STRESS DEPENDENCE OF THE FERMI SURFACE OF LEAD

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STRESS DEPENDENCE OF THE FERMI SURFACE OF LEAD

W. Joss, W. van der Mark

To cite this version:

W. Joss, W. van der Mark. STRESS DEPENDENCE OF THE FERMI SURFACE OF LEAD.

Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1093-C6-1094. �10.1051/jphyscol:19786484�.

�jpa-00217966�

JOURNAL DE PHYSIQUE Colloque C6, suppl6ment au no 8, Tome 39, aofit 1978, page C6-1093

STRESS DEPENDENCE OF THE FERMI SURFACE OF LEAD

W. Joss and W. van der Mark

Eidgenijsische Technische HochschuZe, Laboratoriwn fi@ Festkisrperphysik, 8093 Ztirich, Switzerland

~ s u m 6 . - Les ddpendances en tension uniaxiale de plusieurs sections extrdmales de la surface de Fermi du plomb ont dtd ddtermindes 2 l'aide d'une mesure simultande de l'amplitude de la magnd- tostriction et du couple magndtique de Haas-van Alphen. Les ddrivdes par rapport h la tension des aires des sections extrhales sont compardes avec les valeurs thdoriques ddduites d'un calcul par la mdthode du pseudopotentiel utilisant 8 ondes p1,anes orthogonalisdes. Dans Le present mod~le'les pentes av/aq du facteur de forme prises aux vecteurs approprids du rdseau rdciproque sont utilisdes c o m e paramstres ajustables. Les parambtres optimalisds reproduisent aussi bien nos rdsultats en tension uniaxiale que les rdsultats en pression hydrostatique de Anderson et al. /l/.

Abstract.- The uniaxial stress dependence of several extremal cross-sections of the Fermi surface of lead has been determined using the results of simultaneous measurements of the amplitude of the oscillatory magnetostriction and the Haas-van Alphen torque. The stress derivatives of areas of extremal orbits are compared with the theoretical predictions of an 8-OPW (orthogonalized-plane- wave) pseudopotential calculation. In the model used, the slopes av/aq of the form-factor at the reciprocal lattice vectors of interest are treated as fitting parameters. The fitted values explain both our uniaxial stress data and the hydrostatic pressure data of Anderson et al. /l/.

Lead is the simple metal whose Fermi surface has been the most intensively studied, and the question of the interp,retationof the experimental data is still open. On the one hand Anderson et al.

/ l / have to assume a non-local dependent

pseudopotential in their 4-OPW calculation to re- produce their results. On the other hand Van Dyke 121 uses a local pseudopotential but has to go up to 90-OPW's to achieve convergence. In order to settle the question we made an experimental and theoretical investigation of lead. Our conclusion is that the full stress dependence of the Fermi surface of lead can be explained without having to introduce these, partly unphysical, refinements.

The uniaxial stress dependence of various Fermi surface cross sections was determined by com- bining the amplitude of the oscillatory magnetos- triction E = AE/k and the Haas-van Alphen torque T

131. The stress dependence for an extremal orbit of area A is given by the relation

(I/A)dA/da = -(e/~)(l/A)dA/d$, where the stress a is parallel to a crystallographic axis, and is the angle between the stress a and the directionof the magnetic field. The sign of the stress depen- dence is determined by the relative phase between the two oscillatory quantities BE/& and T. Both ef- fects were simultaneously measured with a dilator- quemeter 141. The measurements were done at 1.3 K in magnetic fields up to 10.5 Tesla.

Our results on the uniaxial stress dependence are summarized in Table I. Table I1 shows that the hydrostatic pressure dependence calculated from the uniaxial stress dependences are in good agreement with the direct measurements /l/.

Table I

Experimental and theoretical stress derivatives of extremal cross sectional areas of the Fermi surface of lead. (Units 1 0 - ~ bar1) Orbit

For the calculation of the Fermi surface we used an 8-OPW matrix which is the minimum size re- quired to describe the stress dependence under a general homogeneous strain. The spin-orbit inter- action was not included and all matrix elements Vs = V(Gs) for reciprocal lattice vectors (recips) Gs with modulus larger than the Fermi wave number kF were set equal zero. The stress derivative of an extremal cross-sectional area A is given by 131:

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786484

where alnA/ao is a partial derivative taken at constant EF and Vs. All the partial derivatives of A may be calculated analytically /4/, and depend only on the zero pressure parameters EF=0.35085a;u., V l l l = -0.04769 a.u., V200= -0.02409a.u. The

Table I1

Hydrostatic pressure derivatives for lead determined from (a) uniaxial stress data (b) direct measurements by Anderson et al. /I/ (c) theoretical

calculation. (Units 1 0 - ~ bar-').

Orbit dlnA/dp

(a) (b) (C)

partial derivatives of the Fermi energy are calcu- lated by computing the density of states and its derivatives with the Gilat and Raubenheimer method /5/. For the evaluation of dV /do we used /3/ :

where v is the form factor. av /alnkF is calculated by assuming a local expression for the form factor and av/alnq is considered as the only fitting para- meter. A fit to the uniaxial stress and hydrostatic pressure data leads to :

In parenthesis we give the slopes computed from the form factor of Appapillai and Heine /6/ for compa- rison.

values to those obtained from the Appapillai and Heine form factor shows that the non-locality ari- sing from folding back the secular equation / 7 / as well as the neglect of matrix elements for recips longer than kF do not play an important role forthe stress dependence of the Fermi surface. The uniaxial stress dependences calculated with the fitted form factor slopes are listed in Table I, and the results for hydrostatic pressure are given in Table 11. The root-mean-square deviation of the hydrostatic pres- sure dependences corresponding to our fit

( M S = 0.31 X 1 0 - ~ bar-') is significantly better than for the existing model of Anderson et al. /l/

using the local potential approximation

( M S = 1.29 X I O - ~ bar1). It is also better than that obtained by van Dyke /2/ (RMS = 0.67 X 10-~bar-') by means of a 90 OPW band structure calculation and the non-local 4-OPW calculation of Anderson et al.

/l/ (RMS = 0.82 X 1 0 - ~ bar-l).

This work was supported financially by the Schweizerischer Nationalfonds zur Fgrderung der wissenschaftlichen Forschung.

References

/l/ Anderson, J.R., O'Sullivan, W.J. and Schirber, J.E., Phys. Rev. (1972) 4683 /2/ van Dyke, J.P., Phys. Rev. (1973) 2358 /3/ Griessen, R. and Sorbello, R.S., J. Low Temp.

Phys.

2

141 Griessen, R., Lee, M.J.G. and Stanley, D.J., Phys. Rev.

B16

151 Gilat, G. and Raubenheimer, L.J., Phys. Rev.

144 (1966) 390

/6/ Appapillai, M. and Heine, V., Tech. Rep.

2,

Solid State Theory Group, Gavendish Laboratory, Cambridge (1972)

/7/ Heine, V., in Solid State Physics

14

In contrast to van Dyke's approach, our fit- ting procedure uses no constraint on the slopes of the form factor. Furthermore, the closeness of our