Phenomenological two-carrier model for oxide superconductors

P~SICA

Physica C 192 (1992)315-327 North-Holland

IIIII I

Phenomenological two-carrier model for oxide superconductors Shigenori Tanaka Advanced Research Laboratory. Research and Development Center, Toshiba Corporation. 1 Komukai Toshiba-cho, Saiwai-ku. Kawasaki 210, Japan Received 29 November 1991

A phenomenological two-carrier model for high-T¢ oxide superconductors is proposed on the basis of experimental results for optical conductivity. A dielectric response function is formulated for the system consisting of Drude and hopping carriers, in which ~elaxation and strong-correlation effects are taken into account. Given an appropriate choices for the values of effective masses, carrier concentrations and relaxation times, this dielectric function can well reproduce the experimental results for optical conductivity in La2_xSrxCuO4 and YBaaCu3OT_6. The superconducting transition temperature T¢ is then estimated, regarding the hopping compor, ent as a dielectric medium producing an attractive pairing force between the Drude carriers. This model can consistently account for a number of experimental facts, such as (i) optical anomalies in the mid-infrared region, (ii) high transition temperature, (iii) correlations between Tc and a number of physical parameters, (iv) the significance of low dimensionality.

1. Introduction

A lot of experimental results on copper-oxide superconductors has been gathered, suggesting [ 1-4] that the high-temperature superconductivity is due to the Cooper pair formed by the doped carriers in the two-dimensional CuO: planes. The problem then is to determine what the o~igin is for the attractive pairing force between the carriers. It is natural to adopt the electron-phonon interaction as a promising candidate to produce the force on the analogy of conventional superconductors. Extensive investigations, however, seem to have shown [ 5-7 ] that it is difficult to achieve a high transition temperature T~ beyond the liquid-nitrogen temperature (77 K) by the phonon-mediated force alone. Extending the scenario which relies exclusively on the electron-phonon interaction, the idea that the high Tc may be associated with charge fluctuations other than the phonon has been proposed [ 8,9]. Examples include the viewpoints that the plasmon [ 1012 ] or the exciton [ 13-15 ] may play an important role in realizing high Tc superconductivity. It is remarkable in this connection that strong density fluctuations in the mid-infrared region in optical reflectivity measurements [ 1,16,17 ] and Raman scattering experiments [ 18,19 ] have been observed. There ex-

ists a theoretical work [ 14,15 ] trying to explain the high T¢ in the copper oxides on the basis of these density fluctuations. Experimental results for optical conductivity [ 1,16,17 ] in the superconducting copper oxides suggest that there exist at least two species of carriers contributing to the electrical conduction in the CuO2 planes over the infrared and the lower frequency regimes in the light of the conductivity sum rule [ 1,20]. This observation has led the present author to a phenomenological calculation of the optical conductivity based on a two-carrier model [ 21 ], which could satisfactorily reproduce the experimental results including the mid-infrared anomaly. In that work, the effect that the carriers are scattered by external excitations has been taken into account through frequency-dependent relaxation times; the physical implication of the effect has also been discussed. Developing the previous scheme, this paper presents a quantitative description of the copper-oxide superconductors on the basis of a more sophisticated two-carrier model. A dielectric response function for the two-component charged particle system is constructed, and physical properties such as the optical responses and the transition temperature are discussed in this framework. Special attention is paid in this model to the inclusion of the relaxation effect of

0921-4534/92/$05.00 © 1992 Elsevier Science Publishers B.V. All lights reserved.

3! 6

S. Tanaka / Phenomenoiogical two-carriermodel

the motion of the carders and the strong-correlation effect between the carders. In the next section, a theoretical framework to calculate the dielectric response function for the two-dimensional two-component charged partide system is presented; a scheme to estimate the superconducting transition temperature T~ based on the dielectric function is also formulated. Section 3 shows the results for the optical conductivity and the transition temperature in La2_~SrxCuO4 and YBa2CusOT_,s and compares them with experimental results. In section 4, a number of remarks on the results are presented and the validity of the present model for describing the physical properties of the oxide superconductors is discussed. Section 5 condudes with a summary and points out future problems.

polarizabilities and local-field corrections, respectively. ZuZ.v(k) mean interpaxticle potentials, where we adopt

v(k)=

2~e 2

(4)

assuming that the charged carriers are confined to the two-dimensional CuO2 planes [ 1 ]. In eqs. (2) and (3), the arguments regarding wavenumber k and frequency to have been omitted for brevity. As shown in the previous study [21 ], the inclusion of the frequency-dependent relaxation times for the carders is essential to account for the anomalous density fluctuations observed in the mid-infrared region in optical conductivity and Raman scattering amplitude. We therefore express the modified freeparticle polarizabilities as [23,24]

co)

2. Formalism Let us henceforth consider a two-component charged Fermi system with the effective masses m~,, the electric charges Z~,e and the number densities n, (lt= 1, 2). We refer to the component for reasons which will be mentioned later. The Drude component dominates the magnitude of optical conductivity in the low-frequency limit, and the hopping component dominates the magnitude in the midinfrared regime. Assuming a spatially homogeneous system, the longitudinal dielectric response function for the system is given by [ 22 ] ~/oo

e(k, co) - 1 +v(k) i.t,t, ~, Z~,Z,L,~(k , co),

( 1)

X,, --21°~[ 1 - Z~v~.~°~( I -G22) ]/D,

[co+Jr.(co) ];C~°'(k, co+ ~.(to) ) = w+iZu(co)X~°~ (k, co+ iZ.(co) )/XJ°' ) (k, 0 ) '

(5) taking account of the relaxation effects. Here Z~,(co) are frequency-dependent damping functions, whose wavenumber dependence is ignored; Z~°} (k, z) represent usual free-particle polarizabilities given by

f.(q) - f . ( lg+kl ) hz+e~,(q)-eu( Iq - / t I ) '

X~u°~( k, 2) ~ 2

whereet,(q)=h2q 2 /2m,,*

andfu(q ) is the Fermi distribution function. Note that 2~°} (k, O)=;(~o)(k, 0 )

in eq. (5). It is hereafter assumed that the charged particle system is of complete Fermi degeneracy. For the twodimensional system, we obtain [25] from eq. (6),

X22=2~°)[ 1 - Z ~ w 2 1 ° ~ ( 1 - G , t ) ] / D ,

-Z~v(k) R e z : ) ( k , z ) =

X~2= Z, Z2 v2~°~2:~°~ ( 1 - G, 2 ) / D , Z21 = Z , Z2/2"~ ~l°')~°l ( 1 - G 2 , ) / D ,

{

(2)

• [ sgn(v_.) ~ x v l - 2x/~x~

)-

"3

X v2

D = [ I - - , Z~2z ,.-':.~o ~/~ I - G , , ) ]

(6)

~.

-"

1+

3:;

v-_.

2

X~,

X [I - Z~ ~7~°' ( l -G22) ] 7272,

2~,0~,(0)

--~1"-'2 ~ X!.~2 ( I - G I 2 ) ( 1 - G 2 1 )

•

(3)

H e r e e~, £~°~(k, co) a n d Gu~(k) refer to a high-freq u e n c y dielectric constant, modified free-particle

x5

-~

2v/~x ~

[( +

)2 4 C PT+~

2

X,~

x2 Tt2),/21

2

X#

,

(7)

S. Tanaka / Phenomenological two-carriermodel

- Z 2 v ( k ) Imz~°~(k,z) -~x. I t

+ -

[(

1

1 + X-~u-- t~_"

1 + x~

high-frequency (co/k--,oo) limit for the longitudinal dielectric function ( 1 ), we find

l + ¢__~.2- v 2 x~ -"

.

X-'~u

2 2 Zun~k

¢(k, co).¢~, - , 1 - v ( k ) 1~, m~,w[co+i27u(os) ] , ,,all

(12)

~

where n, refers to the two-dimensional carrier concentrations in the CuO2 planes. The form (12) is common also to the transverse dielectric response function. We thus obtain an expression for the optical conductivity as

- z'z+~

[( For a complex frequency z--z' +iz". Here we have set x,=k/2k~,, ~,+_u=x,, + ~lu/xu, qu=hz' / 4E~u and

~,=hz"/4Ev,,. kv,=(2nn~,),/a, Evu_ h _ 2kv,/2m~,2 . r,*u = au/a~ u, au= (xnu) - ~/2 and a~ = 2 * 2 2 h ¢ J m u Z u e refer to the Fermi wavenumbers, the Fermi energies, the effective Coulomb coupling constants, the average interparticle spacings and the effective Bohr radii. The frequency-dependent relaxation rates Z'u(to ) are expressed as

Z',,(~) =.~,,(O) + [Z',,(oo) -27,(0) l [-icoE,(co) ]

(9)

in terms of memory functions Fu(co ) [26,27]. Here 27~,(0) = l/%u and Zu(oo)= 1/z~. are the low- and high-frequency limits of the relaxation rates, respectively. In order to satisfy the requirements of causality, the memory functions are expressed as 7

Fu(oJ) = J dtf'u(t) exp(io~t),

(10)

o

where/~u(t=0) = 1 and/~u(t= co) =0. We adopt in the present study Gaussian forms [ 26,27 ]: P A t ) =exp( - t ~/ r ;", )

317

(11 )

for the memory functions; z¢. imply characteristic relaxation times. Among the three types of characteristic times which appeared abe.,e, r0, and T~u represent the lifetimes of quasi-particles, due to the scattering by external fields. On the other hand. rcu are regarded as those associated with the motions of the scatterers. There are no constraints on the ordering in magnitude among those three relaxation times. When we consider the long-wavelength (k-*0) and

a(co)= lim W [)2 ~] k lkm ~¢(k'o

(13)

for the electric field parallel to the CuO2 planes. Here l refers to the number density of the CuO2 planes along the c-axis. Equations ( 12 ) and (13) show that the optical conductivity is given as a sum of the contributions of the two components. We shall choose the values of number densities nu, effective masses rtlt~ * and relaxation times Tou, %u, zcu so as to reproduce the experimental results for the optical conductivity, assuming that the forms of the damping functions -ru(to ) are common between the longitudinal and transverse dielectric functions. The local-field corrections G~,~(k) in eqs. ( 2 ) and (3) represent the strong interparticle correlations beyond the mean-field approximation [ 22 ]. As will be shown in the next section, the effective Coulomb coupling constants r ~ for oxide superconductors are usually larger than unity because of the lowness in the carrier densities. This means that the system is in strong-coupling regime, which makes the inclusion of the strong-correlation effects essential in estimating the superconducting transition temperature. We here evaluate the local-field corrections as follows. We first adopt the approximations that G~lG,.2-GI2G2~=O and GII+G2z-G~,.-G,.,=O. They are appropriate in the cases that the two components have nearly the same values of carder concentrations, electric charges and effective masses, and that the Coulomb coupling in the system is strong. Equations ( 1 )- ( 3 ) are then reduced to

e(k, co) = ]----Z2 t~°~ ( 1 - G ~ ) - Z.2~ ° ' ( 1 -Gzz)"

(14)

318

S. Tanaka / Phenomenologicaltwo-carriermodel

We next approximate the local-field corrections in eq. (14) by those in the one-component system: Gr,~,(k)=Gj,(k ) .

(15)

Noting that the wavenumbers are confined to the region less than 2kw in the following discussion, Gu(k) are further approximated by their long-wavelength limits: k Gu(k) =Tu kFu "

~ u _ l - v / 2 r:~yu

(17)

Kp

with the aid of the compressibility sum rule [22,28,291. Here Xou refers to the compressibilities in the noninteracting systems. The compressibilities are, on the other hand, expressed thermodynamically as 8

1.04(1+2) ] T o - (to> 1.2 exp -- 2--~1+---~.622)J

~-2-;..2 E(r:.) drsu r*u ors~/

(18)

where E(r:u ) is the ground-state energy of the onecomponent system in rydberg. Given an analytic expression for E(r:~, ), we thus obtain the values of 7~. We here adopt the expression given by Tanatar and Ceperley [ 29 ]:

instead of solving the Eliashberg equation in order to clarify the physical meaning of the parameters encountered. In eq. (21), ( t o ) , ~. and #* refer to a characteristic frequency of the pairing-mediation excitations, a coupling constant between the excitations and the superconducting carriers, and an effective Coulomb repulsion between the superconducting carders, respectively [8], whose definitions will be specified in the following. It should be noted that eq. ( 2 1 ) gives an accurate solution to the Eliashberg equation for 2~< 1.5 [30]. We assume an isotropic s-wave [4,31 ] for the superconductivity pairing. The coupling constant 2 and the characteristic frequency
E ( r'~u)

1 2~.2 rs~

8v/2 +Ec(r*u) 3rcr~.

(19)

-~7 tanh

K(~,~')A(~')

(22)

--EFI

in the case of the present model. Here A(~) is a gap function and the kernel is given by

Ec(r:u) = -0.3568

(21)

(16)

The coefficients 7u in eq. (16) are related with the compressibilities r , as

x,~

the Drude (/z= 1 ) component is regarded as a collection of carders responsible for the superconductivity and that the hopping (p = 2) component is regarded as that producing a pairing force between the Drude carders instead of the phonons in conventional superconductors. The estimate of Tc is then made by the Allen-Dynes formula [ 30 ],

1 + 1.1300R

1 + 1.1300R+O.9052R2+O.4165R 3' (20)

with R = (r*u) 1/2, which is based on the result of the quantum Monte Carlo simulation for the two-dimensional electron gas. The dielectric response function which will be used in the following has thus been determined. We next proceed to an estimate of the superconducting transition temperature Tc based on the dielectric response function. It will hereafter be assumed that ka=fi= 1 and Z 2 = 1. In the present work, we set up a modeling such that

K(~, ~') =/t(~, ~') - 2 ; dto t o v ( ~ '+t lo~)' l "

(23)

o

It should be noted that there exists arbitrariness in the way the kernel K(~, ~') can be divided into the sum of two terms. In this work we make the choice that all the contributions from the Drude (p= 1 ) component are to be pushed into the Coulomb repulsion term/t(~, ~') and that the remaining contributions are to be described in terms of v(~, ~', to). A difficult problem, concerning what pairing potential should be used in an indistinguishable many-body Fermi system [ 12,33-35 ], is thus avoided.

S. Tanaka/Phenomenological two-carriermodel Let us introduce spectral functions as p(k, to)=

1 Im coo - ~ e(k, ¢o)'

pt(k, t o ) = -

1 Im

~oo ~l(k, to)'

p2(k, to) =p(k, to) - p i ( k , to).

(24)

(25) (26)

The dielectric response function et (k, to) found in eq. (25) is obtained by neglecting the contributions of the hopping (/z=2) component in eq. (14) for the two-component system. We then find v(~, ~',to) =N(~')v( IP-q)P2( IP--ql, to),

319

1/x/1 - ( 1 - k 2 / 2 k 2 1 )2 has appeared owing to the fact that the system is two-dimensional; it works to enhance the value of ;t by emphasizing the contributions arising from the neighborhood of 2kvt and from the long-wavelength regime in comparison with the three-dimensional case [ 32 ]. This may imply the significance of two-dimensionality for achieving a high To. The characteristic frequency (to) of the pairingmediation excitations is estimated as [ 32 ] 2kFt

,o

Em

(27)

dto [" dk 1 k3o to Jo 2 k r t x / l - ( l ' k 2 / 2 k ~ t ) 2

°

x In ~-~ o(k, co)

and obtain an expression for the coupling constant

]

as oo

EFt

2=2 ~ dto to v(0, 0, to)

(28)

0

through estimation on the Fermi surface. In eq. (27), we have set ~ = h 2 p 2 / 2 m T - E F l , ~'=h2q2/2mT EFt and N(~')=d2q/(2n)2/d~'=rnT/2nh2; the overline in eq. (27) implies an angular averaging between vectors p and q. Carrying out this angular averaging, we find x/~ r*, 2= - ~:ZkF,

0

dto -~

dk x/1 - ( l -k"/2k2F, )2 ~oo

× Im[e (~,~to)

et (k-,,to)]

(29,

Witi~ the aid of the Kramers-K.ronig relation, .'q. (29) reduces to 2krt

r~*t ! dk 2=- x/~xkF ' ~/l_(l_k2/2k2Ft) 2 6oo

× c(k,,0)

Eoo

]

e,(k, 0) "

2kFI

,

×

-~-

0

2kF, x/l - (l - k 2 1 2 k 2,

] to)

-- !

.

(31) Here the upper limit EFt in the frequency integrations has been derived from the lower limit in the frequency integration in eq. (22). The reason why the weight function p(k, to) has been used instead of p2(k, to) is that the latter is not positive definite. The theoretical estimate of the Coulomb pseudopotential/t* for a one-component charged Fermi system is by itself a formidable task and is still controversial [12,33,35-37]. The value of a* is therefore chosen between 0 and 0.1 [351 as a parameter in the present study. The value of Tc obtained by eq. (21 ) with the values of (to), z,/~* estimated as above provide a theoretical estimate of superconducting transition temperature in the present model. The effect of the renormalization factor missing in the KMK gap (eq. (22)) is taken into account in the framework of the Allen-Dynes formula, eq. (21 ).

(30)

Two remarks are maie about eq. (30) for coupling constant ,t. First, the value of 2 is determined by the static (to=0) dielectric functions and does not depend on the values of the dielectric functions at finite frequencies; this means that lhe coupling constant is not related with the relaxation effects in the present model. Second, lhe factor

3. Results In this section the results of the calculations mainly for La2_xSr~CuO4 and supplementarily for YBa2Cu3Ov_a will be shown. The reason why the Laz_ ~SrxCuO4 system has bec, chosen is that the estimation of the carrier-concentration dependence ot

320

S. Tanaka/Phenomenological two.carrier model

optical spectra has been accurately investigated [ 1 ] over a wide range of doping concentrations. We henceforth consider only holes as the carriers (i.e.,

z,,=l). First, some comments are made on the magnitudes of the parameters introduced in the preceding section. The value of the high-frequency dielectric constant coo is supposed to lie approximately between 3 and 5 in the copper-oxide superconductors [ 20 ]. Therefore, we hereafter fix this value as coo= 4.0 for simplicity, unless otherwise specified. The values for the effective masses m~, take account of the band effects and of the scattering effects due to the phonon, the magnon and other excitations. We choose this parameter in the range of m ~ , / m = 1-5 [38-41 ] in the present study, where m is the bare mass of electron. It is noted in this connection that the effective mass m.*, for the hopping component is difficult to determine through the de Haas-van Alphen experiments [39,40] because of the shortness of the mean free path. The carrier concentrations n j, in the two-dimensional CUP., planes are estimated as follows. On the basis of the crystal structure data for La,.CuOa [ 42 ], the three-dimensional hole concentration in La2_.,Sr,CuO4 is 1.041X lO'-'-:c cm -3, assuming x holes per unit formula. When we further assume that all the holes homogeneously enter into the CuP2 planes, we obtain a two-dimensional total carrier concentration, n = n t + n2 = 6.882 × 10 ~4x cm-", using the density of the CuOz planes, != 1.513× 107 c m - i . The values of nt and n., are then determined if the ratio of the Drude component, p = n t / n , is specified: this ratio is supposed to be roughly p ~ 0.5 for the compositions showing the superconductivity [ 1 ]. The estimation of the carrier concentrations in YBa.,Cu3OT_~ is, on the other hand, not so unambiguous [43]. Assuming that one hole per YBa2Cu307 unit enters into the C u P , planes, we find 1= 1.713× 107 cm -t and n = 3.369× 10 ~4cm--" based on the crystal structure data [44]. The optical conductivity given by eqs. (12) and (13) sensitively depends on the values of the relaxation times [ 21 ]. The quasi-particle relaxation times r01 and ro~l for the Drude component dominating the low-frequency behavior of the optical conductivity represent the lifetime of the free carriers having a well-defined Fermi surface; they are roughly esti-

mated so that hlzol ,.-hlroo, ~ 2 - S k B T [20 ]. The energy h/rct characterizes the motion of the excitations which scatter the free carriers; it is supposed to fall somewhere in the range of 0.01-0.5 eV since the excitations may be substantiated by the phonon, the magnon or the hopping component. The relaxation rates h/to: and ~//zoo2 for the hopping component dominating the mid-infrared behaviors o f the optical conductivity, on the other hand, are roughly of the order of the intersite hopping energies in the CuP: planes [ 21 ]; their magnitudes are thus estimated to be approximately 0.1-1 eV [45-47] and are expected to decrease with the carrier doping. The lattice system a n d / o r the localized spin system are to be considered to interact with this hopping component; the characteristic energy h/rc~, is thus supposed to fall somewhere in the range of 0.01-0.5 eV. Figures 1-3 show the results of the calculation of the optical conductivity for La2_.,Sr,.CuO4 (x=0.1, 0.15, 0.2 ). "1he values for the carrier concentrations, the effective masses and the relaxation times adopted in the calculation are listed in table 1. If we add the contribution from the interband transition eorrespoading to the charge-transfer gap [ 1,17 ] which is neg.ected in the present model, the calculational resuits satisfactorily reproduce the experimental resuits [ 1 ] obtained for La2_,Sr,CuO4 single crystals

1.0

Lal. s Sro, lCuO 4 ( T =300K )

'E 0

r c~ 0 ' 8

L

0 "" >- 0.6 Ito 0.4 o 7" o 0.2 to 0

../ \~ "...',,, 0

2000 4000 6000 8000 F R E Q U E N C Y (cm -1 )

Fig. I. Calculated result for optical conductivity or(co) tbr La~oSroICuO4at T=300 K. The dashedcurve refers to the contribution from the Drude (/Jr 1) component;the dotted curve, that from the hopping (tt=2) component;the solid curve, the sum of both contributions.

S. Tanaka / Phenomenological two-carrier model

,E 1.2

T

Lo,.~S~o.,~C~O~

!~

% 1.0

( T "300 K )

e,,w

)I> I-L)

0.8 0.6

:::) Z 0 0

0.4 - / \ -

..,: \ -

0.2 0

Fig.

2.

0

2000 4000 6000 8000 FREQUENCY (cm "1)

Calculated

result

for

optical

conductivity for

Lat.ssSr0.tsCuO~.Otherwisethe same as in fig. 1.

1.8

1.6 ~ 1.4 ~

0

"o, 8 s r o , c - o . ( T 30OK)

1.2 1.0 0.8 0.6 ./~.~i}~ 0.4 0.2 .. ,,,, 0

2000 4000

321

is relatively emphasized. Figure 5, on the other hand, corresponds with the result at T= 100 K obtained by Schlesinger et ai. [481, in which the contributions of the two components coalesce and the optical conductivity looks one~'omponentdike. Next, the superconducting transition temperature T¢ has been calculated, using the same parameters as adopted in the calculation of the optical conductivity. It is noted that the relaxation times rot and r®~ of the Drude component for La2_~Sr~CuO4 have been scaled from T= 300 K to T= 30 K with the aid of an empirical relation, Tro,, Troo,=constant [49,50]. Table 2 shows the calculated results of effective Coulomb coupling constants r*,, Fermi energies E~,, characteristic frequency
4. Discussion 6000

8000

FREQUENCY (cm-!) Fig. 3. Calculated result for optical conductivity for La~sSro,,Cu04. Otherwisethe same as in fig. 1.

at room temperature. The experimental results on the optical conductivity for YBa:Cu3OT_~ are somewhat controversial especially with respect to the interpretation of the contribution of the Cu-O chain [ 16,43,48]. It has therefore been tried to reproduce the experimental results of two groups by adopting two kinds of parameters choices for the relaxation times of the hopping component. Figure 4 corresponds to the results at T = 100 K obtained by Thomas et al. [ 16,43 ], in which the peak structure in the mid-infrared regime

In the preceding presentations, we have not clearly substantiated the two components of the carriers assumed. Let us first discuss this issue on the basis of the relaxation times adopted to reproduce the experimental results o f the optical conductivity. It is experimentally suggested [ l, 17 ] that a midgap state is formed between the O 2p band and the Cu 3d upper Hubbard band with the carrier doping and that this state entails a Fermi surface of the cartiers responsible for the superconductivity. It is accordingly natural that the Drude componem introduced above be identified with the carriers in the mid-gap state. It may then be supposed that the molecular orbital associated with this component is predominantly of the oxygen character [ 51 ]. As mentioned in section 3, we may enumerate phonons, magnons and other electronic charge fluctuations [49] as excitations interacting with the Drude com-

,5;. Tanaka/Phenomenological two-carriermodel

322

Table 1 Values of parameters used in the calculations of optical conductivity and superconducting transition temperature, n refers to the twodimensional total carder eom'enlra~o,'; .~ to the ratio of the carder concentration of the Drude (/~= 1) component; m~ to the effective masses for the component #; Zo,, z=,, re, to the relaxation times. System

n (cm -2)

p

mTIm

m~/m

hlrol ( c m - ' )

1

Lat.gSro.iCuO4

6.88 X l0 ta

2 3 5 6

Eat.ssSroasCuO4 Lat.sSro.2CuO4 YBa2CuaOT_a YBa2CuaO7 fictitious

1.03 X I0 t4 1.38 X 10t4 3.37 X l0 ta 3.37 X 10t4 4.00 X 10t4

0.5 0.5 0.5 0.5 0.6 0.7

1.5 i.5 1.5 2.0 2.0 1.0

2.0 2.0 2.0 4.0 4.0 5.0

650 800 900 300 300 1000

No.

h/Too, (cm -t)

htr~t ( c m - ' )

h/to2 ( c m - ' )

til%o: (cm -t)

hlr~: (cm -s )

1 2 3 4 5 6

900 1000 1000 400 400 ~500

500 500 500 500 500 500

2500 2000 1500 4000 2000 4000

1000 1000 800 2000 1000 2000

500 500 500 1000 500 1000

No.

4

A

YBo2CuaOr-~

,-°8

( T=IOOK }

...,. 'E ..o 8

YBoz CuaO-t (T=IO0 K)

o

>_ 6 I>

>- 6 I-

4

F-- 4 L.) D £3

o z

o2i-

(..)

.X

6 .... 2 0 0 0 4 0 0 0

6000 8000 FREQUENCY (crn"1)

Oo

2000

4000

6000

8000

FREQUENCY (cm-I)

Fig. 4. Calculated result for optical conductivity for YBa2Cu30?_a at T= 100 K. The parameters used in the calculation are listed as no. 4 in tables 1 and 2. Otherwise the same as in fig. I.

Fig. 5. Calculated re:~,ultfor optical conductivity for YBa:Cu307 at T= 100 K. The paranaeters used in the calculation are listed as no. 5 in tables 1 and 2. Otherwise the same as in fig. I.

ponent; the relaxation time r¢~ is regarded as a quantity which characterizes the m o t i o n o f the excitations. It is, however, difficult to accurately determine the m a g n i t u d e o f re, from the c o m p a r i s o n between the experimental and calculational results because the calculational values for the optical conductivity do not sensitively depend on the values of re,.. It is therefore a formidable problem to specify which ex-

citation is the most important for scattering the D r u d e c o m p o n e n t . A representative value, h~ r~ = 500 c m - t , has thus been taken for all the cases considered in the present study. The relaxation times associated with the hopping c o m p o n e n t satisfy a relation that ! / r o z > l / r ~ 2 > 1/ re2. Th~s relation represents such a situation that the m o t i o n o f the scatterers cannot afford to follow that

S. Tanaka / Phenomenological two-carriermodel

323

Table 2 Values of parameters associated with the calculation of superconducting transition temperature, r~*~refer to the effective Coulomb coupling constants for the component/t; Eru to the Fermi energies; (to) to the characteristic frequency of the pairing-mediation excitations: 2 to the coupling constant for the superconducting pairing; To(#°) to the superconducting transition temperature with the effective Coulomb repulsion #*; Tc(experiment) to the experimental value of the superconducting transition temperature [ I, 16,48 ]. The highfrequency dielectric constant ~®was chosen to be 4.0 for the systems 1-5 and 3,0 for the system 6. The meaning of the system numbers is the same as in table 1.

No.

r~*,

r~

Evl (cm- I)

1 2 3 4 5 6

6.82 5.57 4.82 3.75 3.75 2.12

9.09 7.42 6.43 9.19 9.19 16.22

443 664 886 1951 1951 5406

E ~ (cm- J) 332 498 664 650 650 463

(co) (K )

,I

174 246 317 607 599 1221

! .427 !.132 0.967 !.519 1.519 2.192

No.

Tc (#*=0) (K)

T, (#*=0.05) (K)

T¢ (/~*=0.1) (K)

T¢ (exp.) (K)

1 2 3 4 5 6

24.7 28.9 31.9 90.2 89.2 223.8

21.8 24.6 26.4 80.2 79. i 205.3

18.9 20.4 21.0 70.0 69.1 186.4

18 27 22 50 90 -

of the carriers [ 21 ], corresponding to a state called an incoherent motion [ 52,53 ] in the context of the t-J model [45 ]. It is observed in this case that a peak structure in the optical conductivity appears in the mid-infrared region and that the frequency-dependent effective mass [20] rapidly decreases [ 1,17 ] accompanied with lm ~2 (co) > 0. It is, however, difficult to specify the excitations which strongly interact with the hopping carders based on the comparison between the calculational and experimental results as in the case of the Drude component. The fact that the relaxation times Zo2 and zoo2satisfy a condition that h/ro2, ~l/'Co~2~EF2 reflects that the motion of the carders is hopping-like. The fact that the mean free path of the hopping carders in LaLssSro ~5CuO4 estimated from %2 and the Fermi velocity vvz=hkvz=m~=l.04× l 0 7 cm/s, is 2.8 A, may imply that the motion corresponds to the intersite hoppings in the CuO2 planes. Thus, the Fermi surface of the hopping component almost loses its sense. When we estimate the magnitude of the pseudogap Eg associated with the transition between the mid-gap state and the Cu 3d upper Hubbard band from the position of the mid-infrared peak in the optical conductivity [17], we find h/ro2, h/zooz>Eg,

suggestmg tbat the motion is excited between the midgap state and the Cu 3d upper Hubbard band. It is further supposed that this excitation is of the chargetransfer type in the sense that it is associated with both the orbitals of Cu and O, and that the molecular orbital of the hopping component is mainly of the copper character and is localized as compared with the Drude component. As seen in fig. 3, it becomes difficult to separate the contributions to the optical conductivity between the two components in the heavily doped region. Under such a situation, the two components coalesce into a one-component Fermi liquid [ i ], invalidating the two-component picture assumed in the present work. The spectral function I m [ - e ~ / ~ ( k , o9)t for LaLssSro tsCuO4 at T = 3 0 K and k/kvt =0.01, 0.05, 2 is depicted in figs. 6-8 in order to specify the most significant excitation determining the characteristic frequency (o9) for the Cooper-pair formation. The peak structure observed in figs. 6 and 7 is essentially ascribed to a two-dimensional plasmon. As compared with the dispersion relation,

324

S. Tanaka I Phenomenological t,~-camer model

2.0 12

Lau~ Sro~eCu04

1.0~

LotS5 Sro.1- CU04 { k-O.O = k F t }

~0

E ""

{k-Zk~'~)

1.5

3 =--~u 8

4

0.5

2 i

500

,000

FREQUENCY

,500

Zooo

3

-

4

""

3

4

LaLSSSro.leCu04 ( k = 0.05 kF1)

|

E 2 I

0

0

500

I 000

500

l 500

t 1000

L

1 500

2000

F R E Q U E N C Y (cm "t )

6 5

0

(cm "! )

Fig. 6. Result for spectral function I m [ - e ~ / f ( k , ~ ) ] for LaLssSro.,sCuO4 at T=30 K and k=0.01kFt.

I--I

0

2000

FREQUENCY (cm-I) Fig. 7. Result for spectral function l m [ - ( = / ~ ( k , ~ ) l for Lal ssSro.tsCuO4 at T= 30 K and k=0.05kr,.

Fig. 8. Result for spectral function I m [ - ¢ ~ / ¢ ( k , (o)] for La,.85Sl'o.i $CuO 4 at T=40 K and k=2krl.

vicinity of k=2kv~, on the other hand, (electronhole) pair excitations dominate the spectrum. The value of the characteristic frequency ( t o ) is thus determined cooperatively by the plasmons, the pair excitations and the mid-infrared fluctuations, and does not exclusively depend on the peak position of the mid-infrared fluctuations in the optical conductivity. The two kinds of results for YBa2Cu307_ a shown in section 3 clearly support this point of view. The present model therefore markedly differs from the one proposed by Tanaka et al. [ 14,15 ]. Taking account of the interlayer interaction between the carders confined in the two-dimensional CuO2 planes, we may expect that the plasmon peak found in figs. 6 and 7 would broaden up to approximately the three-dimensional plasma frequency:

4he (32) "~

Eoo

2/

derived from eq. (12), the peak position somewhat shifts to the lower-frequency side owing to the strongcoupling and relaxation effects of the carriers. It is to be remarked that such an appearance of the x~elldefined plasmon peak is due to the small values of the damping rates, l/ro~ and l / z ~ . For larger damping rates and smaller wavenumbers, the plasmon would become damped and the spectrum would be dominated by the relaxation times [21 ]. In the

Such a contribution to the spectrum from the plasmon band [ 54,55 ] combined with that from the midinfrared fluctuations may account for those strong density fluctuations ranging over the mid-infrared regime with the carrier doping, as observed in the Raman scattering experiments [ ! 8, ! 9 ]. In fact, the estimated value Wp3=4520 cm-~ for La~.ssSro.tsCuOa is consistent with the experimental results [ 18,19 ]. On the other hand, not the two-dimensional but the three-dimensional plasmon alone is expected to

s. Tanaka / Phenomenologicaltwo-carriermodel contribute to the spectral function derived from the optical reflectivity at finite frequencies, since the matrix element with k = 0 is observed. A possibility may then be pointed out that the peak structure of the spectral function observed at to,,,0.8 eV in La2_~SrxCuO4 (0.1 ~
-Ira

1-2 X/~n~-to), to

(34)

= 4~tr(to) '

(35)

it could be actually observed that the reflectivity rapidly decreases and the spectral function shows a maximum when the optical conductivity a(cJ) takes a minimum accompanied with a preceding maximum [151. We find in table 2 that the characteristic frequency (o9) of the pairing-mediation excitations is approximately proportional to the Fermi energy Er~ of the Drude component, as may be seen from eq. (31 ). If we assume that the values of the coupling constant 2 are of the same order of magnitude among a series of copper-oxide superconductors with the optimized Tc, this correlation implies that the superconducting transition temperature T¢ is approximately proportional to the Fermi energy Es, of the superconducting carriers. The present model is thus consistent with the fact discovered through muon-spin relaxation (~tSR) experiments [56]. In the vicinity of the parameter regions chosen for La2_~Sr~CuO4, the characteristic frequency (to) is an increasing function and the coupling constant 2 is a decreasing function with the doping content x or with the total hole concentration n. This fact accounts for the experimental results that the superconducting transition temperature Tc shows a maximum as a function of the hole concentration [57 ]. In the actual calculation, the maximum in T~ is observed at x ~ 0.2. Also, a tendency that the lransition temperature T~ increases both with the ratio of the Drude component, p = n , / n , and with the mass ratio, m~/m'~ has been found. This implies that for the enhance-

325

ment of Tc it is favorable on the one hand to increase the Fermi energy Er, of the Drude component and on the other hand to keep the Coulomb coupling constant r*2 of the hopping component large. It should be remarked in this connection that we find r~*~,> 2.03 for all the cases listed in tables 1 and 2. Under these circumstances, the inverse of the compressibility and the long-wavelength limit of the inverse of the static dielectric function in the one-component system become negative [ 29 ], favoring the occurrence of the superconductivity [ 32,58,591. In addition, the tendency that T¢ increases with the value of n~/n2 may correspond to the empirical law [60,61 ] that T¢ becomes higher as the crystal structure sustains more and more holes in the oxygen orbitals relative to the copper orbitals in the p-type superconductors. We may thus enumerate a number of conditions to enhance the value of Tc as: (i) an appropriate carrier concentration n in the CuP2 planes; (ii) a high ratio of the carrier concentration of the Drude component, n,/n; (iii) a high mass ratio between the hopping and Drude corrponents, m'~/rnT; (iv) a small value of the high-frequency dielectric constant e~ to enhance the Coulocob coupling constant. A fictitious system has thus been assumed in tables [ and 2 so that it optimizes the above conditions, where e~ = 3.0 is assumed only in this case. We then find that the value of Tc in this system may exceed 180 K. Additional conditions would be required in order to realize such high-To materials like this, including (v) appropriate degeneracy and hybridization of the energy bands; (vi) nearness to the metal-insulator transition, to sustain a low-density two-component system with the hopping component. It may also be necessary that the system should possess (vii) low dimensionality, to enhance the value of the coupling constant 2. In connection with the last condition, it is interesting to make comparative studies between the cuprate superconductors and the bismuthate superconductors such as Ba~_.~KxBiO3 (To~30 K) which shows analogous behavior of the optical conductivity in the mid-infrared regime [62]. The model adopted in the present study is so idealized that it extracts only the essential contributions to describe the physical properties of the oxide superconductors. In order to carry out a more complete comparison on the optical and the super-

326

S. Tanaka/Phenomenological two-carriermodel

conducting properties between theory and experiment, i*. is therefore necessary to include the effects brought about by the phonons, the magnons, the highfrequency excitons and other excitations [810,12,58,59]. Nevertheless, it has been found that this simple model may account for a considerable part of the experimental results for the optical spectra and the superconducting transition temperature. The calculated values of the characteristic frequency ( t o ) and the coupling constant it are also consistent with the experimental results for the superconducting gap, the isotopy effect coefficient and the electron-phonon coupling constant [ 9 ]. The good agreement between the calculational and experimental values of T¢, however, may be somewhat accidental, leaving more accurate analyses on the electron-correlation effects [ 11,12,33,35,37,63] as future problems. It may be possible that the two-component model proposed in the present study would also be applicable to wider varieties of exotic superconductors [56] such as the bismuthate superconductors. The model would then afford a guide for the investigation of the possibility that makes nonsuperconducting materials superconducting with the aid of the experimental means such as optical absorption, Raman scattering, photoemission and laSR. The optical studies on layered 3d transition-metal oxides [64,65 ] are interesting in this sense.

component and the pairing-mediation component, respectively. The proposed model may, in spite of its simple structure, provide a consisteat account for a number of experimental facts, such as: (1) the anomalous structure of the optical conductivity in the mid-infrared region; (2) the high value of the superconducting transition temperature To; (3) the maximum in T¢ as a function of the carrier concentration; (4) the correlation between T¢ and the Fermi energy of the superconducting carders; (5) the correlation between Tc and the crystal (or band) structure; (6) the significance of the low dimensionality of the system. Therefore, this two-component model could be a promising candidate for describing the physical properties of copper-oxide superconductors. The following problems still remains unsolved and are to be studied in the future, to (a) microscopically elucidate the origin of the two-component carders, (b) develop more accurate many-body analyses on the estimate of the transition temperature, (c) consider the interlayer coupling between the twodimensional carders, (d) takes account of the contributions arising from the excitations neglected in the present analysis, in order to further investigate the high-T~ oxide superconductivity on the basis of the present model.

Acknowledgement 5. Conclusion In the present study, we have investigated the extent to which the optical and superconducting properties of the oxide superconductors can be described in terms of a two-carrier ~nodel consisting of the Drude and hopping components. A two-component dielectric response function has been formulated, in which the relaxation and strong-correlation effects are appropriately taken into account; the unknown parameters therein have been so determined as ~.o reproduce the experimental results of the optical conductivity for La2_~SrxCuO, and YBa2CU3OT_6. We have then made an estimate of the superconducting transition temperature based on the dielectric function, regarding the Drude component and the hopping ccr~',ponent as the superconducting

The authors thank Noburu Fukushima for continuing discussions on this and related topics.

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