VOLUME 72, NUMBER 4 PHYSICAL REVIE%' LETTERS 24 JANUARY 1994

D(k) =dk

r '(k)— 1

Jk (tk +const)

as a function of k, and therefore the averaged diffusioncoefficient monotonically increases with decreasing tem-perature below T;..

The shape of the thermal conductivity of near 2D bo-sons below their condensation temperature, Fig. 1 in Ref.[I], is controlled by the simultaneous increase ofdiffusivity (due to the screening of the scattering poten-tial by the Bose condensate and the long-wave singularityof the group velocity dru/dk-k 't [I]) and decrease ofheat capacity C- T . Therefore there is no disagreementwith the experimental data [2].

In general we share the doubts by Yu, Salamon, andLu [3] about a purely phononic explanation of the

Alexandrov and Mott Reply: In our Letter [I] we found

an infinite thermal conductivity below T, o.f near two-

dimensional bosons for any short-range scattering poten-tial. The maximum value of the thermal conductivitybelow T, is .restricted by the long-range (Coulomb)scattering or by the 3D corrections to the energy disper-sion, but its position only slightly depends on the scatter-ing mechanisms or three dimensional corrections in awide range of the single parameter g. The maximum lies

approximately at T~0.4T, (see .Fig. I of our Letter).Contrary to the statement of the Comment by Fishman

and Kino [2] we find that their experimental result forthe "diffusivity" D =K/C actually supports our explana-tion of the thermal conductivity of high-T, copper oxides.To show this we present in Fig. 1 the temperature depen-dence of the diffusivity of near 2D charged bosons belowTc' &

(I —t)' " x'dxDK =D,,p (I)

sinh'(x) [x'+ ri(l t ) '/t—'] '

which is obtained using our formula for K, Eq. (24) in

Ref. [I] and the temperature dependence of the heatcapacity of 2D Coulomb Bose gas well below T„C-T.Here D,p is temperature independent. The theoreticalcurve does not show any maximum and is in qualitativeagreement with the experiment of Fishman and Kino (seeFig. I in Ref. [2]). There is also no maximum in adiffusion coefficient of the Bogoliubov excitations with aparticular rnomenturn k

r 2

1 10

oCl

10

A

0.1

0.010 0.2 0.4 0.6 0.8

t=T /' Tc

FIG. 1. Temperature dependence of the "diAusivity" of near2D charged bosons in the superconducting state, g =0.01.

A. S. Alexandrov and N. F. MottInterdisciplinary Research Centre in SuperconductivityUniversity of CambridgeMadingley RoadCambridge, CB3 OHE, United Kingdom

Received 1 November 1993PACS numbers: 74.20.Mn, 74.25.Fy, 74.72.—h

[I] A. S. Alexandrov und N. F. Mott, Phys. Rev. Lett. 7l,1075 (1993).

[2] I. M. Fishman and G. S. Kino, preceding Comment,Phys. Rev. Lett. 72, 588 (1994).

[3) R. C. Yu, M. B. Salamon, and J. P. Lu, Phys. Rev. Lett.7l, 1658 (1993).

thermal conductivity. They arise from low values of the caxis K in a superconductor and of the in-plane K in an in-

sulator, and from the near equality in superconductingand insulating samples above 100 K. These features of Kare difficult to explain with phonon transport without anunrealistic choice of five or more parameters. On the oth-er hand, our charged Bose-liquid model, which is a natu-ral extension of the BCS theory to the strong-couplinglimit explains all features of the thermal conductivity ofhigh-T, . oxides with only one parameter [I].

We acknowledge the assistance of William Beere.

0031-9007/94/72 (4)/589 ( I )$06.001994 The American Physical Society

589