Incommensurate Mott Insulator in One-Dimensional Electron Systems close to Quarter Filling

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by  Hideo Yoshioka

Typeset with jpsj2.cls <ver.1.2> Letter
Incommensurate Mott Insulator in One-Dimensional Electron Systems close to
Quarter Filling
Hideo Yoshioka1∗, Hitoshi Seo2 and Hidetoshi Fukuyama3
5 1Department of Physics, Nara Women’s University, Nara 630-8506
0 2 Non-Equilibrium Dynamics Project, ERATO-JST, c/o KEK, Tsukuba 305-0801
0 3International Frontier Center for Advanced Materials (IFCAM), IMR, Tohoku University, Sendai 980-8577
2
(ReceivedFebruary2,2008)

n
A possibility of a metal-insulator transition in molecular conductors has been studied for
a
systems composed of donor molecules and fully ionized anions with an incommensurate ratio
J
close to 2:1 based on a one-dimensional extended Hubbard model, where the donor carriers
1
areslightlydeviatedfromquarterfillingandunderanincommensurateperiodicpotentialfrom
1
the anions. By use of the renormalization group method, interplay between commensurability

energyonthedonorlatticeandthatfromtheanionpotentialhasbeenstudiedandithasbeen
]
l foundthatan“incommensurateMottinsulator”canbegenerated.Thistheoreticalfindingwill
e
- explain themetal-insulator transition observed in (MDT-TS)(AuI2)0.441.
r
t
s KEYWORDS: Mott insulator, incommensurate potential, organic conductors, metal-insulator transition
.
t
a
m
Mostofthe conducting molecularcrystalsarerealized the other hand, it is difficult to understand the strong-
- by combining two kinds of molecules, A and B, with couplingnature ofthe insulating groundstate in(MDT-
d
n commensurate composition ratios. Typical examples are TS)(AuI2)0.441, deduced from Tρ 6= TN, which is to be
o the well-studied 2:1 compounds, i.e., A2B, which show explored in this Letter; if it is the weak-coupling spin-
c a variety of phases such as Mott insulator, chargeorder, density-wave state due to the nesting of the Fermi sur-
[ superconductivity and so on.1,2 In the compounds, the face, Tρ =TN would be expected.1

1 molecule B is usually fully ionized either as −1 or +1 We consideraone-dimensional(1D)modelinorderto
v to form a closed shell, and as a consequence the energy capture the essence of the MI transition in a more con-
4
band formed by HOMO or LUMO of A is quarter-filled trolled way than considering a 2D model relevant to ex-
7
as a whole in terms of holes or electrons, respectively. periments.Our1DmodelconsistsofN donormolecules
2 L
1 Recently, molecular conductors with incommensurate coupled with N anions both forming regular lattices, as
0 (IC) composition ratios close to 2:1 have been synthe- shown in the inset of Fig. 1. The donors are modeled by
5 sized based on new donor molecules.3–5 (MDT-TSF)X the 1D extended Hubbard model, known to be relevant
n
0 and (MDT-ST)X (X = I , AuI or IBr , n ≃ 0.42 – for typical 2:1 systems,6,7 and the small potential from
/ n 3 2 2
t 0.45)showmetallicbehaviorandundergosuperconduct- the anions is added, which is crucial for the insulating
a
m ing transitionataboutTc = 4 Kat ambientpressure.3,4 state to appear. The Hamiltonian is written as follows,
In contrast, in (MDT-TS)(AuI ) a metal-insulator
d- (MI)crossoveroccurswherethe2te0m.4p41eraturedependence H = −t (c†j+1,scj,s+h.c.)+ U2 nj,snj,−s
n of resistivity displays a minimum at T = 85 K.5 In ad- j,s j,s
ρ X X
o dition, anantiferromagnetictransitiontakesplace atT
c = 50 K. When pressure is applied to this compound, TN + V njnj+1+ vjnj, (1)
: ρ
v decreases and the superconducting phase appears above Xj Xj
Xi Pc = 10.5 kbar (Tc = 3 K). All these compounds are where t, U and V are respectively the transfer energy
isostructural with alternating donor and anion layers. between the nearest-neighbordonor sites, the on-site re-
r
a Since the anions are fully ionized as X− as in the 2:1 pulsive interaction and the nearest-neighbor repulsion;
compounds,3 the electronic properties can be attributed the creation operator at the j-th site with spin s=± is
tothedonorswiththeICband-fillingslightlylargerthan denoted as c†j,s, nj,s = c†j,scj,s and nj = snj,s. Since
3/4 for the HOMO bands. The extended Hu¨ckel scheme the fully ionized anions form a regular lattice, the an-
P
predicts two-dimensional (2D) Fermi surfaces which are ion potential at the j-th site, vj, can be expressed as
similar to each other.4,5 It should be noticed that the vj = NL−1 ∞m=−∞v(mQ)eimQja where Q = 4kF, kF =
anionsinthesecompoundsarenotrandomlydistributed πn/(2a)is the Fermiwavenumber,n=N/NL is the car-
P
in the layers, but are found by the X-ray scattering ex- rierdensity(wetakethe holepicture inthe following)in
periments to form regular IC lattices with a different thedonorchainandaisthespacingbetweendonorsites.
periodicity from the donors.3,4 In the following, we consider only the relevant ±Q com-
Themetallicstateobservedinthesecompoundsisnat- ponent of the potential, v(±Q) ≡ v0e±iχ. This can lead
urally expected from the IC band-filling since the sys- toagap,2v0,at±2kFinthenon-interactingband,which
tem would avoid insulating states due to strong cor- weassumetobesmallcomparedtothebandwidth.Then
relation such as Mott insulator or charge order. On the system becomes effectively half-filled in reference to
the IC lattices as is seen in Fig.1.
∗E-mailaddress:[email protected]
2 J.Phys.Soc.Jpn. Letter HideoYoshioka,HitoshiSeoandHidetoshiFukuyama
Ua
g = −4δ( +Vacos2k a)
2 donor 3⊥ 2 F
anion
Ua
K)/t0 2v /t + 4δD2( 2 +Vacos4kFa)
0
(
ε ×(Ua+Vacos2k a+Vacos6k a), (6)
F F
−2 Ua
g′′ = −4δD ( +Vacos4k a)
3⊥ 2 2 F
−1 −k a/π 0 k a/π 1
F F
Ka/π ×(Ua+2Vacos2kFa), (7)
2
Fig. 1. Energy dispersion in the presence of the anion potential Ua Ua
g = +Va−2D +Vacos4k a , (8)
v0=0.1twheretheoccupiedone-particlestatesareexpressedby 4⊥ 2 2 2 F
the thick curve. The figureiswritten inthe case of n=0.33to (cid:18) (cid:19)
clarifythecharacteristicsofthepresentmodel.Inset:aschematic g = Va−2D (Vacos4k a)2, (9)
4|| 2 F
representationofourmodel.
X
g = , (10)
1/4 2(πα)2(ǫ(3k )−ǫ(k ))2
F F
WederiveaneffectiveHamiltonianforlowenergyscale
in terms of phase variables following the just quarter- X = 2 (2Vacos2kFa)2(Ua+2Vacos4kFa)
filling case.7 To the lowest order of the normalized an- n
+ (Ua+2Vacos2k a)2(Ua+4Vacos4k a)
ion potential δ ≡ v0/{ǫ(3kF)−ǫ(kF)} with ǫ(K) = F F
−2tcosKa,the phaseHamiltonianisobtainedasHeff = + (2Vacos2k a)(Ua+2Vacos2k a)
F F
H +H +H .HereH ,H andH arerespectivelythe
ρ σ ρσ ρ σ ρσ
chargepart, the spin part and the term mixing both de- ×(Ua+Vacos2k a+Vacos6k a)
F F
grees of freedom. The spin part, H has the same form
σ o
as that of the 1D Hubbard model, so the spin excita- + (Ua+2Vacos2kFa)2
tion becomes gapless.8 The term H is expressed by
ρσ ×(Vacos2k a+Vacos6k a), (11)
F F
the product of the non-linear terms seen in H and H ,
ρ σ
and then has a larger scaling dimension. Hence we can with
neglectit.9 Thereforethepropertiesofthechargedegree 1
D =
of freedom are determined by Hρ, expressed as 1 4πv(3kF)
v 1 ǫ(3k )−ǫ(k )+v(3k )k
H = ρ dx (∂ θ )2+K (∂ φ )2 ×ln F F F F, (12)
ρ x ρ ρ x ρ
4π K ǫ(3k )−ǫ(k )−v(3k )k
Z (cid:26) ρ (cid:27) F F F F
g 1 2k
+ 3⊥ dxcos(2θ +3χ) D = F , (13)
(πα)2 ρ 2 4πǫ(3k )−ǫ(k )
Z F F
g′′ where v(3k )=2tasin3k a.
+ 3⊥ dxcos(2θ +5χ−q x) F F
(πα)2 ρ 0 In eq.(2), there are three non-linear terms. First, the
Z
half-fillingUmklappterm,g ,isgeneratedbytheanion
g 3⊥
+ 1/4 dxcos(4θ +8χ−q x). (2) potential δ because the band is effectively half-filled as
2(πα)2 ρ 0
Z seen in Fig.1. This can lead to a Mott insulator, as we
Hereq =2π/a−8k =2π(1−2n)/aisthemisfitparam- will later show explicitly. We call the state as IC Mott
0 F
eter, v =2tasink a, v =v B A and K = B /A insulator since it has a periodicity not matching with
F ρ ρ ρ ρ ρ ρ
with A = 1+(g +g +g +g −g )/(πv) and the donorsbutwiththeanions.However,itisnottrivial
ρ 4|| 4⊥ p2|| 2⊥ 1|| p
B =1+(g +g −g −g +g )/(πv). α−1 is the whether this IC Mott insulator can be realized, and if
ρ 4|| 4⊥ 2|| 2⊥ 1||
ultra-violet cut-off (α ∼ a). The interaction parameters so, in which condition it is stabilized, in contrast to the
are written as half-filled Hubbard model where infinitesimal on-site re-
pulsion stabilizes the Mott insulator.8 This is because of
g = Vacos2k a
1|| F the presence of the other two non-linear terms in eq.(2),
− 4D (Vacos2k a)(Vacos4k a), (3) the“quarter-filling”Umklappterm,g ,withthemisfit
1 F F 1/4
which is present even without the anions owing to the
Ua
g2⊥ = 2 +Va−2D1 proximity to a quarter filling on one hand, and the g3′′⊥
term,the combinationofbothcommensurabilitiesofthe
2 2
Ua Ua donors and the anions on the other hand.
× +Vacos2k a + +Vacos4k a ,
2 F 2 F In the present calculation, it is crucial to fix the car-
((cid:18) (cid:19) (cid:18) (cid:19) )
riernumberatthevaluedeterminedbytheaniondensity.
(4) However,itistobenotedthatifoneevaluatesthequan-
tity based on eq.(2), it results in a deviation of the car-
g = Va
2||
rier number from the value in the non-interacting case,
2 2
− 2D (Vacos2k a) +(Vacos4k a) ,(5)
1 F F
n o
J.Phys.Soc.Jpn. Letter HideoYoshioka,HitoshiSeoandHidetoshiFukuyama 3
∆N =(1/Lπ) dxh∂ θ i, as 1 2
e x ρ U/t=3.0, V/t=0.0, n=0.441
∆Ne/L =R 2KπρGα′3′⊥2 Zα∞ dαr (cid:16)αr(cid:17)2−4KρJ1(q0r) Kρ 01/4(q −q
+4KρG21/4 ∞ dr r 2−16KρJ1(q0r)6=0, (14) δδ==00..000015 3" )α
πα α α
Zα (cid:16) (cid:17)
where G′3′⊥ =g3′′⊥/(πvρ), G1/4 =g1/4/(2πvρ), and Jn(x) 00 10 20−2
l
is the the Bessel function of the first kind. The origin of
the deviation is the existence of the misfit parameter.10 Fig. 2. ThesolutionsoftheRGequations,Kρand(q1/4−q3′′)α(
Therefore, we must add the term, −(µ/π) dx∂ θ and themisfitparameteroftheG3⊥-term)forU/t=3.0,V/t=0.0
x ρ
keep∆N to zero.To the lowestorder of G′′ and G , andn=0.441.Thecasesofδ=0.001and0.005aredenotedby
e R3⊥ 1/4 thesolidanddotted curves,respectively.
the chemical potential µ is given as,
∞ dr r
qµα = −4KρG′3′⊥2 Zα α (α)2−4KρJ1(q0α) −4KρG23⊥J˜1(q1/4α−q3′′α)
∞ dr r −4K G′′2J˜(q′′α), (20)
−8KρG21/4 α (α)2−16KρJ1(q0α),(15) ρ 3⊥ 1 3
Zα d
G = (2−2K )G
where qµ =4Kρµ/vρ. dl 3⊥ ρ 3⊥
To determine the low energy behavior of this effective −G′′ G J (|(q′′α+q α)/2|),(21)
Hamiltonian, we derive the renormalization group (RG) 3⊥ 1/4 0 3 1/4
equations by rewriting the action Sρ corresponding to d G′′ = (2−2K )G′′
the Hamiltonian, H −(µ/π) dx∂ θ , as dl 3⊥ ρ 3⊥
ρ x ρ
S = 1 d2~r (∂ θR˜ )2+(∂ θ˜ )2 −G3⊥G1/4J0(|q1/4α−q3′′α/2|), (22)
ρ x ρ y ρ
4GπKρ Z n o ddl G1/4 = (2−8Kρ)G1/4
+ 3⊥ d2~rcos 2θ˜ +3χ−(q −q′′)x
πα2 ρ 1/4 3 −G G′′ J (|q′′α−q α/2|), (23)
Z n o 3⊥ 3⊥ 0 3 1/4
+ Gπα′3′⊥2 d2~rcos 2θ˜ρ+5χ−q3′′x ddl qµα = qµα+4KρG23⊥J˜1((q1/4−q3′′)α)
Z n o
+ G1/4 d2~rcos 4θ˜ +8χ−q x , (16) +4KρG′3′⊥2J˜1(q3′′α)
πα2 ρ 1/4
Z n o +8KρG21/4J˜1(q1/4α). (24)
whereθ˜ =θ −q x/2,G =g /(πv ),q =q −2q
ρ ρ µ 3⊥ 3⊥ ρ 1/4 0 µ Eqs.(18)-(23) are obtained for the condition of the ac-
and q′′ = q −q . The condition, ∆N =0, leads to the
3 0 µ e tion, eq.(16), being invariant under RG transformation,
following self-consistent equation,
whereas eq.(24) is obtained from the condition of the
qµα = −4KρG23⊥ ∞ dαr(αr)2−4KρJ˜1((q1/4−q3′′)α) citheims ischaolwpnotetnhtaital,theqe.(q1u7a).nFtirtoymqe,qsw.(h1o9s)e,(d20im)eannsdio(n24i)s,
Zα 0
∞ dr r (length)−1, is scaled as q0(l)=q0el. This shows the fact
−4KρG′3′⊥2 α (α)2−4KρJ˜1(q3′′α) that the carriernumber is indeed conservedwithout any
Zα effects from the interaction.
−8KρG21/4 ∞ dαr(αr)2−16KρJ˜1(q1/4α), (17) forTUyp/itca=l fl3o.w0saonfdthVe/RtG=e0q.u0atwiohnesrearteheshcoawrnrieinr nFuigm.2-
Zα
ber is fixed as n = 0.441 taken from the actual mate-
where J˜(x) = sgn(x)J (|x|). Eqs.(16) and (17) lead to
1 1 rial (MDT-TS)(AuI ) . In the case of δ = 0.005, K
the following RG equations, 2 0.441 ρ
tendstozeroimplyingthatthegroundstateisaninsula-
d K = −8K2G2 J (|q α|) tor,duetothecommensurabilityinthe half-filling,G3⊥.
dl ρ ρ 1/4 0 1/4 ThiscanbeseenintheRGequationssincethemisfitpa-
−2Kρ2G23⊥J0(|q1/4α−q3′′α|) rameter in the G3⊥-term, (q1/4−q3′′)α=−qµα vanishes
(see Fig.2) and then G affects the renormalization of
3⊥
−2Kρ2G′3′⊥2J0(|q3′′α|), (18) Kρ through eq.(18) while those in the G1/4- and G′3′⊥-
terms, q α and q′′α, tend to ∞ and then these effects
d 1/4 3
dl q1/4α = q1/4α−16KρG21/4J˜1(q1/4α) become negligible due to the oscillating behavior of the
Bessel function. Hence the origin of this insulating state
−8KρG23⊥J˜1(q1/4α−q3′′α) is nothing but the commensurate potential of the effec-
tivehalf-fillinggeneratedbytheanionpotential.Namely,
−8K G′′2J˜(q′′α), (19)
ρ 3⊥ 1 3 the insulating state is indeed the IC Mott insulator.
d On the other hand, in the case of smaller potential
q′′α = q′′α−8K G2 J˜(q α)
dl 3 3 ρ 1/4 1 1/4 due to the anions, δ = 0.001, a metallic state with fi-
4 J.Phys.Soc.Jpn. Letter HideoYoshioka,HitoshiSeoandHidetoshiFukuyama
0.1 scheme provides transfer integrals, i.e., the band width,
n=0.441
of the MDT-TSF families larger than that of the MDT-
V/t=0.0
TS compound,4,5 which is consistent with our results
δ Incommensurate that the decrease of the bandwidth lead to an MI tran-
−Mott Insulator sition, as seen in Fig.3. In our 1D model the spin de-
gree of freedom is essentially that of the 1D Heisenberg
Metal
model showing no magnetic order. However, in the IC
0
0 1 2 3 Mott insulating state in the actual 2D material we gen-
U/t
erallyexpectthatantiferromagneticorderappearsatlow
Fig. 3. Thephasediagramontheplaneofδ andU/tinthecase
temperature due to the three dimensionality, as in fact
ofV/t=0andn=0.441.
observed.5 In this case, the magnetic ordered moment
should be large, compared to, e.g., that of the spin-
0.001 density-wave state due to the nesting of the Fermi sur-
n=0.441 face.
Metal
U/t=3.0
In conclusion, we investigated the electronic state of
the one-dimensional extended Hubbard model close to
δ quarter-fillingunderanincommensurateanionpotential.
Incommensurate
We found that a transition between the metallic state
−Mott Insulator
andanincommensurateMottinsulatorcanoccur,whose
0 originisthe interplaybetweenthe commensurabilityen-
0 1 2 3
V/t ergy generated by the anion potential and that in the
Fig. 4. Thephasediagramontheplaneofδ andV/tinthecase donor lattice. To the authors’best knowledge this is the
ofU/t=3.0andn=0.441. first theoretical study of a “Mott transition” generated
by such interplay between different commensurabilities.
It wouldbe interestingto investigatethe criticalproper-
nite value of K is realized. Here in eq.(18) the effects
ρ ties of this transitionin the actual compounds and com-
of the G -, G′′ - and G -terms on K disappear at
3⊥ 3⊥ 1/4 ρ pare with the “usual” Mott transition seen in the typ-
the low energy since all the misfit parameters are diver-
ical 2D A B molecular conductors, κ-(BEDT-TTF) X,
2 2
gent. This metallic state is not realizedif we set G1/4 to which is recently attracting interests.12
zero. Therefore we can state that the origin of the MI
The authors would like to thank T. Kawamoto for
transition is the interplay between the different kinds of
sending them his preprint prior to publication. They
commensurabilities.
also acknowledge G. Baskaran, M. Ogata, K. Kanoda,
Next,weshowgroundstatephasediagramsasafunc-
and J. Kishine for valuable discussions and comments.
tion of the model parameters. First, the phase diagram
This work was supported by Grant-in-Aid for Scien-
on the plane of U/t and δ in the case of V/t = 0 is
tific Researchon Priority Area of Molecular Conductors
shown in Fig.3. Since the quantity δ is proportional to
(No.15073213) and Grant-in-Aid for Scientific Research
v /t, the transition from the metallic state to the IC
0 (C) (No. 14540302and 15540343)from MEXT.
Mott insulator occurs when the potential from the an-
ions increases and/or the band width decreases. When
U → ∞, the present system can be mapped onto a 1) T.Ishiguro,K.YamajiandG.Saito:OrganicSuperconductors
non-interacting spinless Fermion system with the Fermi (Springer-Verlag,Berlin,1998)2nded.
wavenumber doubled, as 2k .11 In this case the insulat- 2) Forvariousrecentreviews,Chem.Rev.104(2004) No.11.
F
3) K.Takimiyaet al.,:Angew.Chem.Int.Ed.40(2001) 1122;
ing state is realized by an infinitesimal δ because a gap
Chem.Mat.15(2003) 1225;ibid.3250.
opens at ±2kF (see Fig.1), consistent with Fig.3. 4) T. Kawamoto et al., : Phys. Rev. B 65 (2002) 140508; ibid.
The role of the G1/4-term on the MI transition be- 67(2003)020508(R);Eur.Phys.J.B36(2003)161;J.Phys.
comes clearer when V is varied. It is because the cou- IVFrance114(2004) 517.
pling constant G changes its sign when V increases.7 5) T.Kawamotoet al.,:preprint.
1/4 6) H. Seo, C.Hotta and H.Fukuyama: Chem. Rev.104 (2004)
WeshowthephasediagramontheV/t-δ planeinFig.4
5005.
for U/t = 3.0. At V = V = 0.838t where G = 0, the
c 1/4 7) H. Yoshioka, M. Tsuchiizu and Y. Suzumura: J. Phys. Soc.
IC Mott insulating state is realized for infinitesimal δ. Jpn.69(2000) 651;ibid.70(2001)762.
For V > Vc, the absolute value of G1/4 increases again 8) V.J.Emery: inHighly Conducting One-Dimensional Solids,
and results in a finite metallic region. Therefore, a re- ed.J.Devresse,R.Evrard,andV.VanDoren(Plenum,New
York,1979),p.247.
entranttransition, metal → IC Mott insulator → metal,
9) Thespin-chargecoupledtermplaysanessentialrolewhenthe
occurs when V increases. Note that there is no quali-
coefficients ofthenon-lineartermsofbothdegreeoffreedom
tative difference between the metallic states in the two vanish(seeM.TsuchiizuandA.Furusaki:Phys.Rev.Lett.88
distinct regions. (2002) 056402), whichdoesnothappeninthepresentcase.
Finally let us discuss the relevance of our results to 10) M.Mori,H.FukuyamaandM.Imada:J.Phys.Soc.Jpn.63
(1994) 1639.
the experiments. The difference of the ground state in
11) A.A.Ovchinnikov:Sov.Phys.-JETP37(1973)176;M.Ogata
metallic MDT-TSF and MDT-ST compounds and that andH.Shiba:Phys.Rev.B41(1990)2326.
in the MDT-TS compoundundergoing MI crossovercan 12) S.Lefebvreetal.,:Phys.Rev.Lett.85(2000)5420;F.Kagawa
benaturallyunderstoodasfollows.TheextendedHu¨ckel et al.,:Phys.Rev.B69(2004)064511.