THE EFFECT OF kHBROADENING ON SURFACE LANDAU LEVEL RESONANCES

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THE EFFECT OF kHBROADENING ON SURFACE LANDAU LEVEL RESONANCES

R. Gordon, J. Frandsen

To cite this version:

R. Gordon, J. Frandsen. THE EFFECT OF kHBROADENING ON SURFACE LANDAU LEVEL RESONANCES. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1135-C6-1137.

�10.1051/jphyscol:19786503�. �jpa-00217987�

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JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 8 , Tome 39, aofit 1978, page C6- 1 135

T H E

EFFECT

OF kH-BROADENING O N SURFACE LANDAU

LEVEL

RESONANCES R.A. Gordon and J.B. ~randsen'

+The TeehnieaZ University of Denmark, DK-2800 Lyngby,Dewnark

Computer Research Division, RegnecentraZen Hovedvejen 9, DK-2600 GZostmp, Denmark

Rdsum6.- A partir de calculs numdriques adaptss 1 la surface de Fermi de l'argent on montre l'importance de la distribution de k~ pour ddterminer 1 la fois la position et la forme de raie de la rdso- nance de surface des niveaux de Landau.

Abstract.- The importance of kH-broadening in determining both the position and line shape of Surfa- ce Landau level resonances is demonstrated by numerical calculations for a silver sample.

It has been well established that the resonances observed in the microwave surface impedance of a large number of metals in low magnetic fields are primarily due to resonant transitions between the Surface Landau Levels of electrons executing quan- tized skipping trajectories along the surface of the metal/l/. Experimentally the Surface Landau Level resonances originate from narrow bands of the Fermi surface

(2

^1-2" in width) for which the Fermi velocity is essentially parallel to the sample surface.

The integrated contribution of the various resonance bands along the direction of the magnetic field, H, (so-called kH-broadening) then results in the Surface Landau Level spectra. Such an integration has, however, never been carried out for other than the simplest Fermi surface geometries (e.q. a single ellipsoidal/2/ or circular band/3/) where the effect of kH-broadening is predictably small. It is the purpose of this communication to show that kH broa- dening can have a large effect on Surface Landau Level Resonance Spectra in more general Fermi Surfa- ce geometries. To this end we consider the calculation of a Surface Landau Level resonance spectrum in the

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1101 plane of a noble metal for which suffi- cient experimental data and knowledge of the Fermi surface exist to make such a comparison worthwhile.

This particular Fermi surface geometry is also of considerable interest for the information which such a calculation might yield on the anomalously large Fermi velocity obtained from SLL resonance measure- ments in silver141 where kHbroadening was not explicitly calculated for the relevant crystallogra- phic directions.

The contribution of Surface Landau Level Reso- nances to the low-field surface impedance of a metal

follows from the well-known Prange-Nee theory151

Here Z is the surface impedance for microwave currents flowing perpendicular to the magnetic field H, v~ = v (k

1

H ) is the component of the Fermi velocity parallel to the same microwave currents, a,(B,h)

is the matrix element between the various surface Landau Level states, 5 and

c

are the roots of the

m n

corresponding Surface Landau Level wavefunctions, T

is a field-independent relaxation time between electron scattering events, K = K(kH) is the local radius of curvature of the Fermi surface along narrow

"resonance" bands where the Fermi velocity is parallel to the surface, 6 is the microwave skin depth, while the remaining symbols represent the electronic charge (e), Planks' constant

(H),

and the angular frequency (U). In the case of a d.c. magnetic field along the < 1 10> direction in the

1

^{1 10}

1

plane of a noble metal, the k integration will include contri-

H

butions to dZ/dH from resonance bands perpendicular to the

"

shaft" and "knuckle"/6/ portions of the dog's bone hole orbit as well as a contribution from a band perpendicular to the central bulge of the electron belly orbit (see figure 1 ) . The neces- sity for such a kH integration is illustrated in figure la where it is seen that Surface Landau Level resonance curves calculated using the standard cy- linder approximation (where vl and K are taken to be equal to their extremal values independent o:

k ) disagree with the measured experimental resonan- H

ces both in position and line width regardless of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786503

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Fig. 1 : The derivative of the real part of the surface impedance, R, as a function of magnetic field, H. The solid curves represent experimental Surface Landau Level resonances in silver obtained by Diemel and Doezema (reference 141). The dotted and dashed curves in figure la represent numerically calculated curves using a cylindrical approximation for the Fermi Surface of silver with v,as given by Diemel and Doezema (dotted curve) and by Halse in reference /7/ (dashed curve). Figure l b shows Surface Landau Level resonances calculated using Halse's Fermi Surface geometry for the entire Fermi Surface (dashed curve) and for the central portion of the dog's bone orbit (dotted curve) alone. The insert in figure la shows represen- tative points from the resonance bands contribu- ting to the Surface Landau Level resonances (indi- cated by the arrows) from the dog's bone orbit (solid curve) and from the belly orbit (dashed curve). kp denotes the Fermi radius.

whether the generally well established value of v*

given by Halse/7/ or a revised value given by Diemel and Doezema/4/ is employed. It is important to note that the displacement of such approximate curves

(Q 0.3 G) suggested by the latter authors on the ba- sis of a kH integration performed for a simple circular resonance band131 is insufficient to give agreement with the position of the experimental resonance curve. A more realistic indication of the large effect of kHbroadening on the form of the Surface Landau Level resonances for the relevant experimental geometry is shown in figure Ib where the k integration is limited to the central portion

H

of the dog's bone orbit (the dotted curve) and also extended over the entire range of kH. The calculated

der to explicitly show the effects of the kH integration alone. Thevalue of

B

arid, to a lesser extent, the values of w~ will of course also be functions of kH SO that the form of.the resonance:spectra will al- sb be a function of kH. We have carried out such a k integration where

B

is a function of kH with the

H

minimumvalue of B(kH) chosen to be 0.5. The resul- ting resonance curve, after normalization to the experimental resonance differs only slightly in position (< - 0.2 G) and line width from the Surface Lan- dau Level resonance curve calculated for the single value of

B

= 0.5. The position of the integrated curves are nevertheless sensitive functions of both

B

and UT as the use of higher values of

B

and w~ in our calculations have shown (e.g. the integrated Surface Landau Resbnance maximum shifts to 19.4G for B = 0.625) and it must therefore be concluded that the choice of 8, wr, and v l which gives the best agreement with the experimental Surface Landau Level resonance curve can only be obtained by a comparison with numerically calculated resonance curves for a

systematic variation of B, w r 181, and v~ as parameters. Such numerical calculations would necessari- ly involve a good deal of computer work but the essential point to be made here is that until such a thorough investigation can be made, the presently- quoted values of Fermi surface parameters obtained from simpler analyses of the experimental Surface Landau Level resonances must be taken with a reaso- nable amount of caution. The same considerations will of course, also apply to similar investigations in the other noble metals where the Fermi surface geometry is almost identical to that of silver and will also be applicable in general to any Fermi surface geometry where the weight factor vL(kH)y(kH) in equation (1) is a much more slowly-varying function of kH than for the ellipsoidal and cylindrical resonance bands calculated heretofore.

curves in figure I b has been obtained using Halse's Fermi surface model171 with the experimental values for wr and

B

suggested by Diemel and Doezema in or-

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References

/ l / Nee,T.W., Koch,J.F. and Prange,R.E., Phys. Rev.

174 (1968) 758

-

/2/ Koch,J.F. and Jensen,J.D., Phys. Rev. +(1969) 643

Doez.ema,R.E. and Koch,J.F., Phys. Rev.

3866

141 Dieme1,P. and Doezema,R.E., Phys. Rev. (1974) 4897

/ 5 / Prange,R.E. and Nee,T.W., Phys. Rev.

168

⁽¹⁹⁶⁸⁾

779

161 Henningsen,J.O., Phys. Rev. (1971) 3180 /7/ Halse,M.R., Philos. Trans. R. Soc.

265

⁽¹⁹⁶⁹⁾

507

181 The effects of curvature broadening as well as field-dependent scattering might also have to be considered in r if a fitting of the calculated resonance curves to the experimental curve were required over the entire range of magnetic fields.