PHYSICAL REVIEW B VOLUME 38, NUMBER 16 1 DECEMBER 1988

Out-of-plane transport mechanism in the high-T, oxide Y-Ba-Cu-0

D. Y. Xing, M. Liu, and C. S. TingDepartment of Physics and Texas Center for Superconductivity U. niversity of Houston, Houston, Texas 77004

(Received 6 July 1988)

The out-of-plane transport properties of the high-T, oxide compounds, including conductivity,

thermoelectric power, and Hall coefficient, are extraordinarily anomalous. In this Rapid Com-

munication, we show that the sign and temperature dependence of these transport coefficients can

be explained by a phonon-assisted hopping mechanism of electrons between the neighboring two-

dimensional Cu-0 planes.

Strong anisotropy is one of the most important proper-ties of the new high-T, oxide compounds. Apart fromtheir anisotropic superconducting properties, the normal-state electronic transport of these oxide compounds is alsohighly anisotropic. In the study of single crystals ofYBa2Cuq07 (Y-Ba-Cu-O), Tozer et al. ' reported that theratio of the out-of-plane to in-plane resistivity, p, /p, b, is30 at room temperature and increases to about 80 as tem-perature is lowered. Recently, larger values for the an-

isotropy (50-300) have been estimated by Hagen et al. 2

Even anisotropies as high as 103-10 have been reportedfor the new Bi-Ca-Sr-Cu crystals. ~ The anisotropy is alsoapparent in the measurements' of the Hall coefficientRH and the thermoelectric power (TEP) S of single crys-tals of Y-Ba-Cu-0. Since the in-plane transport dom-inates the polycrystalline samples, the early observed (p),(RH), and (S) in ceramics of Y-Ba-Cu-0 have similartemperature dependence to the corresponding in-planequantities in single crystals. Both p, b and (p) are almostlinearly T dependent from T, to room temperature. 'Both 1/RH' in single crystals, with the magnetic field

along the c axis, and (1/RH) in ceramics are positive andalmost linearly T dependent. The S,b vs T profile issimilar to (S) in ceramic samples. However, the out-of-plane transport coefficients have many remarkable behav-iors which are very different from those of the correspond-ing in-plane quantities. First, the Hall coefficient RH' orRH' in single crystals, with the magnetic field appliedparallel to the Cu-0 planes, is negative or electronlike,while the sign of the thermopower along the c axis is posi-tive, indicating holelike conduction according to usualtransport theory. ' Second, the c-axis TEP exhibits me-tallic behavior while the c-axis conductivity appears to besemiconducting or insulating and it satisfies the empiricalformula A/T+BT proposed by Anderson and Zou. " Al-though the contradiction on the signs of both the out-of-plane TEP and Hall coefficient may be interpreted by theband-theory calculation of Allen, Pickett, and Krahau-er, ' their predictions on the sign of S,b, the magnitudesand temperature dependence of both S,b and S, disagreewith the recent experimental data. ' To our knowledge,no unified theory to date can explain all the anisotropictransport properties including p, RH, and S in the high-T,oxide compounds, particularly their extraordinary anoma-lous temperature dependence. Very recently, we have ex-plained the linear temperature dependence of both the in-

plane resistivity and 1/RH' of Y-Ba-Cu-0 by assuming atwo-band model' for the Cu-0 planes with an exteriorcarrier source. In this model, which is based upon theband-structure calculation of Herman, Kasowski, andHsu, ' both electrons and holes contribute to the in-planetransport. In this paper, we further assume that the out-of-plane transport originates from a phonon-assisted hop-ping process' in which the electrons, not the holes, aredominantly scattered from layer to layer with emission orabsorption of phonons. In terms of the two-band modelproposed in Ref. 13 and the hopping transport of electronsbetween the neighboring two-dimensional Cu-0 planes,we show that (i) RH' (or RH' ) (0 and S, & 0; (ii) theout-of-plane conductivity o, and thermoelectric power S,are linearly T dependent in the temperature regime'b

p » T & T* with T =—2skF (s denoting the sound veloci-ty and kF the two-dimensional momentum) and p thechemical potential; and (iii) the Hall coefficient RHb" (orRH' ) is almost independent of T. These results are inqualitative agreement with the experimental data' 7 ofthe out-of-plane transport properties in single crystals ofY-Ba-Cu-0. These results, together with the calcula-tions' ' for the in-plane transport properties, thus pro-vides a unified, although possibly not unique, explanationfor the anisotropic transport behavior of the high-T, oxideY-Ba-Cu-O.

Both structure analysis' and band-structure calcula-tions' ' indicate the essentially two-dimensional natureof electron dynamics in Y-Ba-Cu-0. The system underconsideration can be assumed to be composed of two-dimensional Cu-0 planes of which each is coupled weaklywith its neighboring ones. The carriers (electrons orholes) can move in free motion within the two-dimen-sional Cu-0 planes, but have localized behavior along thec direction. In such a model, the in-plane transportcomes from the traditional electron (hole) -phonon orelectron(hole)-impurity scattering. It has been shown' 'that the linear T dependence of the in-plane resistivity p,b

can be obtained by the electron(hole)-phonon scatteringmechanism. However, strong anisotropy indicates thatthe out-of-plane transport mechanism must be verydifferent from the in-plane one. In this paper, we assumethat the out-of-plane transport originates from a phonon-assisted hopping process in which the electrons are dom-inantly scattered, not within the layer, but from layer tolayer with emission or absorption of phonons. In both

11 992 1988 The American Physical Society

OUT-OF-PLANE TRANSPORT MECHANISM IN THE HIGH-T, . . . 11993

electric and temperature fields along the c axis, the tunneling rate for electrons from layer i to neighboring layer j is

8n(i j)/8t 4mg I M J(q, k, ) I [f;(q,)[1 —fj(~+q)]n(toq)b(& ~+z+toq+eV)k,q, A,

+f;(q)[l fJ(q —q)][1+n(toq)]b(et, q q-a)q+ev)[. (1)

Here, f;(~)=1/[exp[(~ —p)/T;]+1] is the usual Fermi function, T; is the carrier temperature for layer i, ~ k k/2mis the carrier energy, and k and q are the two-dimensional momenta. n(toq) = I/[exp(toq/T) —1] is the Bose functionwith toq the phonon energy, and T (T;+TJ)/2 is the average temperature. eVis the difference of the electric potentialbetween layer i and j, and M~J(q, k.) is the hopping matrix element of the electron-phonon interaction from layer i to jwith A, the phonon branch index. The two terms on the right-hand side of Eq. (1) represent, respectively, the processes ofphonon absorption and emission. The inverse tunneling rate corresponding to Eq. (1) is

8n(j i)/8t 4m Q I M~, (q,x) I'[[I -f;(z)]fj(z+q)[1+n(toq)]8(+ ++q+toq+eV)

k,q, A,

+ [1 —f;(q,)]f, (q,-,)n(co, )b(q, —q, —a), +ev)] .

The net tunneling rate dn/dt for electrons from layer i to j is the difference between Eqs. (1) and (2). After some alge-braic work, we have obtained the following expression for dn/dt to the linear order in the electric field E and the thermalgradient VT:

dn/dt Sn Z I M(q, ~) I n(toq) [1+n(toq)] [f(q,+q) —f(et, )]b(et, —~+q+toq) [eEd/T —(~ —tt)dV T/T ] .k,q, A.

In deriving the equation above, we have used the relations that eV eEd and TJ —T; dVT with d the lattice parameteralong the c axis. The hopping current is J edn/dt It th. en follows that

with

J e KaE+(e/T)K~( —VT), (4)

(gnd/T)Z I M(q ~) I n(toq) [1+n(toq)]g (q, —p)'[f(q, +q) —f(q, )]b(q, —q, +q+toq) . (5)q, A,

Using the replacement of Pp by jd k/(2n) in Eq. (5) and performing the integration' over angle of k, we obtain forKt the expression

where

(4md/nT)Z IM(q, &) I 'n(toq) [I+n(toq)]/q «k [—df (ek)/«k] (ek —p) 'p(ek, q, to),q, A,

p(eq, q, to) 8[2m' —(q/2 mto/q) ] [—2mek —(q/2 mto/q) ] '——8[2me —(q/2+mco/q) ] [2me —(q/2+md/q) ] '~, (7)

x T3 g IM(q, z) I 'n(coq)

dek

From the definitions cr e Ko and S (1/eT)K~/Ko wecan easily calculate the conductivity and thermopoweralong the c direction. In usual metals and semiconductorsboth K0 and EC~ are positive so that the sign of the thermo-power depends on the type of the charged carriers. Thethermopower is negative for n type carriers (e &-0) and is

with 8(x) the unit step function and m the effective massof the carriers. When the approximate relation df(ek)/dek= —b(ek —p) is used for Pp»1, Ka and K~ reduceto 20

Ko-(4md/zT)Z I M(q, ~) I 'n(toq)q, A,

x [I+n(toq)]y(p, q, a)q)/q,

positive for p-type carrier (e & 0). But in the present hop-ping transport model the conclusion is quite the contrary.It is easily seen from Eq. (7) that p(p, q, toq) &0 and[dp(ek, q, toq)/dek]„-„& 0 so that K~/Ko & 0. It then fol-lows that S, & 0 for electrons (e & 0) and S, & 0 for holes(e &0). This result is just opposite to the usual case inthe electron-phonon scattering mechanism. The physicalorigin of this effect is that for the hopping conduction pro-cess the electron-phonon interaction does help the out-of-plane transport of electrons, while for the usual case itplays the role of hindering the motion of electrons alorigthe direction of the applied field. Recent measure-ments on S, in single crystals of Y-Ba-Cu-0 have re-ported that the thermopower along the c axis is positive(S, & 0), from which it follows that the carriers dominat-ing the out-of-plane transport are electrons rather thanholes. This conclusion is also consistent with the mea-sured data of the Hall coefficient' which show that RH"(or Rtt' ) is negative or electronlike. Next, we discuss thetemperature dependences of o, and S, from the previousformula. For two-dimensional electron gas, due to the ex-

11 994 D. Y. XING, M. LIU, AND C. S. TING

istence of the e function which appears in expression (7)for p(sk, q, coq), the upper limit of the summation over thephonon energy is T rather than the Debye energy. 's Sowhen the condition T & T* is satisfied, the factorn(coq)[1+n(co~)] is approximately equal to (T/coq) I.tthen follows from Eqs. (8) and (9) that Kocx:T andKt cL T3. Finally, we obtain the linear T dependence ofboth cr, and S,: a, cx:Ko cx: T and S, ts'-Kt/TKo cs T.When the condition lt»T & T is not strictly satisfied,both o, and S, are still nearly linear T dependent from T,to room temperature if the parameters in Ref. 15 are em-ployed. Recently, it has been pointed out that the experi-mental data of the out-of-plane resistivity p, in single-crystal Y-Ba-Cu-0 are well described by the equation

p A/T+BT as proposed by the authors of Ref. 11. Inpractice a measurement of the resistivity along the c axisunavoidably picks up the in-plane component p,b whichcould account for the term of BT. So the linear T depen-dence of cr, ( 1/p, ) obtained from the hopping transportof electrons can be used to explain the experimental dataof the out-of-plane conductivity. At the same time, theobtained linear T dependence of S, is generally consistentwith the measurements ' of the out-of-plane TEP insingle-crystal Y-Ba-Cu-O.

The Hall coefficient R~~" or RH's with the magnetic fieldparallel to the Cu-0 planes is closely related to both thein-plane and out-of-plane transport properties. Accordingto the two-band model'3 in which there exist two types ofcarriers: electrons and holes, it then follows that RP' orRH' can be expressed as

b~z PPpgpc ~gnPnc 1

(pttp+np„)(plt~+nls ) (e(c ' (10)

where p and n are the density of holes and electrons, re-spectively. Jtt, and p„are the in-plane mobility of holesand electrons, respectively, and p~, and Jt are the corre-sponding out-of-plane mobility. c is the speed of light.Since we are proposing in this paper that the carriersdominating the out-of-plane transport are the electronsrather than the holes, we expect p »Jt~. Under thiscondition Eq. (10) reduces to

RH"- —I/[(n+pltp/p„) (e(c].

The measured Hall constant, ' with a magnetic field ap-plied parallel to the a-b plane, is negative and nearly in-dependent of temperature. This result can be explained interms of the present model by Eq. (11) in whichn+ pit~/Jt„ is almost T independent. '

Finally, we wish to point out that the electron-phononscattering, not the phonon-assisted electron hopping, ishighly likely responsible for the in-plane transport of thehigh-T, oxide compounds. It has been shown' 's that thelinear T dependence of the in-plane resistivity p,s from T,to room temperature can be explained by the traditionalelectron-phonon scattering mechanism in a simple single-band modeL Using sets of parameters extracted from theexperimental data for Y-Ba-Cu-0 and La-Sr-Cu-0 andreasonable McMillan's electron-phonon coupling parame-ters, we have obtained' quantitative agreements betweenthe calculated and measured p,b. But in order to explainthe nearly linear T dependence s of the Hall numberI/ecRQ' from T, to room temperature, with the magneticfield along the c axis, the two-band model with an exteriorcarrier source is required. ' Furthermore, using the sametwo-band model, the calculated thermopower S,b is posi-tive and weakly T dependent, ' which is also qualitativelyconsistent with the experimental data of S,b in single-crystal Y-Ba-Cu-0 obtained by Wang and Ong.

In summary, we have proposed that the out-of-planetransport of high-T, oxide compounds comes from thephonon-assisted hopping process of electrons between theneighboring Cu-0 planes. It has been shown that thishopping transport of electrons leads to S, & 0,Rg'(Rg ) (0, the linear T dependence of both cr, andS„and the T independence of Rg" and Rfg . All theseresults are in qualitative agreement with recent measure-ments of the out-of-plane transport coefficients in Y-Ba-Cu-0 single crystals.

We wish to thank T. K. Lee and L. Gao for helpful dis-cussions. This work was supported by the Texas Centerfor Superconductivity at the University of Houston underprime Grant No. MDA 972-8-G-0002 (from the U.S. De-fense Advanced Research Projects Agency and the Stateof Texas) and also a grant from the Robert A. WelchFoundation.

~S. W. Tozer et al. , Phys. Rev. Lett. 59, 1768 (1987).S. J. Hagen, T. W. Jing, Z. Z. Wang, J. Horvath, and N. P.

Ong, Ph'ys. Rev. B 37, 7928 (1988).3J. Wang, G. Chen, X. Chu, Y. Yann D. Zheng, Z. Mai, Q.

Yang, and Z. Zhao (unpublished); S. Martin, T. Fiory, R. M.Fleming, L. F. Schneemeyer, and J. V. Waszczak (unpub-lished).

4T. Penney, S. von Molnar, D. Kaiser, F. Holtzberg, and A. W.Kleinsasser, Phys. Rev. B 3$, 2918 (1988).

5G. S. Grader, P. K. Gallagher, and A. T. Fiory, Phys. Rev. B3$, 844 (1988).

sZ. Z. Wang and N. P. Ong, Phys. Rev. B 3$, 7160 (1988).7M. F. Crommie, A. Zettl, T. W. Barbee III, and M. L. Cohen,

Phys. Rev. B 37, 9734 (1988).SR. C. Yu et al. , Phys. Rev. B 37, 7963 (1988).

9R. J. Cava et al. , Phys. Rev. Lett. 5$, 1676 (1987).'oJ. M. Ziman, Electrons and Phonons (Oxford Univ. Press,

Oxford, 1960)."P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60, 132 (1988).

P. B. Allen, W. E. Pickett, and H. Krakauer, Phys. Rev. B 36,3926 (1987);37, 7482 (19&&).

'3D. Y. Xing and C. S. Ting, Phys. Rev. B 3$, 5134 (1988).~4F. Herman, R. V. Kasowski, and W. Y. Hsu, Phys. Rev. B 36,

6904 (1987).'5D. Y. Xing, M. Liu, and C. S. Ting, J. Phys. C 21, L591

(1988).'6R. Micnas, J. Ranninger, and S. Robaszkiewicz, Phys. Rev. B

36, 4051 (1987).~7D. Y. Xing, M. Liu, and C. S. Ting, Phys. Rev. B 37, 9769

(1988).

OUT-OF-PLANE TRANSPORT MECHANISM IN THE HIGH-T, . . . 11 995

' T. Siegrist, S. Sunshine, D. W. Murphy, R. J. Cava, and S. M.Zahurak, Phys. Rev. B 35, 7137 (1987).

' S. Massidda, J. Yu, A. J. Freeman, and D. D. Koelling, Phys,Lett. A 122, 198 (1987); L. F. Matteiss and D. R. Hamann,Solid State Commun. 63, 395 (1987).I. M. Lifshits, M. Ya. Azbel', and M. I. Kaganov, in ElectronTheory of Metals, edited by Soviet Physics Semiconductors

(Consultants Bureau, New York, 1973), p. 163.2'C. S. Ting and D. Y. Xing, in Proceeding of 1st Asia Pactftc

Conference on Condensed Matter Physics-High Temperature Superconductivity and Other Related Topics, Singa-pore, 1988, edited by K. K. Phau (World Scientific, Singa-pore, in press).