Anomalous c-axis transport in layered metals
D. B. Gutman and D. L. Maslov
Department of Physics, University of Florida, Gainesville, FL 32611, USA
(Dated: November 4, 2018)
Transport in metals with strongly anisotropic single-particle spectrum is studied. Coherent band
transport in all directions, described by the standard Boltzmann equation, is shown to withstand
both elastic and inelastic scattering as long as EF τ ≫ 1. A model of phonon-assisted tunneling via
resonant states located in between the layers is suggested to explain a non-monotonic temperature
dependence of the c-axis resistivity observed in experiments.
PACS numbers: 72.10.-d,72.10.Di
Electron transport in layered materials exhibits a num-
ber of unusual properties. The most striking example is
a qualitatively different behavior of the in-plane (ρab)
and out-of-plane (ρc) resistivities: whereas the temper-
ature dependence of ρab is metallic-like, that of ρc is ei-
ther insulating-like or even non-monotonic. At the level
of non-interacting electrons, layered systems are metals
with strongly anisotropic Fermi surfaces. A commonly
used model is free motion along the planes and nearest-
neighbor hopping between the planes:
εk = k
||/2mab + 2J (1− cos k⊥d) , (1)
where k|| and k⊥ are in the in-plane and c-axis com-
ponents of momentum, respectively, mab is the in-plane
mass, and d is lattice constant in the c-axis direction. For
the strongly anisotropic case (J ≪ EF ), the equipoten-
tial surfaces are “corrugated cylinders” (see Fig.1).
If the Hamiltonian consists of the band motion with
spectrum (1) and the interaction of electrons with poten-
tial disorder as well as with inelastic degrees of freedom,
e.g., phonons, the Boltzmann equation predicts that the
conductivities are given by
σBab = e
2ν〈vavbτtr〉, σBc = 4e2νJ2d2〈sin2 (k⊥d) τtr〉, (2)
where 〈. . . 〉 denotes averaging over the Fermi surface and
over the thermal (Fermi) distribution, ν = mab/πd is the
density of states, and τtr is the transport time, resulting
from all scattering processes (we set h̄ = kB = 1). If
τtr decreases with the temperature, both σab and σc are
expected to decrease with T as well. This is not what
the experiment shows.
The c-axis puzzle received a lot of attention in con-
nection to the HTC materials [1], and a non-Fermi-liquid
nature of these materials was suggested to be responsible
for the anomalous c-axis transport [2]. However, other
materials, such as graphite [3], TaS2 [4], Sr2RuO4 [5], or-
ganic metals [6], etc., behave as canonical Fermi liquids
in all aspects but the c-axis transport. This suggests that
the origin of the effect is not related to the specific prop-
erties of HTC compounds but common for all layered
materials. A large number of models were proposed to
explain the c-axis puzzle. Despite this variety, most au-
thors seem to agree on that the coherent band transport
FIG. 1: Fermi surface corresponding to Eq.(1) with Fermi
velocity vectors at two different points.
in the c-axis direction is destroyed. Although there is no
agreement as to what replaces the band transport in the
”incoherent” regime, the most frequently discussed mech-
anisms include incoherent tunneling between the layers,
assisted by either out-of-plane impurities [8, 10, 11, 12] or
by coupling to dissipative environment [13], and polarons
[14, 15].
The message of this Letter is two-fold. First, we ob-
serve that neither elastic or inelastic (electron-phonon)
scattering can destroy band transport even in a strongly
anisotropic metal as long as the familiar parameter EF τ
is large. Nothing happens to the Boltzmann conductivi-
ties in Eq.(2) except for σBc becoming very small at high
temperatures so that other mechanisms, not included in
Eq.(2), dominate transport. This observation is in agree-
ment with recent experiment [7] where a coherent fea-
ture (angle-dependent magnetoresistance) was observed
in a supposedly incoherent regime. Second, we propose
phonon-assisted tunneling through resonant impurities as
the mechanism competing with the band transport. As
such tunneling provides an additional channel for trans-
http://arxiv.org/abs/0704.0613v1
port, the total conductivity is [8]
σc = σ
c + σres, (3)
where σres is the resonant-impurity contribution. Be-
cause σres increases with the temperature, the band chan-
nel is short-circuited by the resonant one at high enough
temperatures[9]. Accordingly, σc goes through a min-
imum at a certain temperature (and ρc = σ
c goes
through a maximum). We consider phonon-assisted tun-
neling through a wide band of resonant levels distributed
uniformly in space. We show that the non-perturbative
(in the electron-phonon coupling) version of this the-
ory is in a quantitative agreement with the experiment
on Sr2RuO4 [5]. Due to a similarity between phonon-
assisted tunneling and other problems, in which inter-
action leads to the formation of a cloud surrounding the
electron (such as polaronic effect and zero bias anomaly),
many ideas put forward earlier [8, 10, 11, 12, 13, 14, 15]
agree with our picture. Nevertheless, we believe that
only a combination of resonant impurities and electron-
phonon interaction solves the puzzle of c-axis resistivity
and provides a microscopic theory for some of the mech-
anisms considered in prior work. We begin with the dis-
cussion of the breakdown (or lack of it thereof) of the
Boltzmann equation.
One may wonder whether the band transport along
the c-axis breaks down because the Anderson localization
transition occurs in the c-direction whereas the in-plane
transport remains metallic. This does not happen, how-
ever, because an electron, encountering an obstacle for
motion along the c-axis, moves quickly to another point
in the plane, where such an obstacle is absent. More
formally, it has been shown the Anderson transition oc-
curs only simultaneously in all directions [16, 17, 18] and
only if J is exponentially smaller than 1/τ . Therefore,
localization cannot explain the observed behavior.
Refs.[19, 20] suggested an idea of the “coherent-
incoherent crossover”. It implies that the coherent band
motion breaks down if electrons are scattered faster than
they tunnel between adjacent layers, i.e., if Jτ ≪ 1. Con-
sequently, the current in the c-direction is carried via in-
coherent hops between conducting layers. It was noted
by a number of authors that the assumption about inco-
herent nature of the transport does not, by itself, explain
the difference in temperature dependences of σab and σc
[20, 21]: due to conservation of the in-plane momentum,
σc is proportional to τ both in the coherent and inco-
herent regimes. Nevertheless, an issue of the “coherent-
incoherent crossover” poses a fundamentally important
question: can scattering destroy band transport only in
some directions, if the spectrum is anisotropic enough
[22]? We argue here that this is not the case.
Since we have already ruled out elastic scattering, this
leaves inelastic one as a potential culprit. We focus on
the case of the electron-phonon interaction as a source of
inelastic scattering. For an isotropic metal, the quantum
kinetic equation is derived from the Keldysh equations of
motion for the Green’s function via the Prange-Kadanoff
procedure [23] for any strength of the electron-phonon in-
teraction. In this Letter, we apply the Prange-Kadanoff
theory to metals with strongly anisotropic Fermi surfaces,
such as the one in Fig. 1. We show that, exactly as in
the isotropic case, the Boltzmann equation holds its stan-
dard form as long as EF τe-ph ≫ 1. Since this form does
not change between coherent (Jτe-ph ≫ 1) and incoher-
ent (Jτe-ph ≪ 1) regimes, it means that the coherent-
incoherent crossover is, in fact, absent.
We adopt the standard Frölich Hamiltonian for
the deformation-potential interaction with longitudinal
acoustic phonons (ωq = sq)
k+qak
Since tunneling matrix elements are much more sensi-
tive to the increase in the inter-plane distance than the
elastic moduli, the anisotropy of phonon spectra in lay-
ered materials, albeit significant, is still weaker than
the anisotropy of electron spectra (see, e.g., Ref. [24]).
Therefore, we treat phonons in the isotropic approxima-
tion, and assume that the magnitude of the Fermi veloc-
ity is larger than the speed of sound s.
For a static and uniform electric field, the Keldysh
component of the electron’s Green function satisfies the
Dyson equation
L̂GK +
[ReΣR,⊗GK ]− + [ΣK ,⊗ReGK ]−
[ΣK ,⊗A]+ − [Γ,⊗GK ]+
. (5)
Here L̂ = (∂t + v · ∇R + eE · ∇k) is the Liouville op-
erator, A = i(GR − GA) is the spectral function, Γ =
ΣR − ΣA
, and ⊗ denotes the convolution in space and
time. Thanks to the Migdal theorem, the self-energy does
not depend on electron’s dispersion ξk ≡ εk − EF , and
Eq.(5) can be integrated over ξk. This results in an equa-
L̂gK +
[ReΣR, gK ]− = 2iΣ
K − 1
[Γ, gK ]+ (6)
for the “distribution function”
gK(ǫ, n̂) =
GK(ǫ, ξk, n̂)dξk , (7)
where n̂ = vk/ |vk| is a local normal to the Fermi surface.
We consider a linear dc response, when the self-energy
is needed only at equilibrium. Within the Migdal theory,
the Matsubara self-energy is given by a single diagram
Σ(ǫ, n̂) = −
g2 (q)G(ǫ− ω,k− q)D(ω, q) ,
where the dressed phonon propagator
D−1 = D−10 − g
is expressed through bare one
D0(ω, q) = −s2q2/
ω2 + s2q2
and polarization operator Π which, for EF > 2J, is given
by its 2D form
Π(ω, q) = −ν
1− |ω|/
v2F q
‖ + ω
We assume that the electron-phonon vertex decays on
some scale kD shorter than Fermi momentum (kD ≪ kF ).
This assumption allows one to linearize the dispersion
ξk−q ≈ ξk − vk · q and simplifies the analysis without
changing the results qualitatively. As long as J ≪ EF ,
we have |vk| ≈ kF /mab ≈ vF , where kF is the radius of
the cylinder in Fig. 1 for J = 0. Despite the fact that
the electron velocity does have a small component along
the c-axis, its in-plane component is large (cf. Fig. 1).
Since it is the magnitude of vk that controls the Migdal’s
approximation, the problem reduces to the interaction of
fast 2D electrons with slow 3D phonons. With these
simplifications, we find
ReΣR(ǫ, n̂) = −1
ǫ; (8a)
ImΣR(ǫ, n̂) = − ζ
12(1− ζ)2
, (8b)
where ζ = νg2 is a dimensionless coupling constant and
ωD = skD. We see that, despite the strong anisotropy,
the self-energy remains local, i.e., independent of ξk.
Vertex renormalization leads to two types of correc-
tions to the self-energy: those that are proportional to
the Migdal’s parameter (s/vF ) and those that are pro-
portional to ms2/ǫ. The second type of corrections inval-
idates the Migdal’s theory for temperatures below ms2,
which is about 1 K in a typical metal. For metals with
anisotropic spectrum the existence of such a scale is po-
tentially dangerous, since it is not obvious which of the
masses (light or heavy) defines this scale. We find that
the in-plane mass (mab) controls the vertex renormaliza-
tion for the nearly cylindrical Fermi surface. This shows
that the Migdal theory for layered metals has the same
range of applicability as for isotropic metals [25].
The rest of the derivation proceeds in the same way as
for the isotropic case [23], and the resulting Boltzmann
equation assumes its standard form. Since no assump-
tion about the relation between τe-ph and the dwell time
(1/J) has been made, the conductivities obtained from
the Boltzmann equation have the same form regardless
of whether Jτe-ph is large or small. In other words, there
is no coherent-incoherent crossover due to inelastic scat-
tering in an anisotropic metal [29].
The situation changes qualitatively if resonant impu-
rities are present in between the layers. Electrons that
tunnel through such impurities are moving with the speed
controlled by the broadening of a resonant level, i.e.,
much slower than speed of sound. For that reason they
can not be treated within the formalism outlined above
and require a separate study.
To evaluate the resonant-impurity contribution to the
conductivity, we assume that the impurities are randomly
distributed in space with density nimp whereas their en-
ergy levels uniformly distributed over an interval Eb. The
tunneling conductance of a bilayer junction is
G = −e2
dǫdǫ′Wǫ,ǫ′
(1 − n′ǫ) +
, (9)
where Wǫ,ǫ′ is a transition probability per unit time and
nǫ is the Fermi function. To calculateWǫ,ǫ′ , we use the re-
sults of Ref.[30, 31] for the probability of phonon-assisted
tunneling through a single impurity
Wǫ,ǫ′ = ΓLΓR
it1(ǫ
dt2dt3e
i(t2−t3)(ǫ−ǭ0)−Γ(t2+t3) (10)
× exp
|αq|2
|1− e−it3 + eit1
e−it2 − 1
|2 coth
e−it3 + eit2 + eit1(e−it2 − 1)(1− eit3)− c.c.
where αq = −iΛq/
ρωq, Λ is the deformation-potential
constant, ΓL and ΓR are tunneling widths of the resonant
level, Γ = ΓL + ΓR, and ǭ0 is the energy of a resonant
level renormalized by the electron-phonon interaction. In
the limit of no electron-phonon interaction, Eq.(10) re-
produces the well-known Breit-Wigner formula. From
now on, we consider a wide band of resonant levels:
Eb ≫ T ≫ Γ. Averaging Eq.(10) over spatial and en-
ergy positions of resonant levels, one obtains
σres=σel
1−coth
sinh2
dteitǫ−λf(t)
f(t)=
(1−cos(ωt)) coth
+i sin(ωt)
.(11)
Here σel is the conductivity due to elastic resonant tun-
neling and λ ≡ Λ2ω2D/ρs5π2 is the dimensionless cou-
pling constant for localized electrons. In the absence
of electron-phonon interaction, σres is temperature inde-
pendent and given by σel ≃ πe2Γ1nimpa0d/Eb[32], where
a0 is the localization radius of a resonant state and
Γ1 ≃ ǫ0e−d/a0 is its typical width. We note that the
electron-phonon interaction is much stronger for localized
electrons than for band ones: λ/ζ ∼ (kFd) (vF /s) ≫ 1.
Since typically ζ ∼ 1, one needs to consider a non-
perturbative regime of phonon-assisted tunneling. In
that case, resonant tunneling is exponentially suppressed
at T = 0: σres(T = 0) = σele
−λ/2. At finite T , we find
σres = σel
e−λ/2
1 + π
, T ≪ ωD√
, T ≫ λωD.
As T increases, σres growth, resembling the zero-bias
anomaly in disordered metals and Mössbauer effect. At
high temperatures (T ≫ λωD) σres approaches the non-
interacting value (σel). The asymptotic regimes in the
interval ωD/
λ ≪ T ≪ λωD can also be studied but we
will not pause for this here. Notice that, in contrast to
the phenomenological model of Ref.[8], there is no simple
relation between the T -dependences of σBc and σres.
To compare our model with the experiment, we extract
σBc from the low-temperature (between 10 and 50 K) c-
axis resistivity of Sr2RuO4 and extrapolate it to higher
temperatures [5]. The resonant part of the conductivity
is calculated numerically using Eq.(11). The fit to the
data for σel = 43 · 103Ω−1 cm−1, ωD = 41 K and λ = 16
is shown in Fig. 2. The agreement between the theory
and experiment is quite good and the values of the fitting
parameters are reasonable. An immediate consequence of
our model is the sample-to-sample variation of the c-axis
conductivity. Among the layered materials, the largest
amount of data is collected for graphite [3]. Even within
the group of samples with comparable in-plane mobili-
ties, the temperature of the maximum in ρc varies from
40K to 300 K [3, 33].
To conclude, we have shown that the Boltzmann
equation and its consequences are no less robust for
anisotropic metals than they are for isotropic ones. The
only condition controlling the validity of the Boltzmann
equation is the large value of EF τ, regardless of whether
τ comes from elastic or inelastic scattering. Out-of-
plane localized states change the c-axis transport rad-
ically while playing only minor role for the in-plane
one. While ρab remains metallic, an interplay between
100 200 300 400
T,     Kelvin
FIG. 2: ρc vs temperature. Solid: experimental data on
Sr2RuO4; dashed: fit into the phonon-assisted tunneling
model in the non-perturbative regime, Eq.(11)
phonon-assisted tunneling and conventional momentum
relaxation causes insulating or non-monotonic depen-
dence of ρc on temperature. This model is in a good
agreement with the experimental data on Sr2RuO4.
This research was supported by NSF-DMR-0308377.
We acknowledge stimulating discussions with B. Alt-
shuler, A. Chubukov, A. Hebard, S. Hill, P. Hirschfeld,
P. Littlewood, D. Khmelnistkii, N. Kumar, Yu. Makhlin,
A. Mirlin, M. Reizer, A. Schofield, S. Tongay, A.A. Var-
lamov, and P. Wölfle. We are indebted to A. Hebard, A.
Mackenzie, and S. Tongay for making their data available
to us.
[1] S. L. Cooper and K. E. Gray, in Physical Properties
of High Temperature Superconductors, edited by D. M.
Ginsberg, (World Scientific, Singapore, 1994), p. 61.
[2] P. W. Anderson, Science 256, 1526 (1990); P. W. An-
derson and Z. Zou, Phys. Rev. Lett. 42, 2642 (1992); D.
G. Clarke, S. P. Strong, and P. W. Anderson, Phys. Rev.
Lett. 72, 3218-3221 (1994).
[3] see N. B. Brandt, S. M. Chudinov, and Ya. G.
Ponomarev, Semimetals: I. Graphite and its com-
pounds, (North-Holland, Amsterdam, 1988) and refer-
ences therein.
[4] W. J. Wattamaniuk, J. P. Tidman, and R. F. Frindt,
Phys. Rev. Lett. 35 62 (1975).
[5] A. W. Tyler, A. P. Mackenzie, S. NishiZaki, and Y.
Maeno, Phys. Rev. B 58, 10107 (R) (1998).
[6] J. Singleton and C. Mielke, Contemp. Phys. 43, 63
(2002).
[7] J. Singleton et al. cond-mat/0610318.
[8] A. Rojo and K. Levin, Phys. Rev. B 48, 16861 (1993).
[9] V. Fleurov, M. Karpovski, M. Molotskii, A. Palevski, A.
Gladkikh, R. Kris Solid State Comm. 97, 543, (1996).
[10] M. J. Graf, M. Palumbo, D. Rainer, and J. A. Sauls,
Phys. Rev. B 52, 10588 (1995).
http://arxiv.org/abs/cond-mat/0610318
[11] P. J. Hirschfeld, S. M. Quinlan, and D. J. Scalapino,
Phys. Rev. B 55, 12742 (1997).
[12] A. A. Abrikosov, Physica C 317-318, 154 (1999).
[13] M. Turlakov and A. J. Leggett, Phys. Rev. B 63, 064518
(2001).
[14] U. Lundin and R. H. McKenzie,Phys. Rev. B 68,
081101(R) (2003).
[15] A. F. Ho and A. J. Schofield, Phys. Rev. B 71, 045101
(2005)
[16] P. Wölfle and R. N. Bhatt, Phys. Rev. B 30, 3542 (1984).
[17] N. Kumar, P. A. Lee, and B. Shapiro, Physica A 168,
447 (1990).
[18] N. Dupuis, Phys. Rev B 56, 9377 (1997).
[19] N. Kumar and A. M. Jayannavar, Phys. Rev. B 45, 5001
(1992).
[20] P. Moses and R. H. McKenzie, Phys. Rev. B 60, 7998
(1999).
[21] L. Ioffe, A. Larkin, A. Varlamov, and L. Yu, Phys. Rev.
B 47, 8936 (1993).
[22] D. G. Clarke, S. P. Strong, P. M. Chaikin, and E. I.
Chashechkina, Science 279, 2071 (1998).
[23] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323
(1986).
[24] J. Paglione, C. Lupien, W.A. MacFarlane, J.M. Perz, L.
Taillefer, Z.Q. Mao, and Y. Maeno, Phys. Rev. B 65,
220506(R).
[25] The self-energy in Eqs.(8a,8b) diverges at ζ = 1. This di-
vergence –also present for the isotropic case – results from
the renormalization of the sound velocity and is an arte-
fact of the Frölich Hamiltonian. A divergence-free theory
is obtained by applying the adiabatic approximation to
the coupled system of electrons and ions [26, 27, 28].
[26] J. R. Schrieffer, Theory of Superconductivity, (Addison-
Wesley, Redwood City, 1988).
[27] E.G. Brovman and Yu. Kagan, Sov. Phys. JETP 25, 365
(1967).
[28] B. T. Geilikman, Sov. Phys.-Usp. 18, 190 (1975).
[29] Migdal’s theory also rules out models based entirely on
polaronic effects since polarons are stable only if a typical
electron velocity does not exceed the sound one.
[30] L.I. Glazman and R.I. Shekhter, Sov. Phys. JETP 61
163, (1988).
[31] N. S. Wingreen, K. W. Jacobsen, and J. W. Wilkins,
Phys. Rev. Lett. 59, 376 (1987); Phys. Rev. B 40, 11834
(1989).
[32] A.I. Larkin and K. Matveeev, Sov. Phys. JETP 66, 580
(1987).
[33] S. Tongay and A. F. Hebard, private communication.

  Transport in metals with strongly anisotropic single-particle spectrum is
studied. Coherent band transport in all directions, described by the standard
Boltzmann equation, is shown to withstand both elastic and inelastic scattering
as long as $E_F\tau\gg 1$. A model of phonon-assisted tunneling via resonant
states located in between the layers is suggested to explain a non-monotonic
temperature dependence of the c-axis resistivity observed in experiments.

Anomalous c-axis transport in layered metals
D. B. Gutman and D. L. Maslov
Department of Physics, University of Florida, Gainesville, FL 32611, USA
(Dated: November 4, 2018)
Transport in metals with strongly anisotropic single-particle spectrum is studied. Coherent band
transport in all directions, described by the standard Boltzmann equation, is shown to withstand
both elastic and inelastic scattering as long as EF τ ≫ 1. A model of phonon-assisted tunneling via
resonant states located in between the layers is suggested to explain a non-monotonic temperature
dependence of the c-axis resistivity observed in experiments.
PACS numbers: 72.10.-d,72.10.Di
Electron transport in layered materials exhibits a num-
ber of unusual properties. The most striking example is
a qualitatively different behavior of the in-plane (ρab)
and out-of-plane (ρc) resistivities: whereas the temper-
ature dependence of ρab is metallic-like, that of ρc is ei-
ther insulating-like or even non-monotonic. At the level
of non-interacting electrons, layered systems are metals
with strongly anisotropic Fermi surfaces. A commonly
used model is free motion along the planes and nearest-
neighbor hopping between the planes:
εk = k
||/2mab + 2J (1− cos k⊥d) , (1)
where k|| and k⊥ are in the in-plane and c-axis com-
ponents of momentum, respectively, mab is the in-plane
mass, and d is lattice constant in the c-axis direction. For
the strongly anisotropic case (J ≪ EF ), the equipoten-
tial surfaces are “corrugated cylinders” (see Fig.1).
If the Hamiltonian consists of the band motion with
spectrum (1) and the interaction of electrons with poten-
tial disorder as well as with inelastic degrees of freedom,
e.g., phonons, the Boltzmann equation predicts that the
conductivities are given by
σBab = e
2ν〈vavbτtr〉, σBc = 4e2νJ2d2〈sin2 (k⊥d) τtr〉, (2)
where 〈. . . 〉 denotes averaging over the Fermi surface and
over the thermal (Fermi) distribution, ν = mab/πd is the
density of states, and τtr is the transport time, resulting
from all scattering processes (we set h̄ = kB = 1). If
τtr decreases with the temperature, both σab and σc are
expected to decrease with T as well. This is not what
the experiment shows.
The c-axis puzzle received a lot of attention in con-
nection to the HTC materials [1], and a non-Fermi-liquid
nature of these materials was suggested to be responsible
for the anomalous c-axis transport [2]. However, other
materials, such as graphite [3], TaS2 [4], Sr2RuO4 [5], or-
ganic metals [6], etc., behave as canonical Fermi liquids
in all aspects but the c-axis transport. This suggests that
the origin of the effect is not related to the specific prop-
erties of HTC compounds but common for all layered
materials. A large number of models were proposed to
explain the c-axis puzzle. Despite this variety, most au-
thors seem to agree on that the coherent band transport
FIG. 1: Fermi surface corresponding to Eq.(1) with Fermi
velocity vectors at two different points.
in the c-axis direction is destroyed. Although there is no
agreement as to what replaces the band transport in the
”incoherent” regime, the most frequently discussed mech-
anisms include incoherent tunneling between the layers,
assisted by either out-of-plane impurities [8, 10, 11, 12] or
by coupling to dissipative environment [13], and polarons
[14, 15].
The message of this Letter is two-fold. First, we ob-
serve that neither elastic or inelastic (electron-phonon)
scattering can destroy band transport even in a strongly
anisotropic metal as long as the familiar parameter EF τ
is large. Nothing happens to the Boltzmann conductivi-
ties in Eq.(2) except for σBc becoming very small at high
temperatures so that other mechanisms, not included in
Eq.(2), dominate transport. This observation is in agree-
ment with recent experiment [7] where a coherent fea-
ture (angle-dependent magnetoresistance) was observed
in a supposedly incoherent regime. Second, we propose
phonon-assisted tunneling through resonant impurities as
the mechanism competing with the band transport. As
such tunneling provides an additional channel for trans-
http://arxiv.org/abs/0704.0613v1
port, the total conductivity is [8]
σc = σ
c + σres, (3)
where σres is the resonant-impurity contribution. Be-
cause σres increases with the temperature, the band chan-
nel is short-circuited by the resonant one at high enough
temperatures[9]. Accordingly, σc goes through a min-
imum at a certain temperature (and ρc = σ
c goes
through a maximum). We consider phonon-assisted tun-
neling through a wide band of resonant levels distributed
uniformly in space. We show that the non-perturbative
(in the electron-phonon coupling) version of this the-
ory is in a quantitative agreement with the experiment
on Sr2RuO4 [5]. Due to a similarity between phonon-
assisted tunneling and other problems, in which inter-
action leads to the formation of a cloud surrounding the
electron (such as polaronic effect and zero bias anomaly),
many ideas put forward earlier [8, 10, 11, 12, 13, 14, 15]
agree with our picture. Nevertheless, we believe that
only a combination of resonant impurities and electron-
phonon interaction solves the puzzle of c-axis resistivity
and provides a microscopic theory for some of the mech-
anisms considered in prior work. We begin with the dis-
cussion of the breakdown (or lack of it thereof) of the
Boltzmann equation.
One may wonder whether the band transport along
the c-axis breaks down because the Anderson localization
transition occurs in the c-direction whereas the in-plane
transport remains metallic. This does not happen, how-
ever, because an electron, encountering an obstacle for
motion along the c-axis, moves quickly to another point
in the plane, where such an obstacle is absent. More
formally, it has been shown the Anderson transition oc-
curs only simultaneously in all directions [16, 17, 18] and
only if J is exponentially smaller than 1/τ . Therefore,
localization cannot explain the observed behavior.
Refs.[19, 20] suggested an idea of the “coherent-
incoherent crossover”. It implies that the coherent band
motion breaks down if electrons are scattered faster than
they tunnel between adjacent layers, i.e., if Jτ ≪ 1. Con-
sequently, the current in the c-direction is carried via in-
coherent hops between conducting layers. It was noted
by a number of authors that the assumption about inco-
herent nature of the transport does not, by itself, explain
the difference in temperature dependences of σab and σc
[20, 21]: due to conservation of the in-plane momentum,
σc is proportional to τ both in the coherent and inco-
herent regimes. Nevertheless, an issue of the “coherent-
incoherent crossover” poses a fundamentally important
question: can scattering destroy band transport only in
some directions, if the spectrum is anisotropic enough
[22]? We argue here that this is not the case.
Since we have already ruled out elastic scattering, this
leaves inelastic one as a potential culprit. We focus on
the case of the electron-phonon interaction as a source of
inelastic scattering. For an isotropic metal, the quantum
kinetic equation is derived from the Keldysh equations of
motion for the Green’s function via the Prange-Kadanoff
procedure [23] for any strength of the electron-phonon in-
teraction. In this Letter, we apply the Prange-Kadanoff
theory to metals with strongly anisotropic Fermi surfaces,
such as the one in Fig. 1. We show that, exactly as in
the isotropic case, the Boltzmann equation holds its stan-
dard form as long as EF τe-ph ≫ 1. Since this form does
not change between coherent (Jτe-ph ≫ 1) and incoher-
ent (Jτe-ph ≪ 1) regimes, it means that the coherent-
incoherent crossover is, in fact, absent.
We adopt the standard Frölich Hamiltonian for
the deformation-potential interaction with longitudinal
acoustic phonons (ωq = sq)
k+qak
Since tunneling matrix elements are much more sensi-
tive to the increase in the inter-plane distance than the
elastic moduli, the anisotropy of phonon spectra in lay-
ered materials, albeit significant, is still weaker than
the anisotropy of electron spectra (see, e.g., Ref. [24]).
Therefore, we treat phonons in the isotropic approxima-
tion, and assume that the magnitude of the Fermi veloc-
ity is larger than the speed of sound s.
For a static and uniform electric field, the Keldysh
component of the electron’s Green function satisfies the
Dyson equation
L̂GK +
[ReΣR,⊗GK ]− + [ΣK ,⊗ReGK ]−
[ΣK ,⊗A]+ − [Γ,⊗GK ]+
. (5)
Here L̂ = (∂t + v · ∇R + eE · ∇k) is the Liouville op-
erator, A = i(GR − GA) is the spectral function, Γ =
ΣR − ΣA
, and ⊗ denotes the convolution in space and
time. Thanks to the Migdal theorem, the self-energy does
not depend on electron’s dispersion ξk ≡ εk − EF , and
Eq.(5) can be integrated over ξk. This results in an equa-
L̂gK +
[ReΣR, gK ]− = 2iΣ
K − 1
[Γ, gK ]+ (6)
for the “distribution function”
gK(ǫ, n̂) =
GK(ǫ, ξk, n̂)dξk , (7)
where n̂ = vk/ |vk| is a local normal to the Fermi surface.
We consider a linear dc response, when the self-energy
is needed only at equilibrium. Within the Migdal theory,
the Matsubara self-energy is given by a single diagram
Σ(ǫ, n̂) = −
g2 (q)G(ǫ− ω,k− q)D(ω, q) ,
where the dressed phonon propagator
D−1 = D−10 − g
is expressed through bare one
D0(ω, q) = −s2q2/
ω2 + s2q2
and polarization operator Π which, for EF > 2J, is given
by its 2D form
Π(ω, q) = −ν
1− |ω|/
v2F q
‖ + ω
We assume that the electron-phonon vertex decays on
some scale kD shorter than Fermi momentum (kD ≪ kF ).
This assumption allows one to linearize the dispersion
ξk−q ≈ ξk − vk · q and simplifies the analysis without
changing the results qualitatively. As long as J ≪ EF ,
we have |vk| ≈ kF /mab ≈ vF , where kF is the radius of
the cylinder in Fig. 1 for J = 0. Despite the fact that
the electron velocity does have a small component along
the c-axis, its in-plane component is large (cf. Fig. 1).
Since it is the magnitude of vk that controls the Migdal’s
approximation, the problem reduces to the interaction of
fast 2D electrons with slow 3D phonons. With these
simplifications, we find
ReΣR(ǫ, n̂) = −1
ǫ; (8a)
ImΣR(ǫ, n̂) = − ζ
12(1− ζ)2
, (8b)
where ζ = νg2 is a dimensionless coupling constant and
ωD = skD. We see that, despite the strong anisotropy,
the self-energy remains local, i.e., independent of ξk.
Vertex renormalization leads to two types of correc-
tions to the self-energy: those that are proportional to
the Migdal’s parameter (s/vF ) and those that are pro-
portional to ms2/ǫ. The second type of corrections inval-
idates the Migdal’s theory for temperatures below ms2,
which is about 1 K in a typical metal. For metals with
anisotropic spectrum the existence of such a scale is po-
tentially dangerous, since it is not obvious which of the
masses (light or heavy) defines this scale. We find that
the in-plane mass (mab) controls the vertex renormaliza-
tion for the nearly cylindrical Fermi surface. This shows
that the Migdal theory for layered metals has the same
range of applicability as for isotropic metals [25].
The rest of the derivation proceeds in the same way as
for the isotropic case [23], and the resulting Boltzmann
equation assumes its standard form. Since no assump-
tion about the relation between τe-ph and the dwell time
(1/J) has been made, the conductivities obtained from
the Boltzmann equation have the same form regardless
of whether Jτe-ph is large or small. In other words, there
is no coherent-incoherent crossover due to inelastic scat-
tering in an anisotropic metal [29].
The situation changes qualitatively if resonant impu-
rities are present in between the layers. Electrons that
tunnel through such impurities are moving with the speed
controlled by the broadening of a resonant level, i.e.,
much slower than speed of sound. For that reason they
can not be treated within the formalism outlined above
and require a separate study.
To evaluate the resonant-impurity contribution to the
conductivity, we assume that the impurities are randomly
distributed in space with density nimp whereas their en-
ergy levels uniformly distributed over an interval Eb. The
tunneling conductance of a bilayer junction is
G = −e2
dǫdǫ′Wǫ,ǫ′
(1 − n′ǫ) +
, (9)
where Wǫ,ǫ′ is a transition probability per unit time and
nǫ is the Fermi function. To calculateWǫ,ǫ′ , we use the re-
sults of Ref.[30, 31] for the probability of phonon-assisted
tunneling through a single impurity
Wǫ,ǫ′ = ΓLΓR
it1(ǫ
dt2dt3e
i(t2−t3)(ǫ−ǭ0)−Γ(t2+t3) (10)
× exp
|αq|2
|1− e−it3 + eit1
e−it2 − 1
|2 coth
e−it3 + eit2 + eit1(e−it2 − 1)(1− eit3)− c.c.
where αq = −iΛq/
ρωq, Λ is the deformation-potential
constant, ΓL and ΓR are tunneling widths of the resonant
level, Γ = ΓL + ΓR, and ǭ0 is the energy of a resonant
level renormalized by the electron-phonon interaction. In
the limit of no electron-phonon interaction, Eq.(10) re-
produces the well-known Breit-Wigner formula. From
now on, we consider a wide band of resonant levels:
Eb ≫ T ≫ Γ. Averaging Eq.(10) over spatial and en-
ergy positions of resonant levels, one obtains
σres=σel
1−coth
sinh2
dteitǫ−λf(t)
f(t)=
(1−cos(ωt)) coth
+i sin(ωt)
.(11)
Here σel is the conductivity due to elastic resonant tun-
neling and λ ≡ Λ2ω2D/ρs5π2 is the dimensionless cou-
pling constant for localized electrons. In the absence
of electron-phonon interaction, σres is temperature inde-
pendent and given by σel ≃ πe2Γ1nimpa0d/Eb[32], where
a0 is the localization radius of a resonant state and
Γ1 ≃ ǫ0e−d/a0 is its typical width. We note that the
electron-phonon interaction is much stronger for localized
electrons than for band ones: λ/ζ ∼ (kFd) (vF /s) ≫ 1.
Since typically ζ ∼ 1, one needs to consider a non-
perturbative regime of phonon-assisted tunneling. In
that case, resonant tunneling is exponentially suppressed
at T = 0: σres(T = 0) = σele
−λ/2. At finite T , we find
σres = σel
e−λ/2
1 + π
, T ≪ ωD√
, T ≫ λωD.
As T increases, σres growth, resembling the zero-bias
anomaly in disordered metals and Mössbauer effect. At
high temperatures (T ≫ λωD) σres approaches the non-
interacting value (σel). The asymptotic regimes in the
interval ωD/
λ ≪ T ≪ λωD can also be studied but we
will not pause for this here. Notice that, in contrast to
the phenomenological model of Ref.[8], there is no simple
relation between the T -dependences of σBc and σres.
To compare our model with the experiment, we extract
σBc from the low-temperature (between 10 and 50 K) c-
axis resistivity of Sr2RuO4 and extrapolate it to higher
temperatures [5]. The resonant part of the conductivity
is calculated numerically using Eq.(11). The fit to the
data for σel = 43 · 103Ω−1 cm−1, ωD = 41 K and λ = 16
is shown in Fig. 2. The agreement between the theory
and experiment is quite good and the values of the fitting
parameters are reasonable. An immediate consequence of
our model is the sample-to-sample variation of the c-axis
conductivity. Among the layered materials, the largest
amount of data is collected for graphite [3]. Even within
the group of samples with comparable in-plane mobili-
ties, the temperature of the maximum in ρc varies from
40K to 300 K [3, 33].
To conclude, we have shown that the Boltzmann
equation and its consequences are no less robust for
anisotropic metals than they are for isotropic ones. The
only condition controlling the validity of the Boltzmann
equation is the large value of EF τ, regardless of whether
τ comes from elastic or inelastic scattering. Out-of-
plane localized states change the c-axis transport rad-
ically while playing only minor role for the in-plane
one. While ρab remains metallic, an interplay between
100 200 300 400
T,     Kelvin
FIG. 2: ρc vs temperature. Solid: experimental data on
Sr2RuO4; dashed: fit into the phonon-assisted tunneling
model in the non-perturbative regime, Eq.(11)
phonon-assisted tunneling and conventional momentum
relaxation causes insulating or non-monotonic depen-
dence of ρc on temperature. This model is in a good
agreement with the experimental data on Sr2RuO4.
This research was supported by NSF-DMR-0308377.
We acknowledge stimulating discussions with B. Alt-
shuler, A. Chubukov, A. Hebard, S. Hill, P. Hirschfeld,
P. Littlewood, D. Khmelnistkii, N. Kumar, Yu. Makhlin,
A. Mirlin, M. Reizer, A. Schofield, S. Tongay, A.A. Var-
lamov, and P. Wölfle. We are indebted to A. Hebard, A.
Mackenzie, and S. Tongay for making their data available
to us.
[1] S. L. Cooper and K. E. Gray, in Physical Properties
of High Temperature Superconductors, edited by D. M.
Ginsberg, (World Scientific, Singapore, 1994), p. 61.
[2] P. W. Anderson, Science 256, 1526 (1990); P. W. An-
derson and Z. Zou, Phys. Rev. Lett. 42, 2642 (1992); D.
G. Clarke, S. P. Strong, and P. W. Anderson, Phys. Rev.
Lett. 72, 3218-3221 (1994).
[3] see N. B. Brandt, S. M. Chudinov, and Ya. G.
Ponomarev, Semimetals: I. Graphite and its com-
pounds, (North-Holland, Amsterdam, 1988) and refer-
ences therein.
[4] W. J. Wattamaniuk, J. P. Tidman, and R. F. Frindt,
Phys. Rev. Lett. 35 62 (1975).
[5] A. W. Tyler, A. P. Mackenzie, S. NishiZaki, and Y.
Maeno, Phys. Rev. B 58, 10107 (R) (1998).
[6] J. Singleton and C. Mielke, Contemp. Phys. 43, 63
(2002).
[7] J. Singleton et al. cond-mat/0610318.
[8] A. Rojo and K. Levin, Phys. Rev. B 48, 16861 (1993).
[9] V. Fleurov, M. Karpovski, M. Molotskii, A. Palevski, A.
Gladkikh, R. Kris Solid State Comm. 97, 543, (1996).
[10] M. J. Graf, M. Palumbo, D. Rainer, and J. A. Sauls,
Phys. Rev. B 52, 10588 (1995).
http://arxiv.org/abs/cond-mat/0610318
[11] P. J. Hirschfeld, S. M. Quinlan, and D. J. Scalapino,
Phys. Rev. B 55, 12742 (1997).
[12] A. A. Abrikosov, Physica C 317-318, 154 (1999).
[13] M. Turlakov and A. J. Leggett, Phys. Rev. B 63, 064518
(2001).
[14] U. Lundin and R. H. McKenzie,Phys. Rev. B 68,
081101(R) (2003).
[15] A. F. Ho and A. J. Schofield, Phys. Rev. B 71, 045101
(2005)
[16] P. Wölfle and R. N. Bhatt, Phys. Rev. B 30, 3542 (1984).
[17] N. Kumar, P. A. Lee, and B. Shapiro, Physica A 168,
447 (1990).
[18] N. Dupuis, Phys. Rev B 56, 9377 (1997).
[19] N. Kumar and A. M. Jayannavar, Phys. Rev. B 45, 5001
(1992).
[20] P. Moses and R. H. McKenzie, Phys. Rev. B 60, 7998
(1999).
[21] L. Ioffe, A. Larkin, A. Varlamov, and L. Yu, Phys. Rev.
B 47, 8936 (1993).
[22] D. G. Clarke, S. P. Strong, P. M. Chaikin, and E. I.
Chashechkina, Science 279, 2071 (1998).
[23] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323
(1986).
[24] J. Paglione, C. Lupien, W.A. MacFarlane, J.M. Perz, L.
Taillefer, Z.Q. Mao, and Y. Maeno, Phys. Rev. B 65,
220506(R).
[25] The self-energy in Eqs.(8a,8b) diverges at ζ = 1. This di-
vergence –also present for the isotropic case – results from
the renormalization of the sound velocity and is an arte-
fact of the Frölich Hamiltonian. A divergence-free theory
is obtained by applying the adiabatic approximation to
the coupled system of electrons and ions [26, 27, 28].
[26] J. R. Schrieffer, Theory of Superconductivity, (Addison-
Wesley, Redwood City, 1988).
[27] E.G. Brovman and Yu. Kagan, Sov. Phys. JETP 25, 365
(1967).
[28] B. T. Geilikman, Sov. Phys.-Usp. 18, 190 (1975).
[29] Migdal’s theory also rules out models based entirely on
polaronic effects since polarons are stable only if a typical
electron velocity does not exceed the sound one.
[30] L.I. Glazman and R.I. Shekhter, Sov. Phys. JETP 61
163, (1988).
[31] N. S. Wingreen, K. W. Jacobsen, and J. W. Wilkins,
Phys. Rev. Lett. 59, 376 (1987); Phys. Rev. B 40, 11834
(1989).
[32] A.I. Larkin and K. Matveeev, Sov. Phys. JETP 66, 580
(1987).
[33] S. Tongay and A. F. Hebard, private communication.
