60 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

have been used. Equations (2.119)–(2.121) now enter the equation for the quasiparticle self-energy, i.e. (2.28), and are solved self-consistently. The equation for the even-ω part ξ(k, ω) of the self-energy is obtained from (2.119) by changing the sign in front of the term |ψd|2K and replacing the spectral function A0 of the Green’s function proportional to τ0 by the spectral function A3 that refers to the τ3 component. The spin fluctuation interaction Ps = (3/2)U 2Im χ0(1 − U χ0)−1, the spectral functions A0 and A3, and the kernel I are defined in (2.29) and are calculated self-consistently as described in Appendix A. It should be pointed out that our pair propagator K is related to the T -matrix Γ calculated earlier by Hotta et al. [58] by the equation πK = −Im Γ0 = Im φ0[(V −1 − Re φ0)2 + (Im φ0)2]−1, where Γ0 and φ0 are the undressed particle-particle T -matrix and susceptibility, respectively, and V is the BCS pairing constant for a d-wave. It has been pointed out in [58] that the equation 1 = V Re φ0(0, Ω) agrees with that for the binding energy Ω of a Cooper pair. In calculating K from the expression given above we have subtracted 1/V from Re φ0 utilizing the gap equation and expanding the rest in terms of ξ02q2 and τ ω (see (9) and (10) in [54] for ξ02 and τ ). Thus, our contribution from K to the self-energy in (2.119) and the corresponding contribution for ξ agrees with those obtained in [58] if the dressed particle– particle T -matrix Γ is replaced by the undressed one. Of course, we have solved (2.119) for the self-energy component ω [1 − Z(k, ω)] and the corresponding equation for ξ(k, ω) self-consistently by employing a tight-binding energy (k) with nearest and next-nearest neighbor hopping, whose 2D Fermi line has been adjusted to that of underdoped hole-doped cuprates. For example, in (2.4) the chemical potential has been taken to be µ = −1.1, yielding a doping concentration x = 0.08. For the e ective Coulomb repulsion we employed U (q) introduced in [54], with a maximum value U (Q) = U = 3.2t. In that case, in accordance with our calculated phase diagram for hole-doped cuprates (see Fig. 2.4), the value of Tc in the absence of order parameter fluctuations, i.e. K = 0, becomes about 0.03t.

2.3 Derivation of Important Formulae and Quantities

2.3.1 Elementary Excitations

For understanding the high-Tc cuprates, their elementary excitations are of central significance. Their theoretical analysis is based on the Green’s functions for the elementary excitations, which are given in Nambu space [18, 19]. After continuation to the real ω axis, the corresponding spectral density for a fixed temperature is given by

Here, k is the tight-binding energy dispersion on a square lattice introduced in (2.4) and Σ (k, ω) and Σ (k, ω) are the real and imaginary parts, respectively, of the self-energy described earlier in this chapter. We have performed our calculations for the elementary excitations

for various doping concentrations x, where Σ was calculated self-consistently using the 2D one-band Hubbard Hamiltonian for a CuO2 plane and the FLEX approximation. Below Tc the superconducting gap function φ(k, ω) has also been calculated self-consistently. As described earlier the full momentum and frequency dependence of the quantities has been kept and no further parameter has been introduced.

These equations are standard; however, it is important to realize that, owing to the combined e ects of Fermi surface topology and χ(q = Q, ω) at the antiferromagnetic wave vector QAF = (π, π), the k and ω dependences of Σ(k, ω) become very pronounced and change the dispersion ω(k). It is known that the strong scattering of quasiparticles by antiferromagnetic spin fluctuations results in a non-Fermi liquid behavior of the quasiparticle selfenergy for low-lying energy excitations; in particular, in Im Σ ω [66, 67]. Clearly, it follows from (2.123) that the expected doping and momentum dependence resulting from the crossover from Σ ω2 to Σ ω, i.e. to a non-Fermi liquid behavior, can be reflected in ω(k) and A(k, ω). Physically speaking, the change in the ω dependence of the self-energy Σ(k, ω) changes the velocity of the elementary excitations. Thus, as mentioned in the Introduction and as will be discussed later, for a given k-vector, the momentum distribution curve in ARPES experiments shows a “kink” at some characteristic frequency controlled by the spin fluctuation energy ωsf . Regarding the superconducting state, the k and ω dependence of the order parameter ∆(k, ω) is important and yields the feedback of the superconducting state on the elementary excitations.

From the elementary excitations of the quasiparticles mentioned above, we have calculated the spin excitations of the cuprates mainly via the expression

× [A(k + q, ω + ω)A(k, ω ) ± A1(k + q, ω + ω)A1(k, ω )] , (2.124)

k

where f denotes the Fermi distribution function and A1 is the spectral function of the anomalous Green’s function defined in (2.26). The corresponding procedure is described in Appendix A. In order to compare the results with inelastic neutron scattering experiments, we have calculated the imaginary part of the renormalized (RPA) susceptibility, i.e.

62 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

Here, U denotes an e ective Coulomb interaction using the Hubbard Hamiltonian ( (2.1)). We have choosen t = 0.2 in order to describe the Fermi surface topology of both YBCO and LSCO. In general, one finds that the structure of Im χ is determined by Im χ0 if (U Re χ0) = 1 and by (U Re χ0) = 1 if this latter condition can be fulfilled. This will be discussed later in connection with the resonance peak.

2.3.2 Superfluid Density and Transition Temperature for Underdoped Cuprates

In order to calculate the generic phase diagram for hole-doped cuprates (see Fig. 1.4), we have assumed that a rigid-band approximation is valid and have varied only the chemical potential µ in (2.1) in order to change the carrier

corresponds to the local density of states and has been defined in sect. 2.1. f denotes the Fermi distribution function. n = 1 corresponds to half filling. In order to simplify the discussion, we have also fixed the e ective Coulomb interaction U as U = 4t ≈ 1 eV in accordance with the tight-binding energy dispersion k measured in ARPES [68]. No further parameter is introduced.

The bulk transition temperature Tc at which phase coherence of the Cooper pairs occurs was determined by the Ginzburg–Landau free–energy functional ∆F {ns, ∆}, where the superfluid density ns(x, T )/m was calculated self-consistently from the current–current correlation function and from

SS is the value of (2.127) in the superconducting state. Here, we utilize the

f-sum rule for the real part of the conductivity σ1(ω), i.e. ∞ σ1(ω) dω =

0

πe2n/2m, where n is the 3D electron density and m denotes the e ective band mass for the tight-binding band considered. σ(ω) was calculated in the normal and superconducting states using the Kubo formula [20, 69]

where vk,i = ∂ k/∂ki are the calculated band velocities within the CuO2 plane and c is the c-axis lattice constant. The spectral function A1 was defined in (2.26). Vertex corrections have been neglected. Physically speaking, we are looking for a loss of spectral weight of the Drude peak at ω = 0 that corresponds to excited quasiparticles above the superconducting condensate for temperatures T < Tc . The penetration depth λ(x, T ) was calculated within the London theory through λ−2 ns [70] and will be discussed later.

Most importantly, using our results for ns(x, T ), we have calculated the doping dependence of the Ginzburg–Landau-like free-energy change ∆F ≡ FS − FN , [10, 11],

where ∆Fcond α{ns/m}∆0(x) is the condensation energy due to Cooper pairing and ∆Fphase ¯h2ns/2m is the loss in energy due to phase incoherence of the Cooper pairs. α describes the available phase space for Cooper pairs (normalized per unit volume) and can be estimated in the strongly overdoped regime. In the BCS limit one finds α 1/400. ∆0 is the superconducting order parameter at T = 0. Within standard (time-dependent) Ginzburg– Landau theory2, the superfluid density ns can be calculated via n0s/ns =φ(r, t) φ(0) , where φ(r, t) reflects the changes of the spatial and time dependence of the Cooper pair wave function. n0s is the static mean-field value of the superfluid density for a given temperature, calculated with our extended FLEX approximation for the generalized Eliashberg equations.

Owing to the layered structure of the cuprates, in the underdoped case they should behave in accordance with the 2D XY model except in a narrow critical range around Tc where the 3D XY model is more appropriate [71, 72]. The standard theory for the 2D XY model, the Berezinskii–Kosterlitz– Thouless (BKT) renormalization group theory, should thus be a reasonable starting point [73, 74, 75, 76]. The superconducting transition predicted by the BKT theory is due to unbinding of fluctuating vortex–antivortex pairs in the superconducting order parameter. Gaussian phase fluctuations are not important, since they do not shift Tc.3 Furthermore, if one takes the coupling of the phase to the electromagnetic field into account, they become gapped at the plasma frequency (of the order of 1 eV) owing to the Higgs mechanism [77].

2In analogy to a ferromagnet, we expect n0s/ns = Φ(r, t) Φ(0) . Note that it is straightforward to map our electronic theory onto a lattice (Wannier–type

representation) and then to derive from the product of the anomalous Green’s functions, {F · F }, a contribution to the free energy of the form ns cos Θij as used by Chakraverty et al. [51]. Here, Θij is the angle between the phases of

T < Tc and ¯ij = 0 for T > Tc.

Θ

3This is true in three dimensions. Of course, in the 2D case Gaussian fluctuations destroy the long-range order, yielding Tc = 0, but the mean-field transition is still unchanged.

64 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

Let us now turn to a second possibility for the determination of Tc, namely with the help of the BKT theory. It turns out that the relevant parameters for the unbinding of thermally created vortex–antivortex pairs are the dimensionless sti ness K and the vortex core energy Ecore. The sti ness is related to ns by

where β again denotes the inverse temperature and m the e ective mass. d is the average spacing between CuO2 layers. In our calculations we set d to half the height of the unit cell of YBa2Cu3O6+x. One then has to solve the Kosterlitz recursion relations

Here y = e−βEc denotes the vortex fugacity. For the vortex core energy, we have used the approximate result of Blatter et al., i.e.

where κ is the Ginzburg parameter, and l = (r/r0) is a logarithmic length scale which relates K to the strength of the vortex–antivortex interaction. For T > Tc, K tends to zero for l → ∞, so that the interaction at large distances is screened and the largest vortex–antivortex pairs unbind. This destroys the Meissner e ect and leads to dissipation. On the other hand, bound Cooper pairs reduce K and thus ns, but do not destroy superconductivity. After (2.131) and (2.133) have been solved, it turns out that that the renormalization of K is very small [78]. Thus, to a good approximation, one can obtain Tc(x) from the simple criterion [62, 78, 79, 80]4

In order to investigate the dynamical phase sti ness, one has to calculate ns(ω). A dynamical generalization of the BKT theory was developed by Ambegeokar et al. [81, 75]. It turns out that the critical size for a vortex–

4In our FLEX theory, the fluctuations of the ordered antiferromagnetic state in the paramagnetic metallic regime are treated beyond the mean-field level, however, the fluctuations of the superconducting condensate were neglected in earlier treatments. A detailed comparison between ns(ω) in the XY model and in the FLEX approximation is given in [78, 80].