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arXiv:cond-mat/0501654v2 [cond-mat.supr-con] 5 Apr 2005
Surface pinning of fluctuating charge order: an “extraordinary” surface phase
transition.
Stuart E. Brown,1Eduardo Fradkin,2and Steven A. Kivelson3, 1
1Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547
2Department of Physics, University of Illinois, 1110 W. Green St., Urbana, Il linois 61801-3080, USA
3Department of Physics and Astronomy, Stanford University, Stanford CA 94305
(Dated: February 2, 2008)
We study the mean-field theory of charge-density wave (CDW) order in a layered system, including
the effect of the long-range Coulomb interaction and of screening by uncondensed electrons. We
particularly focus on the conditions necessary for an “extraordinary” transition, in which the surface
orders at a higher temperature, and is more likely to be commensurate, than the bulk. We interpret
recent experiments on Ca2−xNaxCuO2Cl2as indicating the presence of commensurate CDW at the
surface that is not present in the bulk. More generally, we show that poor screening of the Coulomb
interaction tends to stabilize incommensurate order, possibly explaining why the CDW order in
La2−xSrxCuO4and NbSe2remains incommensurate to T→0, despite the small magnitude of the
incommensurability.
PACS numbers:
With the advent of high resolution angle resolved
photoemission (ARPES) and scanning tunneling spec-
troscopy (STS), there is increasing interest in looking for
evidence of novel order or incipient order in strongly cor-
related electron systems by studying the electronic struc-
ture of the surface layer – some of the most interesting
recent evidence that charge order plays a critical role in
the physics of the cuprate high temperature supercon-
ductors (HTC) comes from such studies1,2,3,4. A persis-
tent question about such studies arises, “Is the surface
electronic structure the same as the bulk?”
In the present note, we outline some of the possibilities
for transitions to ordered states at the surface of a bulk
system, where the surface order reflects, but in some-
what subtle and indirect ways, the character of the bulk.
In particular, we argue that the “commensurate checker-
board order” recently discovered4in Ca2−xNaxCuO2Cl2
(NaCCOC) is very likely not directly representative of
charge order in the bulk of the sample. However, we de-
scribe suggestive, but not conclusive reasons to believe
that this surface order is a pinned relative of fluctuating
charge order in the bulk - probably related to the fluctu-
ating charge stripe order seen in bulk measurements on
La2−xSrxCuO4(LSCO). This work extends earlier inves-
tigations by two of us5of general strategies for observing
correlations - “fluctuating order” - which reflect the prox-
imity in a generalized phase diagram of a true ordered
state.
When a bulk system undergoes a phase transition to
a broken symmetry state, such as a charge-density wave
(CDW) state, the surface of the system must reflect the
broken symmetry, as well. However, one might expect
order to be weaker at the surface. Nonetheless, there
are known cases of surface phase transitions in which an
“extraordinary transition” occurs6,i.e. a phase transi-
tion in which the surface orders at a higher temperature
than the bulk.
In the present paper, we consider the case of a CDW
in a layered (quasi 2D) material with a smooth surface
obtained by cleaving between two layers, for which:
1) We analyze the circumstances under which an
extraordinary surface phase transition can occur. We
show that if the couplings within each layer, including
the surface layer, are identical, the surface is unlikely
to order before the bulk. However, phonon modes
associated with the motion of atoms transverse to the
layers tend to be softer at the surface than in the bulk.
If the coupling to such modes is sufficiently strong,
an extraordinary surface phase transition occurs. An
example of such a phonon mode is the apical O or
Cl modes in LSCO and NaCCOC, respectively, which
independent studies suggest are strongly coupled to the
charge density in the Cu-O planes.
2) Under circumstances in which an anomalous tran-
sition occurs, one expects from simple Landau-Ginzburg
considerations, the following hierarchy of transition tem-
peratures: TsI ≥TsC and TsI ≥TI, where TsI,TsC ,
and TIare the transition temperatures at which sur-
face incommensurate CDW order, surface commensurate
CDW order, and bulk incommensurate order onsets, re-
spectively. Moreover, as is generally the case, TI≥TC,
where TCmarks the bulk transition to a commensurate
CDW. The extraordinary transition is particularly dra-
matic when TCand even TIare zero, i.e. when the bulk
system is in a quantum disordered phase, but possibly
with a commensurate CDW on the surface layer. This
sequence of transitions is shown schematically in Fig. 1.
3) We interpret the STS experiments4on NaCCOC as
being indicative of a commensurate CDW phase on the
surface (although the large effect of quenched disorder
apparent in the STS images make this conclusion far
from certain). We show that bulk measurements appear
inconsistent with the existence of a commensurate CDW
2
TI
TsI
TC
TsC
SI
SI
SC
SC
SC
T
r0/r
1
Bulk Disordered
Bulk Commensurate
Bulk Incommensurate
FIG. 1: Schematic one-parameter cut of the phase dia-
gram for a CDW system with a surface: rand r0mea-
sure the strength of the quadratic term in the McMillan
free energy in the bulk and on the surface. The full and
broken lines represent the bulk and surface phase transi-
tions respectively: TIand TCare the bulk critical temper-
atures for the disordered-incommensurate and for the bulk
incommensurate-commensurate phase transitions; TsI and
TsC are the corresponding surface phase transitions. The
extraordinary surface transitions for r0/r are shown. SI: sur-
face incommensurate CDW; SC: surface commensurate CDW.
Other surface orderings (e.g. “ordinary”surface transitions)
are possible but are not shown.
in the bulk. Thus, it is possible that either the order on
the surface has no relation to the electronic properties
of the bulk, or that there is an extraordinary surface
phase transition which reflects the existence of bulk
electronic correlations corresponding to a nearly ordered
CDW in the bulk. By comparing the character of the
observed surface order with modulations with similar
periodicity (but rather different character) seen in STS
studies of Bi2Sr2CaCu2O8+δ(BSCCO) surfaces1,2,3 , and
with stripe order and fluctuating order seen in bulk
diffraction studies of LSCO and YBa2Cu3O6+y(YBCO),
we tentatively favor the latter interpretation.
4) In the course of this study, we have been forced to
study the non-local Coulomb interaction. Under most
circumstances – certainly, at any finite temperature
or when the Fermi surface is incompletely gapped by
the CDW – the Coulomb interactions between CDW
fluctuations are screened, with a screening length ξ.
While when ξis small, the usual Landau-Ginzburg
theory is recovered, we have found that poor screening
(large ξ) substantially stabilizes the incommensurate
CDW – it both tends to increase TIand decrease TC.
It is possible that this explains the remarkable stability
of the incommensurate phase in some systems, despite
very small values of the incommensurability. For in-
stance, the ordering wave vector in NbSe2is roughly 1%
incommensurate, but there is apparently no transition
to a commensurate state even in the limit T→0.
The paper is organized as follows. In Section I we
introduce a Landau-Ginzburg theory of quasi-2D CDW
order, including the effects of Coulomb interactions and
screening on the bulk CDW phase transitions. In Section
II we discuss the extraordinary surface phase transition.
In Section III we discuss the role of surface phonons in
the “mechanism.” In Section IV we discuss the case of
NaCCOC and other cuprates. Readers who are exclu-
sively interested in the application of these ideas to the
experimental system can skip directly to this section.
I. LANDAU-GINZBURG MODEL OF QUASI 2D
CDW ORDER
To discuss these issues in the context of an explicit
model, we consider the case of a CDW in a quasi 2D
(layered) system with tetragonal symmetry7. We con-
sider the case in which the local considerations in each
plane favor density wave order with two, mutual orthogo-
nal ordering vectors, ~
Q1and ~
Q2, (|~
Q1|=|~
Q2|since they
are related by rotation by π/2) which lie along a pre-
ferred symmetry axis of the crystal. We then express the
theory in terms of two complex order parameter fields,
ψ1,n and ψ2,n, such that the local charge density ρn(~x)
in plane nis
ρn(~x) = ρN
n(~x) +
X
jhiΛψ∗
j,n(~x)ˆ
Qj·~
∇ψj,n(~x) + ψj,n (~x)ei~
Qj·~x + c.c.i
+higher harmonics (1.1)
where ρN
nis the “normal” component of the charge den-
sity, i.e. the part which is not tied to the CDW, and
ˆ
Qj=~
Qj/|~
Qj|are two unit vectors along the two CDW
ordering directions. The terms represented as “higher
harmonics” refer to components of the density at higher
harmonics of the fundamental periods, n1~
Q1+n2~
Q2with
|n1|+|n2|>1; we will simply assume that these harmon-
ics can be integrated out, and their effects captured by
non-linear couplings to the fundamentals. The dimen-
sionless parameter Λ reflects the change of the density
which results from a compression of the CDW.
We are going to consider the case in which the CDW is
commensurate or weakly incommensurate, and with the
situation relevant to NaCCOC in mind, we have consid-
ered the case in which the CDW is near commensurabil-
ity 4. In this case, it is convenient to take 4 ~
Qj=~
Gj
where ~
Gjis a reciprocal lattice vector. Hence, a uniform
amplitude for the CDW order parameter ψ= const.cor-
responds to a commensurate CDW while ψ∝exp[iδx]
corresponds to an incommensurate CDW, where δ(which
3
will be derived by minimizing the effective Hamiltonian)
is the “incommensurability.” The case in which the CDW
is far from being commensurate involves no new physics,
so we will not discuss it explicitly.
In terms of the order parameters ψ1and ψ2, we can
write a McMillan (Landau-Ginzburg) type effective free
energy functional8,9,10 suitable for a CDW in a layered
system (keeping lowest order terms in gradients and all
terms allowed by symmetry through order ψ4):
F=X
nZd2x{L0+L1+Lc}+Fcoul +FN
L0=1
2X
jnK|(iˆ
Qj·~
∇+δ0)ψj,n(~x)|2
+K′|ˆ
Qj×~
∇ψj,n(~x)|2+r|ψj,n (~x)|2o
L1=u(|ψ1,n(~x)|2+|ψ2,n (~x)|2)2+γ|ψ1,n(~x)|2|ψ2,n (~x)|2
Lc=VcX
jψj,n(~x)4+ c.c.
Fcoul =e2
2ǫX
n,n′Zd2x d2x′[ρn(~x)−¯ρ][ρn′(~x ′)−¯ρ]
p(~x −~x ′)2+a2(n−n′)2
(1.2)
where n= 0,1,... labels the layers with n= 0 being the
surface layer of a semi-infinite system; j= 1,2 labels the
two CDW order parameters. In Eq.(1.2), r,uand γare
phenomenological couplings which depend weakly on the
temperature T. As usual, rchanges from positive to neg-
ative with decreasing temperature, and so is the one pa-
rameter whose temperature dependence will be explicitly
considered, r=α0(T−T0
I). Moreover, near the surface,
the various parameters could also depend on the layer
index, n. For simplicity, we will assume that only the
surface layer (n= 0) is distinct, and that the most im-
portant difference between the surface layer and the bulk
is an additive correction to r0=r+δr =α0(T−T0
sI ),
i.e. a distinct mean-field transition at the surface.
In Eq.(1.2) we have assumed that we can neglect all
inter-plane interactions except the Coulomb coupling,
Fcoul. In Eq.(1.2) ¯ρis a uniform background charge den-
sity, Kand K′are the CDW stiffnesses, and ais the
lattice spacing between planes. Here, the shear stiffness,
K′will play little role in the present discussion. The sign
of γdetermines whether the ordered state is a unidirec-
tional CDW, γ > 0, or an isotropic checkerboard, γ < 0.
To simplify explicit expressions, we will assume that |γ|
is small, although so long as γ > −2u(necessary for sta-
bility). No qualitative results depend on this assumption.
Finally, Lcis the lock-in potential which favors a (period
4) CDW, Vcis the strength of the commensurability, and
e2/ε is the strength of the Coulomb interaction.
We introduced a term in Eq.(1.2) , FN, which governs
the fluctuations of the normal density; for our purposes,
what is important is the preferred density, ¯ρN, and the
small fluctuations about it (which lead to screening), so
we take
FN=
∞
X
n=0 Zd2xκ0
2[ρN
n(~x)−¯ρN]2(1.3)
where κ0is the inverse compressibility of the normal
fluid. Notice that the normal density and the CDW or-
der parameters are coupled through the Coulomb inter-
action in the Landau-Ginzburg effective theory, as given
by Eq.(1.1) and Eq.(1.2). Integrating out the fluctua-
tions of the normal excitations, parametrized by ρN
n(~x),
result in an effectively screened Coulomb interaction with
a screening length ξ= (4πe2/εκ0)−1/2, and with the
density expressed as in Eq.(1.1) with the replacement
ρN
n(~r)→¯ρN.
Before we proceed further, it is worth commenting on
the ways in with the present free energy differs from the
usual Landau-Ginzburg treatment (obtained by taking
the limit κ0→0 in the above), which does not treat
the effects of the non-local Coulomb interactions. For
κ0= 0, the incommensurability is determined directly
by δ=δ0, and so has no interesting temperature de-
pendence that is not put in by hand, and the mean-field
transition to the incommensurate state occurs just where
rchanges sign, TI=T0
I, or at the surface at TsI =T0
sI .
For imperfectly screened Coulomb interactions, however,
the situation is more complex. The optimal value of δ
is determined by a combination of δ0and ∆ ¯ρ≡¯ρ−¯ρN.
In the limit of no screening, κ0→ ∞, the incommensu-
rate state is stable at all temperatures (TI→ ∞), and at
long distances, the incommensurability is set by the con-
straint,δ=−∆¯ρ[2Λ|ψ|2]−1. For intermediate values of
the screening, the behavior is intermediate between these
two limits. Finally, since the Coulomb interaction cou-
ples different planes, the surface transition temperature
is not determined solely by the values of the parameters
in the surface layer, but depends on the coupling between
the surface and the bulk in a non-trivial fashion.
Since our principal focus is on the possibility of sur-
face transitions, we will treat the bulk properties curso-
rily. Just to be specific, we consider the case of a striped
phase (γ > 0) in which ψ2= 0, but the generalization to
checkerboard phases, which occur for γ < 0, is straight-
forward. In order to determine the bulk (mean field)
phase diagram we compare the energies of three different
forms of the order parameter to get a sense of the bulk
phase diagram:
1) The disordered state, ψ1,n = 0. The mean-field free
energy density of the disordered state is
Fdis = (1/2)κ0(∆¯ρ)2.(1.4)
2) The commensurate solution, ψ1,n (~r) = ψ. The free
energy of the commensurate state is
Fc=Fdis + (1/2) [r+Kδ2
0VQ]|ψ|2+ [u−2Vc]|ψj|4,
where VQ= 4πe2/(ǫ[Q2+ξ−2]).
3) The harmonic incommensurate state, ψj,n(~r) =
4
ψjexp[iδ ˆ
Qj·~r]. (We know from the work of McMillan9
that, especially near the commensurate to incommensu-
rate transition, the structure of the incommensurate state
is highly anharmonic, and this anharmonicity has a sig-
nificant quantitative effect on the phase diagram, but one
can understand much of the qualitative physics ignoring
this.) The optimal incommensurability and free energy
of this state are, respectively,
δ=δ0[1 + A−1]h1 + 2A−1|˜
ψ|2i−1
(1.5)
FI=Fdis +r+VQ
2|ψ|2+u|ψ|4
−κ0(∆¯ρ)2
2"1 + 2A− | ˜
ψ|2
1 + A|˜
ψ|2#|˜
ψ|2
(1.6)
where ˜
ψ= [2Λδ0/∆¯ρ]ψand A=Kδ0/2κ0Λ∆ ¯ρ.
One interesting consequence of these expressions is
that they imply a non-trivial temperature dependence
of the incommensurability as the order parameter grows.
This is a general feature of an incommensurate state, but
what is new here is the singular temperature dependence
inherited from the Tdependence of |ψ|2. The other im-
portant observation is that a more poorly screened the
Coulomb interaction (larger κ0), generally tends to sta-
bilize the incommensurate phase. This can be seen from
the fact that, the final term in Eq. (1.6) is generally
negative. Since TIis the first temperature at which the
quadratic term (in powers of ψ) in Eq. (1.6) becomes
negative, it is manifest that TIis an increasing function
of κ0. However, even for finite ψ, this term is negative
so long as |˜
ψ|2<1 + 2A, and so it generally tends to
favor the incommensurate over the commensurate phase,
as well.
This final observation may be significant for under-
standing the remarkable stability of weakly incommen-
surate CDW states. When the Coulomb interaction
is fully screeened (κ0= 0), the commensurate to in-
commensurate transition occurs when the gain in com-
mensurability energy, equals the loss in elastic energy,
2Vc|ψ|4= (1/2)Kδ2|ψ|2. Since both Vcand Kare typi-
cally determined by the electronic structure, unless either
δis large or ψis small (which typically means that Tis
close to TI), we expect universally to see only commen-
surate states. However, where the Coulomb interactions
are poorly screened (κ0large), there is an additional en-
ergetic cost ∼(1/2)κ0(∆ ¯ρ)2for the commensurate state,
which could stabilize the incommensurate state to low
temperatures, even if δ≪1.
In the above we have kept only terms to order ψ4and
lowest order in the density fluctuations. Near a continu-
ous transition between a disordered phase and an incom-
mensurate CDW, this is justified, at least at mean-field
level, on the basis of the small magnitude of the order
parameter. Since the incommensurate to commensurate
transition occurs only when the order parameter exceeds
a critical magnitude, the above treatment is only valid
when δis small enough. More generally, at temperatures
well below TI, the low energy physics, and the commensu-
rate to incommensurate transition in particular, should
be treated in terms of a phase-only model. Thus, as
long as we are comfortably below the mean-field tran-
sition temperature, we can integrate out the amplitude
modes of the CDW order parameters (and the fluctua-
tions of ρN, as well), and concentrate on the low-energy
physics of the phase degrees of freedom. Thus, we set
ψjn (~r) = |ψj|exp[iθjn(~r)]. To begin with, again consider
the stripe case, in which ψ1=ψand ψ2= 0; then, the
effective free energy for the phase degrees of freedom is
Feff [θ] = X
nZd2xnκk
2(∂xθn−δ0)2+κ⊥
2(∂yθn)2−Ucos[4θn]o
+g
2X
n,n′Zd2x d2x′(∂xθn−δ′
0)e
V(~x −~x ′, n −n′) (∂x′θn−δ′
0) (1.7)
where e
V(~x −~x ′, n −n′) is the screened Coulomb interac-
tion potential. In what follows we will find it convenient
to use a form of the screened interaction potential which
is the solution of
−a−2
z△+∇2+ξ−2
se
V(x−x′;n−n′) =
a−1
zδn,n′δ(x−x′)
(1.8)
where we have set △f(n)≡f(n+ 1) + f(n−1) −2f(n),
and ξsis the screening length .
The expression for Feff [θ], the effective free energy for
the phase degrees of freedom, Eq.(1.7), can be derived
from the Landau-Ginzburg theory above, Eq.(1.2), which
results in expressions for the effective stiffness constants
κα, the commensurability potential, U, and the second
incommensurability, δ′
0, in terms of the parameters of
the Landau-Ginzburg model. The only important aspect
of this for our purposes is that U∝ψ4,κα∝Kαψ2,
and g∝(e2/ε)ψ4. It is also important to note that in
the neighborhood of a surface, the parameters in Feff [θ]
5
inherit layer index, n, dependence.
We conclude with a final observation concerning the
checkerboard phase. The Landau-Ginzburg free energy
in Eq.(1.2) has no direct coupling between the phase of
ψ1and that of ψ2. Thus, to this order, Feff [θ] for the
checkerboard phase is simply two, totally independent
copies of the above effective free energy. In the incom-
mensurate phase, this reflects an exact symmetry - the
origin of the two components of the CDW can be shifted
relative to each other with no cost in energy. This has
implications for the fluctuation spectrum of an incom-
mensurate checkerboard phase. (For the commensurate
phase, the phase is locked to the underlying crystalline
lattice, in any case, so although there are higher order
couplings that link the two phases, they are not impor-
tant. )
II. THE NATURE OF THE EXTRAORDINARY
COMMENSURATE-INCOMMENSURATE
TRANSITION
There are two cases of interest. In the first case, we
can envisage a situation in which the coefficient rfor
the surface layer is different than in the bulk, and such
that the surface orders while the bulk remains disordered.
This is the direct analog of the “ordinary extraordinary”
surface phase transition, which has been well studied in
magnetic systems6. (In the next section we give a brief
discussion of the microscopic physics that can lead to this
situation.) A mean-field state of this type has the form
ψn(~x)∼Anexp[i(1 + δn)~
Q·~x] where limn→∞ An= 0
exponentially fast (on a length scale of the order of the
bulk correlation length.) The only difference from the
ordinary case is that, because of the Coulomb interaction,
the incommensurability, δn, varies from plane to plane,
approaching an asymptotic value limn→∞ δn=δ0. It is
straightforward to construct this state using the Landau-
Ginzburg theory of Eq.(1.2).
The second case of interest, which is the focus of
this section, does not have an obvious analog in surface
phase transitions in magnetic systems (although it may
happen in incommensurate spin-density-wave systems as
well.) Here we imagine that the temperature is well be-
low the critical temperature for bulk incommensurate or-
der so both in the bulk and at the surface, the magni-
tude of the order parameter is large and essentially fixed.
We can now ask if it is possible for the commensurate-
incommensurate transition to occur at the surface at a
higher critical temperature than in the bulk. We can dis-
cuss the physics of this state in the simpler phase-only
model of Eq.(1.7). To simplify the analysis, in what fol-
lows we will focus on the special case δ′
0=δ0=δas
this does not change the qualitative properties of the so-
lutions, and it greatly simplifies the algebra.
We construct this inhomogeneous state as follows: We
first note that for the bulk homogeneous commensu-
rate state θn(~x) = 0 everywhere, while for the bulk
homogeneous incommensurate state θn(~x) = δ x ev-
erywhere (assuming stripe order perpendicular to the x
axis). As in the previous section, we have thus neglected
the physics of near-commensurability, i.e. discommen-
surations, within a given plane - including this physics
greatly complicates the analysis without changing the
qualitative conclusions. We will construct an inhomo-
geneous state which is commensurate at the surface, and
hence we set θ0(~x) = 0, but incommensurate everywhere
else, ∂xθn(~x)6= 0 for n= 1,2,....
In short, we need to study the circumstances un-
der which there exists an approximate mean-field state
(which minimizes Feff[θ]) in which (by assumption)
the phase field on each plane has a constant gradient,
∂xθn(~x)≡fn+δ(discommensuration-free), but which
varies from plane to plane. This solution must satisfy
the boundary conditions f0=−δ,i.e. commensurate
at the n= 0 layer, and limn→∞ fn= 0, i.e. the bulk
incommensurate state. It should be stressed that while
approximate, these configurations are upper bounds to
the actual non-linear solutions. The effective free energy
(per unit area) for configurations of this type is readily
found from Eq.(1.7) to be
Feff [f] = κk
2
∞
X
n=0
f2
n−
∞
X
n=0
Un+g
2
∞
X
n,m=0
fnfmG1D(n−m)
(2.1)
where G1D(n−m) is given by
G1D(n−m) = µ
µ2−1µ−|n−m|(2.2)
with
µ=1
2n2 + (a/ξs)2+(2 + (a/ξs)2−41/2o(2.3)
For configurations of this type, the contribution of the
pinning potential vanishes for all n≥1 (as they are
incommensurate), while on the commensurate surface
layer, n= 0, it contributes with the surface value of
the pinning potential, Us. Hence, Un=Usδn,0.
The unique configuration fnwhich minimizes Feff[f]
and satisfies the boundary conditions is
fn=−δ1−γ
µγ−n(2.4)
where γsatisfies the identity
γ+1
γ=µ+1
µ+g
κk
(2.5)
Since γ > µ > 1, the solution decays to the bulk value
on a scale a/ log γshorter than the length scale of the
screened interaction. The free energy (per unit area) of
this solution is
F=Ucrit
s−Us, Ucrit
s≡1
2δ2gγ
µg
κk.(2.6)
6
Thus, if the surface value of the pinning potential Usis
greater than Ucrit
s, the uniform incommensurate CDW
state is unstable at the surface. Moreover, since Ucrit ∝
δ2, for the case of a system which is only weakly incom-
mensurate in the bulk, it requires a very small value of
the surface commensurability coupling to stabilize a com-
mensurate state, there.
This is not quite the whole story. The same analysis
can be applied to look for a bulk, inhomogeneous state, in
which a periodic arrangement of (possibly far separated)
planes are commensurate, while the intervening planes
have incommensurabilities that can be obtained in simi-
lar fashion by minimizing Eq. 2.1. An upper- bound to
the energy of such an inhomogeneous bulk state is given
by using the surface solution we have just described, but
with an arbitrary layer in the bulk taken to be the layer
which is commensurate - it is in fact possible to do some-
what better than this. Thus, there is a critical value,
Ucrit < Ucrit
s, such that when U > Ucrit , there is a bulk
instability of the uniform state. To find a circumstance
in which there is a surface instability, but no bulk in-
stability, it is necessary that Uis larger in the surface
layer than in the bulk, so that Us> U crit
s> Ucrit > U .
As we will see in the next section, this is possible when
a surface phonon results in an enhanced magnitude of
the CDW in the surface layer, and hence an enhanced
tendency toward commensurability.
III. A POSSIBLE PHONON “MECHANISM” OF
AN EXTRAORDINARY TRANSITION
In this section, we discuss a simple model, motivated
by the structure of NaCCOC, in which an electron-
phonon coupling can lead to an enhancement of the sur-
face tendency to CDW order and commensurability. Of
course, there are many possible surface effects, so this
discussion should be taken as illustrative rather than “re-
alistic.”
Each Cu site in NaCCOC sits at the center of an oc-
tahedron, with the apical (out of plane) sites occupied
by a Cl, instead of the O that appears there in LSCO.
It is known that NaCCOC cleaves such that the surface
layer is a Ca-O layer, so the topmost Cl is exposed at the
surface. It is thus highly plausible that the motion of the
apical Cl is less constrained due to the absence of mate-
rial above it. It is known11, moreover, that the motion of
the apical atom is strongly coupled to the charge density
in the copper-oxide plane – the apical O moves 0.013 ˚
A
closer to the Cu in optimally doped LSCO than in un-
doped La2CuO4(in YBCO, the apical O displacement is
even larger12 ∼0.15 ˚
A.)
We therefore consider the effect of coupling to an Ein-
stein phonon corresponding to the motion of the local
charge density in the plane,
Hel−ph =P2
2M+1
2Mω2
0X2+λ X [ρ(~x)−¯ρ].(3.1)
Here Pand Xare the phonon momentum and displace-
ment, ω0is the phonon frequency, λis the electron-
phonon coupling, and ρ(~x) is the local electron density.
If we further assume that the phonon is “fast” (ω0large
compared to the frequency scales of interest), we can in-
tegrate it out to obtain an effective attraction
Heff =−λ2
2Mω2
0
[ρ(~x)−¯ρ]2(3.2)
when we further substitute the expression in Eq. (1.1) for
ρin terms of the CDW order parameter, we see that the
electron-phonon coupling leads to a renormalization of
the various parameters that enter the Landau-Ginzburg
free energy functional, but most importantly, it leads to
a negative additive shift of r
r→r−2λ2
2Mω2
0
(3.3)
or equivalently, to an upward renormalization of the
mean-field ordering temperature. If the surface phonon is
softer than in the bulk (i.e. the elastic constant k=Mω2
0
is smaller on the surface), as the above discussion sug-
gests, then this renormalization is larger at the surface
than in the bulk. This implies both that the ordering
temperature at the surface is enhanced, and at a given
temperature, the magnitude of the order parameter is in-
creased, thereby increasing the chance of a commensurate
lock-in.
It is important to note that a large shift in the CDW
ordering can occur for rather small displacements of
the apical Cl positions. To make a dimensional esti-
mate of the expected magnitude of this displacement,
we note that the contribution of this interaction to
the condensation energy is Eel−ph ∼M ω2
0(∆X)2; this
must be less than or equal to the full condensation en-
ergy, and hence we can make an upper-bound estimate,
Eel−ph ∼ρ(EF)T2
c. If we further crudely estimate that
Mω2
0∼EF/a2, we find that ∆X∼a(Tc/EF), which is
generally small.
IV. EXPERIMENTS IN THE CUPRATES
Dramatic evidence of CDW order in a high tempera-
ture superconductor was recently obtained from low tem-
perature (T= 4K) STS experiments of Hanaguri and
coworkers4on NaCCOC with x= 0.08,0.10, and 0.12.
All of these doping levels are less than the optimal value
x= 0.15, where Tcreaches 20K; for x= 0.08 there is no
bulk superconductivity at all. For all doping levels, the
tunneling conductances exhibit a pseudogap structure for
energies less than about 150meV , but no sign of the co-
herence peaks at lower energies that have been identified
with the superconducting gap in earlier STS studies of
BSCCO2,13. Although high-energy topographic maps do
not exhibit any periodic modulations other than those
associated with the underlying crystal structure, within
7
the pseudogap, STS reveals large amplitude (order 1)
spatial modulations of the local density of states (LDOS)
with a checkerboard pattern. Fourier transforms of the
STS maps reveal peaks corresponding to a commensurate
modulation with a 4a0×4a0periodicity and peak widths
of order of one tenth of the Brillouin Zone dimensions.
The large amplitude LDOS modulations observed at
low energies are reminiscent of those seen in conven-
tional CDW systems such as the dichalcogenides14. In
those systems, the CDW also shows up in the topographic
maps. Correspondingly, in NaCCOC, one might have ex-
pected a signal from the height modulations of the sur-
face Cl atoms to show up in the high-energy topographs
as a result of CDW-induced atomic displacements. How-
ever, according to the estimates in the previous section,
for a range of plausible Tc/TF∼10−1−10−2, the ex-
pected magnitude of these displacements is in the range
0.1−0.01 ˚
A, which probably would be undetectable on
the grey-scale maps in Ref. 4. Another difference with
conventional CDW behavior is the doping independence
of the ordering vector in NaCCOC. In conventional CDW
systems, the CDW ordering vector changes as the loca-
tion of the Fermi surface changes. Finally, whereas the
STS modulations seen in the dichalcogenides are highly
coherent, the correlation length of about 10 lattice con-
stants found in NaCCOC is less than 3 periods, making
any definitive statements about the character of the order
difficult.
To determine whether the observed modulations are
indicative of bulk CDW order, we need to consider what
other signatures of CDW order would be expected. There
are presently no high resolution STS studies on NaCCOC
at higher temperatures, so little is directly known about
the thermal evolution of the checkerboard order. X-ray
or neutron diffraction are the traditional sources of defini-
tive evidence for charge order; the in-plane components
of the wavevector are well-defined by the STS experi-
ments, but the body-centered tetragonal coordination of
the the Cu atoms gives little reason to suspect signif-
icant coherence between planes. As a result, peaks in
the scattering intensity should form rods at wave-vectors
Q= (2π/a)(1/4,0, ℓ). However, we are not aware of any
reports of any such diffraction peaks in NaCCOC.
A phase transition to a density wave state is expected
to affect the resistivity, in general, by removing some part
of the Fermi surface (FS) and by modifying scattering
rates on the remaining FS. This expectation is generally
realized in conventional CDW systems. The effect on the
resistivity is especially strong in cases in which the CDW
order is sufficiently strong that there is a low tempera-
ture commensurate lock-in. In contrast, the temperature
dependence of the resistivity of NaCCOC15 does not ex-
hibit any distinct features we can associate with a phase
transition below 300K. Moreover, the magnitude of the
in-plane resistivities are close to what is reported16 for
LSCO and YBCO, where there is little or no static CDW
order.
More evidence against the existence of a bulk CDW
comes from an examination of the electromagnetic re-
sponse. Optical conductivities of conventional CDW
systems show a shift of (typically most of the) os-
cillator strength to energies above the single-particle
gap, whereas there is no evidence for such a shift in
NaCCOC15. If there were bulk CDW order in NaC-
COC with an ordering temperature below 300K, surely
it would have produced a detectable feature in the elec-
tromagnetic response. If the ordering temperature were
above 300K, surely it would have produced a large quan-
titative change of the resistivity. (Note: it is usual in
CDW systems that the ratio of the CDW gap to Tcis
large, 2∆/kTc∼10, see Ref. 17. For a gap size of
∆∼150 meV , one might therefore expect a bulk order-
ing temperature of around 300K.)
The one caveat on this argument is that, already in
earlier studies of the onset of stripe order in LNSCO, it
was observed that while there is a characteristic signa-
ture in the temperature dependence of the resistivity18
and a “localization” like suppression19 of the low fre-
quency optical conductivity associated with the onset of
charge order, these features are considerably more muted
than in conventional CDW systems. Presumably, the dif-
ference reflects the different origins of the charge order.
In conventional CDW systems, the ordering is at least
loosely associated with Fermi surface nesting, and hence
the fluctuations above Tccause singular scattering of the
quasi-particles across these nested portions, and a gap
is opened on the Fermi surface in the low temperature
phase. In contrast, in the cuprates, the charge order-
ing is a strong coupling effect5, not directly associated
with any identifiable Fermi surface nesting vector, and
hence the effect of the onset of order on the low energy
quasiparticle dynamics is much more subtle. (The fact
that the quasiparticles are always relatively short-lived,
in any case, may exacerbate this effect.) It is thus pos-
sible that in NaCCOC, the effect of the CDW ordering
on the electrodynamics is simply so small as to have es-
caped detection. However, given that the order is com-
mensurate (while that in LNSCO is incommensurate and
weak ), and that the charge order in NaCCOC apparently
produces a pseudo-gap in which the density of states es-
sentially vanishes at zero energy, we conclude that it is
unlikely that this order, if present in the bulk, would not
produce a detectable signature in the electrodynamics.
We conclude that it is likely that the commensurate
CDW order in NaCCOC resides only on the surface as
an extraordinary state of the type described in Section
II. This conclusion is suggested by the absence of any
evidence of bulk density wave order, the absence of the
expected evolution of the CDW periodicity with dop-
ing level, and the absence of superconducting coherence
peaks at the surface20 .
Is the checkerboard order, then, simply a surface arti-
fact, from which we learn nothing about a bulk tendency
toward charge order? We think a more plausible inter-
pretation is that density wave order, already incipient in
the bulk, is stabilized at the surface, most probably by a
8
softening of a surface phonon.
In the first place, modulations of the low energy LDOS
with similar period, although somewhat incommensurate
and much smaller in magnitude than those seen in NaC-
COC, have been documented1,2,3 on BSCCO surfaces,
with a long correlation length of the order of 80 ˚
A, and
interpreted (rightly we believe) as being induced by the
disorder pinning of some form of incipient (fluctuating)
CDW order. In the case of BSCCO, the relevant Cu-O
planes are not exposed on cleaving the crystal, but are
rather buried under a Bi-O layer. Thus, there is more
reason to hope that the surface electronic structure is
similar to that in the bulk. Moreover, clear signatures
of the superconducting gap have been reported in STS
studies of BSCCO surfaces13 , again suggesting that the
bulk electronic structure is well preserved at the surface.
The fact that the periodicity of the observed modulations
is similar (although not equal) to those in NaCCOC, sug-
gests that they are related phenomena, and so supports
the notion that they both reflect interesting bulk corre-
lations.
Secondly, neutron scattering studies of LSCO and
YBCO reveal ubiquitous evidence of fluctuating stripe
order5,21,22,23,24,25,26 (that is, a strong enhancement
of the dynamical structure factor at low ωand at
the characteristic, stripe ordering wave-vector, ~qstripe[x],
smoothly dependent on the doping fraction x) and a weak
(possibly extrinsically stabilized) tendency toward static
spin-stripe order. In LSCO, ~qstripe[x] is incommensurate
in the relevant range of doping, a fact which can be in-
ferred from its absolute magnitude, its continuous xde-
pendence, and the small rotation from the Cu-O direc-
tion induced by the weak orthorhombicity of LSCO27.
Nevertheless, in the relevant range of doping, the im-
plied periodicity is only slightly greater than 4a0. Sim-
ilar statements apply to the bulk properties of YBCO,
with the difference that there is still less tendency to-
ward static CDW order. The fact that the structures
seen in diffraction correspond to unidirectional density
wave order (stripes) while the checkerboards preserve the
point-group symmetry of the tetragonal lattice, would
seem to differentiate these two phenomena. However, it
is a rather subtle energy which leads to the selection of
stripe vs checkerboard order, while the basic tendency to
charge order and the characteristic ordering wave-vector
is considerably more robust. This is certainly the case,
as stressed previously5,22 , when the charge order results
from a form of Coulomb frustrated phase separation.
Finally, we turn to speculation concerning possible
surface phenomena in BSCCO. BSCCO is highly mi-
caceous, and the top bilayer is concealed below a Bi-O
layer. Thus, there is good reason to think that the
properties of this top bilayer resemble the properties of
the bulk. However, there is one effect, which even if
weak, may be important since it breaks a symmetry of
the bulk. In the bulk, the two layers of the bilayer are
equivalent, so that in the absence of bilayer splitting
there should be a single, doubly degenerate band. At
the surface, this symmetry is broken, in that the upper
layer is closer to the surface. Thus means that, even
absent bilayer splitting, there should be two distinct
bands. This effect may need to be considered when
discussing evidence of bilayer splitting from ARPES
studies of BSCCO28.
Acknowledgments
We are grateful to Peter Armitage, J.C. S´eamus Davis,
T. Hanaguri, Aharon Kapitulnik, Dung-Hai-Lee, Subir
Sachdev, John Tranquada, Ali Yazdani and Shoucheng
Zhang for many illuminating conversations. We are also
grateful to Adrian del Maestro and Subir Sachdev for
communicating their unpublished results with us. This
work was supported in part by the National Science
Foundation through the grants NSF DMR-04-42537 at
the University of Illinois (EF), NSF DMR-04-21960 at
UCLA/Stanford (SAK), NSF DMR-02-03806 at UCLA
(SEB), and NSF PHY-99-07949 at the Kavli Institute for
Theoretical Physics, UCSB, where EF and SAK where
participants at the KITP Program on Exotic Order and
Criticality in Quantum Matter. EF and SAK thank
KITP Director David Gross for his kind hospitality.
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