Superconducting correlations

4.4 Effect of interlayer couplings

4.4.2 Superconducting correlations

Earlier we have mentioned about the non-monotonic variation of the phase boundary between the SC and normal state or equivalently the variation of critical hole concentration, x_c where the SC phase vanishes, with interlayer hopping. For the d-wave state in the J_⊥ = 0.10 and 0.20 phase planes, x_c increases initially with increasing t_⊥, reaches a maximum at t_⊥ ∼ 0.3 and then decreases sharply. Here we try to explain this feature on the basis of energetics. We find that increasing t_⊥ can have two contrasting effects on the d-wave SC correlations. Let us look the momentum distribution, n(k) for the d-wave state in the overdoped region. It is calculated as,

n(k)= 1 2

X

σ

hc^†_kσckσi (4.1)

In Fig.4.9, we plot the two dimensional projection of the momentum distribution function in the kz =0 and kz=πplanes for two different interplanar hopping parameters in the overdoped region of the J_⊥ = 0.10 phase diagram. The values of the hopping parameter are chosen to

0 π

0

0.2 0.3 0.4 0.5

π

0.6 t_⊥ = 0.10

t_⊥ = 0.30

F4.9: Plots of n(k) for the d-wave state at hole concentration, x = 0.28, projected on a two dimensional (kx,ky) plane for kz = 0 and kz = πmarked in the figures. The top panel

shows n(k) for t_⊥=0.10 while the bottom panel corresponds to t_⊥=0.30. J_⊥=0.10.

be, t_z = 0.10 and 0.30 and hole concentration, x = 0.28. Around this doping, we see the SC-normal phase boundary extending towards higher doping region as we go from t_z = 0.10 to tz = 0.30. The plots in the figure show a significant redistribution of n(k) weights in the two momentum planes as t_⊥ is increased. Comparing the plots for the two different values of t_z, we see that the spectral weights shift from the k_z = π plane to the k_z = 0 plane as t_⊥ is increased from 0.10 to 0.30. Thus for larger t_⊥, occupation of the pairs with k_z = 0 is higher in the overdoped region. This favours the stability of the SC phase, as the k_z = 0 pairs are planar and contribute to the development of the SC pairings which for the d-wave state are essentially planar. The kz = πpairs are interplanar and hence are not expected to be important as far as the d-wave state is concerned. We believe that this transfer of weight from k_z = π to k_z = 0 plane as t_⊥ is increased helps in stabilizing the SC order in the overdoped region for small values of t_⊥. For larger t_⊥ however, hopping frequency of the electrons across the planes increases and consequently the probability of breaking a planar pair becomes increasing greater. Thus we see a rapid fall of x_c for larger t_⊥. To provide evidence for this, we look at the interlayer hopping energy and planar exchange energy for the d-wave state as a function of t_⊥. This is shown in Fig.4.10. In Fig.4.10(a), we plot the interplanar hopping energy, E_⊥^t (= hH_⊥^tiper site) scaled by t_⊥ as a function of t_⊥. The lowering of this quantity

-0.3 -0.2 -0.1 0

0 0.5 1

Et ⊥/t⊥

t_⊥

x = 0.16 x = 0.25

J_⊥ = 0.10

(a)

-0.26 -0.22 -0.18

0 0.5 1

EJ ||

t_⊥

x = 0.16 x = 0.25 J_⊥ = 0.10

(b)

F 4.10: (a) Interplanar hopping energy scaled by t_⊥, E^t_⊥/t_⊥ and (b) planar exchange energy, E_k^J for the d-wave SC state as a function of t_⊥. The values of hole concentrations are

shown in the figures. J_⊥=0.10.

with increasing t_⊥as observed in the figure, is an indication of occurrence of higher interlayer hopping frequency of electrons for larger t_⊥. The disruption of planar pairings by the enhanced interplanar hopping is confirmed by the loss of planar exchange energy, E^J_k (=hH_k^Jiper site) for larger t_⊥, which is shown in Fig. 4.10(b). On the other hand, for the case of d_z-wave

state in the J_⊥ = 0.35 phase plane, we see that x_c decreases uniformly with increasing t_⊥, except for small t_⊥where variation of x_c is negligible. This is probably due to the fact that the mechanism for enhancement of planar SC correlations by increased t_⊥mentioned above is not operative in this case. However the pair breaking effect of interlayer hopping of electrons continues to exist for the interlayer pairs as indicated by the loss of interlayer exchange energy in the d_z-wave SC state with increasing t_⊥in Fig.4.11.

-0.18 -0.14 -0.1

0 0.5 1

EJ ⊥

t_⊥

x = 0.25

x = 0.34

J_⊥ = 0.35

F4.11: Interplanar exchange energy, E_⊥^J for the d_z-wave state as a function of t_⊥for two different hole concentrations. J_⊥=0.35.

To investigate the effect of interlayer couplings on the superconducting correlations, we calculate the SC order parameter,Φdefined as (see§3.3.4),

F_α,β(r−r^′)→ ±Φ² (4.2)

where F_α,β(r−r^′)=D

B^†_rαB_r′β

E. The operator B^†defined as B^†_rα= ¹

2(c^†_r↑c^†_r+α↓−c^†_r↓c^†_r+α↑) creates a singlet pair at two nearest neighbour sites r and r+α. The unit vectorsα,βcan be either ˆx, ˆy or ˆz. We further define,Φ≡ Φ_dwhen bothαandβlie in a single plane with the+(−) signs in Eq. (4.2) referring to the case whenαparallel (perpendicular) toβ. For α,βboth equal to ˆz, we defineΦ≡Φ_dz which denotes the order parameter corresponding SC correlations across the planes.

The SC pair-pair correlations, F shows very different behaviour in the d- and dz-wave states. For the d-wave state, the SC correlations in plane are strong and is long range or-

are found to be very weak, roughly an order of magnitude smaller than that of the planar correlations. The order parameter,Φ_d corresponding to the planar correlations in the d-wave state is shown in Fig. 4.12(a) as a function of hole doping for different values of interlayer hopping. While the dome shaped structure of Φ_d has been observed and explained before

0 0.02 0.04 0.06

0 0.1 0.2 0.3 0.4

Φd

x t_⊥ 0.10 0.30 0.60

J_⊥ = 0.10

(a)

0 0.1 0.2

0.1 0.3 0.5

Φdz

x t_⊥

0.10 0.30 0.60

J_⊥ = 0.35

(b)

F4.12: SC order parameter as a function of hole doping at different values of interlayer hopping. (a) Order parameter, Φ_d corresponding to planar correlations for the d-wave SC state (J_⊥ = 0.10). (b) Order parameter,Φ_dz corresponding to interplanar correlations for the d_z-wave SC state (J_⊥= 0.35). Interplanar (planar) SC correlations for the d-wave (d_z-wave)

state are negligibly small.

for planar systems[142], here it is interesting to note the effect of t_⊥on the correlations. The figure shows that single electron interlayer hopping reduces the SC correlations at all values of doping, except for in the overdoped region where superconductivity enhances for moderate values of t_⊥. This should be contrasted with the variation of the SC gap, ˜∆_{S C}, which increases with t_⊥in the underdoped region as shown earlier. It may be mentioned that whileΦ_dscales with the transition temperature, Tc of cuprate superconductors, the gap ˜∆_{S C} scales with T^∗, the temperature scale for the pseudogap generic to the underdoped cuprates. On the other hand, the superconducting correlations in the d_z wave state are found to be predominantly interplanar. This is a consequence of large J_⊥ which make the formation of interlayer pairs energetically favourable. The corresponding order parameter, Φ_dz for J_⊥ = 0.35 is shown Fig. 4.12(b). It is remarkable that the interplanar dz-wave correlations is almost an order of magnitude stronger than the planar d-wave correlations as seen in the figure. The correlations between planar pairs in this case are very weak and are not shown. Another appealing feature is that in spite of the very different nature of the SC correlations, we observe the same dome

like structure of Φ_dz as a function of x which is interesting. Also to be noted is the fact that the interlayer hopping t_⊥reduces SC correlations similar to the d-wave case.