1. Phase diagrams of (La,Y,Sr,Ca)14Cu24O41: switching between the ladders and chains
T.Vuletic, T.Ivek, B.Korin-Hamzic, S.Tomic
B.Gorshunov, M.Dressel
C.Hess, B.Büchner
J.Akimitsu
Institut za fiziku, Zagreb, Croatia
1.Physikalisches Institut, Universität Stuttgart, Germany
Leibniz-Institut für Festkörper- und Werkstoffforschung, Dresden, Germany
Dear colleagues, my intention is to provide you with an overview of the quasi-one-dimensional cuprates. This work was done in close cooperation of Zagreb dielectric spectroscopy group and Stuttgart group, where quasi-optical microwave and far infrared mesurements are performed. We also thank to colleagues which provided samples. Our detailed charge response study, allowed us to assemble phase diagrams for the materials. It seems that our contribution was necessary for completion, besides the plethora of data produced in the field in last 10 years.
Dept.of Physics, Aoyama-Gakuin University, Kanagawa, Japan
B. Gorshunov et al., Phys.Rev.B 66, (R) (2002)
T. Vuletic et al., Phys.Rev.B 67, (2003)
T. Vuletic et al., Phys.Rev.Lett. 90, (2003)
T. Vuletic et al., Phys.Rev.B 71, (2005)
T. Vuletic et al., submitted to Physics Reports (2005)

2. Motivation (La,Y,Sr,Ca)14Cu24O41 : Task:
q1D materials:
proximity (competition and/or coexistence)
of superconductivity & magnetic/charge ordered phases
(La,Y,Sr,Ca)14Cu24O41 :
spin chain/ladder composite q1D cuprates
strongly interacting q1D electron system
Task:
In general, in q1D materials sc competes with magnetic/charge orders which render materials insulating. This competition appears also in q1D cuprates. Another interesting issue here is that this is also a strongly interacting q1D electron system, and later it will show that some concepts we know from other q1D, mostly density wave systems do not apply here. So our task could have been recognized as to catalyze discussions on
the nature of superconductivity and charge-density wave and their relationship with the spin-gap, which is an important property of these materials.
assemble phase diagrams to catalyze discussions on
the nature of superconductivity and charge-density wave
and their relationship with the spin-gap

3. t-J(-t’-J’) model for ladders
E.Dagotto et al., PRB’92
ladders map onto 1D chain
with effective U<0 for hole pairing!
chain
layer
U<0, t≠0
V>0: 2kF CDW
V<0: singlet SC
V.J.Emery, PRB’76
In fact, the theory quite early recognized that the spin gap which occurs for spins arranged in ladders might lead to superconductivity (or CDW, that is) if the ladders are hole doped. Pure ladder materials were soon after sinthetized, however doping did not lead to SC.
pairing of the holes
 superconducting or CDW correlations
spin gap
doped holes enter O2p orbitals  form ZhangRice singlet with Cu spin

4. Crystallographic structure (La,Y,Sr,Ca)14Cu24O41
b=12.9 Å
a=11.4 Å
A14
Cu2O3 ladders
CuO2 chains cC
Chains: Ladders: cC=2.75 Å cL=3.9 Å
10·cC≈7·cL≈27.5 Å cL
Only when good crystals of composites of chains and ladders were synthetized SC was discovered below 10-12K, however, only when more than 3GPa were applied. The structure of the composite is arranged in a following manner. The chains and ladders separately form layers. Metal A atoms are coordinated to ladders, as they fit into the openings of ladders. Then the two systems are stacked in alternating manner.
cL/cC=√2

5. Bond configurations and dimensionality
Cu2O3 ladders
CuO2 chains
90o - FM, J<0
180o - AF, J>0
Cu-O-Cu bonds cC
Chains: Ladders: cC=2.75 Å cL=3.9 Å
10·cC≈7·cL≈27.5 Å cL
{{{{{It is important to note that the subsystems are incomensurate with 1-to-sqrt2 ratio. Due to this, the bonds formed are never 90 degree, and Copper and its four coordinated oxygens form irregular tetragons and even distorted tetrahedrons. Obviously the intrinsic disorder in the material is strong.}}}}}
Here we can recognize more easily why ladders are regarded as q1D system and not 2D. Ladders are decoupled by 90 degree bonds and also bytriangular, frustrated arrangement of Coppers between neighboring atoms.

6. 2D AF dimer / charge order
Magnetic structure and holes distribution
(La,Y)y(Sr,Ca)14-yCu24O41
y≠0, all holes in chains
No charge ordering
Sr14-xCaxCu24O41
x=0, around 1 hole in ladders
increases for x≠0
backtransfer to chains at low T
2D AF dimer / charge order
Cu2+  spin ½
holes O2p orbitals

7. Physics of chains: dc transport in (La,Y)y(Sr,Ca)14-yCu24O41
T> Tco 
Nearest neighbor hopping
Tco = 300 K
y=3
y=3
T< Tco
Mott’s variable range hopping
Tco = 330 K
y=5.2
Dimensionality of hopping: d=1

8. ac response in y=3 No collective response
nco- crossover frequency: ac hopping overcomes dc
Quasi-optical microwave/FIR: hopping in addition to phonon
Hopping dies out
THere is no collective response in radio frequency, no pinnedmode.It was interesting to observe how the achopping contribution eventually died out

9. Chains Phase Diagram (La,Y)y(Sr,Ca)14-yCu24O41
localized holes, hopping transport, D disorder driven insulator
Unresolved issues...
- Transport switches: chains  ladders
in 1<y<0 range?
- A phase transition:
La-substituted 
La-free materials?
Matsuda et al., PRB’96-98
So the chains were recognized as a 1D disordered insulator. Hey are also a paramagnet which at low T orders magnetically.Chains are ferro (when there are no holes) but organize antiferomagnetically. With more holes, less La content AFdimer pattern starts to apear.
But unresolved issue is how the transport switches from chains to ladders

10. Chains in Sr14-xCaxCu24O41 : AF dimers/charge order
ESR signal due to Cu2+ spins in the chains
Kataev et al., PRB’01
only SRO for x=8&9
AF order for x≥11
nh=4
nh=5
nh=6
for T>T* broadening of ESR line due to
the thermally activated hole mobility
Slope of DH*(T) vs T is approximately
the same for all x of Sr14-xCaxCu24O41 and
for La1Sr13Cu24O41 (nh=5, all in chains)
That is chains in fully doped materials keep the same number of holes, so the additional one is gradually doped to ladders. We also note another AF order in the chains at high x in these materials.
LRO below T*
Upper limit: 1 hole transferred
into the ladders for all x

11. Phase diagram for chains
Principal results:
merges with ph.diagram of chains in underdoped materials
suppression of T*
2D ordering inferred from magnetic sector results
AF dimers order vanishes with holes transferred to ladders
AF Néel order for x≥11
Kataev et al., PRB’01; Takigawa et al., PRB’98; Ohsugi et al., PRL’99; Nagata et al., JPSJ’99; Isobe et al., PRB’01

12. What is important in the ladders in Sr14-xCaxCu24O41?
Now we are finished with the chains, what is important in the ladders. There is CDW suppressed by Ca content, not by worsened nesting and dimensionality. It is somewhat strange CDW. SC is obtained only in the presence of pressre, which increses dimensionality, and seemingly gives a regular HTC SC, despite the presence of spingap and Dagoto’s elegant ideas.

13. CDW W0= Phason: Elementary excitation associated with
spatio-temporal variation of the CDW phase F(x,t)
phason response to dc & ac field governed by: free carrier screening
and pinning potential V0 of impurities or commensurability
W0=
Experimental fingerprints: mode at pinning frequency
broad radio-freq. modes centred at
enhanced effective mass
nonlinear dc conductivity above sliding threshold ET=2kFV0/r0
1/t0=
We identified CDW through its phason response, charaterized by pinned mode and broad radiofreq modes. There should also be some nonlinearity.
We also estimated effective mass from these results.
Littlewood, PRB ‘87

14. Ladders: charge response
1300K

15. Ladders: charge response
radio-frequency ac response:
similar to phason response
Charge Density Wave
Dielectric response:
Generalized Debye function
De 104–105
t01/sz
t∞  0.1 ns D
1300K

16. 2D phason response in ladder plane
radio-freq. mode
enhanced effective mass
m*=
pinned mode W0
Kitano et al.,
EPL’01

17. Ladders: Non-linear conductivity
good contacts and no unnested voltage
No ET, negligible non-linear effect
bad contacts, or large unnested voltage
No ET, “large”non-linear effect

18. “large” non-linear effect
Ladders: Non-linear conductivity
small non-linear effect
Maeda et al., PRB’03
“large” non-linear effect
Blumberg et al., Science’02

19. Phase diagram for ladders (corresponds to chains ph.diag.)
TCDW
suppresion of Tc and charge gaps
2D CDW in ladder plane
unique to ladders? or common
to low-D systems with charge order?
order in ladders: CO of localized or CDW of itinerant electrons?
analogy with AF/SDW
half-filled ladder in Hubbard model:
CDW+pDW
in competition with d-SC
Suzumura et al., JPSJ’04

20. FIN
CDW SC

21. Physics of ladders (Sr14-xCaxCu24O41): gapped spin-liquid
NMR/NQR: Takigawa al., PRB’98; Kumagai et al., PRB’97; Magishi et al., PRB’98;
Imai et al., PRL’98, Thurber et al., PRB’03
Inelastic neutron scattering: Katano et al., PRL’99, Eccleston et al., PRB’96
for x=0: Cross-over between paramagnetic and spin-gapped phase
T*  200 K
Polycrystalline
} Single crystal
spin gap
Spin gap is present even for x=12, where SC sets-in

22. Physics of ladders (Sr14-xCaxCu24O41): superconductivity
Nagata et al., JPSJ’97
x=11.5
SC: x≥10 i T<12K, p= 3-8 GPa
pressure removes insulating phase
x=0
Motoyama et al., EPL’02
no superconductivity
NMR under pressure x=0 & x=12, p=3.2 GPa
in x=12 pressure only decreases spin gap, low lying excitations are present (Korringa behavior in T1-1)
Piskunov et al., PRB’04
Fujiwara et al., PRL’03
in x=0, the same, but no low-lying excitations and no SC!

23. Physics of ladders (Sr14-xCaxCu24O41): insulating phase(s)
Experiment:
temperature range: 2 K -700 K
dc transport, 4 probe measurements:
lock-ins for 1 mW-1 kW
dc current source/voltmeter 1 W-100 MW
2 probe measurements:
electrometer in V/I mode, up to 30 GW
lock-in and current preamp, up to 1 TW
ac transport – LFDS
(low-frequency dielectric spectroscopy)
lock-in and current preamp, 1 mHz-1 kHz
impedance analyzers, 20Hz-10MHz
Zagreb
x<11: insulating behavior
D: decreases with x
sc(300 K): (Wcm)-1
x≥11 i T>50K : metallic
Gorshunov et al., PRB’02
Vuletic et al., PRL’03
Vuletic et al., submitted to PRB’04

24. Anisotropic ac-response: 2D charge order in ladder plane
Broad screened relaxation modes E||a, E||c, x≤6
E||b, no response for any x
same Tc for E||a, E||c
same t0-1for E||a, E||c
Dec/Dea  10  sc/sa
No response along rungs for x=8,9
&
Transition broadening
Increase in sc/sa at low-T for x=9
Long-range order in planes is destroyed

25. The nature of H.T. insulating phase is the key for CDW suppression
nesting: strong e-e interactions – the concept not applicable, in principle
– dimensionality change also contradicts
disorder: renders Anderson insulator – but, this wouldn’t be removed by pressure
Ca-substitution Pressure
Increase ladder/chain coupling
increase dhchange V, U b
Increase W change U/W
CDW suppresed due to changes in U/W, V, dh (and disorder)
decrease lattice parameters
HT phase: Mott-Hubbard insulator (1/2 filling, U/W>1)
on-site U, inter-site V, hole-doping dh, bandwidth W=4t
Pachot et al., PRB‘99 a c
H.T. phase persists – “disorder resistant”

26. Instabilities in 1D weakly interacting Fermi gas
4 possible scattering processes
instabilities in the system
proximity of SC and magnetic/charge order
a-backward, q=2kF, short range interactions (Pauli principle or on-site U)
b-forward, q=0, long range interactions
c-Umklapp, q=4kF, in a half-filled band, lattice vector equals 4kF and cancels scattering momentum transfer
d-forward, q=0

27. Electronic structure (La,Sr,Ca)14Cu24O41
local density approximation
Osafune et al., PRL’97
inter-ladder hopping:
5-20% of intra-ladder
optics: Mott-Hubbard gap 2 eV EF pulled down by doped holes
M. Arai et al.,
PRB’97
top of lower Hubbard band of ladders
finite DOS on EF
bands at 3 & 5.5 eV
T. Takahashi et al., PRB’97

28. Strong coupling limit for cuprates
two band model (oxygens!)
copper: strong on-site repulsion U
Zhang and Rice., PRB’88
Cu (3d9) and O (2p6) form the structure
charge-transfer limit
1 electron, S=½ per copper site
doped holes enter O2p orbitals  form ZhangRice singlet with Cu spin
chain
ladder
layer
Hamiltonian reduces to Heisenberg spin ½ model + effective hopping term for ZR singlet motion
lattice formed of 1 kind of sites

29. Physics of chains: Sr14-xCaxCu24O41
2cC
2cc
X-ray difraction Cox et al., PRB’98
5 holes
T=50K
INS Regnault et al., PRB’99; Eccleston et al., PRL’98
XRD Fukuda et al., PRB’02
NMR/NQR Takigawa al., PRB’98
2cC
3cC
6 holes T=
5-20K
Chains: AF dimer / charge order complementarity
spin-gap
Holes are localized in chains of fully hole doped Sr14-xCaxCu24O41

30. Ladder plane dc conductivity anisotropy vs. Temperature
anisotropy: approximately 10 for all x and temperatures
increase at lowest T for x>8
more instructive picture if anisotropy is normalized to RT value

31. Unconventional CDW in ladder system
recently derived (extended Hubbard type) model for two-leg ladder with both on-site U and inter-site V|| along and V across ladder.
Suzumura et al., JPSJ’04
CDW +p-DW
CDW +p-DW
hole transfer
dh=2.8 dh
decrease V, increase the doping dh, destabilizes CDW and p-DW, in favor of d-SC state.

32. (TMTTF)2AsF6 Charge order vs. CDW in ladders
D.S.Chow et al., PRL’00
NMR detects charge disproportionation
In the vicinity of CO transition dielectric constant follows Curie law
F. Nađ et al., J.Phys.CM’00
Zagreb
Relaxation time is temperature independent – not phason like

33. CDW in the Ladders versus CO in the Chains
Takigawa al., PRB 1998
chain
ladder
No splitting of 63Cu NMR line
Splitting of 63Cu NMR line
Charge disproportionation

34. Phason CDW dielectric response
Fukuyama, Lee, Rice
Periodic modulation of charge density
Random distribution of pinning centers
Local elastic deformations (modulus K) of the phase f(x,t)
Damping g
Effective mass m*»1
External AC electric field Eex is applied
Phason: Elementary excitation associated with
spatio-temporal variation of the CDW phase f(x,t)
in the third term, which is the elastic energy associated with the local deformations
Random distribution of pinning centers taken into account in the fourth term
Phason dielectric response governed by: free carrier screening, nonuniform pinning

35. Phason CDW dielectric response
Max. conductivity close to the pinning frequency W0
pinned mode - transversal
g0- weak damping
in the third term, which is the elastic energy associated with the local deformations
Random distribution of pinning centers taken into account in the fourth term
Littlewood

36. Phason CDW dielectric response
Max. conductivity close to the pinning frequency W0
pinned mode - transversal
g0- weak damping
plasmon peak
longitudinal
Screening:
Low frequency tail extends to
1/t0=
strong damping
g»g0
in the third term, which is the elastic energy associated with the local deformations
Random distribution of pinning centers taken into account in the fourth term
Longitudinal mode is not visible in diel. response since it exists only for e=0!
Littlewood

37. Experiments detect two modes
Phason CDW dielectric response
Nonuniform pinning of CDW gives the true phason mode a mixed character!
W0=
1/t0=
in the third term, which is the elastic energy associated with the local deformations
Random distribution of pinning centers taken into account in the fourth term
Longitudinal response mixes into the low-frequency conductivity
Experiments detect two modes
Littlewood

38. m* - CDW condensate effective mass Sr14-xCaxCu24O41
W0 & t0 are related:
t0 & sz – from our experiments
r0 – carriers condensed in CDW (holes transferred to ladders = 1·1027 m-3 = 1/6 of the total)
Microwave conductivity measurements (cavity perturbation)  peak at W0=60 GHz  CDW pinned mode Kitano et al., 2001.
in the third term, which is the elastic energy associated with the local deformations
Random distribution of pinning centers taken into account in the fourth term
CDW effective mass m*≈100 m*

39. Complex dielectric function
Generalized
Debye function ∞
Debye.fja

40. Complex dielectric function
Generalized
Debye function ∞
relaxation process strength  = (0) - ∞
0 – central relaxation time
symmetric broadening of the relaxation time distribution 
Debye.fja

41. ere eim We analyze real & imaginary part of the dielectric function
We fit to the exp. data in the complex plane
We get the temp. dependence De, t0, 1-a
eim
ere
Eps im eps re