PHONONS AND THE DE HAAS-VAN ALPHEN EFFECT

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PHONONS AND THE DE HAAS-VAN ALPHEN EFFECT

B. Watts, I. Chapman

To cite this version:

B. Watts, I. Chapman. PHONONS AND THE DE HAAS-VAN ALPHEN EFFECT. Journal de

Physique Colloques, 1978, 39 (C6), pp.C6-1086-C6-1088. �10.1051/jphyscol:19786481�. �jpa-00217963�

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JOURNAL DE PHYSIQUE Col10,pe C6, supplkment au no 8, Tome 39, aoGt 1978, page ~ 6 - 1 0 8 6

PHONONS AND THE DE HAAS-VAN ALPHEN EFFECT

B.R. Watts and I.A. Chapman

School of Mathematics and Physics, University of East Anglia, Norwich NR4 7TJ, U.K.

Rdsum6.- L'interaction dlectron-phonon est calculde pour ddterminer l'effet de Haas-van Alphen et dvalu6e dans le cas de l'orbite B du mercure. Nous regardons ensuite si cette action serait assez grande pour Ltre perceptible si elle Btait substitude dans la formule normale de Dingle.

Abstract.- Electron-phonon scattering is calculated for the de Haas-van Alphen effect, and evaluated for the

B

orbit in mercury. Assuming this scattering rate were substituted into a conventional Dingle term, the question is considered as to whether it would be large enough to be observed.

Palin/]/ has reported de Haas-van Alphen expe- riments on the

B

orbit in mercury up to temperatures of about 15 K. By puttilig the phonon scattering rate, estimated from thermal conductivity, into the conventional Dingle temperature term, he deduced that he should have been able to observe the effects of the T~ like phonon scattering. However he was un- able to observe any effect.

E = COS

h

^(2-zO)7

-

^{( 1 )}

Dephasing theory requires the strain to be integra- ted round the de Haas-van Alphen orbit of interest.

Suppose this orbit to be circular, radius ad

,

^and

orientated perpendicular to a general direction de- termined by polar coordinates (B,$). If dl is an element of length on the orbit, it is straightfor- ward to obtain the result

Explanations have been given by Engelsberg and

g

dl= 2ngoad cos(qz ) S (qa sine)

I

⁰ ⁰ ^d ⁽²⁾

Simpson/2/, Mueller and ~ ~ r o n / 3 / and also by Gantma-

where qz is the phase of the phonon at the orbit kher/4/, involving more sophisticated considerations.

center. --.- - - ^~ O

However, it may be noted that Palin's expectations

The phase error $ produced in the orbit is were based on scattering rates averaged over the

taken to be proportional to the average strain seen whole Fermi surface. Therefore a more trivial expla-

by the orbiting electron ; to be precise it is assu- nation of his observation might be that the actual

med that phonon scattering rate on the orbit (a

9

small

part of the Fermi surface) is substantially smaller than the average.

We have therefore made a calculation, applica- ble in principle to any de Haas-van Alphen orbit, and deduced a preliminary result for the

B

^{orbit in}

mercury which does, however, tentatively suggest that the phonon scattering rate ought to be large enough to produce measurable effects if the simple theory were correct.

The dephasing method of calculation has been described by Watts/5/. In this method the interaction of the electron with the strain field of the

where @ is the total phase of the orbit, which de- pends on the de Haas-van Alphen frequency F and the magnetic field strength B according to

c $ = - 2aF

B ( 4 )

and 1.1 is the sensitivity of the orbit to strain

The reduction of de Haas-van Alphen amplitude due to this one phonon is given by cos $

.

However, since

$ is extremely small in any finite sized crystal. and tends to zero as the crystal becomes infinite, we phonon (supposed frozen) is characterized by a para-

may write (I

-

in place of cos $ without any meter which is in effect the strain dependence of

significant approximation. Thus, due to this one the orbit in question ; this may either be measured

phonon, the amplitude reduction R I , is given by or, as in this case, calculated.

Suppose that there is a single phonon of angu- lnRl =

- -

€ 0 @ ~ cos(qzo)Jo(qad~in8)

2

'C - 1'

⁽⁶⁾

lar frequency w, and wavenumber q propagating along

the z direction. The strain field produced will be The next step is to add together the effects of all

of the form phonons. Because phonons are independent this is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786481

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simple in principle, and the result is obtained by summing 1nR. over all phonons. One naturally repla- ces the sum by an integral, which would be exact for an infinite crystal. The integral must be performed over all directions 8 of the phonon, over all z and over all q taking due regard to the density of modes. For the time being we assume one (unspecified) polarisation for the phonon and assume that p is constant. We shall return to this assumption later.

The integral over z simply introduces another fac- tor of

-,

1 and that over 8 involves the following

2 integral ;

Unfortunately we cannot do this integral exactly.

However, we note that I = 1 for small qad. For large qa we may assume the asymptotic form d

Jo(z) + ($)*cos(=-

T )

A (8 1 to obtain the result for large qa that

1 ^C1

I N - It is therefore plausible to assume the following approximate form for I :

With this approximate analytic form it is now possi- ble to integrate over all q (or U). The density of modes is

N(w) =

~a

^R

l?iZiiJ

⁽¹⁰⁾

where Q is the crystal volume and V the phonon velo- city. As well as the w dependent Bose-Einstein fac- tor, there is also an w dependence in

S; =

-

- QCij (1 1 )

where C is the stiffness constant relevant to the i j

phonon. This will be treated as a constant in the integration. Therefore the reduction of de Haas-van Alphen amplitude is given by

where w = V/(2ad )

,

B = H/@T), and the upper frequency in the second integral taken to be infinite without appreciable error. At any temperature abo- ve 0.1 K, Bwo<< 1 even for the smaller de Haas-van Alphen orbits. Therefore, for T

&

0.1 K, the second integral in (12) will dominate, and its lower limit set to zero without significant error. Evaluating this standard integral, one obtains

The approximation in (13) may now be seen to be ve- ry slight, in fact, because the approximate form as- sumed for I (9) is rather close to the true form when w>>wo, from where most of the contribution to

the integral in (12) comes.

In order to evaluate (13) for a given de Haas- van Alphen orbit we write both @ and ad (which ap- pears in wo) in terms of F and B, and write V in terms of Cij and p

,

the density. Thus,

1nR 2.

-

^3.9^X1 0 - ~ U 2 ~ 3 / 2 p ~ 3

(14) The familiar T~ dependence is seen, as well as the normal dependence on B. The effect of phonons is obviously largest in heavy, soft metals. In small orbits .I? is generally large, but F is small, so there is no obvious conclusion as to how p 2 ~ 3 b will vary from orbit to orbit. However, since small orbits are visible up to much higher temperatures, the best chance of seeing the effect of phonons would be in such as the B orbit of mercury. Since C is

i j relatively small for shear, we assume transverse phonons dominate. Putting into (14) the appropriate values of p and Cij for mercury together with the value of F fir the B orbit, and a value of B = 2T, as might well be appropriate in a typical experiment yields

1nR = 1 .l3 X ~ o - ~ ~ ~ T ~ (15) If we put T = 15 K, around the upper temperature used by Palin, then

Which may be compared with 1nR

- ^-

13 due to the normal temperature dependence of the de Haas-van Alphen effect. As a rough estimate one might therefore expect the effect of phonons to be unobserva- ble unless p2

lo3.

The value of

u2

used should be the average of over all phonon directions and polarisations. At present we are making detailed calculations of this quantity. Approximate calculations however give a value of about 5 X

lo3 ,

which should be treated as provisional.

Therefore, we conclude tentatively that the absence of any observable effect does probably need a sophisticated explanation.

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References

/ l / Palin,C.J., Proc. R. Soc. Lond. A

329

^{(1972) 17}

/2/ Engelsberg,S. and Simpson,G., Phys. Rev. (1970) 1657

/ 3 1 Mueller,F.M. and Myron,H.W., Commun. Phys.

1

(1976) 99

/ 4 / Gantmakher,V.E., Sov. Phys. JETP Lett.

16

(1972) 180

/ 5 / Watts,B.R., Phys. Condens. Matter.

19

(1975) 125