Angular dependence of magnetoresistance and Fermi-surface shape in quasi-2D metals

# Angular dependence of magnetoresistance and Fermi-surface shape in quasi-2D metals

P.D. Grigoriev L.D. Landau Institute for Theoretical Physics, Chernogolovka, Russia
July 19, 2019
###### Abstract

The analytical and numerical study of the angular dependence of magnetoresistance in layered quasi-two-dimensional (Q2D) metals is performed. The harmonic expansion analytical formulas for the angular dependence of Fermi-surface cross-section area in external magnetic field are obtained for various typical crystal symmetries. The simple azimuth-angle dependence of the Yamaji angles is derived for the elliptic in-plane Fermi surface. These formulas correct some previous results and allow the simple and effective interpretation of the magnetic quantum oscillations data in cuprate high-temperature superconducting materials, in organic metals and other Q2D metals. The relation between the angular dependence of magnetoresistance and of Fermi-surface cross-section area is derived. The applicability region of all results obtained and of some previous widely used analytical results is investigated using the numerical calculations.

Fermi surface, magnetoresistance, quasi-2D, layered, metals, AMRO
###### pacs:
72.15.Gd,73.43.Qt,74.70.Kn,74.72.-h

## I Introduction

The layered quasi-two-dimensional (Q2D) compounds attract great attention for their novel physical properties and promising technical applications. High-temperature cuprate superconductors,HTc () organic metals,IYS () heterostructures,Heterostructures () intercalated graphitesIntercalatedGraphitesReview2002 () are the examples of these compounds. The knowledge of quasiparticle dispersion in these compounds is very important for understanding their properties and electronic phase diagram. The traditional and powerful tools to determine the Fermi surface (FS) geometry and the electron dispersion in various metals are the magnetic quantum oscillations (MQO)Shoenberg () and the angular dependence of magnetoresistance (ADMR)MarkReview (). There is a huge amount of publications, devoted to the experimental determination of the FS geometry and electron dispersion in high-temperature cuprate superconductorsMQOHighTc (); Bergemann (), in MgB,MgB2MQOReview () in organic metals (see refs. MarkReview (),MQORev () for reviews) and in many other Q2D metals. The interpretation of the MQO data is, usually, based on the detailed comparison with the band-structure calculations, which is a complicated and often ambiguous procedure. The interpretation of ADMR is also based on fitting by the numerical calculations with a large number of fitting parameters.HusseyNature2003 (); AbdelPRL2007AMRO (); McKenzie2007 () The quick and effective extraction of the FS geometry and of electron dispersion from the experimental data on MQO and on ADMR requires reliable and simple theoretical formulas.

The general form of the electron dispersion in Q2D  compounds with monoclinic or higher crystal symmetry can be expressed as the Fourier series in cylindrical coordinates:

 ε(k)=∑ν≥0,μ=evenϵμν(k)cos(νkzc∗)cos(μϕ+ϕμν), (1)

where the integers the electron momentum (we put ), is the interlayer lattice constant, is the absolute value of the in-plane momentum and is the in-plane angle (i.e., the azimuth angle in spherical coordinates). In triclinic crystals the only symmetry constraint on the electron dispersion is , and the electron dispersion (1) may also contain the additional terms

 Δε(k)=∑ν>0∑μ=oddϵμν(k)sin(νkzc∗)cos(μϕ+ϕμν). (2)

For simplicity, below we only consider the case of monoclinic or higher crystal symmetry, where the terms (2) are absent.

Usually, it is sufficient to keep only the first few terms in the infinite series (1). For example, if the interlayer transfer integral of conducting electrons, , is much smaller than the in-plane band width , the tight-binding approximation can be used, and one keeps only the terms with and :

 ε(k)=ε(k,ϕ)−2tc(ϕ)cos(kzc∗). (3)

The FS, being given by the equation , is a warped cylinder in Q2D compounds. If magnetic field is applied along the -axis of the Q2D metals with the electron dispersion (3), there are two extremal FS cross-section areas encircled by the closed curves . Hence, the two close fundamental frequencies appear in MQO, giving the beats of MQO.Shoenberg () The temperature dependence of the MQO amplitude gives the cyclotron mass for the extremal orbit: . The beat frequency gives the interlayer transfer integral: . The difference of the two extremal cross-section areas and, hence, the beat frequency depend on the magnetic field direction. In the first order in the interlayer transfer integral and for the axially symmetric FS, this dependence is given byYam ()

 ΔAext∝J0(c∗kFtanθ), (4)

where is the Bessel function, is the in-plane Fermi momentum and is the tilt angle of magnetic field with respect to the -axis (the polar angle of ). Eq. (4) was first derived geometrically by Yamaji Yam () to explain the oscillating angular behaviorExpAMROMark () of interlayer magnetoresistance in Q2D organic metals. As the difference between the two extremal cross-section areas is proportional to the interlayer transfer integral , Eq. (4) suggests that the interlayer transfer integral has the similar angular dependence:

 tc(θ)≈tc(0)J0(c∗kFtanθ), (5)

which gives a strong angular dependence of interlayer magnetoresistance . Eq. (5) was later confirmed by the quantum-mechanical calculation of the amplitude of interlayer electron tunnelling in tilted magnetic field.Kur () The angles , for which the Bessel function has zeros:

 J0(c∗kFtanθm)=0, (6)

are called the Yamaji angles and used to determine the in-plane Fermi momentum . At these angles both the interlayer magnetoresistance and the amplitude of MQO have maxima. Usually, the in-plane electron dispersion is anisotropic, and Eqs. (4),(5) acquire a -dependent correction, where is the azimuthal angle of the magnetic field direction, . There is a considerable practical need of the simple analytical formula for the -dependence of AMRO and MQO, which can be used to extract the in-plane electron dispersion from the experimental data.

The widely used analytical result for the -dependence of the FS cross section, derived by Bergemann et al.Bergemann () and given by Eq. (24) below, takes the FS corrugation only in the first order, which is not enough to obtain correctly even the main -dependent term in the angular dependence of the cross-section area. Another simple and widely usedMark92 (); Nam1 (); HousePRB1996 () analytical result for the -dependence of AMRO maxima (Yamaji angles),

 tanθn≈π(n−1/4)/pmaxBc∗, (7)

with being the maximum value of the Fermi momentum projection on the in-plane magnetic field direction, was derivedMark92 () from the Shockley tube integralZiman () using the saddle point approximation. This approximation assumes that the -component of the electron velocity oscillates rapidly when the electron moves along its closed classical orbit in the momentum space in magnetic field. This is valid only at high tilt angles of magnetic field, when , and only in the very clean samples with , where is the cyclotron frequency and is the electron mean free time. For the first Yamaji angle this derivation is too approximate because . Below we show that Eq. (7) is valid only for the elliptical FS in the limit . With some small error it can also be applied to the FS, which is close to elliptical. However, Eq. (7) gives completely wrong result for the -dependence of Yamaji angles when the in-plane FS has tetragonal (as in cuprate high-Tc superconductors) or hexagonal (as in MgBMgB2MQOReview () or intercalated graphitesIntercalatedGraphitesReview2002 ()) symmetry.

The aim of the present paper is to derive the new suitable analytical formulas for the -dependence of the FS cross-section area, Yamaji angles and magnetoresistance, which can be used to extract the FS parameters from the experimental data. The applicability region of some previous and widely used results will also be studied.

In Sec. II we write down the relation between the dispersion (1) and FS harmonic expansion. In Sec. III we find the main -dependent correction to the FS cross-section area for the anisotropic dispersion  in Eq. (3), when the in-plane anisotropy of the FS is weak. As will be shown, this result has wide applicability region and can also be applied to almost square-shaped in-plane FS as in the high-Tc cuprate superconductors. In Sec. IV we derive the exact expression for the Yamaji zeros for the elliptical FS shape. The deviations from this result for non-elliptic FS will also be studied. In Sec. V we derive the relation between the -dependence of the cross-section area and magnetoresistance in the clean samples, where . This relation shows, that the geometrical and resistivity Yamaji angles coincide in the limit . The discussion and summary of the results is given in Sec. VI.

## Ii Fermi surface parametrization

The dependence of the Fermi momentum on the polar angle and the momentum component can be expanded in the Fourier series:Bergemann ()

 kF(ϕ,kz) = ∑ν≥0kν(ϕ)cos(νkzc∗) (8) = ∑μ,ν≥0kμνcos(νkzc∗)cos(μϕ+ϕμ). (9)

The Fermi momentum satisfiesthe equation

 ε[kF(ϕ,kz),ϕ,kz]=EF, (10)

where is the Fermi energy. The coefficients in Eq. (1) are related to the coefficients in the FS parametrization (9) through the equation

 ∑ν≥0,μ=evenϵμν[kF(ϕ,kz)]cos(νkzc∗)cos(μϕ+ϕμν)=EF. (11)

This equation on can be solved by the iteration procedure, assuming that the warping coefficients are small and fall down rapidly with increasing and . In the first order, each term in the series (9) comes only from the term in (1) with the same indices :

 k(1)μν=−ϵμν(kF)/ϵ′00(kF). (12)

In the second order in , the coefficients come from the interference of the infinite number of the terms and in the dispersion (1), such that and .

For simplicity, we take the dispersion (3) and assume that the dependence of the energy is weak, i.e. the interlayer transfer integral . The solution of equation

 ε(k,ϕ)=EF+2tccos(kzc∗),

in the first order in the interlayer transfer integral gives the FS shape in the cylindrical coordinates:

 kF(ϕ,kz)=k0(ϕ)+k1(ϕ)cos(kzc∗), (13)

where satisfies  and

 k1(ϕ)=2tc(ϕ)/[∂ε(k,ϕ)/∂k]|k=k0(ϕ). (14)

The partial derivative is the projection of the Fermi velocity on the line, connecting the point on the FS with the coordinate origin . It depends on the electron dispersion and on the azimuthal angle .

For the quite general form of the electron dispersion,

 ε(k,ϕ)=kαg(ϕ), (15)

where is an arbitrary function and is also arbitrary, the derivative

 [∂ε(k,ϕ)/∂k]|k=k0(ϕ)=EF/k0(ϕ). (16)

The superelliptic dispersion

 ε(kx,ky)=(kx/k1)α+(ky/k2)α, (17)

which includes both linear and quadratic dispersions, is only a particular case of the dispersion (15). With the relation (16), Eq. (13) simplifies to

 kF(ϕ,kz)=k0(ϕ)[1+2tc(ϕ)EFcos(kzc∗)]. (18)

However, the relation (16) may violate in some compounds, and the application of the simplified formula (18) instead of Eqs. (13),(14) requires additional proof.

## Iii Cross-section area

If the magnetic field is applied at polar and azimuthal angles and , the Fermi surface cross-sectional area , cutting the -axis at and perpendicular to the field, is given by the integral

 A(kz0,θ,φ)=∫2π0dϕ′k2F(φ+ϕ′,kz)/(2cosθ), (19)

where is the angle in the - plane between the direction of magnetic field and a point on the FS, and at this FS point satisfies the equation

 kz=kz0−kF(φ+ϕ′,kz)tanθcosϕ′. (20)

Eqs. (19) and (20) allow to find the cross-section area numerically for any given FS, determined by the function or, equivalently, by the coefficients in the expansion (9). In practice, one usually solves the inverse problem of the extraction of FS parameters from the experimental data on MQO or AMRO. Then, the direct procedure of fitting the experimental data by the parameters in the expansion (9) is rather ambiguous because of too large number of fitting parameters. Usually, the coefficients fall down rapidly with increasing and . Therefore, it is useful to fit only the first few terms in the similar harmonic expansion of the cross-section area

 (21)

keeping only the first few terms in the expansion (9). The first coefficients can be found analytically in the main order in . The analytical formula for the coefficients of the cross-section area is especially useful because of their rather complicated dependence on . In Sec. V it will be shown that the coefficient is directly related to the angular dependence of magnetoresistance at and .

In the zeroth order in coefficients in the expansion (9), i.e. for cylindrical FS neglecting any warping and in-plane asymmetry, one obtains the trivial result , where . In the first order in these coefficients , one can neglect the dependence in Eq. (20) and substitute to Eq. (9). Then, substituting Eq. (9) to Eq. (19), one obtains the first order correction to :

 A(1)(kz0,θ,φ)=∫2π0dϕ′k2F(φ,ϕ′,kz0)−k2002cosθ ≈∫2π0k00dϕ′cosθ∑μ,ν≥0′kμνcos[μ(φ+ϕ′)+ϕμ] ×cos[ν(kz0−kFtanθcosϕ′)c∗],

where the sum does not include the term . Since is even, one can replace in the integrand (here and later we introduce the notation )

 cos[ν(kz0−kFtanθcosϕ′)c∗]→cos[νkz0c∗]cos[νκcosϕ′]. (22)

One can also replace in the integrand

 cos[μ(φ+ϕ′)+ϕμ]→cos[μφ+ϕμ]cos[μϕ′], (23)

because all odd terms vanish after the integration over . Then, after the integration over , the correction in the first-order in writes down as

 A(1)=2πk00cosθ∑μ,ν≥0′(−1)2μkμνcos[μφ+ϕμ]cos(νkz0c∗)Jμ(νκ) (24)

in agreement with Eq. (2) of Ref. Bergemann (). Since for , all terms vanish in (24). This is natural, because in the zeroth order in the cross-section area

 A(0)(kz0,θ,φ)=∫2π0dϕ′k20(φ+ϕ′)2cosθ (25)

is independent of . Hence, to extract any information about the -dependence of the FS, one needs to consider the first order in , i.e. to find . Thus, the -dependence of the cross-section area starts from the term , which is of the same order as the second order term [see Eq. (18)]. Since Eq. (24), or Eq. (2) in Ref. Bergemann (), is derived only in the first order in , it does not give the correct -dependence of the cross-section area even in the lowest -dependent order. This is illustrated below in Figs. 1,2. The extraction of the higher harmonics using Eq. (24) is even more incorrect.

Let us calculate more accurately the lowest-order -dependent term in the cross-section area, which is given by the coefficient in the Fourier expansion. To calculate this coefficient in the main order in FS warping, it is sufficient to use the FS shape in the first order in , given by Eq. (13). Then, in the same order, Eq. (19) rewrites

 A(kz0,θ,φ)≈∫2π0dϕ′k20(φ+ϕ′)2cosθ[1+2k1(φ+ϕ′)k0(φ+ϕ′)cos(kzc∗)], (26)

and substituting Eqs. (20) and (22), we obtain the following expression for correction to :

 A(1)=cos[c∗kz0]cosθ∫2π0dϕ′k0(ϕ′)k1(ϕ′)cos[c∗k0(ϕ′)tanθcos(ϕ′−φ)]. (27)

Here we have also changed the integration variable: .  The Yamaji formula (4) is easily obtained from (27) after taking where the dispersion-dependent constant

 C1≡(EF/kF)/(∂ε00/∂k)|k=kF∼1, (28)

and the integration over , resulting to

 A01(θ)=2tcEFC12πk2FcosθJ0(κ). (29)

The lowest-order -dependence of the cross-section area is determined by the Fourier coefficient , given by

 Am1(θ)=∫2π0cos(mφ+φm1)dφπcosθA(1)(kz0,θ,φ)cos[c∗kz0]. (30)

Performing the integration over , we obtain

 Am1(θ)=2(−1)m/2cosθ∫2π0dϕ′k0(ϕ′)k1(ϕ′)cos(mϕ′+φm1)Jm[c∗k0(ϕ′)tanθ]. (31)

To go further, we need to specify the functions and . We distinguish two symmetries of electron dispersion, namely, with straight and -dependent (corrugated in the main order) interlayer transfer integral.

### iii.1 Straight interlayer hopping

When the in-plane FS anisotropy is weak, one can keep only the first -dependent term in the Fourier expansion of the functions and . If the crystal symmetry allows the -independent (straight) interlayer coupling, these functions expand as

 k0(ϕ) ≈ (1+βcosmϕ)kF, (32) k1(ϕ) ≈ 2tcC1EF(1+β1cosmϕ)kF,

where is an even integer number, and the constant is given by Eq. (28). Now we expand in the small parameter up to the first order (for the first Yamaji angle ), and the integral over in Eq. (31) simplifies to

 ∫2π0dϕ′(β+β1)cos(mϕ′)cos(mϕ′)Jm(κ) +∫2π0dϕ′cos(mϕ′)J′m(κ)κβcos(mϕ′) = πβ{Jm(κ)(1+β1/β)+J′m(κ)κ},

where the derivative

 J′m(κ)=dJm(κ)dκ=mκJm(κ)−Jm+1(κ).

Hence, in the first order in we obtain

 Am1(θ)=(−1)m/24πk2FC1βtcEFcosθ[Jm(κ)(1+β1β+m)−κJm+1(κ)]. (33)

Combining the results (21),(29) and (33), we obtain the cross-section area

 A(kz0,θ,φ)≈πk2Fcosθ+4πk2FtcC1EFcosθcos[c∗kz0]× (34) ×{J0(κ)+β(−1)m/2[(1+β1/β+m)Jm(κ)−κJm+1(κ)]cos(mφ)}.

The ratio , entering this formula, can also be expressed via the FS parametrization, given by Eq. (9): , . The constant is equivalent to the renormalization of and does not influence the Yamaji angles. However, it changes the beat frequency of the magnetic quantum oscillations. It also changes the amplitude of the -dependent term in the cross-section area. For the dispersion of the form (15), i.e. if the relation (16) satisfies, the ratio and the constant . For arbitrary dispersion, and .

The difference between two analytical results, given by Eqs. (34) and (24), is very strong: the factor in Eq. (24) is replaced by the completely different factor in Eq. (34). First, the -dependence of the cross-section area, predicted by Eq. (34), is stronger approximately by a factor than that of Ref. Bergemann (). Second, it may have different -dependence due to the term, especially for high tilt angles . To illustrate the above statement, we plot the results of Eqs. (19), (34), (24) and (45) in Figs. 1 and 2 for the dispersions with monoclinic and tetragonal symmetries.

### iii.2 Strongly ϕ-dependent interlayer hopping

In some compounds, e.g. in the high-temperature superconductors SrRuO and TlBaCuO,Bergemann (); HusseyNature2003 (); AbdelPRL2007AMRO (); McKenzie2007 () the body-centered tetragonal symmetry of the crystal leads to the -dependent lowest-order interlayer transfer integral, , in the all or some parts of the FS. Then, instead of Eq. (32), we have

 k0(ϕ) ≈ (1+βcos2mϕ)kF, (35) k1(ϕ) ≈ 2tcEFkFC1sin(mϕ)(1+β1cos2mϕ).

Substituting this into Eq. (31) we obtain the main -dependent term, determined by the coefficient

 Am1(θ)≈2πk2F(−1)m/2cosθ2tcC1EFJm(κ) (36)

in agreement with the first-order result, given by Eq. (24). This term does not depend on the in-plane FS anisotropy . To extract this anisotropy in the first order in the -dependent interlayer transfer integral, one needs to consider harmonic in the cross-section area. For this we replace in Eq. (31) , substitute Eq. (35) and perform the calculation, similar to that in the derivation of Eq. (33). Then, in the lowest order in we obtain

 A3m 1(θ)=(−1)3m/2cosθ2πtcC1EFβk2F[(1+β1β+3m)J3m(κ)−κJ3m+1(κ)]. (37)

This result differs from Eq. (33) by the replacement (note, that for SrRuO and TlBaCuO), and the prefactor before the square brackets is two times smaller. The difference between the first-order result of Eq. (24) and the new formula (37) for the harmonic is even stronger than in the case of Eq. (33). The total -dependence of the cross-section area in the case of -dependent interlayer coupling, given by Eq. (35), writes down as

 A(kz0,θ,φ)≈πk2Fcosθ+4πk2FtcC1EFcosθcos[c∗kz0]× (38) ×{Jm(κ)sin(mφ)+β2(−1)3m/2[(1+β1β+3m)J3m(κ)−κJ3m+1(κ)]sin(3mφ)}.

This formula can be applied to analyze the experimental data in high-temperature superconductors SrRuO, TlBaCuO, where in Eq. (38), and to some other layered compounds with the appropriate symmetry. Note, that Eqs. (34) and (38) were derived under condition , which is fullfilled in the compounds of tetragonal or hexagonal symmetry at not very high tilt angle of magnetic field. At very high tilt angle, , above derivations are not valid also because of the multiple intersections of the FS by the cross-section plane.

### iii.3 Analysis of magnetic quantum oscillations

The well-resolved magnetic quantum oscillations in quasi-2D metals give two close frequencies and , corresponding to the maximum and minimum of the FS cross-section area. To extract the -dependence of the FS, as follows from Eqs. (34) and (38), one needs to measure the -dependence of the difference between these two close frequencies (the beat frequency), which is harder because requires the resolution of MQO in the wider interval of magnetic field. The observation of the beat frequency itself is important, because it means the existence of the 3D Fermi surface, i.e. of the coherent interlayer electron transport. Eqs. (34) and (38) can be used to determine the optimal orientation of magnetic field for the observation of the beat frequency. For straight interlayer electron coupling as in Eq. (32), the beat frequency has maximum value when magnetic field is perpendicular to the layers, i.e. at polar angle . However, for the -dependent interlayer coupling at the beat frequency is zero, as follows from Eq. (38). Hence, in this case to observe the beat frequency one needs to incline the magnetic field. The angular dependence of the beat frequency is given by the function in the curly brackets in Eq. (38). The first term in the curly brackets is much larger than the second. Its maximum gives the optimal orientation () of magnetic field for the observation of MQO beat frequency. For , as in SrRuO, TlBaCuO and some other high-Tc compounds, the factor has maximum at and , where is the interlayer lattice constant. Note, that the spin factor of MQO also depends on the angle .Shoenberg ()

If the beat frequency of MQO cannot be resolved (in dirty materials or at high temperature), the minima of the beat frequency, i.e. the Yamaji angles, can be detected from the increase of the amplitude of MQO. This increase of MQO amplitude happens because at the Yamaji angles the MQO from both extremal electron orbits have the same phase.Shoenberg (); MarkReview () The Yamaji angles can be much easier distinguished from the angular dependence of background magnetoresistance (AMRO) (see Sec. IV). To determine the -dependence of the Yamaji angles one can again use Eqs. (34) and (38).

If the -dependence of MQO beat frequency is clearly resolved, one can obtain the information about the in-plane FS. Eqs. (34) and (38) again can be used to determine the optimal magnetic field orientation for the observation of this -dependence. In the case of straight interlayer electron hopping, this -dependence has the maximum amplitude when the factor in Eq. (34) has maximum. For typical value this factor as function of for and is plotted in Fig. 3. The function has first maximum at for and (see Fig. 3). It is reasonable to use only the first maximum, because at high tilt angle of magnetic field the cyclotron mass is large and the amplitude of MQO is too small.

In the case of straight hopping, already the lowest-order harmonic in the -dependence of the MQO frequency gives the relative amplitude  of the same harmonic in the -dependence of the in-plane Fermi momentum [see Eq. (34)]. In the case of -dependent interlayer electron hopping, given by Eq. (35), in the main order, the -dependence of MQO frequency comes from the -dependence of the interlayer transfer integral and does not give information about the in-plane FS. To determine the shape of the in-plane FS, one needs to study higher harmonics in the MQO frequency. The amplitude of the term in MQO frequency is given by the function in Eq. (38) and has first maximum at . This determines the optimal polar angle , at which this -dependence is most easily observed. According to Eq. (38), this dependence gives the amplitude of the harmonic modulation of the in-plane FS.

## Iv Elliptic Fermi surface

Now we derive the analytical formula for the Yamaji angles for the elliptic in-plane dispersion

 ε(kx,ky)≡k2x/2mx+k2y/2my=ε(k)[1+βcos2ϕ], (39)

where and . The shape of the FS for this dispersion is, of cause, also elliptical. The ellipse can be obtained from the circle by the dilation along one in-plane direction (along the -axis): . Consider the cross-section area of the FS by the plane, cutting the -axis at the point , and perpendicular to the magnetic field direction , where the unit vector

 n=(nx,ny,nz)=(sinθcosφ,sinθsinφ,cosθ). (40)

For the circular in-plane FS this cross-section area is independent of the angle . In the first order in , it is also independent of at special directions corresponding to the Yamaji angles , given by Eq. (6). After the dilation , the direction of magnetic field, which is perpendicular to the cross-section plane, also changes:

 n→Λx(n)=(nx/λ,ny,nz)√(nx/λ)2+n2y+n2z. (41)

However, the cross-section area perpendicular to remains independent of , if it was independent before the dilation. Hence, the direction corresponds to the new Yamaji angle . The polar and azimuthal angles are related to the components of the vector as

 tanθ1=√n21x+n21yn1z, tanφ1=n1yn1x. (42)

Combining above equations we obtain the relation between the old and new Yamaji angle

 tanθ∗YamtanθYam=√n2x/λ2+n2ynztanθYam=√cos2φλ2+sin2φ.

The angle here is the angle before the dilation . It is related to the angle after the dilation as

 tanφ1=λtanφ.

Then, after simple trigonometric algebra, we obtain

 tanθ∗YamtanθYam=cosφλ√1+tan2φ1=√1+tan2φ1√λ2+tan2φ1=1√λ2cos2φ1+sin2φ1. (43)

For the elliptic dispersion (39) the maximum value of the Fermi momentum projection on the in-plane magnetic field direction is given by

 pmaxB=√(p1cosφ)2+(p2sinφ)2, (44)

where and . The r.h.s. of Eq. (43) coincides with . Hence, the generalization of the Yamaji zeros to the elliptic dispersion (39) writes down as

 J0[c∗pmaxB(φ)tanθn]=0. (45)

Approximately, Eq. (45) coincides with Eq. (7), derived for the interlayer conductivityMark92 (); MarkReview () from the Shockley tube integralZiman (). The saddle point approximation, used in Ref. Mark92 () to derive Eq. (7), assumes that the -component of electron velocity oscillates rapidly when the electron moves along its closed orbit in the momentum space. This is valid only at high tilt angles of magnetic field, when , and only in very clean samples with . The reason, why Eq. (7) describes well some experimental data,MarkReview () comes from its coincidence with the exact geometrical expression (45) for the Yamaji angles for the elliptic Fermi surface, which according to Eq. (55) gives the maxima of magnetoresistance.

## V Magnetoresistance

To calculate magnetoresistance as function of the direction of magnetic field one can use the quasi-classical Boltzmann transport equation for electrons moving along the closed orbits in magnetic field. This approach gives the Shockley-Chambers formulaZiman (), which at zero temperature expresses conductivity tensor via the integral over the Fermi surface:

 σαβ(θ,φ) = e24π3ℏ2∫dkz0m∗Hcosθ/ωH1−exp(−2π/ωHτ) ×∫2π0∫2π0vα(ψ,kz0)vβ(ψ−ψ′,kz0)e−ψ′/ωHτdψ′dψ.

Here the momentum space is parametrized by the momentum component along the magnetic field, by energy and by the angle of the rotation in the cross-section plane. The effective cyclotron mass of the orbit is given by

 m∗H≡12π∂A∂E=12πε′(kF)∂A∂kF, (47)

where is the area of the FS cross-section perpendicular to the magnetic field at the momentum . The cyclotron frequency of the orbit , and is the component of the electron velocity on the FS. Generally, the mean scattering time in the integrand (V) may also depend on the position on the Fermi surface. However, we neglect this dependence because in the simplest theory of spin-independent short-range impurity scattering depends only on the density of states at the Fermi level.

For dispersion (3) the electron velocity component along the -axis is a function of only:

 vz(kz)=(2c∗tz/ℏ)sin(c∗kz). (48)

The coordinate of the FS point satisfies the equation (20), where is the azimuthal angle of the projection of the FS point on the - plane. Approximately, this equation can be solved by the iteration procedure. In the zeroth order

 k(0)z=kz0−kFcos(ϕ′)tanθ, (49)

and in the next orders

 k(i+1)z=kz0−kF(ϕ′+φ,k(i)z)cos(ϕ′)tanθ. (50)

The angle entering the Shockley-Chambers formula (V) corresponds to the increment of the cross-section area at a given increment of energy:

 dψ=1m∗Hdkv⊥=dk∂k⊥m∗H∂E.

Generally, is different from the angles and of the rotation in the cross-section and in the - planes. The cross-section area multiplied by is equal to the area of the projection in the - plane, and is related to the angle of the rotation in the - plane as

 dψdϕ=kF(ϕ,kz)m∗Hcosθ∂kF(ϕ,E)∂E. (51)

For cylindrical FS one has , and coincides with the angle .

Now we show that for the minima of , given by Eq. (V), coincide with the minima of the mean-square value of the derivative , i.e. with the geometrical Yamaji angles. At the exponent , and Eq. (V) gives

 σαα(θ,φ)=e24π3ℏ2∫dkz0m∗Hcosθ/ωH1−exp(−2π/ωHτ)(∫2π0vα(ψ,kz0)dψ)2. (52)

Using Eq. (51) we transform the integral

 I ≡ ∫2π0dψvz(ψ,kz0) (53) = ∫2π0dϕkF(ϕ,kz)m∗Hcosθ∂kF(ϕ,E)∂E∂E∂kz = ∫2π0dϕkF(ϕ,kz)m∗Hcosθ∂kF(ϕ,kz)∂kz.

The derivative

 ∂kF(ϕ,kz)∂kz=∂kF[ϕ,kz(kz0,ϕ)]∂k