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Impurity-induced localization in cuprate superconductors
Solid State Communications,
104, No. 3, pp. 155-160, 1997 0 I997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.Mk.00
Pergamon
Vol.
PII: !!%038-10!%3(97)ooo88-4
IMPURITY-INDUCED
LOCALIZATION
IN CUPRATE
Y.M. Malozovsky Department
of Physics Southern University (Received
17 September
SUPERCONDUCTORS
and J.D. Fan
and A&M College, Baton Rouge, LA 70813, U.S.A.
1996; accepted 27 February
1997 by B. Lundqvist)
The localization in a disordered metal by means of the perturbation approach is considered. It is shown that in the case of a given impurity potential a small disorder leads to localization in 2D as well as in 1D and causes the weak localization in 3D. In contrast, in the case of the impurity potential screened by electrons (interacting electrons) the disorder effect on polarizability can significantly be suppressed by the Coulomb vertex corrections to the bare impurity vertex as shown. The application of the results to ID-3D and layered 2D cases are discussed. 0 1997 Elsevier Science Ltd Keywords: A. disordered systems, high-T, superconductors, localization, electron-electron interactions, phase transition.
Among a number of unusual properties of layered 2D cuprates is the impurity-doping effect which cannot be understood based on the conventional theory of the disordered Fermi system [ 1, 21. The substitution for Cu by the divalent impurities in Cu-0 planes (like diamagnetic Zn and magnetic Ni) causes strong localization and suppression of i”, [3] which cannot be explained based on the weak localization [ 1,2,4, 51 and Abrikosov-Gorkov theory [6] of the pair-breaking effect, while doping of the other elements (magnetic or nonmagnetic) practically does not change T, nor exhibit any appreciable localization effect until a quite high dopant density [3]. We think that such an unusual behavior seems to be related to two different views on the interaction of an electron with impurities. In the first case scattering of an electron on impurities is described by a given potential where other electrons do not affect the scattering amplitude. The second case corresponds to the scattering of an electron on an impurity potential screened by other electrons. We consider two aspects of the electron localization in a disordered system. The resummation of the diagrams for the three-point vertex part shows that the backward scattering incorporated in the calculation of the charge and spin density response functions produces the strong localization not only in 1D but also in 2D, whereas in 3D such an interference effect leads to a weak localization. As a result, in the system appears the localized magnetic moments. In contrast, the
D. quantum
incorporation of screening leads to a strong suppression of the localization effect in any dimension. The consideration of the electron localization is based on two different aspects of the problem. The first is the problem of the Anderson localization, which deals with a single electron scattering on a random potential of impurities. The second is the interaction among electrons in the presence of the random potential. First let us turn to the problem of the Anderson localization. Applying perturbation theory to the scattering on the random potential v(r) and averaging each term of the expansion, the electron Green’s function averaged over the impurity distribution has the form [7] G-‘(p,~) = E - {a - C(p,c), where tp = er, - p and C(p, E) is the single-particle selfenergy. In the first Born approximation C(p, 6) has the form
E~(pvE) = ni =-
(V(p -
p')12G(p', e)d3P’l(2r)3
i sign ~12~a,
where ni is the scatterer density and v(q) is the Fourier component of the impurity potential and 70 is the elastic collision time. To consider the vertex corrections to the irreducible charge and spin density responses we consider the higher order corrections to C(p, c) by following the method adopted in the previous work by the authors 18, 91. In 155
order to make the use of the conventional frequencydependent (time-dependent) perturbation theory it is convenient to introduce the impurity localized propagator as I- \ e~.2/_\rr I _P, ._.I___._ ..2/_\ ._ 1_./^\12 :_ WIICIC uo(q, = q,v\q,, 15
Dimptqv 0) the
=
“bare”
LUOlqjILW
t
loJ3
electron-impurity
interaction
constant.
Renlacirw hv II;_ (0. 01, ~~ in the -, _ ‘,,,F,_, .~~~ahove _ equation _‘-T_--___c) n,lv(nN2 __,, \I,l
for C(p, E), the single-particle self-energy incorporating the vertex correction can be written as C(p)=i
$$G@ I . ,
- k)Dimp(k)riq@, k),
where
It is seen from equation (3) that K&,(p, p’) corresponds to the first order interaction, whereas two others K$$p, p’) and K$,Jp,p) correspond to the second -..A__ VU0I- jT/ -r\) SL~ULUJ _r..-2.. c-.. rL_ C--r --A-- :- rL_ Z-r,.. UIUCL. R2iqply, IV1 l.l1G LILbL UIUGI 111 U1G IIIIGLaction incorporating the vertex correction of the first order. K&& p’) stands for so-called “crossed” diagram and is very important because it represents the interaction in the so-called “Cooper channel”. The scattering amplitude due to such diagrams is divergent for p + p’ = 0. This can be interpreted as a singular backward scattering which is essentiai in the iocaiization issue. This was first shown by Langer and Neal [lo]. WP . . ” ran .,-. write 1.. .._ 2. CPY;PC v_.._y nf .,_ the ..._ mavimallv -“--s”‘.+“J
x R”i,,iy,p’)c((‘)G((’
-
k)p&,
k)
(2)
r+ncrwl v..,yyII cliaotamr ..‘..b.-..’
to evaluate K&p, p’), the effective particle-hole propagator. K&I, p’) becomes a ladder diagram in the particle-particle propagator P(p, p’) if we twist all hole lines and make a series of the ladder graphs. To carry out the ladder summation of aii the graphs we replace the first two terms in equation (3) by the irreducible kernel AP 1nAAn.. r.*mmnt:,Y.. the P@, p’). AS a iXsU!t ,.F “1 tuti IauuGI au.IIIIIau”‘I, equation for P(p, p’) can be written as n/- -I, ~~%_ _I\ ‘VVY J ==wVY !
is the scalar three-point impurity vertex part and Ki,ip,p’j denotes the kemei of the integrai equation for the vertex part and describes the multiple scattering IVYVPQEPP The LP~PI intemrtinn K. In rn’j , s&fies -_____ nf __ ________.____ --‘mpu, r-‘.,.o”.w. -m._ __ Ward’s identity and can therefore be written as a functional derivative of the self-energy with respect to the electron Green’s function [7], Kz&,p’) = - iSC,(p)lGG,~(p’) which depends, in general, on the spin indices. To evaiuate this kemei we use the iteration procedures and take I”,,@, k) = 1 in equation (1) as fi+nt (4) the rr~lJ’“,X”“ULX”“. Using ?he ;rl,=nt;trr ‘U”L’L’CJ .Xn UI ,Zlnnmv;mot;nn 6G,@)/6G,,(p’) = 6(p - p’)6,,, the expression for the Assuming that P(p, w; p’, a’) = 2&p - p’)/[w - w’ + ia] kernel in the zeroth-approximation has the form and P(p, u; p’, 0’) = 2n(p, 0; p’, @‘)![O - 0’ + ir;], Ki’&,(p,p’) = Dimp(p- ~‘)a,,,,!. Substituting this kernel equation (4) for II(p + p’,Q), the particle-particle into equation (2) and then using equation (I), we obtain propagator, can be written in the form of the Dyson the self-energy incorporating the vertex correction in equation the first order of perturbation beyond the first Born l-&p + p’, n, = u* + u*l-I& + p’, Q)rI(p + p’, nj, (5) ^^^_,..:-,..:,rT,.:-^_^ ‘IIVIF: -^_^ lL11‘K :.-app1”MlUan”U. “alug ,X,,...J,““LUUJ :A,.._.:&.. IUE;III‘Iy“llC for the single-particle self-energy that incorporates the where u2 = (u:(q)) is the mean-square impurity potential vertex correction in the first step of perturbation and and II& + p’? a) = C, @(q, w)GR(p + p’ -q: o -l- !J) noting that D,,,(k) is the impurity propagator for an is the bare particle-particle propagator. We obtain the impurity with a given potential, the kernel of interaction solution to equation (5) as below [5, 1l] up to the second order can be expressed as follows K:;(P,
P’)
=
KP,$P,
P’)
+ K;&(P,p’)
+ fG&(P,
I&p + p’, !d) = u270 ‘/[-if2 + D,-,(p+ P’)~I, .-L-_n = v$r($d is the CiifhSiOiiw&kierii vane ~0
P’),
(3)
qEo(q)Gm@’ -p + q> , (34
iri d-
dimension. The __.irrechxihle including the imnmitv --_.-__ rnnlarixahilitv -.-_---__._, _‘________~ -__ ____r____, vertex part can thus be written in the form g(k) = -2i
c r14n =G@)G@ (2*)4 J
- k)fi,(p,
k),
(7)
r..hnra **llblb
and K&@,
(6)
p’) = iSO,,
Fi,(p,
k) =
l+iJ c $$Yp.p’)ii(p’)c(p’
-kjfi,@‘,
k)
(8) is the rearranged three-point vertex part. The diagram representation for I=‘,,@, k) is given in Fig. 1. In
is the free-particle polarizability including scattering and (rimp(p, w; k, w))~~~=,+= ~0 ‘I[- io + n(l,,lr2,
)fcr/( = ~~--jit:p?p*) on impurity .
.-
_\L,__ ~.
---
Finally, we have
(b)
Fig. i. Diagram representation for fim&, kj, the threepoint vertex part (a) and P(p, p’), the particle-particle propagator (b). equations (7) and (8) &) satisfies the Dyson equation: Zipj = Gipj -i- mjZipj@j, where Gipj is the zerothapproximation Green’s function as before and &) is the cPif_PnPtuV ““.A W.‘_‘b, ri3i LA_, %hirb_ hzr the fern_
Dok’
R(k 0) =
_ jw + Dcojk2,
(12)
where D(w) is the frequency-dependent effective diffusion coefficient which is equal, for 3D, to &D(O)
Do
==
Wa)
9
i+a-
&D(W)
Combining this equation with the zeroth-approximation of EJn inverse scager;_fi,o I_p-- c/37., _, _. “, the ___ &&eve -“\I-, 61 -I z - i- cian time can be written as
-NF
J 8+$
in (w70j
Do
= 1+--
i
i
DID(~)
’
=
Do
1 + w-27~2’
2+70 WrO
respectively, where a! = 3 In (4eFra)/(8&ri). As seen from equations (13 j, the irreducible polarizability has a diffusive form of D(w), the frequency-dependent
i’lfki\ t---1
for 3D and
(lob)
-=_1 7(E) i
1 r.,+_ 70 L ir
1
/w21 -
GA
7@) = 70 L ’ + [2@-p,?)2
(10~)
1 1’
where Lo = G is the diffusion length and the correlation length [ was introduced, which can also be
diffusion __________ coefficient:
It
foiinws
from
equation (13a) that a weak localization exists even in 3D. In 2D (in analogy to 1D) even a small disorder (eFr0 Z+1) leads to a quite strong localization when o - 0. It means that all the states are localized both in 2D and in iD caused by a smaii disorder due to the substitutional doping of impurity. Nevertheless, in the r3r.a nf the Ir_r_ W.&Y” “I L.1” XIIPIL .I”.... rlirnwler U.Y”I..I. \LP ,” S c l\ ‘, the 1.1” lwnlivatinn .V”....Y....Y.. effect in 2D much harder to be observed than that in 1D. Using the Kubo formula, the conductivity of a disordered system [5, 121 can be evaluated from the equation
a(wj =
E)J. i
effective
e2N,cD(oj.
Using equations (13) we can see that the conductivity in iD
his
the
weii-lmnwn
f~_~_
for
an
inwllntnr __--_- ___, i.e.
for w * 1/7o, where aa = e2~70/rn* is the Drude conductivity. In 2D for for 07~ < 1, a(w) = U~ow7(3/(W70+ EF 17; ‘12) - w w Q l/(2& and therefore it has an insulating behavior a(w) =
aow27~/(1
+ w27$
-
co2
as well. The frequency region of the localized states in 2D (0 < 7; !i(t+70jj for cF~O S i is much narrower than that in 1D (o < 11~~). In 3D for WT~< 1, Q(oj = e&l +a-? a In l,.w_\/lQ,?.~?\l fcx a.1\yY, “,# \v-p I “,J - ina (@- ‘j w < 7;
’
exp (-8&i/3). Thus, the conductivity has the weak localization behavior that is actually difficult to observe in the case of such a small disorder that +70 % 1. In the model with a given impurity potential g(k, w) = i,(k, co)= g,(k, w), the density response function, given by equation (12), where z,(k, w) and - 11 A-..“:+.. ..a”.._..,.,5 I~~.~IIJ~ XS[“, W) arc the charge and Spiii ur;u~lry functions of the noninteracting electrons, respectively. Thus, &(k, w) for w of order w - l/r+ - T represents the temperature-dependent spin susceptibility of the diffusive electrons, where 7+ is the phase relaxation time which is usually introduced to provide the cutoff and is of order r+ - r, with 7, being the energy relaxation &I~._ TZ 1*, TL r-*1-.- _ c.-__ _____l’___ /,n\ umc:
LJ,
111.
II
IUIIUWS
Irum rqua~lurls (ILJ arid (i3bj
that &o(T) = - N,D,/D(T) = - NFl(2e&T), the spin s1usceptibility of the localized electrons for a 2D case. Thus, the localization in a 2D system can lead to the appearance of the localized magnetic moment and Curie type susceptibility xS = C/T with C = n?ln, being the Curie constant, where XPl(2&) xp = &iv; is the Pauii susceptibiiity, ni is the impurity density and n, is the 2D electron density. It seems that cnrh ““I..
lnrali~atinn .I l”I..l.Y..L.V..
c.ffm-t hat in lm,~rd WII.dWC a...” hm.n “IVI. nhcenrm4 VYYVl,“.. 1.1 ‘..,.~LVU
2D cuprates where the substitution for Cu by the divalent like diamagnetic Zn and magnetic Ni impurities in Cu-0 planes causes the strong localization and suppression of the superconducting transition temperature. Magnetization studies of cuprates with Zn and Ni doping show a Curie type behavior of susceptibility. A surprising fact is tl.nt lLa P..&;P tn:.. .-..‘.,..m.*;l.:1:,.. :, n.. -..:,4e...,w. Fn.. ll., “,PL LUG LUIIC, LLIIII 111 m.mkLp’““‘LJ 1J (111 LYI”~IILG 1”, LIIL;
which is determined
I$tp,k)=i +i
from the solution of the equation
I$$ \I
X K,C~,p’)G@‘)G(p’-k)ri,,(p’, W,c@‘,k),
(1% where _Kcln ir the J w)r n’\, _" ---_ lwrn~l _~_.~__. nf _. the _-_ Pnnlnmh -VIA-..-l
intetartinn .l..~.~...~V..
in the spin channel. It can be shown [8, 91 that in the lowest order of perturbation KS@, p’) coincides with the screened Coulomb interaction, i.e. @(p,p’) = V&p - p’), where V,,,(k) = V(k)/#) is the screened . ___ . Couiomb with V(k) being the bare Couiomb and e(k) is the dielectric response function in the random phase ~.nrrw;m~t;~n IDDA\ th.n os..r~a..s4 SL~~I”*U.IuLI”II \&\A‘L,. Tn 111nrlrl;t;r\n LLuUICI”LI, LIICJ~Iti,cIII~UP.w.lsw.xl. ~“U,“‘I,” interaction is practically frequency-independent in the region of the particle-hole excitations (w < vck). In this case we can replace KFtp,p’) = V&p - p’, 0) in equation (15) by the average over the Fermi surface, i.e. K,$@,p’) = (V&p - p’, 0)j = 2V, where V is the screened exchange interaction constant. LT^__. *s ^^ a,, ^- C~illU~K, ^_.^--,a __.^ ^^- ~“~USIK _....l.._r_ ”,,:-11,aI I__.___> I’IVW, WC‘ill, ,aye,cu 2D (L2D) Fermi gas with the isotropic electron spectrum in any of layers, i.e. eF = p2/2m*, the 2D sinple-aarticle -4-m r~--mm-energy. Note that the bare Coulomb interaction V(k) here in an L2D system is given by
V(k) = V&J
sinh (ckll) cash (ck!!j - cos (ck: j
= Vo(W(kj,
(16)
where kll and kZ are the momentum components in the ..,n..,,..A :.. ,X..a,,:-.. ..,.-,.1 /1. \ _= ~__2/..lS ~‘P”C allU 111,La 1‘1GUIIS~U”LL U”LllUll +,. I” :,II) ,I“O[&,,, L71r If&K,, is the bare Coulomb interaction in a pure 2D case with the dielectric constant K of the lattice background and flk) is the form factor of an L2D system. Thus, we have
localized magnetic moments induced by nonmagnetic Zn. In addition. it was found that the localized magnetic moment induced by Zn substitution depends on the carriers (hole) density. For example, for optimally T/C 2* (x = 0.93) doped YBCO the magnetic moment correx 9; d4 Vscr(~-p’)~~~,=,~,,=~,= $(a. sponds to -0.2~~ per Zn, whereas for underdoped - T/T I.. fi L?\ .____I_ AL_ 1___11__> _____._. ..__ C_~._> Am 0 (.X.-.=U.WJ) biUlll.“C UK IUCiillLLXl lIlUllltXll Wkib 1UUllU LO increase up to -0.8 ,.&Bper Zn, where pB is the Bohr magneton [13]. Such a behavior seems to be in agreewhere ment with the dependence of the Curie constant C - l/n, on the carrier density n,. cix The spin density response function of the interacting F(cx,0 = jelectrons in the presence of the scattering on impurities o J( 1 - x2)(X2 + 2aLxcoth3sr + oL*’
I I
0,
(17)
(18)
-with a given impurity poieniiai foiiows from the equation
where ri,,Jp, k) is the impurity vertex part as before and I’:@, k) is the Coulomb vertex part in the spin channel
and the use of kif = lpi1- ~‘111~= 2p$(lcos 4) is made9 ry = P’/Kv,is the interelertmnir -- p ---_ -_____-_________interaction ~QCstant with vF = pFlm* and is related to the dimensionless density parameter t-s by the relation r, = 6 and { = 2pFc with the 2D Fermi momentum pF = e and the interlayer spacing c.
Vol. 104, No. 3
LOCALIZATION IN CUPRATE SUPERCONDUCTORS
It follows from equation (15) that I’S@, k), the Coulomb vertex part, in the case when K&x p’) is repiaced by u, the exchange interaction constant, depends o;c(t~ momentum transfer only, namely .FCI.. I\ 1 I r1 I 11C1L\l . . ..A c,IL\ &..,m krr ‘s\Y,A]=‘s\n/=II [’ T “=\a,, VT1111 A\&, g;l.tiu “J equation (12). Thus, taking into account this F:(k) in equation (14) the equation for the spin density response function is derived as _-.,I.
xs(k 0) =
+“;;;,
1
n
.\
w) =
-NF -
iw
1.2
"0" + Ds(w)kp
159
for a given impurity potential we introduce the impurity localized propagator which can, in this case, be written as
uo(s> *
1 0 + i6 sign w’
( >
d&(q, wj = 2 e(q*w) where 6i&q,
(21)
o) is the screened impurity propagator,
i?(a)-I = ni!v(a)12 is the “bare” electron-impurity inter. ”
__
.
action constant as before. Substituting equation (21) into equation (1) and using Ward’s identity we can get the perturbation series for the self-energy beyond the first Born approximation. In terms of Ward’s identity the __.._._--I-_ ior c-. AL. I-- -1 .~.__,..J. ._ ,-1, *a.- __~~_ .L~~Af~ .~ expression me Kernel in~mumg du me contnmmons up to the second order can thus be derived as - o81 - GB’ Ki,(p*p’j =Klimp(P,P’j + &&(p,p’j
where D,(w) = D(o) - UNFDO is the effective spin diffusion coefficient with D(w) given by equation (13). It follows from equation (19) that the spin density ran..-..,.c....,-.+:-.. ,,..r..:..c (1 ,, UlllUDl”‘, ,I:4X...:-.. Y”lG ..,.,a .WLII&,II ..k..nl.G”I,G_ ,.,-._A -c8 --xmpu~r k?! In n’G?P’ (n lM?? In n’119 1331 I--4rmpvyr I--5lmpvvr \--I IG~tJ”11Jz; I”II~LI”IIb,“LIIalIIJ sponds to the diffusive paramagnon. In addition, where k$,(p,p’) = Bi,(p - p’)6,,, with B+(k) given equation (19) leads to the Curie-Weiss type spin L.. __.._r:_- (LIJ 1-1, iUlU -_J uy eyua11u11 susceptibility (w - T) as follows n
k;&@,
c
xs =xp$j=
P’) =
(20)
T-T;’
where xp = p;NF is the Pauli susceptibility as before and C = xps/(2&) is the Curie-Weiss constant and with xps = &Nr/[l - UNF] being the paramagnetic susceptibility including the Stoner enhancement factor and ?-* rr..ln\ll?, z\ ;0 .**I&L1 . ..tr. c -- “~~“,,\L.C,?,,,, 13 &.P ,,,C r..r;a L”11ti tnm..&rot*.r~ &L.+kILILUIC x(O) = NFI[ 1 - UNF]. It should be noticed that equation (20j is actually valid for a L2D system. Thus, the interaction in the system of localized electrons can lead to the appearance of fetromagnetism. Such an effect of localization can also lead to the coupling of quasiparticles with the diffusive paramagnons and therefore -l__~~__.-~l-. f_____-_Al__ _..__~__-.1_1_ __^^ --A esselluiilly -___-r:_,,_. slgnlll~““uy IIIGIC~ IlIt:yuiis1paruc1t: llliixi iulU decrease T,, the superconducting transition temperature. The consideration __ an __. electron ______.. with _-__-_-_--__-.. of -_ the interaction of the diffusive paramagnon is out of the scope of the present paper. Now, consider the model with charged impurities interstitially doped between layers. In this case the impurity potentiai screened by eiectrons is determined by the equation u(q) = v(q)le(q,O), where v(q) is the hwta imnnritv “WV ““y”L”J
nntmwial y”“‘~.......
nf Y”IL.W rnm~ I.&.... kid (a chnrt_mnoc= “L \.A “...,L. ..e.‘b.. nr Y.
Coulomb-type potential) and e(q, co) is the dielectric response function: E(Q,w) = 1 - V(q)j&(q, o), where V(q) is the matrix element of the bare Coulomb interaction and z,(q, w) is the irreducible charge density response function which includes the vertex corrections due to the impurity scattering and Coulomb interaction. hTnnlnnr;nn 1.C~.c1~1111~
rhn U1b
~ra..+~v .b-I,“.%
,.a-ot;nnr C”ll~W~l”IIU
tn b”
th.P . ..w
&w.rlnrihl~ IanIVVI.“II
response, i.e. assuming ji,(q,u) = xo(q,o), the bare particle-hole bubble (the Lindhard function) corresponds to the well-known RPA. In analogy to the above model
_2i
(224 I d4q ~Di,@-qjV,,,@-qjG,(qjG,,@’
dc \L?r,
+P-4)~ (22b)
and k&&p, p’) and ~~&p,p’) are similar to equations (3a) and (3b) respectively, except that in equations (3a) _-J (JU) ,?I_\ Uiw(K) n IL\ SUUlUU _I__..,~ L_ ___,____IL_. iF ,I.\ ,L_ i(llU Ut: l’SylilGCU Uy ui,pp(K), UK renormalized
impurity propagator. In equation (22)
V.__tkl is the SCKYXCX! Co&& .x.r\~-,~-
interaction
in the RP.4
as before. The diagrams for these interactions are given in Fig. 2. It is seen from Fig. 2 that the first diagram corresponds to the first order screened interaction, whereas two others appear as a result of the Coulomb interaction. Tlne second diagram corresponds to the second order direct screened interaction (so called the Uatirpp rliaoram\ The diaoram fnr mmntinn 133hl -I ic yeq ---..II --..e.---,. a------ -1 ______ \---, important because it represents the interaction in the Hartree-“Cooper” channel [ll] (this diagram in fact differs from the Cooperon’s one since it has an opposite sign with respect to the Cooperon diagram and its initial and final momenta in scattering are t‘he samej. For the model with a given interaction there are no such ,-i;~i,~nm~Thn. this cmvwwi mtvbl whirh .arm,,ntr fnr IIIUY) Lll” Y.,““.l.. I..““_. I.... v.. U”“V”...” 1-1 V’Upum’.U. the interaction between electrons in the RPA includes more contributions than the first for a given impurity potential. Substituting equation (22) into the equation for
LOCALIZATION IN CUPRATE SUPERCONDUCTORS
160
Hame diagrams
p0 po *’
p’d
p0
/~“‘,p~d , *
2 _I_ PO
PO pa *.
+
i
u P’d
‘.
-. k fl’ ,“, ‘.
PC
’
PO
P’ 6 .* +...
,’
p’ d PO Cooperon diagram
Fig. 2. Diagram for &$.(p, p’), the kernel of interaction, up to the second order. The wavy lines stand for the screened Coulomb interaction, X corresponds to an impurity and thick dashed lines stand for a screened impurity propagator. the self-energy
where K&,&p’) = k&,@.p’) + K$p,p’) is the spinsymmetric kernel of interaction with K$,(p, p’) given by equation (22), we can also get a series for the singleparticle self-energy. The comparison to the model with a given impurity potential shows that the diagrams which appear as a result of the screened Coulomb interaction can compensate the diagrams for the backward scattering. It is meant that the contributions coming from the Coulomb interactions can significantly reduce the backward scattering effect if the screened Coulomb interaction is quite strong. Such a compensation effect will exist in the higher orders of perturbations as can be shown. In this case the diagram of the first-order perturbation (the first Born approximation) is sufficient for consideration of the disorder effect. The ordinary diffusion ladder is important in this case only and the irreducible density response function is given by equation (12) with D(w) = Du, the frequency-independent diffusion coefficient as before. The consideration of the problem in this case is given in [ 111. In the case of the high density limit the screening effect is very strong and the contribution of diagrams shown in Fig. 2 does not play an important role in superconductivity and localization effect. In contrast,
Vol. 104, No. 3
the high-T, superconductors (cuprate compounds) correspond to the case of a dilute Fermi gas and therefore the screening effect in cuprates is weak. In this case the Coulomb interaction between carriers can not be ignored and the contribution of diagrams shown in the Fig. 2 plays a significant role in the localization issue and superconductivity. Thus, our results at least can qualitatively explain both the unusual strong localization and suppression of superconductivity in cuprates induced by the substitution of Zn and Ni and much weaker effect of the substitution of Fe, Co, Al, Ga, etc. Acknowledgements-This work is supported by DOE with the grant No. DE-FC48-95RB10542 through TCSUH. REFERENCES 1. Lee, P.A. and Ramakrishnan, T.V., Rev. Modern Phys., 57, 1985, 287.
2. Belitz, D. and Kirkpatrick, T-R., Rev. Modern Phys., 66, 1994, 261. 3. Greene, L.H. and Bagley, B.G., “Oxygen Stoichiometric Effects and Related Atomic Substitutions in the High-T, Cuprates”, ibid. Vol. II, pp. 509-569, 1990. 4. Abrahams, E., Anderson, P.W., Licciardello, D.C. and Ramakrishnan, T.V., Phys. Rev. Left., 42,1979, 673.
5. Gorkov, L.P., Larkin, A.I. and Khmelnitskii, D.E., Soviet JETP Lett., 30, 1979, 228.
6. Abrikosov, A.A. and Gorkov, L.P., Sov. Phys. JETP 10, 1960,593;
6, 1961, 1243.
7. Abrikosov, A.A., Gorkov, L.P. and Dzyaloshinskii, I.E., Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, New Jersey, 1964. 8. Malozovsky, Y.M. and Fan, J.D., Supercond. Sci. Technol., 9, 1996, 622.
9. Malozovsky, Y.M. and Fan, J.D., J. Phys. C (published in 1996). 10. Langer, J.S. and Neal, T., Phys. Rev, Lett., 16,1966, 984.
11. Altshuler, B.L. and Aronov, A.G., in ElectronElectron Interaction in Disordered Systems (Edited by M. Pollak and A.L. Efros). North-Holland, Amsterdam, 1985. 12. Maleev, S.V. and Toperverg, B.P., Sov. Phys. JETP, 42, 1976,734.
13. Mizuhashi, K., Takenaka, K., Fukuzumi, Y. and Uchida, S., Phys. Rev., B52, 1995, R3884.
104, No. 3, pp. 155-160, 1997 0 I997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.Mk.00
Pergamon
Vol.
PII: !!%038-10!%3(97)ooo88-4
IMPURITY-INDUCED
LOCALIZATION
IN CUPRATE
Y.M. Malozovsky Department
of Physics Southern University (Received
17 September
SUPERCONDUCTORS
and J.D. Fan
and A&M College, Baton Rouge, LA 70813, U.S.A.
1996; accepted 27 February
1997 by B. Lundqvist)
The localization in a disordered metal by means of the perturbation approach is considered. It is shown that in the case of a given impurity potential a small disorder leads to localization in 2D as well as in 1D and causes the weak localization in 3D. In contrast, in the case of the impurity potential screened by electrons (interacting electrons) the disorder effect on polarizability can significantly be suppressed by the Coulomb vertex corrections to the bare impurity vertex as shown. The application of the results to ID-3D and layered 2D cases are discussed. 0 1997 Elsevier Science Ltd Keywords: A. disordered systems, high-T, superconductors, localization, electron-electron interactions, phase transition.
Among a number of unusual properties of layered 2D cuprates is the impurity-doping effect which cannot be understood based on the conventional theory of the disordered Fermi system [ 1, 21. The substitution for Cu by the divalent impurities in Cu-0 planes (like diamagnetic Zn and magnetic Ni) causes strong localization and suppression of i”, [3] which cannot be explained based on the weak localization [ 1,2,4, 51 and Abrikosov-Gorkov theory [6] of the pair-breaking effect, while doping of the other elements (magnetic or nonmagnetic) practically does not change T, nor exhibit any appreciable localization effect until a quite high dopant density [3]. We think that such an unusual behavior seems to be related to two different views on the interaction of an electron with impurities. In the first case scattering of an electron on impurities is described by a given potential where other electrons do not affect the scattering amplitude. The second case corresponds to the scattering of an electron on an impurity potential screened by other electrons. We consider two aspects of the electron localization in a disordered system. The resummation of the diagrams for the three-point vertex part shows that the backward scattering incorporated in the calculation of the charge and spin density response functions produces the strong localization not only in 1D but also in 2D, whereas in 3D such an interference effect leads to a weak localization. As a result, in the system appears the localized magnetic moments. In contrast, the
D. quantum
incorporation of screening leads to a strong suppression of the localization effect in any dimension. The consideration of the electron localization is based on two different aspects of the problem. The first is the problem of the Anderson localization, which deals with a single electron scattering on a random potential of impurities. The second is the interaction among electrons in the presence of the random potential. First let us turn to the problem of the Anderson localization. Applying perturbation theory to the scattering on the random potential v(r) and averaging each term of the expansion, the electron Green’s function averaged over the impurity distribution has the form [7] G-‘(p,~) = E - {a - C(p,c), where tp = er, - p and C(p, E) is the single-particle selfenergy. In the first Born approximation C(p, 6) has the form
E~(pvE) = ni =-
(V(p -
p')12G(p', e)d3P’l(2r)3
i sign ~12~a,
where ni is the scatterer density and v(q) is the Fourier component of the impurity potential and 70 is the elastic collision time. To consider the vertex corrections to the irreducible charge and spin density responses we consider the higher order corrections to C(p, c) by following the method adopted in the previous work by the authors 18, 91. In 155
order to make the use of the conventional frequencydependent (time-dependent) perturbation theory it is convenient to introduce the impurity localized propagator as I- \ e~.2/_\rr I _P, ._.I___._ ..2/_\ ._ 1_./^\12 :_ WIICIC uo(q, = q,v\q,, 15
Dimptqv 0) the
=
“bare”
LUOlqjILW
t
loJ3
electron-impurity
interaction
constant.
Renlacirw hv II;_ (0. 01, ~~ in the -, _ ‘,,,F,_, .~~~ahove _ equation _‘-T_--___c) n,lv(nN2 __,, \I,l
for C(p, E), the single-particle self-energy incorporating the vertex correction can be written as C(p)=i
$$G@ I . ,
- k)Dimp(k)riq@, k),
where
It is seen from equation (3) that K&,(p, p’) corresponds to the first order interaction, whereas two others K$$p, p’) and K$,Jp,p) correspond to the second -..A__ VU0I- jT/ -r\) SL~ULUJ _r..-2.. c-.. rL_ C--r --A-- :- rL_ Z-r,.. UIUCL. R2iqply, IV1 l.l1G LILbL UIUGI 111 U1G IIIIGLaction incorporating the vertex correction of the first order. K&& p’) stands for so-called “crossed” diagram and is very important because it represents the interaction in the so-called “Cooper channel”. The scattering amplitude due to such diagrams is divergent for p + p’ = 0. This can be interpreted as a singular backward scattering which is essentiai in the iocaiization issue. This was first shown by Langer and Neal [lo]. WP . . ” ran .,-. write 1.. .._ 2. CPY;PC v_.._y nf .,_ the ..._ mavimallv -“--s”‘.+“J
x R”i,,iy,p’)c((‘)G((’
-
k)p&,
k)
(2)
r+ncrwl v..,yyII cliaotamr ..‘..b.-..’
to evaluate K&p, p’), the effective particle-hole propagator. K&I, p’) becomes a ladder diagram in the particle-particle propagator P(p, p’) if we twist all hole lines and make a series of the ladder graphs. To carry out the ladder summation of aii the graphs we replace the first two terms in equation (3) by the irreducible kernel AP 1nAAn.. r.*mmnt:,Y.. the P@, p’). AS a iXsU!t ,.F “1 tuti IauuGI au.IIIIIau”‘I, equation for P(p, p’) can be written as n/- -I, ~~%_ _I\ ‘VVY J ==wVY !
is the scalar three-point impurity vertex part and Ki,ip,p’j denotes the kemei of the integrai equation for the vertex part and describes the multiple scattering IVYVPQEPP The LP~PI intemrtinn K. In rn’j , s&fies -_____ nf __ ________.____ --‘mpu, r-‘.,.o”.w. -m._ __ Ward’s identity and can therefore be written as a functional derivative of the self-energy with respect to the electron Green’s function [7], Kz&,p’) = - iSC,(p)lGG,~(p’) which depends, in general, on the spin indices. To evaiuate this kemei we use the iteration procedures and take I”,,@, k) = 1 in equation (1) as fi+nt (4) the rr~lJ’“,X”“ULX”“. Using ?he ;rl,=nt;trr ‘U”L’L’CJ .Xn UI ,Zlnnmv;mot;nn 6G,@)/6G,,(p’) = 6(p - p’)6,,, the expression for the Assuming that P(p, w; p’, a’) = 2&p - p’)/[w - w’ + ia] kernel in the zeroth-approximation has the form and P(p, u; p’, 0’) = 2n(p, 0; p’, @‘)![O - 0’ + ir;], Ki’&,(p,p’) = Dimp(p- ~‘)a,,,,!. Substituting this kernel equation (4) for II(p + p’,Q), the particle-particle into equation (2) and then using equation (I), we obtain propagator, can be written in the form of the Dyson the self-energy incorporating the vertex correction in equation the first order of perturbation beyond the first Born l-&p + p’, n, = u* + u*l-I& + p’, Q)rI(p + p’, nj, (5) ^^^_,..:-,..:,rT,.:-^_^ ‘IIVIF: -^_^ lL11‘K :.-app1”MlUan”U. “alug ,X,,...J,““LUUJ :A,.._.:&.. IUE;III‘Iy“llC for the single-particle self-energy that incorporates the where u2 = (u:(q)) is the mean-square impurity potential vertex correction in the first step of perturbation and and II& + p’? a) = C, @(q, w)GR(p + p’ -q: o -l- !J) noting that D,,,(k) is the impurity propagator for an is the bare particle-particle propagator. We obtain the impurity with a given potential, the kernel of interaction solution to equation (5) as below [5, 1l] up to the second order can be expressed as follows K:;(P,
P’)
=
KP,$P,
P’)
+ K;&(P,p’)
+ fG&(P,
I&p + p’, !d) = u270 ‘/[-if2 + D,-,(p+ P’)~I, .-L-_n = v$r($d is the CiifhSiOiiw&kierii vane ~0
P’),
(3)
qEo(q)Gm@’ -p + q> , (34
iri d-
dimension. The __.irrechxihle including the imnmitv --_.-__ rnnlarixahilitv -.-_---__._, _‘________~ -__ ____r____, vertex part can thus be written in the form g(k) = -2i
c r14n =G@)G@ (2*)4 J
- k)fi,(p,
k),
(7)
r..hnra **llblb
and K&@,
(6)
p’) = iSO,,
Fi,(p,
k) =
l+iJ c $$Yp.p’)ii(p’)c(p’
-kjfi,@‘,
k)
(8) is the rearranged three-point vertex part. The diagram representation for I=‘,,@, k) is given in Fig. 1. In
is the free-particle polarizability including scattering and (rimp(p, w; k, w))~~~=,+= ~0 ‘I[- io + n(l,,lr2,
)fcr/( = ~~--jit:p?p*) on impurity .
.-
_\L,__ ~.
---
Finally, we have
(b)
Fig. i. Diagram representation for fim&, kj, the threepoint vertex part (a) and P(p, p’), the particle-particle propagator (b). equations (7) and (8) &) satisfies the Dyson equation: Zipj = Gipj -i- mjZipj@j, where Gipj is the zerothapproximation Green’s function as before and &) is the cPif_PnPtuV ““.A W.‘_‘b, ri3i LA_, %hirb_ hzr the fern_
Dok’
R(k 0) =
_ jw + Dcojk2,
(12)
where D(w) is the frequency-dependent effective diffusion coefficient which is equal, for 3D, to &D(O)
Do
==
Wa)
9
i+a-
&D(W)
Combining this equation with the zeroth-approximation of EJn inverse scager;_fi,o I_p-- c/37., _, _. “, the ___ &&eve -“\I-, 61 -I z - i- cian time can be written as
-NF
J 8+$
in (w70j
Do
= 1+--
i
i
DID(~)
’
=
Do
1 + w-27~2’
2+70 WrO
respectively, where a! = 3 In (4eFra)/(8&ri). As seen from equations (13 j, the irreducible polarizability has a diffusive form of D(w), the frequency-dependent
i’lfki\ t---1
for 3D and
(lob)
-=_1 7(E) i
1 r.,+_ 70 L ir
1
/w21 -
GA
7@) = 70 L ’ + [2@-p,?)2
(10~)
1 1’
where Lo = G is the diffusion length and the correlation length [ was introduced, which can also be
diffusion __________ coefficient:
It
foiinws
from
equation (13a) that a weak localization exists even in 3D. In 2D (in analogy to 1D) even a small disorder (eFr0 Z+1) leads to a quite strong localization when o - 0. It means that all the states are localized both in 2D and in iD caused by a smaii disorder due to the substitutional doping of impurity. Nevertheless, in the r3r.a nf the Ir_r_ W.&Y” “I L.1” XIIPIL .I”.... rlirnwler U.Y”I..I. \LP ,” S c l\ ‘, the 1.1” lwnlivatinn .V”....Y....Y.. effect in 2D much harder to be observed than that in 1D. Using the Kubo formula, the conductivity of a disordered system [5, 121 can be evaluated from the equation
a(wj =
E)J. i
effective
e2N,cD(oj.
Using equations (13) we can see that the conductivity in iD
his
the
weii-lmnwn
f~_~_
for
an
inwllntnr __--_- ___, i.e.
for w * 1/7o, where aa = e2~70/rn* is the Drude conductivity. In 2D for for 07~ < 1, a(w) = U~ow7(3/(W70+ EF 17; ‘12) - w w Q l/(2& and therefore it has an insulating behavior a(w) =
aow27~/(1
+ w27$
-
co2
as well. The frequency region of the localized states in 2D (0 < 7; !i(t+70jj for cF~O S i is much narrower than that in 1D (o < 11~~). In 3D for WT~< 1, Q(oj = e&l +a-? a In l,.w_\/lQ,?.~?\l fcx a.1\yY, “,# \v-p I “,J - ina (@- ‘j w < 7;
’
exp (-8&i/3). Thus, the conductivity has the weak localization behavior that is actually difficult to observe in the case of such a small disorder that +70 % 1. In the model with a given impurity potential g(k, w) = i,(k, co)= g,(k, w), the density response function, given by equation (12), where z,(k, w) and - 11 A-..“:+.. ..a”.._..,.,5 I~~.~IIJ~ XS[“, W) arc the charge and Spiii ur;u~lry functions of the noninteracting electrons, respectively. Thus, &(k, w) for w of order w - l/r+ - T represents the temperature-dependent spin susceptibility of the diffusive electrons, where 7+ is the phase relaxation time which is usually introduced to provide the cutoff and is of order r+ - r, with 7, being the energy relaxation &I~._ TZ 1*, TL r-*1-.- _ c.-__ _____l’___ /,n\ umc:
LJ,
111.
II
IUIIUWS
Irum rqua~lurls (ILJ arid (i3bj
that &o(T) = - N,D,/D(T) = - NFl(2e&T), the spin s1usceptibility of the localized electrons for a 2D case. Thus, the localization in a 2D system can lead to the appearance of the localized magnetic moment and Curie type susceptibility xS = C/T with C = n?ln, being the Curie constant, where XPl(2&) xp = &iv; is the Pauii susceptibiiity, ni is the impurity density and n, is the 2D electron density. It seems that cnrh ““I..
lnrali~atinn .I l”I..l.Y..L.V..
c.ffm-t hat in lm,~rd WII.dWC a...” hm.n “IVI. nhcenrm4 VYYVl,“.. 1.1 ‘..,.~LVU
2D cuprates where the substitution for Cu by the divalent like diamagnetic Zn and magnetic Ni impurities in Cu-0 planes causes the strong localization and suppression of the superconducting transition temperature. Magnetization studies of cuprates with Zn and Ni doping show a Curie type behavior of susceptibility. A surprising fact is tl.nt lLa P..&;P tn:.. .-..‘.,..m.*;l.:1:,.. :, n.. -..:,4e...,w. Fn.. ll., “,PL LUG LUIIC, LLIIII 111 m.mkLp’““‘LJ 1J (111 LYI”~IILG 1”, LIIL;
which is determined
I$tp,k)=i +i
from the solution of the equation
I$$ \I
X K,C~,p’)G@‘)G(p’-k)ri,,(p’, W,c@‘,k),
(1% where _Kcln ir the J w)r n’\, _" ---_ lwrn~l _~_.~__. nf _. the _-_ Pnnlnmh -VIA-..-l
intetartinn .l..~.~...~V..
in the spin channel. It can be shown [8, 91 that in the lowest order of perturbation KS@, p’) coincides with the screened Coulomb interaction, i.e. @(p,p’) = V&p - p’), where V,,,(k) = V(k)/#) is the screened . ___ . Couiomb with V(k) being the bare Couiomb and e(k) is the dielectric response function in the random phase ~.nrrw;m~t;~n IDDA\ th.n os..r~a..s4 SL~~I”*U.IuLI”II \&\A‘L,. Tn 111nrlrl;t;r\n LLuUICI”LI, LIICJ~Iti,cIII~UP.w.lsw.xl. ~“U,“‘I,” interaction is practically frequency-independent in the region of the particle-hole excitations (w < vck). In this case we can replace KFtp,p’) = V&p - p’, 0) in equation (15) by the average over the Fermi surface, i.e. K,$@,p’) = (V&p - p’, 0)j = 2V, where V is the screened exchange interaction constant. LT^__. *s ^^ a,, ^- C~illU~K, ^_.^--,a __.^ ^^- ~“~USIK _....l.._r_ ”,,:-11,aI I__.___> I’IVW, WC‘ill, ,aye,cu 2D (L2D) Fermi gas with the isotropic electron spectrum in any of layers, i.e. eF = p2/2m*, the 2D sinple-aarticle -4-m r~--mm-energy. Note that the bare Coulomb interaction V(k) here in an L2D system is given by
V(k) = V&J
sinh (ckll) cash (ck!!j - cos (ck: j
= Vo(W(kj,
(16)
where kll and kZ are the momentum components in the ..,n..,,..A :.. ,X..a,,:-.. ..,.-,.1 /1. \ _= ~__2/..lS ~‘P”C allU 111,La 1‘1GUIIS~U”LL U”LllUll +,. I” :,II) ,I“O[&,,, L71r If&K,, is the bare Coulomb interaction in a pure 2D case with the dielectric constant K of the lattice background and flk) is the form factor of an L2D system. Thus, we have
localized magnetic moments induced by nonmagnetic Zn. In addition. it was found that the localized magnetic moment induced by Zn substitution depends on the carriers (hole) density. For example, for optimally T/C 2* (x = 0.93) doped YBCO the magnetic moment correx 9; d4 Vscr(~-p’)~~~,=,~,,=~,= $(a. sponds to -0.2~~ per Zn, whereas for underdoped - T/T I.. fi L?\ .____I_ AL_ 1___11__> _____._. ..__ C_~._> Am 0 (.X.-.=U.WJ) biUlll.“C UK IUCiillLLXl lIlUllltXll Wkib 1UUllU LO increase up to -0.8 ,.&Bper Zn, where pB is the Bohr magneton [13]. Such a behavior seems to be in agreewhere ment with the dependence of the Curie constant C - l/n, on the carrier density n,. cix The spin density response function of the interacting F(cx,0 = jelectrons in the presence of the scattering on impurities o J( 1 - x2)(X2 + 2aLxcoth3sr + oL*’
I I
0,
(17)
(18)
-with a given impurity poieniiai foiiows from the equation
where ri,,Jp, k) is the impurity vertex part as before and I’:@, k) is the Coulomb vertex part in the spin channel
and the use of kif = lpi1- ~‘111~= 2p$(lcos 4) is made9 ry = P’/Kv,is the interelertmnir -- p ---_ -_____-_________interaction ~QCstant with vF = pFlm* and is related to the dimensionless density parameter t-s by the relation r, = 6 and { = 2pFc with the 2D Fermi momentum pF = e and the interlayer spacing c.
Vol. 104, No. 3
LOCALIZATION IN CUPRATE SUPERCONDUCTORS
It follows from equation (15) that I’S@, k), the Coulomb vertex part, in the case when K&x p’) is repiaced by u, the exchange interaction constant, depends o;c(t~ momentum transfer only, namely .FCI.. I\ 1 I r1 I 11C1L\l . . ..A c,IL\ &..,m krr ‘s\Y,A]=‘s\n/=II [’ T “=\a,, VT1111 A\&, g;l.tiu “J equation (12). Thus, taking into account this F:(k) in equation (14) the equation for the spin density response function is derived as _-.,I.
xs(k 0) =
+“;;;,
1
n
.\
w) =
-NF -
iw
1.2
"0" + Ds(w)kp
159
for a given impurity potential we introduce the impurity localized propagator which can, in this case, be written as
uo(s> *
1 0 + i6 sign w’
( >
d&(q, wj = 2 e(q*w) where 6i&q,
(21)
o) is the screened impurity propagator,
i?(a)-I = ni!v(a)12 is the “bare” electron-impurity inter. ”
__
.
action constant as before. Substituting equation (21) into equation (1) and using Ward’s identity we can get the perturbation series for the self-energy beyond the first Born approximation. In terms of Ward’s identity the __.._._--I-_ ior c-. AL. I-- -1 .~.__,..J. ._ ,-1, *a.- __~~_ .L~~Af~ .~ expression me Kernel in~mumg du me contnmmons up to the second order can thus be derived as - o81 - GB’ Ki,(p*p’j =Klimp(P,P’j + &&(p,p’j
where D,(w) = D(o) - UNFDO is the effective spin diffusion coefficient with D(w) given by equation (13). It follows from equation (19) that the spin density ran..-..,.c....,-.+:-.. ,,..r..:..c (1 ,, UlllUDl”‘, ,I:4X...:-.. Y”lG ..,.,a .WLII&,II ..k..nl.G”I,G_ ,.,-._A -c8 --xmpu~r k?! In n’G?P’ (n lM?? In n’119 1331 I--4rmpvyr I--5lmpvvr \--I IG~tJ”11Jz; I”II~LI”IIb,“LIIalIIJ sponds to the diffusive paramagnon. In addition, where k$,(p,p’) = Bi,(p - p’)6,,, with B+(k) given equation (19) leads to the Curie-Weiss type spin L.. __.._r:_- (LIJ 1-1, iUlU -_J uy eyua11u11 susceptibility (w - T) as follows n
k;&@,
c
xs =xp$j=
P’) =
(20)
T-T;’
where xp = p;NF is the Pauli susceptibility as before and C = xps/(2&) is the Curie-Weiss constant and with xps = &Nr/[l - UNF] being the paramagnetic susceptibility including the Stoner enhancement factor and ?-* rr..ln\ll?, z\ ;0 .**I&L1 . ..tr. c -- “~~“,,\L.C,?,,,, 13 &.P ,,,C r..r;a L”11ti tnm..&rot*.r~ &L.+kILILUIC x(O) = NFI[ 1 - UNF]. It should be noticed that equation (20j is actually valid for a L2D system. Thus, the interaction in the system of localized electrons can lead to the appearance of fetromagnetism. Such an effect of localization can also lead to the coupling of quasiparticles with the diffusive paramagnons and therefore -l__~~__.-~l-. f_____-_Al__ _..__~__-.1_1_ __^^ --A esselluiilly -___-r:_,,_. slgnlll~““uy IIIGIC~ IlIt:yuiis1paruc1t: llliixi iulU decrease T,, the superconducting transition temperature. The consideration __ an __. electron ______.. with _-__-_-_--__-.. of -_ the interaction of the diffusive paramagnon is out of the scope of the present paper. Now, consider the model with charged impurities interstitially doped between layers. In this case the impurity potentiai screened by eiectrons is determined by the equation u(q) = v(q)le(q,O), where v(q) is the hwta imnnritv “WV ““y”L”J
nntmwial y”“‘~.......
nf Y”IL.W rnm~ I.&.... kid (a chnrt_mnoc= “L \.A “...,L. ..e.‘b.. nr Y.
Coulomb-type potential) and e(q, co) is the dielectric response function: E(Q,w) = 1 - V(q)j&(q, o), where V(q) is the matrix element of the bare Coulomb interaction and z,(q, w) is the irreducible charge density response function which includes the vertex corrections due to the impurity scattering and Coulomb interaction. hTnnlnnr;nn 1.C~.c1~1111~
rhn U1b
~ra..+~v .b-I,“.%
,.a-ot;nnr C”ll~W~l”IIU
tn b”
th.P . ..w
&w.rlnrihl~ IanIVVI.“II
response, i.e. assuming ji,(q,u) = xo(q,o), the bare particle-hole bubble (the Lindhard function) corresponds to the well-known RPA. In analogy to the above model
_2i
(224 I d4q ~Di,@-qjV,,,@-qjG,(qjG,,@’
dc \L?r,
+P-4)~ (22b)
and k&&p, p’) and ~~&p,p’) are similar to equations (3a) and (3b) respectively, except that in equations (3a) _-J (JU) ,?I_\ Uiw(K) n IL\ SUUlUU _I__..,~ L_ ___,____IL_. iF ,I.\ ,L_ i(llU Ut: l’SylilGCU Uy ui,pp(K), UK renormalized
impurity propagator. In equation (22)
V.__tkl is the SCKYXCX! Co&& .x.r\~-,~-
interaction
in the RP.4
as before. The diagrams for these interactions are given in Fig. 2. It is seen from Fig. 2 that the first diagram corresponds to the first order screened interaction, whereas two others appear as a result of the Coulomb interaction. Tlne second diagram corresponds to the second order direct screened interaction (so called the Uatirpp rliaoram\ The diaoram fnr mmntinn 133hl -I ic yeq ---..II --..e.---,. a------ -1 ______ \---, important because it represents the interaction in the Hartree-“Cooper” channel [ll] (this diagram in fact differs from the Cooperon’s one since it has an opposite sign with respect to the Cooperon diagram and its initial and final momenta in scattering are t‘he samej. For the model with a given interaction there are no such ,-i;~i,~nm~Thn. this cmvwwi mtvbl whirh .arm,,ntr fnr IIIUY) Lll” Y.,““.l.. I..““_. I.... v.. U”“V”...” 1-1 V’Upum’.U. the interaction between electrons in the RPA includes more contributions than the first for a given impurity potential. Substituting equation (22) into the equation for
LOCALIZATION IN CUPRATE SUPERCONDUCTORS
160
Hame diagrams
p0 po *’
p’d
p0
/~“‘,p~d , *
2 _I_ PO
PO pa *.
+
i
u P’d
‘.
-. k fl’ ,“, ‘.
PC
’
PO
P’ 6 .* +...
,’
p’ d PO Cooperon diagram
Fig. 2. Diagram for &$.(p, p’), the kernel of interaction, up to the second order. The wavy lines stand for the screened Coulomb interaction, X corresponds to an impurity and thick dashed lines stand for a screened impurity propagator. the self-energy
where K&,&p’) = k&,@.p’) + K$p,p’) is the spinsymmetric kernel of interaction with K$,(p, p’) given by equation (22), we can also get a series for the singleparticle self-energy. The comparison to the model with a given impurity potential shows that the diagrams which appear as a result of the screened Coulomb interaction can compensate the diagrams for the backward scattering. It is meant that the contributions coming from the Coulomb interactions can significantly reduce the backward scattering effect if the screened Coulomb interaction is quite strong. Such a compensation effect will exist in the higher orders of perturbations as can be shown. In this case the diagram of the first-order perturbation (the first Born approximation) is sufficient for consideration of the disorder effect. The ordinary diffusion ladder is important in this case only and the irreducible density response function is given by equation (12) with D(w) = Du, the frequency-independent diffusion coefficient as before. The consideration of the problem in this case is given in [ 111. In the case of the high density limit the screening effect is very strong and the contribution of diagrams shown in Fig. 2 does not play an important role in superconductivity and localization effect. In contrast,
Vol. 104, No. 3
the high-T, superconductors (cuprate compounds) correspond to the case of a dilute Fermi gas and therefore the screening effect in cuprates is weak. In this case the Coulomb interaction between carriers can not be ignored and the contribution of diagrams shown in the Fig. 2 plays a significant role in the localization issue and superconductivity. Thus, our results at least can qualitatively explain both the unusual strong localization and suppression of superconductivity in cuprates induced by the substitution of Zn and Ni and much weaker effect of the substitution of Fe, Co, Al, Ga, etc. Acknowledgements-This work is supported by DOE with the grant No. DE-FC48-95RB10542 through TCSUH. REFERENCES 1. Lee, P.A. and Ramakrishnan, T.V., Rev. Modern Phys., 57, 1985, 287.
2. Belitz, D. and Kirkpatrick, T-R., Rev. Modern Phys., 66, 1994, 261. 3. Greene, L.H. and Bagley, B.G., “Oxygen Stoichiometric Effects and Related Atomic Substitutions in the High-T, Cuprates”, ibid. Vol. II, pp. 509-569, 1990. 4. Abrahams, E., Anderson, P.W., Licciardello, D.C. and Ramakrishnan, T.V., Phys. Rev. Left., 42,1979, 673.
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