Dielectric response of modified Hubbard models with neutral-ionic and Peierls transitions

0.32 MB·English

by  Zoltan G. Soos

Dielectric response of modified Hubbard models with neutral-ionic and Peierls
transitions
Zoltan G. Soos, Sharon A. Bewick
Department of Chemistry, Princeton University, Princeton
Andrea Peri, Anna Painelli
4 Dip. di Chimica GIAF Universit`a di Parma, and INSTM-UdR Parma, 43100 Parma, Italy
0 (Dated: February 2, 2008)
0
2 The dipole P(F) of systems with periodic boundaryconditions (PBC) in a static electric field F
isappliedtoone-dimensionalPeierls-Hubbardmodelsfororganiccharge-transfer(CT)salts. Exact
n
results for P(F) are obtained for finite systems of N= 14 and 16 sites that are almost converged
a
toinfinitechainsin deformable lattices subject toaPeierls transition. Theelectronic polarizability
J persite, αel =(∂P/∂F)0, of rigid stacks with alternating transfer integrals t(1±δ) diverges at the
5 neutral-ionic transition for δ = 0 but remains finite for δ > 0 in dimerized chains. The Peierls or
1
dimerization mode couples to charge fluctuations along the stack and results in large vibrational

contributions, αvib, that are related to ∂P/∂δ and that peak sharply at the Peierls transition.
]
l The extension of P(F) to correlated electronic states yields the dielectric response κ of models
e with neutral-ionic or Peierls transitions, where κ peaks > 100 are found with parameters used
-
r previouslyforvariableionicityρandvibrationalspectraofCTsalts. Thecalculated κaccountsfor
t the dielectric response of CT salts based on substituted TTFs (tetrathiafulvalene) and substituted
s
. CAs (chloranil). The role of lattice stiffness appears clearly in models: soft systems have a Peierls
t
a instability atsmall ρandcontinuouscrossover tolargeρ,whilestiff stackssuchasTTF-CA havea
m first-order transition with discontinuous ρ that is both a neutral-ionic and Peierls transition. The
transitions are associated with tuning the electronic ground state of insulators via temperature or
-
d pressure in experiments, or via model parameters in calculations.
n
o
c I. INTRODUCTION polarizability of Peierls-Hubbard models for organic CT
[ salts and vary GS parameters to induce transitions. We

obtainexplicitly electronic andvibrationalcontributions
1 Torrance et al.1 reportedthe first neutral-ionic transi-
v to κ for correlated states. Systems with continuous ρ
tion(NIT)intheorganiccharge-transfer(CT)saltTTF-
1 at the κ peak and ρ < 1/2 throughout are of particu-
CA. Tetrathiafulvalene (TTF) is a potent π-electron
6 lar interest, since they do not fit the simple paradigm of
donor (D), while chloranil (CA) is a strong π-acceptor
2 neutral-regular or ionic-dimerized.
1 (A). As confirmed subsequently by detailed vibrational
Thedielectricconstant,κ,istheGSpolarizabilityα,or
0 and structural studies,2,3,4 the 300 K structure contains
induced dipole moment, per unit cell in the crystal. The
4 mixed regular stacks ..Dρ+Aρ−Dρ+Aρ−... with face-to-
moderntheoryofpolarizationininsulators12hasresolved
0 face overlap and ionicity ρ 0.3. Equal intermolecu-
/ ∼ the problem of a position or dipole operator in systems
at lar interactionwith each neighbor follows from inversion withperiodicboundaryconditions(PBC).Resta13 intro-
symmetryatthecentersofTTFandCA.Belowthefirst-
m duced the following expression for the expectation value
ordertransitionatT =81K,thedimerizedionicstacks
- have ρ 0.7 and altcernately short and long intermolec- of the dipole moment, P:
d ∼
n ular spacing. The ionic lattice is unstable to a transition 1 2πM 1
o that becomes a spin-Peierls transition5 in the limit ρ P = Imln ψ exp(i )ψ ImlnZ (1)
→ 2π h | N | i≡ 2π
c 1. In TTF-CA and other CT salts with discontinuous ρ
v: at Tc, the neutral-ionic and Peierls transitions coincide. Here ψ is the exact electronic GS, M is the conven-
| i
i Systems with neutral-ionic or valence transitions are tional dipole operator and PBC are ensured via super-
X
necessarily rare: two ground states (GS) with differ- cells. Polarization in insulators is related to the phase
r
a ent charge densities must be almost degenerate in or- ratherthanthe amplitude ofψ andcanbe formulatedas
der to switch between them by changing temperature or a Berry phase. P applies to correlated as well as non-
pressure. Horiuchi et al.6,7,8,9,10,11 have recently stud- interacting systems and provides a unified approach to
ied NITs in a series of substituted TTFs andCAs whose previous treatments of polarization and GS localization
static dielectric responses have large peaks (κ > 100). ininsulators.14 ThemagnitudeofZ isfiniteininsulators
Their interpretation is a quantum phase transition be- and zero in conductors. The phase or twist operator in
tween a neutral-regular and an ionic-dimerized GS con- Eq. (1) also appears in other contexts.15 We will work
trolled by an external parameter such as temperature or with P for the exact GS of modified Hubbard models
pressure.11 No modeling of κ is proposed, however, and with N sites.
no modeling is possible without the static polarizability Although a static electric field F is not compatible
ofextendedsystems. Inthispaperwecomputethestatic with PBC, Nunes and Gonze16 showed recently how to
2
combine F and P. They propose to define P(F) with II. STATIC ELECTRONIC POLARIZABILITIES
ψ(F) in Eq. (1) and to minimize a functional that, in a
one-dimensional system, is Quantum cell models typically have a single Wannier
orbital φ per site that, in principle, can incorporate
p
E(ψ ,F)=E(ψ ) NFP(ψ ) (2) intra-site correlations.23 On-site repulsion U > 0 is kept
F F F
−
inHubbardmodelsandlong-rangeCoulombinteractions
Severalgroups17,18,19,20 have applied Eq.(2) to the elec- inthePariser-Parr-Poplemodel.24TotalspinS isusually
conserved, as in models for CT salts, but this condition
tronic dielectric constants of metal oxides and silicates,
can be relaxed. The defining feature of cell models is a
using Wannier functions for ψ(F) and perturbation the-
large but finite many-electron basis in real space that is
ory to find corrections to ψ(0). As shown below, cor-
complete for a finite number oforbitals φ . Slater deter-
related models with finite N are simpler because the p
minants of φ ’s can be used in general for eigenstates of
basis is finite, which allows exact solution for ψ(F) in p
S , butlinearcombinationsofdeterminantsarerequired
Eq. (2). The procedure in Section 2 differs from the z
for the smaller basis when S is conserved. Easily visual-
model-exacttreatmentofstaticordynamicnonlinearop-
tical responses,21 however, since the dipole operator M ized valence bond (VB)25 diagrams are then convenient
and used below.
is now in the exponent.
The normalized singlet (S = 0) diagram k,q speci-
The optical phonon of the mixed regular stacks | i
fies the occupation number n = 0,1 or 2 at each site
..Dρ+Aρ−Dρ+Aρ−... describesarigiddisplacementofei- pk
p = 1,2,...N and the spin pairing of all singly-occupied
ther sublattice, and in the hypothesis of linear electron-
site. The index q is introduced below for the eigenvalue
phonon(e-ph)couplingleadstoalternatingCTintegrals:
M of the dipole operator. The VB basis is orthogonal
t(1+δ),t(1 δ). Weassigntheopticalphononaharmonic k
potential w−ith frequency ω at ρ = 0. With increasing withrespecttocharge,butthere is overlapSkk′ between
P diagrams with identical n and different spin pairing.
ρ, e-ph coupling generates large anharmonicities22 and pk
The operator M for N sites with uniform spacing a= 1
an overall softening of the vibrational frequency, which
and e=1 is
vanishes at the Peierls transition. A static electric field,
F, strongly affects the lattice degrees of freedom at in- N N
termediate ρ, especially near the Peierls transition. The M = pq = p(z n ) (4)
p p p
−
polarizabilitypersitehasthensubstantialelectronicand Xp=1 Xp=1
vibrational contributions,
Hereq =z n isthechargeoperator,withz =0atA
p p p p
−
andz =2atD,andtherearetwoelectronsperDAunit
dP ∂P ∂P ∂δ p
= + =α +α (3) cell. TheneutralGShasregularspacingaalongthestack
el vib
dF ∂F ∂δ ∂F
while the dimerized ionic GS has alternating spacing
a u. ForsimplicityweuseM inEq.(4)forbothregular
TheP derivativesaboveillustratethatphysicalprocesses ±
anddimerizedGS, thereby neglecting the motionof eρ
are associated with changes14 of P and underscore the ±
charges;correctionsfor u canreadilybe included.22 M
centralroleofP forthedielectricresponse. Inthe vibra- ±
is diagonal in the k,q basis. Its eigenvalues are the in-
tionalpart, ∂P isrelatedtotheIRintensityofthePeierls | i
∂δ tegers, modulo N, M = 0, 1, 2,.., N/2. We take
mode22 and ∂δ to the field-induced change in dimeriza- k ± ± −
∂F them as q = 2πMk/N for π q < π in the first Bril-
tion. Asanticipatedfromnon-interactingsystems,22αvib louin zone. The matrix elem−en≤t in Eq. (1) is then22
has a peak at the Peierls transition that is further en-
hanced by the frequency softening. The electronic part, 2πM
ψ expi ψ = W expiq Z +Z (5)
by contrast, diverges at the NIT of the rigid regular lat- h | N | i q ≡ x y
tice but varies slowly near the Peierls instability of de- Xq
formable lattices.
where W 0 is the weight of basis vectors k,q with
q
The paper is organized as follows. Section 2 presents q =2πM /≥N. Diagramswithq 0havegreat|estwieight
k
the exact solution of P(F) using Eq. (2) for quantum on the neutral side and lead to∼Z 1, while diagrams
x
cell models with a large but finite basis. The electronic with q π and Z 1 domina∼te on the ionic side.
x
polarizability of modified Hubbard models with fixed al- The reg∼ula±r chain has∼rea−l Z by symmetry and Z =0 at
ternation is obtained in Section 3. The roles of lattice the NIT.
vibrations are introducedin Section 4 and developedex- Since the VB basis is complete, the GS can be ex-
plicitly for the dimerization mode and its softening at panded as
the Peierls transition. Section 5 contains polarizability,
ionicity and related results for Peierls-Hubbard models ψ(F)= c (F)k,q (6)
kq
withequilibriumdimerization. TheκpeakatthePeierls Xk,q | i
transition is due to α and increases with the lattice
vib
stiffness. We discuss in Section 6 the applicability of The linear coefficient c are real and known at F = 0
kq
model results to dielectric peaks in organic CT salts. by hypothesis. Substitution of Eq. (6) into Eq. (2) and
3
varying the c leads to a secular determinant. As dis- operatorversionofM isoftenusedtoestimatetransition
kq
cussed elsewhere,25 sparse-matrix methods that exploit dipole moments for extended (PBC) systems and is con-
completenessaremuchmoreconvenientforlarge(>106) sistent with the real-space results.27 Susceptibilities ob-
basis sets. We consequently seek an operator whose ma- tained with the familiar sum-over-states (S0S) approach
trixelementscorrespondtoP(F)inEq.(2). Thepartial requireallexcitedstates,adifficulttaskforsystemswith
derivative of P(F) with respect to ck′q is largecorrelatedbasis,butcanberelatedtoGSresponses
withcorrectionvectors.21ThecalculationofP(F)andits
∂P(F) Z (F)sinq Z (F)cosq
∂ck′q =Xk,q ck,q(F)Sk′k(cid:26) x2π[Zx(F)−2+Zyy(F)2] (cid:27) derivatives only requires the GS.
(7)
We define the quantity in ... as µ and interpret it as III. ELECTRONIC POLARIZABILITY OF
kq
{ } MODIFIED HUBBARD MODELS
the induced dipole of diagram k,q . Since precisely the
| i
same result is obtained by introducing diagonal energies
NFµ for each diagram, the desired induced-dipole To illustrate the PBC polarizability of rigid lattices,
kq
−
operator is we consider Hubbard models that have been applied to
theelectronicstructureofbothorganicCTsalts28,29 and
N exp(i2πMˆ/N) transition-metal oxides30. Face-to-face π-overlap in CT
∆M(F)= Im (8)
2π Z(F) salts leads to transfer integrals t between sites with en-
ergy ∆atDand∆atA.ThemodifiedHubbardmodel
It follows immediately that |k,qi is an eigenfunction of is28 −
∆M witheigenvalueNµ ,asgivenbyEq.(7). Thelarge
kq
response or polarizability for small Z(F) is also evident.
H = [1 δ( 1)p](a† a +H.c.)
TheminimizationoftheenergyfunctionalinEq.(2)at Hu − − − p,σ p+1,σ
Xp,σ
finiteF isequivalenttofindingtheGSoftheHamiltonian
n (n 1)
H(F)=H F∆M(F) (9) + ∆( 1)pnp+U p p− (10)
(cid:20) − 2 (cid:21)
− Xp
Given the action of H(F) on any diagram, we obtain
ψ(F) as usual25 except for one complication. Since we We take t=1 as the unit of energy,consider alternation
know Z(0) from Eq. (1), but need Z(F) in Eq. (8), it- 0 < δ < 1 along the chain, and assume equal on-site
erations are needed. We start with ∆M(0) to obtain U > 0 for A and D sites. Coulomb or other interactions
Z(1)(F) and hence ∆M(1)(F), which is then used in can be added to HHu without increasing the correlated
Eq. (9) to find Z(2)(F) and ∆M(2)(F). We repeat un- basis. SuchmodelsconserveS,haveC symmetryand
N/2
til Z(F), ψ(F) and P(F) have converged. The GS with require comparable effort to obtain the exact GS.
∆M(0) requires the same computational effort as ψ(0) On physical grounds, high-energy D2+ and A2− sites
and suffices for the linear response, α = P′(0). More can be neglected to obtain a restricted basis that is use-
el
precisely,we use ∆M(0)inEq.(8)to find ψ(F) atsmall fully smaller. Formally, we define Γ=∆ U/2 and take
−
F, compute P(F) in Eq. (1) and evaluate P′(0). We the limit ∆,U in Eq. (10) while keeping finite Γ.
→ ∞
have ψ(F)∆M(0)ψ(F) =FP′(0)and,asseendirectly The electronic problem for rigid lattices simplifies to
h | | i
from Eq. (7), ψ(F)∆M(F)ψ(F) = 0 The interpre-
tation of Eq. (8h) as t|he induc|ed-diipole operator is now H0(δ,Γ) = − [1−δ(−1)p](a†p,σap+1,σ+H.c.)
clear: ∆M(0) generates the linear correction to P(F). Xp,σ
We note that the Berry-phase formulation of P(F) in + Γ( 1)pn (11)
p
Eq. (2) differs from standard expressions for finite sys- −
Xp
tems with open boundary conditions. In finite systems,
the interaction with a field is H FM instead of the The GS for Γ>>0 is the neutral (ρ 0) lattice with N
− ∼
PBC Hamiltonian in Eq. (9). The dipole operator M is electronspairedonN/2donors,whiletheGSforΓ<<0
diagonal in the VB basis, with Mk = ppqpk for dia- is the ionic lattice (ρ 1) with one electron per site and
∼
gram k,q . Now Φ(F) is the GS of HP FM and the overall singlet pairing. The regular (δ =0) stack has an
| i −
electronic polarization is Φ(F)M Φ(F) . In contrast NIT as a function of ∆ in Eq. (10) or Γ in Eq. (11) that
h | | i
to PBC, there is no iteration and M is not modulo N. has been studiedby differentmethods,29 including exact
k
The theoreticalideas and computational difficulties that GS energies and ρ up to N = 16 for Eq. (10) and up to
motivated the PBC formulation of P in Eq. (1) and its N = 22 for Eq. (11).
coupling to F in Eq. (2) do not arise in quantum cell In the present work, we take Ψ(0) in Eq. (6) to be
models,whoselimitedbasisis unphysicalinthis respect. the exact GS of H (δ,Γ). P(0) in Eq. (1) is the relevant
0
But extrapolations of linear and nonlinear optical coef- GS dipole per site in units of ea. Then, using ∆M(0)
ficients to infinite systems have been widely discussed26 in Eq. (8), we obtain Ψ(F) exactly, compute Z(F) and
for both noninteracting (Hu¨ckel) and correlated (Hub- P(F), and evaluate α = P′(0) numerically. Figure 1a
el
bard, Pariser-Parr-Pople) models, especially in connec- andbshow,respectively,α andZ(F)forδ =0stacksof
el
tion with frequency-dependent responses. The velocity N = 14 (circles) and 16 (crosses). To understand these
4
FIG. 2: Logarithm of the linear electronic susceptibility, αel,
vs Γ for a 16 site stack with δ =0.02, 0.05, 0.1, 0.4 and 0.99.
Thearrowsmarkthedirectionofincreasingδ,andthedashed
line for δ= = 0 diverges at theNIT.
independent dimers,
FIG. 1: The logarithm of the linear electronic susceptibility, αel =(Γ2+8)−23 (12)
αel,andthemodulusofZvsΓcalculatedfortheHamiltonian
in Eq. (11) with δ = 0 for stacks of 14 and 16 sites (circles whenthereisnoelectronicdelocalization. Thelargevari-
and crosses, respectively). The arrows mark the NIT of the ation of α (δ,Γ) in Fig. 2 is understood in terms of lo-
el
infinitestrand. calizationaroundΓ 0 due to the dimerizationgapand
∼
for Γ >> 0 due to charge localization for any δ. The
| |
pronouncedasymmetryofα (δ,Γ)withrespecttoΓ=0
el
is due to strong correlations in H (δ,Γ) leading to quite
results, we note that regular stacks have real Z(0) by 0
different GS on the neutral and ionic sides.
symmetry and hence P(0) = 0 for any N. We have
WeillustrateinFig. 3theiterativesolutionofEq.(9).
Z (0) 1 at large positive Γ where neutral diagrams
x ∼ Circlesrefertothefirstiterationwith∆M(0)forN=16
with q 0 dominate in Eq. (5) and Z (0) 1 at
∼ x ∼ − andδ =0.1. Crossesandsquaresreferto the secondand
largenegativeΓwhereionicdiagramswithq π dom-
inate. Z (0) vanishes at some Γ 0, as see∼n±for both third iteration, respectively. The results for αel = P′(0)
x ∼ donotchange,sincethelinearperturbationinF appears
N = 14 and 16. The Jahn-Teller instability of 4n rings
fromtheoutsetanditerationsintroducehigher-ordercor-
appears as a symmetry change of Ψ(0)29,31 where Z(0)
rections. Finite δ is requiredforfiniteP′′(0)andF2 cor-
vanishes and changes discontinuously in Fig. 1b. The
rections must be included. The second iteration more
arrow at Γ = 0.666 marks the NIT of the extended
c − than doubles P′′(0), but the third iteration produces no
regular stack as found from extrapolations of symme-
additional change. Taylor expansion of ∆M(F) through
try crossovers.29 Finite-size effects in Z(0) are reduced
the linear term suffices for F2 contributions. Nonlinear
considerably in Nln(Z(0)2), which is the proper size-
independentqua−ntity.14| The|α resultsinFig. 1anearly polarizabilities based on the PBC formulationin Eq. (2)
el or (9) raise interesting issues for rigid lattices. Here we
coincide for N = 14 and 16 except around Z 0. In
∼ focus instead on vibronic contributions in Eq. (3) to the
Section 5 we report almost identical α for N = 14 and
el linear polarizability of models with Peierls transitions.
16 over the entire Γ range for stacks with equilibrium
dimerization and Z =0.
6
Figure2showsα (δ,Γ)forN=16. Finitealternation, IV. POLARIZABILITY NEAR THE PEIERLS
el
δ >0, strongly suppresses the divergence at the NIT, as TRANSITION
seen on comparing δ = 0.02 and 0 (dashed line), and
shifts the peak to Γ=0 at δ = 1. Regular stacks with a TheSu-Schrieffer-Heeger(SSH)model32forpolyacety-
finite potential against dimerization have finite polariz- lene,(CH) ,describesintheadiabaticapproximationthe
x
ability for Γ Γ >Γ . The polarizability of dimerized Peierls instability of Hu¨ckel chains with U = ∆ = 0 in
P c
≥
stacks with Γ < Γ remains finite because δ > 0 opens Eq. (10) and linear e-ph coupling α = tδ/u. Simi-
P e−ph
a gap in the energy spectrum. The δ = 0.99 curve fol- lar approaches have been applied to Peierls transitions
lows closely the simple analytical result for δ = 1 and in segregated (∆ = 0) stacks of π-electron donors or
5
FIG. 4: Equilibrium dimerization, δ, softened frequency ΩP
inunitsofthereferencefrequencyωP,andinfrared oscillator
FIG. 3: The linear electronic susceptibility, αel, and the sec- strength, calculated for Eq. (13) with ǫd = 0.28 and 0.64
ondderivativeoftheGSpolarizationontheelectricfield,P′′, (left and right panels, respectively.) Circles refer to N = 14,
calculated as a function of Γ for a 16 site stack with δ=0.1. crosses to N = 16.
Circle,squaresandcrossesrefertoresultsobtainedatthefirst
second and third iteration, respectively (see text).
forastackofdecoupleddimerswhoseGSiseasilyfound.
The top panel of Fig. 4 shows the equilibrium dimeriza-
acceptors,33ortouncorrelated(U =0)chains34withsite tion of H (δ,Γ) at F =0 for ǫ = 0.28 and 0.64 in units
0 d
energies ∆. The stability of mixed stacks described by of t. Since results for N = 14 (crosses) and 16 (circles)
±
H0hasbeenstudiedthroughtheresponseofaregularlat- now coincide almost exactly, we conclude that finite-size
tice to the SSH coupling.31 In the adiabatic approxima- effectshavebecomenegligibleindeformablelatticeswith
tion,theGSpotentialenergysurfaceisdefinedbyadding δ > 0.1 around Γ 0 that precludes strong delocaliza-
the elastic energy for lattice motion to the electronic en- tion. ∼
ergy. Forharmonicpotentialsandlinearelectron-phonon
Previousmodeling of TTF-CA used the estimate ǫ
d
coupling,the elastic energyofthe Peierlsmode is δ2/2ǫd 0.15 0.3.31,35 The ǫ = 0.64 stack is less stiff, wit∼h
d
persite,whereǫ =α2 /kisthesmallpolaronbinding −
d e−ph a three-fold increase in the maximum dimerization to
energyandk isthelattice forceconstant.22 Wetherefore δ 0.3 around Γ 0. The Peierls instability occurs
eq
define the Peierls-Hubbard model by adding the lattice at (∼∂2 /∂δ2) = 0∼, where the curvature 1/ǫ of the
T 0 d
E
energy to H0(δ,Γ) in Eq. (11). The electronic energy in lattice potential and the curvature of the electronic en-
anapplied fieldis givenby Eq.(2). The totalGS energy ergy cancel exactly. The stack is regular for Γ > Γ (ǫ )
P d
per site is and dimerized for Γ < Γ (ǫ ). As expected on general
P d
grounds, the neutral lattice with Γ >> 0 is a band in-
δ2
= (δ,Γ,F)+ (13) sulator that is conditionally stable. The divergence of
T 0
E E 2ǫd (∂2 /∂δ2) at Γ ensures a Peierls instability at some
0 0 c
E
The same development holds for arbitrary U,∆ in Γ > Γ that is model dependent. Dimerization de-
P c
Eq. (10) or for other cell models with PBC. creases but persists as Γ becomes more negative.31 The
Eq. (13) fully defines the GS potential energy surface Γ<<0 limit with ρ 1 maps into the spin-1/2 Heisen-
with one additional parameter, the lattice stiffness 1/ǫ , berg antiferromagnet∼ic chain36 whose GS is unstable to
d
besides those entering the electronic Hamiltonian. The a spin-Peierls transition.5
equilibriumdimerizationisobtainedbyminimizingofthe ThecurvatureoftheGSpotentialenergysurfaceisthe
total energy, frequency of the Peierls mode Ω2 = (∂2 /∂δ2) . At
P ET eq
F =0theratioΩ /ω attheequilibriumδ isafunction
∂ (δ,Γ,F) P P
δ = ǫ E0 (14) of Γ:
eq d
− ∂δ
The limiting inversestiffness is ǫd =√2, which yields an ΩP 2 =1+ǫ ∂2E0(δ,Γ) (15)
equilibrium dimerization in Eq. (14) of δ = 1 at Γ = 0 (cid:18)ω (cid:19) d(cid:18) ∂δ2 (cid:19)
P eq
6
ThemiddlepanelofFig. 4showstheevolutionofΩ /ω
P P
for the correlated model H (δ,Γ) in Eq. (11). The soft-
0
eningofthePeierlsmodeontheneutralsideanditssub-
sequent hardening are observed for TTF-QBrCl whose
3
continuous dimerizationtransition has been investigated
by far-IR spectroscopy.10 The temperature evolution of
a combinationband in the mid-IR spectrum of TTF-CA
for T > T , also supports the presence of a soft lattice
c
mode.37 Thedataareinbothcasesconsistentwithω
P
100 cm−1, a typical frequency for lattice vibrations i∼n
crystals with molecular masses of 100 AMU. FIG.5: Electronicandvibrationalcontributionstothestatic
∼ polarizability (dotted and continuous lines, respectively), for
The Peierls mode is IR active by symmetry in mixed
N = 16 and the same parameters as in Fig. 4. The three
stacks andborrowshuge IR intensity from electronic de-
greesoffreedom.22Inthepresentapproximation,withM linesforαvib have,fromtoptobottom,forγ/ωP =0.01, 0.1,
0.2.
in Eq. (4) for fixed sites, the IR intensity of the Peierls
mode is entirelydue to chargefluctuations induced byδ.
The corresponding oscillator strength is:
The small effective mass associated with dimerization
at the Peierls transition implies strong mixing of elec-
2m Ω µ 2 m a2ω2ǫ ∂P 2
f = e P| IR| = e P d (16) tronic and vibrational degrees of freedom. Lattice con-
IR e2¯h t (cid:18)∂δ (cid:19) tributions to the polarizabilitycanbe substantial,as an-
eq
ticipated for α in Eq. (3). To demonstrate this, we
vib
Heremeandearetheelectronicmassandcharge,respec- differentiatebothsidesofEq.(14)withrespecttoF and
tively,|µIR|2isthesquaredtransitiondipolepersite,and use P =−(∂E0/∂F) to obtain
ǫ isinunitsoft. OnceagaintheIRintensityisgoverned
d
by P-derivatives and ǫd. However ωP, a and t enter the ∂δeq ǫd ∂P
= (17)
expressionforthedimensionlessfIR. Thebottompanels ∂F 1+ǫ (∂2 /∂δ2) ∂δ
of Fig. 4 show f for typical parameters,35 a = 3.7 ˚A, d E0 eq
IR
ω = 100 cm−1, t = 0.2 eV.
P WethenuseEq.(15)towritethevibrationalpolarizabil-
The intensity of the Peierls mode has a pronounced
ity of the Peierls mode at δ :
eq
peak at Γ where the electronic charges are maximally
P
mobile.22 Dimerization localizes charges and lowers f
IR ω 2 ∂P 2 ǫ ω2 ∂P 2
for Γ < ΓP. We have fIR = 0 when the electronic flux α =ǫ P d P (18)
induced by dimerization reverses from D A on the vib d(cid:18)ΩP(cid:19) (cid:18)∂δ (cid:19) → Ω2P +γ2 (cid:18)∂δ (cid:19)
→
neutral side to A D on the ionic side. The neutral-
→
ionic interface of deformable lattices can be identified as This expressionis exact when the vibrationalkinetic en-
f = 0, which occurs near Γ = Γ for regular stacks or ergyis neglected, which is a goodapproximationinview
IR c
Z (δ,Γ)= 0 in dimerized stacks.22 The actual IR inten- of the low frequencies involved.39 It is equivalent to the
x
sity does not vanish due to the motion of the molecular sum-over-statesexpression,39αvib =2µIR 2/¯hΩP,based
| |
siteswithcharges eρ,butfrozen chargecontributions22 on IR transition moments and energies.
±
areneglectedinEq.(4). Inmaterialsthatdimerizeinthe Large α is expected for the dimerization mode due
vib
neutralside, the N-I interface could, in principle, appear to its large transition dipole and low frequency. Indeed,
experimentally as a dip in fIR. However, this dip does ΩP = 0 at the Peierls transition gives a divergent po-
not mark a phase transition. There is no change in sym- larizability. We suppress the divergence by introducing
metry andallpropertiesofthe systemvary continuously damping γ thatrepresentsthe lifetimes oflattice modes,
at the interface.31,38 anharmonic potentials, etc. Damping is introduced em-
The IR intensity of the stiff (ǫ = 0.28) stack in Fig. pirically by changing (Ω /ω )2 as shown in the second
d P P
4 is an order of magnitude larger than that of the ǫ = equality in Eq. (18). Figure 5 reports α for the ǫ =
d vib d
0.64 stack. While contrary to the ǫ factor in Eq. (16), 0.28and0.64systemsshowninFig. 4fordampingγ/ω
d P
this is readily understood in terms of greater delocaliza- = 0.01, 0.1 and 0.2. The dotted line is the electronic
tion in stiff lattices whose Γ approaches Γ and whose polarizability P′(0) of the deformable lattice. The indi-
P c
∂P/∂δ diverges in the limit ǫ 0. At the Peierls tran- cateddampingcorrespondstoγ =1,10and20cm−1 for
d
sition, the large oscillator stre→ngth of the dimerization an estimated ω = 100 cm−1. Quite predictably, damp-
P
mode corresponds to effective masses m∗ = m /f ing affects α in a narrowregionabout Γ and is more
e IR vib P
∼
1000 and 6000 for ǫ = 0.28 and 0.64, respectively. So pronounced in the softer lattice. We note that typical
d
m∗ atthetransitionisroughlyaprotonmassandisfully bandwidths of lattice modes in molecular crystals are in
two orders of magnitude smaller than molecular masses. the 1-10 cm−1 range. We will use γ = 0.1ω to model
P
This justifies a posteriori the neglect of frozen charge the dielectric response of CT salts and cannot specify
contributions to f near the Peierls transition.22 α peaks more accurately than shown in Fig. 5.
IR vib
7
FIG. 6: The ionicity, ρ, and thedielectric response, κ, calcu-
lated for the same parameters as in Fig. 4. Circles refer to
N = 14, crosses to N = 16. Dotted lines in the bottom pan-
els show the dielectric response obtained by neglecting the
vibrational contribution to α.
FIG. 7: The equilibrium dimerization amplitude, δ, the GS
V. DIELECTRIC ANOMALY OF ionicity,ρ,andthedielectricresponse,κ,forthesameparam-
PEIERLS-HUBBARD MODELS eters as in Fig. 4 except for V =2. Circles and crosses refer
to N =14 and 16, respectively, with continuous and dashed
We now combine the electronic and vibrational polar- linesjoiningexactandmfresults. Dottedlinesinthebottom
izabilities of the Peierls-Hubbard model in Eq. (11) to panels are theelectronic mf dielectric response for N =16.
obtain the dielectric constant of the equilibrium lattice.
In SI units,
not the case for the ionicity ρ shown in the upper panel
α +α ofFig. 6. We properlyhaveρ 0.40atthe Peierlstran-
κ=κ + el vib (19) ∼
∞ sition of the ǫ = 0.28 stack and ρ 0.25 in the softer
ǫ v d
0 ∼
stack, but ρ(Γ) hardly shows any sign of a transition.
Here ǫ0 is the vacuum permittivity constant, κ∞ 3 is By contrast,the measuredionicity has a kink in systems
∼
theusualcontributionfrommolecularexcitedstatesthat withcontinuoustransitionsaroundρ 0.25.6,7,8,9,10 The
∼
are not being modeled, and v is the volume per site. We basic model H (δ,Γ) in Eq. (11) is deficient in this re-
0
are interested in remarkably large peaks with κ > 100. spect, just as its continuous NIT in rigid lattices can-
To get proper dimensions for κ the polarizabilities (αel not account for discontinuous ρ at Tc 81 K in TTF-
and αvib) are multiplied by e2a2/t. We adopt typical CA. Several extensions of the model ha∼ve long been rec-
TTF-CA values, v = 206˚A3, a = 3.7 ˚A, and t = 0.2 eV, ognized. Coulomb interactions occur both within and
and in the lower panels of Fig. 6 we show the resulting between stacks.28,40 Coupling to molecular modes is an-
κ=3+60(αel+αvib)forN =14and16stackswithequi- othergeneralphenomenon.31Inpractice,mean-field(mf)
librium dimerization and ǫd= 0.28 and 0.64. The dotted approximationsarenecessaryforinter-stackinteractions.
lines are the electronic contribution to κ. The dielectric Whileobservingaρanomalyatthe Peierlstransitionre-
peak is clearly associatedwith the Peierlstransition and quiresgoingbeyondH (δ,Γ),theappropriateextensions
0
vibrations. The divergence of αel in Fig. 1 at the NIT and parameters remain open.
(Γc =-0.666)oftheδ =0stackisstronglyattenuatedby To illustrate a simple extension that is suitable for ei-
dimerizationand appears as a small bump aroundΓP in ther exact or mf analysis, we introduce nearest-neighbor
Fig. 6. The electronic fluxes leading to the IR intensity Coulomb interactions V along the stack,
of the Peierls mode are also responsible for the large κ
peaksof 1000and100estimatedforǫ =0.28and0.64, δ2
∼ d HV =H0(δ,Γ)+V qpqp+1+N (20)
respectively. We note that neglecting the softening, i.e. 2ǫ
Xp d
by imposing Ω = ω in Eq. (18), lowers the κ peak by
P P
anorderofmagnitude,whileneglectingdamping(γ =0) Since V is diagonal in the VB basis, all GS properties
gives a divergence. We took γ = ω /10 = 10 cm−1 in are found as before. The solid lines in Fig. 7 are exact
P
Fig. 6. N = 14 and 16 results for the equilibrium dimerization,
Both the height and shape of the κ peaks as a func- ionicity anddielectric anomalyofstackswith V = 2 and
tion of temperature compare favorably to the dielectric otherwise the same parameters as in Figs. 4 and 6. The
data, discussed below, of Horiuchi et al.6,7,8,9,10 That is dashed lines are the mf approximation to the V term in
8
FIG. 9: Mean-field results for the softening of the Peierls
modeanditsIRintensity,calculatedforthesameparameters
as in Fig. 8. Circles refer to N = 14, crosses to N = 16.
effectsofthiscouplingonvibrationalspectraofCT salts
FIG.8: ThesameasFig. 7,forV =3; forthesakeofclarity are well known.42,43,44 In regular stacks, ts vibrations
onlymfresultsarereported,withcirclesandcrossesreferring only appear in Raman spectra, whereas in dimerized
to N = 14, and 16, respectively. Dotted lines in the lowest stacks they acquire large IR intensity, proportional to
panels show the electronic contribution to κ. (∂P/∂Γ)2, through their coupling to electronic degrees
of freedom.45 In the adiabatic approximation, Holstein
couplingplaysexactlythe sameroleasV withinmf.31,45
Eq. (20). The mf results for V = 2 are similar, with ThusmfresultsinFigs. 7and8representaneffectivepa-
somewhat sharper ρ at ΓP as expected for an interac- rameterV whose interpretationis model dependent. Al-
tion that stabilizes adjacent ion-pairs. Both ǫd = 0.28 thoughthelargeIRintensityofHolsteinmodesindimer-
and 0.64 now have a kink at ρ(ΓP) that qualitatively re- izedlatticessuggeststhatalsotheycontributetoκ,their
sembles ρ(T) data. The κ peak is lower by a factor of high frequency ( 1000 cm−1) makes this contribution
∼
two, while the maximum dimerization hardly changes. negligible in comparisonto the Peierls modes.
Charges are kept together for V > 0, thereby reducing
the polarizability.
The first-order transition of TTF-CA at 81 K has VI. DISCUSSION
a ∆ρ 0.2 and concomitant neutral-ionic and Peierls
transit∼ions.2,3,4 The ionicity is a suitable order parame- The dipole P for insulators with PBC and its ex-
terinthiscase. Discontinuousρhaslongbeentreatedin tension in Eq. (2) to an applied field, make it possible
mf theory,28 and mf results for V= 3 are shown in Fig. to model the electronic and vibrational polarizability of
8 for the same quantities and parameters as in Fig. 7. quantumcellmodels. Wefoundtheinduced-dipoleoper-
Now ǫd = 0.28 produces a ∆ρ jump in stacks of N = ator∆M inSection2forquantumcellmodelsingeneral
14 and 16 that resemble previous results for smaller N and applied it to a modified Hubbard model, H (δ,Γ)
0
up to 12.31 The kink in ρ for ǫd = 0.64 is stronger and in Eq. (11). We report model-exact electronic polariz-
still larger V will make it discontinuous. The maximum abilities in Fig. 2 for the NIT of regular (δ = 0) and
dimerization remains almost independent of V. The κ dimerizedstacks. WethenincludethePeierlsmodewith
peak now appears at the NIT and is reduced for both frequencyω intheadiabaticapproximationanditssoft-
P
ǫd = 0.28 and 0.64. The reason for reduced vibrational ening. The IR intensity of the Peierls mode in Fig. 4
contributions is that a discontinuous NIT interrupts the and the dielectric peaks in Figs. 6 and 7 are related to
softening of ωP. This is shown in Fig. 9 for the same Peierls transitions in stacks with stiffness 1/ǫd. We have
parameters, together with fIR. chosenǫd =0.28asaplausibleestimateforTTF-CAand
Figure 7 contrasts exact and mf results for small V = a softer stack with ǫ = 0.64 whose Peierls transition is
d
2, while Fig. 8 shows a first-order mf transition for V = around ρ 0.3 on the neutral side. The choice of t =
∼
3 in the stiff stack. To summarize the various roles of V 0.2eVcomesfromTTF-CAopticaldata. Themodelpa-
in Eq. (20), we note that in mf it also accounts for the rameters are not arbitrary,but typical of values used for
coupling of electrons with molecular vibrations. Totally- otherproperties.31,35 Thesameanalysisholdsforvarious
symmetric(ts)molecularvibrationsmodulateon-siteen- extensions of H (δ,Γ) for organic CT salts. All results
0
ergies, and hence Γ, leading to Holstein coupling.41 The are derived from the Γ-dependence of the GS properties
9
of stacks described by H (δ,Γ), and at the equilibrium of the Peierls mode around the Peierls transition22 and
0
dimerization for fixed ǫ in Eq. (13) and fixed nearest- its softening are responsible for α >> α in Eq. (3).
d vib el
neighbor interactions V in Eq. (20). Available spectroscopic data10 in the far-IR region for
The dielectric data of Horiuchi et al. are taken as a TTF-QBrCl3 support this picture.22 It is interesting to
function of temperature at ambient pressure,6,7,8,9,10 or, compare the Peierls transition occurring in this system
morerecently,asafunctionofpressure,upto 10kbar, atρ 0.3withthediscontinuousNIToccurringinTTF-
at constant temperature.11 Decreasing volum∼e on cool- CA a∼t similar ρ: in TTF-CA the NIT occurs before the
ing or compression stabilizes the ionic phase by increas- complete softening of the Peierls mode and a reduced κ-
ing the crystal’s Madelung energy. Cooling corresponds peak is observed. Whereas far-IR data are not available
to decreasingΓ andincreasing ρ in models, but the rela- for TTF-CA, the incipient softening of the dimerization
tionbetweentemperatureorpressureandΓisnotknown mode has been recently extracted from a detailed study
at present. Cooling also increases DA overlaps that en- of combination bands in the mid-IR region.37 A more
ter in t and ǫ for one-dimensional models. Overlap be- systematicanalysisoffar-IRspectraanddielectricprop-
d
tweenstacksalsoincreases,butalthoughtheimportance erties of mixed stack materials is certainly desirable as
of inter-stack interactions has been noted,4 their model- it will confirm the important connection between lattice
ing is still rather qualitative. degrees of freedom and materials properties.
The crystals studied by Horiuchi et al. in refs. 9, 11 Organic CT crystals have DA repeat units along the
have D = DMTTF (dimethyl-TTF) and the quinones A stack and, in contrast to most inorganic salts, are quasi-
= QBrnCl4−n with n = 0-4; n = 0 and 4 are CA and one-dimensional systems. Rapid convergence, often as
bromanil (BA), respectively. The acceptors have two- 1/N2, is typical for GS properties of systems with PBC
fold disorder in Br/Cl in the crystal lattice for n = 1, 2 and small repeat units. It is very convenient for model-
and3. TherearetwoQBr2Cl2 isomers,a centrosymmet- ing that P(F) is a GS property. As found throughout,
riconecalled2,5andapolaronecalled2,6. Then=1,3 all N = 14 and N = 16 results coincide in deformable
and2,6complexesareformallypolar,butonlyweaklyso: lattices with δ > 0.1 at Γ 0. Larger N is not required
Theirdielectricpeaksaresimilarbutareshiftedtolower for CT salts, but is needed∼for the NIT of δ =0 stacks.
temperature. As there is no disorder in the prototypical
system, TTF-CA, the best candidates with continuous Discontinuous ρ at the NIT is fundamentally differ-
transitions are DMTTF-CA and DMTTF-2,5QBr Cl . ent from continuous ρ at the Peierls transition of softer
2 2
The dielectric peak of DMTTF-CA at 65 K is κ 220 lattices. We havefocusedonthePeierls-Hubbardmodel,
at 30 kHz; ρ increases quickly from 0.32 to ∼0.42 H0(δ,Γ)inEq.(11),whichhascontinuousρandaPeierls
between 65 and 60 K and then slowly∼to 0.48∼at 10 instability on the neutral side for large ǫd. The order
K. The DMTTF-2,5QBr Cl peak at 50 K∼is κ 170 at parameter δ for the second-order transition breaks the
2 2
30 kHz, with ρ increasing from 0.29 to 0.36 on∼cooling reflectionsymmetryof the regularchain. Oncethe stack
from50to10K.TheκpeakofDMTTF-QBrCl at 55 hasdimerized, there isno further transitionand, indeed,
3
K closely resembles the 2,5 system, but ρ has not ∼been hardlyanyremnantsoftheprominentneutral-ionictran-
reported. Pressure-induced κ peaks are reported11 at 5 sition of the δ =0 stack. Models with continuous ρ have
K in DMTTF-BA (n = 4) and in DMTTF-2,6QBr Cl , Peierls rather than neutral-ionic transitions. Similar re-
2 2
both with ρ< 0.3 at 5 K and 1 atm. sults are obtained for stacks with not too large V-like
interactions: The Peierls or structural transition occurs
The observedpeak heights of κ are consistentwith di-
at small ρ, and V promotes a more prominent kink in ρ.
electric peaks of the model at the Peierls transition. We
Increasingfurther V (or Holstein)interactionsleads to a
conclude that a κ peak marks the Peierls transition in
discontinuous NIT, where the on-site charge reorganiza-
systems with continuous ρ. The narrowerκ peak in Fig.
tionisthedrivingevent,ρistheproperorderparameter,
8formodelparametersleadingtoafirst-ordertransition
is also seen in the narrow κ = 600 peak of TTF-CA.8 and dimerization just follows from the unconditional in-
stability of the ionic phase.
Thepeakshapesandheightsfollowqualitativelythecal-
culatedpatternofincreasedpolarizabilityinstifflattices The increasing number of CT salts undergoing phase
with Peierls transitions (continuous ρ) occurring at ΓP transitions demonstrates46 far more diversity than the
closetoΓc. Morequantitativeanalysisrequiresarelation neutral-regularandionic-dimerizedGS ofTTF-CA.The
between temperature and Γ. As illustrated in Section lattice stiffness 1/ǫ is an important parameter for dis-
d
5, accurate modeling of ρ(Γ) near ΓP depends on such tinguishingbetweenPeierlsandneutral-ionictransitions.
specifics as damping and intra and inter-stack interac- Disordered D or A in single crystals raises different is-
tions. By contrast, the Peierls transition of deformable sues. ThemicroscopicmodelingofCTsaltshasprimarily
stacks and NIT of rigid stacks are general features of been in terms of Hubbard-type models with many open
Hubbard and related models. questions about parameters. We anticipate that accu-
The dielectric anomaly observed in mixed stack CT rate modeling of P(F) and its applications to dielectric,
salts is clearly associated with lattice degrees of free- vibrational and other data will lead to more quantita-
dom and the Peierls transition in systems with either tive descriptions of quantum phase transitions in these
continuous or discontinuous NIT. The huge IR intensity materials.
10
Acknowledgments in Parma was partly supported by the Italian Ministry
of Education (MIUR) through COFIN-2001, and by IN-
Z.S. thanks R.Resta for fruitful discussions, and A.P. STM through PRISMA-2002.
likewise thanks S. Ramasesha and F. Terenziani. Work
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