The role of the Lifshitz point in high-Tc superconductivity

Physica C 153-155 (1988) 1181-1182 North-Holland, Amsterdam

THE ROLE OF THE LIFSHITZ POINT IN HIGH-T c SUPERCONDUCTIVITY

R.S. MARKIEWICZ

Physics Department,

Northeastern University,

Boston, MA 02115 T USA

The importance of the two-dimensional van Hove singularity for high-temperature superconductivity is discussed. A modified Bilbro-McMillan hamiltonian is introduced, in which an "excitonic" pairing process arises naturally, the resulting gap equation has highly non-BCS solutions, with the possibility of 2A(O)>>3.53kBT c.

I. THE LIFSHITZ POINT Every two-dimensional energy band has a particular van Hove singularity (Lifshitz point (I)) at which the carriers cross over from electron-like to hole-like conduction. At this singularity, the density-of-states (dos) has a logarithmic divergence. Band structure calculations suggest that the Fermi energy E F in the new high-T c superconductors lies close to this dos peak (2,3). The ease with which the superconductors can be prepared is evidence for this. In La2_xSrxCuO 4 (LSCO) it has been shown that samples prepared with x~0.15 are two-phase mixtures, with one phase at the composition of maximum Tc(X=0.15) (4,5). Such an effect (a spinodal decomposition) was in fact predicted for this system, due to the high stability associated with the Lifshitz point (6). Evidence of the role of the Lifshitz point in YBa2Cu307_ 6 (YBCO) is more indirect. The Hall effect is strikingly different than in LSCO,

1.0 Vo 0.5 LSCO O.OiIo0

Fig.

I.

.....

260 T(K)300

Proposed electron and hole carrier densities in YBCO (8) compared to LSCO (data of Penney, et al., Ref. 7). V 0 = unit cell volume.

tPresent Address:

Department of Physics,

0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

with the Hall coefficient varying nearly linearly with temperature (7). However, this can easily be explained in a two-band model, with holes in the Cu-O 2 planes nearly identical to those in LSCO, and electrons on the Cu-O chains (Fig. I) (8). This assignment is consistent with the band structure calculations of Whangbo, et al. (3). The similarity of normal state resistivity in these two materials is the further evidence that in both compounds E F is near a Lifshitz point (9). 2. MODEL FOR HIGH-T C SUPERCONDUCTIVITY Within single-phase samples, the high dos associated with the Lifshitz point will drive a competition between superconductivity and some other collective state - a charge or spin density wave (CDW/SDW) for instance (c-axis dispersion may round off the singluarity in the dos, but a large finite peak is adequate for the effects discussed here. A more important role of the weak three-dimensional coupling lies in fluctuation effects, discussed below.) This competition is very similar to that which occurs in the A15 compounds, and can be described using the formalism introduced by Bilbro and McMillan (BM) (10-12). It is probable that the superconductivity in the BaPbxBi1_xO 3 compounds, in which a well-defined CDW state has been found, can be explained solely in terms of this m o d e l (11). However, the higher Tc'S in LSCO and YBCO, accompanied by a weak isotope effect, seem to require some additional, non-phonon coupling. This additional coupling arises quite naturally within the BM hamiltonian, In this formalism, the electrons are divided into two groups, depending on their position in momentum space, group I is associated with carriers near the dos peak; group II with carriers in the remainder of the Brillouin zone. The CDW(SDW) transition is associated with carriers of group I: the condensate is stabilized by splitting the large dos peak and driving the occupied peak

Boston University,

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02215 USA

R.S. Markiewicz / Role of the Lifshitz point

I 182

~=c2+A(c)2+W26~,1 I

I

I

IOO

A(c) =

8O A,W (meV) 60

................................

i

40

"

\

20

"',

O~ Fig. 2.

20

i 40 kBT(meV)

Superconducting and CDW gaps for modified BM model. Solid line=W(Vx=O); dotted line=W, dashed line=A(VxN2=0.5) ; VpNI=I, hm1=50meV.

below the Fermi level. Superconductivity is associated with gapping the remainder of the Fermi surface. The gap produced by the group I collective mode introduces a well defined electronic excitation which can couple to the group II carriers to produce a non-phonon attractive pairing (13). This can be modelled by adding an "excitonic" term to the BM hamiltonian: H=HBM+Hx

(I)

where HBM is the BM hamiltonian (10) and

Hx

=

-

Vx N

- -

~' kk'

t c

t c

c

c

(2)

k'÷ -k'+ -k+ k*

where ctk~ are the usual creation operators for electrons with energy ek, V x is a constant effective attractive pair potential, and the rime limits the sum to k-values such that Ck,,I ~W, where W is the CDW gap. this hamiltonian leads to the coupled gap equations:

I~kl,

~

i

h~ I dgp I (c) W=Vp !hml ~

A=vBcs

[ -hm~ Ih~ dc

+V x ~ Iw -w de p~(s) ~ with

W

p~(E)

A

(5)

I

~I (s) tanh 2kBT

~

A

u

(3)

%(~)

tanh 2kB--T

c~(E____~) tanh 2kBT

(4)

A, Ic o I~

~ h~£ > h~

,

(6)

P2~N2 , P1~N1 £n (to/C), 8 t o is the width of the 2D band, and m1' m2 are appropriate phonon frequencies. Figure 2 shows an example of the solution of Eqs. 3,4 in the special case VBcs=O. Both the CDW and superconducting gaps show a very non-BCS temperature dependence. This arises both from the energy dependence of the dos PI' and from the form of the excitonic term, Eq. 2 (e~W(T)). In the particular example given, 2A(O)~7.9kBT c. There are many questions which must be answered in applying this model to the known high-T c superconductors. Chief among them is the identification of the CDW/SDW transition. While in LSCO, it is tempting to identify this with the tetragonal-orthorhombic transition (12,13), there is evidence of another, lower temperature transition which may play the dominant role. This may be connected with evidence for oxygen dimerization (14). The above calculations have ignored the posible role of fluctuations, which should be important in a 2D system. It may turn out that the CDW transition is restricted by fluctuations to short-range ordering, and that long-range order arises as a result of the superconducting transition itself (15). REFERENCES I. I.M. Lifshitz, Zh. Eksp. Teor. Fiz. 3_88 (1960) 1569 [Sov Phys. JETP11 (1960) 1130]. 2. J.H. Xu, T.J. Watson-Yang, J. Yu, and A.J. Freeman, Phys. Lett. A 120 (1987) 489. 3. M.H. Whangbo, M.Evain, M.A. Beno, and J.M. Williams, Inorg. Chem. 26 (1987) 1832. 4. J.D. Jorgensen, to be published. 5. R.J. Cava, MRS Meeting, Boston, 1987. 6. R.S. Markiewicz, Mod. Phys. Lett.B1(1987)187. 7. T. Penney, M.W. Shafer, B.K. Ols~n, and T.S. Plaskett, Adv. Ceram. Mat. 2 (1987) 577; Z.Z. Wang, J. Clayhold, N.P. Ong, J.M. Tarascon, L.H. Greene, W.R. McKinnon, and G.W. Hall, Phys. Rev. B36(1987) 7222. 8. R,S. Markiewicz, unpublished. 9. P.A. Lee and N. Read, Phys. Rev. Lett. 58 (1987) 2691. 10. G. Bilbro and W.L. McMillan, Phys. Rev. B14 (1976) 1887. 11. A.M. Gabovich and A.S. Shpigel, J. Phys. F14 (1984) 3031. 12. T.C. Choy and H.X. He, Phys Rev. B36 (1987) 8807. 13. R.S. Markiewicz, unpublished. 14. D.D. Sarma and C.N.R. Rao, J. Phys. C20 (1987) L659; B. Dauth, T. Kachel, P. Sen, K. Fischer, and M. Campagna, Z. Phys. B68 (1987) 407. 15. Z. Tesanovich, Phys. Rev. B36 (1987) 2364.