonance peak as a dominant effect the following relation could be found:

C_J^res≈dE_J^res/dT=-(3hJ/8π²µ²_B) d(∫dω χ”_res(ω)/[1-exp(hω/k_BT)])/dT

                                 = -(J/2µ²_B) d<m²_res>/dT

By J=125±20 meV for x=0.6 the pattern of this calculation is comparable with the experimental results of the C^el(x,T). The difference of the calculated C_J^res and the experimental one could be reduced by considering the other parts of temperature dependence of the magnetic fluctuation, spin-gap and pseudogap, and the part related to the kinetic energy of the t-J Hamiltonian. Finally, this agreement suggests that a large part of the specific heat near T_c and its associated entropy are due to resonant spin fluctuation.

Ref.: P. Dai, H.A. Mook, S.M. Hayden, G. Aeppli, T.G. Perring, R.D. Hunt, F. Dogan, Science 284 (1999) 1344.      

Special Relationship between Real and Momentum Spaces

Scientists from the USA, China and Japan have studied the Furrier transform scanning tunneling spectroscopy (FT-STS) of Bi-2212 crystals to find the relation between the features of r-space and k-space electronic structures of HTSCs: such as the modulation of the electronic local density of states (LDOS~g(r,ω)) observed by STM, and coherence peak, Fermi surface and d-wave gap determined by ARPES. The banana like shape of the constant quasiparticle energy near the Fermi surface of the HTSCs suggest that we should have 16 distinct ±q pairs of maximal intensity in the Fourier transform of r-space local density of states, corresponding to each vector connected the end of each banana (Fig. 1). To detect such a pattern the atomic resolution is requires for all measurements and the filed of view at the measurements should be L>450 Å (∆q=2π/L to be enough small). Fig. 2 shows such a resolution.

From the analysis of the data, it has been concluded that the Fermi surface and the amplitude and angular dependence of the gap are agreement with the ARPES results (Fig. 3b&c). This indicates the clear link between the r-space and k-space characteristics of the electronic states, because the matrix elements for tunneling and photoemission are quite different. ∆(θ_k)=∆₀[0.818 cos(2θ_k)+0.182 cos(6θ_k)] has been derived from this study with the gap amplitude of ∆₀=39.3 meV.

The authors have discussed some complementary results from this study:

• The g(r,ω) data are more relevant to the proposals that the LDOS modulation might result from the existence of a charge-density-wave order parameter with fixed q-vector, (for example stripes) rater than the quantum interference models.

• Although the g(q,ω) are not completely symmetric relative to the bias potential (for example ω=+14 and -14 in Fig. 2), the deduced ∆(k) from the same positive and negative bias energy is indistinguishable within errors. This is provides evidence that the quasiparticles of this system are particle-hole superposition, consistent with the Bogoliubov description.

• The strong response in g(q,ω) for all ω at q<0.15 may reflect long-wavelengths inhomogeneity  in the integrated LDOS. This reveals an obvious candidate for weak but ubiquitous potential scattering that could produce the LDOS modulations.

• For a given ω the Umklapp LDOS modulation signal is localized to the nanoscale regions where ω is equal to the local gap value (Fig. 4) whish is different from the pattern for usual crystals. This implies strong nanoscale spatial variation in the quasiparticle dispersion near k=(π/a,0), and therefore significant scattering.

Ref.: K. McElroy, R. W. Simmonds, J. E. Hoffman, D.-H. Lee, J. Orenstein, H. Eisaki, S. Uchidak & J. C. Davis; Nature 422 (2003) 592.

Is Continuum Background More Important Than Resonance Mode?

The observed sharp ‘kink’ in the ARPES has been related to the phonon or magnetic mode as a possible signature of the force that create the superconducting state in HTSCs. A group of scientists from Canada and the USA have studied this situation by Fourier-transform infrared (FTIR) spectroscopy of optimally doped (T_c=96 K) and highly overdoped (T_c=82, 65, and 60 K) Bi2212 crystals. The optical self energy derived from FTIR (4πσ(ω)=-iω[ε(ω)-ε_H]=-iω_p²/[2Σ^op(ω)-ω];  Σ^op(ω)= Σ₁^op(ω)+iΣ₂^op(ω)) is in close related with the quasiparticle self energy (Σ^qp(ω))  from ARPES. 

Most important results of this study:

• At high frequencies the scattering rate (1/τ(ω)=-2Σ₂^op(ω)) has a linear frequency dependence originate from marginal Fermi-liquid behavior. The overall scattering rate decreases as the doping increases, and drop suddenly below 700 cm^-1 at low temperatures (Fig. 1a-d).

• There is a sharp peak in Σ₁^op(ω), optical resonance mode, around 700 cm^-1 which clearly separated from the broad continuum. This peak tracks the depressions in 1/τ(ω) in both frequency and amplitude, and weakens by increasing the doping and temperature (Fig. 1e-h). 

• Σ₁^op(ω) has been derived from FTIR and ARPES include qualitative similar features (sharp peak and broad continuum in Fig. 2), although part of the difference in the amplitude and frequency of the peaks could be related to the deriving method.

• The amplitude of the resonance peak weakens with doping and disappears completely at critical hole concentration p=0.225±0.010, where the superconductivity is still large enough to show with T_c=55 K.  

• The center frequency of the mode is proportional to T_c (Ω_res^op≈8 k_BT_c from FTIR and 5.6 k_BT_c from ARPES) and reaches the maximum at the optimally doped phase for both FTIR and ARPES.   

Discussion:

The authors have discussed that neither the continuum nor the peak resonance features could originate from the phonons: the continuum’s spectral weight extends beyond the cut-off frequency of phonons; and the sharp peak appear at T_c of overdoped and slightly above T_c in the underdoped samples which is the characteristics of magnetic resonance mode. The amplitude of the coupling of optical resonance mode to charge carriers (Fig. 3c and 1a-d) extrapolates to zero at a critical doping of p=0.23, where the superconductivity is still exist. This study rules out both the magnetic resonance mode and phonons as the principal cause of high-T_c superconductivity, suggest the universal broad background as a good candidate signature of the ‘glue’ that make pairs.

Ref.: J. Hwang, T. Timusk, and G.D. Gu, Nature 427 (2004) 714.

  

High Energy Magnetic Excitations in YBCO

Dramatic changes in the magnetic excitation spectrum by going to the superconducting state of HTSCs, known as the “resonance” peak, have been considered as a strong evidence for the magnetic pairing mechanism of superconductivity in these systems. The resonance peak occurs at E_res=41 and 34 meV [at q=(0.5, 0.5) in the CuO₂ reciprocal lattice units (rlu)] respectively for optimal doped YBCO_6..93 and underdoped YBCO_6.6, and below the E_res the peaks are formed at the incommensurate positions (0.5±δ,0.5) and  (0.5±δ,0.5), δ≈0.105 for YBCO_6.6. It is difficult to believe the resonance peak as the only origin of the electron pairing because it has only small part of the total spectral magnetic weight. To know about higher-energy excitations, a group of scientists from the UK and the US have measured the energy and momentum dependence of the spin fluctuation by neutron scattering measurements in different temperature and energy (E>E_res) in YBCO_6.6 (T_c=62.7 K, E_res=34 meV).

Their most important results summarized as:

• At E=85 meV, there is a single peak at T=300 K centered at q=(0.5,0.5) but with decreasing the temperature a quartet of peaks is formed at Q_ε=(0.5±ε,0.5±ε) and  (0.5±ε,0.5-±ε), with ε=0.12±0.01 rlu (Fig. 1a-c). These peaks have different characteristic with the low energy incommensurate peaks, because they are rotated by 45 degree relative to low energy peaks (0.5±δ,0.5). The coherency of excitations develops in temperature between 300 and T_c which could be related to the pseudogap temperature.

• By reducing the temperature to 66 K (T_c+3.3 K), the magnetic response becomes strong near (0.5,0.5) and E_res=34 meV, and then (T=10 K) the resonance becoming stronger and incommensurate peaks are developed below E_res (Fig.1 g-i). In contrast, the high-energy incommensurate peaks are fully developed at T_c. Both the low-energy and the high-energy incommensurate peaks develop through a loss of spectral weight near (0.5,0.5) and at their respective energies as the temperature is lowered.
• There is a square structure with the vertices of the square pointing along the (110) and (1-10) directions in the E=66-105 meV region (Fig. 2a-q). The incommensurate peaks are most developed in the 70-90 meV energy range. These data rule out a spin-wave like dispersion for higher-energy excitations, because we expect the continuous rings of scattering at the constant energy (rather than square shape in this study) for a spin-wave.
• The value of <m²> has been derived to be 0.12±0.02 μ²_B/f.u. (formula unit) for the resonance and sub-resonance structure (24<E<44 meV) and 0.26±0.05 μ²_B/f.u. for high energy scattering (60<E<120 meV). The significantly greater contribution (~<m²&gt of the higher-energy excitation to the total fluctuating moment than the resonance structure, these excitations should be considered in a magnetically mediated pairing mechanism of HTSCs.

The authors have discussed that existence of the similar pattern for high energy magnetic excitations in La_1.875Ba_0.125CuO₄ indicates a universality of this phenomenon for HTSCs compounds. Candidates for the common origin of this phenomenon include an incipient spin-charge separation leading to unidirectional stripes and the underlying electronic structure.

 Ref.: S.M. Hayden, H.A. Mook, P. Dai, T.G. Perring & F. Dogan; Nature 429 (2004) 531.  

   

Electron-Boson Interaction: Phonons or Magnetic Modes?

Determining of bosons (phonon or magnetic modes) which play an important role in the electron-boson interaction (EBI) and making pairs in HTSCs has been complicated, because different aspects of experimental evidences such as ARPES and optical spectroscopy could be interpreted by one of these bosons. Recently atomic-resolution STM study of Bi-2212 has been done by a group of scientists from the USA and Japan indicates phonon is the boson. They have studied the dI/dV and d²I/dV² spectra for Bi2212 in different doping value (0.12-0.24 of hole concentration) which give information about the superconducting gap ∆ and boson energy Ω. Because ∆ is inhomogeneous in atomic scale, the atomic scale information of Ω and their correlation also give valuable information.

Their most important results could be summarized as:

•  Ω(r)=Π(r)-∆(r) [Π(r) is the position of peak in  the d²I/dV² and r is the position on the surface] is heterogeneous at ~2nm scale in the range of 40-65 meV (Fig. 1a&b). The mean value of Ω(r) is < Ω(r)>=52 meV with a statistical spread of ±8 meV in consistent with the value for antinodal EBI in ARPES.  

• There is no spatially periodic structure in unprocessed d²I/dV², but this is maybe due to the strong special disorder of ∆(r). However the Γ(r)=∑_ω=40 ⁶⁵ d²I/dV²(r,ω) [ω(r)= E-∆(r) and integrated is because of enhancing the their spatial contrast] shows modulations parallel to the Cu-O bond direction with wavelength ~5a and correlation length of ~50 Å (Fig. 2). The Fourier transform Γ(q) determines the modulation wavevector as P₁≈2π/a[(0.2,0);(0,0.2)]±0.15.

• The modulation of d²I/dV² can be calculated by considering scattering of electronic states due to atomic-scale variations in the pair potential and considering phonons for bosons, which is consistent with the experimental data.     

• Although both <Π(r)> and <∆(r)> change rapidly with doping, the <Ω(r)>=52±1 meV is almost constant with doping (Fig. 4). Changes occur in spatial correlation of Ω(r), but both of the Γ(r) modulation and <Ω(r)> are independent of hole doping.

• Completely substituted of ¹⁶O by ¹⁸O in some different samples indicate that the histogram of Ω(r)-∆(r) shifted downward by several meV (Fig. 5a). Quantitative analysis indicates that this value is -3.7±0.8 meV (~6 % of <Ω(r)&gt which is near the value expected for lattice vibrational modes involving the O atom (1-√16/18). This effect happens for both filled E=-(∆+Ω) and empty E=(∆+Ω) states, as expected from strongly coupling theory of superconductivity.

• Whereas the zero-displacement correlation O(r):∆(r)≈+0.35 [O(r) is the dopant atom location] and Ω(r):∆(r)≈-0.30 (Fig. 3), Ω(r) and O(r) are uncorrelated (Fig. 5c). Therefore correlation between Ω(r) and ∆(r) cannot be occurring trivially, through a similar effect of dopant disorder on both.

The authors have discussed that all the experimental evidences in this study such as the value of < Ω(r)>=52 meV, its doping independence and the isotope effects are in agreement with the phonon scenario. Resonant spin-1 magnetic excitation mode is inconsistent with these data because its energy is 43 meV in Bi2212 and is strongly doping dependent. The incommensurate dispersive spin density wave mode is also inconsistent because of their characteristic strong energy or doping dependence.

The data present some intriguing new possibilities: 1) ∆(r) disorder is a consequence of heterogeneity in the pairing potential caused by disorder in the frequencies and coupling constant of Ω(r), but is inconsistent with these data where the Ω(r) and ∆(r) have strongly different doping dependence. 2) The d²I/dV² features are unconnected to pairing related EBI, and perhaps occurs because of inelastic simulation of Ω(r) in the tunneling data or because of non-pairing-related electron lattice interaction. But the ubiquitous anticorrelation between Ω(r) and ∆(r) cannot be explained trivially within such scenario. 3) The d²I/dV² features represent electron-lattice interactions related to a competing electronic ordered state and the anticorrelation between Ω(r) and ∆(r) occurs because of this competition. More studies are necessary to distinguish between them.

Ref.: J. Lee, K. Fujita, K. McElroy, J. A. Slezak, M. Wang, Y. Aiura, H. Bando, M. Ishikado, T. Masui, J.-X. Zhu, A. V. Balatsky, H. Eisaki, S. Uchida & J. C. Davis; Nature 442 (2006) 546. 

 

   

Strong Electron-Phonon Interaction in HTSCs

One of the most important issues related to the mechanism of superconductivity in HTSCs is the role of electron-phonon (el-ph) interaction in this mechanism for such a strongly correlated electron systems. A group of scientists from the US and Japan have revealed a strong experimental evidence for existence of the strong electron-phonon interaction in these systems, however its role in the mechanism of superconductivity has not been cleared yet. They have measured ARPES spectra of different HTSCs families LSCO, Pb-Bi2212 (Pb doped) and Pb-Bi2201 in the nodal direction (0,0)-(π,π) in a vide range of temperature and doping concentration.

Experimental results:

• There is a clear kink in the electronic dispersion curves around 50-80 meV (Fig. 1) for all the samples in different values of doping and temperature (below and above T_c). This kink clearly persists above T_c but a thermal broadening is present.

• The energy dispersive curves (EDC) of Bi2212 in the nodal direction and for different doping are in good agreement with the corresponding data for Be (0001) surface which has strong el-ph interaction, and simulated spectra (Fig. 2).

• There is also a dip feature in all the spectra in Fig. 2 near 55-70 meV. Fig. 3b sows the doping dependence of this dip energy in the (0,π) direction.

• The quasiparticle width decrease more rapidly below the relevant energy (Fig. 3a), which corresponds to the kink position in the dispersion.

Discussion:

The change of electron self-energy below ~70 meV (i.e. kink) indicates electrons have an interaction with a collective boson mode with this energy. The similar energy scales in systems with different energy gap ranging from 10-20 meV for LSCO and Bi2201, to 30-50 meV for Bi-2212, rules out the superconducting gap as the origin. Also coupling with the magnetic mode at 41 meV can be ruled out, because the kink has been observed in LSCO where the magnetic mode dose not exist and also well above T_c in all samples, while the magnetic mode sets at T_c for optimal and overdoped systems.

These experimental evidences leave the phonon as the only possible candidates, although there are also some reasons that this kink could be originated from el-ph interaction: 1) there is bond stretching phonon mode with energy ~50-80 meV (highest energy phonon mode) in nearly all HTSCs, 2) neutron scattering of LSCO indicates strong coupling of this phonon mode to charge, 3) the kink persist in all temperature but thermally broaden in higher temperature, 4) the EDC curves are very similar to Be (0001) surface which has strong el-ph with a dip energy which could be related to phonon energy, 5) the rapidly decreasing of quasiparticle width below phonon energy is also in consistent with el-ph coupled systems.

By estimating el-ph coupling strength (λ) by simple model (the ratio of the velocities below and above the kink in Fig. 1 ~ λ+1) the doping (Fig. 1f) and momentum (Fig. 4a) dependence of λ’ can be determined. λ’ is different but proportional to λ, and is an overestimate for λ because there is also strong el-el interaction in these systems. Fig. 4a shows that λ’ is not very anisotropic (change by a factor of <2) and not temperature dependence, both in consistent with phonon interpretation of the data.  

The authors have addressed that the magnetic mode (41 meV) explanation of the kink and dip in EDC not only are inconsistent with the data presented here, but also has some serious weakness such as: 1) the small percent of this mode in the total spectral spin fluctuation weight is not enough to produce large changes in EDC above and below T_c, 2) calculations show strong anisotropy for λ’ near (0,π) which is inconsistent with these data (inset of Fig. 4a). A problem with considering strong el-ph in HTSCs is that there are not clear resistivity dropping or saturation near phonon frequency Ω and T≈(0.3-0.5)Ω, respectively.

In Summary, this study indicates existence of strong el-ph interaction in HTSCs and suggests including it in any microscopic theory of superconductivity.

Ref.: A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu, T. Yoshida, H. Eisaki, A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Nodak, S. Uchidak, Z. Hussain & Z.-X. Shen, Nature 412 (2001) 510.

Are HTSCs Quantum Protectorate!?

“Quantum protector” (QP) is a macroscopic quantum (for example quantum Hall effect and flux quantization of superconductivity) that is usually unaffected by imperfection, impurities and thermal fluctuations. In this state the excitations are collective excitations of the whole system rather than usual elementary excitations in many-body systems such as in Fermi liquid. P.W. Anderson believes that all phases of HTSCs (Fig. 1), not only the superconducting phase, seem to be QP and its source is the spin-charge separation (SCS). In SCS systems the elementary excitations are not consist of quasiparticles having both charge and spins, but for example in phase I of Fig. 1 there is a large charge gap (~2 eV) while there is no spin gap and in other phases there is neither a charge nor spin gap. In one dimensional, SCS always occurs but two dimensional is the critical dimension for this separation.   

Reasons for QP in different phases:

• Phase I: there are no phonon contributions to the electron self-energy and no extraneous energy scale in the simple scaling relation of conductivity, σ=ωF(T/ω)     which means that there are no interaction and scattering. Not even conventional electron-electron scattering would show the striking linear rise of scattering rate 1/τ ~ ω above the Debye frequency. Resistivity saturation due to strong phonon scattering is also absent in this phase. The strong scattering of the Zn and Ni substituted at the Cu in the CuO₂ planes, could be related to the Kondo scattering of the spin degrees of freedom not charge, and behaves completely different from other impurities.  

• Phase II: the pseudogap which is a one-electron gap in antinodal direction, is the most striking evidence for SCS, whereas there is not evidence for a gap for charge excitations (except in systems with static stripes).

• Phase III: almost all cuprate superconductors are strongly self-doped (~10-20%), but there is no evidence that T_c is even affected by the degree of purity or by phonon scattering which is pair breaking for d-wave. It also seems that high value of T_c can be achieved in a QP in which scattering dose not affect the collective state.

• Phase IV: the spin waves which are Goldstone bosons are weakly scatted by phonons and conventional impurities and are not scattered at all in the limit of ω and Q→0. They are in QP because the spin and charge dynamics are independent, and perturbations that interact primarily with charge and create large charge gap do not much affect spins.

Anderson discusses there are two sources for QP which are not entirely independent but physically distinct. 1) Spinons are relatively weakly scatter because they are the “Goldstone fermions” that express fundamental symmetries of the spin system. 2) In the SCS state, as observed by ARPES, the one-electron density of states vanishes at ω=0 as a power law, i.e. N(ω)~ω^p with 0.5<p<1. Thus, any perturbation that couples to electrons, in particular any time reversal invariant other than substitution in the copper site renormalize to zero at low frequencies. This discussion holds for phases I and II and more direct than the No. 1.

Finally, he notes that although the precise mechanism of superconductivity in HTSCs is of course still in question, reduction of the frustrated kinetic energy of the system could determine T_c. 

Ref.: P.W. Anderson, Science 288 (2000) 480.