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Weakly screened Coulomb interaction and properties of HTSC's
PHYSICA Physica C 222 (1994) 191-196
ELSEVIER
Weakly screened Coulomb interaction and properties of HTSC's A.A. Abrikosov Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA Received 2 December 1993
Abstract In order to explain the anisotropy of the energy gap observed in direct photoemission experiments on the basis of the "extended saddle-point singularities" in the electron-energy spectrum lying close to the Fermi level, the assumption is made that the Coulomb interactions are weakly screened, i.e. the Debye screening radius is much larger than the lattice period; this makes the electron interaction anisotropic. Another consequence of this assumption is that at low temperatures the normal-state resistivity is mostly defined by electron-electron scattering, and its temperature dependence is poc T. Some aspects of the model are discussed.
1. Introduction. Screening radius In the recent publication by Abrikosov et al. [ 1 ] it was demonstrated that the electron-energy spectrum of several high-T~ superconductors has an "extended saddle-point" singularity close to the Fermi energy, and this can provide an explanation o f the high value o f T¢. However, this model alone could not explain the large value o f 2 A / T ¢ ~ 6 observed in these substances. In a subsequent publication by the present author [2 ] it was demonstrated that this large ratio could be due to the anisotropy o f A as a function o f the m o m e n t u m . This anisotropy was really observed in direct photoemission experiments in BISCO [ 3 ]. In the usual superconductors, even with an anisotropic basic energy spectrum, the gap is rather isotropic. This is due to the isotropy o f the interaction between electrons transmitted by phonons. Even in the case o f a high state density in definite regions o f the quasi-momentum space the equation o f the type ,d(p) = ~ K ( p , p ' ) j ~ , d ( p ' ) ] d3p'/(2rQ 3
(1)
will connect J at a n y p with the singular region, and
hence will not permit A(p) to be anisotropic, except in the case ifK(p, p ' ) is anisotropic. In principle this could be achieved by spin-fluctuation exchange, if their relevant m o m e n t a were concentrated within narrow intervals around some particular values. However, we consider much more natural the idea of a small m o m e n t u m transfer. Although the high-T~ materials are certainly metals, they are in some sense close to ionic crystals. This can be traced from the experimental observation that in the infrared as well as in Raman measurements the phonon peaks are very high (see e.g. refs. [4] and [ 5 ] ) . It has also to be considered that structural models consisting o f differently charged ions are o f much help, not less than band-structure calculations. The cross-over from metals to ionic crystals can be understood if one imagines that the Coulomb forces, which are the basis o f all real interactions, are poorly screened, i.e. the Debye screening radius is not of atomic size, as in good metals, but much larger. The usual expression for the square o f the reciprocal Debye redius is Ic2---4n(eE/eoo)v(fl), where eo~ is the part o f the dielectric constant due to ionic cores, and u(fl) the state density. If we assume that the most important is the
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A.A. Abrikosov /Physica C 222 (1994) 191-196
contribution from the extended saddle-point singularities, where the state density is much higher, we obtain at T = 0 and at zero frequency (see e.g. ref. [61) K2~
8xe__~ 2 ~ nv(p-k/2)-nF(p+k/2) ~ J ~(p-k/2)-~(p+k/2)
d3p (2x) 3
transferred momentum (we assume that K>> x). This is of course an assumption but it finds some justification in the model of strongly compressed matter (see ref. [ 8 ] ) where x is also small. The kernel of the BCS self-consistency equation ( 1 ) is in this case K2
o)2(k)
K(k) = - g k2+~¢2 (8~)2 0)2(k) 16nNse2mxPvo
- ~k--~
ln~l [2mx(/*-~o)]l/2-kx/21]" (2)
Here we inserted the electron spectrum (=p2/ 2 m x - (/Z-eo) (see ref. [ 1 ] ), d is the period along the c-axis, Pyo the extension of the 1D region alongp r (in ref. [ 1 ] we used the notation kyo), and N~ is the number of singular points per Brillouin zone. If k:,<< [2mx(/Z-Co) ]1/2, X2 tends to a constant: x2(0) = 2x/~NseZmlx/ZPyo
nEood(lt_Eo) l/2 .
(3)
where g has the usual value of the order of (pom/ n2) - 1, 8~ is the change of electronic energy, and co(k) is the phonon frequency.
2. Anisotropy of J(p) If we insert the kernel (5) with k = p - p ' we obtain ~J(r) = f
With increasing kx, after a logarithmic singularity at k~=2[2m~(/~-eo) ]1/2, x2 decreases as kx 2. From this formula it can be seen that x2(0) is small only in case of large e~. The experimental measurements of the dielectric constant as a function of frequency [ 7 ] give very different values, sometimes larger than 1000, but in case of a complicated energy spectrum it is not easy to decide how ¢~ has to be extracted, and so we will make the assumption that E~ >> 1 ~1. There appear several m o m e n t u m scales in the problem. For simplicity we consider the extreme case [ 2 m x ( / t - Co) ]1/2 <(< x << Pyo << K
(4)
(K is the reciprocal lattice period), although in reality other cases are also possible. There is no unique way to choose the model interaction. We will assume that the square of the electron-phonon interaction matrix element entering the electron-electron interaction, as well as the scattering probability, is multiplied by K2 kE+K2 ,
(6)
[ [2mx(#-Eo)]l/2+k,,/2
(5)
where K is the reciprocal lattice period and k the ~ It can be demonstrated that in the case of uniaxial anisotropy, e~o entering our formulas has to be replaced by ~ , and hence at least one component of e~ik has to be large.
--g ×
K2/I ( p , )
092 ( p - - p ' )
(p_p,)2+x2 [~(p)-~(p')lZ-ogZ(p-p ') tanh{ [ (E(p') - / t ) Z + A 2 ( p ') ] 1/2/2T} d3p ' [ (~ (p,) _/t)2 + 3 2 ( p , ) ] 1/2 (2X) 3" (7)
Usually, without the factor (4), most important are large values of IP-P'[. This corresponds to the Debye frequency of phonons, and hence the kernel is taken as equal to g with the condition 18~1 < coD. In the case under consideration the integral depends rather strongly on the external m o m e n t u m p , and this leads eventually to the anisotropy of A (p). If we assume that the reciprocal lattice period in the c-direction, namely 2n/d (d is the layer spacing), is larger than any momentums scale of the problem (except K), d ( p ) will not depend on Pz. Let us consider first the momentum p in the vicinity of the saddle-point singularity. The electron-energy difference in Eq. (7) will have the magnitude of the largest o f # - % , 3, T (see ref. [ 1 ] ). We assume, as in ref. [ 1], that the phonon frequency involved will be much larger than that. Therefore the factor depending on (n(k) in Eq. (7) can be replaced by - 1. The important kx=px-p'~ will be of the order of [2m~,(/~- Co) ] 1/2 (or with T or A instead o f / ~ - ~o). As we will see these values are much less than the important values of ky and kz. Since the electron energy
A.A. Abrikosov/Physica C 222 (1994) 191-196
193
in the region of the singularity depends on kx we may integrate the kernel over kz and obtain
same type of interaction we obtain in the right-hand side a term
nK 2 K(p-p')--*g [ (py __py)2_l_ R72] 1/2"
K2 g31 ( ~ ) 2 v In npo r ,m 2090 ~ ,
(8)
The integrals over p~ and p " are performed separately, since the energy ~(p') depends only on p " which is not contained in the kernel. The limits of the integration over p~ are defined by the boundaries of the region where the spectrum is quasi-one-dimensional. Performing this integration we get with logarithmic accuracy
where too is some phonon frequency between to(x) and too, Po and v are the Fermi momentum and velocity, and A~ means A far from the singular region; the latter we denote by Ao. In c a s e / z - % >> Ao the equation for d (p) far from the singular region (there are likely to be several of them) has the form
Al(py) K(p-p')-,grtK 2ln(-~).
(9)
(12)
A0
g(2mx) t/zK2 Pyo l n 8 ( / t - % ) Ao
-- 87~2(~--~0) 1/2 IPyl gK 2 .
xpo Z]l (py) , 2090 Ao m A - ~ y ) "
After that, the right-hand side depends no more on p, and hence A(p) can be assumed independent of momentum in the whole region of the singularity. This is of course true only in the crude logarithmic approximation. Thus the problem is reduced to the one considered in ref. [ 1 ] with the replacement
The first term in the right-hand side with a factor ln(Pyo/X) instead of Pyo/Ipyl defined Ao, and so it can be rewritten in the form (Pyo/IPyl)/ln(Pyo/X). After that Eq. (10) becomes
( 2 n ) 2 d - , ~n2 In
Al(py) ( 1 - gK2 lr'P° In 2o90 Ao 4--x-2vIn ~c A,(py)]
.
(10)
The essential values ofpy-p'y and Pz-P'~ are defined by the inequalities x<< IPy-P~,l ~ IPz -P'zl << Pyo • Hence the assumption that the energy difference I~(P) - ~ ( P ' ) I << t o ( P - P ' ) means actually / z - ~o<< c IP y - P ~ I, c being the sound velocity. This is a somewhat more restrictive condition than # - % << taD but it is plausible. Now suppose that the m o m e n t u m p is far from the saddle point. In Eq. (7) there are two regions of integration over p'. One is in the vicinity of the singularity. I f IPyl >> eyo the integral over p~ of the kernel (8) is approximately
K(p--p') --,gxK 2 Pyo Ipy I '
( 11 )
where py is measured from the corresponding coordinate of the singularity in m o m e n t u m space. There is, however, another region of integration, where p~ is far from the singularity. For the sake of simplicity here we will use the isotropic model. Considering the
+ 4n2----vm x
Pyo/ Ipy l - l n ( Pyo/X) "
(13)
(14)
This means a decrease of A with departure from the singular region and its approach to the solution of a regular BCS type at sufficiently small values of Pyo/ Ipyl. All these qualitative features correspond to the observations [ 3 ].
3. Resistivity in the normal state The linear temperature dependence of the normalstate resistivity of high-Tc cuprates was always a puzzle for theorists, and one of the checks for the theory to be correct was this dependence. Unfortunately, people always managed to get the linear dependence in their theories based on completely different assumptions, e.g. RVB (Anderson and Zou ) [ 9 ], nested Fermi surfaces (Virosztek and Ruvalds) [ 10], spin fluctuations (Morita et al.) [ 1 1 ], oxygen chains (Abrikosov and Falkovsky) [ 12 ]. Therefore such a result can neither prove nor disprove a theory. The
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A.A. Abrikosov /Physica C 222 (1994) 191-196
easiest way to obtain a linear temperature dependence is the assumption that the electrons interact mainly with some optical mode having a low frequency, and the T comes from the Bose distribution [ exp( coo/T) - 1 ] -’ at TB oo. This sort of explanations is likely to be wrong because the high-frequency resistivity a(o) -’ at w 3 T varies linearly with w (see e.g. refs. [ 12]), i.e. o replaces T, this can happen only if T, w << oo, the limiting energy of the quasiparticles transferring the interaction. We are not going to make an exception and will also obtain the linear resistivity in the framework of the theory presented above. We will show that scattering of electrons from electrons at low temperatures is much larger than scattering from phonons; therefore we will consider it first (the same was true for the model of nested Fermi surfaces [ lo] ). In the previous section we have assumed that the interaction between electrons due to exchange of phonons is stronger than the Coulomb repulsion (actually they are of the same order of magnitude). Here, however, the situation can be different. The matrix element of the Coulomb interaction is 4xe2/e, k2+K2
(15)
’
whereas the interaction gK2
o(k)’
via phonons
is (16)
1 -=27c =,,
n,(l-n{)(l-n$)S(e,+c,-t;-ti)(l-cos.8) [(P,--P:)2+K2]2
d3p;d3pz
x
(2x)6
’ (17)
where 8 is the angle between the velocities before and after scattering, and ni are the Fermi functions. First consider the situation when the second electron is in the vicinity of the same singularity as the first one. Then the energies depend only on the xcomponents of the momenta, and we can integrate over the y- and z-components. Assuming, as before, that KC Pyowe obtain after this integration 1 -= =e,
-
cose)n2(l-n;)(l-n;)
(Pl,
(18)
-~;x)~l W,x bx.
+pzx
The argument of the &function has zeros at pi, =plX andp’,, =p2X. For the first case cos 13~0. For the second case, integrating over p’,, we obtain 2e4m,PN --1 ~c?.Y &rc2d x
In the following the integral over k will require k- K, and since St will be of the order of T, the second matrix element will be much less than the first one in the caseT%~(~)=~~.IfonthecontraryT~w(Ic),the second factor in Eq. ( 16 ) becomes - 1, the same as in the gap equation of the previous section. Then the interaction acquires the same form as a pure Coulomb (15) with the replacement 4rce2/t,+ -gK2. The sign is of no importance, since only the square of the interaction enters the scattering probability, and the order of magnitude of both interactions is likely to be the same. Therefore we will write the interaction in the form ( 15 ) with the replacement 4rre2/ e,-+ -gK2 in case T+z O(K). We obtain
4ae2 2 t L>
s
(1-c0se)n2(1-n2)(1-n,)
@2x
2lPLc-P2XI
(19)
The factor n2( 1 - n2) requires lplxl to be close to [ 2m,(p- co) ] ‘j2, and lplxl is also close to the same value. Due to the factor ( 1 -cos 0) only the case Pi, =P2x- - -pIx is important. Integrating we obtain the final expression 1 e4Pfl m, T 1 -=---2rr: &d rc2 ,u-to L5 1 =m
e2m:12
T
~c3(p-~o)1’2
(here we have substituted the formula The reason why we get in this case pendence instead of the usual quadratic a 3D or 2D case the J-function fixes tween the momenta whereas in the 1D
(20) (3 ) for K2). a linear T deone is that in the angle besituation there
A.A. Abrikosov/Physica C222 (1994) 191-196 is no angle to fix, and the J-function reduces two momentum integrations to one. The coefficient in the linear dependence (20) would be large (e2/VF ~ 1 for an ordinary metal, and here VF is much smaller) if not for coo>> 1. Therefore this requirement is actually the condition for the Landau Fermi-liquid theory to be applicable to the layered cuprates under consideration. In the case when the second electron is in the vicinity of another saddle point the momenta will be interchanged: Px will be the "long" one, and py the "short" one. Due to momentum conservation we will have in this case a restriction for four components of momenta instead of two, as in the previous case, and that will contribute a factor T 2. A more exact calculation gives the relative estimate T2[ (/.t--~o) (Pyox/ mx)] -x. Now we calculate the scattering probability from phonons; it is given by the expression 1
Tcv
-2rig
×
j K2m(k)J[~(p)-e(p-k) ]
× (1 - c o s 0 ) d3k/(2n) 3
× [ (kE+xE){exp[to(k)/T] - 1}] - l
(21) where 0 is the angle between the velocities v= 8~/8p before and after scattering. For electrons in the vicinity of the saddle-point singularity E~)--~(p--k)~
p~k~ mx
p~
(px-k~) ~
2rn~
2mx
- -
1
195
gK2mx ?
Zep -
ck 2 dk
nlpxl oJ [ e x p ( c k / ~ - - - ~ (k2+/(2)
- gK2mx Tin T . nlpxl
(22)
cx
For the case, when T<< cx, the result is
1
ggEmx ~
ck 2 dk
reD -- ~ IPxlx 2 J [exp(ck/T) -- 1 ] 0
2((3)gK2mx T 3
-
n(cK)2lPxl
(23)
The ratio of the probabilities for T>> cx has the order of magnitude
z~l Z~eI
gK2e°° In T e2
(24)
C/¢ "
This can be large but also can be of the order of unity, since most probably the constant g, being of Coulomb origin, contains also a factor ~ 1 . However, for T<< cx the electron-electron scattering is dominant. This was actually mentioned also in ref. [ 10 ], where the linear dependence was obtained as a result of nesting. The expression for the conductivity due to electrons in the vicinity of the singular point (with two spin projections) has the form
2e2pyo ~f ( p x ~ 2 ( _ Onv~ , , \ m x / \ -~E /Z(px) dpx
axx= (2n)2d
4Ns Pyo ~ ( l t - e o ) T
-- ~,2 d m ~
(25)
k~ 2rex "
The zero of this difference is either at kx=O, or at k~= 2p~. Only the latter value is important in view of the factor 1 - c o s 0. Hence the electron has to jump from one side of the Fermi surface to the other one. We will assume this value k~= 2 [2m~(/~-e0)]1/2 to be small compared to the relevant ky and k~. If the temperature is much larger than cx, we are left with an integral (2 comes from 1 - c o s 0)
Now one has to take into account that there possibly exist four singular points in the plane being symmetric with respect to a n/2 rotation (or two per Brillouin zone). In this case a will be isotropic and equal to the calculated a~. T h e resistivity will be equal to 1/tr, and hence, introducing the necessary power of h and putting Ns = 2, we get in usual units
~,2 rn~d T P= 8 hPyo(#-eo) co~ "
(26)
The coefficient in this linear dependence c9ntains a large factor as well as a small one; therefore more must be known in order to say whether it fits the experi-
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A.A. Abrikosov /Physica C 222 (1994) 191-196
mental high values ( 1 5 0 - 1 8 0 I . t ~ c m K - I at r o o m t e m p e r a t u r e for Y B C O [ 14 ] which is a p p r o x i m a t e l y one h u n d r e d times larger than the resistance o f C u ) b u t the linear d e p e n d e n c e gives hope that this will happen. At the same t i m e we have to exclude the possibilities T>> c k a n d T > > / ~ - eo where the temperature d e p e n d e n c e will be very different (in the latter case p ~ 1/x//T).
4. Conclusions The crucial idea for all results o b t a i n e d in the foregoing is the a s s u m p t i o n ¢~ >> 1. A p a r t from that we have p r e s u m e d a m o d i f i c a t i o n o f the e l e c t r o n p h o n o n interaction (square o f the m a t r i x e l e m e n t ) which is described by f o r m u l a ( 5 ) . In principle a higher power o f the d e n o m i n a t o r could also be possible. The characteristic p h o n o n frequency would be t o ( x ) , a n d in o r d e r to have no significant isotope effect the c o n d i t i o n Tc,,
A n i m p o r t a n t issue is the d e p e n d e n c e o f the gap anisotropy on the content o f impurities. Experimentally it was f o u n d that samples with a smeared-out resistive transition have a m o r e isotropopic gap [ 3 ]. This could be explained in terms o f isotropic scattering from i m p u r i t i e s which could m a k e the effective interaction between electrons m o r e isotropic. Such a scattering could be due only to neutral defects since scattering from charged defects would be at small angles a n d would hardly change essentially the nature o f the effective interaction. All this has to be considered in future studies.
Acknowledgement This work was s u p p o r t e d by the D e p a r t m e n t o f Energy u n d e r the contracts # W-31- 109-ENG-38.
References [ 1] A.A. Abrikosov, J.C. Campuzano and K. Gofron, Physica C214 (1993) 73. [2] A.A. Abrikosov, Physica C 214 (1993) 103. [3] Z.-X. Shen et al., Phys. Rev. Lett. 70 (1993 ) 1553; H. Ding et al., unpublished. [4 ] A.M. Pao et al., Phys. Rev. B 42 (1990) 193. [ 5 ] M. Boekholt, M. Hoffmann and G. Guentherodt, Physica C 175 (1991)42. [6]A.A. Abrikosov, L.P. Gor'kov and I.E. Dzyaloshinsky, Quantum Field Theoretical Methods in Statistical Physics (Pergamon, Oxford, 1965 ). [ 7 ] G.P. Mazzara et al., Phys. Rev. B 47 ( 1993 ) 8119; G. Cao et al., Phys. Rev. B 47 (1993) 11510. [8]A.A. Abrikosov, Sov. Phys. JETP 12 (1961) 1254; 14 ( 1962 ) 408. [ 9 ] P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132. [ 10] A. Virosztek and J. Ruvalds, Phys. Rev. B 42 (1990) 4064. [ 11 ] T. Morita, Y. Takahashi and K. Ueda, J. Phys. Soc. Jpn. 59 (1990) 2905. [ 12 ] A.A. Abrikosov and L.A. Falkovsky, Physica C 168 (1990) 556. [ 13 ] Y. Watanabe et al., Phys. Rev. B 40 (1989) 6884; R.T. Collins et al., Phys. Rev. B 39 (1989) 6571; Phys. Rev. Lett. 64 (1990) 84; J.H. Kim et al., Phys. Rev. B 41 (1990) 7251. [ 14] T.A. Friedmann et al., Phys. Rev. B 42 (1990) 6217. [ 15 ] S. Chakravarty et al., Science 261 (1993) 337. [ 16 ] M. Randeria et al., Phys. Rev. Lett. 69 (1992) 2001; C. Sa de Melo, M. Randeria and J. Engelbrecht, Phys. Rev. Lett. 71 (1993) 3202.
ELSEVIER
Weakly screened Coulomb interaction and properties of HTSC's A.A. Abrikosov Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA Received 2 December 1993
Abstract In order to explain the anisotropy of the energy gap observed in direct photoemission experiments on the basis of the "extended saddle-point singularities" in the electron-energy spectrum lying close to the Fermi level, the assumption is made that the Coulomb interactions are weakly screened, i.e. the Debye screening radius is much larger than the lattice period; this makes the electron interaction anisotropic. Another consequence of this assumption is that at low temperatures the normal-state resistivity is mostly defined by electron-electron scattering, and its temperature dependence is poc T. Some aspects of the model are discussed.
1. Introduction. Screening radius In the recent publication by Abrikosov et al. [ 1 ] it was demonstrated that the electron-energy spectrum of several high-T~ superconductors has an "extended saddle-point" singularity close to the Fermi energy, and this can provide an explanation o f the high value o f T¢. However, this model alone could not explain the large value o f 2 A / T ¢ ~ 6 observed in these substances. In a subsequent publication by the present author [2 ] it was demonstrated that this large ratio could be due to the anisotropy o f A as a function o f the m o m e n t u m . This anisotropy was really observed in direct photoemission experiments in BISCO [ 3 ]. In the usual superconductors, even with an anisotropic basic energy spectrum, the gap is rather isotropic. This is due to the isotropy o f the interaction between electrons transmitted by phonons. Even in the case o f a high state density in definite regions o f the quasi-momentum space the equation o f the type ,d(p) = ~ K ( p , p ' ) j ~ , d ( p ' ) ] d3p'/(2rQ 3
(1)
will connect J at a n y p with the singular region, and
hence will not permit A(p) to be anisotropic, except in the case ifK(p, p ' ) is anisotropic. In principle this could be achieved by spin-fluctuation exchange, if their relevant m o m e n t a were concentrated within narrow intervals around some particular values. However, we consider much more natural the idea of a small m o m e n t u m transfer. Although the high-T~ materials are certainly metals, they are in some sense close to ionic crystals. This can be traced from the experimental observation that in the infrared as well as in Raman measurements the phonon peaks are very high (see e.g. refs. [4] and [ 5 ] ) . It has also to be considered that structural models consisting o f differently charged ions are o f much help, not less than band-structure calculations. The cross-over from metals to ionic crystals can be understood if one imagines that the Coulomb forces, which are the basis o f all real interactions, are poorly screened, i.e. the Debye screening radius is not of atomic size, as in good metals, but much larger. The usual expression for the square o f the reciprocal Debye redius is Ic2---4n(eE/eoo)v(fl), where eo~ is the part o f the dielectric constant due to ionic cores, and u(fl) the state density. If we assume that the most important is the
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A.A. Abrikosov /Physica C 222 (1994) 191-196
contribution from the extended saddle-point singularities, where the state density is much higher, we obtain at T = 0 and at zero frequency (see e.g. ref. [61) K2~
8xe__~ 2 ~ nv(p-k/2)-nF(p+k/2) ~ J ~(p-k/2)-~(p+k/2)
d3p (2x) 3
transferred momentum (we assume that K>> x). This is of course an assumption but it finds some justification in the model of strongly compressed matter (see ref. [ 8 ] ) where x is also small. The kernel of the BCS self-consistency equation ( 1 ) is in this case K2
o)2(k)
K(k) = - g k2+~¢2 (8~)2 0)2(k) 16nNse2mxPvo
- ~k--~
ln~l [2mx(/*-~o)]l/2-kx/21]" (2)
Here we inserted the electron spectrum (=p2/ 2 m x - (/Z-eo) (see ref. [ 1 ] ), d is the period along the c-axis, Pyo the extension of the 1D region alongp r (in ref. [ 1 ] we used the notation kyo), and N~ is the number of singular points per Brillouin zone. If k:,<< [2mx(/Z-Co) ]1/2, X2 tends to a constant: x2(0) = 2x/~NseZmlx/ZPyo
nEood(lt_Eo) l/2 .
(3)
where g has the usual value of the order of (pom/ n2) - 1, 8~ is the change of electronic energy, and co(k) is the phonon frequency.
2. Anisotropy of J(p) If we insert the kernel (5) with k = p - p ' we obtain ~J(r) = f
With increasing kx, after a logarithmic singularity at k~=2[2m~(/~-eo) ]1/2, x2 decreases as kx 2. From this formula it can be seen that x2(0) is small only in case of large e~. The experimental measurements of the dielectric constant as a function of frequency [ 7 ] give very different values, sometimes larger than 1000, but in case of a complicated energy spectrum it is not easy to decide how ¢~ has to be extracted, and so we will make the assumption that E~ >> 1 ~1. There appear several m o m e n t u m scales in the problem. For simplicity we consider the extreme case [ 2 m x ( / t - Co) ]1/2 <(< x << Pyo << K
(4)
(K is the reciprocal lattice period), although in reality other cases are also possible. There is no unique way to choose the model interaction. We will assume that the square of the electron-phonon interaction matrix element entering the electron-electron interaction, as well as the scattering probability, is multiplied by K2 kE+K2 ,
(6)
[ [2mx(#-Eo)]l/2+k,,/2
(5)
where K is the reciprocal lattice period and k the ~ It can be demonstrated that in the case of uniaxial anisotropy, e~o entering our formulas has to be replaced by ~ , and hence at least one component of e~ik has to be large.
--g ×
K2/I ( p , )
092 ( p - - p ' )
(p_p,)2+x2 [~(p)-~(p')lZ-ogZ(p-p ') tanh{ [ (E(p') - / t ) Z + A 2 ( p ') ] 1/2/2T} d3p ' [ (~ (p,) _/t)2 + 3 2 ( p , ) ] 1/2 (2X) 3" (7)
Usually, without the factor (4), most important are large values of IP-P'[. This corresponds to the Debye frequency of phonons, and hence the kernel is taken as equal to g with the condition 18~1 < coD. In the case under consideration the integral depends rather strongly on the external m o m e n t u m p , and this leads eventually to the anisotropy of A (p). If we assume that the reciprocal lattice period in the c-direction, namely 2n/d (d is the layer spacing), is larger than any momentums scale of the problem (except K), d ( p ) will not depend on Pz. Let us consider first the momentum p in the vicinity of the saddle-point singularity. The electron-energy difference in Eq. (7) will have the magnitude of the largest o f # - % , 3, T (see ref. [ 1 ] ). We assume, as in ref. [ 1], that the phonon frequency involved will be much larger than that. Therefore the factor depending on (n(k) in Eq. (7) can be replaced by - 1. The important kx=px-p'~ will be of the order of [2m~,(/~- Co) ] 1/2 (or with T or A instead o f / ~ - ~o). As we will see these values are much less than the important values of ky and kz. Since the electron energy
A.A. Abrikosov/Physica C 222 (1994) 191-196
193
in the region of the singularity depends on kx we may integrate the kernel over kz and obtain
same type of interaction we obtain in the right-hand side a term
nK 2 K(p-p')--*g [ (py __py)2_l_ R72] 1/2"
K2 g31 ( ~ ) 2 v In npo r ,m 2090 ~ ,
(8)
The integrals over p~ and p " are performed separately, since the energy ~(p') depends only on p " which is not contained in the kernel. The limits of the integration over p~ are defined by the boundaries of the region where the spectrum is quasi-one-dimensional. Performing this integration we get with logarithmic accuracy
where too is some phonon frequency between to(x) and too, Po and v are the Fermi momentum and velocity, and A~ means A far from the singular region; the latter we denote by Ao. In c a s e / z - % >> Ao the equation for d (p) far from the singular region (there are likely to be several of them) has the form
Al(py) K(p-p')-,grtK 2ln(-~).
(9)
(12)
A0
g(2mx) t/zK2 Pyo l n 8 ( / t - % ) Ao
-- 87~2(~--~0) 1/2 IPyl gK 2 .
xpo Z]l (py) , 2090 Ao m A - ~ y ) "
After that, the right-hand side depends no more on p, and hence A(p) can be assumed independent of momentum in the whole region of the singularity. This is of course true only in the crude logarithmic approximation. Thus the problem is reduced to the one considered in ref. [ 1 ] with the replacement
The first term in the right-hand side with a factor ln(Pyo/X) instead of Pyo/Ipyl defined Ao, and so it can be rewritten in the form (Pyo/IPyl)/ln(Pyo/X). After that Eq. (10) becomes
( 2 n ) 2 d - , ~n2 In
Al(py) ( 1 - gK2 lr'P° In 2o90 Ao 4--x-2vIn ~c A,(py)]
.
(10)
The essential values ofpy-p'y and Pz-P'~ are defined by the inequalities x<< IPy-P~,l ~ IPz -P'zl << Pyo • Hence the assumption that the energy difference I~(P) - ~ ( P ' ) I << t o ( P - P ' ) means actually / z - ~o<< c IP y - P ~ I, c being the sound velocity. This is a somewhat more restrictive condition than # - % << taD but it is plausible. Now suppose that the m o m e n t u m p is far from the saddle point. In Eq. (7) there are two regions of integration over p'. One is in the vicinity of the singularity. I f IPyl >> eyo the integral over p~ of the kernel (8) is approximately
K(p--p') --,gxK 2 Pyo Ipy I '
( 11 )
where py is measured from the corresponding coordinate of the singularity in m o m e n t u m space. There is, however, another region of integration, where p~ is far from the singularity. For the sake of simplicity here we will use the isotropic model. Considering the
+ 4n2----vm x
Pyo/ Ipy l - l n ( Pyo/X) "
(13)
(14)
This means a decrease of A with departure from the singular region and its approach to the solution of a regular BCS type at sufficiently small values of Pyo/ Ipyl. All these qualitative features correspond to the observations [ 3 ].
3. Resistivity in the normal state The linear temperature dependence of the normalstate resistivity of high-Tc cuprates was always a puzzle for theorists, and one of the checks for the theory to be correct was this dependence. Unfortunately, people always managed to get the linear dependence in their theories based on completely different assumptions, e.g. RVB (Anderson and Zou ) [ 9 ], nested Fermi surfaces (Virosztek and Ruvalds) [ 10], spin fluctuations (Morita et al.) [ 1 1 ], oxygen chains (Abrikosov and Falkovsky) [ 12 ]. Therefore such a result can neither prove nor disprove a theory. The
194
A.A. Abrikosov /Physica C 222 (1994) 191-196
easiest way to obtain a linear temperature dependence is the assumption that the electrons interact mainly with some optical mode having a low frequency, and the T comes from the Bose distribution [ exp( coo/T) - 1 ] -’ at TB oo. This sort of explanations is likely to be wrong because the high-frequency resistivity a(o) -’ at w 3 T varies linearly with w (see e.g. refs. [ 12]), i.e. o replaces T, this can happen only if T, w << oo, the limiting energy of the quasiparticles transferring the interaction. We are not going to make an exception and will also obtain the linear resistivity in the framework of the theory presented above. We will show that scattering of electrons from electrons at low temperatures is much larger than scattering from phonons; therefore we will consider it first (the same was true for the model of nested Fermi surfaces [ lo] ). In the previous section we have assumed that the interaction between electrons due to exchange of phonons is stronger than the Coulomb repulsion (actually they are of the same order of magnitude). Here, however, the situation can be different. The matrix element of the Coulomb interaction is 4xe2/e, k2+K2
(15)
’
whereas the interaction gK2
o(k)’
via phonons
is (16)
1 -=27c =,,
n,(l-n{)(l-n$)S(e,+c,-t;-ti)(l-cos.8) [(P,--P:)2+K2]2
d3p;d3pz
x
(2x)6
’ (17)
where 8 is the angle between the velocities before and after scattering, and ni are the Fermi functions. First consider the situation when the second electron is in the vicinity of the same singularity as the first one. Then the energies depend only on the xcomponents of the momenta, and we can integrate over the y- and z-components. Assuming, as before, that KC Pyowe obtain after this integration 1 -= =e,
-
cose)n2(l-n;)(l-n;)
(Pl,
(18)
-~;x)~l W,x bx.
+pzx
The argument of the &function has zeros at pi, =plX andp’,, =p2X. For the first case cos 13~0. For the second case, integrating over p’,, we obtain 2e4m,PN --1 ~c?.Y &rc2d x
In the following the integral over k will require k- K, and since St will be of the order of T, the second matrix element will be much less than the first one in the caseT%~(~)=~~.IfonthecontraryT~w(Ic),the second factor in Eq. ( 16 ) becomes - 1, the same as in the gap equation of the previous section. Then the interaction acquires the same form as a pure Coulomb (15) with the replacement 4rce2/t,+ -gK2. The sign is of no importance, since only the square of the interaction enters the scattering probability, and the order of magnitude of both interactions is likely to be the same. Therefore we will write the interaction in the form ( 15 ) with the replacement 4rre2/ e,-+ -gK2 in case T+z O(K). We obtain
4ae2 2 t L>
s
(1-c0se)n2(1-n2)(1-n,)
@2x
2lPLc-P2XI
(19)
The factor n2( 1 - n2) requires lplxl to be close to [ 2m,(p- co) ] ‘j2, and lplxl is also close to the same value. Due to the factor ( 1 -cos 0) only the case Pi, =P2x- - -pIx is important. Integrating we obtain the final expression 1 e4Pfl m, T 1 -=---2rr: &d rc2 ,u-to L5 1 =m
e2m:12
T
~c3(p-~o)1’2
(here we have substituted the formula The reason why we get in this case pendence instead of the usual quadratic a 3D or 2D case the J-function fixes tween the momenta whereas in the 1D
(20) (3 ) for K2). a linear T deone is that in the angle besituation there
A.A. Abrikosov/Physica C222 (1994) 191-196 is no angle to fix, and the J-function reduces two momentum integrations to one. The coefficient in the linear dependence (20) would be large (e2/VF ~ 1 for an ordinary metal, and here VF is much smaller) if not for coo>> 1. Therefore this requirement is actually the condition for the Landau Fermi-liquid theory to be applicable to the layered cuprates under consideration. In the case when the second electron is in the vicinity of another saddle point the momenta will be interchanged: Px will be the "long" one, and py the "short" one. Due to momentum conservation we will have in this case a restriction for four components of momenta instead of two, as in the previous case, and that will contribute a factor T 2. A more exact calculation gives the relative estimate T2[ (/.t--~o) (Pyox/ mx)] -x. Now we calculate the scattering probability from phonons; it is given by the expression 1
Tcv
-2rig
×
j K2m(k)J[~(p)-e(p-k) ]
× (1 - c o s 0 ) d3k/(2n) 3
× [ (kE+xE){exp[to(k)/T] - 1}] - l
(21) where 0 is the angle between the velocities v= 8~/8p before and after scattering. For electrons in the vicinity of the saddle-point singularity E~)--~(p--k)~
p~k~ mx
p~
(px-k~) ~
2rn~
2mx
- -
1
195
gK2mx ?
Zep -
ck 2 dk
nlpxl oJ [ e x p ( c k / ~ - - - ~ (k2+/(2)
- gK2mx Tin T . nlpxl
(22)
cx
For the case, when T<< cx, the result is
1
ggEmx ~
ck 2 dk
reD -- ~ IPxlx 2 J [exp(ck/T) -- 1 ] 0
2((3)gK2mx T 3
-
n(cK)2lPxl
(23)
The ratio of the probabilities for T>> cx has the order of magnitude
z~l Z~eI
gK2e°° In T e2
(24)
C/¢ "
This can be large but also can be of the order of unity, since most probably the constant g, being of Coulomb origin, contains also a factor ~ 1 . However, for T<< cx the electron-electron scattering is dominant. This was actually mentioned also in ref. [ 10 ], where the linear dependence was obtained as a result of nesting. The expression for the conductivity due to electrons in the vicinity of the singular point (with two spin projections) has the form
2e2pyo ~f ( p x ~ 2 ( _ Onv~ , , \ m x / \ -~E /Z(px) dpx
axx= (2n)2d
4Ns Pyo ~ ( l t - e o ) T
-- ~,2 d m ~
(25)
k~ 2rex "
The zero of this difference is either at kx=O, or at k~= 2p~. Only the latter value is important in view of the factor 1 - c o s 0. Hence the electron has to jump from one side of the Fermi surface to the other one. We will assume this value k~= 2 [2m~(/~-e0)]1/2 to be small compared to the relevant ky and k~. If the temperature is much larger than cx, we are left with an integral (2 comes from 1 - c o s 0)
Now one has to take into account that there possibly exist four singular points in the plane being symmetric with respect to a n/2 rotation (or two per Brillouin zone). In this case a will be isotropic and equal to the calculated a~. T h e resistivity will be equal to 1/tr, and hence, introducing the necessary power of h and putting Ns = 2, we get in usual units
~,2 rn~d T P= 8 hPyo(#-eo) co~ "
(26)
The coefficient in this linear dependence c9ntains a large factor as well as a small one; therefore more must be known in order to say whether it fits the experi-
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A.A. Abrikosov /Physica C 222 (1994) 191-196
mental high values ( 1 5 0 - 1 8 0 I . t ~ c m K - I at r o o m t e m p e r a t u r e for Y B C O [ 14 ] which is a p p r o x i m a t e l y one h u n d r e d times larger than the resistance o f C u ) b u t the linear d e p e n d e n c e gives hope that this will happen. At the same t i m e we have to exclude the possibilities T>> c k a n d T > > / ~ - eo where the temperature d e p e n d e n c e will be very different (in the latter case p ~ 1/x//T).
4. Conclusions The crucial idea for all results o b t a i n e d in the foregoing is the a s s u m p t i o n ¢~ >> 1. A p a r t from that we have p r e s u m e d a m o d i f i c a t i o n o f the e l e c t r o n p h o n o n interaction (square o f the m a t r i x e l e m e n t ) which is described by f o r m u l a ( 5 ) . In principle a higher power o f the d e n o m i n a t o r could also be possible. The characteristic p h o n o n frequency would be t o ( x ) , a n d in o r d e r to have no significant isotope effect the c o n d i t i o n Tc,,
A n i m p o r t a n t issue is the d e p e n d e n c e o f the gap anisotropy on the content o f impurities. Experimentally it was f o u n d that samples with a smeared-out resistive transition have a m o r e isotropopic gap [ 3 ]. This could be explained in terms o f isotropic scattering from i m p u r i t i e s which could m a k e the effective interaction between electrons m o r e isotropic. Such a scattering could be due only to neutral defects since scattering from charged defects would be at small angles a n d would hardly change essentially the nature o f the effective interaction. All this has to be considered in future studies.
Acknowledgement This work was s u p p o r t e d by the D e p a r t m e n t o f Energy u n d e r the contracts # W-31- 109-ENG-38.
References [ 1] A.A. Abrikosov, J.C. Campuzano and K. Gofron, Physica C214 (1993) 73. [2] A.A. Abrikosov, Physica C 214 (1993) 103. [3] Z.-X. Shen et al., Phys. Rev. Lett. 70 (1993 ) 1553; H. Ding et al., unpublished. [4 ] A.M. Pao et al., Phys. Rev. B 42 (1990) 193. [ 5 ] M. Boekholt, M. Hoffmann and G. Guentherodt, Physica C 175 (1991)42. [6]A.A. Abrikosov, L.P. Gor'kov and I.E. Dzyaloshinsky, Quantum Field Theoretical Methods in Statistical Physics (Pergamon, Oxford, 1965 ). [ 7 ] G.P. Mazzara et al., Phys. Rev. B 47 ( 1993 ) 8119; G. Cao et al., Phys. Rev. B 47 (1993) 11510. [8]A.A. Abrikosov, Sov. Phys. JETP 12 (1961) 1254; 14 ( 1962 ) 408. [ 9 ] P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132. [ 10] A. Virosztek and J. Ruvalds, Phys. Rev. B 42 (1990) 4064. [ 11 ] T. Morita, Y. Takahashi and K. Ueda, J. Phys. Soc. Jpn. 59 (1990) 2905. [ 12 ] A.A. Abrikosov and L.A. Falkovsky, Physica C 168 (1990) 556. [ 13 ] Y. Watanabe et al., Phys. Rev. B 40 (1989) 6884; R.T. Collins et al., Phys. Rev. B 39 (1989) 6571; Phys. Rev. Lett. 64 (1990) 84; J.H. Kim et al., Phys. Rev. B 41 (1990) 7251. [ 14] T.A. Friedmann et al., Phys. Rev. B 42 (1990) 6217. [ 15 ] S. Chakravarty et al., Science 261 (1993) 337. [ 16 ] M. Randeria et al., Phys. Rev. Lett. 69 (1992) 2001; C. Sa de Melo, M. Randeria and J. Engelbrecht, Phys. Rev. Lett. 71 (1993) 3202.