Effect of Fermi-liquid interactions on the low-temperature de Haas - van Alphen oscillations in quasi-two-dimensional conductors
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Effect of Fermi-liquid interactions on the low-temperature de Haas-van Alphenoscillations in quasi-two-dimensional conductors
Natalya A. Zimbovskaya
Department of Physics and Electronics, University of Puerto Rico-Humacao, CUH Station, Humacao, PR 00791and Institute for Functional Nanomaterials, University of Puerto Rico, San Juan, PR 00931 (Dated: November 6, 2018)In this work we present the results of theoretical analysis of the de Haas-van Alphen oscillations inquasi-two-dimensional conductors. We have been studying the effect of the Fermi-liquid correlationsof charge carriers on the above oscillations. It was shown that at reasonably low temperatures andweak electron scattering the Fermi-liquid interactions may cause noticeable changes in both ampli-tude and shape of the oscillations even at realistically small values of the Fermi-liquid parameters.Also, we show that the Fermi-liquid interactions in the system of the charge carriers may causemagnetic instability of a quasi-two-dimensional conductor near the peaks of quantum oscillationsin the electron density of states at the Fermi surface, indicating the possibility for the diamagneticphase transition within the relevant ranges of the applied magnetic fields.
PACS numbers: 71.18.+y, 71.20-b, 72.55+s
I. INTRODUCTION
Magnetic quantum oscillations [1, 2, 3] have been rec-ognized as one of the major tools to map Fermi surfaces(FS) in metals. Analysis of the experimental data isnow a well established procedure and is based on theclassical paper by Lifshitz and Kosevich (LK) [4] whofirst layed out a quantitative theory of the de Haas-vanAlphen effect. The LK expression for the oscillating partof the thermodynamic potential and magnetization ofmetallic electrons in a strong (quantizing) magnetic field B was derived assuming that conduction electrons arenoninteracting quasiparticles in a periodic crystal poten-tial. This potential determines the electron dispersion, E ( p ) ( p being the electron quasimomentum), and there-fore, the effective mass of conduction electrons m ∗ andtheir FS. In fact, conduction electrons interact with eachother. Studies of modifications of the LK results arisingdue to electron-electron interactions within the generalmany-body quantum field theoretical approach startedin early sixties in the works of Luttinger [5], and con-tinued through the next three decades [6]. It was shownthat electron-electron interactions may bring noticeablechanges in the de Haas-van Alphen oscillations whichmakes further analysis worthwhile.One of the oldest and still powerful method to dealwith electron-electron interaction is the Landau Fermi-liquid (FL) theory [7, 8, 9]. It is important to realize thatwhile the phenomenological Fermi liquid theory and themicroscopic many-body perturbation theory (and, whenapplicable, the exact density functional theory) by def-inition lead to the same observable quantities, such asresponse to an external field, both zero approximationand its renormalization depend on the taken approach.A discussion to this effect in application to the dielectricresponse of metals can be found, for instance, in Ref. [10].It is always instructive to look at the same phenomenon from different points of view. Respecting the value ofthe many-body perturbation theory as applied to quan-tum oscillations [6], we emphasize that the Fermi-liquidtheory has provided important insights in such areas, rel-evant for the de Haas-van Alphen physics, as high fre-quency collective modes in metals [11, 12, 13, 14, 15, 16],or oscillations of various thermodynamic observables inquantizing magnetic fields [17, 18, 19, 20].An advantage of this phenomenological theory is thatit enables to describe the effects of quasiparticles in-teractions in such a way that makes the interpretationof the results rather transparent, as compared with thefield-theoretical methods. At the same time the many-body approach brings general but cumbersome results,and usually it takes great calculational efforts and/orsignificant simplifications to get suitable expressions forcomparison with experimental data. Which is more im-portant, adopted simplifications may lead to omission ofsome qualitative effects of electron-electron interactionsin quantum oscillations, as we show below.In the last two decades an entire series of quasi-two-dimensional (Q2D) materials with metallic type conduc-tivity has been synthesized. These are organic conduc-tors belonging to the family of tetrathiafulvalene salts,dichalcogenides of transition metals, intercalated com-pounds and some other. At present, these materials at-tract a significant interest. Their electronic properties areintensively studied, and the de Haas-van Alphen effect isemployed as a tool in these studies [2, 3]. Correspond-ingly, the theory of this effect in the Q2D materials iscurrently being developed [21, 22, 23, 24, 25, 26]. Thepresent analysis of the effect of Fermi-liquid interactionson the de Haas-van Alphen oscillations contributes tothe above theory. Also, the analysis is motivated by thespecial features in the electron spectra in Q2D materialsproviding better opportunities for the Fermi-liquid effectsto be manifested, as was mentioned in some earlier works[17, 20].In this paper we show how renormalizations of con-duction electron characteristics arising from FL interac-tions affect quantum oscillations, and we express them inthe form appropriate for comparison with experiments.Also, we show that a magnetic phase transition lead-ing to emergence of diamagnetic domains may happenat low temperatures when the cyclotron quantum ~ ω is very large compared to the temperature expressed inthe energy units: ( ~ ω ≫ k B T ) . We emphasize thatthe analyzed effects are different from the many-bodyrenormalizations of the band structure. Within the phe-nomenological theory the latter are already included inthe ground state of conduction electrons. II. FERMI-LIQUID RENORMALIZATIONS OFTHE CONDUCTION ELECTRONCHARACTERISTICS
Within the phenomenological Landau FL theory singlequasiparticle energies get renormalized, and the renor-malization is determined by the distribution of excitedquasiparticles. Accordingly, the energy of a “bare” (non-interacting) quasiparticle E ( p ) moving in the effectivecrystal potential is replaced by the renormalized energydefined by the relation [7]: E σ ( p , r ,t ) = E ( p ) + X p ′ σ ′ F ( p , s ; p ′ , s ′ ) δρ ( p ′ , s ′ , r ,t ) . (1)Note that E ( p ) here is the energy spectrum in the ab-sence of any excited quasiparticles, that is, at equilib-rium and at zero temperature, while δρ ( p ′ , s ′ , r ,t ) rep-resents the nonequilibrium part of the electrons distri-bution function, which may depend on both position r of the quasiparticle and time t. Also, s , s ′ are spinPauli matrices, ( σ is the spin quantum number), and F ( p , s ; p ′ , s ′ ) is the Fermi-liquid kernel (Landau corre-lation function), which describes additional renormaliza-tion of the quasiparticle spectrum due to interaction withother excited quasiparticles (but not with all electronsin the system, which is included in E ( p ) ). Neglectingspin-orbit interactions, the Landau correlation functionmay be written as: F ( p , s ; p ′ , s ′ ) = ϕ ( p , p ′ ) + 4 ψ ( p , p ′ )( ss ′ ) . (2)As follows from the Eq. 1, the conduction electronvelocity v = ∇ p E differs from the bare velocity v = ∇ p E . To proceed in our analysis we need to bring ve-locities v and v into correlation. For brevity we donot explicitly write out the variables r ,t in the Eq. 1in further calculations. This omission does not influencethe results. Differentiating the Eq. 1 we obtain: v σ ( p ) = v ( p ) + ∇ p X p ′ σ ′ F ( p , s ; p ′ , s ′ ) δρ ( p ′ , s ′ ) . (3) The electron distribution function ρ ( p , s ) is the sum ofthe equilibrium part ρ (the latter coincides with theFermi distrubution function for quasiparticles with singleparticle energies E ( p )) and the nonequlibrium correc-tion δρ. Multiplying both parts of the Eq. 3 by ρ ( p , s )and performing summation over p , σ we get: X p σ ρ ( p , s ) v σ ( p ) = X p σ ρ ( p , s ) v ( p ) + X p σ ρ ( p , s ) ∇ p × X p ′ σ ′ F ( p , s ; p ′ , s ′ ) δρ ( p ′ , s ′ ) . (4)The second term on the right hand side of the Eq. 4could be converted to the form − X p σ X p ′ σ ′ F ( p , s ; p ′ , s ′ ) δρ ( p ′ , s ′ ) ∇ p ρ ( p , s ) . (5)Keeping only the terms linear in δρ (which is sup-posed to be small compared to the equilibrium part ofthe distribution function), we may approximate ∇ p ρ as v σ ( p ) ∂f p σ ∂E p σ . Here, f is the Fermi distribution functionand the quasiparticle energies E σ ( p ) correspond to thelocal equilibrium of the electron liquid. Also, assumingthat the FS of a considered metal possesses a center ofsymmetry, we get X p σ ρ ( p , s ) v σ ( p ) = X p σ δρ ( p , s ) v σ ( p ) , X p σ ρ ( p , s ) v ( p ) = X p σ δρ ( p , s ) v ( p ) . (6)Then, using the well known relation F ( p , s ; p ′ , s ′ ) = F ( p ′ , s ′ ; p , s ) and carrying out the replacement p , s ⇋ p ′ , s ′ in the sums included in the Eq. 5, we couldrewrite Eq. 4 as X p σ δρ ( p , s ) " v σ ( p ) + X p σ ∂f p ′ σ ′ ∂E p ′ σ ′ F ( p , s ; p ′ , s ′ ) v σ ′ ( p ′ ) = X p σ δρ ( p , s ) v ( p ) . (7)Solving this for v , we obtain v σ ( p ) = v ( p ) − X p ′ ,σ ′ ∂f p ′ σ ′ ∂E p ′ σ ′ F ( p , s ; p ′ , s ′ ) v σ ′ ( p ′ ) . (8)Exact expressions for the functions ϕ ( p , p ′ ) and ψ ( p , p ′ ) are of course unknown. The simplest approx-imation is to treat them as constants. This approxima-tion is reasonable as long as the interaction of quasipar-ticles (located at r and r ′ , respectively), is extremelyshort range, so that the interaction can be approximatedas V ( r , r ′ ) = Iδ ( r − r ′ ) . Using this approximation onecaptures some FL effects but in general case it is notsufficient.As a next step, one may expand the Fermi-liquid func-tions in the Eq. 2 in basis functions respecting the crystalsymmetry, such as Allen’s Fermi surface harmonics [27]: ϕ ( p , p ′ ) = d X j =1 d j X m =1 ϕ j ( p, p ′ ) R jm ( θ, Φ) R ∗ jm ( θ ′ , Φ ′ ) ,ψ ( p , p ′ ) = d X j =1 d j X m =1 ψ j ( p, p ′ ) R jm ( θ, Φ) R ∗ jm ( θ ′ , Φ ′ ) . (9)Here, we introduce spherical coordinates for p : p =( p, θ, Φ); d is the order of the point group; index j labelsirreducible representations of the group; d j is the dimen-sion of the j -th irreducible representation; { R jm ( θ, Φ) } is a basis of the j -th irreducible representation including d j functions.For an isotropic metal the spherical harmonics Y jm can be used as the basis. Including orbital moments up to j = 2 we have, for a cubic symmetry (cubic harmonics): (cid:18) ϕ ( p , p ′ ) ψ ( p , p ′ ) (cid:19) = (cid:18) ϕ ψ (cid:19) + (cid:18) ϕ ψ (cid:19) ( p x p ′ x + p y p ′ y + p z p ′ z )+ (cid:18) ϕ ψ (cid:19) ( p z p ′ z p x p ′ x + p z p ′ z p y p ′ y + p x p ′ x p y p ′ y )+ (cid:18) ϕ ψ (cid:19) ( p x − p y )( p ′ x − p ′ y )+ 13 (cid:18) ϕ ψ (cid:19) (2 p z − p x − p y )(2 p ′ z − p ′ x − p ′ y ) . (10)The coefficients ϕ, ψ are material dependent constants.A common feature of Q2D metals is their layered struc-ture with a pronounced anisotropy of the electrical con-ductivity. In such materials electron energy only weaklydepends on the quasimomentum projection p = pn onthe normal n to the layers plane. In further considera-tion we assume n = (0 , ,
1) and we neglect the asymme-tries of the electron spectrum in the layers planes. Thenthe relevant Fermi surface is axially symmetrical.For systems with an axial symmetry this expression(10) needs to be correspondingly modified. For instance,in the first order we have (cid:18) ϕ ( p , p ′ ) ψ ( p , p ′ ) (cid:19) = (cid:18) ϕ ψ (cid:19) + (cid:18) ϕ ψ (cid:19) p z p ′ z + (cid:18) ϕ ψ (cid:19) ( p x p ′ x + p y p ′ y ) . (11)This expression will be used from now on in the presentpaper.When an external magnetic field B = (0 , , B ) is ap-plied the spin degeneracy of the single electron energiesis lifted, and we can write: E σ ( p ) = E ( p ) + σgβ B ≡ E ( p ) + ∆ E (12) where E ( p ) does not depend on the electron spin, g isthe electron Lande factor, and β = e ~ / m c is the Bohrmagneton ( m is the free electron mass). The nonequi-librium correction to the electron distribution functionsatisfies the equation [7]: δρ ( p , s ) = δρ ( p , s ) + ∂f p σ ∂E p σ X p ′ ,σ ′ F ( p , s , p ′ , s ′ ) δρ ( p ′ , s ′ )(13)where δρ ( p , s ) describes the deviation of the electronliquid from the state of local equilibrium. When the de-viation arises due to the effect of the applied magneticfield δρ = − ( ∂f p σ /∂E p σ ) gβσB. Substituting Eq. 11into Eq. 13 and using the result in Eq. 1 we get:∆ E = ∆ E − b ∗ σgβ B ≡ σgβ B b . (14)where b ∗ = b / (1 + b ) , and b is a dimensionlessparameter describing FL interactions of the conductionelectrons, namely: b = − ν (0) ψ where ν (0) is thedensity of states of noninteracting conduction electronson the Fermi surface in the absence of the magneticfield. This is nothing but the standard Stoner renor-malization of the paramagnetic susceptibility. Note thathere ψ plays the role of the Stoner parameter I = (cid:10) δ E xc /δρ ↑ δρ ↓ (cid:11) in the density functional theory, or ofthe contact Coulomb interaction in the many-body the-ory.The Luttinger theorem dictates the Fermi surface vol-ume. Therefore, the radius and the cross-sectional areasof the Fermi sphere associated with an isotropic Fermiliquid remain unchanged due to quasiparticles interac-tions. In realistic metals whose conduction electronsform anisotropic Fermi liquids one may expect some mi-nor changes in the FS geometry to appear. Such effectscould be considered elsewhere. In the present work we ne-glect them. So, in further consideration we assume thatFL interactions do not affect the FS geometry. Thenthe cross sectional areas of the Fermi surface A ( p z ) cutout by the planes perpendicular to the magnetic field B do not change when electron-electron interactions are ac-counted for. However, the cyclotron masses of conductionelectrons undergo renormalization due to the electron-electron interactions. The cyclotron mass is defined as: m ⊥ = 12 π ∂A∂E (cid:12)(cid:12)(cid:12)(cid:12) E = µ ≡ I dlv ⊥ . (15)Here, dl = q dp x + dp y is the element of length alongthe cyclotron orbit in the quasimomentum space, v ⊥ = q v x + v y , v α = ∂E/∂p α ( α = x, y ) , and µ is the chem-ical potential of conduction electrons.Substituting Eqs. 11 into the Eq. 8 we get v ⊥ = v ⊥ / (1+ a ) where v ⊥ = q v x + v y , and a is relatedto the FL parameter ϕ as follows: − ν (0) p ϕ / a (16)where p is the maximum value of the longitudinal com-ponent of quasimomentum. So we get: m ⊥ = m ⊥ (1 + a ) , (17) m ⊥ being the cyclotron mass of noninteracting quasi-particles. In the case of isotropic electron system thecyclotron mass m ⊥ coincides with the crystalline ef-fective mass m ∗ Therefore, our result agrees with thestandard isotropic FL theory.Other quantities, such as the chemical potential of con-duction electrons and their compressibility, may expe-rience different renormalizations, as well. The latter,for instance, is renormalized by a factor 1 / (1 + a ) =1 / [1 − ν (0) ϕ ] , and the former by the factor (1 + a )[7]. So, the renormalized density of states ν (0) appearsin the expressions for the electron compressibility and thevelocity of sound in metals.The model of the extremely short range (contact)Coulomb interaction between quasiparticles is often em-ployed, while applying the many-body theoretical ap-proach to study de Haas–van Alphen effect (see e.g. Ref.[6]). Within the phenomenological FL theory this modelresults in the approximation of the functions ϕ ( p , p ′ )and ψ ( p , p ′ ) by constants ϕ and ψ , respectively.Such approximation enables us to get the Stoner renor-malization of the paramagnetic susceptibility, and elec-tron compressibility as shown above. However, it missesFermi-liquid effects associated with the subsequent FLcoefficients included in the Eqs. 9–11, which could besignificant in renormalizations of other parameters char-acterizing the charge carriers such as their cyclotronmasses. III. QUANTUM OSCILLATIONS OF THELONGITUDINAL VELOCITY OFCHARGE-CARRIERS IN Q2D CONDUCTORS
In further calculations we adopt the commonly usedtight-binding approximation for the charge carriers spec-trum in a quasi-two-dimensional metal. So, when a quan-tizing magnetic field is applied, the charge carriers ener-gies may be written in the form: E ( n, p z , σ ) = ~ ω (cid:18) n + 12 (cid:19) + σ ~ ω − t cos (cid:18) π p z p (cid:19) . (18)where ~ ω is the spin splitting energy, t is the inter-layer transport integral, and p = π ~ /L where L is theinterlayer distance. This expression (18) describes singleparticle energies of noninteracting quasiparticles. Now,the relation of matrix elements of renormalized v νν ′ and bare v νν ′ velocities in accordance with Eq. 8 takes onthe form [9]: v νν ′ = v νν ′ − X ν ν f ν − f ν E ν − E ν F ν ν νν ′ v ν ν . (19)Here, E ν is the quasiparticle energy including the correc-tion arising due to the FL interactions, and ν = { α, σ } is the set of quantum numbers of an electron in the mag-netic field. The subset α includes the orbital num-bers n, p z and x (the latter labels the positions ofthe cyclotron orbits centers). Also, F ν ν νν ′ = ϕ α α αα ′ +4 ψ α α αα ′ ( ss ) are the matrix elements of the Fermi-liquidkernel.For an axially symmetrical FS the off-diagonal matrixelements of the longitudinal velocity vanish and we ob-tain: v νν = v νν − X ν df ν dE ν F ν ν νν v ν ν (20)Substituting the expression for the Fermi-liquid kernelinto the Eq. 20, we get: v νν = v αα δ σσ ′ + σv sαα (21)where both v αα and v sαα only depend on p z , so inthe further calculations we will use the notation v αα ≡ v ( p z ) , v sαα ≡ v s ( p z ) . These matrix elements could befound from the system of equations that results from Eqs.20 and 21: v ( p z ) = h v ( p z ) − X α ϕ α α αα (cid:0) v ( p z )Γ α α + v s ( p z )Γ sα α (cid:1)i , (22) v s ( p z ) = − X α ψ α α αα (cid:0) v ( p z )Γ sα α + v s ( p z )Γ α α (cid:1) . (23)Here,Γ α α = X σ df α σ dE α σ , Γ sα α = X σ df α σ dE α σ σ . (24)The de Haas-van Alphen oscillations are observed in mag-netic fields when the Landau levels spacing is small com-pared to the chemical potential of electrons ( ~ ω ≪ µ ) . Under these conditions we may approximate the Fermi-liquid kernel by its expression in the absence of the mag-netic fields (Eq. 2). Using the above-described approx-imation of the Fermi-liquid functions ϕ ( p , p ′ ) ψ ( p , p ′ ) ,Eqs. 11, we can solve Eqs. 22, 23. To this end, we needsome averages over the Fermi surface, namely: R = − X α Γ αα v ( p z ) p z = − π ~ λ × X n,σ Z df ( E n,σ ( p z )) dE n,σ ( p z ) v ( p z ) p z dp z . (25) R ′ = − X α Γ sαα v ( p z ) p z . (26)The Fermi-liquid effects enter the system 22 23 throughthe averages A, A ′ , B, B ′ closely related to the Fermi-liquid parameters: A = − ϕ X α Γ αα p z , A ′ = − ϕ X α Γ sαα p z . (27)The expressions for B, B ′ could be obtained replacing ϕ by ψ in Eqs. 27.Applying the Poisson summation formula, ∞ X n =0 ϕ ( n ) = ∞ X r = −∞ Z ∞ exp(2 πirn ) ϕ ( n ) dn. (28)to Eq. 24, we get: R = − π ~ λ X σ Z dn Z dp z df ( E n,σ ( p z )) dE n,σ ( p z ) v ( p z ) p z × ( ∞ X r =1 exp(2 πirn ) ) . (29)So, we see that oscillating terms appear in the expres-sions for R and other averages over the Fermi surfaceincluded in Egs. 22, 23. Due to this reason, an oscillatingterm occurs in the resulting formula for the renormalizedlongitudinal velocity v σ ( p z ) . This oscillating term origi-nates from the Fermi-liquid interactions between chargecarriers, and it appears only when the FL coefficients ϕ and ψ are taken into account, that is, beyond thecontact approximation for the Coulomb interaction.We get the following results for the oscillating parts of R, R ′ :˜ R = 2 N (cid:18) BF (cid:19) ∆ , R ′ = 2 N (cid:18) BF (cid:19) ∆ s , (30)where N is the electron density, and the functions ∆and ∆ s have the form:∆ = ∞ X r =1 ( − r πr D ( r ) sin (cid:18) πr FB (cid:19) cos (cid:18) πr ω ∗ ω ∗ (cid:19) × J (cid:18) πr t ~ ω ∗ (cid:19) , (31)∆ s = ∞ X r =1 ( − r πr D ( r ) cos (cid:18) πr FB (cid:19) sin (cid:18) πr ω ∗ ω ∗ (cid:19) × J (cid:18) πr t ~ ω ∗ (cid:19) . (32)Here, F = cA / π ~ e, A is the FS cross-sectional areaat p z = ± p /
2; and the cyclotron quantum ~ ω ∗ andspin-splitting energy ~ ω ∗ are renormalized according toEqs. 14, 17. The damping factor D ( r ) describes theeffects of the temperature and electron scattering on the magnetic quantum oscillations, and J ( x ) is the Besselfunction. The simplest and well known approximationfor D ( r ) equates it to the product R T ( r ) R τ ( r ) where R T ( r ) = rx/ sinh( rx ) ( x = 2 π k B T / ~ ω ∗ ) is the temper-ature factor and R τ ( r ) = exp[ − πr/ω ∗ τ ] is the Dinglefactor describing the effects of electrons scattering char-acterized by the scattering time τ. The temperature fac-tor appears in the Eqs. 31,32 as a result of standardcalculations repeatedly described in the relevant worksstarting from the LK paper [4]. The Dingle factor can-not be straightforwardly computed starting from the ex-pessions like 25, 26. This term is phenomenologicallyincluded in the Eqs 31,32 in the same way as in theShenberg’s book [1]. Under low temperatures requiredto observe magnetic quantum oscillations, the value of τ is mostly determined by the impurity scattering.Using the microscopic many-body perturbation the-ory it was shown that in this case the Dingle term re-tains its form, and the corresponding relaxation timecould be expressed in terms of the electron self-energypart Σ arising due to the presence of impurities, namely: τ − = 2ImΣ / ~ . In strong magnetic fields the self-energyΣ gains an oscillating term which describes quantum os-cillations of this quantity [23, 26]. So, the scatteringtime becomes dependent of the magnetic field B . A thor-ough analysis carried out in the earlier works of Cham-pel and Mineev [26] and Grigoriev [23] shows that theoscillating correction to the scattering time could be ne-glected when the FS of a Q2D metal is noticeably warped(4 πt > ~ ω ∗ ) . In such cases one may treat τ as a phe-nomenological constant. However, when the FS is veryclose to a pure cylinder (4 πt ≪ ~ ω ∗ ) the scattering timeoscillations must be taken into consideration in studiesof the de Haas-van Alphen effect. These oscillations maybring some changes in both shape and magnitude of themagnetization oscillations but we do not discuss the issuein the present work. In further analysis we assume that4 πt > ~ ω ∗ . One may notice that the oscillating function ∆ hasexactly the same form as that describing the magneti-zation oscillations in Q2D metals when the Fermi-liquideffects are omitted from the consideration (see e.g. Refs.[23, 24]). Also, the Fermi-liquid terms included in the ex-pression (27) exhibit oscillations in the strong magneticfield. For instance, applying the Poisson summation for-mula to the expressions (27) we can convert these expres-sions to the form: A = a (1 + δ ) , A ′ = a δ s where theoscillating functions δ and δ s are δ = ∞ X r =1 ( − r D ( r ) cos (cid:18) πr FB (cid:19) cos (cid:18) πr ω ∗ ω ∗ (cid:19) S (cid:18) πrt ~ ω ∗ (cid:19) − ∞ X r =1 ( − r D ( r ) sin (cid:18) πr FB (cid:19) cos (cid:18) πr ω ∗ ω ∗ (cid:19) Q (cid:18) πrt ~ ω ∗ (cid:19) , (33) δ s = ∞ X r =1 ( − r D ( r ) sin (cid:18) πr FB (cid:19) sin (cid:18) πr ω ∗ ω ∗ (cid:19) S (cid:18) πrt ~ ω ∗ (cid:19) + ∞ X r =1 ( − r D ( r ) cos (cid:18) πr FB (cid:19) sin (cid:18) πr ω ∗ ω ∗ (cid:19) Q (cid:18) πrt ~ ω ∗ (cid:19) . (34)The factors S and Q entered in Eqs. 33 and 34 areexpressed in the series of the Bessel functions: S ( x ) = J ( x ) + 32 π ∞ X m =1 ( − m m J m ( x ) , (35) Q ( x ) = 6 π ∞ X m =0 ( − m (2 m + 1) J m +1 ( x ) . (36)The expressions for B, B ′ are similar to those for A, A ′ and we may get to former by replacing the factor a byanother constant b . The oscillating function δ behaveslike the function describing quantum oscillations of thecharge carriers density of states (DOS) on the FS of aQ2D metal (see Appendix). As for the parameters a , b we can define a = − ν (0) p ϕ / b is similarlydefined, namely: b = − ν (0) p ψ / . We remark thatthe parameter a differs from a which enters the ex-pression for the cyclotron mass (see Eq. 17). This reflectsthe anisotropy of electron properties in Q2D conductors. IV. QUANTUM OSCILLATIONS IN THEMAGNETIZATION
To compute the longitudinal magnetization M || , westart from the standard expression: M || ( B, T, µ ) ≡ M z ( B, T, µ ) = − (cid:18) ∂ Ω ∂B (cid:19) T,µ (37)Here, the magnetization depends on the temperature T and on the chemical potential of the charge carriers µ, and H is the external magnetic field related to the field B inside the metal as B = H + 4 π M . When the mag-netic field is directed along a symmetry axis of a highorder we may assume that the fields B and H are par-allel. One may neglect the difference between B and H when the magnetization is weak. Otherwise the uni-form magnetic state becomes unstable, and the Condondiamagnetic domains form, with the alternating signs ofthe longitudinal magnetization [28]. We will discuss thispossibility later. Now, we assume H by B in the Eq.37. To incorporate the effects of electron interactions weassume, in the spirit of the FL theory, that the thermo-dynamic potential Ω has the same form as for noninter-acting quasiparticles, but with the quasiparticle energiesfully renormalized by their interaction:Ω = − k B T X ν ln (cid:26) (cid:20) µ − E ν k B T (cid:21)(cid:27) . (38) In this expression E ν is the quasiparticle energy includ-ing the correction arising due to the FL interactions, and k B is the Boltzmann’s constant.Accordingly, we rewrite Eq. 38 as follows:Ω = − k B T π ~ λ X n,σ Z ln (cid:26) (cid:20) µ − E n,σ ( p z ) k B T (cid:21)(cid:27) dp z (39)where λ = ~ c/eB is the squared magnetic length. Per-forming integration by parts, Eq. 39 becomes:Ω = − π ~ λ X n,σ Z f ( E n,σ ( p z )) v σ ( p z ) p z dp z . (40)Applying the Poisson summation formula, we get:Ω = − π ~ λ X σ Z dn Z dp z f ( E n,σ ( p z )) v σ ( p z ) p z × ( ∞ X r =1 exp[2 πirn ] ) . (41)So we see that the expression for the thermodynamicpotential includes two oscillating terms. One originatesfrom the oscillating part of v σ ( p z ) . The second term in-side the braces in the Eq. 41 gives another oscillatingcontribution.The effects of temperature and spin splitting on themagnetic oscillations are already accounted for in the Eq.41. Assuming 4 πt > ~ ω, we take into account the effectof electron scattering adding an imaginary part i ~ / τ tothe electron energies [1]. After standard manipulations,we obtain the following expression for the oscillating partof the longitudinal magnetization:∆ M || = − N β ω ∗ ω ∗ (1 − a ∗ ) × ∆ − a ∗ + b ∗ ) δ + 3 b ∗ (∆ δ − ∆ s δ s ) − a ∗ b ∗ ( δ − δ s )1 + 3( a ∗ + b ∗ ) δ + 9 a ∗ b ∗ ( δ − δ s ) . (42)where a ∗ = a / (1 + 3 a ); b ∗ = b / (1 + 3 b ) . This isthe main result of the present work. It shows that theFermi-liquid interactions may bring significant changesin the de Haas-van Alphen oscillations. Below, we an-alyze these changes. If we may neglect the oscillatingcorrections proportional to a ∗ , b ∗ , then our result for∆ M || reduces to the usual LK form with some renor-malizations arising from the quasiparticle interactions.The cyclotron mass m ⊥ differs from the bare Fermi liq-uid cyclotron mass before the quasiparticle interaction istaken into account (cf. Eq. 17), and ~ ω ∗ includes theextra factor (1 + b ) − . Also, the factor (1 − a ∗ ) mod-ifies the magnetic oscillations magnitudes. As for theoscillations frequencies, they remain unchanged by theFL interactions, as expected. V. DISCUSSION
Comparing our result (42) with the corresponding re-sult reported by Wasserman and Springfield [6], we seethat these results agree with each other. A seeming dif-ference in the expressions for the oscillations frequenciesarises due to the fact that in Ref. [6] the frequenciesare expressed in terms of the chemical potential of elec-trons µ instead of the cross-sectional areas of the Fermisurface. It is worth reiterating that “unrenormalized”mass in the FL theory is already renormalized (sometimesstrongly) from fully noninteracting (or density functional- calculated) mass. Again, we remark that the presentanalysis was carried out assuming noticeable/significandFS warping (4 πt > ~ ω ∗ ) , so, we may neglect the mag-netic field dependence of the electrons scattering τ treat-ing the latter a constant phenomenological parameter.This results in a simple form of the Dingle damping fac-tor R τ ( r ) describing the effects of electrons impurityscattering.The LK form of the expression for the longitudi-nal magnetization is suitable to describe de Haas-vanAlphen oscillations in conventional three dimensionalmetals within the whole range of temperatures. However,this is not true for quasi-two-dimensional conductors.The Fermi surface of such a conductor is nearly cylindri-cal in shape, therefore the oscillating term in the denom-inator of Eq. 42 significantly increases. The oscillationsof the denominator of Eq. 42 occur due to the functions δ and δ s . These functions are presented in the Fig. 1,and we see that at low temperatures and weak scatteringln( ~ ω ∗ /k B T ∗ ) > t/ ~ ω ∗ ( T ∗ = T + T D , T D = ~ / πk B τ is the Dingle temperature) the peak values may be ofthe order of 1 , especially for a rather weakly warpedFS ( t/ ~ ω ∗ ∼ . ÷ . . So, quantum oscillations in themagnetization in the electron Fermi-liquid in quasi-two-dimensional metals may have more complicated struc-ture than those in the electron gas described by theLK formula [4]. The effect of the Fermi-liquid interac-tions on these oscillations depends on the values of theFermi-liquid parameters a ∗ , b ∗ and on the damping fac-tor D ( r ) = R T ( r ) R τ ( r ) included in the expressions forthe oscillating functions. The FS shape determined bythe ratio t/ ~ ω is important as well.The most favorable conditions for the changes in themagnetization oscillations to be revealed occur when theoscillating terms in the denominator of the Eq. (42) maytake on values of the order of unity at the peaks of oscil-lations. We may estimate the peak values δ m and δ sm ofthe functions included into the above denominator usingthe Euler-Macloren formula. The estimations depend onthe shape of the FS of the Q2D conductor. When the FSis significantly crimped ( t ≫ ~ ω ) we obtain: δ m , δ sm ∼ ( ~ ω/t ) / ( k B T ∗ ) − / . So, we may expect the Fermi-liquid interaction to be distinctly manifested in the mag-
FIG. 1: The magnetic field dependencies of the functions δ (solid lines) and δ s (dashed lines). The curves are plottedat B = 10 T, F/B = 300 , π θ ∗ / ~ ω ∗ = 0 . t/ ~ ω ∗ = 2(left panel) and t/ ~ ω ∗ = 0 . netization oscillations when | a ∗ | ( ~ ω/t ) / ( k B T ∗ ) − / ∼ | b ∗ | ( ~ ω/t ) / ( k B T ∗ ) − / ∼ | a ∗ | , | b ∗ | ≪ . Nevertheless, the changes in the deHaas-van Alphen oscillations arising due to the Fermi-liquid effects may occur at k B T ∗ ≪ . It is worthwhileto remark that due to the character of the electron spec-tra, the Q2D conductors provide better opportunities forobservations of the Fermi-liquid effects in the de Haas-van Alphen oscillations than conventional 3D metals. Inthe latter the peak values of the oscillating functions δ, δ s have the order ( ~ ω/µ ) / ( k B T ∗ ) − / . Typical values ofthe transfer integral t are much smaller than those ofthe chemical potential µ, therefore significantly smallervalues of k B T ∗ and/or greater values of the parameters a ∗ , b ∗ are required for the Fermi-liquid effects to be re-vealed in 3D metals.Due to the special character of the electron spectra inthe Q2D metals, the variations in the magnetization os-cillations may be noticeable at reasonably small values ofthe Fermi-liquid constants. In the Fig. 2 we compare theoscillations arising in a gas of the charge carriers (top leftpanel) with those influenced by the Fermi-liquid interac-tions between them. All curves included in this figure areplotted within the limit t > ~ ω. We see that both magni-tude and shape of the oscillations noticeably vary due tothe Fermi-liquid effects. When t/ ~ ω ∗ ∼ . ÷ . M, this bring some changes in the magnitudeand shape of the oscillations. These changes are moresignificant when ( a ∗ + b ∗ ) < . The most important manifestation of the Fermi-liquideffects occurs in very clean conductors at low tempera-tures when T ∗ is reduced so much that ( a ∗ + b ∗ ) δ m isgreater than 1 . Then the denominator of the Eq. 42
FIG. 2: The effect of the Fermi-liquid interactions on the deHaas-van Alphen oscillations in a Q2D metal at t/ ~ ω ∗ = 2 . The curves are plotted using Eq. 42, M = 2 Nβ.
Calcu-lations are carried out for a ∗ = b ∗ = 0 (top left panel) a ∗ = b ∗ = 0 .
02 (top right panel), a ∗ = b ∗ = − . − . becomes zero at some points near the peaks of the DOSoscillations provided that ( a ∗ + b ∗ ) < . This is illus-trated in the Fig. 4. Correspondingly, ∆ M diverges atthese points which indicates the magnetic instability ofthe system.This means that the condition for the uniform mag-netization of the electron liquid is violated near the os-cillations maxima, and the diamagnetic domains couldemerge. It is known that both crystal anisotropy and de-magnetization effects originating from the shape of themetal sample, could modify the relation between B and H and cause magnetic instability which results in theoccurence of the diamagnetic domains [28]. Our resultdemonstrates that the interactions of conducting elec-trons also may play an important role in magnetic phasetransitions. In principle, such magnetic instabilities mayappear in 3D metals as well which was shown earlier ana-lyzing quantum oscillations in the longitudinal magneticsusceptibility of the isotropic electron liquid (see Refs.[18, 29]). However, we may hardly expect these tran-sitions to appear in conventional metals for the require-ments on the temperature and intensity of scattering pro-cesses are very strict. Estimations made in the earlierworks [20, 29] show that the temperature (including the FIG. 3: The effect of the Fermi-liquid interactions on the deHaas-van Alphen oscillations in a Q2D metal at t/ ~ ω ∗ = 0 . . The curves are plotted using Eq. 42 at a ∗ = b ∗ = 0 (top leftpanel), a ∗ = b ∗ = 0 .
02 (top right panel), a ∗ = b ∗ = − . a ∗ = b ∗ = − .
04 (bottom rightpanel). The remaining parameters are the same as in thefigure 1. Dashed lines represent the oscillations in the gas ofcharge carriers.
Dingle correction) must be about 10 mK or less for thesediamagnetic phase transitions to emerge in conventionalmetals. On the contrary, the special (nearly cylindri-cal) shape of the FS in Q2D conductors gives grounds toexpect the above transitions to appear in realistic exper-iments.To summarize, in the present work we theoreticallyanalyzed possible manifestations of the FL interactions(that is, residual interactions of excited quasiparticles) inthe de Haas-van Alphen oscillations in Q2D conductors.The same approach can be easily applied for a metalwith the crystalline lattice of arbitrary symmtery, usingthe appropriate basis for expanding the FL functions.So, the phenomenological Fermi-liquid theory becomesmore realistic and suitable to analyze effects of electroninteractions in actual metals.We showed that the residual quasiparticle interactionsaffect all damping factors inserted in the LK formulathrough the renormalization of the cyclotron mass. Thespin splitting is renormalized as well, in a manner similarto the so-called Stoner enhancement. The frequency ofthe oscillations remains unchanged for it is determinedwith the main geometrical characteristics of the Fermisurface, which probably are not affected by electron- FIG. 4: The plot of the function Y = ( a ∗ + b ∗ ) δ + a ∗ b ∗ ( δ − δ s ) near the peak of the DOS quantum oscillations at t/ ~ ω ∗ = 0 . a ∗ = b ∗ = − . . The remaining pa-rameters have the same values as used in the figure 1. electron interactions. However, the shape and magni-tude of the oscillations are affected due to the FL effects,and their changes may be noticeable. Also, the obtainedresults indicate that (under the relevant conditions) theelectron interactions may break down the magnetic sta-bility of the material creating an opportunity for the dia-magnetic phase transition. The discussed effects may beavailable for observations in realistic experiments bring-ing extra informations concerning electronic properties ofquasi-two-dimensional metals.Finally, we want to emphasize once again that therenormalizations, most importantly, mass renormaliza-tion, are in addition to what is conventionally called“mass renormalizatin”, namely, renormalization of thespecific heat coefficient compared to band structure cal-culations. It is usually implicitely assumed that theweighted average of the de Haas-van Alphen mass reno-ramization is exactly equal to the specific heat renor-malization, i.e., the FL effects are small. In many casesthis is a good approximation, but one can never excludea possibility that in some materials these two massesmay be different, namely, the de Haas-van Alphen massmay be larger. A curious example when one would haveneeded to exercise caution, but did not, is given by Ref.[30], where quantum oscillations in a highly uncoven- tional metal, Na x CoO , were measured, and it wastaken for granted that the large mass renormalizationfound in the experiment should be fully accounted forin specific heat. Based on this assumption, a naturaland straightforward interpretation of the data was aban-doned and counterintuitive explanation, requiring someunverified assumptions was accepted. It is possible thatthe results reported in the Ref. [30] give a case whereadditional mass renormalization discussed in this paperis significant. Hopefully, at some point we will see a care-ful and accurate experimental study on various materialsthat would compare the de Haas-van Alphen masses withthe thermodynamic masses and give us a quantitative an-swer on how different may the two be in real life. APPENDIX
Here, we analyze the expressions for the oscillatingfunctions δ, δ s within the limits of significant ( t ≫ ~ ω )FS warping. We may use the standard asymptotics forthe Bessel functions , at x ≫ , namely: J k ( x ) ≈ r πx cos (cid:20) x − πk − π (cid:21) . (43)Substituting these aproximation into Eqs. 34, 35 we ob-tain: S ( x ) = r πx cos h x − π i ( π ∞ X m =1 m ) , (44) Q ( x ) = 6 π r πx sin h x − π i ∞ X m =0 m + 1) , (45)where ∞ X m =1 m = π , ∞ X m =0 m + 1) = π . (46)So, we have: S ( x ) = 54 r πx cos h x − π i , (47) Q ( x ) = 34 r πx sin h x − π i . (48)Using these results we may write the following expressions for δ, δ s at t ≫ ~ ω : δ = 54 (cid:18) ~ ω ∗ π t (cid:19) / ∞ X r =1 ( − r √ r D ( r ) cos (cid:20) πr FB (cid:21) cos (cid:20) πrt ~ ω ∗ − π (cid:21) cos (cid:20) πr ω ∗ ω ∗ (cid:21) − (cid:18) ~ ω ∗ π t (cid:19) / ∞ X r =1 ( − r √ r D ( r ) sin (cid:20) πr FB (cid:21) sin (cid:20) πrt ~ ω ∗ − π (cid:21) sin (cid:20) πr ω ∗ ω ∗ (cid:21) . (49)0Or: δ = (cid:18) ~ ω ∗ π t (cid:19) / ∞ X r =1 ( − r √ r D ( r ) cos (cid:20) πr ω ∗ ω ∗ (cid:21) (cid:26) cos (cid:20) πr F max B − π (cid:21) + 14 cos (cid:20) πr F min B + π (cid:21)(cid:27) . (50)Likewise, we obtain for δ s : δ = (cid:18) ~ ω ∗ π t (cid:19) / ∞ X r =1 ( − r √ r D ( r ) cos (cid:20) πr ω ∗ ω ∗ (cid:21) (cid:26) sin (cid:20) πr F max B − π (cid:21) + 14 sin (cid:20) πr F min B + π (cid:21)(cid:27) . (51)Here, F max , F min correspond to the maximum and minimum cross-sectional areas of the FS, respectively. ACKNOWLEDGMENTS
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