ELSEVIER Physica C 282-287 (1997) 1919-1920

PHYSICA ;.,,

E L E C T R O N - E L E C T R O N A N D E L E C T R O N - P H O N O N I N T E R A C T I O N S IN A L K A L I

M E T A L D O P E D F U L L E R E N E S : I M P L I C A T I O N S F O R T H E R E S I S T I V I T Y

Dinesh Varshney a* and R.K. Singh b

aSchool of Physics, Devi Ahilya University, Khandwa Road, Indore 452 001, India.

blnstitute of Physics, Guru Ghasidas University, Bilaspur 495009, India.

Temperature dependence of resistivity of K3C60 is theoretically investigated on the basis of Ziman's formula. Estimated phonon contribution when subtracted from the single crystal data infers a quadratic temperature dependence over the most of temperature range (20_<T<100K). Quadratic temperature dependence is understood in terms of3D electron-electron inelastic scattering.

1. I N T R O D U C T I O N 2. F O R M A L I S M

Within recent years, much attention has been given in the study of alkali metal doped fullerenes M3C60

(M=K, Rb) as they are the highest Tc isotropic three dimensional (3D) superconductors presently known [1]. Inelastic neutron scattering measurements [2] suggest that the vibrational spectrum of K3C60

involves the intermolecular (2.5 to 25 meV) as well as intramolecular (25 to 200 meV) phonons. As regards of pairing mechanism is concerned it is argued that the coupling of electrons with the high frequency intramolecular phonons (O)ra) will describe the

superconducting state [3]. Measurements of normal state transport properties as resistivity will provide a means to understand which molecular phonons are participating in these materials and can be well described within the Bloch Boltazman transport theory [4] based on Migdal approximation which is valid if the electron-phonon scattering process dominates i.e., on the existence of small parameter [N(o) h co(q)]. Previous measurements of resistivity on single crystal of K3C60 has shown a linear temperature

dependence above Tc [5]. In the present work we have first estimated the resistivity due to electron- intramolecular phonon interaction (Pe-ph) and then

subtracted it from single crystal data. A clear T 2 dependence of p is depicted which is understood in terms of 3D electron-electon scattering.

* corresponding author.

The contribution to the normal state resistivity due to electron-phonon scattering is calculated at high temperatures using the resistivity formula [4]

8rt2KB ~. Pe-ph(7) = ~ ~ T ( 1 )

where O)p is the bare plasma frequency,

=

Here, N(o) and V F are the density of states and Fermi velocity, respectively.

The dimensionless electron-phonon coupling strength (L) is expressed as

= N(o) <12> / M <~2> (3)

with <I2> is

<I2> = ~Krq3lo V(q)12dq/ ~ KI" qdq (4)

Here, V(q) is the screened Coulomb potential and is V°(q)/e(q) with V°(q) represents bare Coulomb potential. The static dielectric function is e(q) and in the low (q) limit we write it as e(q) = 4meZKF / (xh2q2). Thus, we have V(q) as

4xe2/q2%o V(q) = 4mezKF /~2q2

0921-4534/97/$17.00 © Elsevier Science B.V. All rights reserved. PII S0921-4534(97)01012-5

1920 D. Varshney, R.K. Singh/Physica C 282-287 (1997) 1919-1920

= h2x2/rnKl~oo (5)

Using (4) and (5), one obtained

<I2> = ~ (6)

to get the coupling strength (%) as

~. = 2N(o) (h27t2/m) 2 (7)

M~oo <.Q2>

with <f22> v2 is the renormalized phonon frequency

and M denotes the ionic mass.

3. RESULTS AND DISCUSSION

Treating K3C60 as a diatomic chain and using the experimental data on bulk modulus as well as lattice parameters, we deduce the intermolecular vibrational frequency coer=60 cm'l [6]. For this the mass of 3 alkali metal is taken as 0.1947 x 10 -24 gm and charge carrier density is estimated as 4.12x1021 cm :3. We use the

band structure mass value m as 2 m e to obtain KF=4.95 x 107cm" 1, VF = 0.285 x 108 cm see" 1 ,gF = 0.4 eV, N(o)

= 1.6 eV/spin cell and cop = 1.2eV. With these realistic

parameter based on experimental data, we estimates the coupling strength (~.) for both the inter- and intramolecular phonons from (7). Taking O)er into considerations, we find %er as 1.47, while cora

(=1600cm "l) leads to a value of Z.ra=0.22. This is

consistent with the earlier reports [7]. Looking to the validity of BIoch Boitazman theory we find that parameter N(o)h co(q) is two small when coer is taken into account in comparison to cora. Thus, it is appropriate to consider the electron-intramolecular phonon coupling strength for fitting the metallic resistivity of K3C60. Using the extrapolated value of

p(o) as 0.18 m[2cm from single crystal measurement

[5] together with estimated Pe-ph is plotted, a clear

difference is seen from Tc to room temperature. Subtraction of estimated 9(-519o + Pe-ptO with the

measured 19 value yields a quadratic temperature dependence. We deduce Pe_e=4xl0 "4 f~cm at T=100

K. It is appropriate to modelled the metallic resistivity of K3C60 as a three component expression of the

form 9(7) = 9o + 9e-ph(T) + Pe-e(T), where 9o is zero

temperature residual resistivity which is a temperature independent contribution due to disorder whose magnitude is determined from the experimental data. Pe-ph was evaluated using the Ziman's formula

considering the contributions from intramolecular

phonons and extra term Pe-e was indeed required to

understand the measured resistivity data, since attempts to fit the data with Po + Pe-ph were not

successful. Nevertheless, if T 2 contribution to p is indeed due to 3D electron-electron scattering then its magnitude depend conventionally on the carrier concentration as n 5/3 or on the plasma frequency as

cop j%. Furthermore from the slope of resistivity

ot = dPe_ph/dT we find electron-phonon scattering

time'Ce_phaS 1.1xl0-14sec. WeuseL = VF~ toestimate

mean free path as 31.35 A ° which suggest that K3C60 is neither in clean nor in dirty limit. This is consistent with the earlier reports [8]. In conclusion the analysis presented here utilizes the Bloch Boltazman theory to estimate the contribution to resistivity (9) from electron-intramolecular phonon scattering and iden t i f i ed the ex t ra co n t r i b u t i o n from 3D electron-electron scattering. In addition, we also succeeded in establishing a correlation in magnitude of Pe-e and the plasma frequency COp of 3D isotropic

K3C60 superconductors.

ACKNOWLEDGEMENT : DV thanks Madhya Pradesh Council of Science and Technology, Bhopal, India for financial assistance.

REFERENCES

1. R. C. Haddon, et al., Nature 350 (1991) 320.

2. K. Prassides, et al., Nature 354 (1991) 462.

3. C. M. Verma, et al., Science 254 (1991) 989.

4. J. M. Ziman, Electrons and Phonons (Oxford Univ. Press, New York) (1987).

5. X. D. Xiang, et ai., Science 256 (1992) 1190; 352 (! 992) 1900.

6. Dinesh Varshney and R.K. Singh, Jour. of Super- conductivity (1996) Commun.

7. V.Z. Kresin, Phys. Rev. B46 (1992) 14833.

8. J.G. Hou, et al., Sol. Stat. Commun. 86 (1993) 643