Chemical potential oscillations mix individual-band frequencies of the de
Haas-van Alphen (dHvA) and Shubnikov-de Haas (SdH) magneto-oscillations in
canonical low-dimensional multi-band Fermi liquids. We predict a similar mixing
in canonical single-band Fermi liquids, which Fermi-surfaces have two or more
extremal cross-sections. Combination harmonics are analysed using a single-band
almost two-dimensional energy spectrum. We outline some experimental conditions
allowing for resolution of combination harmonics

A. S. Alexandrov

D. Schoenberg

J. M. Lifshits

M. V. Kartsovnik

R. Kubo

V. V. Kabanov

Y. Kurihara

English

arXiv

Chemical potential oscillations mix individual-band frequencies of the de Haas–van Alphen and
Shubnikov–de Haas magnetooscillations in canonical low-dimensional multiband Fermi liquids. We predict a
similar mixing in canonical single-band Fermi liquids, which Fermi surfaces have two or more extremal cross
sections. Combination harmonics are analyzed using a single-band almost two-dimensional energy spectrum.
We outline some experimental conditions allowing for resolution of combination harmonics

A.S. Alexandrov (7170284)

V.V. Kabanov (7170560)

Loughborough University Institutional Repository

   This item was submitted to Loughborough’s Institutional Repository (https://dspace.lboro.ac.uk/) by the author and is made available under the following Creative Commons Licence conditions.      For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/  Combination quantum oscillations in canonical single-band Fermi liquidsA. S. Alexandrov1 and V. V. Kabanov21Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom2Josef Stefan Institute 1001, Ljubljana, SloveniaReceived 3 August 2007; published 5 December 2007Chemical potential oscillations mix individual-band frequencies of the de Haas–van Alphen andShubnikov–de Haas magnetooscillations in canonical low-dimensional multiband Fermi liquids. We predict asimilar mixing in canonical single-band Fermi liquids, which Fermi surfaces have two or more extremal crosssections. Combination harmonics are analyzed using a single-band almost two-dimensional energy spectrum.We outline some experimental conditions allowing for resolution of combination harmonics.DOI: 10.1103/PhysRevB.76.233101 PACS numbers: 72.15.Gd, 75.75.a, 73.63.NmMagnetic quantum oscillations of magnetization deHaas–van Alphen dHvA effect and resistivityShubhikov–de Haas SdH effect are unequivocal hall-marks of the Fermi liquid, providing most reliable and de-tailed Fermi surfaces,1 in particular, in layered organicmetals2,3 and almost two-dimensional 2D superconductorssuch as Sr2RuO4.4 An interesting feature of dHvA and/orSdH oscillations is a difference between canonical and grandcanonical ensembles. While it is negligible in three-dimensional metals,5 the quantum fluctuations of two ther-modynamically identical ensembles are very different in 2Dmultiband metals. In particular, there are combination fre-quencies in dHvA and/or SdH oscillations of a 2D multibandmetal with fixed electron density ne canonical ensemble,predicted by Alexandrov and Bratkovsky AB,6 and studiednumerically6–11 and analytically.12–14 The effect has beenexperimentally observed in different low-dimensionalsystems.3,15,16 Obviously, there are no mixing of the funda-mental single-band frequencies, if the chemical potential  isfixed by a reservoir grand-canonical ensemble. Importantly,samples are normally placed on nonconducting substrateswith no electrodes attached, so the system is closed in actualdHvA experiments.As it happens the fundamental frequency mixing due tothe chemical potential oscillations AB effect may be ob-scured by mixing due to the magnetic breakdown17 MB ef-fect, as discussed by Kartsovnik.3 The MB effect is theswitching of two close electron orbits in different bands onthe Fermi surface FS at sufficiently strong magnetic fields.Here, we predict a mixing of two or more fundamental fre-quencies in a canonical single-band Fermi liquid with a fewextremal FS cross sections, where the MB is nonexistent.To illustrate the point, we consider an anisotropic singleband, with the dispersion Ek in zero magnetic field,Ek =2k22m− 2t coskd , 1which is a fair approximation for a band in layered metals.2,3Here, k and k are the in-plane and out-of-plane quasimo-menta, t is the interplane hopping integral, and d is theinterplane distance.When the magnetic field B is applied, the spectrum Eq.1 is quantized as18Enk = cn + 1/2 − 2t coskd ± gBB/2, 2where c=eB cos /m is the cyclotron frequency n=0,1 ,2 , . . . , t= tJ0kFd tan J0x is the Bessel func-tion,  is the angle between the field and the normal to theplanes, g is the electron g factor, and B is the Bohr magne-ton. The spectrum Eq. 2 is perfectly 2D at the Yamajiangles19 found from J0kFd tan=0, where kF= 2m1/2 is the Fermi momentum in pure 2D case, butotherwise there are two extremal semiclassical orbits. Theygive rise to beats in dHvA/SdH oscillations with twofundamental Fourier-transform FT frequencies, Fl,s=Al,s /2e cos, revealing modulations of the cylindricalFS along the perpendicular direction Fig. 1, as observed,e.g., in Sr2RuO4.4,16Since there are no different bands, one might expect nei-ther AB nor MB mixing of the fundamental frequencies Fland Fs in the single-band model Eq. 1, in contrast withcanonical multiband systems.6,17 Actually, as we show below,Fl and Fs turn out mixed, if ne is constant, so that a combi-nation frequency F+=Fl+Fs appears similar to the AB com-bination frequency6 in two-band canonical Fermi liquids. Us-ing conventional Poisson’s summation and integrals,1 thegrand canonical potential per unit volume, = −kBTeB cos42 n −/d/ddk ln1 + e−Enk/kBT ,3is given by =˜ −m2 /2d2, whereAl As k┴θBFIG. 1. Color online Large, Al=2m+2t /2, and small,As=2m−2t /2, extremal cross sections of a layered-metalFermi surface.PHYSICAL REVIEW B 76, 233101 20071098-0121/2007/7623/2331014 ©2007 The American Physical Society233101-1˜ =e2B2 cos243md r=1RTrcos rgmme cos	 − Rrr2J04rtc	cos2rc 4is its quantum part with the conventional temperature,RTr= 22rkBT /c sinh22rkBT /c, and Dingle colli-sion damping factor R, as derived in Ref. 13 0RTr ,R1. Differentiating  with respect to the magnetic field atconstant , one obtains the oscillating part of the magneti-zation, M˜ =−˜ /B,M˜ =e cos22d r=1− RrRTrrJ04rtcsin2rc ,5where we neglect small terms of the order of 2t /1, andtake g=0 for more transparency.We are interested in the regime c4t, where three-dimensional corrections to the spectrum are significant,rather than in the opposite ultraquantum limit,13 where thequantized spectrum is almost 2D. In our intermediate-fieldregime, one can replace the Bessel function in Eq. 5 by itsasymptotic, J0x2 /x1/2 cosx− /4 at large x to obtainM˜ =e cos42d  2BB1/2r=1− RrRTrr3/2sin2rFlB − 4 + sin2rFsB + 4 	 , 6where B4mt /e cosB.Naturally, the FT of Eq. 6 yields two fundamental fre-quencies in the grand-canonical ensemble, where  is fixedFig. 2. However, the chemical potential oscillates with themagnetic field in the canonical system,1,6 which affects quan-tum corrections to magnetization. Using ne=− /, onecan find the oscillating component, ˜zc /2=˜ /, ofthe chemical potential, =0+ ˜, where 0=d2ne /m isits zero-field value andz =  B2B1/2r=1− RrRTrr3/2sinrz + 2rf lB − 4 + sinrz + 2rfsB + 4 	 . 7Here, the “bare” fundamental frequencies, f l,s=m0±2t /e cos, are now field independent. Remark-ably, apart from a normalizing factor, the dimensionlessquantum correction z to the chemical potential Eq. 7 turnsout identical to the magnetization quantum correction M˜ Eq.6, which is not the case in a two-band canonical Fermiliquid.10To get insight regarding the FT of zB or M˜ B Eq. 7,we first apply an analytical perturbation approach of Refs. 10and 12 expanding z in powers of R up to the second order,zz1+z2+zmix, wherez1 = − RRT1 B2B1/2sin2f lB − 4 + sin2fsB + 4 	 8yields two first fundamental harmonics with the frequenciesf l and fs identical to those of the grand-canonical system,z2 R2RT223/2  B2B1/2sin4f lB − 4 + sin4fsB + 4 	 9yields two second fundamental harmonics with the frequen-cies 2f l and 2fs as in the grand-canonical system, andFIG. 2. Color online Upper panel: magnetization as functionsof the inverse magnetic field 1 /B for a grand-canonical and bcanonical single-band Fermi liquids, and c their difference. Lowerpanel: magnetization FTs for grand-canonical solid line and ca-nonical dashed line single-band Fermi liquids. Here, 2f=f l+ fs=6500 T, B=300 T, R=1, and T=g=0.BRIEF REPORTS PHYSICAL REVIEW B 76, 233101 2007233101-2zmix = R2RT21B2Bsin2F+B  10is the mixed harmonic with the frequency F+= f l+ fs, whichis a specific signature of the canonical ensemble. Its ampli-tude is small compared with the first-harmonic amplitudes asB /2B1/2 in contrast with multiband systems, where themixed-harmonic amplitudes have roughly the same order ofmagnitude as the fundamental-harmonic amplitudes at R=RTr=1.6,12 Also, there is no F−= f l− fs frequency in theFT spectrum of the single-band canonical system, differentfrom the multiband canonical systems.7,8To assess an accuracy of the analytical approximationEq. 10 and some experimental conditions, allowing forresolution of the mixed harmonic, we present numericallyexact magnetization and their FTs in Fig. 2 at T=0. Sinceconvergence of the sum in Eq. 7 is poor at T=0, one canuse its integral representation in numerical calculations asz =  B2B1/2Ie−i/4Li3/2− Reiz+2f l/B+ ei/4Li3/2− Reiz+2fs/B , 11where Li3/2x= 2 /1/20dtt1/2 / et /x−1 is the polyloga-rithm. The analytical amplitudes Eqs. 8–10 prove to bepractically exact with the relative error below 10% at any RFig. 3, as the amplitudes of the analytical theory of dHvAeffect in canonical multiband systems.10,12 Another importantfeature of the numerical FT of the solution of Eq. 11 is thatthe resolution of the mixed central peak in the middle be-tween two fundamental second harmonics Fig. 2 lowerpanel essentially depends on the magnetic-field windowused in FT Fig. 4. Since the mixed amplitude is relativelysmall as B /2B1/21, the window affects its experimen-tal resolution. We believe that a relatively small interval ofthe magnetic fields, used in FT, has prevented so far thesingle-band combination frequency to be seen in layeredmetals.2–4,16 Importantly, since the characteristic field, BJ0kFd tan /cos, is an oscillating function of thetilting angle , the combination amplitude also oscillates asa function of the angle, which could be instrumental in itsexperimental identification. We notice that the angle depen-dence of the second fundamental harmonics has been clearlyobserved in Sr2RuO4.16There is also mixing in the SdH quantum oscillations oftransverse and longitudinal conductivities. For example, thelongitudinal conductivity is given by the Kubo formula,20 = − e2 dEfEETrE − HvE − Hv ,12where v=2td sinkd / is the longitudinal component ofthe velocity operator, H is the single-particle Hamiltonianincluding the impurity scattering, and fE=1 / expE− /kBT+1 is the Fermi-Dirac distribution function. Aver-aging over random impurity distributions and approximatingthe scattering rate by a constant , one obtains the trace inEq. 12 as vIE−Enk− i−12 in the ladder approxi-mation. Then, applying Poisson’s summation, one canreadily obtain a quantum correction ˜ to the classical con-ductivity for detail see Refs. 21–23, which is at T=0˜  Br=1− RrrJ14rtccos2rc . 13The asymptotic of the Bessel function, J1x2 /x1/2 cosx+ /4, yields FTs of ˜ very similar tothose of magnetization Fig. 2, with the combination har-monic in the canonical system. Generally, the scattering ratedepends on the magnetic field,22 so that its oscillations re-quire more thorough analysis of the SdH effect, but mixingshould be robust. Interestingly, some mixing of fundamentalfrequencies may occur even in grand-canonical multi- orsingle-band layered systems, if there is an interband or inter-extremal cross-section scattering by impurities.FIG. 3. Relative errors of analytical harmonic amplitudes Eqs.8–10 with respect to numerically exact amplitudes.FIG. 4. Effect of the magnetic field window on the Fouriertransform of magnetization. Decreasing the window increases thewidth of FT harmonics obscuring the mixed harmonic.BRIEF REPORTS PHYSICAL REVIEW B 76, 233101 2007233101-3In conclusion, we have found the combination frequencyin the quantum magnetic oscillations of the single-band ca-nonical layered Fermi liquid. The difference between quan-tum oscillations of the canonical and grand-canonical en-sembles is tiny Fig. 2, but not obscured by the MB effect,which is absent in the single-band case in contrast with themultiband systems. We have also shown that the analyticalperturbation FT amplitudes are numerically accurate evenat zero temperature and in clean samples i.e., for R=1 asthey are in the multiband analytical theory.10,12 A widemagnetic-field window is essential for experimental resolu-tion of the combination dHvA/SdH frequency.We greatly appreciate valuable discussions with IorwerthThomas and support of this work by EPSRC UK GrantNo. EP/D035589.1 D. Schoenberg, Magnetic Oscillations in Metals Cambridge Uni-versity Press, Cambridge, 1984.2 J. Singleton, Rep. Prog. Phys. 63, 1111 2000.3 M. V. Kartsovnik, Chem. Rev. Washington, D.C. 104, 57372004 and references therein.4 A. P. Mackenzie, S. R. Julian, A. J. Diver, G. J. McMullan, M. P.Ray, G. G. Lonzarich, Y. Maeno, S. Nishizaki, and T. Fujita,Phys. Rev. Lett. 76, 3786 1996.5 J. M. Lifshits and A. M. Kosevich, Zh. Eksp. Teor. Fiz. 29, 7301955 Sov. Phys. JETP 2, 636 1956.6 A. S. Alexandrov and A. M. Bratkovsky, Phys. Rev. Lett. 76,1308 1996.7 M. Nakano, J. 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Combination quantum oscillations in canonical single-band Fermi liquids

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Combination quantum oscillations in canonical single-band Fermi liquids

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