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Dynamical conductivity of high-Tc superconductors
PhysicaC 165 (1990) l-7 North-Holland
DYNAMICAL
CONDUCTIVITY
K.F. RENK, H. ESCHRIG
OF HIGH-T, SUPERCONDUCTORS
*, U. HOFMANN,
J. KELLER,
J. SCHUTZMANN
and W. OSE
Fakultiit ftir Physik, Universitiit Regensburg, 8400 Regensburg, Fed. Rep. Germany Received 29 August 1989 Revised manuscript received 30 October 1989
We present an analysis of the far- and mid-infrared conductivity of YBa2Cu30,_, We suppose that two types of low-lying electronic excitations in high-T, superconductors are present, namely mobile quasi holes in the Cu02 planes and almost localized doping electron states in the CuOp chains. We explain the far-infrared conductivity at low temperature (T-x T,) with a two-band model derived from the two-layer structure of YBaaCusO,_, This model leads to two gap values, well above and below the BCS
1. Introduction On the way towards an understanding of high-T, superconductivity, a crucial task is the experimental determination and the theoretical interpretation of the excitation spectra in the normal and superconducting states. Strong correlation effects are generally believed to be present in the electronic excitation spectra so that ordinary bandstructure theory is insufficient. Nevertheless, electron band states of predominantly oxygen hole character crossing the Fermi level are markedly indicated in recent experiments [ l-31. An experimental tool for investigating the low-lying part of the electronic excitation spectra is the measurement of the far-infrared and mid-infrared dynamical conductivity. Remarkably unusual features are found in both frequency regions [4-81. Several attempts of a theoretical understanding of the conductivity in the normal state or the superconducting state exist [ 4- 12 1. Among different proposals a generalized Drude description with a frequency dependent effective mass [ 6 1, a mid-infrared oscillator [ 9 1, hopping conductivity [ 10 1, strong electron-phonon coupling behavior [ 111 and one-band combined with inter-band transitions [ 121 have been discussed. In this paper we present a new analysis, taking account l
On leave from ZPW Dresden, German Dem. Rep.
0921-4534/90/$03.50 (North-Holland)
0 Elsevier Science Publishers B.V.
of electronic coupling between the two CuOz layers within a unit cell and also coupling of the layers to the Cu03 chain. These coupling effects, expected from band structure calculations [ 13 1, have not yet been taken into account [ 4- 12 ] but may play an important role. An important microscopic point of view which we think has not yet been sufficiently considered in the theoretical analysis of the dynamical conductivity of layered HTSCs is the situation with intrinsic doping 6 between the [ CuOz] - 2+6 planes and a second layered anionic complex, namely the [CUO~] -3-26 chains of the REBa2Cu30,_, system, the [Bi204]-2-_(i+n)6, [T1204] -‘--(‘+“)‘, and [T1(-j3]-3-(l+n)6 in complexes the Bi&zCa$ui +n06+2n, T12Ba2CanCuI +n06+2n, and T1Ba2Ca,Cu l+n05+2n systems (n=O, 1, 2) respectively, or the oxygen of the La0 planes in La2_,AXCu04 (A=Ca, Sr, Ba) [15-171. In this doping process another band of doping states around the Fermi level is assumed to develop which overlaps the conduction band of the [ Cu02] -‘+’ planes responsible for the metallic conductivity and for superconductivity. Since there is experimental evidence for considerable disorder in the anionic complexes, we assume the corresponding electron states to be localized but with a quasicontinuous density of states around the Fermi level and that these states contribute to the dynamical conductivity. In the case
2
K.t;. Renk et al. / Dynamcal
conductivit_v
of REBa2Cu@_ ,, this contribution should furthermore be expected to be highly anisotropic with respect to the light polarization directions in the u-il plane of the crystal. High anisotropy of the reflectivity in the visible and near-infrared has indeed been regions of observed in untwinned crystal YBaZCu307_,. crystals [ 18.191. Except for Laz_,A,CuO,. all other layered HTSCs contain two CuOz planes, or two [CuOz] , +,Can sandwiches, in one unit cell of the structure, related to each other by a mirror plane or a glide mirror plane, respectively, perpendicular to the c-direction. This symmetry causes a splitting of the conduction bands of the two planes into a symmetric and an antisymmetric combination. For symmetry reasons, only the symmetric combination can couple to the localized doping states of the second anionic complex placed near the (glide) mirror plane of the structure. This situation puts some perspective on a combination of a two-band model for superconductivity [20] with an earlier proposed interlayer coupling mechanism via doping charge transfer [ 2 11.
2. Electronic band structure model and dynamical conductivity All layered high-T, superconductors form ionic crystals consisting of individual cations and anionic complexes extending through the lattice and carrying the conductivity. A two-dimensional [ CuOZ] -‘+’ anionic complex is present in all these materials and is generally believed to be responsible for superconductivity. In the undoped case (6= 0) its conduction band is half-filled and, due to correlation, a charge transfer gap opens transforming the material into an (antiferromagnetic) insulator. By doping (6> 0) the Fermi level is assumed to shift into an unconventional rudimentary 0-2~ conduction band [ 2 11. The full antibonding dpo* band of the [CuOZ] -’ plane is split by correlation into an upper and a lower Cu3d subband with the remainder 0-2~ subband inbetween, the latter forming the conduction band. The doping takes place because of overlap of a band of quasilocalized states from a second anionic complex varying from material to material. In YBa,Cu,O,_, it is formed by the [CUO~]-~-~~ chains, in the Bi and Tl based materials by Biz04 and Tl,O,+, layers,
qfhigh-7;
superconduc’tor.c
respectively, and in La2_,Sr_,Cu0, the oxygen ions of the La0 layers are most likely doping. This general picture (and most of the model discussions below) apply to all high-T= materials. We will. however. in the following, specialize the considerations to YBa2Cu@_ L,. The density of states of the bands crossing the Fermi level. projected onto the Cu02 planes and CuO, chains, respectively. is drawn schematically in fig. 1. Hole conductivity is found from Hall conductivity experiments and is attributed basically to the planes. In our picture (fig. 1 ) the number of holes per unit cell in the planes must be equal to the number of electrons per unit cell in the chains. The chain states are probably localized due to lattice imperfections (oxygen vacancies), but form a quasicontinuous density of states, the width of which corresponds to that of the unperturbed tight-binding band (1-2 eV). Therefore, in the analysis of the contribution from these chain states to the dynamical conductivity, we will formally use an effective bond mass tii to characterize the width of the quasicontinuous density of states. We treat the conductivities in the planes and along the chains as additive quantities and attribute therefore the conductivity a, measured for an electric field E polarized parallel to the a-axis to the plane conductivity aP and assume, for this plane conductivity. isotropy within the plane. We then write (for Ella)
‘VP'
-m(m)
D
chains
I
It(w)--iwm(w)/m(co)
D
--~
(1)
planes
Fig. I Electron density of states of YBa2Cu,07_, level (EF) projected onto the [Cu02]-‘+” [ Cu03] --3-26 chains, respectively.
near the Fermi planes and
K.F. Renk et al. /Dynamical conductivity of high-T, superconductors
where N is the number of holes per unit cell in the planes, and m(w) and y(o) frequency dependent mass and scattering rate of the holes. Correspondingly, the conductivity for E parallel to the b-axis is the sum of the plane conductivity and the chain conductivity a,,,, i.e., for E/lb, d~)=fJp(~)
+Gl(~)
.
(2)
We assume for the chain conductivity
a Drude form (3)
where N is the number of electrons per unit cell in the chains, A the effective mass and 7 the scattering rate of the chain electrons. Corresponding to our model of intrinsic doping we assume IV= N. We describe the reflectivity of a mosaic with dimensions small compared to the wavelength of the radiation as a(o)=a,(o)+fa,h(w).
(4)
This equation will be applied for an analysis of flectivity data for epitaxial films with the a- and axis in the film plane, with alternation of a- and direction along a specific direction because twinning.
rebbof
3. Analysis of the infrared conductivity in the normal state Fig. 2 shows
experimental
infrared
reflectivity
3
curves for single domain crystals [ 191 for E(la (R,) and El1b (Rb) together with the reflectivity (RFILM) of an epitaxial thin film [ 71; the curves are for room temperature, the infrared reflectivity remains almost unchanged if the temperature is lowered. The reflectivity along the b-axis is larger than along the aaxis. We attribute the difference between R, and Rb to chain conductivity as discussed also in ref. [ 19 1. The reflectivity of the film lies between R, and Rb. At small frequencies the reflectivity of the film is slightly larger than the reflectivities of the single crystal, indicating a higher dynamical conductivity; this is consistent with a lower DC resistivity of the film ( 180 $2 cm at 300 K) compared to that of the single crystal (possibly 400 @2 cm at 300 K). In the far-infrared the reflectivity (fig. 3) increases with decreasing temperature. Since the quantity (1-R)’ is a measure of the conductivity, the experimental curves show that the far-infrared dynamical conductivity is strongly dependent on temperature. There is no phonon structure in the farinfrared reflectivity spectra (fig. 3) indicating that phonons with elongations in the a-b plane do not influence the reflection behavior; we suggest that due to the high conductivity in the a-b plane the phonons are screened and have too small oscillator strengths to be observed. We have performed an analysis according to section 2 which leads to reflectivity curves (figs. 2 and 3) that describe the experimental curves in a satisfactory way. Deviations between the calculated curves and the curves for the untwinned crystal (fig. 2) may
EXPERIMENT EXPERIMENT
v (IO3cm-’) Fig. 2. Infrared reflectivity of YBazCu307_, for a single domain crystal for ElIa (R,) and El1b (Rb) according to ref. [ 161 and of an epitaxial film according to ref. [ 7 ] (upper part). Besides the experimental curves calculated curves are shown.
0
500 v (cm-‘)
1000
Fig. 3. Far-infrared reflectivity of the epitaxial YBa&u@_, thin film according to ref. [ 71 (upper part) and fitted curves for the normal state.
K.F. Renk et al. /Dynamical conductivity ofhigh-7; superconductors
4
be due to inhomogeneous hole distributions and therefore due to a distribution of plasma frequencies in the single crystal. Our analysis leads to conductivities shown in fig. 4. We have assumed that the high-frequency dielectric constant (E,= 3.7) is isotropic and for the chain conductivity we took the values of ref. [ 191, namely R/Hr=4X 102’ cmw3/m, (m,=free electron mass) and p=4400 cm-‘, or a scattering time of the order of lo-l5 s. The corresponding mean free path of the charge carriers is of the order of few lattice constants; thus, this analysis is consistent with quasilocalized chain states. The plane conductivity, on the other hand, shows a more complicated behavior: it is strongly frequency and temperature dependent at small frequencies. A description of the plane conductivity according to eq. (1) leads to N/ rn(c0)=8x 102’ cm-3/m0, and no frequency dependent effective masses and scattering rates shown in fig. 5. With increasing frequency the effective mass decreases and the scattering rate increases. At low
v ( IO' cm-‘) Fig. 4. Dynamical
0'
0
frequencies the effective mass and scattering rate are strongly frequency dependent. The effective mass changes strongly at low frequencies (Y < 100 cm- ’ ). This gives evidence for a mass change due to interaction with low energy excitations, with an effective interaction parameter /?z 1 for 100 K and i z 0.3 for 300 K. With N=N (section 2) we have r?i=2m(co), i.e. the chain mass is about twice the high-frequency plane mass. Assuming N=N=8x 102’ cm-3 (0.7 holes per CuOz ) we have m (co ) = m,, m (0 ) = 4~ at 100 K and r?r= 2m,. The scattering rate of the holes in the planes increases at high frequencies almost linearly with frequency and corresponds, at very high frequencies, to about a quarter of the value for the chain charge carriers, i.e., the mean free path is also only few lattice constants as for the charge carriers in the chain. At low frequencies the scattering rate reaches a small value at low temperature; the effectivescatteringrateat 10OKisy*(O)=~(O).m(~0)/ m( 0) z 50 cm-’ and the effective scattering time is about 1O-l3 s and corresponds to a mean free path of the order of 100 A. At high frequencies the plane conductivity is almost proportional to w-’ (fig. 6). This behavior is consistent with the assumption of a hopping conductivity in the planes as proposed in [ lo]: for the hopping parameter of ref. [ lo] a value of the order of 0.1 eV would follow. Thus our analysis gives evidence for hopping conductivity in the chains, and for high frequencies also in the planes while a more band like behavior for low energy excitations in the planes is concluded.
conductivity.
I
1
2 3 ” (103 cm-‘1
Fig. 5. Effective mass and scattering
I
rate.
5
v (cm-‘1 Fig. 6. Dynamical
conductivity
in the plane
K.F. Renk et al. /Dynamical conductivity of high-T, superconductors
4. Low-frequency and DC conductivity From our analysis we can derive the low-frequency dynamical conductivity and compare it with the DC conductivity, or correspondingly, the DC resistivity. We find that the low-frequency resistivity, p(O)=a-l(O), given by o(()
T)=++(O) Ne2
coincides with the DC resistivity (fig. 7 ). This shows that the dynamical conductivity at small far-infrared frequencies is equal to the DC conductivity. According to eq. (5) the linear temperature dependence of the DC resistivity is due to a linear temperature dependence of the scattering rate y(O). The scattering rate in the normal state, or the energy hey(O), is of the order of the thermal energy k,T; we find for the scattering energy hey - 3k,T for h v << k,T and -$hv for hv>kT (fig. 5). We suggest that the conductivity in the chains delivers a negligibly small contribution to the DC conductivity; most likely, the chain conductivity is very small for v-0 because the chains are interrupted by defects.
5. Dynamical conductivity in the superconducting state In a recent publication [20] we have proposed a pairing model which is based on the two-layer structure of YBa2Cu30,_,, and explains in a simple manner the occurrence of two distinct energy gaps. We
"0
Fig. 7. Low-frequency
100 TEMPERATURE far-infrared
200
resistivity
300
(K 1 and DC resistivity.
5
assumed that the two layers are coupled by a tightbinding hopping t; furthermore we assumed two different pairing interactions g, and gb for electrons within the same plane and for electrons in different planes (detailed models for such pairing interactions have been developed by several authors, see for instance refs. [ 2 1 ] and [ 22 ] ). This leads to two different order parameters A, and Ab. The tight-binding coupling causes a splitting of the energy band ep (k) for the electronic motion along the planes into two bands e,,(k)=cp(k)-t and cp2(k)=cp(k)+t corresponding to electronic states with wave functions which are antisymmetric and symmetric with respect to the two planes. In the superconducting state the two bands acquire different energy gaps A, =Aa-Ab, A2=Aa+Ab. If one of the pairing interactions g, or gb is repulsive one easily obtains ratios 2Ai/k,Tc much smaller and much larger than the BCS value 3.5. These two gaps also appear in the dynamical conductivity as distinct steps at Vi= 2Ai/hc. Two-band models for superconductivity have been discussed already shortly after the invention of the BCS theory [ 231. In the present case we start from a simple ansatz in real space and transform the results into a two-band model. This gives us a direct connection between the parameters of the bandmodel and the interaction parameters in real space. The occurrence of two distinct gaps in the excitation spectrum depends on the assumption that the coupling between different double-layers is relatively weak. Otherwise one obtains a gap-distribution with a maximal gap A2 and a minimal gap Al [ 241. In view of the recent experiments and the above theoretical considerations we have extended the model to include the electronic states of the chains between the two layers. As the electrons in the chains also contribute to the conductivity we assume an additional electronic band e,,,(k) which also crosses the Fermi surface, but at a different k-vector. The type of band structure we have in mind is shown in fig. 8. As before we assume that the two layers are coupled by a tight-binding hopping t which now occurs in two steps from one plane to the chain and then to the next plane. This results in a hybridization of the chain band with the electronic bands of the plane (note that only the band whose electronic wave function is symmetric in the two planes couples to the chain as already mentioned in section 1). In the
K.fi.. Kenk et al. /Dynamrcal
conductwrty
oj’hlgh-1;
superconduc,ior.\
Conclusions
Wavevec tor Fig. 8. Excitations energies for electrons with wave vector lel to the chains in the normal state (dashed lines) and superconducting state (solid lines). d, and & are the gaps electrons in the planes, dl is the mini-gap for the electrons chain.
paralin the for the in the
case where t/K<< 1, where ~=t~(k~~)--t~,,(k~~). we find a band-splitting of the form E,, (k) =cp(k) for the antisymmetric band and tP2 (k) = c,(k) + 2f ‘/K for the symmetric band. For the pairing interaction we use the same model as before, i.e. we do not take into account a pairing interaction for chain-electrons. In the superconducting state the electronic bands of the plane acquire two different energy gaps d, and d2. In the limit t/Kc 1 they are given by similar selfconsistency equations as in the previous model. For the chain band, however, only a mini-gap ‘A2 is found. This means that for all finite &h=(t/K) frequencies the electrons in the (ideal) chain behave as normal electrons and may contribute to the dynamical conductivity in the normal as well in the superconducting state. The reason for the relative decoupling between the plane states and the chain states with respect to their superconducting properties is the large energy difference K between the chain-band and the electronic bands of the planes at a k-vector where the latter crosses the Fermi surface. Note that the two gaps discussed here both occur for excitations in the u-b plane and should not be interpreted as a gap-anisotropy for excitation in the u-b plane and c-direction. The extension of the model to include the chain-states shows that with simple model assumptions superconducting and normal electrons may coexist, even if both bands cross the Fermi surface.
Midfar-infrared conductivitics in and YBa2Cu307_1. are interpreted in terms of two bands of mobile holes in the CuOz layers and of a quasi continuous band of almost localized states in the CuOi chains. According to our analysis it is just these states which dope the Cu02 layers. After subtracting from the experimental conductivity a chain conductivity assumed to be Drude like the remaining dynamical conductivity shows a CL) ~ ’ dependence in the mid-infrared and a more Drude like behavior, however with a mass enhancement, in the far-infrared. For the superconducting state we find evidence for two superconducting energy gaps. These can be interpreted in terms of a two-band model where the symmetric and the antisymmetric two-layer bands are split by the hybridization of the symmetric band with the states in the chain. If one of the pairing interactions is repulsive. this model leads to an enhanced energy gap well above the BCS value of 33/ li,7, and a reduced energy gap below the BCS value. We believe that the interplane pairing interaction is attractive and may be mediated again by the doping states between the planes. We show also that. under certain conditions, normal conductivity in the chains can coexist with superconductivity in the planes. For a final decision about the applicability of our model further experiments would be necessary. especially far-infrared experiments on single domain crystals of YBa2Cu,0,_b (~3~0) and on single domain crystals containing different oxygen concentrations; crystals available at present are too small for far-infrared experiments.
References [ I ] T. Takahashl,
H. Matsuyama, H. Katayama-Yoshida. 1’. Okabe, S. Hosoya. K. Seki. H. Fujimoto, M. Sato and H. Inokuchi, Nature 334 (1988) 691: ibid.. Phys. Rev. B 39 (1989) 6636: T. Takahashi, Proc. IBM Japan Intern. Symp. on Strong Correlation and Superconductivity. Fuji, 1989. [2] C.G. Olson, R. Liu, A.-B. Yang, D.W. Lynch, A.J. Arko. R.S. List, B.W. Veal, Y.C. Chang, P.Z. Jiang and A.P. Paulikas. preprint. [3] R. Claessen. R. Manzke. H. Cartensen. B. Burandt. T. Buslaps. M. Skibowski and J. Fink, preprint.
K.F. Renk et al. /Dynamical conductivity ofhigh-T, superconductors [4] J. Orenstein
and D.H. Rapkine,
Phys. Rev. Lett. 60 (1988)
965. [ 51 K. Kamaras, C.D. Porter, M.G. Doss, S.L. Herr, D.B. Tanner, D.A. Bonn, J.E. Greedan, A.H. O’Reilly, C.V. Stager and T. Timusk, Phys. Rev. Lett. 60 ( 1988) 969. [ 61 G.A. Thomas, J. Orenstein, D.H. Rapkine, M. Capizzi, A.J. Millis, R.N. Bhatt, L.F. Schneemeyer and J.V. Waszczak, Phys. Rev. Lett. 61 (1988)
1313.
[ 7 ] J. Schiitzmann,
W. Ose, J. Keller, K.F. Renk, B. Roas, L. Schultz and G. Saemann-Ischenko, Europhys. Lett. 8 ( 1989) 679.
[ 8 ] R.T. Collins, Z. Schlesinger, F. Holtzberg, P. Chandhari
and C. Feild, Phys. Rev. B 39 (1989) 6571. [ 91 T. Timusk and D. Tanner, in: Infrared Properties of HighT, Superconductors, ed. D.M. Ginsberg (World Scientific Publishing Company, 1989).
[lo] T.M. Rice and F.C. Zhang, Phys. Rev. B 39 (1989) 815. [ 111 W. Lee, D. Rainer and W. Zimmerman, preprint. [ 121 S.T. Chui, R.V. Kasowski 61 (1988) 885.
and W.Y. Hsu, Phys. Rev. Lett.
[ 131 L.F. Mattheiss and D.R. Hamann, (1987)
395.
Solid State Commun.
63
7
[ 141 For energy band structure see also H. Eschrig and G. Seifert, Solid State Commun. 64 ( 1987) 52 1; A. Bansil, R. Pankaluoto, R.S. Rao, P.E. Mijnarends, W. Dlugosz, R. Prasad and L.C. Smedskjaer, Phys. Rev. Lett. 61 (1988) 2480. [ 15 ] J. Zaanen, J. Jepsen, 0. Gunnarsson, A.T. Paxton, O.K. Andersen and A. Svane, Physica C 153-155 ( 1988) 1636. [ 161 H. EschrigandG. Seifert, PhysicaC 153-155 (1988) 1243. [17] H. Eschrig, PhysicaC 159 (1989) 545. [ 181 V.K. Vlasko-Vlasov, M.V. Indenbom and Yu.A. Ossipyan, PhysicaC 153-155 (1988) 1677. [ 191 B. Koch, H.P. Geserich and T. Wolf, to be published in Solid State Commun. [20] U. Hofmann, J. Keller, K.F. Renk, J. Schiltzmann and W. Ose, Solid State Commun. 70 ( 1989) 325. [ 2 1 ] H. Eschrig and G. Seifert, Phys. Scripta T 25 ( 1989) 88. [22] A.R. Bishop, R.L. Martin, K.A. Miiller and B. Tesanovic, Z.Phys.B76(1989) 17. [23] H. Suhl, B.T. Matthias and R.L. Walker, Phys. Rev. Lett. 3 (1959) 522. [24] T. Schneider, H. de Raedt and M. Fricke, to be published in Z. Phys. B 76 ( 1989) 3.
DYNAMICAL
CONDUCTIVITY
K.F. RENK, H. ESCHRIG
OF HIGH-T, SUPERCONDUCTORS
*, U. HOFMANN,
J. KELLER,
J. SCHUTZMANN
and W. OSE
Fakultiit ftir Physik, Universitiit Regensburg, 8400 Regensburg, Fed. Rep. Germany Received 29 August 1989 Revised manuscript received 30 October 1989
We present an analysis of the far- and mid-infrared conductivity of YBa2Cu30,_, We suppose that two types of low-lying electronic excitations in high-T, superconductors are present, namely mobile quasi holes in the Cu02 planes and almost localized doping electron states in the CuOp chains. We explain the far-infrared conductivity at low temperature (T-x T,) with a two-band model derived from the two-layer structure of YBaaCusO,_, This model leads to two gap values, well above and below the BCS
1. Introduction On the way towards an understanding of high-T, superconductivity, a crucial task is the experimental determination and the theoretical interpretation of the excitation spectra in the normal and superconducting states. Strong correlation effects are generally believed to be present in the electronic excitation spectra so that ordinary bandstructure theory is insufficient. Nevertheless, electron band states of predominantly oxygen hole character crossing the Fermi level are markedly indicated in recent experiments [ l-31. An experimental tool for investigating the low-lying part of the electronic excitation spectra is the measurement of the far-infrared and mid-infrared dynamical conductivity. Remarkably unusual features are found in both frequency regions [4-81. Several attempts of a theoretical understanding of the conductivity in the normal state or the superconducting state exist [ 4- 12 1. Among different proposals a generalized Drude description with a frequency dependent effective mass [ 6 1, a mid-infrared oscillator [ 9 1, hopping conductivity [ 10 1, strong electron-phonon coupling behavior [ 111 and one-band combined with inter-band transitions [ 121 have been discussed. In this paper we present a new analysis, taking account l
On leave from ZPW Dresden, German Dem. Rep.
0921-4534/90/$03.50 (North-Holland)
0 Elsevier Science Publishers B.V.
of electronic coupling between the two CuOz layers within a unit cell and also coupling of the layers to the Cu03 chain. These coupling effects, expected from band structure calculations [ 13 1, have not yet been taken into account [ 4- 12 ] but may play an important role. An important microscopic point of view which we think has not yet been sufficiently considered in the theoretical analysis of the dynamical conductivity of layered HTSCs is the situation with intrinsic doping 6 between the [ CuOz] - 2+6 planes and a second layered anionic complex, namely the [CUO~] -3-26 chains of the REBa2Cu30,_, system, the [Bi204]-2-_(i+n)6, [T1204] -‘--(‘+“)‘, and [T1(-j3]-3-(l+n)6 in complexes the Bi&zCa$ui +n06+2n, T12Ba2CanCuI +n06+2n, and T1Ba2Ca,Cu l+n05+2n systems (n=O, 1, 2) respectively, or the oxygen of the La0 planes in La2_,AXCu04 (A=Ca, Sr, Ba) [15-171. In this doping process another band of doping states around the Fermi level is assumed to develop which overlaps the conduction band of the [ Cu02] -‘+’ planes responsible for the metallic conductivity and for superconductivity. Since there is experimental evidence for considerable disorder in the anionic complexes, we assume the corresponding electron states to be localized but with a quasicontinuous density of states around the Fermi level and that these states contribute to the dynamical conductivity. In the case
2
K.t;. Renk et al. / Dynamcal
conductivit_v
of REBa2Cu@_ ,, this contribution should furthermore be expected to be highly anisotropic with respect to the light polarization directions in the u-il plane of the crystal. High anisotropy of the reflectivity in the visible and near-infrared has indeed been regions of observed in untwinned crystal YBaZCu307_,. crystals [ 18.191. Except for Laz_,A,CuO,. all other layered HTSCs contain two CuOz planes, or two [CuOz] , +,Can sandwiches, in one unit cell of the structure, related to each other by a mirror plane or a glide mirror plane, respectively, perpendicular to the c-direction. This symmetry causes a splitting of the conduction bands of the two planes into a symmetric and an antisymmetric combination. For symmetry reasons, only the symmetric combination can couple to the localized doping states of the second anionic complex placed near the (glide) mirror plane of the structure. This situation puts some perspective on a combination of a two-band model for superconductivity [20] with an earlier proposed interlayer coupling mechanism via doping charge transfer [ 2 11.
2. Electronic band structure model and dynamical conductivity All layered high-T, superconductors form ionic crystals consisting of individual cations and anionic complexes extending through the lattice and carrying the conductivity. A two-dimensional [ CuOZ] -‘+’ anionic complex is present in all these materials and is generally believed to be responsible for superconductivity. In the undoped case (6= 0) its conduction band is half-filled and, due to correlation, a charge transfer gap opens transforming the material into an (antiferromagnetic) insulator. By doping (6> 0) the Fermi level is assumed to shift into an unconventional rudimentary 0-2~ conduction band [ 2 11. The full antibonding dpo* band of the [CuOZ] -’ plane is split by correlation into an upper and a lower Cu3d subband with the remainder 0-2~ subband inbetween, the latter forming the conduction band. The doping takes place because of overlap of a band of quasilocalized states from a second anionic complex varying from material to material. In YBa,Cu,O,_, it is formed by the [CUO~]-~-~~ chains, in the Bi and Tl based materials by Biz04 and Tl,O,+, layers,
qfhigh-7;
superconduc’tor.c
respectively, and in La2_,Sr_,Cu0, the oxygen ions of the La0 layers are most likely doping. This general picture (and most of the model discussions below) apply to all high-T= materials. We will. however. in the following, specialize the considerations to YBa2Cu@_ L,. The density of states of the bands crossing the Fermi level. projected onto the Cu02 planes and CuO, chains, respectively. is drawn schematically in fig. 1. Hole conductivity is found from Hall conductivity experiments and is attributed basically to the planes. In our picture (fig. 1 ) the number of holes per unit cell in the planes must be equal to the number of electrons per unit cell in the chains. The chain states are probably localized due to lattice imperfections (oxygen vacancies), but form a quasicontinuous density of states, the width of which corresponds to that of the unperturbed tight-binding band (1-2 eV). Therefore, in the analysis of the contribution from these chain states to the dynamical conductivity, we will formally use an effective bond mass tii to characterize the width of the quasicontinuous density of states. We treat the conductivities in the planes and along the chains as additive quantities and attribute therefore the conductivity a, measured for an electric field E polarized parallel to the a-axis to the plane conductivity aP and assume, for this plane conductivity. isotropy within the plane. We then write (for Ella)
‘VP'
-m(m)
D
chains
I
It(w)--iwm(w)/m(co)
D
--~
(1)
planes
Fig. I Electron density of states of YBa2Cu,07_, level (EF) projected onto the [Cu02]-‘+” [ Cu03] --3-26 chains, respectively.
near the Fermi planes and
K.F. Renk et al. /Dynamical conductivity of high-T, superconductors
where N is the number of holes per unit cell in the planes, and m(w) and y(o) frequency dependent mass and scattering rate of the holes. Correspondingly, the conductivity for E parallel to the b-axis is the sum of the plane conductivity and the chain conductivity a,,,, i.e., for E/lb, d~)=fJp(~)
+Gl(~)
.
(2)
We assume for the chain conductivity
a Drude form (3)
where N is the number of electrons per unit cell in the chains, A the effective mass and 7 the scattering rate of the chain electrons. Corresponding to our model of intrinsic doping we assume IV= N. We describe the reflectivity of a mosaic with dimensions small compared to the wavelength of the radiation as a(o)=a,(o)+fa,h(w).
(4)
This equation will be applied for an analysis of flectivity data for epitaxial films with the a- and axis in the film plane, with alternation of a- and direction along a specific direction because twinning.
rebbof
3. Analysis of the infrared conductivity in the normal state Fig. 2 shows
experimental
infrared
reflectivity
3
curves for single domain crystals [ 191 for E(la (R,) and El1b (Rb) together with the reflectivity (RFILM) of an epitaxial thin film [ 71; the curves are for room temperature, the infrared reflectivity remains almost unchanged if the temperature is lowered. The reflectivity along the b-axis is larger than along the aaxis. We attribute the difference between R, and Rb to chain conductivity as discussed also in ref. [ 19 1. The reflectivity of the film lies between R, and Rb. At small frequencies the reflectivity of the film is slightly larger than the reflectivities of the single crystal, indicating a higher dynamical conductivity; this is consistent with a lower DC resistivity of the film ( 180 $2 cm at 300 K) compared to that of the single crystal (possibly 400 @2 cm at 300 K). In the far-infrared the reflectivity (fig. 3) increases with decreasing temperature. Since the quantity (1-R)’ is a measure of the conductivity, the experimental curves show that the far-infrared dynamical conductivity is strongly dependent on temperature. There is no phonon structure in the farinfrared reflectivity spectra (fig. 3) indicating that phonons with elongations in the a-b plane do not influence the reflection behavior; we suggest that due to the high conductivity in the a-b plane the phonons are screened and have too small oscillator strengths to be observed. We have performed an analysis according to section 2 which leads to reflectivity curves (figs. 2 and 3) that describe the experimental curves in a satisfactory way. Deviations between the calculated curves and the curves for the untwinned crystal (fig. 2) may
EXPERIMENT EXPERIMENT
v (IO3cm-’) Fig. 2. Infrared reflectivity of YBazCu307_, for a single domain crystal for ElIa (R,) and El1b (Rb) according to ref. [ 161 and of an epitaxial film according to ref. [ 7 ] (upper part). Besides the experimental curves calculated curves are shown.
0
500 v (cm-‘)
1000
Fig. 3. Far-infrared reflectivity of the epitaxial YBa&u@_, thin film according to ref. [ 71 (upper part) and fitted curves for the normal state.
K.F. Renk et al. /Dynamical conductivity ofhigh-7; superconductors
4
be due to inhomogeneous hole distributions and therefore due to a distribution of plasma frequencies in the single crystal. Our analysis leads to conductivities shown in fig. 4. We have assumed that the high-frequency dielectric constant (E,= 3.7) is isotropic and for the chain conductivity we took the values of ref. [ 191, namely R/Hr=4X 102’ cmw3/m, (m,=free electron mass) and p=4400 cm-‘, or a scattering time of the order of lo-l5 s. The corresponding mean free path of the charge carriers is of the order of few lattice constants; thus, this analysis is consistent with quasilocalized chain states. The plane conductivity, on the other hand, shows a more complicated behavior: it is strongly frequency and temperature dependent at small frequencies. A description of the plane conductivity according to eq. (1) leads to N/ rn(c0)=8x 102’ cm-3/m0, and no frequency dependent effective masses and scattering rates shown in fig. 5. With increasing frequency the effective mass decreases and the scattering rate increases. At low
v ( IO' cm-‘) Fig. 4. Dynamical
0'
0
frequencies the effective mass and scattering rate are strongly frequency dependent. The effective mass changes strongly at low frequencies (Y < 100 cm- ’ ). This gives evidence for a mass change due to interaction with low energy excitations, with an effective interaction parameter /?z 1 for 100 K and i z 0.3 for 300 K. With N=N (section 2) we have r?i=2m(co), i.e. the chain mass is about twice the high-frequency plane mass. Assuming N=N=8x 102’ cm-3 (0.7 holes per CuOz ) we have m (co ) = m,, m (0 ) = 4~ at 100 K and r?r= 2m,. The scattering rate of the holes in the planes increases at high frequencies almost linearly with frequency and corresponds, at very high frequencies, to about a quarter of the value for the chain charge carriers, i.e., the mean free path is also only few lattice constants as for the charge carriers in the chain. At low frequencies the scattering rate reaches a small value at low temperature; the effectivescatteringrateat 10OKisy*(O)=~(O).m(~0)/ m( 0) z 50 cm-’ and the effective scattering time is about 1O-l3 s and corresponds to a mean free path of the order of 100 A. At high frequencies the plane conductivity is almost proportional to w-’ (fig. 6). This behavior is consistent with the assumption of a hopping conductivity in the planes as proposed in [ lo]: for the hopping parameter of ref. [ lo] a value of the order of 0.1 eV would follow. Thus our analysis gives evidence for hopping conductivity in the chains, and for high frequencies also in the planes while a more band like behavior for low energy excitations in the planes is concluded.
conductivity.
I
1
2 3 ” (103 cm-‘1
Fig. 5. Effective mass and scattering
I
rate.
5
v (cm-‘1 Fig. 6. Dynamical
conductivity
in the plane
K.F. Renk et al. /Dynamical conductivity of high-T, superconductors
4. Low-frequency and DC conductivity From our analysis we can derive the low-frequency dynamical conductivity and compare it with the DC conductivity, or correspondingly, the DC resistivity. We find that the low-frequency resistivity, p(O)=a-l(O), given by o(()
T)=++(O) Ne2
coincides with the DC resistivity (fig. 7 ). This shows that the dynamical conductivity at small far-infrared frequencies is equal to the DC conductivity. According to eq. (5) the linear temperature dependence of the DC resistivity is due to a linear temperature dependence of the scattering rate y(O). The scattering rate in the normal state, or the energy hey(O), is of the order of the thermal energy k,T; we find for the scattering energy hey - 3k,T for h v << k,T and -$hv for hv>kT (fig. 5). We suggest that the conductivity in the chains delivers a negligibly small contribution to the DC conductivity; most likely, the chain conductivity is very small for v-0 because the chains are interrupted by defects.
5. Dynamical conductivity in the superconducting state In a recent publication [20] we have proposed a pairing model which is based on the two-layer structure of YBa2Cu30,_,, and explains in a simple manner the occurrence of two distinct energy gaps. We
"0
Fig. 7. Low-frequency
100 TEMPERATURE far-infrared
200
resistivity
300
(K 1 and DC resistivity.
5
assumed that the two layers are coupled by a tightbinding hopping t; furthermore we assumed two different pairing interactions g, and gb for electrons within the same plane and for electrons in different planes (detailed models for such pairing interactions have been developed by several authors, see for instance refs. [ 2 1 ] and [ 22 ] ). This leads to two different order parameters A, and Ab. The tight-binding coupling causes a splitting of the energy band ep (k) for the electronic motion along the planes into two bands e,,(k)=cp(k)-t and cp2(k)=cp(k)+t corresponding to electronic states with wave functions which are antisymmetric and symmetric with respect to the two planes. In the superconducting state the two bands acquire different energy gaps A, =Aa-Ab, A2=Aa+Ab. If one of the pairing interactions g, or gb is repulsive one easily obtains ratios 2Ai/k,Tc much smaller and much larger than the BCS value 3.5. These two gaps also appear in the dynamical conductivity as distinct steps at Vi= 2Ai/hc. Two-band models for superconductivity have been discussed already shortly after the invention of the BCS theory [ 231. In the present case we start from a simple ansatz in real space and transform the results into a two-band model. This gives us a direct connection between the parameters of the bandmodel and the interaction parameters in real space. The occurrence of two distinct gaps in the excitation spectrum depends on the assumption that the coupling between different double-layers is relatively weak. Otherwise one obtains a gap-distribution with a maximal gap A2 and a minimal gap Al [ 241. In view of the recent experiments and the above theoretical considerations we have extended the model to include the electronic states of the chains between the two layers. As the electrons in the chains also contribute to the conductivity we assume an additional electronic band e,,,(k) which also crosses the Fermi surface, but at a different k-vector. The type of band structure we have in mind is shown in fig. 8. As before we assume that the two layers are coupled by a tight-binding hopping t which now occurs in two steps from one plane to the chain and then to the next plane. This results in a hybridization of the chain band with the electronic bands of the plane (note that only the band whose electronic wave function is symmetric in the two planes couples to the chain as already mentioned in section 1). In the
K.fi.. Kenk et al. /Dynamrcal
conductwrty
oj’hlgh-1;
superconduc,ior.\
Conclusions
Wavevec tor Fig. 8. Excitations energies for electrons with wave vector lel to the chains in the normal state (dashed lines) and superconducting state (solid lines). d, and & are the gaps electrons in the planes, dl is the mini-gap for the electrons chain.
paralin the for the in the
case where t/K<< 1, where ~=t~(k~~)--t~,,(k~~). we find a band-splitting of the form E,, (k) =cp(k) for the antisymmetric band and tP2 (k) = c,(k) + 2f ‘/K for the symmetric band. For the pairing interaction we use the same model as before, i.e. we do not take into account a pairing interaction for chain-electrons. In the superconducting state the electronic bands of the plane acquire two different energy gaps d, and d2. In the limit t/Kc 1 they are given by similar selfconsistency equations as in the previous model. For the chain band, however, only a mini-gap ‘A2 is found. This means that for all finite &h=(t/K) frequencies the electrons in the (ideal) chain behave as normal electrons and may contribute to the dynamical conductivity in the normal as well in the superconducting state. The reason for the relative decoupling between the plane states and the chain states with respect to their superconducting properties is the large energy difference K between the chain-band and the electronic bands of the planes at a k-vector where the latter crosses the Fermi surface. Note that the two gaps discussed here both occur for excitations in the u-b plane and should not be interpreted as a gap-anisotropy for excitation in the u-b plane and c-direction. The extension of the model to include the chain-states shows that with simple model assumptions superconducting and normal electrons may coexist, even if both bands cross the Fermi surface.
Midfar-infrared conductivitics in and YBa2Cu307_1. are interpreted in terms of two bands of mobile holes in the CuOz layers and of a quasi continuous band of almost localized states in the CuOi chains. According to our analysis it is just these states which dope the Cu02 layers. After subtracting from the experimental conductivity a chain conductivity assumed to be Drude like the remaining dynamical conductivity shows a CL) ~ ’ dependence in the mid-infrared and a more Drude like behavior, however with a mass enhancement, in the far-infrared. For the superconducting state we find evidence for two superconducting energy gaps. These can be interpreted in terms of a two-band model where the symmetric and the antisymmetric two-layer bands are split by the hybridization of the symmetric band with the states in the chain. If one of the pairing interactions is repulsive. this model leads to an enhanced energy gap well above the BCS value of 33/ li,7, and a reduced energy gap below the BCS value. We believe that the interplane pairing interaction is attractive and may be mediated again by the doping states between the planes. We show also that. under certain conditions, normal conductivity in the chains can coexist with superconductivity in the planes. For a final decision about the applicability of our model further experiments would be necessary. especially far-infrared experiments on single domain crystals of YBa2Cu,0,_b (~3~0) and on single domain crystals containing different oxygen concentrations; crystals available at present are too small for far-infrared experiments.
References [ I ] T. Takahashl,
H. Matsuyama, H. Katayama-Yoshida. 1’. Okabe, S. Hosoya. K. Seki. H. Fujimoto, M. Sato and H. Inokuchi, Nature 334 (1988) 691: ibid.. Phys. Rev. B 39 (1989) 6636: T. Takahashi, Proc. IBM Japan Intern. Symp. on Strong Correlation and Superconductivity. Fuji, 1989. [2] C.G. Olson, R. Liu, A.-B. Yang, D.W. Lynch, A.J. Arko. R.S. List, B.W. Veal, Y.C. Chang, P.Z. Jiang and A.P. Paulikas. preprint. [3] R. Claessen. R. Manzke. H. Cartensen. B. Burandt. T. Buslaps. M. Skibowski and J. Fink, preprint.
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and D.H. Rapkine,
Phys. Rev. Lett. 60 (1988)
965. [ 51 K. Kamaras, C.D. Porter, M.G. Doss, S.L. Herr, D.B. Tanner, D.A. Bonn, J.E. Greedan, A.H. O’Reilly, C.V. Stager and T. Timusk, Phys. Rev. Lett. 60 ( 1988) 969. [ 61 G.A. Thomas, J. Orenstein, D.H. Rapkine, M. Capizzi, A.J. Millis, R.N. Bhatt, L.F. Schneemeyer and J.V. Waszczak, Phys. Rev. Lett. 61 (1988)
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[ 7 ] J. Schiitzmann,
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[ 8 ] R.T. Collins, Z. Schlesinger, F. Holtzberg, P. Chandhari
and C. Feild, Phys. Rev. B 39 (1989) 6571. [ 91 T. Timusk and D. Tanner, in: Infrared Properties of HighT, Superconductors, ed. D.M. Ginsberg (World Scientific Publishing Company, 1989).
[lo] T.M. Rice and F.C. Zhang, Phys. Rev. B 39 (1989) 815. [ 111 W. Lee, D. Rainer and W. Zimmerman, preprint. [ 121 S.T. Chui, R.V. Kasowski 61 (1988) 885.
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Solid State Commun.
63
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[ 141 For energy band structure see also H. Eschrig and G. Seifert, Solid State Commun. 64 ( 1987) 52 1; A. Bansil, R. Pankaluoto, R.S. Rao, P.E. Mijnarends, W. Dlugosz, R. Prasad and L.C. Smedskjaer, Phys. Rev. Lett. 61 (1988) 2480. [ 15 ] J. Zaanen, J. Jepsen, 0. Gunnarsson, A.T. Paxton, O.K. Andersen and A. Svane, Physica C 153-155 ( 1988) 1636. [ 161 H. EschrigandG. Seifert, PhysicaC 153-155 (1988) 1243. [17] H. Eschrig, PhysicaC 159 (1989) 545. [ 181 V.K. Vlasko-Vlasov, M.V. Indenbom and Yu.A. Ossipyan, PhysicaC 153-155 (1988) 1677. [ 191 B. Koch, H.P. Geserich and T. Wolf, to be published in Solid State Commun. [20] U. Hofmann, J. Keller, K.F. Renk, J. Schiltzmann and W. Ose, Solid State Commun. 70 ( 1989) 325. [ 2 1 ] H. Eschrig and G. Seifert, Phys. Scripta T 25 ( 1989) 88. [22] A.R. Bishop, R.L. Martin, K.A. Miiller and B. Tesanovic, Z.Phys.B76(1989) 17. [23] H. Suhl, B.T. Matthias and R.L. Walker, Phys. Rev. Lett. 3 (1959) 522. [24] T. Schneider, H. de Raedt and M. Fricke, to be published in Z. Phys. B 76 ( 1989) 3.