The hall coefficient for isotropic electron scattering

NF 22

Phrsica I08B (1981) 899-900 North.ltolland Publishi,g Company

THE HALL COEFFICIENT FOR ISOTROPIC ELECTRON SCATTERING

B.L. Gallagher and D. Greig

Physics Department University of Leeds Leeds LS2 9JT, U.K. The Hall coefficient, R H, of the ternary alloys Cu-Au-Ge and Cu-Ni-Ge is found to be independent of temperature from room temperature down to (at least) 5 K . Above about lOOK the values of RH are in close agreement with those of pure copper. From these two observations we deduce that, in accordance with predictions from de Haas van Alphen measurements, the scattering for both electron-phonon and electron-impurity processes is isotropic. For free electron metals the Hall coefficient, RH, is given very simply by RHF = -I/ne, where n is the electron density and e the electronic charge. On this first level of approximation the Hall coefficient of a pure metal is therefore expected to be independent both of temperature and residual impurity. In practice, however, R H is independent of T only between room temperature and about ;OOK . At lower temperature the values can change by as much as 100% and are completely determined by the type of impurity present. Furthermore as regards the numerical values of R H these are only close to RHF in a few metals such as K and Rb at the higher temperature end of the range. For a number of polyvalent metals R H is even of the 'wrong' (positive) sign. For the noble metals R H is certainly negative but at room temperature the values are about 50% smaller than IRHFI. The reason for these observations is that R H is only equal to RHF when all the conduction electrons are alike. Otherwise R H depends on (i) the curvature of the Fermi surface -- a 'geometrical' effect -- and (ii) the anisotropy of the relaxation time over the Fermi surface -- a scattering effect E l i . Quite generally we argue that at higher temperatures where phonon wavevectors are large the electron scattering is isotropic and the measured values of R H differ from RHF for reason (i). In the case of copper, for example, the negative curvature ~f the necks must lead to a reduction in the Hall voltage from its free electron value, and it is possible to use an expression first derived by Tsuji E 1,2 ~ to calculate R H to within a few percent of the experimental value. At low temperatures, on the other hand, the dominant scattering is always by 'impurities' (accidental or otherwise) so that the scattering is normally quite anisotropic with R H determined by a combination of (i) and (ii). In these circumstances it is considerably more difficult to calculate R H although here again, for very dilute alloys, the anisotropy of relaxation time is sufficiently well known to make a detailed calculation possible I_--3,4~.

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Several years ago Poulson et al. E 4 ] made the proposal that in the ternary alloys Cu-Au-Ge and Cu-Ni-Ge the impurity scattering should be isotropic. They argued that, whereas for Ge the scattering is greatest in the neck regions of the Fermi surface, for both Au and Ni belly scattering predominates with the scattering most nearly isotropic when the atomic ratios of Au:Ge and Ni:Ge are equal to 15.9.:I and 5"12:1 respectively. For such alloys R H should be determined solely by the curvature of the Fermi surface and should equal the room temperature value throughout the temperature range. At that time we published a note E 5 ~ giving values in the two alloys at a few selected temperatures showing that R H did indeed appear to be independent of T. However the data which were measured on very simple equipment were subject to uncertainties ~ 10% and could only be regarded as preliminary. In particular the values of ]RHI appeared to be ~ 15% greater than the room temperature value in pure copper E ] , 6 ] whereas a calculation for Cu-Ni based on the de Haas van Alphen data of T-eempleton and Coleridge E 7 ~ showed that the change in RH from effect (i) was likely to be extremely small. We now have available a much more sensitive apparatus incorporating a 4T split-pair superconducting magnet and gas-flow cryostat supplied by Thor Cryogenics and have measured RH down to helium temperatures on a new set of specimens. As before the specimens contained a total of 2 atomic percent solute so that for the Cu-Au-Ge alloy in particular the Ge content is extremely small. This time we have also included measurements on the Cu-Au binary alloy to emphasise the effect of adding the Ge. The samples were made by first preparing A u-Ge and Ni-Ge alloys and then adding these to copper. Apart from Ni-Ge which had to be melted in an argon arc furnace, all the alloys were prepared in sealed quartz tubes and then rolled down to foils ~ 0 . 3 m m thick. An important part of the hygiene of this process is to etch with nitric acid after each pass.

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Temperature (K) Figure I : Temperature variation of R H. O

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C u + 2 at % Au;

Cu + 2 at % (Ni-Ge).

The measurements shown in figure 1 give ample support for 3 major points. (i) The values of R H in both ternary alloys are indeed independent of T. (ii) The addition of the very small amount of Ge makes a spectacular difference to the low temperature measurements on -Cu-Au. (The results on this binary alloy --m are in excellent agreement with earlier measurements by Dugdale and Firth E 8 ] " (iii) In contrast to our preliminary measurements there is virtually no change in the value of [RH[ at room temperature from the value in pure copper. The whole experiment therefore gives valuable confirmation both of the theory of RH and of the experimental determination of scattering anisotropies.

REFERENCES

E5]

Greig, D., Lawson, G.M., Martin, M.F. and Morris, L.K.E., J.Phys. F: Metal Physics, 6 (1976) L28]-3.

We are most grateful to Mr M.J. Walker for preparing the specimens, to Dr I.M. Templeton for providing a computer program containing his results and to a former colleague, Mr G.M. Lawson for his general analysis of this problem.

[6]

Fletcher, R., Friedman, A.J., and Stott, M.J., J.Phys. F:Metal Physics, 2 (1972) 729-41.

E 7~

Templeton, I.M. and Coleridge, P.T., J.Phys.F:Metal Physics, 5 (1975) 1307-16.

[ I~

Hurd, C.M., The Hall effect in metals and alloys (Plenum, New York, 1972).

[2~

Tsuji, M., J.Phys. Soc. Japan 13 (1958) 979-86.

E3~

Coleridge, P.T., J.Phys.F:Metal Physics, 2 (1972) 1016-32.

4]

8~

Poulsen, R.G., Randles, D.L. and Springford, M., J.Phys. F:Metal Physics, 4 (]974) 98]-98.

Dugdale, J.S. and Firth, L.T., J.Phys. C: Solid State Physics, 2 (1969) ;272-84.