Coulombian disorder in Charge Density Waves

0.1 MB·English

by  Alberto Rosso

Pr1-1
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1 Coulombian disorder in Charge Density Waves
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] Alberto Rosso1, Edmond Orignac2, R. Chitra3 and Thierry Giamarchi4
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1 Laboratoire de physique th´eorique et mod`eles statistiques CNRS UMR8626
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s Baˆt. 100 Universit´e Paris-Sud; 91405 Orsay Cedex, France; e-mail: [email protected]
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d 2 Laboratoire de physique th´eorique de l’E´cole Normale Sup´erieure,
. CNRSUMR8549,24,RueLhomond,75231ParisCedex05,France;e-mail: [email protected]
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a 3 Laboratoire de physique th´eorique de la mati`ere condens´ee CNRS UMR7600
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Universit´e de Pierre et Marie Curie, 4, Place Jussieu 75252 Paris Cedex 05 France; e-mail:
- [email protected]
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n 4 Universit´e de Gen`eve, DPMC, 24 Quai Ernest Ansermet, CH-1211 Gen`eve 4, Switzerland;
o e-mail: [email protected]
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[

1
v Abstract
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3
When unscreened charged impurities are introduced in charge density wave system the
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1 longwavelengthcomponentofthedisorderislongrangedanddominatesstaticcorrelation
0 functions. Wecalculatethex-rayintensityandfindthatitisidenticaltotheoneproduced
6 by thermal fluctuationsin a disorder-free smectic-A. Wesuggest that thiseffect could be
0 observed in the blue bronze material K0.3MoO3 doped with charged impurities such as
/ Vanadium.
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a
m
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d 1 INTRODUCTION
n
o
c Quasi-onedimensionalmaterialformedofweaklycoupledchainsareknowntoposseslowtem-
: peratureinstabilitytowardsastatewithanincommensuratestaticmodulationoftheelectronic
v
charge and of the lattice displacement known as charge density wave (CDW) phase. CDWs
i
X behaveaselastic systems[1,2]providedCoulombinteractionarescreened. WhenCoulombin-
r teraction are not sufficiently screened elasticity becomes non-local [3,4] and disorder becomes
a
long ranged correlated. An important question is whether the competition of the non-local
elasticitywiththe long-rangerandompotentialresultsinthe completedestructionofthe crys-
tallineorder,orwhethersomeremnantsoftheunderlyingperiodicstructureremainobservable.
Apowerfulexperimentaltechnique abletocharacterizethestructuralpropertiesofCDWwith
impurities is affordedby x-raydiffraction. Inthis paper we derivethe static displacementcor-
relationfunctionsandx-rayintensityoftheCDWwithchargedimpuritiesandwehighlightthe
similitude of the latter to the x-ray intensity of smectic-A liquid crystals subjected to thermal
fluctuations. Finally, we discuss the experimental significance of our result and suggest that
the smectic-like correlations should be observable in experiments on KMo1−xVxO3.
1-2
2 CHARGE DENSITY WAVE WITH CHARGED IMPURITIES
We consider an incommensurate CDW, in which the electron density is modulated by a wave
vectorQincommensuratewiththeunderlyingcrystallattice. Inthisphase,theelectrondensity
has the following form [1,2]:
ρ
ρ(r)=ρ + 0Q φ(r)+ρ cos(Q r+φ(r))), (2.1)
0 Q2 ·∇ 1 ·
whereρ ,istheaverageelectronicdensity. ThesecondterminEq.(2.1)isthelongwavelength
0
density and corresponds to variations of the density over scales larger than Q−1 . The last
oscillating term describes the sinusoidal deformation of the density at a scale of the order of
Q−1 induced by the formation of the CDW with amplitude ρ and phase φ.
1
The low energy properties of the CDW are described by an elastic Hamiltonian for the
fluctuations of the phase φ. When the Coulomb interactions are unscreened this Hamilto-
nian becomes non-local. Choosing the x axis aligned with Q the total Hamiltonian in three
dimensions reads [3,4]:
1
H = H +H = G−1(q)φ(q)2, (2.2)
el. 0 C
2 | |
Z
n ¯hv v2 q2
G−1(q) = c F ⊥ x +q2
2πv2 q2ξ2 ⊥
x (cid:20) 0 (cid:21)
where v is the Fermi velocity and n is the number of chains per unit surface that crosses
F c
a plane orthogonal to Q. The velocity of the phason excitations parallel to Q is v =
x
(me/m∗)1/2vF with m∗ the effective mass of the CDW and me the mass of an electron. v⊥
denote the phason velocities in the transverse directions. The lengthscale ξ is defined by:
0
n ¯hv v2
ξ2 = c F ⊥Q2ǫ. (2.3)
0 2πv2e2ρ2
x 0
where wehaveassumedforsimplicity anisotropicdielectric permittivity ǫ ofthe hostmedium
and e is the charge of the electron. When a non-zero concentration of charged impurities is
present, the Hamiltonian becomes:
e2 ρ (r)ρ(r′)
H = H + d3rd3r′ imp. ,
el. 4πǫ r r′
Z | − |
(2.4)
where ρ (r) is the density of impurities at position r. We assume Gaussian distributed
imp
impuritieswithρ (r)ρ (r’)=Dδ(r r’),whereD isthemeasureofthedisorderstrength.
imp imp
−
Using the expressionofthe density in (2.1), the interactionwith disordercomprises twoparts,
one that describes the interaction with long wavelength fluctuations of the density (forward
scattering),andasecondpartthatdescribesinteractionsofimpuritieswithdensityfluctuations
of wavelength Q−1 (backward scattering). In [5] we have shown that even in the case of long
rangedisorder,thebackwardscatteringtermsbehaveessentiallyliketheirshortranged(neutral
impurities) counterparts. In that case, it is known that in three dimensions, the contribution
of the backward scattering to the phase correlations grows slowly as log(logr). [6,7] The
contribution of the forward scattering term is obtained from the Hamiltonian:
d3q G−1(q) iρ e2q
H = φ(q)2+ 0 xρ ( q)φ(q) ,
(2π)3 2 | | Qǫq2 imp. −
Z (cid:20) (cid:21)
(2.5)
Pr1-3
and reads in the limit of r⊥,x :
→∞
B(r) = (φ(r) φ(0))2
h − i
r2 +4xξ
⊥ 0
= κln | | . (2.6)
Λ−2
(cid:18) ⊥ (cid:19)
where Λ⊥ is a momentum cutoff and:
DQ2 DQev 1
x
κ= = . (2.7)
16πξ0ρ20 8ρ0v⊥ √nchvFǫ
As a result, in the presence of charged impurities, the forward scattering terms generate the
leading contribution to the phase correlations.
3 X-RAY SPECTRUM AND ANALOGY WITH SMECTICS A
Itis wellknownthatthe presence ofa CDWin acompoundis associatedwith the appearance
of two satellites at positions q K Q around each Bragg peak. [8] The intensity profiles
∼ ±
of these satellites give access to the structural properties of the CDW. The main contribution
to the x-ray satellite intensity is given by I , which is symmetric under inversion around the
d
Bragg vector K. [7,9–12] The intensity I is given by the following correlation function:
d
I (K+Q+k) = u2f2K2 d3re−ik·rC (r),
d 0 d
Z
C (r) = ei(φ(r/2)−φ(−r/2)) (3.1)
d
D E
where f is an averagex-rayscattering amplitude and u is the amplitude of the lattice modu-
0
lation induced by the presence of the CDW. Using the result given in Eq.(2.6) we extract the
small k behavior of the intensity peak:
(k )κ−2 for k2ξ k ,
I (K+Q+k) | x| ⊥ 0 ≪| x| (3.2)
d ∼((k⊥ )2(κ−2) otherwise.
| |
The intensity I (K+Q+k) is divergent for κ < 2 but is finite for κ > 2, i.e. for strong
d
disorder.
We note that these intensities are remarkably similar to those of a disorder-free smectic-A
liquid crystal [13] at positive temperature. In fact, the expression of the exponent κ Eq. (2.7)
is analogousto the expression(5.3.12)in [14],with the disorderstrength D playing the role of
the temperature T in the smectic-A liquid crystal.
Let us turn to an estimate of the exponent κ appearing in the intensities to determine
whether such smectic-like intensities are indeed observable in experiments. A good candidate
is the blue bronze material K MoO which has a full gap so that Coulomb interactions are
0.3 3
unscreened at low temperature. We use the parameters of [15]:
n =1020chains/m2, (3.3)
c
v =1.3 105m.s−1, (3.4)
F
×
ρ =3 1027e−/m3, (3.5)
0
×
Q=6 109m−1, (3.6)
×
1-4
and a relative permittivity of ǫ = 1, so that ǫ is equal to the permittivity of the
K0.3MoO3
vacuum. For the doped material K0.3Mo1−xVxO3, we find that the disorder strength can be
expressed as a function of the doping and obtain:
#(Mo atoms/unit cell)
D =x(1 x) . (3.7)
− a (b c)
· ×
Forthecrystalparameters,a=18.25˚A,b=7.56˚A,c=9.86˚A,β =117.53o[16],with20Molybde-
numatomsperunitcell,andadopingx=3%,thedisorderstrengthD =4.8 1026m−3. More-
over, using the experimental bounds of the velocities: 3.6 102m.s−1 <v⊥×<1.6 103m.s−1
andv =3.7 103m.s−1,wefindthatκisintherange[0.1×6 0.8]. Therefore,the×smectic-like
x
× −
order should be observable in x-ray diffraction measurements on this material.
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