Nature of the intensification of a cyclotron resonance in potassium in a normal magnetic field N. A. Zimbovskaya and V. I. Okulov Institute of Metal Physics, Ural Science Center, Academy of Sciences of the USSR 6 0 Thecyclotron-resonancepeakwhichhasbeenobserved[G.A.Baraffet al.,Phys. Rev. Lett. 22, 0 2 590 (1969) ] in potassium in a magnetic field directed perpendicular to the surface may be due to an effect of zero-curvature pointsof theFermi surface on the cyclotron orbit of effective electrons. n a PACSnumbers: 71.18.+y,71.20-b,72.55+s J 6 1 Recent experimental data [2, 3] have altered the pic- vanishing of the curvature along the basic part of the ture of the Fermi suface of potassium as a closed, nearly Fermi surface. Since there are many gaps, the points of ] i spherical surface. That picture had been drawn on the zero curvature are distributed quite widely and fall on c basis of an analysis of de Haas-van Alphen oscillations the cyclotron orbit of the effective electrons over consid- s - [4]. Coulter and Datars [2] have observed open orbits erable intervals of the field orientation. There will then l r for several directions of the magnetic field. Jensen and be a markedincreaseinthe effective masscorresponding t m Plummer [3] have discussed photoemission data which to motion away from the boundary and thus in the time imply the existence of small energy gaps in potassium, spent by electrons in the skin layer; the effect will be to . t a as in sodium. According to the explanation offered for intensify the resonance. The deviations from a spherical m these results [2, 5, 6], there are charge density waves in shape of the Fermi surface are, on the whole, small and - the ground state of the conduction electrons in potas- donotsubstantiallychangethecyclotronfrequency. The d sium. These waves lead to discontinuities of spherical observed cyclotron-resonance peak can be described on n Fermi surface at several planes. the basis of arguments of this sort. o c The observed distortions of the Fermi sphere should In calculating the conductivity σ for a resonant cir- [ cular polarization of an alternating field of frequency ω be accompanied by the occurrence of transitions from a underconditionscorrespondingtotheanomalousskinef- 1 positive curvature to a negative curvature, i.e., by the v presenceofpoints orlines ofzerocurvature. Suchpoints fect, the deviations ofthe Fermisurface from a spherical 4 shape are important only for the component of the elec- or lines will lead to characteristic effects in the disper- 5 sion and absorption of short-wave sound [7, 8]. They tronvelocitywhichrunsnormaltotheboundary, vz. We 3 can accordingly write the conductivity in a normal field 1 should also affect the frequency dependence of the sur- as follows [9]: 0 face impedance and the cyclotron resonance in a normal at/06 fitaieoclnydco[l9off]te.rroeIndt-rfisoesrtohtnhuaesnecnxeepcpeeesrasimakrewynattsaolorrbeesesexuralvtmesdionfienRtaehfep.oe1tx,apwslsahiunermae- σ = 8πi3e¯h23q Z dψZ dpzw−mvv⊥2z((ppzz,)ψ) ≈ 4πe¯h23qp2F(1+(s1)), m plateinanormalfield. Thetheoryforaresonanceofthis where m and v⊥(pz) are the cyclotronmass andtrans- sort, which is based on the assumption that the Fermi - verse velocity component on a spherical Fermi surface; d surface is spherical [10], explains neither the compara- n tively large amplitude of the peak nor its shape. It can w = (ω −Ω +iν)/q; Ω is the cyclotron frequency; ν o is the collision frequency; and q is a wave vector. The nowbesuggestedthatanintensificationoftheresonance c integrationin (1)is carriedoutoverthe momentumpro- : occursbecausezero-curvaturepointsfallonthecyclotron v jection pz and over the angle ψ = Ωt (t is the time of orbit of the effective electrons, as predicted in Ref. [9]. i motionalongthe orbit). The asymptotic behaviorofthe X In offering an explanation for the intensification of the integral is calculated in the limit w 0. For a spheri- r resonance, Lacueva and Overhauser [11] assumed that → a this resonance stems from a small part of the Fermi sur- cal Fermi surface, vz is independent of ψ, and we have s = 0. The quantity s describes the contribution of a face, with the shape of a right cylinder, which has been point of zero curvature if it correspondsto effective elec- split off by a gap. However, the assumption that there trons with vz(pz,ψ)=0. The minimum of the function exists a finite, strictly cylindrical part of the Fermi sur- face, with the same cyclotron frequency as on a sphere, vz(pz,ψ) under the conditions pz = pz0 and ψ = ψ0 wouldbe one possibility forapointofthis type. Nearit, isnotjustified. Furthermore,aresonancewouldoccurin thatmodelonlyifthe magneticfieldwereorientedalong for small values of pz −pz0, ψ−ψ0, we can write the axis of the cylinder. This point makes it difficult to 2 vz(pz,ψ) = a(pz pz0) +2b(pz pz0)(ψ ψ0) explain the experimental data obtained with a polycrys- − − − 2 talline sample. Thereis astrongercasefor regardingthe + c(ψ ψ0) ; a>0; − 2 amplification of the resonance as resulting from a local ac b >0. (2) − 2 a positive, asymmetric, resonance peak with an abrupt low-field cutoff and a slow decay with distance from the point of the resonance. The peak observed in Ref. [1] hasspecificallythisqualitativeshape. Aparticularlyim- portant point is that the height of the peak, which is not explained in the model of a spherical Fermi surface, canbe reconciledwith (4)ata plausiblevalue of η. The scale of the variations and the height of the peak near the resonance (∆ 0.1) are close to those observedat valuesof η and|γ|o≤ntheorderof 10−1 10−2 (Fig. 1). ÷ A resonance may also be caused by the appearance of zero-curvature points of other types. For example, a narrower peak arises in the case in which we have ac b2 0 inEq. (2). Simulatingthefunction vz(pz,ψ) by−the→functionaldependence vz(pz,ψ)=a[pz pz0(ψ)]2 − over a small but nonzero interval of ψ for this case, we find s = η w0/w. The ∆-dependence factor in the FIG. 1: Curve 1 is relative magnitude of the real part of the term proporptional to η in Eq. (4) takes the form impedanceversusthemagneticfield(theratio Ω/ω =H/Hr) Re 1/√∆+iγ +(2+−√3)Im 1/√∆+iγ . near the resonance found in the experiments by Baraff et al [1]; curve2 is plot of expression (4) with γ2 =2×10−4 and (cid:8) (cid:9) (cid:8) (cid:9) Expression (4) was derived without allowance for the η = 0.03. The origin of the scale for the measured values Fermi-liquid interactionor the surface scattering of elec- of R(H)/R(0), whichwasnotdeterminedexperimetally,has been chosen in a way different from that in thefigurein Ref. trons. These factors may influence the position and in- [1] – in such a way that the resonant values are identical for tensity of the resonance peak, so this peak may shift curves 1 and 2. The cyclotron frequency Ω has been de- slightlyandchangeinshapewhenthesefactorsaretaken termined for a resonance in a field parallel to the boundary. into consideration. However,the basic resultwill remain A possible explanation for the slight difference between the in force. A cyclotron resonance is amplified in a normal positions of the peaks on curves 1 and 2 is that the effec- field because of the existence of zero-curvature points. tive electrons have a slightly shifted resonant frequency in a normal field. Acknowledgments: NAZ thanks G. M. Zimbovsky for help with the manuscript. The corresponding value of s is s=η(π ilnw0/w), (3) − [1] G. A. Baraff, C. C. Grimes, and P. M. Platzman, Phys. where the parameters η and w0 characterizethe prop- Rev. Lett.22, 590 (1969). erties of the Fermi surface near the point of zero cur- [2] P. G. Coulter and W. R. Datars, Can. J. Phys. 63, 159 vature. In particular, η is, in order of magnitude, the (1985). relativesize ofthe regioninwhichdependence (2)holds. [3] J.JensenandE.W.Plummer,Phys.Rev.Lett.55,1912 Since the distortions of the Fermi sphere are small, we (1985). [4] D. Shoenberg and P. J. Stiles, Proc. R. Soc. London have η 1; in calculating the surface impedance from ≪ A281, 62 (1964). (1)–(3) we can therefore expand it in powers of η. The [5] A. W. Overhauser,Can. J. Phys. 60, 687 (1982). resonant increment in which we are interested turns out [6] A. W. Overhauser,Phys. Rev.Lett. 55, 1916 (1985). to be small, in accordance with the experimental results [7] G.T.Avanesyan,M.I.Kaganov,andT.Yu.Lisovskaya, of Ref. [1]. We write, in the linear approximation in η, Zh.Eks.Teor.Fiz.75,1786(1978)[Sov.Phys.JETP48, the result calculated for the ratio of the real part of the 900 (1978)]. impedance in a magnetic field, R(H) to that without [8] V. M. Kontorovich and N. A. Stepanova, Zh. Eks. Teor. Fiz. 76, 642 (1979) [Sov. Phys.JETP 49, 321 (1979)]. the field, R(0) (this ratio was measured in Ref. [1]): [9] N. A. Zimbovskaya, V. I. Okulov, A. Yu. Romanov and V. P. Silin, Phys. Met. and Metalloved. 62, 1095 (1986) R(H) η γ =1 √3ln ∆2+γ2+sign∆arctan , (in Russian). R(0) −3h p ∆2+γ2i [10] A. V. Kobelev and V. P. Silin, Trudy FIAN 158, 125 p (4) (1985) (in Russian). where ∆ = 1 H/Hr, Hr is the resonant value of [11] G. Lacueva and A. W. Overhauser, Phys. Rev. B 33, − the field, and γ = ν/ω 1. Expression (4) describes 3765 (1985). ≪