c-axis transport in highly anisotropic metals: role of small polarons

0.19 MB·English

by  A. F. Ho

c-axis transport in highly anisotropic metals: role of small polarons
A. F. Ho1 and A. J. Schofield1
1School of Physics and Astronomy, Birmingham University, Edgbaston, Birmingham B15 2TT, U.K.
(Dated: February 2, 2008)
We show in a simple model of interlayer hopping of single electrons, that transport along the
weakly coupled c-axis of quasi-two-dimensional metals does not always probe only the in-plane
5
electron properties. Inour model where thereis astrong coupling between electrons and abosonic
0
mode that propagates in thec direction only, we find a broad maximum in the c-axis resistivity at
0
a temperature near the characteristic energy of the bosonic mode, while no corresponding feature
2
appears in the ab plane transport. At temperatures far from this bosonic energy scale, the c-axis

n resistivity does track the in-plane electron scattering rate. We demonstrate a reasonable fit of our
a theory to the apparent metallic to non-metallic crossover in the c-axis resistivity of the layered
J ruthenateSr2RuO4.
0
1 PACSnumbers: 71.27.+a,71.38.Ht,72.10.Di

]
l Many metallic systems of current interest have highly non-Fermiliquid7,8 orotherwisereflectsunusualin-plane
e
- anisotropic electronic properties with one or more direc- physics9 such as superconducting fluctuations10. These
r
t tionsmuchmoreweaklyconductingthantheothers. Ex- approaches have generally been argued to apply to the
s amples include the cuprate metals1, some ruthenates2, cuprate metals, but our motivation comes from the lay-
.
t and many organics3, where a measure of this anisotropy ered strontium ruthenate compounds where the appeal
a
m is the resistivity ratio, ρc/ρab, which can be of order 103 to unusual in-plane physics is less well grounded. Alter-
to105. (Henceforth, cwilldenotethe weaklycoupleddi- nativelythec-axiscouplingmayconsistofmorethansin-
-
d rectionwithaandbthemoreconducting.) Manyofthese gle electron tunneling: with disorder and boson-assisted
n materialshaveunusualmetallicorsuperconductingprop- hopping11 or perhaps inter-plane charge fluctuations12.
o erties which have been attributed to strong correlation
Inthispaperwealsogobeyondsingleparticlehopping
c
effects in the ab plane. Although the theoretical under-
[ astheonlycouplingbetweentheplanesandconsiderelec-
standing of strongly correlated electrons is incomplete,
trons coupled to a bosonic mode. However, in contrast
2 the electron anisotropy can be used advantageously to to the above work, we use a canonical transformation13
v giveinsightsinto thein-planephysics. Byassumingthat
to treat the electron-boson coupling, and thus can deal
9
the coupling up the c-axis occurs only by weak single-
5 with the strong coupling regime. Our assumptions are,
electron tunneling, many workers4,5,6 have shown that
0 first, that there is a well defined separation of energy
7 c-axis resistivity depends only on the in-plane electron scales: t ≫ ω ≫ t , where t is the in-plane ki-
ab 0 c ab
0 Green’s function. netic energy scale, ω is the characteristic energy of the
0
4 This insight makes all the more puzzling the obser- bosonicmode(e.g. theDebyetemperature)andt is the
0 c
vation that in many of these materials the resistivity- single particle tunneling between layers. This assump-
/
t anisotropy ratio is strongly temperature dependent with tionis certainlywellfounded for Sr RuO , as shown,for
a 2 4
m a cross-over from metallic behavior to insulating be- example, in Ref. [14]. With this assumption it is suf-
havior in the out-of-plane direction only. For example, ficient to compute conductivity to leading order in t .
d- Sr2RuO42showsT2resistivitybothintheplaneandout- Our second assumption is that electron-electron correlac-
n of-plane for T <20K but ρc shows a broad maximum at tions dominate overthe in-plane coupling to the bosonic
o around 130K and above this temperature the resistivity mode such that the formation of a small polaron in the
c startstodecreasewithincreasingtemperature(seeFig.2 planes is suppressed by the electronic interactions. We
:
v below). Inmarkedcontrast,thein-planeresistivityshows will therefore be considering a limit where the quasipar-
i ∂ρ/∂T >0 for all temperatures. Evidently the low tem- ticle excitations of the system are insensitive to bosonic
X
perature metal is a 3D Fermi liquid, so this begs the modes whose wavelength in the plane is short compared
ar questionhow the out-of-plane transportis being blocked tothesizeofaquasiparticle(assumedtobelarge). With
whilethein-planetransportremainsmetallic. Inthispa- theseassumptionsweshowthatanisotropiccouplingcan
per we consider how anisotropic coupling of the in-plane leadtodifferences betweenthe in-planeandout-of-plane
electrons to bosonic degrees of freedom can give rise to conductivities that are consistent with the experiments.
this effect. We show that if the electron-boson coupling Abroadmaximumcanappearinthec-axisresistivityρ
c
is stronger in the inter-plane direction then, at the char- asafunctionoftemperature,whilenosuchfeatureexists
acteristic energyscale of the bosonmode, there canbe a inthe in-planeresistivityρ . Onlyawayfromthebroad
ab
maximum in the c-axis resistivity with no corresponding maximum, does ρ trackthe intrinsic in-plane scattering
c
transport feature for currents in the plane. rate. We shallalsoshowthatour modelcalculationscan
Otherapproachestothis puzzlehavebeenofferedpre- fit reasonably the Sr RuO data of Tyler et al.2, using
2 4
viously. One possibility is that the in-plane metal is a physically plausible parameters.
2
Strictly our assumption that the electrons couple Even with this limiting electron-phonon coupling,
strongly only to out-of-plane bosonic modes should be interlayer hopping can affect in-plane correlations by
regardedasphenomenological,andinthispaperwecom- spreading the in-plane correlation from one plane to an-
pute the consequences of such an approachand compare other, especially since the in-plane correlations are sup-
with experiment. Nevertheless we can offer some justifi- posedto be strongfor a pure single layerin the cuprates
cation for it. In a quasi-2D metal, the anisotropy of the andalsotheruthenates. However,thisrequirestheinter-
crystalstructurehasimportantconsequencesforboththe layer hopping to be coherent, which means the temper-
electronic structure and the bosonic modes the electrons ature must be low compared to t : otherwise, the c-axis
c
can couple to. Usually the anisotropyin the characteris- transport is diffusive and cannot propagate correlations
ticenergiesofphononmodesaremuchsmallerthanthose between the layers. In this paper, we have assumed a
than the electronic structure: the ruthenate that we will phenomenological form for the in-plane spectral weight,
beconcernedwithisthoughttobeanionic(notcovalent- andso we cannotaddressthe issue ofthe stronglycorre-
bonded) crystal. Where significant anisotropy can arise lated three dimensional state at T ≪t .
c
is in the strength of the electron-phonon couplings for
The Hamiltonian for the c-axis bosons, H =
these different phonons. Krakaueret. al.15 have found b
ω a†a , defines a dispersion ω . In this paper we
in an LDA calculation that the electron phonon cou- q q q q q
shall look closely at the Einstein phonon ω = ω . This
pling is much stronger when atomic displacements are P q 0
is both because optical phonons tend to have little dis-
perpendicular to the Cu-O plane in LSCO, while Kim
persion relative to acoustic ones, and also we are able to
et. al.16 and Grilli and Castellani17 found that strong
obtain analytic results for this case. We shall also dis-
in-plane electronic correlation suppresses in-plane elec-
cuss, somewhat more qualitatively, the opposite case of
tron phonon coupling. Usually strong electron-phonon
a generic dispersion where the phonon density of states
coupling leads to the well-studied small polaron regime,
does not have any sharpfeatures. For the data fitting to
where electronic motion becomes affected by the accom-
Sr RuO ,wewilluseasimpleformthatallowsustotest
panying large cloud of phonons13,18. (For a review, see 2 4
if the putative mode does indeed disperse.
[19].) In this paper, our assumptions will lead to a small
polaron forming only between the layers. Of course electron-phonon systems have been well
Our model Hamiltonian is then: studied, but our findings will be shown to be clearly dif-
ferent to ones found in the small polaron models stud-
H = H(n)+Hc+He−b+Hb , ied in the classic papers13,18,19: small anisotropy in
Xn electron-bosoncouplingcannotleadto the propertieswe
H = d2x t c† (~x)c (~x)+H.c.. (1) shall demonstrate below. In particular, at temperatures
c c n+1 n
greater than the temperature of the maximum in ρ (T),
Z n c
X our model has ρ (T) tracking the intrinsic in-plane scat-
c
The model system consists of a stack of 2D planes de- tering rate while in standard small polaron theory (even
scribed by H(n) containing arbitrary intra-planar inter- if generalized to some degree of anisotropic coupling),
actions. In this work it will be sufficient to characterize ρ (T)keepsondecreasing(exponentially)withincreasing
c
these planes by a phenomenological 2D Green’s func- T. In short, in our model, the small polaronbroadmax-
tion. In Hc, electrons are assumed to hop from one imum in ρc(T) is “grafted” ontop of the backgroundin-
plane directly to a neighboring plane only, at the same plane scattering. We note that Kornilovitch20 has stud-
in-plane coordinate ~x. For simplicity, we have omitted ied a similar anisotropic model: he did not calculate the
the spin index on the electron operators cn(~x). Many d.c. conductivity, and, with the effective mass approach
approaches4,5,6 assume only this coupling between the employed, he cannot access the non-metallic regime [be-
planes in a quasi-2D system. We go beyond that by in- yond the broad maximum of ρ (T)]. After completion
c
cluding a bosonic mode that also couples the layers of our theory, we became aware of the related work of
LundinandMcKenzie21whoalsostudiedasmallpolaron
He−b = Mqexp(iqnc)ρn aq+a†−q . (2) model for anisotropicmetals. Their model is different to
Xn Xq (cid:16) (cid:17) oursinanessentialway: theytakethebosonmodetobe
uncorrelated from one layer to another, while we have a
Our analysis can be applied to any gapped neutral
boson mode that propagate coherently along the c-axis.
bosonic mode with long wavelength in the plane, such
as a magnetic collective mode, but we have in mind The key physics we are considering is the effect of
the (strong) coupling to a phonon mode with displace- strongelectron-phononcoupling onthe chargetransport
ment in the c direction and long (infinite) wavelength in the weakest direction, the c-axis. Physically, the mo-
in the plane. Here q is along the c-axis, with c the tion of the electron is accompanied by the emission and
inter-layer distance. In this limit, only the total charge absorption of a large number of phonons due to the
ρn = d2xc†n(~x)cn(~x)ofthen’thplanecouplestothebo- strong coupling, forming the so-called small polaron18.
soncreationandannihilationoperatorsa†,a so,by con- We treat H exactly by the canonical transformation,
structRion,thesebosonsdonotdominatechqarqgetransport H¯ =exp(−Se−)Hb exp(S), a straightforwardgeneralization
within the plane. of the transformation well known in the small polaron
3
problem13,19, conductivity being no longer simply a convolutionof the
in-plane electron Green’s functions and we explore the
M
S =− d2x qeiqncc†(~x)c (~x) a −a† . (3) consequences in the following.
Z Xn,q ωq n n (cid:16) q −q(cid:17) calToin-mpalakneefuelretchterronprGogrreeesns’swfeunucsteiona pinhetnhoemsepneoclotrgai-l
Then, H¯ = nH(n)+H¯c+Hb, and representation: G(n2D)(~kk,ωm) = dzA(i2ωDm)(−~kkz,z), and
H¯ = −P |Mq|2ρ dUo(nνen.)W=e shdazlliνbtn(az−)kze,tthheenin-tphleanMeaetlesRcutbraornasspuemctsraclafnunbce-
c n
n,q ωq tion to haRve the form: A(2D)(~k,ω) = 1 Γ(ω,T) ,
X π(ω−ǫ~k)2+Γ(T,ω)2
+ d2xt c† (~x)c (~x)X† X +H.c.,(4) where we assume that the scattering rate Γ(T,ω) has no
c n+1 n n+1 n
in-plane momentum dependence. Then, integrating over
n Z
X ~k ,thezero-frequency,zero-momentumc-axisconductiv-
k
ity becomes:
M
where Xn =exp(Xq ωqqeiqnc(cid:16)aq−a†−q(cid:17)) . (5) σc(T,B)= eπ2NcTt2cRe dωdνnF(ω)−nνF(ω+ν)
Z
D(ν,T)
Thus strong electron-phonon coupling leads to (1) a
× ,(9)
renormalization of the in-plane chemical potential (first [(v q )2−(ν+iΓ(T,ω)+iΓ(T,ω+ν))2]1/2
F B
termofH¯ ),whichweshallhenceforthignore,and(2)an
c
effective vertex correction for the c-axis hopping tc (sec- where D(ω,T) = ωb(ω)nB(ω)[1+nB(ω)], nF,B are the
ondtermofH¯ ). Attemperatureslowerthanthecharac- Fermi and Bose distributions and N is the electron den-
c
teristicenergyoftheboson,electronscanhopcoherently sity. D(ω,T) can be calculated23 following Ref.19. We
fromoneplanetoanother,withsomesuppressiondue to now analyze this equation in detail for Einstein modes
the “dragging” of the boson cloud. With increasing T, and, more qualitatively, for a general dispersing mode.
the hopping electron gets inelastically scatteredby more For Einstein modes, ωq = ω0, the function D(ω,T)
and more bosons: inter-layer hopping is now more diffu- is made up of delta functions at the harmonics ω =
sive. Hence a crossoverto non-metallic behavior occurs. nω0, n=0,±1,±2,...19. Itcanbeshownthatforexper-
We consider the d.c. conductivity and the orbital imental temperatures, one needs only the first few har-
magnetoconductivity for fields in the ab plane. For monics. We now discuss the three temperature regimes
the conductivity, since the charge in the n’th plane is (T ≪ω0,T <∼ω0,T ≫ω0)separately,andshowthatthe
Qc =e d2xc†(~x)c (~x), the current in the c-direction is low and high temperature regimes basically tracks the
n n n
just jc = ∂ Qc = −i[Qc,H]. After the canonical trans- in-plane scattering rate, just like in previous studies of
formantioRn, jtc n=−i[Qc,nH¯ ], and so: interlayer transport4,5,6. Only in the regime where the
n n c
temperature is near the boson scale ω can the electron-
0
hjci=iet d2x c† (~x)c (~x)X† X −h.c. .(6) boson scattering dominate over the in-plane scattering
n c n+1 n n+1 n contribution, and instead, leads to a small polaron like
Z Dh iE broad maximum in resistivity.
The orbital effect of a magnetic field can be included At low temperatures (T ≪ ω ), the asymptotic form
0
using a Peierls substitution22. The mathematical details for the c-axis resistivity is the same as for band elec-
of our derivation follow our analysis reported in Ref. 23. trons, but with an effective hopping parameter teff =
tUhseincgontdhuecKtivuibtoy,fio.rem.,utlhaealnindeaerxpreasnpdoinngsettooOa(nt2ca)pgpilvieeds tce−∆2/2ω02, where ∆2 = q2|Mq|2(1−cosq) characcter-
izes the strength of the electron-phonon interaction:
electric field in the c-direction σc =jc/Ec: P
e2Nt2
σ (B) = lim e2t2c d2k dτ σc ≃ 2cπΓ(Tc,0)e−(∆/ω0)2 (T ≪ω0). (10)
c Ω→0 Ω k
n Z Z
×eiΩτU(τX)G(2D)(~k ,τ)G(2D)(~k +~q ,τ)(,7) For T ≫ ω0, then σc ∼ T−η−12, using the scattering
n k n+1 k B rate Γ(T)∝ Tη. (η =2 for a Fermi liquid, while for the
marginalFermiliquid,η =1.) Thusthelowtemperature
B = 0 conductivity directly probes the in-plane scatter-
whereU(τ)= X† (τ)X (τ)X†(0)X (0) , (8) ing,whilethehightemperatureconductivitystillreflects
n+1 n n n+1 Hb the in-plane scattering with an extra T−1/2 factor24.
D E
However,beyondacriticalphononcoupling(∆ )there
G(n2D)(~kk) is the in-plane (dressed) Green’s function for is an intermediate temperature region where ac broad
plane n at in-plane momentum ~k , and ~q = e~c × B~ maximum appears in the c-axis resistivity at tempera-
k B
with~c being the inter-plane lattice vector. The temper- ture T (Fig. 1). Beyond T , the resistivity dips to
max max
ature dependent factor U(τ) in Eq. 7 leads to the c-axis a (broad) minimum at T before finally joining onto
min
4
2
the asymptotic T ≫ ω0 regime mentioned above . We and F = Mq 2(1−cosq).
estimate,byusingonlythezerothharmonicsn=0,that q ωq
if the in-plane scattering rate has the form Γ(T)=αTη, For illus(cid:12)(cid:12)trat(cid:12)(cid:12)ion, we consider ωq = ω0[1−0.4cos(q)],
then the maximum appears when and set t(cid:12)he (cid:12)electron-phonon coupling Mq to be q-
independent. Eq. 13 is then evaluated numerically and
plotted in the inset to Fig. 1. ∆¯ = F ω /2 charac-
∆>∆ ≈ω η/0.4. (11) q q q
c 0
terizesthe strengthofthe electron-phononinteractionin
P
The position of the resistivityppeak for ∆>∼∆c is found this case. The result is qualitatively similar to that of a
non-dispersing mode.
tobeT ≈ω /3.0. For∆onlyslightlylargerthan∆ ,
crit 0 c
Now we consider the magnetic field dependence of the
T and T is given by:
max min
conductivity for the Einstein mode. At low tempera-
turesT ≪ω wefindthatthemagnetoresistancereflects
ω0 ω0 η ω0 2 ω0 2 the usual cr0oss-over from quadratic to linear field de-
≈ ± − . (12)
2Tmmainx 2Tcrit vuu0.2"(cid:18)∆c(cid:19) (cid:16)∆(cid:17) # peleencdtreonnces2c7atatteraingscraaletedetermined only by the in-plane
t
Note that both T and ∆ depend mainly on phonon
max c ∆ρ =[ρ (B)−ρ (0)]/ρ (0)= 1+[ω /2Γ(T,0)]2,
parameters; one can show that the in-plane scattering c c c c c
(14)
rateentersonlyintheformoftheexponentη. Hencethe p
where ω =v ecB is the cyclotron frequency. At higher
widthofthebroadmaximumisgovernedbythescaleω0. temperacturesF(T >∼ω0),theweak-fieldmagnetoresistance
isquadraticinfield∆ρ ∝(ω /Γ )2 ,anddefinesanew
c c eff
60 2.2 6 scattering rate, Γ , depending on both the electronic
2.0 eff
s) nits) 5 scattering and the phonon frequency. Taking into ac-
nit 50 b. u 4 count only up to n = 1 harmonics, with Γ0 = Γ(T,0),
. u 40 r(arc23 1.7 and Γ1 =Γ(T,ω0), we find
b 1 + 4Γ1
r(arc30 2.0 1 0.0 1.3 1.0 1.5 T/w02.0 Γ2eff ≈ 161Γ302Γ+0 2Γ(144(ΓΓ42121Γ++21ωω−00223)ω302) . (15)
20
Asforthezerofieldresistivity,themagnetoresistancefor
the dispersing mode case is qualitatively similar to that
10
1.0
fortheEinsteinmodeandispositiveforalltemperatures.
We now consider in more detail how this model might
0.5 1 1.5 2 apply to real materials and, in particular, the ruthen-
T/w ate systems. The 2Delectronic nature ofthese materials
0
reflects the crystal structure and so it is natural that
the electron-phonon interaction should be anisotropic.
FIG.1: Thezero-fieldc-axisresistivitywithin-planemarginal
While there are to date no calculation for the electron-
FermiliquidscatteringΓ=ωcoth(ω/T)forelectronscoupled
phononinteractionparametersfor the ruthenates,in the
to c-axis Einstein phonons for different values of electron-
phononcoupling(∆/ω0). Inset: asabovebutforadispersing iso-structural LSCO cuprate family, Krakauer et. al.15
mode ωq =ω0[1−0.4cos(q)]. have calculated that there is a strong electron-phonon
coupling only for modes corresponding to atomic dis-
placements perpendicular to the Cu-O plane, (partly)
For a more general dispersing bosonic mode, D(ω,T)
can be calculated approximately19 assuming strong because of weak screening of the resulting electric fields
inthisdirection. Moreoverithasbeenarguedthatinthe
electron-phonon interactions. Using the usual approxi-
perovskite structure the coupling to c-axis vibrations is
mation [n (ν)−n (ν+ω)]/ω ≃−∂n /∂ν ≃δ(ν) valid
F F F enhanced20. Thereexistopticalphononswiththeappro-
at low T when Γ(T,ω) varies slowly, we find
priate symmetry for c-axis transport28 in Sr RuO , and
2 4
e2Nt2 dν ν experimentallythebroadmaximuminthec-axisresistiv-
σc(T,B)= πcTce−C(T) πγ2(T)sinh ν ity has been linked to a structural phase transition29 in
Z 2T Ca Sr RuO ataroundthebroadmaximumtempera-
1.7 0.3 4
exp −pν2 ture. Thusoneshouldconsiderthepossibilityofelectron-
4γ2(T)
×Re , (13) phononinteractionaffectingthec-axistransport. Inpar-
[(vFqB)2−((cid:16)ν+2iΓ(T(cid:17),ν))2]1/2 ticular, both the Sr2RuO4 and Ca1.7Sr0.3RuO4 systems
exhibit2,29 qualitatively this broad maximum structure
where
found in our simple model, in the c-axis resistivity near
2sinh2 ωq F |ω |2 to their characteristic (c-axis) phonon energy.
C(T)= q Fq sinh(cid:0)2ωTq4T(cid:1), γ2(T)= q 2sinhq(ωqq/2T) , scoApefuollfqthueanstimitaptliivfieedmmodoedleinlgproefsSenr2tReduOhe4reis.bDeeysopnidtetahlel
X X
5
only lead to some quantitative changes in the fit param-
30 eters, because the calculation of ρ (Eq. 7) involves an
c
integral over in-plane momenta and thus averages out
)25 such dependences.
m
c20 We now show in Fig. 2 a fit to the Sr2RuO4 data of
Ω Tyler et al.2. The theoretical curve is the thin continu-

ousline,thedataarethepoints. Thetheoreticalcurveis
m15
generated as follows: the in-plane scattering rate is ap-
(
proximatedasbeingproportionaltothein-planeρ (also
c10 ab
ρ taken from the data of [ 2]): Γ(T,ω) ≈ A(ρ (T)+ρ ),
ab 0
5 where A and ρ0 are fitting parameters. We take a sim-
ple optical boson dispersion ω(q)= ω (1−λcosq) with
0
the boson bandwidth λ and the characteristic energy
200 400 600 800 1000 1200
ω as fitting parameters. Γ(T,ω) and ω(q) are then
0
T (K)
fed into Eq. 13, where the overall scale of ρ is found
c
by fitting ρmax ∼ 33mΩ cm to the peak of the theory
c
FIG. 2: Fitting the zero-field c-axis resistivity to the data of curve. Despite the simplicity of the model and the lack
Tyler et.al.2. See thetext for theexplanation of the theoret- of knowledge of the exact boson dispersion form, the
ical curve(thin line) and theparameters used for thefitting.
theory curve is almost indistinguishable from the data
The data are plotted as points: for clarity, we have plotted points,exceptathighT.32 Thefittingisrelativelyinsen-
only 3% of thetotal numberof available datapoints.
sitive to the precise values of the parameters: some fine
tuning (mainly, λ and ∆¯) is needed to get the positions
of the maximum, minimum and final upturn correctly.
these simplifying assumptions, our model does a reason- Furthermore, the parameters used are physically plausi-
able fit to the resistivity data of Sr2RuO4 as we shall ble: λ ≈ 0.67,ω0 ≈ 515K, ∆¯/ω0 ≈ 1.4,A/ω0 ≈ 0.1(mΩ
show, perhaps indicating some degree of universality of cm)−1,ρ ≈ 0.17 mΩ cm. The value of ω is consistent
0 0
themechanismforc-axistransportstudiedinthispaper. with the available phonon data of Braden et.al.28, while
Here for reference, we list the simplifications we haveas- thesizeofthebosonbandwidthsuggeststhatthebosons
sumed in our modeling: 1) the in-plane scattering rate do disperse in the interplane direction. Note that the
can be deduced directly from ρab, 2) the main interlayer dispersion ω(q) always enter the conductivity expression
coupling is single particle hopping from one layer to an- under a q-integral (see Eq.13), and is the reason why
otherand3)asimplephonondispersionisemployed(see oursimplified bosondispersioncanstill modelthe broad
later). maximum in the c-axis resistivity successfully. The size
For 1), this amounts to ignoring vertex corrections to oftheelectron-bosoncouplingparameter∆¯ indicatethat
in-plane transport. In Fermi liquids, the vertex cor- Sr2RuO4isjustslightlyabovethresholdforobtainingthe
rection leads to an extra cosine of the angle between maximum in ρc.
in- and out-going electrons thereby correctly penalizing The c-axis magnetoresistance seen in Sr RuO is un-
2 4
back-scattering in conductivity. However, the qualita- usual30, becoming negative for both transverse (B||ab)
tive trend is still correct. As mentioned already,we have and longitudinal (B||c) fields above 80K and maximally
taken an in-plane spectral function that has no in-plane so at around 120K—coinciding with the peak position.
momentum dependence. This ignores the complicated In this paper we have only considered orbital magne-
multiple Fermi surfaces in Sr RuO . For 2), this means toresistance and, as might be expected, found it to be
2 4
we ignore interlayer coulomb interaction. Also, we ig- positive31. This is not inconsistent with the data since
nored the multiband nature of Sr RuO and the depen- the transverse magnetoresistance is always less negative
2 4
dence of t on in-plane momenta. For 3), we have taken than the longitudinal one. If the change of sign is linked
⊥
the limiting case of an optical phonon mode where the totheoriginoftheresistivitymaximumthen, withinthe
atomicdisplacementsareperpendiculartotheplane,and frameworkpresentedhere,itindicatesthatthefrequency
all the atoms in the plane move together. We envis- of, or coupling to, the bosonic mode is field dependent.
age that just as in LSCO, there will be strong electron- At present, we have no microscopic picture of how this
phonon coupling only for phonons propagating mainly might occur.
in the c direction15. Now in reality, because of the non- Inconclusion,wehaveshownthatc-axistransportina
trivialperovskitestructureofSr2RuO4,thesemodesthat quasi-2D metal does not always probe only the in-plane
can affect c-axis transport will have relative atomic dis- electron properties. Strong coupling between electrons
placements both in-plane and out of plane. But this and a bosonic mode polarized in the c-direction in a
makes electron hopping from one to another plane even highlyanisotropicmetalcanleadtoabroadmaximum—
more difficult, as the polaron has to create disturbances anapparentmetallictonon-metalliccrossover—inthec-
both in-plane and out-of-plane. axis resistivity with no corresponding feature in the ab
To some extent, these simplifying assumptions may plane properties. The position in the temperature axis,
6
and the shape of this broad maximum are determined We are pleased to acknowledge useful and stimulat-
mainly by boson parameters and not on the magnitude ing discussionswithM. Braden,C.Hooley,V. Kratvsov,
of the in-plane scattering rate. We have discussed the P. Johnson, M. W. Long, Y. Maeno, A. P. Mackenzie,
potential application of the model to Sr RuO and its G. Santi, I. Terasaki and Yu Lu. A.F.H. was supported
2 4
relativeCa Sr RuO . Despite certainsimplifying fea- by ICTP, Trieste where this work was initiated, and by
1.7 0.3 4
tures of the model, the qualitative (and even quantita- EPSRC (UK). A.J.S. thanks the Royal Society and the
tive) properties of this crossoverin these layeredruthen- Leverhulme Trust for their support.
ates are captured succinctly.
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094510 (2001). Sr2RuO4, the mode that can couple to c-axis transport is
7 P. W. Anderson, The theory of superconductivity in the the axial Oz mode, at (0,0,2π/c)). It has the strongest
high-Tc cuprates (Princeton University Press, Princeton, electron-phonon coupling in the whole zone and behaves
New Jersey, USA,1997). like an Einstein phonon: C. Falter, M. Klenner, and W.
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15 H. Krakauer,W. E. Pickett,and R.E. Cohen, Phys. Rev. tance within this model. It requires either a phonon func-
B 47(2), 1002 (1993). tion D(ω) that peaks at a finite frequency, or an in-plane
16 J. H. Kim, K. Levin, and R. Wentzcovitch, Phys. Rev. B spectral weight where the scattering rate is smallest at a
40(16), 11378 (1989). finite frequency. Either of these possibilities suppress the
17 M. Grilli, and C. Castellani, Phys. Rev. B 50(23), 16880 ω = 0 contribution to σc (see Eq. 9.) Intriguingly, after
(1994). this work was completed, we learnt from P. Johnson that
18 T. Holstein, Ann.Physics 8, 343 (1959). the in-plane spectral weight near and above Tmax may be
19 G.D.Mahan,Many-Particle Physics (PlenumPress,New anomalous, in contrast to the spectral weight observed in
York,1990), 2nd ed. thelowtemperatureFermiliquidstate.Workisinprogress
20 P. E. Kornilovitch, Phys. Rev. 59(21), 13531 (1999). to takethis into account within the present framework.
21 U. Lundin and R. H. McKenzie, Phys. Rev. B 68, 081101 32 Other processes may intervene at high temperature, but
(2003), cond-mat/0211612. one source of discrepancy at high T is that the experi-
22 A. Altland, C. H. W. Barnes, F. W. J. Hekking, and mental data is the resistivity at constant pressure while
A.J.Schofield,Phys.Rev.Lett.83(6),1203(1999),cond- ourcalculation isforconstantvolume.Non-trivialthermal
mat/9907459. expansion at higher T will contaminate thedata.
23 A. F. Ho and A. J. Schofield, Small polarons and c-axis
transport in highly anisotropic metals, cond-mat/0211675.