Dielectric response in high Tc superconductors

Physics C 162-164 (1989) 759-760 North-Holland

DIELECTRIC RESPONSE IN HIGH T~ SUPERCONDUCTORS F. LOpez-Aguilar(~), J. Costa-Quintana(f), S. Balle(*), F. Febrero(f), M. T. AureU(t), A. Sgnchez(f) and J. S. Mufioz(f). (t)Departamento de Fisica, Grupo de Electromagnetismo, Universidad AutOnoma de Barcelona. BeUaterra E08193 Barcelona, Spain. (*)Departament de Fisica, Unlversitat de les Illes Balears, E-07071 Palms de Mallorca; Balears, Spain. The random phase approximation integral equation is solved for obtaining the dynamically screened interaction between d electrons (Wd(w)), p electrons (Wp(w)) and p/d electrons (W,d(W)). Some characteristic divergences of this last interaction correspond to the plasmon frequencies which one inn relate with the effective masses of the p and d electrons close to EF by means of the relation w~,l/w,,2 = (m2/ml) 1/2. Another feature:of this Wpd(w) interaction is the attractive character for low frequencies. The lowest frequency for which epd = 0 decreases when the localization of states close to EF arising from the z 2 - y2 symmetry increases. Experimental results of ac Josephson effect 1 in high

can be obtained from the random phase approximation

T~ superconductors show that the charge unit of the superconducting current is twice the electron charge. This implies that the supercurrent is constituted by

(RPA) integral equation applied to systems described by a multiband Hubbard hamiltonian. This hamilto-

electron or hole pairs. The electronic structure of these oxide superconductors presents, as in the heavy fermion systems, a large localization of the states close to EF and this is consistent with strong electronic correlations detected by direct and inverse X-ray photoemision spectroscopy 2. Therefore, the e-e or h-h coupling may have a similar origin to those of organic superconductors a n d / o r heavy fermion systems. The main feature of the crystai structure of these copper oxides is the two dimensional CuO layer structure 3. The d states of Cu and p states of O corresponding to atoms of the CuO layers are close to EF 4's. The d,~_y~ symmetry of Cu and the pz/py symmetry of O present hybridization through a large transfer integral s and their orbitais d ~ and p~ are strongly localizeds. Some models for the h-h coupling consider that the pairing is due to an effective (attractive) interaction coming from the ,dynamically screened interatomic bare repulsion (Vpd)4'7. This requires negative response for low frequencies. In this work, we analyze the dynamical screening of the repulsive interaction between two holes localized in a d~2_~ level of Cu and p~ or py levels of O. A first approximation to the response of the electron gas

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nian reads IV,

U

ct

c.

ct

c.

,

(i)

z~rLn

where i runs over the lattice sites; m and r~ label either a copper d state or an oxigen p state corresponding to a primitive cell; Upd is the interatomic bare repulsion between d,2_y2 orbitals and p= or py orbitals belonging to neigbouring Ca and O atoms. Udd is the Coulomb correlation energy for d=2_y2 states and Uvp is the U for p= or p~ states. The integral RPA equation in a system described by (1) leads to the following approximated equations

Wa(w) ~ Usa[1 - Xd(w)Uad] -1 Wp(w) ~

U,,[I

- Xv(w)U,,]-'

Wpd(w) ~ Vpd[1 - Xd(w)Ud,z] -1 [1 -- X~,(t,,,,)Upp] -1

The dependence of W,,(,,,) %(,.,,) and %,,(,,,) on k is dropped by considering average values over the first Brillouin zone, (i.e. Wd(w) : function X is defined as

-~

Wd(k,~)); the

XdCp)(w)= Fdo,)(w)+ Fdcp)(--w) being

rE.

rhwo Nd(v)(e)Ndo,)(g)ded d

F. L6pez-Aguilar et al. / Dielectric response in high T~ superconductors

760

(we2) depends on the same parameters corresponding

(a) 3

1

to the p orbitals of O. It is established 8 that, for Hubbard band systems, Wpl/Wp~ = ( m 2 / m l ) 1/2, rn~ and mx being the effective masses assigned to the states p

~o

and d respectively. The masses of the d states are be-

[ .,11

....

np =.80 I

tween 2.5 and 5 times larger than those of the p states

nd--.60

I

I

for the range of parameters analysed in this work. An-

I

other characteristic feature of the epd(w) functions is 3

1

the negative sign at low frequencies which implies an

~o

attractive interaction Wpd(w). The first zero of epd(w)

-I -2

i , . . I

I

dp •.50

6 d •.15

(~ 0.004 Ry) becomes closer to EF as the width of

np =.70

n d =.50

the band arising from the d=~_~ orbital decreases.

I

I

I

Therefore, the energy interval where Wpd is attractive 3

is smaller for an increasing localization of the d~_~2

1

~o -I -2

states. However, this localization implies a decrease of

--,i 0.02

=. I

0.04

:; ;; I

0.06

dd

=.15

n d

----.60

.I

....

0.08

the bare interatomic interaction Vpd. In conclusion, we have deduced a dynamically screened effective interaction between the a states of Cu and p of O correspond-

I ....

0.1

0.12

(Re)

ing to the CuO layer and we show that the effective masses of the d states are larger than those of O. In addition, Wpd is negative from a frequency value close

FIGURE 1 Dielectric response for different band parameters. The baadvidth ip (Sd) of the p (d) orbital is given in eV, Allcurves correspond to Uppffi0,35Ry and Udd=0,40R y

to zero and this value decreases when the correlation between the d~2_y2 states increases. REFERENCES

where Nd(p)(e) is the density of states (DOS) corresponding to the d(p) orbitals calculated by a local density band hamiltonian.

1.-J.S. Tsai, Y. Kubo and J. Tabuchi, Phys. Rev. Left. 58 (1987) 1979.

In our case, as in Emery's

model 4, we have only the DOS corresponding to the

2.-A.J. Viescas, J.M. Tranquada, A.R. Modenbaugh and D.D. Johnson, Phys. Rev. B 37 (1988) 3738.

symmetries d~2_y2 of Cu and p= of O. In fig. 1 we represent the dielectric response cor-

3.-R. Beyers and T.M. Shaw, Sol. Stat. Phys. 42 (1989) 135.

responding to interatomic Upd , epd( W) = Upd/ Wpd( w ), for different values of the occupation of the orbitals and

4.- V.J. Emery, Phys. Rev. Left. 58 (1987) 2794.

the widths of the partially occupied bands. The study

5.- K.C. Hass, Sol. Stat. Phys. 42 (1989) 213.

of the variations of epd(W) respect to band parameters is performed elsewhere. However, we wish to remark that a common feature of all these curves is the existence of two zeros of the function &d(W) for which this function present positive derivative respect to w. These zeros correspond to plasmon poles (see ref. 8). The location of the low frequency pole (wpl) is governed by the occupation, width and Coulomb correlation energy of the orbital d~2_~2, while the high frequency plasmon pole

6.- J. Costa-Quintana, F. L6pez-Aguilar, S. Balle R. Salvador, Phys. Rev. B. 39 (1989) 9675. 7.-C.M. Varma, S. Schmitt-Rink and E. Abrahams, Novel superconductivity ed. S.A. Wolf and V.Z. Kresin (Plenum Press, New York 1987) p295. 8.- A.M. Awasthi, W.P. Beyermann, J.P. Carini and G. Gruner, Phys. Rev. B 39 (1989) 2377.