d-Wave Pairing

It is plausible that the notion of Cooper pairing and BCS theory may still be applicable to the high-temperature superconductors, yet the nature of the pairing mechanism may be something other than the phonon-induced electron-electron interaction. The formation of a bound state can be achieved by any attractive interaction capable of overcoming the natural Coulomb repulsion between two electrons. Several alternative sources for this attractive force which are compatable with conventional BCS theory have been proposed. One such mechanism, which has received much attention in recent years, is an electron-electron interaction mediated by magnetic spin fluctuations [51,52,53]. The concept is not entirely new. A similar process is believed to help facilitate p-wave spin-triplet pairing (L=1, S=1) in superfluid ³He, and to lead to other pairing states in certain organic superconductors and heavy fermion systems such as UPt₃ [49,53].

The antiferromagnetic state of the parent materials such as $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6}$ }$ and the anomalous normal-state properties of the high-T_c superconductors provide the inspiration for attempts at describing the superconducting properties in terms of a spin-fluctuation exchange mechanism [52,53,82]. A logical starting point for such a theory is to suggest that the physical origin of those normal-state features which differ from normal metals may somehow be responsible for superconductivity in the cuprates. It has been suggested that the measured anomalous normal-state properties of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$stem from strong antiferromagnetic correlations of spins, and these same antiferromagnetic spin fluctuations are also responsible for superconductivity in the cuprates [1]. NMR measurements of the normal state have been successfully modelled with a nearly antiferromagnetic Fermi liquid [55,56,57].

Some insight into the origin of possible antiferromagnetic spin fluctuations in the superconducting phase of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$ may be obtained by examination of the antiferromagnetic insulating compound $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6}$ }$. The structures of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6}$ }$ and $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$ appear in Fig. 2.9 and the phase diagram for YBa₂Cu₃O_x (6 < x < 7) appears in Fig. 2.10. In reference to Fig. 2.9(a), the Cu(1) or Cu-O chain layer of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6}$ }$consists entirely of Cu¹⁺ ions. The singly ionized Cu ions have no magnetic moment. Oxygen doping places O ions along the b-axis, resulting in a progressive conversion of Cu¹⁺ into Cu²⁺ with the development of holes in the 3d-shell of the Cu ions [59].

  

Figure 2.9: (a) The structure of the insulator $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6}$ }$ and (b) the structure of the superconductor $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$ [18].

  

Figure 2.10: Phase diagram for YBa₂Cu₃O_x, as a function of oxygen formula concentration x. Also shown is the phase diagram for a Zn-doped sample [58].

The Cu(2) or CuO₂ planes of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6}$ }$have predominantly Cu²⁺ ions. Each Cu gives up two electrons; an electron from the 4s-shell and the other from the 3d-shell. The absence of an electron in the 3d-shell (a hole) results in a net magnetic moment (spin) on the Cu ions in this layer. Oxygen cannot easily be removed or added to the CuO₂ planes. The oxygen concentration can be varied appreciably only in the Cu-O chains. As mentioned, adding oxygen converts the copper ions in the Cu-O chains from Cu¹⁺ to Cu²⁺. Beyond $x \approx 6.5$it is believed that adding oxygen is equivalent to adding holes to the CuO₂ planes. The oxygen which is randomly added to the chains becomes O^2- by trapping two electrons which are believed to originate from the creation of two holes in the oxygens of an adjacent CuO₂ plane. However, Hall coefficient measurements suggest that holes may also be forming in the chains [23].

Neutron difffraction and muon precession experiments indicate that the Cu moments are antiferromagnetically aligned in $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6}$ }$below the Neel temperature T_N (see Fig. 2.10). The Cu²⁺ spins (i.e. spin 1/2 holes) in the CuO₂ planes are coupled antiferromagnetically through a superexchange process with the oxygen ions. The O^2- ions themselves have no net magnetic moment. At sufficiently low temperatures, the Cu ions in the chains also become antiferromagnetically ordered and couple with the Cu²⁺ ions in adjacent CuO₂ planes [18]. In view of this, the Cu¹⁺ labelling of the chain layer Cu ions may not be entirely accurate. Neutron and Raman scattering experiments suggest that the exchange interactions within the CuO₂planes are much greater than the coupling between adjacent layers. This is likely due to the greater separation between Cu ions in the vertical direction and a lack of O^2- ions between adjacent CuO₂planes. The difference between strengths of the interplanar and intraplanar couplings means that $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6}$ }$ exhibits a quasi two-dimensional magnetic behaviour. Furthermore, the measured maximum magnetic moment on the Cu ions is substantially smaller than what one would expect for a localized Cu²⁺ ion. This may be due to the enhanced thermal fluctuations associated with a two-dimensionally ordered system [18].

It is a widely accepted belief that the electrons responsible for conduction in the copper-oxide superconductors are more or less confined to the CuO₂ planes [52]. If this is the case then it is plausible that these electrons are paired by way of a two-dimensional system of antiferromagnetic spin fluctuations. The next obvious question to ask is,

``Do these spin fluctuations persist in the superconducting phase?''

Raman and neutron scattering measurements suggest that the magnetic fluctuations do indeed survive into the superconducting state. The spin-correlation length is substantially diminished in the superconducting state, but the amplitude of the magnetic moments is not greatly diminished [18]. Furthermore, NMR data taken above T_c indicate the presence of two-dimensional antiferromagnetic spin fluctuations arising from the nearly localized Cu²⁺ d-orbitals in the CuO₂ planes.

It is difficult to give an intuitive description of pairing due to spin-fluctuation exchange. It is clear that a single hole will help destroy the antiferromagnetic order. However it is less clear whether it will attract or repel a second hole with the same or opposite spin. The answers to these questions seem to depend very much on the regions of k-space and r-space considered [60].

Weak-coupling calculations of the normal and superconducting state properties have been carried out [1,57,82] for an antiferromagnetic spin-fluctuation induced interaction between quasiparticles on a two-dimensional square lattice. Such a two-dimensional model is at best an approximation to the behaviour of the three-dimensional $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$ compound. Nevertheless, these calculations yield a value of the transition temperature T_c which is near 90K and a superconducting pairing state with d_{x² - y²} symmetry. For this pairing state the energy gap is of the form:

$$\Delta (\vec{k}, T) = \Delta_{\circ} (T) \mid \cos (k_{x}a) - \cos (k_{y}a) \mid$$ (31)

where $\Delta_{\circ} (T)$ is the maximum value of the energy gap at temperature T and a is the lattice constant or distance between nearest neighbor Cu atoms in the plane. The angular momentum and spin of a Cooper pair is L=2 and S=0, respectively (i.e. singlet d_x²-y² pairing).

The superconducting gap originating from the spin-fluctuation mediated interaction has a momentum (or k) dependence, in contrast to the phonon-frequency dependence of the gap associated with an electron-phonon interaction. It is clear from Eq. (2.31) that the excitation gap vanishes when $\vert \hat{k}_{x} \vert = \vert \hat{k}_{y} \vert$. Fig. 2.11 shows the four nodes which result along the diagonals in the Brillouin zone at the Fermi surface. In three-dimensional k-space the gap vanishes along four nodal lines running parallel to the k_z-axis for a cylindrical Fermi surface, or along four nodal lines joining the north and south poles for a spherical Fermi surface (see Fig. 2.12). Because of these nodal lines, there will be considerably more quasiparticle excitations at low temperatures compared to conventional s-wave superconductors. Thus even at T=0K there is a quasiparticle contribution to the supercurrent [61]. For $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$ which is highly anisotropic, a cylindrical Fermi surface with no gap in the k_z direction seems like a plausible description. However, it has been suggested that the cross-section of the Fermi surface in $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$ is not a perfect circle. If this is the case, then the pairing state can have d-wave symmetry but not give rise to nodes in the gap [49].

 

 

To gain a qualitative understanding of the symmetry in Fig. 2.11, consider the two-dimensional square lattice of antiferromagnetically ordered spins depicted in Fig. 2.13. In real space nearest-neighbor spins are separated by the lattice constant a. Such an arrangement is a simplified model of the CuO₂ planes in $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$. In the $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$ compound, $a \approx b$, so that the CuO₂ planes are almost square and the localized spins of Fig. 2.13 correspond to the antiferromagnetically-correlated spin fluctuations associated with the Cu²⁺ d-orbitals. In the derivation of Eq. (2.31), a spin-spin correlation function (electronic spin susceptibility) was chosen which gave a good quantitative fit to NMR measurements of the Knight shift and the spin-lattice relaxation rates of ⁶³Cu, ¹⁷O and ⁸⁹Y nuclei in $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$[82]. The electronic spin susceptibility is representative of the strength of the spin-fluctuation-mediated pairing potential. This function is strongly peaked at the nesting wave vector $\vec{Q} = ( \pm \frac{\pi}{a}, \pm \frac{\pi}{a} )$ in the first Brillouin zone. For an s-wave gap, the electronic spin susceptibility is suppressed at $( \pm \frac{\pi}{a}, \pm \frac{\pi}{a} )$ [2].

  

Figure 2.13: A square lattice of antiferromagnetically arranged spins.

To understand the significance of $\vec{Q}$, consider first the phase-space restrictions on the electron-electron scattering rate for a conventional electron gas assuming a cylindrical Fermi surface. At T=0K the Fermi cylinder is full and there are no electrons to scatter from such that energy and momentum are conserved. For T > 0K, an excited electron with energy different than E_F can scatter into a shell of partially occupied levels centered about E_F in $\vec{k}$-space. That is to say, the range of momenta available to the scattered electron is proportional to the temperature T [47]. This situation is depicted in two dimensions in Fig. 2.14(a), where the incoming quasiparticle momenta are $\vec{k}$ and $- \vec{k}$. Any interaction between electrons changes the momenta of the quasiparticles such that $\vec{k}^{\prime} \approx \vec{k} \pm \vec{q}$. Any orientation of the wave vector $\vec{q}$ in Fig. 2.14(a) will yield the same available phase space for electron scattering near the Fermi surface at a temperature T.

 

Consider now a nearly antiferromagnetic Fermi liquid with nesting vector $\vec{Q}$ as shown in Fig. 2.14(b). In this nested-Fermi liquid, two electrons with momenta ( $\vec{k}, -\vec{k}$) near the Fermi surface exchange the antiferromagnetic spin fluctuation which has a sharp peak at $\vec{Q}$. The two electrons are subsequently scattered (by the oscillating potential set up by the corresponding spin density wave) to states with wave vectors ( $\vec{k}^{\prime}, -\vec{k}^{\prime}$) near opposite sides of the Fermi surface. In this case $\vec{k}^{\prime} \approx \vec{k} \pm \vec{Q}$. As illustrated in Fig. 2.14(b), the range of momenta available to the scattering electron is greater than in the conventional Fermi liquid of Fig. 2.14(a). For all $\vec{k} - \vec{k}^{\prime} \equiv \vec{Q}$ parallel to the wave vectors $( \pm \frac{\pi}{a} , 0 )$ and $( 0, \pm \frac{\pi}{a} )$, the available phase space for scattering is the same. As $\vec{k} - \vec{k}^{\prime} \equiv \vec{Q}$ is rotated away from these directions, the range of momenta available to the scattering electron decreases so that it is smallest when $\vec{Q} = ( \pm \frac{\pi}{a}, \pm \frac{\pi}{a} )$. The evolution of the wave vector $\vec{k} - \vec{k}^{\prime}$ in Fig. 2.14(b) maps out the gap function of Fig. 2.11. This is demonstrated in Fig. 2.15. As $\vec{k} - \vec{k}^{\prime}$ is rotated away from $( \pm \frac{\pi}{a} , 0 )$ or $( 0, \pm \frac{\pi}{a} )$, the range of momenta available to scatter into decreases and so does the magnitude of the energy gap.

It should be noted that there are theories which predict d_x²-y²-wave pairing which are not based on spin fluctuations [28]. Considering these, it seems appropriate to discuss d_x²-y² symmetry as it pertains to the Fermi-surface geometry, rather than to introduce the details of the theoretical calculations which predict d_x²-y²-wave pairing from antiferromagnetic spin fluctuations. The orientation of the Fermi surface in Fig. 2.14(b) and Fig. 2.15 is that used to explain commensurate peaks at $( \pm \frac{\pi}{a}, \pm \frac{\pi}{a} )$ in neutron experiments involving $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$ [62]. The corners of the Fermi surface are actually more rounded than they appear in these figures. The Fermi surface for La_2-xSr_xCuO₄ has more curvature in the sides and is rotated $45^{\circ}$ from that of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6.95}$ }$ so that incommensurate peaks are observed at $( \pm \frac{\pi}{a}, \pm \frac{\pi}{a} )$ in neutron experiments.

 

Returning to $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6.95}$ }$ and the notion of spin fluctuations, if one assumes that a given oxygen nucleus is coupled predominantly to the spins on its two nearest-neighbor Cu sites, then the Cu spin-density of states is greatest near the Brillouin zone corners $( \pm \frac{\pi}{a}, \pm \frac{\pi}{a} )$ [56]. Also since the O nucleus is resting between two oppositely directed Cu spins, the transferred hyperfine field from the Cu moments cancels at the O site so that the spin density vanishes there. Thus the Cu spins relax the O nuclei so that the dominant contribution to the spin susceptibility comes from the Cu²⁺ d-orbital spin states. Thus low temperature excitations may result from the influence of the Cu spins on the superconducting carriers.

Opponents of the spin-fluctuation mechanism have argued that the measured quasiparticle lifetimes in $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$ are much too short for the quasiparticles to take advantage of this sort of interaction [53]. Strong-coupling calculations (which normally imply a short quasiparticle lifetime) have been carried out. Results show that one still obtains a T_c of 90K for $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$, with a d_x²-y²pair gap [53]. Subsequent calculations of the anomalous normal-state quasiparticle properties in the corresponding strong-coupling regime have also been done [53,63]. Results are consistent with experimental measurements, indicating that the strong-coupling calculations are reasonable.

Theoretically a d-wave pairing state is appealing because it avoids the strong on-site Coulomb repulsion which is inherent in an s-wave pairing state. Also a spin-fluctuation pairing mechanism for the high-T_c superconductors would not lead to a lattice instability as may occur for an extremely strong electron-phonon interaction. Unfortunately it is yet to be shown if the d_x²-y² symmetry evolves out of calculations for oxygen concentrations less than in $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$. Also a complete microscopic theory for d_x²-y²pairing is still unavailable. On the experimental front, attempts at determining the pairing state in the copper-oxide superconductors have been conflicting and inconclusive.