Experimentally observed extended saddle point ... - Science Direct

Physica C 214 (1993) 73-79 North-Holland

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Experimentally observed extended saddle point singularity in the energy spectrum of YlaECU306.9 and YBaECU408 and some of the consequences A.A. Abrikosov a, J.C. Campuzano a,b and K. G o f r o n a,b • Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA b University oflllinois at Chicago, Chicago, IL 60680, USA

Received 29 April 1993 Revised manuscript received 21 June 1993

Angle-resolvedphotoemission experiments show that the high temperature superconductors YBa2Cu306.9and YBa2Cu40s have a spectral density which exhibits an "extended" saddle point singularity along the F-Y symmetry direction, centered on the Y point. This, in turn, leads to a van Hove singularity in the density of states with a divergence stronger than the well-known logarithmic one. We examine the consequences of this singularity in several limiting cases on the superconducting transition temperature, the ratio 22(0)/Tc, and the isotope effect. We conclude that this singularity alone, although it can possibly lead to sufficiently high transition temperatures and a vanishingly small isotope effect, does not explain the ratio of 22(0)/T¢ ~ 6 observed experimentally.

1. Introduction Many mechanisms have been proposed to explain the high transition temperatures in the perovskite superconductors, including the presence o f a van Hove singularity near the Fermi energy [ 1-4 ]. With this motivation, we have examined the electronic structure in YBa2Cu306.9 and YBa2Cu4Os by angleresolved photoemission, and find an "extended" saddle point singularity in the energy spectrum in the neighborhood o f the Y point. At the Y point, this singularity occurs in the band derived from the CuO2 planes at a binding energy o f less than 30 meV. Our angle-resolved photoemission measurements show that not only is there a saddle point behavior in the band dispersion, but that this behavior has an extended character along the F - Y symmetry line.

2. Experimental data Energy-distribution curves ( E D C ' s ) were measured for various orientations o f the analyzer, corresponding to different angles (0, ~) o f the pho-

toemitted electron beam relative to the surface normal. A given orientation (0, ~) selects initial states with a particular value o f k I at a given kinetic energy E according to k I = ( 2 m E / h 2)1/2 sin 0. All o f the data presented here were taken with polarization o f the photon beam in the I ' - Y plane, along the baxis o f the crystal. The crystals were aligned by Laue diffraction to an accuracy o f < 1 °. The position o f the Fermi energy was carefully determined with the aid o f a platinum sample. Figure l ( a ) shows the observed experimental E D C ' s obtained along the F-Y symmetry line in the first and second BZ's, as indicated. It can be seen that a band disperses towards the Fermi energy, but does not cross it, remaining instead at a very small binding energy o f less than 30 meV, as determined from the position o f the highest intensity in the peak. Since this is the limit of our resolution, we cannot state exactly what the binding energy is, but fitting to the data indicates that it is ~ 10 meV. The peak width is very narrow, being limited entirely by the instrumental resolution, again indicating that at these values o f k ~ ky, the band bottom is dose to EF. We would like to point out that we cannot measure the dis-

0921-4534/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

A.A. Abrikosov et al. / Extended saddle point singularity

74

S_ 1.46-~ 1.32_ '~"~-~/

~ 1.171 ,.Ol

"

~

~

~

~- 0.85~- 0.68 ~- 0.52_ 0.35

0.17 i

i

i

,

1.5

I

b

i

L

1

.

.

.

.

Notice that our experiments show the peak to have the largest spectral density in all of the valence band spectra. Special care was taken to determine the band crossing in the Y-S direction (fig. 1 ( b ) ) , since the band lies closer to the Fermi energy than our experimental resolution. Therefore, the band crossing the Fermi surface is manifested by a drastic loss of intensity in the feature. The experimental band structure, e(k), is shown in fig. 2. Along the F - Y direction, fig. 2(a), the plane band exhibits an extended maximum. Along the Y S direction, fig. 2 (b), the same band exhibits a minimum. In these materials the photoemission intensity depends very strongly on the photon energy. This could be due to several factors, such as final state effects. However, the same dispersion is observed at

i ~

0.5

0

-0.5

Binding energy (eV)

50

;> ';~

'

'

~

. . . .

'

. . . .

'

'

~F

o

' ib)'

-50 ¢¢ -100

ea~ -150

~ O.ll

;~

-200

~

-250 -300

(a) I

F

Y

Momentum

i

1.5

i

i

,

I

1

i

,

i

i

I

0.5

i

i

i

i

I

0

,

ii:l ~

i

S -0.5 i

50 >

o

30

Binding energy (eV) Fig. 1. (a) EDC'salongtheF-Y symmetry line in Y124 at h~=28 eV. The values of the momentum at Er are displayed alongside each EDC; (b) EDC's along the Y-S symmetry line in Y124 at hp=28 eV.

persion along k~ since the position o f the peak does not change with photon energy, even though in our experiments we change the photon energy to cover more than two BZ's along kz. This peak has been observed before b y Manzke et al. [ 5 ] and Tobin et al. [ 6 ], and we havecharacterized it in great detail [ 7 ].

EF ~:m -10 ~

1

-30

(b) -50

I

S

Y

S

Momentum Fig 2. (a) Experimental band structure along the F-Y symmetry line; (b) experimental band structure along the Y-S symmetry line.

A.A. Abrikosov et al. / Extended saddle point singularity

several photon energies in the first and second BZ's [ 7 ]. Some of the data have been already published [8] and are not reproduced here. Tobin et al. [6] show this peak at additional photon energies (in their fig. 6) at the same binding energy, again indicating small dispersion with kz along the F-Y direction. This peak is not due to an alteration of the spectral density due to superconductivity, as it persists unchanged at temperatures above T¢, as shown in fig. 3. This figure deafly shows the two non-degenerate plane bands associated with the two CuO2 planes in Y 123 and Y 124. Here we will not concern ourselves with the deeper lying band, as it probably does not affect the superconducting properties. The experiment thus shows that the band dispersion exhibits an extended maximum from about half of the I-'-Y symmetry line to the Y-point. On the other hand, the band exhibits a minimum in the Y S direction. Thus, there is a saddle point singularity located on the F - Y symmetry line and somewhere between the middle of the F - Y symmetry line and the Y point at a binding energy smaller than 30 meV, limited by our experimental resolution, as shown in fig. 4. At the same time, the quasi-particle spectrum e(k~, ky) is almost independent ofkz due to the quasi2D nature of these superconductors. The idea that

104K

91K 4iK

12K 0.8

0.6

0.4

0.2

0

-0.2

: '

Binding energy (eV) Fig. 3. Temperaturedependanceof the bands in Y123. The data wereobtained from a twinnedsample.

75

Fig. 4. 3D view of the experimentalband structure of Y123 and Y124 in the vicinityof E~ based on the data offi& 1 and in refs. [7] and [8]. the van Hove singularities arising from such a saddle point in e(kx, ky) increases the density of states, and therefore To, is not new (see for example refs. [ 14 ] ). But there are other possible consequences which have not been fully explored, such as ( l ) the effect of an "extended" saddle point singularity on T¢; (2) suppression of the isotope effect, due to the fact that the integration limits in this case are determined not by the Debye frequency, but by the limits of the singularity, i.e. bY some electronic energy scale; (3) a change in the ratio 2A(0)/T~ and other superconducting characteristics, or (4) unusual properties of the normal state, which will be the subject of a future paper. Here we examine the consequences of the, experimentally observed spectral density on the fn'st three points, T~, the isotope effect and the ratio 2~(O)/Tc. In the case under consideration, the spectral density becomes almost one-dimensional. Therefore, a careful analysis is necessary to reveal the possible consequences to superconductivity and comparisons to the experimental data. 3 . Density of states

We consider a 2D energy spectrum and imagine

76

A.A. Abrikosov et al. I Extended saddle point singularity

the simplest possible model of an "extended" saddie-point singularity, k2 2M' k2

k~

% + 2m~ k2

e=

et + 2m---~- 2rn~, '

ky < k~o ,

2L v ( e ) = v -~ - ' I n (2n)2 d ~/2my(~- %) " If

(1)

(~-~o) k~o,

(9)

2my '

where M>>my>_.mx. The matching condition at

then the density of states becomes

g~=k~o is ¢1=%+

m4m~M

k~o _ k~o ~ C o + k~o

2my

(8)

2M

(2)

P(~) = ~ 2 ~

2k~o

(10)

in x/2M(e-eo) "

2my " If e < %, we obtain

The density of states for a particular spin orientation is given by

Lzmy - ( % - ~ ) ] v ( e ) - (2n)2d

dkxdky6[e-e(kx, kr)l .

(3)

r,/~xM

xl

ln

k,,,+Uk~o--EM('o--')

,/2M(,o-,)

In the case where e > %, + m v / - m~ ~ in 2L } (2x)2 d kyo+x/k~o+2m~,(e,-e)

v(,)-(2n)2d{dkj,(._.~/k22,2M) L

~,/2

1 Idk( + (2~)2----d k~o

+ 0 [% _ ~_ 2k__~M]mx/~my in

rex12 *tE--%+kJl2m]

(11)

mx/--m~M, kyo + ~/k~o + 2M ( e - % ) - (2r02-----~m x / 2 M ( e _ %)

2L

+ " - " In • 2 (27t)2d kyo+~/kyo+2my(e-E,) '

The limiting cases are (2/~) 2 d

k~o

~

(2x)2 d q T

1 E ~ - ~% _ .

(6)

we obtain (see eq. (2))

(7)

(12)

v ( e ) = mv/-~myln . L (2n)2d x/2my(el - c ) ' k~o

(13)

Without the flattening of the spectral density (kyo= 0 or m r = M), we obtain the well-known simple saddle point singularity with a density of states which diverges logarithmically with the energy, I mv/m~m~ln 4E v(~)= ~ (2x)2d 1~-%---~'

If, on the other hand (~_ %) >> k~o 2my '

¢o- e %. The most interesting case arises if 2k_~M O. The integral (21 ) is convergent, i.e. the essential values of y are defined by the conditions y,-, max ( / t - Co,A, T). Assuming that these quantities are less than k~o/2my, which in turn can be assumed to be of the order of several eV, the upper limit of the integral (21 ) can be made equal to infinity. With the substitutions z=x/Y/ (lt-eo) , 5=d/ (lZ-Co), r = T/(lz- %) we obtain

- 8Y (/z-%)exp ( - ~

f tanh [ (Z2 - 1 )2+52]

"~ =

J

-

-[ ( - - ~ _ ~ - F ~

0

with

'/2/2"~

=,k mxl

4nSd 2 v2(----~exp

=E

(23)

We now consider limiting cases of 5(0) and r~. Assuming that 0 = 0 and z=c¢