Magnetoconductivity and quantum oscillations in intercalated graphite CaC6 with the Fermi

surface reconstructed by the uniaxial charge density wave

P. Grozić,1 A.M. Kadigrobov,2 Z. Rukelj,1 I. Kupčić,1 and D. Radić*1
1
Department of Physics, Faculty of Science, University of Zagreb, Bijenička 32, Zagreb 10000, Croatia
2
Ruhr-Universität Bochum, Theoretische Physik III,
Universitätsstraße 150, Bochum D-44801, Germany
(Dated: January 6, 2025)
arXiv:2501.01526v1 [cond-mat.mes-hall] 2 Jan 2025

We report a magnetoconductivity tensor σ for the intercalated graphite CaC6 , in the ground state
of the uniaxial charge density wave (CDW), under conditions of coherent magnetic breakdown due
to strong external magnetic field B perpendicular to the conducting plane. The uniaxial charge den-
sity wave reconstructs initially closed Fermi surface into an open one, accompanied with formation
of a pseudo-gap in the electron density of states around the Fermi energy. The magnetoconductiv-
ity tensor is calculated within the quantum density matrix and semiclassical magnetic breakdown
approach focused on modification of the main, so-called ”classical” contribution to magnetoconduc-
tivity by magnetic breakdown, neglecting the higher order corrections. In the presence of magnetic
breakdown, in spite of open Fermi surface configuration, all classical magnetoconductivity compo-
nents, the one along the CDW apex σxx ∼ B −2 , perpendicular to the CDW apex σyy ∼ const, as
well as the Hall conductivity σxy ∼ B −1 , undergo strong quantum oscillations vs. inverse magnetic
field. Those oscillations do not appear as a mere additive correction, but rather alter the classical
result becoming an inherent part of it, turning it to essentially non-classical.

I. INTRODUCTION toconductivity tensor manifest an onset of quantum os-

cillations appearing due to magnetic breakdown, periodic
The intercalated graphite compounds (GIC) [1], in inverse magnetic field.
among which in this work we set our focus to CaC6 , have The paper is organized in the following way: after the
been known and studied for several decades. The inter- introduction in the first section, in the second section we
calating atoms, mostly alkali metals from the first group, present the model within which we describe the CaC6 sys-
but also metals from the second and even third group, tem; the third section contains calculation of the disper-
are located between graphene sheets in graphite. Beside sion law, spectrum and wave functions under conditions
chemical doping of π-bands of graphene sheets leading of magnetic breakdown; in the fourth section we calculate
to formation of Fermi pockets, the intercalating atoms the magnetoconductivity tensor and present the results;
form the superlattice which introduces new periodicity the final section contains concluding remarks and discus-
on top of graphene honeycomb. Although mostly known sion.
and studied for their superconducting properties [2], the
research field of certain GICs was recently widened by ex-
perimental observation of charge density waves (CDW) II. THE MODEL
in CaC6 [3, 4]. The origin of the CDW ground state ap-
pears to be quite controversial, since the rather isotropic We model the CaC6 system as a 2D graphene sheet,
Fermi surface does not possess property of nesting [5], chemically doped by electrons from intercalating atoms
fulfilling physical assumptions for paradigmatic model of to provide a finite electron pocket at the Fermi surface.
the CDW instability based on it, i.e. on the Peierls in- The underlying Ca-lattice is of hexagonal symmetry,
stability [6]. In our recent paper, we proposed the model comprising three carbon primitive cells into the new
of the CDW instability in CaC6 based on the topological CaC6 supercell. This periodic potential folds the original
reconstruction of the Fermi surface, from closed pockets carbon Brillouin zone (BZ) to the new one, three times
to the open contours [7, 8]. smaller, with Fermi pockets, originally located at 6
In this paper we focus on the magnetotransport proper- graphene K and K’ points, falling to the center of the
ties in such reconstructed geometry of the Fermi surface. new zone (Γ point). The Fermi surface, for the matter
The spacing between the open contours in the reciprocal of presentation and simplicity, can be approximated
space, due to the CDW, corresponds to the energy scale with the 6-fold degenerate circle, while the details of
of the order of 102 K. Therefore, in strong magnetic field, the shape of the Fermi pockets can be addressed in
effects of magnetic breakdown are expected to be pro- the conductivity calculations as parameters appearing
nounced, profoundly affecting properties of electron spec- as effective carrier concentrations. The uniaxial CDW
trum and transport properties. Those properties consti- is formed with peaks along the graphene armchair
tute the core of this paper, in particular the way in which direction, with periodicity that triples the CaC6 cell.
magnetic breakdown modify the main, so-called ”classi- This further, uniaxial reduction of the BZ brings the
cal” contribution to magnetoconductivity. We show that, Fermi pockets to touching or slight overlap, leading in
otherwise non-oscillating, classical components of magne- turn to the reconstruction of the Fermi surface due to
 2

finite CDW order parameter acting as the gap parameter trajectories I, II with branches ± (see Fig. 1c), bB =
√
in electron spectrum. The Fermi surface is topologically e~B is the ”magnetic length” (in momentum space) for
reconstructed: from the closed pockets, it is turned into electron with charge −e, ~Ky is the conserved general-
set of open sheets. To study the magnetoconductiv- ized momentum of the semiclassical motion of electron
ity, the system is put into an external homogeneous in the used gauge, ε is the eigenvalue of energy. The
magnetic field B, perpendicular to the sample plane. semiclassical eigenfunctions are
The configuration of the real and reciprocal space is "
C± ~2 kx ± ′
Z
schematically shown in Fig. 1.
G± (kx , Ky ) = q exp i 2 ky (kx ; εF )
|vy± | bB
The zero-field electron spectrum in the CDW ground-
state attains the well known form (see Ref. [8] for details) −Ky ) dkx′ ] (3)
1h analogous for both regions I, II (we omit writing
E± (k) = ε(k − Q Q
2 ) + ε(k + 2 ) ±
2 these indices for simplicity here), where C± are the
corresponding coefficients, vy± ≡ vy (kx ; ky± (kx , ε)) are
r #
 2
Q Q
ε(k − 2) − ε(k + 2) + 4∆2 , (1) the group velocity components of v = ~1 ∇k ε(k) along
electron semiclassical trajectories at energy ε. Coeffi-
where ε(k) = ~vF |k| is the initial electron spectrum - the cients C± are found by matching the wave functions
Dirac-like electron dispersion with the Fermi velocity vF (3) at the MB points. The integral in the exponent is
and electron wave vector k = (kx , ky ), while Q = (Q, 0) the semiclassical phase (area enclosed by the trajectory
is the CDW wave vector. Here, the origin of the recipro- in the reciprocal space, i.e. the semiclassical action)
cal space is conveniently chosen at the crossing point of with lower limit determined by the starting point of
the initial electron bands (the edge of the reconstructed the trajectory along the ky± (kx ; εF ) at the Fermi energy
BZ). Due to finite CDW order parameter, ∆, the degen- εF in each region I, II. Note that these trajectories
eracy in the band crossing region is lifted, leading to the in the presented procedure are found from the equa-
reconstruction of the FS, as shown schematically in Fig. tion ε± (kx , ky ) = εF , i.e. from the initial electron
1c. dispersion with gap parameter ∆ neglected (dotted
trajectories in Fig. 1c). Therefore the solutions are
valid far from the MB-regions. In thepconsidered case
III. ELECTRON SPECTRUM AND WAVE FUNCTIONS this dependence is simply ky± (kx ; ε) = ± (ε/~vF )2 − kx2 .
IN FINITE MAGNETIC FIELD
The semiclassical solutions GI,II
± in regions I and II (see
I,II
To obtain the electron spectrum in external magnetic Fig. 1c), characterized by coefficients C± , are connected
field B perpendicular to the sample, under conditions in the ”MB-junction” by the MB-scattering matrix that
of magnetic breakdown (MB), we utilize a semiclassical relates pairs of incoming and outgoing electron waves
technique based on the Lifshitz-Onsager Hamiltonian  I    II 
[9, 10] which describes semiclassical motion of electrons C− iθ t r C−
= e . (4)
between the MB regions. The necessary assumption, C+II −r∗ t∗ C+I

required to formulate magnetic breakdown problem Here t(B) and r(B), fulfilling the unitarity condition
beyond the mere perturbative contribution of magnetic |t|2 + |r|2 = 1, are the complex probability amplitudes
field, is that the field is strong enough to provide the for electron to pass through the MB-region and to get re-
Larmor radius of electron motion much smaller that the flected on it, respectively, while θ is the phase determined
mean free path of scattering on impurities. The further by the problem-specific boundary conditions. It has been
assumption is the absence of dislocation fields, required shown [13–16] that, in the configuration originating form
to provide conditions for so-called coherent magnetic the very slight overlap of semiclassical trajectories, the
breakdown [11] which is in the focus of this paper. The probability of passing through the MB-region is
limit of so-called stohastic magnetic breakdown [12] is
∆2 3 εF
 r 
not a subject of this work. 2
|t(B)| ≈ 1 − exp − , (5)
~ωc εF ~ωc
Choosing the Landau gauge of the vector potential
A = (0, Bx, 0), the Lifshitz-Onsager Hamiltonian leads where ωc ≡ eB/m∗ is the cyclotron frequency for electron
to the Schrödinger equation in the reciprocal space with effective cyclotron mass m∗ (~ωc is then ”magnetic
energy”). Here, the latter physical quantities are just in-
b2 d
 
troduced as terms while their specific forms and energy
εν kx , Ky − i B2 Gν (kx , Ky ) = ε Gν (kx , Ky ), dependence will be elaborated later, in the next section.
~ dkx
(2) In our consideration the limit |t| = 1 accounts for the
total transparency of the ”MB junction” with zero reflec-
where εν (kx , ky ) is the initial electron dispersion shifted tion, i.e. the absence of (or negligible) magnetic break-
in the reciprocal space to the position corresponding to down. Regime |t(B)| < 1 accounts for finite magnetic
 3

FIG. 1. Schematic presentation of a 2D layer in CaC6 in real and reciprocal space. (a) In the real space, carbon atoms form a
2
honeycomb lattice with unit vectors are a1,2 (a ≡ |a1,2 | ≈ 2.5Å, the area
√ of the cell is AC ≈ 5.41Å ). Ca atoms (circles) form
the hexagonal superlattice with unit vectors are b1,2 (b ≡ |b1,2 | = 3a ≈ 4.32Å, the area of the cell is ACaC6 ≈ 16.16Å2 ).
The CDW charge stripes (red-shaded along the CDW peaks) are formed along the armchair direction, creating the uniaxial
periodic structure along the zig-zag direction, characterized by the vector W = 3b1 that triples the CaC6 cell along b1 . The
new primitive cell CaC6 × 3 is shown as the dashed orange rhombus. (b) In the reciprocal space, the carbon Brillouin zone
(BZ) is depicted by the dashed hexagon. The Ca-superlattice, with reciprocal unit vectors b∗1,2 (b∗ ≡ |b∗1,2 | ≈ 1.68Å−1 ), folds
the carbon BZ to a three times smaller CaC6 BZ (solid hexagon). All 6 Fermi pockets, from carbon K and K’ points, fall
into the Γ point (shaded), approximated by a circle of the same area SF 0 depicted by the dashed blue circle. The chemical
doping of ξ ≈ 0.2 electrons per carbon atom [3] is related to the area of the Fermi pocket S0 = 2π 2 ξ/ACaC6 ≈ 0.244Å−2 ,
which gives an average Fermi wave number kF 0 ≈ 0.28Å−1 . The CDW potential, with the wave vector Q k b∗1 of periodicity
Q = b∗ /3 ≈ 0.56Å−1 , folds the CaC6 BZ, bringing the FSs into touch (or slight overlap). The corresponding unit cell in
reciprocal space is marked by dashed orange rhombus. (c) The Fermi surface reconstructed by the CDW potential, forming
the open sheets in kx -direction. Arrows show the direction of semiclassical motion of electrons in external magnetic field B
perpendicular to the sample. Magnetic breakdown (MB) affects the semiclassical motion causing electrons to pass through
I,II
the MB-junction (shaded) with probability amplitude t(B), or get reflected from it with probability amplitude r(B), C± are
±
coefficients denoting the branches of semiclassical wave functions corresponding to trajectories ky (kx ; ε).

I,II
breakdown, the activation of over-gap tunneling assisted four coefficients C± . They constitute, together with (4),
by magnetic field. It is worth mentioning that, despite a homogeneous system of two algebraic equations for two
I I
its name, there is no typical breakdown with some finite unknowns C+ and C− , i.e.
threshold field, but rather exponential activation of the
tunneling at any finite field. One immediately notices
that the exponent in Eq. (5) has an additional large fac- 
I
h
~2
i
C exp −i (S + QK )
 I  
tor, i.e. the third root of ratio of the Fermi energy and C− t r −
h bB2
2 − y
= eiθ i .
I −r∗ t∗

magnetic energy, compared with the standard Blount’s C+ C+I
exp −i b~2 (S+ − QKy )
result [17] obtained for arbitrary large overlap of trajec- B

tories. It is result of the peculiar band topology in the (6)

reconstruction region. Dependence of |t(B)|2 is shown in
Fig. 2.
RQ R0
Periodic boundary conditions, imposed upon the semi- Here, S+ (ε) = 0 ky+ (kx ; ε)dkx , S− (ε) = Q ky− (kx ; ε)dkx
classical solutions by the CDW, i.e. G± (kx , Ky ) = are the semiclassical actions along the corresponding elec-
G± (kx + Q, Ky ), yield two additional relations between tron trajectories. The determinant of that system, taken
 4

tives
~2 Q ~2 QKy
 
∂D
= |t| 2 sin +µ ,
∂Ky ε bB b2
 B2 
∂D πε πε
= − 2 2 sin +θ , (8)
∂ε Ky vF bB 2vF2 b2B

which will be used in the section that follows.

Electron spectrum εn (Ky ) is determined from the dis-
persion equation

FIG. 2. The MB parameter |t(B)|2 according to Eq. (5), D(ε, Ky ) = 0. (9)

where B is scaled in terms of magnetic energy, i.e. B e ≡
~ωc /εF . Cyclotron frequency is taken at the Fermi energy, For example, in large enough field to produce a very
while the gap parameter is set to the typical order of magni- strong magnetic breakdown |t(B)| → 0, |r(B)| → 1,
tude of ∆/εF = 0.01. Dotted curve is the Blount’s result [17] the dominant electron motion is along the closed or-
plotted for comparison. Left panel shows dependence on B, e bits (dotted circles in Fig. 1c) due the maximized
e −1
the right panel on B . over-gap tunneling between open trajectories ky+ (kx ) and
ky− (kx ). Orbital effect of magnetic field is then a mere
Landau quantization of closed orbits. Dispersion law
reduces to cos(~2 S0 (ε)/2b2B + θ) = 0 which, assum-
ing the initial spectrum ε = ~vF |k| and S0 = π|k|2 ,
at arbitrary energy ǫ, reads
yields the Landau-quantized
p spectrum in magnetic field
εn = ±vF bB 2(n + 1/2 − θ/π), n = 0, 1, 2, .... In con-
 2   2  trast to the monolayer graphene, where the nontrivial
~ S0 (ε) ~ QKy geometric (Berry) phase φB = π appears leading to spec-
D(ε, Ky ) = cos + θ − |t| cos +µ ,
2b2B b2B trum with Landau level εn=0 at zero energy, in graphite
(7) the geometric phase is trivial [18]. The |t| = 0 case in
graphite yields p θ = 0 [13, 14], finally resulting in spec-
trum εn = ±vF 2e~B(n + 1/2), which we adopt in our
where S0 (ε) = S+ (ε) + S− (ε) is area of reciprocal space consideration although, for the sake of modelling, we use
enclosed by electron trajectory (dotted circle in Fig. 1c), the two-dimensional formalism.
assuming mirror symmetry of ky+ (kx ) and ky− (kx ) along The spectrum for arbitrary |t(B)| can be obtained in
the kx -axis, i.e. S+ = S− . Phase µ, appearing from the closed form for the considered case, reading
t = |t| exp(iµ), is, along with the phase θ, determined by √

the boundary conditions of the problem specific for the 1 θ
εn (Ky ) = ±vF 2e~B n + (1 − (−1)n ) −
particular MB configuration depending on magnetic field. 2 π
They are obtained by matching the semiclassical solution (−1)n
  2
~ QKy
 21
to the asymptotic form of exact quantum-mechanical so- + arccos |t| cos + µ (10)
π b2B
lution within the MB region [13, 14]. Although in elec-
tron spectrum and some related quantities these phases for n = 0, 1, 2, ..., shown in Fig. 3.
indeed play a role, in the problem of magnetoconductiv- Besides the dispersion law and spectrum, the system
ity that we consider they appear to be irrelevant. We (6) also determines relation between coefficients, i.e.
will keep them in this section for the sake of ”bookkeep-
ing”, but in the calculation of magnetoconductivity they |r| exp [i(φ − κy + θ)]
will be omitted since they are integrated out anyway C− = C+ , (11)
1 − |t| exp [i(φ + κy + θ)]
in expansions of periodic functions. Generally speaking,
in problems involving magnetic breakdown determining where κy ≡ ~2 QKy /b2B and φ ≡ S+ = S− .
electron spectrum can be very challenging, if possible at
The complete semiclassical wave function is
all. In some cases, spectrum can have very complicated
structure, for example possessing fractal properties. In " #
C+ ~2 kx + ′
Z

′
that respect to deal with number of quantities depending Gη (kx ) = q exp i 2 ky (kx ) − Ky dkx
on electron spectrum, such as magnetoconductivity that |vy+ | bB 0
we will explore in the next section, methods to utilize de- " #
C− ~2 kx − ′
Z
terminant D(ε, Ky ) Eq. (7) instead of electron spectrum
′
+ q exp i 2 ky (kx ) − Ky dkx .
were developed (see for example Ref. [11]). Determi- bB Q
|vy− |
nant D is function of electron energy ε and its conserved
momentum Ky yielding the corresponding partial deriva- (12)
 5

to the linear correction to the equilibrium conditions (see

Ref. [19] for the derivation details), with the general form
of it reading

2ζe2 X hη|v̂α |η ′ ihη ′ |v̂β |ηi df (ε)

σαβ = − ,
Lx Ly ′ ~i (εη′ − εη ) + τ1 dε ε=εη
η,η 0

(15)

where α, β ∈ {x, y} account for directions along the real

space (see Fig. 1a) containing a sample of the size Lx by
Ly , factor 2 accounts for the spin degeneracy and ζ = 6
for the CaC6 degeneracy (2 graphene valleys and tripling
of the unit cell). Function f (ε) is the Fermi distribu-
tion function at temperature T . We denote an operator
by ”hat” upon it, while η denotes a complete set of all
quantum numbers, in our problem {n, Ky }. In this ex-
FIG. 3. Spectrum (10) in the nondispersive |t| → 0 limit, pression, τ0 is the relaxation time due to electron scat-
essentially consisting of Landau levels (left panel), and for
tering on impurities. We emphasize again that results to
|t| = 0.9 where magnetic bands are formed due to magnetic
breakdown (right panel). Energy εn (Ky ), n = 0, 1, ..., 7 is
be presented are derived in the limit in which the im-
√ purity scattering rate is much smaller comparing to the
scaled to vF 2e~B and displayed along the ”magnetic zone”
of width K = 2πb2B /~2 Q = 2πeB/~Q. For the matter of cyclotron frequency, i.e. ωc ≫ τ0−1 . The next impor-
presentation we set phases θ = µ = 0. tant physical scale, within the regime of coherent mag-
netic breakdown, is the width of magnetic bands W (B)
compared to the broadening of level due to scattering on
Constants C+ and C− are also related by the normaliza- impurities. W (B) depends on magnetic field, essentially
tion condition of the wave function hGη (kx ) | Gη (kx )i = being controlled by the tunneling probability amplitude
1, where η = {Ky , n} is set of all good quantum numbers. |t(B)| (5) through Eq. (10). In the limit W (B) ≪ ~τ0−1
After neglecting the fast-oscillating cross-terms, sum of the structure of magnetic bands and accompanying in-
their absolute squares reduces to terference effects disappear and physics reduces to the
!−1 merely Landau level physics. On the contrary, in the
2 2 Lx Q dkx
Z opposite limit W (B) ≫ ~τ0−1 , we expect the full-scale
|C+ | + |C− | = , (13) MB effects to be pronounced, these being the goal of this
2π 0 |vy (kx )|
paper.
where Lx is the length of the sample in x-direction. Here The 2 × 2 magnetoconductivity is anisotropic. It con-
we used the fact that due to symmetry of the ± trajec- tains: (1) σxx component along the CDW peaks in real
tories, the velocity components are equal by the absolute space and perpendicular to the open electron trajecto-
value, i.e. |vy | ≡ |vy+ | = |vy− |. For further procedures, it ries in the reciprocal space; (2) σyy component along
turns convenient to normalize the wave function to pe- the CDW periodicity direction in real space and along
riod of the cyclotron motion around the semiclassical or- the open electron trajectories in the reciprocal space; (3)
bit. Assuming this motion to be governed by the Lorentz σxy = −σyx are the Hall conductivity components.
force, i.e. ~dk/dt = −ev×B, and from there substituting
dkx = − ~1 eBvy dt into Eq. (13), we obtain
A. Diagonal magnetoconductivity σxx
2π~ 2
|C+ |2 + |C− |2 = . (14)
eBLx T (ε) The diagonal magnetoconductivity along x-direction
has to be calculated directly from Eq. (15) by eval-
Here T (ε) is period of electron motion around circular
uating the matrix element hη|v̂x |η ′ i due to vanishing
semiclassical trajectory at energy ε, i.e. T /2 corresponds
semiclassical group velocity along that direction at the
to the integral over kx from 0 to Q in Eq. (13). T is re-
apex of the corresponding trajectory (see Fig. 1c, ky -
lated to the cyclotron frequency ωc in the standard way,
direction). Using the above-mentioned expression for
T = 2π/ωc . |C+ |2 and |C− |2 are determined by the sys-
tem of equations (11,14). the Lorentz force, ~k̇y = evx B, and equation
i of motion
˙
h
i
for the momentum operator k̂y = ~ Ĥ, k̂y , we obtain
h i
i
IV. MAGNETOCONDUCTIVITY v̂x = eB Ĥ, k̂y , where Ĥ is Hamiltonian of the sys-
tem with eigenvectors |ηi and corresponding eigenvalues
The magnetoconductivity tensor of a 2D system is ob- εη [11, 20]. The sought for matrix element can be di-
i
tained using the quantum density matrix formalism up rectly evaluated, i.e. hη|v̂x |η ′ i = eB (εη − ε′η )hη|k̂y |η ′ i.
 6

Inserting this expression in Eq. (15) the η = η ′ con- δ(x − xl )/|g ′ (xl )|, yielding
P
of its argument, δ(g(x)) = l
tributions in double summation vanish. For η 6= η ′ we the identity

−1
expand fraction ~i (εη′ − εη ) + τ0−1 under assumption
−1
X ∂D
|εη − εη | ≫ ~τ0 up to the second term in Taylor series.
′ δ(ε − εn (Ky )) = δ(D(ε, Ky )). (19)
After performing one summation over the complete set n
∂ε
|η ′ i and using hη|k̂y |ηi = 0 for a symmetric trajectory,
Eq. (15) reduces to Partial derivative is known from Eq. (8), while delta
function is expanded in Fourier series in the following
2ζe2 ~4 X df (ε) way
σxx = − 4 hη|k̂y2 |ηi . (16)
Lx Ly τ0 bB η dε εη X  2
~ QKy

δ(D(ε, Ky )) = Al (ε) exp il . (20)
b2B
l
To evaluate the matrix element hη|k̂y2 |ηi, we use semiclas-
sical wave functions (12), i.e. |ηi = Gn,Ky (kx ), yielding Contribution from l = 0, i.e. A0 (ε), is what we call the
main contribution or often the ”classical result”, while

2
Lx
Z Q ky+ (kx , εn (Ky )) l 6= 0 contributions represent the fast-oscillating correc-
hη|k̂y2 |ηi = dkx tions to it. In our consideration, we are interested in
2π 0 |vy (kx , εn (Ky ))|
effects of magnetic breakdown to the main contribution
|C+ (Ky , εn (Ky ))|2 + |C− (Ky , εn (Ky ))|2 .


× and eventual modification of otherwise non-oscillating
(17) classical result. Therefore, in expression (20) we keep
only the l = 0 contribution, i.e.
Fraction in Eq. (17) can q be further simplified using re- 2πb2
B
lations ε(kx , ky ) = ~vF kx2 + ky2 and ~vy = ∂ε/∂ky , i.e.
Z
~2 Q
A0 (ε) = δ(D(ε, Ky ))
ky2 /|vy | = m∗ (ε)ky /~, where m∗ (ε) is effective cyclotron 0
mass introduced in the previous section (although not 1 Θ |t|2 − cos2 φ(ε)


everywhere written explicitly for the sake of convenience, = p , (21)
π |t|2 − cos2 φ(ε)
m∗ (ε) and ωc (ε) are functions of energy ε and are treated
as such in further calculations). It is a well-known phys- where φ(ε) = ~2 S(ε)/(2b2B ) = πε2 /(2vF2 b2B ) =
~2
ical quantity, i.e. m∗ (ε) ≡ 2π dS(ε)/dε, where S(ε) is πε/(2~ωc(ε)) and Θ(..) is the Heaviside theta func-
the area enclosed by electron trajectory in the reciprocal tion. Corrections are neglected in our consideration.
space at energy ε. In the case of graphene and graphite Finally, the considered integration over Ky evaluates
it is m∗ (ε) = ε/vF2 for low enough energy to preserve R Km
to 0 y δ(D(ε, Ky )) ≈ A0 (ε)Kym . Maximal value of
linear dispersion. Inserting Eq. (17) into Eq. (16) and
conserved momentum Kym is determined by the stan-
changing the variable R εn → ε by inserting the integral dard condition in the Landau gauge that the electron
over delta function dε δ(ε − εn ) we obtain
wave package centred at x0 lies within the sample, i.e.
0 < x0 < Lx , yielding Kym = ~1 eBLx . Taking it all to-
2ζe2 ~4
 ∗
m (ε) df (ε)
Z
σxx = − 4 dε gether, we obtain the temperature-dependent expression
Lx Ly τ0 bB ~ dε for magnetoconductivity
Z Q
Lx
× dkx ky+ (kx , ε) sin φ(ε)
2π 0 ζ df (ε) 3
Z
Z m σxx = − dε |ε| p
L y Ky 2π 2 ~2 τ0 vF4 B 2 dε |t|2 − cos2 φ(ε)
dKy |C+ (Ky , ε)|2 + |C− (Ky , ε)|2


×
2π 0 ×Θ |t| − cos2 φ(ε) .
 2 
!
X (22)
× δ(ε − εn (Ky )) } . (18)
n
In the absence of magnetic breakdown, i.e. |t| = 1, the
oscillating terms in Eq. (22) reduce to 1. The zero-
The integral over kx (the second row of Eq. (18)) is ap-
temperature result, with df (ε)/dε = −δ(ε − εF ) and
proximately evaluated to the half-size of electron pocket
RQ Sommerfeld correction of the order of (kB T /εF )2 ne-
at energy ε, i.e. 0 ky+ (kx , ε)dkx ≈ S0 (ε)/2. At en- glected, reads
ergy equal to Fermi, ε = εF , it gives the size of the
Fermi surface which determines the number of carriers m∗F n0
per spin projection. The integral over Ky (the third σxx = , (23)
τ0 B 2
and fourth row of Eq. (18)) is calculated taking into ac-
count the normalization condition (14), which evaluates where m∗F ≡ m∗ (εF ) is an effective cyclotron mass and
to |C+ |2 + |C− |2 = 2~/(Lxm∗ (ε)), and utilizing the well- n0 = 2ζS0 (εF )/(2π)2 is surface concentration of carri-
known decomposition of delta function over zero-points ers (electrons of both spin projections), both taken at
 7

the Fermi energy. Magnetoconductivity has ∼ B −2 de- Taking the partial derivatives Eq. (8) in the expression
pendence and it does not contain the MB tunneling am- above and using Eqs. (20) and (21), we obtain
plitude t(B). It is equal to the non-oscillating classical Z ∞ p
result for closed orbits at low temperatures [21]. ζe2 τ0 Q2 vF2 df (ε) 1 |t|2 − cos2 φ(ε)
σyy = − dε
With finite magnetic breakdown in action, i.e. |t| < 1, π3 −∞ dε |ε| sin φ(ε)
the oscillating terms in Eq. (22) remain, causing it to
×Θ |t| − cos2 φ(ε) .
 2 
oscillate. By Fourier (re)expansion of the oscillating part
under the integral and performing the integration in the (28)
complex plane, the characteristic exponential factor ap-
pears, i.e. exp (−π 2 εF kB T /2vF2 b2B ). The argument of The zero-temperature result in the absence of the MB-
exponential function is equal to −(π 2 /2)(kB T /~ωc ). It induced over-gap electron tunneling (|t| = 1) reduces to
defines the temperature scale ∼ ~ωc , up to which the a constant,
oscillations are visible, in the similar way as Lifshitz- ζ e2 τ0 Q2
Kosevich formula for Shubnikov - de Haas oscillations σyy = , (29)
[10]. For temperatures above this scale, oscillations are π 3 m∗F
exponentially suppressed and what remain is the classical independent of magnetic field, which coincides with the
result related with the non-oscillating zeroth coefficient classical result along the open trajectories proportional
in the above-mentioned expansion. For temperatures sig- to the relaxation time τ0 [21].
nificantly lower than the mentioned scale, formally taken The zero-temperature result with finite magnetic
in the T = 0 limit again with df (ε)/dε ≈ −δ(ε − εF ), breakdown, obtained from Eq. (28), attains the form
the Eq. (22) reduces to r  
εF

εF

2 2 |t|2 − cos2 π2 ~ω
m∗F n0 sin π2 ~ω ζ e τ0 Q c

σxx =
c σyy = 3
π m∗F
 
π εF
τ0 B 2 sin 2 ~ωc
r  
εF
|t|2 − cos2 π2 ~ω c   
2 2 π εF
 
π εF
 ×Θ |t| − cos . (30)
×Θ |t|2 − cos2 . (24) 2 ~ωc
2 ~ωc

C. Hall magnetoconductivity σxy

B. Diagonal magnetoconductivity σyy
As in the case of σxx , due to vanishing group velocity
The diagonal magnetoconductivity along y-direction vx , the corresponding contribution must be accounted
(along the open trajectories in reciprocal space, see Fig. through the matrix element hη|v̂x |η ′ i which is expressed
1c, kx -direction) is determined by the nonvanishing group ˙
in terms of the Lorentz force operator proportional to k̂y .
velocity vy . Therefore, Eq. (15) can be expressed in the Expression (15) in the case of Hall conductivity reduces
form to
2ζe2 τ0 X Ly df (ε)
Z
2ζe X 
σyy = − dKy vy2 (Ky , εn (Ky )) σyx = − ~hη|k̂y v̂y |ηi
Lx Ly n 2π dε εn (Ky ) Lx Ly B η
(25)

∂εn (Ky ) df (ε)
− hη|k̂y |ηi , (31)
∂Ky dε εη
in which, using procedure elaborated in Ref. [11], we
express velocity in terms of the dispersion law D(ε, Ky ), where |ηi = |n, Ky i. Two matrix elements, hη|k̂y v̂y |ηi
i.e.
and hη|k̂y |ηi are evaluated as follows. The first one reads
∂D(ε,Ky )
1 ∂εn (Ky ) 1 ∂Ky
!
Lx Q |C+ |2 ky+ vy+ |C− |2 ky− vy−
Z
vy = = − ∂D(ε,K . (26) hη|k̂y v̂y |ηi = dkx + .
~ ∂Ky ~ y)
∂ε 2π 0 |vy+ | |vy− |
Using the same procedure to change the variable εn → ε (32)
as in the previous subsection, as well as Eq. (19), Eq.
Since ky− = −ky+ and |vy+ | = |vy− |, it reduces to
(25) can be expressed as
Lx Q
Z
2
dkx ky+ (kx , εn (Ky )) |C+ |2 + |C− |2


2
∂D hη|k̂y v̂y |ηi =
ζe τ0 df (ε) 2π
Z Z
∂K y
ε 0
σyy =− 2 dKy dε δ(D(ε, Ky )).
π~ Lx dε ∂D Lx S0 (εn (Ky )) 2 2


∂ε = |C + (K y )| + |C− (K y )| .
Ky 2π 2
(27) (33)
 8

The second term reads (with s = l = 0), i.e. F (φ1 , φ2 ) ≈ F̄ , where

Z π
dφ1 π dφ2 |t| − cos φ1 cos φ2 + sin φ1 sin φ2
!
|C+ |2 ky+ |C− |2 ky−
Z
Lx Q
Z
hη|k̂y |ηi = dkx + F̄ =
2π 0 |vy+ | |vy− | −π 2π −π 2π 1 − |t|(cos φ1 cos φ2 − sin φ1 sin φ2 )
sin φ2
Lx Q ky+ × p δ (cos φ1 − |t| cos φ2 ) , (41)
Z
dkx + |C+ |2 − |C− |2 ,


= (34) 1 − |t|2 cos2 φ2
2π 0 vy
neglecting the fast-oscillating corrections with s 6= 0,
where the negative sign in front of |C− |2 appears due to l 6= 0. The δ-function inside the integral in Eq. (41)
∂ε v2 k+
ky− = −ky+ . Using vy = ~ ∂k y
= εF ~ky , and vy+ = ~vε2 , is evaluated as a sum of δ-functions over all zeroes in the
y F
we obtain domain of integration, finally yielding F̄ = 0. Therefore,
the second contribution to the main part of Hall conduc-
II I
Lx Q εn tivity evaluates to σyx ≈ 0 leaving the term σyx as the
Z
dkx 2 |C+ |2 − |C− |2


hη|k̂y |ηi = contributing one.
2π 0 ~vF
The zero-temperature limit in the absence of magnetic
L x Q εn
|C+ (Ky )|2 − |C− (Ky )|2 .(35)


= 2
breakdown (|t| = 1) yields the result
2π ~vF
en0
σxy = −σyx = − (42)
Similarly to expression (14), for |C+ |2 + |C− |2 , using Eqs. B
(11) and (14), one can obtain an expression for |C+ |2 − written in terms of the total 2D electron concentration
|C− |2 , i.e. n0 = 2ζS0 (εF )/(2π)2 . This result corresponds to the
2π~ 2 |t| − cos(φ1 + φ2 ) classical one in a strong field [21].
|C+ |2 − |C− |2 = |t| , The zero-temperature result with finite magnetic
eBLx T (ε) 1 − |t| cos(φ1 + φ2 ) breakdown, obtained from Eq. (28), attains the form
(36)  
εF
en0 sin π2 ~ω
2 ~2 QK c
where φ1 ≡ 2vπε2 2 and φ2 ≡ b2B
y
. σxy = −σyx = −
F bB
r
B 
εF

Now we can evaluate Hall conductivity consisting of |t|2 − cos2 π2 ~ω c
I II
two contributions, i.e. σyx = σyx + σyx . The first contri-   
I
bution σyx , corresponding to Eq. (33), is evaluated in the π εF
×Θ |t|2 − cos2 . (43)
analogous way as σxx by changing the variable εn → ε 2 ~ωc
and using Eqs. (8), (19), (20) and (21), yielding
Both in expressions for magnetoconductivity compo-
ζe
Z
df (ε) 2 sin φ(ε) nents in the presence of finite magnetic breakdown at
I
σyx =− dε ε p finite temperature, Eqs. (22), (28), (37), and those at
2π~2 vF2 B dε |t|2 − cos2 φ(ε)
T = 0, Eqs. (24), (30), (43), the Heaviside theta func-
×Θ |t| − cos2 φ(ε) .
 2 
tion insures that argument of oscillating function inside
(37) expressions is within the (magnetic) band. Simple anal-
ysis of the zero-temperature results shows zero values of
II
The second contribution σyx , corresponding to Eq. magnetoconductivity at fields for which εF /~ωc is equal
(35), contains the expression to an even integer. Around these values there are fi-
nite ”gaps” of zero-conductivity, determined by the afore-
∂εn εn Q ∂D
|C+ |2 − |C− |2 δ(D), mentioned Heaviside function, which correspond to gaps


− hη|ky |ηi = 2 (38)
∂Ky ~vF ∂Ky in the energy spectrum between magnetic bands (see Fig.
3). The widths of these gaps depend on magnetic field
where D is defined in expression (7). Eq. (38) is a fast
also through the dependence of t(B) (see Fig. 2). Finite
oscillating function containing the expression
values of magnetoconductivity outside of these gaps, ap-
|t| − cos (φ1 + φ2 ) pearing periodically with B −1 with period proportional
F (φ1 , φ2 ) = sin φ2 , (39) to S0 (εF ), essentially represent the onset of quantum os-
1 − |t| cos (φ1 + φ2 )
cillations, the sharper, the temperature is lower (see Fig.
which is periodic with 2π in φ1 and φ2 . We expand ex- 4). Temperature is one of usual channels to ”smooth” the
pression (39) in the Fourier series, i.e. oscillations, appearing with oscillating integrand under
the integral in the temperature-dependent expressions.
This particular channel is present in our model while
X
F (φ1 , φ2 ) = Fs,l exp [i (sφ1 + lφ2 )], (40)
s,l the other one, the relaxation time (or analogous self-
energy), is ruled out by our starting condition ωc ≫ τ0−1
where Fs,l are expansion coefficients for all integer s and required to achieve the coherent MB. Therefore, in or-
l. Again, we keep just the main contribution F0,0 ≡ F̄ der to observe these oscillations, one needs to provide
 9

FIG. 4. Quantum oscillations of magnetoconductivity components σij in the presence of finite magnetic breakdown, depending
on magnetic field and temperature. Components σ exx , σ
eyy and σ
eyx are plotted according to expressions (22), (28) and (37) scaled
to n0 m∗F /(τ0 B 2 ), ζe2 τ0 Q2 /(π 3 m∗F ) and en0 /B, respectively in each expression. Graphs represent (scaled) energy integrals of
oscillations-generating functions, embedded with the derivative of Fermi function, vs. inverse magnetic field B −1 (scaled to
magnetic energy i.e. B e ≡ ~ωc /εF ). Graphs for σ exx and σ eyx are on this scale indistinguishable to the eye, thus are presented on
the same figure, while both figures present a characteristic pattern of oscillations with respect to the characteristic scale (period
determined by the area of electron trajectory S0 (εF ) vs. B −1 ). The temperatures (scaled to kB /εF ) in figures are: 0 (blue),
0.0001 (purple), 0.0005 (orange), 0.001 (red). The low-temperature ”gaps” of zero conductivity are closed more and more at
higher temperatures until the oscillations get suppressed at temperatures higher that ~ωc scale. The energy gap parameter in
spectrum is everywhere ∆/εF = 0.01 i.e. of the characteristic order of magnitude of 102 K.

clean samples and low temperatures. It is evident that in due to scattering on impurities, which is not the case in
both, temperature-dependent and zero-temperature ex- the limit of very narrow magnetic bands tending towards
pressions for magnetoconductivity components σij , the Landau levels (|t| = 0).
finite MB effect modifies otherwise non-oscillating main
(”classical”) contribution in terms of emerging quantum
oscillations. Those oscillations do not appear as the ad- V. CONCLUSIONS
ditive small correction to the classical part, but rather
as inherent to it, forcing it to oscillate from zero to its What makes the CaC6 system in the CDW ground
maximal value (at low enough temperatures). As men- state gainful to study effects of magnetic breakdown is
tioned earlier, apart from the onset of oscillations, the an exact match of natural scales in the problem required
MB affects the magnetoconductivity through the field- for them to be pronounced. In particular, the observed
dependent MB transmission probability amplitude t(B), uniaxial CDW [3, 4] reconstructs the closed Fermi pock-
which modifies the magnetic band width and affects the ets into open sheets with characteristic spacings between
amplitude of quantum oscillations at different fields. We them at the Brillouin zone edges approaching values de-
illustrate it in Fig. 5 on example of σyy . Its ”classi- termined by the gap parameter of the order of 102 K [8]
cal” low field value, when |t| → 1 and electron trajec- (experimental values: critical CDW temperature is 250K,
tories are open, is otherwise constant. With increasing pseudogap width from differential conductivity is 475 mV
field the over-gap MB tunneling increases (|t| gets smaller [3]). Those exactly match the MB requirement for mag-
than 1), reconstructing the closed electron trajectories netic fields of the order of B ∼ 10 T [11]. In the sam-
and effectively increasing electron localization. Emerg- ples clean enough to provide mean free path significantly
ing oscillations drop in amplitude with increasing field as longer comparing to the Larmor radius in those fields,
|t| decreases. We have to mention that, using presented with absence of dislocation fields, and temperature low
results, one cannot analytically perform a crossover to enough to neglect relaxation channels other than scatter-
|t| = 0 limit (closed orbits) due to entirely different struc- ing of carriers on impurities, we predict properties of the
ture of starting velocity operators to be used in magne- magnetoconductivity tensor, with the influence of mag-
toconductivity calculation in that case. Besides that, we netic breakdown to the main, so-called its ”classical part”
repeat that validity of our description holds until mag- in focus, and neglecting additive corrections to it.
netic band width is larger than electron level broadening Magnetoconductivity is calculated within quantum
 10

approach we kept the first order term in that interference,

neglecting terms of higher orders. These oscillations are
not just a mere additive correction appearing on top of
the classical part, which start to oscillate from zero to
maximal value, they are an inherent part of it thus turn-
ing it to essentially non-classical. These are not standard
Shubnikov - de Haas (SdH) oscillations, appearing in sys-
tems with closed Fermi surface due to Landau quantiza-
tion where the onset of oscillations lies in modifications
of DOS that pass through the Fermi level as the field is
changed. It generates an additive oscillating correction
to the principal classical result (at low enough fields be-
fore the so-called ultra-quantum limit takes place). In
CaC6 system under CDW, with open sheets of the Fermi
surface, standard SdH type of oscillations does not ex-
ist, and the only oscillatory behavior may appear due to
magnetic breakdown.
What is usually measured in experiments is the magne-
FIG. 5. The upper envelope of fast-oscillating σyy vs. B −1
toresistivity related to magnetoconductivity by inversion
eyy is scaled to ζe2 τ0 Q2 /(π 3 m∗F ),
at different temperatures. σ
e of its tensor, i.e. ρ=σ −1 . Although the low-temperature
while B is scaled as B ≡ ~ωc /εF . The temperatures (scaled
to kB /εF ) in figure are: 0 (blue), 0.0001 (purple), 0.0005
oscillatory behavior is of the same manner in the sense
(orange), 0.001 (red). The energy gap parameter in spectrum of frequency vs. B −1 , the classical field-dependent ”en-
is ∆/εF = 0.01. velope” of each magnetoresistivity component depends
on all magnetoconductivity components, i.e. ρxx/yy =
2
σyy/xx /(σxx σyy + σxy ). Taking the low-temperature val-
density matrix approach and semiclassical approximation ues of magnetoconductivity components from expressions
based on the Lifshitz-Onsager Hamiltonian, using specific (23), (29) and (42) we obtain expressions for magne-
technique developed for magnetic breakdown [11]. Under toresistivity envelopes ρxx = ζτ0 Q2 B 2 /(π 3 n20 m∗F Υ) ∼
the circumstances, electron spectrum consists of so-called B 2 , ρyy = m∗F /(e2 τ0 n0 Υ) ∼ const, where Υ ≡ 1 +
magnetic bands, wider than average level broadening due ζQ2 /(π 3 n0 ). In contrast to systems with closed Fermi
to impurity scattering. It means that the limit of closed surface, in which there appears characteristic crossover
electron trajectories, to which the system would tend in from low-field ∼ B 2 behavior into high-field saturation
the limit of extreme magnetic breakdown (which would to constant magnetoresistivity, here the ∼ B 2 behavior
restore closed trajectories from the open ones and re- of ρxx persists (within validity of the semiclassical ap-
duce the picture to the Landau quantization physics), is proximation).
not covered within our description of magnetoconductiv- To our knowledge, experimental values for magneto-
ity. In the absence of magnetic breakdown, i.e. when conductivity of CaC6 in the CDW groundstate (open
the transmission probability between neighboring cells Fermi surface) are not available so far. There are, how-
|t|2 → 1 at low fields, preserving the picture of open tra- ever, measurements of magnetoresistance in CaC6 in nor-
jectories, the components of magnetoconductivity tensor mal groundstate (absence of the CDW) with closed Fermi
attain their classical values [21]. These zero-temperature surface [22]. The magnetoresistance shows quadratic in
values are: σxx ∼ B −2 for component along the CDW field dependence as expected in low, which crosses over to
crests which is perpendicular to the open semiclassical linear in strong fields instead of saturation which would
electron trajectories in reciprocal space, σyy ∼ const. for be expected metallic behavior. There is a rather old pa-
component along the open trajectories, and σxy ∼ B −1 per by A. A. Abrikosov [23] which relates linear in field
for Hall conductivity. On the other hand, in the regime of magnetoresistance with linearity of electron spectrum,
finite magnetic breakdown |t|2 < 1, all three magnetocon- but in the strong field vs. low doped system with closed
ductivity components start to manifest strong quantum Fermi surface and Dirac point in spectrum, in which only
oscillations vs. inverse magnetic field at low tempera- one Landau level (the first one) one is engaged. In that
tures. They are most pronounced at zero-temperature respect, CaC6 with chemical doping of 0.2 electrons per
and get gradually ”smoothed out” as temperature in- carbon atom and consequently the Fermi energy of the
creases, finally being exponentially suppressed at temper- order of electronvolt is quite highly doped comparing to
atures higher than magnetic energy scale ~ωc . The onset magnetic fields of several Tesla where linear in field be-
of oscillations is a feature of coherent magnetic break- havior is observed. Although we took the linearity of elec-
down, in which large number of electrons takes part. It tron spectrum into account and our result for σxx should
is due to the interference of huge semiclassical phases correspond to one for the system with closed Fermi sur-
(comparing to the characteristic magnetic scale ∼ b2B ) face, it needs to be stressed that the semiclassical picture
characterizing the semiclassical wave functions. In our is valid in the limit of large values of (occupied) Landau
 11

band indices. It is exactly opposite to Abrikosov’s limit ernment and European Union through the European Re-
and corresponds more to the limit in which the observed gional Development Fund - the Competitiveness and Co-
magnetoresistivity is quadratic in field. In that sense ex- hesion Operational Programme (Grant PK.1.1.02). The
planation of observed linear in field magnetoresistance authors are grateful to dr. I. Smolić for constructive dis-
remains an open question. cussions.

ACKNOWLEDGEMENTS

This work was supported by the QuantiXLie Centre of

Excellence, a project co-financed by the Croatian Gov-

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