Table Of ContentSliding conduction by the quasi one-dimensional charge-ordered state in
Sr Ca Cu O
14−x x 24 41
∗
A. Maeda, R. Inoue, and H. Kitano
Department of Basic Science, University of Tokyo
3-8-1, Komaba, Meguro-ku, Tokyo, 153-8902, JAPAN
N. Motoyama, H. Eisaki, and S. Uchida
3
Department of Advanced Materials Science, University of Tokyo
0
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8586, JAPAN
0
(Dated: Received June 5, 2002)
2
n Nonlinear conduction (NLC) of thetwo-leg spin ladder, Sr14−xCaxCu24O41,was investigated for
a thex=0,1and12materials. Althoughinsulatingmaterials (x=0and1)exibitedtheNLCbothin
J theladder-andrungdirections,theNLCintheladderdirectionofthex=0-materialwasfoundtobe
8 veryspecial. Weconsidered this to bedueto thesliding motion of thecharge ordered state, which
was responsible for the resonance at microwave frequencies. We discussed possible candidates for
] thecharge ordered state responsible for the NLC, including Wigner crystal in quasi one dimension
el (4kF-CDW).
-
r
t
.s Sr14−xCaxCu24O41 (SCCO) has two kinds of quasi dimensionalmaterials[9]. It alsoreporteda definite non-
t one-dimensional structures constructed from Cu and linear electrical conduction as a function of electric field
a
m O[1]. One is the CuO2 chain andthe other is the Cu2O3 in a rather weak field region. Quite recently, a similar
ladder, which can be regarded as the one-dimensional NLCwasalsoobservedbyGorshunovetal[10]. ThisNLC
-
d offshootofthe two-dimensionalCuO planes. Thus, this also suggested the collective nature of the phenomena.
2
n material has been thought to be a good reference for These phenomena are quite reminiscent of those estab-
o
high-T cupratesuperconductors,wherethesquareCuO lishedin the CDW systems. However,a verysmalloscil-
c c 2
plane is essentialfor high-T superconductivity. Dagotto lator strengthof the collective resonancein SCO showed
[ c
and Rice[2] predicted that carrier doping into such a aseriousdiscrepancyinquantitativeaspectsoftheinter-
1
ladder structure of a strongly correlated system leads pretationasbeingaresonanceofthepinnedCDW.Thus,
v
to superconductivity, through a mechanism which they it has been suggested that the collective mode found in
2
9 thoughtwassimilarto thatofhigh-Tc superconductivity SCO might be a mode characteristic of a strongly corre-
0 inquasitwo-dimensionalcuprates. Infact,superconduc- lated system.
1 tivitywasrealizedunderhighpressureinamaterialheav- To investigate the details of this collective mode, we
0
ily dopedby Ca[3]. Accordingto anopticalstudy[4],the have studied the nonlinear conduction as a function of
3
0 role of Ca substitution, which does not change the for- dc electric field atvarioustemperatures,in samples with
/ malvalenceofCu,wasthoughttotransferholesfromthe differentcarrierconcentrations,andalongdifferentdirec-
t
a chain to the ladder. Thus, the superconductivity in this tions of the crystal. We found the nonlinear conduction
m system is considered to be realized when the quasi one- to be present in almost all experiments (samples, tem-
- dimensionalityofholesontheladderbecomessufficiently peratures, directions). However, the nonlinear conduc-
d weakened[5]. tion found in the c-direction of SCO was very different
n
o The opposite limiting caseoflowhole concentrationis from all other results. Thus, we conclude that the col-
c also interesting. The c-axis (ladder-direction) dc con- lective mode has a quasi one-dimensional nature, and
v: ductivity of the parent material, Sr14Cu24O41 (SCO), is characteristic of samples with low hole concentration.
i is insulating with an activation gap of ∼ 2000 K[6] at We discussed possible candidates for the charge ordered
X
least below 300 K. An NMR/NQR measurement sug- state responsible for the NLC, including Wigner crystal
ar gested the occurrence of some charge order below 150 (WC)[11] in quasi one dimension (4kF-CDW).
K[7]. A recent study of the microwaveelectrical conduc- Crystals were prepared by the traveling-solvent-
tivity found adramaticincreaseofconductivity below∼ floating-zonemethod[6]. Forelectricalconductivitymea-
170 K[8]. Detailed study of the frequency dependence surement,small pieces were cut fromthe boul, with typ-
of the conductivity showed a small but sharp resonance ical dimensions of 3 × 1 × 0.5 mm3. The longest di-
around 50 GHz (∼ 2.4 K) in this temperature region. mensionwasinthe c-axis. Electricalcontactsweremade
Since the peak frequency is much smaller than the ther- byevaporatingsilveronthe sample,abovewhichelectri-
mal energy per carrier, this strongly suggested the reso- cal leads were attached by EPOTEK-H20E silver paste.
nance of some collective charge excitation,similar to the Post annealing under an oxygen flow atmosphere at 200
pinned phase mode of charge-density wave (CDW) sys- degrees centigrade decreased the contact resistance re-
tems and spin-density wave (SDW) system in quasi-one markably. Typicalcontactresistancewas104 Ωat100K
2
(x=0), which was about 20 % of the intrinsic resistance on temperature below 150 K, either. This is remarkable
of the sample. So, it is important to check the ohmic because the ohmic conductivity, σ , changed about two
0
nature of the electrical contact, which was confirmed in ordersofmagnitudefrom120Kto160K,aswasalready
eachmeasurement. Theelectricalconductivity wasmea- showninFig.1. Thismeansthattheconductivityinthis
sured by four-probe methods. To avoid the temperature temperature region was expressed by a formula,
rise due to Joule heating, short rectangular pulses gen-
σ(E,T) E
erated by an HP8116A pulse function generator, ampli- =f( ), (1)
fied by an NF4105 amplifier, were applied to the sam- σ0(T) E0
ple, and the response was measured by an SRS-SR250
where f(z) is a universal function of z. That is, a scal-
boxcar averager or a TDS420A digital oscilloscope. For
ing relation holds between the nonlinear conductivity,
both cases,an NF5305 differential preamplifier was used
σ(E,T), and the ohmic conductivity, σ (T). At higher
if necessary. By repeating the measurements for pulses 0
with different repetition rates (10∼1 kHz), duty cycles temperatures of200K,the Joule-heatingeffect hindered
(1/10 ∼ 1/1000), and widths (5∼ 500 µsec), we checked detailed studies of the nonlinearity. However, it seems
thattheincreaseofthenormalizedconductivitybecomes
whether or not any changes in conductivity were due to
lessened.
thesimpleJouleheatingeffects. Atweakelectricfields,a
Onthe otherhand, forthe a-axisnonlinearconductiv-
conventional lock-in technique was also used for the dif-
ity, the quantitative behavior is rather different. First,
ferential conductivity measurement. For measurements
in the a direction, as shown in the inset of Fig. 3, the
at various temperatures, an Oxford He-flow type cryo-
existence of the characteristic field, E , for the onset of
stat was used. To avoid the direct contact of the sample 0
nonlinearconductionislessclear. Withinourresolution,
with flowing gas, we fabricated the modified insert. By
the nonlinearity exists even at the lowest field. Second,
using this insert, temperature stability of about 50 mK
thereis notemperatureregionwherethe scalingrelation
was achieved throughout the measurements.
holds. Such quantitative differences between the c-axis
Figure1showsthelow-fieldconductivity,σ ,asafunc-
0 and a-axis suggest that the mechanism of the NLC is
tion of inverse temperature for several different carrier
different between these two directions.
concentrations. Except for the x=12 sample, the mate-
For further comparison, we empirically express the
rials are insulating with similar activation energies of ∼
differential-conductivitydatausingpowerlawsasfollows,
2500Kand1400K aboveandbelow160K,respectively.
since the nonlinearity is not so strong.
Figure2showsthechordalconductivity,σ,ofSCOinthe
twodifferent directions(ladder(c-axis)andrung(a-axis)) σ′(E) E
−1=A(T)( −1)γc (c axis), (2)
aIns athfeunincsteiot,ntohfeedleicffterriecnfitiealldcfoonrdvuacrtiiovuitsy,teσm′p≡eradtjur,eiss. σ0 E0(c)
dE
also shown for comparison. All the data in this figure and
were normalized by the ohmic conductivity, σ , at each
0 σ′(E) E
temperature. In both directions, we observed nonlinear- −1=B(T)( )γa (a axis), (3)
ity in the conductivity as a function of electric field. As σ0 E(c)
0
we already reported previously, the c-axis conductivity
increases with increasing field. However, the amount of where E(c) is the characteristic field for the onset of
0
the conductivity increase is very small (at most 3 % of nonlinearity along the c-axis, σ is the conductivity at
0
the ohmic value at 10 V/cm). On the other hand, in the the lowest field in each direction, γ , γ are exponents,
c a
a-direction (rung direction), the conductivity increase is A(T) is an almost temperature-independent constant,
rather distinct. At the lowest temperature measured (∼ and B(T) is a temperature-dependent constant. If we
100 K), the conductivity increase is more than 17 % of fit the data to these formulae, as is shown in the in-
the low field value. set of Fig. 4, we obtain the temperature dependence of
First, we concentrate on the c-axis conductivity. Al- the exponents, γ and γ , as shown in Fig. 4 (a), and
c a
though it is hard to define a definite threshold field for those of A(T) and B(T), as shown in Fig. 4 (b). As was
the onset of nonlinearity, the nonlinearity seems to start anticipated, A(T) and γ , and also γ , did not depend
a a
between 0.1 V/cm and 2 V/cm, as shown in the inset ontemperature,whereasB(T)dependedontemperature
of Fig. 3. To be quantitative, we define a characteristic strongly.
field, E , for the onset of NLC as the intersection of two Figure5showstheconductivityinthec-direction(lad-
0
extrapolatedstraightlinesinthedifferentialconductivity der direction) as a function of electric field for different
data, as shown in the inset of Fig. 3. In the main panel Ca concentrations, x=1 and x=12. For the x=1 sample,
ofFig. 3, E wasdisplayedasa function of temperature. the basic features of the NLC are very similar to those
0
E was found not to depend very greatly on tempera- in the a-direction of the x=0 sample. That is, the con-
0
ture. The conductivity normalized by the ohmic value ductivityincreaseislarge(morethan15∼20%at100K)
at each temperature, σ (T), was found not to depend withoutanydefinitethresholdnorcharacteristicfieldsfor
0
3
theNLConset,andtherewasnosignofascalingrelation the CDW,
such as eqs. 1 and 2. On the other hand, for the x=12
sample, the conductivity remained field-independent up e∗E ≃m∗ω2λ, (5)
T 0
to 30 V/cm within a resolution of 0.3 %.
The above results showed that the NLC was observed
wheree∗ andm∗ aretheeffectivechargeandthemassof
only in materials with a semiconducting (or insulating)
thecollectivemode,E isthethresholdfieldoftheNLC,
groundstate. Furthermore,the NLCinthe ladderdirec- T
and ω is the pinning frequency of the collective mode,
tion of the x =0 material showed special behavior when 0
providede∗=e=1.6×10−19Candm∗=m=9.1×10−31kg.
compared with the data in the rung direction and with
This suggests that the collective mode does not contain
those from samples with different Ca contents. In the
any lattice deformations as far as the charge dynamics
previous paper[8], we suggested that the NLC is due to
are concerned. Therefore, we suggest that the collective
the collective motion of some charge ordered state that
mode responsible for the NLC in the ladder direction of
alsoexhibitedasharpresonanceinthemicrowaveregion.
the x=0 material is the charge-orderd state of the con-
For the NLC due to the collective motion of the charge
duction electrons, such as the WC[11] in one dimension.
ordered state such as CDW, if the weak field conductiv-
ityshowssemiconductingtemperaturedependence,ithas Formationofthe WC hasbeenconsideredinquantum
beenknown[9]thattheextraconductivityduetothecol- Hall systems and He systems at an interface[14, 15, 16].
lective motion of the condensate, ∆σ , is proportional Bothofthemaretwodimensionalsystems,wherethelow
col
tothequasiparticleconductivity(conductivityduetothe density is essential to from the WC[11, 17].
uncondensed carrier), σqp, However,inquasione-dimensionalsystems,ithasbeen
∆σ ∝σ . (4) suggested theoretically[18] that the long-range nature of
col qp
the Coulomb interaction always leads to a very slow de-
Thisisbecausethedeformationofthedensitywave(DW) cay of the 4k (k is Fermi wave number) components
F F
isscreenedbytheuncondensedquasiparticles. Thec-axis of the density-density correlation function of the elec-
conductivity of the x=0 sample was found to obey the trons,suggestingthestabilizationoftheWC(4k -CDW)
F
scalingrelation(eq.1). This is exactlywhatis described across a wide range of material parameters.
in eq. 4. Thus, as far as the c-axis NLC of the x=0
As for the dynamics in the ordered state, the sliding
materialisconcerned,thisNLCisconsistentwithsliding
motion of the WC was considered theoretically in one
conduction due to the charge ordered condensate.
dimension[18] and in two dimensions[14, 19], and exper-
On the other hand, in the a-direction, the qualita-
imentally only in two dimensions[16, 20, 21, 22, 23, 24,
tively different behavior of the NLC suggest that the
25]. If the NLC reported here is due to the WC, this is
NLC in this direction is not due to the collective mo-
the first example of the sliding conduction of the WC in
tionofthechargeorderedstate. Thissuggestsastrongly
one-dimensional systems.
one-dimensionalnature ofthe conductiondue to the col-
Anotherpossiblecandidateforthechargeorderedstate
lective condensate. This is, again, consistent with the
is the ordered state on the CuO chain. Indeed, the ex-
sliding motion of the charge ordered state such as the 2
istence of the superstructure at low temperatures on the
CDW in the x=0 material.
chains was reported in neutron[27] and X-ray[28] exper-
Theabsenceofthecharacteristicfield,strongtempera-
iments.
ture dependence of the prefactor,B(T), andthe absence
oftheresonanceinthemicrowaveconductivity[8]suggest In conclusion, the NLC of the two-leg spin ladder,
thattheNLCintheadirectionisduetothehoppingcon- Sr14−xCaxCu24O41, was investigated for the x=0, 1 and
ductionofuncondensedquasiparticlesbetweentheneigh- 12materials. Althoughinsulating materials(x=0and1)
boringladders. Thepower-lawfielddependencesuggests exibitedtheNLCbothintheladder-andrungdirections,
delocalizationby the electric field from the weaklylocal- the NLC in the ladder directionofthe x=0-materialwas
ized state of the quasiparticles[12, 13]. foundtobeveryspecial. Weconsideredthistobedueto
Now,letusmoveontothe x=1material. TheNLCof theslidingmotionofthechargeorderedstate,whichwas
the x=1 material in the c-direction is quite reminiscent responsible for the resonance at microwave frequencies.
of the a-axis NLC of the x=0 material, and is probably We discussed possible candidates for the charge ordered
caused by the delocalization of the quasiparticles by the state responsible for the NLC, including Wigner crystal
electricfield. Thissuggeststhatthechargeorderedstate in quasi one dimension (4kF-CDW).
showing the sliding conduction is characteristic of mate- We thank E. Kim, B. Gorshunov, and M. Dressel for
rial with small carrier concentration, and is very fragile helpful discussions. This research was partially sup-
forcarrierdoping. Thus,itcanbesaidthatthecollective ported by Grant-in-Aid for Scientific Research from the
mode is closely related to electron-electron correlation. Ministry of the Science, Sports and Culture of Japan.
Previously[8],wediscussedthatthecharacteristicfield, Oneoftheauthors(RI)thankstheJapanSocietyforthe
E satisfied a relationship expected for the depinning of Promotion of Science for financial support.
0
4
[28] D. E. Cox et al., Phys.Rev.B57, 10750 (1998).
∗
E-mail [email protected]; Web 103
http://maeda3.c.u-tokyo.ac.jp
[1] E.M.McCarronetal.,Mater.Res.Bull.23,1355(1988), 102 x(Ca) = 12, c-axis
x(Ca) = 1, c-axis
[[23]] EMT...DSUiaeegghoraitsrttaoeeetttaaall.l..,,,iJPb.ihdPyh2s.y3sR,.1eS4vo9.c9B.(J41p95n8,.856)7.54,42(7169492(1).996). -1-1Ωcm) 110001 xx((CCaa)) == 00,, ca--aaxxiiss
[4] T.Osafune et al.,Phys.Rev. Lett.78, 1980 (1997). σ (0 10-1
[5] T. Nagata et al.,Phys. Rev.Lett., 81, 1090 (1998).
[6] N.Motoyama et al.,Phys. Rev.B 55, R3386 (1997). 10-2
[7] M. Takigawa, N. Motoyama, H. Eisaki, and S. Uchida, 10-3
Phys.Rev.B 57, 1124 (1998).
[8] H.Kitanoetal.,Europhys.Lett.56,434(2001),Physica 10-4
C 341-348, 463 (2000). 4.0 6.0 8.0 10.0
[9] G.Gru¨ner,Rev.Mod.Phys.60,1129(1988),G.Gru¨ner, 1000 / T (K-1)
Rev.Mod. Phys. 66, 1 (1994).
[10] B.Gorshunovetal.,Phys.Rev.B66,060508(R)(2002). FIG.1: Lowfieldohmicdcconductivity,σ0,asafunctionof
[11] E. Wigner, Phys. Rev. 46, 1002 (1934). temperature for x=0, 1, and 12 samples.
[12] H.FukuyamaandK.Yoshida,J.Phys.Soc.Jpn46,102
!
(1979).
[13] S. Uchida and S. Tanaka, Physica 117B & 118B, 673 (a) c-axis ! (b) a-axis
1.04 1.06 1.2 1.2
(1983).
[[1145]] HE..FYu.kAunydamreaieatnadl.P,.PAh.yLs.eeR,ePvh.yLse.tRt.ev6.0B,12876(519(7189)886)2.45. s ' / 011..0024 ! ss ' / 01.1
[[1176]] JK..MSh.irZaihmaamna, eTthaelo.,ryPhoyfs.SRoleidvs. L(e2tntd., e7d9.,),42(1C8am(1b9r9i9d)g.e ss / 01.02s 1.000.1 1 10 ss / 0!1.1 1.00.1 1 10
UniversityPress, Cambridge, 1972) p.p. 168-170. E (V/cm) E (V/cm)
[18] H.J. Schulz,Phys.Rev. Lett. 71 (1993) 1864.
[19] R. Chitra, T. Giamarchi, and P. Le Doussal, Phys. Rev. 1.00 1.0 100 K
120 K 120 K
Lett.80, 3827 (1998). 140 K Sr Cu O 140 K Sr Cu O
160 K 14 24 41 160 K 14 24 41
[20] V.J.Goldman,M.Santos,M.Shayegan,andJ.E.Cun-
0.1 1 10 0.1 1 10
ningham, Phys.Rev.Lett. 65, 2189 (1990). Electric Field (V/cm) Electric Field (V/cm)
[21] Y. P. Li, T. Sajoto, L. W. Engel, D. C. Tsui and M.
Shayegan,Phys. Rev.Lett. 67, 1630 (1991). FIG.2: Normalizedchordalconductivityofthex=0material
[22] Y.P.Lietal.,SolidStateCommun.96,379(1995),ibid, as a function of electric field at various temperatures. (a) c
99, 255 (1996). (ladder) direction (b) a (rung) direction. The inset shows
[23] C. C. Li et al.,Phys.Rev. Lett.79, 1353 (1997). the differential conductivity, σ′ ≡ dj at the corresponding
dE
[24] Y.P. Li et al.,Solid State Commun. 95, 619 (1995). temperatures. Horizontal dashed lines are guides for eye to
[25] L.W.Engeletal.,SolidStateCommun.104,167(1997). show theohms’s law.
[26] K.Shirahama et al.,Physica B284-288, 277 (2000).
[27] M. Matsuda et al., Phys. Rev. B 54, 12199 (1996), ibid,
56, 14499 (1997).
5
!
1.06 T = 140 K
c-axis
1.5
1.04 a-axis
0
m) ss ' / 11..0002
V/c 1.0 0.98 E (c) !
(c)E (0 0.1 1 0
E (V/cm)
0.5 Sr Cu O
14 24 41
100 110 120 130 140 150 160
Temperature (K)
FIG. 3: The characteristic field, E0, for the NLC onset as a
functionoftemperature. Dashedlinesareguidesforeye. The
inset shows how E0 was derived, where the a-axis data were
also shown.
2.0 ! 10 2.0 ! 10
(a) c-axis (b) a-axis
0.1 ! 8 8
1.5 -100.01 ! 1.5
gc1.0 ss' / 0.0001.1(E / E10(c) -1)10 46-2A(T) (10)ga1.0 46 -2B(T) (10)
1 !
0.5 2 0.5 ss' / -1000.0.011.1 1 10!2
E / E(c)
0.0 0 0.0 0 0
100 120 140 160 100 120 140 160
Temperature (K) Temperature (K)
FIG.4: Exponentsandtheproportionalconstantsappeared
in eqs. 2 and 3. The inset shows the raw condeuctivity data
fitted bythese equations.
1.6 0.23
(a) Sr CaCu O c-axis (b) SrCa Cu O c-axis
13 24 41 2 12 24 41
1.5
100 K
1.4 120 K
ss / 011..23 111246800000 KKKK -1-1Wcm) ( 00.22
s0.21 100 K
1.1 80 K
60 K
1.0 40 K
20 K
0.9 0.20
0.1 1 10 0.1 1 10 100
Electric Field (V/cm) Electric Field (V/cm)
FIG. 5: C-axis chordal conductivity as a function of electric
field in thex=1 and x=12 materials.
