SURFACE ROUGHNESS KAPITZA CONDUCTANCE : DEPENDENCE ON MATERIAL PROPERTIES AND PHONON FREQUENCY

(1)

HAL Id: jpa-00221326

https://hal.archives-ouvertes.fr/jpa-00221326

Submitted on 1 Jan 1981

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

SURFACE ROUGHNESS KAPITZA

CONDUCTANCE : DEPENDENCE ON MATERIAL

PROPERTIES AND PHONON FREQUENCY

N. Shiren

To cite this version:

(2)

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZIrnent au n o 12, Tome 42, ddcernbre 1981 page C6-816

SURFACE ROUGHNESS K A P I T Z A CONDUCTANCE : DEPENDENCE ON MATERIAL PROPERTIES AND PHONON FREQUENCY

N.S. Shiren

IBM Thornas J. Watson Research Center, Yorktown Heights, IVY 10598, U.S.A.

Abstract.- A recent calculation of the surface roughness contribution t o Kapitza conductance is compared with measurements on several different solid materials in contact with He 11.

In a recent publication' I have presented some results of a first order perturbation calculation of the effect of a statistically rough surface on Kapitza resistance. These

3

results were compared with previously published data on Cu

-

He , Cu

-

He 11, and NaF-He I1 interfaces. Excellent agreement in magnitude, temperature dependence, and pressure dependence, was found at temperatures above 0.2 K for reasonably realistic values of the parameters (r.m.s. amplitude and correlation length) characterizing the roughness.

In this paper I compare the calculated Kapitza conductances for several materials with measured values compiled by ~ n ~ d e r , ~ and also discuss the frequency dependence of the scattered phonons.

Let M indicate the wave polarization in the solid; M = I for longitudinal waves; M = a , n for waves polarized perpendicular or parallel to the plane of scattering, respectively. Then, from reference 1, for an incident power per unit surface area per unit solid angle per unit frequency, d9'i/dQi, a t frequency o in the liquid, the fractional differential scattered flux into mode M in the solid is,

-D +

kll and k$ are the components of the wave vectors in the liquid and solid, respectively, parallel to the interface. The F, are dimensionless expressions which are functions of all the various scattering and input angles as well as the velocities and densities. db,/dQM is the scattered power per unit surface area per unit solid angle per unit frequency.

(3)

G(K) is the spectral density of the roughness modes with wave vector K and amplitude

lK.

For a simple exponential surface correlation function with correlation length

t

and mean square amplitude

<12>

(<

>

indicates an average over an ensemble of representations of the surface), G(K) is given by

4

The frequency dependence of DM is completely determined by the factor kMG. Let

II

H = (

I

kM-k

I

)/kM, then H is independent of frequency, and

Thus DM is directly proportional to the mean square of the perturbation parameter kM{, and ! appears as a scale factor on the frequency.

The total power per unit surface area scattered into the solid is

Its contribution to the Kapitza conductance is

h k = aQ,/dT , and the total thermal resistance of the interface is

Fig. 1 shows the frequency dependence of Q, for single frequency inputs. At the

4 2

low frequency end Q,m(kP) (<12>/p ) as expected from Eq. (2). However, for large (k!) small values of H are emphasized and Q, tends to vary as the integral of Eq. (2)

2 2

(over HdH), i.e. Q,oc(kt) (<12>/t' ). Since, as shown by Fig. 1, the scattering probabili- ty increases monotonically with frequency, for thermal inputs the scattered frequency distribution is shifted towards higher frequencies. However, thru H the amount of the shift is angle dependent; large angle inputs (from the helium) are shifted less than small angle inputs. These results are not dependent on the specific form of G(K) used here. They hold for any normalized G(K) which decreases monotonically with increasing K.

(4)

~ 6 - 8 18 JOURNAL DE PHYSIQUE

values) because it is more likely t h a t h s ( > > h i M . T h e integrations required for evaluation o f Eq. ( 5 ) were all computed numerically assuming Bi proportional t o a Debye density of states a n d equilibrium phonon populations. Of course, actual values of

<c2>

and P a r e

2 not known s o some choice has t o b e made. I have used the same values of <12>/p and P/vt (v, is the shear wave velocity) for all the materials.

Fig. 2 shows the relative values of hkplotted against relative hKmax (from snyder2). Obviously, we cannot expect exact agreement because of t h e unknown quantities

<12>

and P, however t h e general trend is correct.

F i g . : Relative values of total transmitted intensity ( Q , X P ~ / < { ~ > ) for H e II t o NaF, versus ktP. kt is the shear wave vector in the solid.

Fig. 2 : Relative values of scattering contrib- 2

-

ution t o Kapitza conductance,

h i ,

versus

largest measured values2 of h K for the indi- 1 ₁ ₂ ₄ ₁₀ ₂₀ ₄₀ cated materials in contact with H e 11. h, (kw/rn2~)- EX^.

References 1. N. S. Shiren, Phys. Rev. Letters, submitted.

2 . N. S. Snyder. NBS Technical Note 3 8 5 (U. S. Govt. Printing Office, Washington, 1969).