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Submitted on 1 Jan 1978
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PHONON-DISLOCATION SCATTERING AND THE
LOW TEMPERATURE ELECTRICAL RESISTIVITY
OF POTASSIUM
R. Taylor, C. Leavens, M. Duesbery, M. Laubitz
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, suppldment au no 8, Tome 39, aotit 1978, page C6-1058
PHONON-DISLOCATION SCATTERING AND THE
LOWTEMPERATURE ELECTRICAL RESISTIVITY OF
POTASS
I
UM
R. Taylor, C.R. Leavens, M.S. Duesbejr and M.J. Laubitz
Physics Division, National Research Council of Canada, Ottawa K I A OR6
R6sumd.- Nous avons calculg la rbsistivitd Clectrique du potassium entre 1 K et
4
K, en utilisantla solution de l'dquation de Boltzmann pour les phonons, dans le cas oii interviennent simultandment les diffusions phonon-electron et phonon-dislocation. Nous avons considgrg un large domaine de densitds de dislocations raisonnables. Les rssultats fournissent une explication possible pour les
importantes ddviations h la rhgle de Matthiessen, observges rdcemment.
Abstract.- We have calculated the electrical resistivity of potassium for 1 K
<
T<
4 K using thesolution of the phonon Boltzmann equation for the case of mixed phonon-electron and phonon-dislo- cation scattering. A wide range of reasonable dislocation densities was considered. The results provide a possible explanation for the recently observed large deviations from Matthiessen's Rule.
Recently two groups/l,2/ measured the low
temperature ( T < 4
-
K) ideal electrical resistivity,p.(T), of high-purity potassium before and after a
long anneal period (80 days /l/ and 25 days 121). As expected, the residual resistivity, po, was lower after annealing but, remarkably, pi(T) also showed a large relative decrease. This change in pi(T) is difficult to understand in terms of the usual de- viations from Matthiessen's Rule (DMR's) arising from electron-impurity scattering. In these experi- ments the samples were extruded wires which must
have had very high densities of dislocations (N 'L
d
10'' cm-'). After a long anneal period Nd can easily
decrease by one or two orders of magnitude and it appears very likely that this change in Nd is in some way responsible for the large observed DMR's. We believe that the operative mechanism is phonon- dislocation scattering. The basic idea is that for high density of dislocations the phonon-dislocation
-
1scattering rate (T ) is sufficiently large to
pd
dominate other phonon scattering processes including that due to electrons. Under these conditions phonon drag is strongly quenched and pi(T) will be close to
the Bloch limit (BL). But with a drop of, say, two
orders of magnitude in T-' the quenching of phonon
pd
drag is much weaker resulting in a large drop in pi(T) towards the phonon drag limit (PDL).
To put the above on a more quantitative basis
we have calculated pi(T) allowing for possible phonon
drag effects by solving the phonon Boltzmann equation for the special case of mixed phonon-electron and
phonon-dislocation scattering 1 3 1 . For the latter we
have used Klemens1/4/ formula for the scattering of
phonons by the elastic strain field of the disloca-
t ion.
where b is the dislocation Burgers vector,y the Gru-
neisen constant and w(qX) is the phonon frequency
-
for wave vector q and branch
-
X.
In figure 1 are dis-played our results for a range of values of Nd. They approach the BL for Nd very large and the PDL for Nd very small as they must. We have also plot-
ted the experimental data for pi(T) from samples 2b
(before annealing) and 2c (after an 80 day anneal)
of van Kempen et a1 /I/. Sample 2b has been plotted
in two ways : (i) exactly as presented in reference
/l/ and (ii) with an increase in p of about 1 part
in 10'
.
For sample 2c we have made a similar adjus-tment to po. These changes were made to illustrate the point that experimentally it is difficult to determine the precise functional form of pi(T) at very low temperatures and to show that these data resemble the calculated curves.
The dislocation densities that would be needed to fit our calculations to the data are not at all unreasonable when we consider that the con- tribution of the dislocation core to the phonon scattering has been ignored in Klemens' formula. Until recently this contribution has generally been dismissed, on the grounds that the rigid-cylinder model of the dislocation core gives rise to only a very small temperature-independent thermal resis- tance /4/. However, recent work on the detailed atomic structure of the dislocation core (for a review see Vitek 151) has shown that the coreregion
References
I I I l the play in po, we see no clear-cut need to invoke
electron-electron scattering to explain the quali- tative features of the results of reference /l/. However, it may eventually turn out that a small
-
amount of electron-electron scattering is needed toobtain quantitative agreement between theory and experiment. Using an improved treatment of electron gas screening /7/ we find that the estimate provi- ded by Lawrence and Wilkins /8/ for this effect in K is almost certainly at least one order of magni- tude too high. Nevertheless, the results of Duwury et a1 /2/ which extend down to T 0,s K clearly show the presence of an additional contribution to pi(T) which cannot be absorbed by an adjustment of po. At the moment we have no explanation for the
-
4 x 1 0 9 T " ~ behaviour that they observe.In conclusion our results show that the ef- 4 x 1 0 8 fe'ct of phonon-dislocation scattering can have a
dramatic effect on the low temperature electrical resistivity whenever phonon drag is likely to be
Fig. 1 : The low temperature ideal electrical resis- /I/ van Kempen, H., Lass, J.S., ~ i b o t , J.H.J.M., tivity of potassium calculated for a range of densi- and Wyder, P., Phys.Rev.Lett. 37 (1976) 1574
-
-
ties of dislocations. The experimental points are /2/ Duwury, C., Rowland, J.A. and Woods, S.B., J. taken from van Kempen et a1 /l/ ; open circles, Phys. F: Metal Phys. (1978) to be published sample 2b ; solid circles, sample 2b adjusted (see
text); crosses, sample 2c adjusted (see text). / 3 / Bailyn, M., Phys.Rev.
112
(1958) 1587 present. The predictions of the model presented here can easily be tested by comparison of well- annealed and heavily deformed specimens./4/
Klemens, P.G., Thermal Conductivity ed, R.P, Tye, Vol,1,
p. 1 (Academic Press 1969)is wider than previously assumed. Further analysis
/5/ Vitek, V., Crystal Lattice Defects
2
(1974) 1 (to be reported at a later date) shows that the core/6/ Ziman, J.M., Electrons and Phonons (Oxford can give rise to a significant frequency-dependent University Press 1960)
contribution to T-' In addition to providing a
pd ' /7/ Geldart, D.J.W. and Taylor, R., Can. J. Phys.
48
further quenching of the phonon drag component of (1970) 167
the electrical resistivity, this will generate an 181 Lawrence, W - E - and Wilkins, J . W . , Phys.Rev. B7 important temperature-dependent contribution to the (1973) 2317
lattice thermal conductivity.
We should point out that we.have made no use of the so-called "fluttert' mechanism proposed by Ziman /6/ in which the dislocation acts as a damped oscillator in the phonon stress field and which, because of the different frequency dependence, could cause significant complications at higher tempera- tures.
In view of the crudeness of the present treat- ment of phonon-dislocation scattering and the uncer-
