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Quasi-One-Dimensional Organic Metals: Theory and Experiment
L. Gor’Kov
To cite this version:
L. Gor’Kov. Quasi-One-Dimensional Organic Metals: Theory and Experiment. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.1697-1710. �10.1051/jp1:1996183�. �jpa-00247275�
J. Phys. I France 6 (1996) 1697-1710 DECEMBER1996, PAGE1697
Quasi.One.Dimensional Organic Metals: Theory and Experiment
L.P. Gor'kov (*)
National High Magnetic Field Laboratory, Florida State University, Tallahassee,
Florida 32306-4005, USA and
L.D.Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117334 Moscow,
Russia
(Receii.ed 25 June 1996, received in final form 5 August 1996, accepted 13 August 1996)
PACS.71.20.Rv Polymers and organic compounds
PACS.71.27.+a Strongly correlated electron systems; heavy fermions PACS.15.30.Fv Spin-density waves
Abstract. Given the success of the weak coupling nesting model in explaining thermody-
namical properties of the Bechgaard salts at low temperatures and in magnetic fields, we first
concentrate on its implications to kinetics in the metallic phase. The model results in nonuni-
versai temperature dependencies of resistivity and magnetoresistance due to proximity of the metallic and the spin density wave phases, which are in a qualitative agreement with the avail- able experimental data. We then analyze whether the phenomenological nesting model can be
justified in frameworks of a more general model of electron-electron interactions in the
one-
dimensional system improved by three-dimensional effects of the interchain hopping. Properties
of the Bechgaard salts look consistent with the Hubbard model with a weak repulsion. Con- siderable high temperature variation of the magnetic susceptibility is ascribed to localization of electrons by quasi-elastic scattering on thermal phonons. The fact that these materials corre-
spond ta the half-filled (hale) band was crucial for the analyses. Except for a new energy scale, introduced by the temperature of a spin density wave transition, no other electronic correlation effects stem from the analysis.
1. Introduction
The reader should not expect to find below a detailed comparison between theory and the
experimental data. In our opinion, such a comparison would be a premature attempt, at least, if a more broad view is taken. Some experimental facts important for our discussion are not available yet, or the corresponding experiments still remain m a preliminary stage. In this presentation we discuss whether remarkable and diverse low temperature properties of quasi- one-dimensional (QID) conductors (especially, those of the Bechgaard's salts family) can be made consistent with predictions of the Fermi hqmd theory. The issue is about the rote played by electron-electron correlations m the system.
The Mass of materials has recently attracted agam much of attention, both of experimental-
ists and theorists. There are important reasons for that. On the theory side, the compounds
(* e-mail: gorkovslmagnet.fsu.edu
© Les Editions de Physique 1996
realize trie best known so far example of trie so-called "quasi-one-dimensional" materials with
a pronounced chain structure. As for trie theory of electron-electron interactions in a one- dimensional (ID) system, it bas been elaborated in great detail years ago Il,2] (see also trie
book [3] for a few exact results). Trie strictly one-dimensional system would indeed possess
numerous unusual interesting features, sometimes dramatically diflerent from properties of elec-
trons m ordinary metals described in frameworks of trie Landau Fermi hquid theory. On trie
other hand, trie discovery of high temperature superconductivity in cuprates, for instance, with ail their unexpected properties, has produced the high demand for new ideas and theoretical
approaches. Therefore, unusual features of trie gas of interacting one-dimensional electrons are
now very often viewed as an illustrative example, a "prototype" of nontrivial new physics, which may also be pertinent to many other systems, such as the 2D electron gas, heavy fermions, high Tc oxides, and so forth, remarkable for importance of electronic correlations (see, e.g. [4]).
In what follows, we concentrate our attention on properties of the selenium-based salts, (TMTSF)2X. The detailed experimental information is available for these materials regarding
their thermodynamic, transport and galvanomagnetic properties, especially, for the two of the most studied compounds, (TMTSF)2PF6 and (TMTSF)2Cl04. At very lo~N. temperature (+~ K) the materials may become superconductors (for reviews see [5]).
Taken as a whole, numerous existing low temperature experimental data are consistent with the idea of a strongly anisotropic three-dimensional metallic behavior in these systems, although
with a few reservations. Namely, among many striking features characteristic of these materi- ais, there is a proximity to some transition, a structural transition, such as the anion ordering (AD) in Cl04, or a spin density wave (SDW) instability, which occur at rather low tempera-
tures (10-20 K) m (TMTSF)2X with a symmetric anion, X. It is well established theoretically,
that electron-electron interactions in the one-dimensional conductors reveal a tendency to pro-
duce varions instabilities [1,2]; the instabilities then may wmd up as a thermodynamical phase
transition [fil, if additional three-dimensional features, such as interchain electron tunneling,
or Coulomb interactions between electrons on diflerent chains, were added to the purely one-
dimensional analysis. However, as the low temperature properties of the Bechgaard salts are concerned (T < jo K), it seems, at least, at first sight, that for the selenium-based compounds, (TMTSF)2X, there is no immediate necessity to invoke the one-dimensional physics at ail. Ail
phase transitions in the materials mentioned above are rather well defined mean field transi- tions with critical fluctuations important only in a narrow enough vicimty of trie transition
temperature. Trie mechanism for trie SDW state itself, its suppression by externat pressure, and trie pecuhar phenomenon of restoration of trie SDW-like states m high magnetic fields,
in particular, can be described even quantitatively in a simple and plausible model of weekly interacting electrons with trie metallic Fermi surfaces possessing an approximate "nesting"
property, as trie starting point I?i.
Some suifur analogs of trie Bechgaard salts, such as (TMTTF)~PF6, however, reveal some- what more pronounced one-dimensional behavior (see discussion [5d]). A number of them show
a tendency to mstabilities at considerable higher temperature, sometimes exceeding estimated values for their transverse tunneling integrals. Therefore m these materials one may anticipate
a more important rote of the on-chair interactions, although, again, it seems that low tem-
perature properties of both the sulfur- and selemum-based materials qualitatively are rather similar and can be mapped on trie top of each other by applymg externat pressure.
As it is well known, trie Landau Fermi liquid theory is nothing but a self-consistent ap- proach based, in particular, on trie assumption that in metals there is only one energy scale
(typically, of an atomic order) which characterizes both the bare electromc spectrum itself and the strength of effective electron-electron interactions. The QID materials are remarkable for strongly amsotropic conducting properties. This fact atone means additional scales for
N°12 QID ORGANIC METALS: THEORY AND EXPERIMENT 1699
the electron spectrum. As for electron-electron interactions, there are no special reasons for them to be strong, especially because high polarizabilities of the constituting organic molecules may essentially reduce the Coulomb forces. The spin density wave (SDW) transition, say, m (TMTSF)2PF6, occurs at TSDv/
+~ 12 K. This scale is almost three order of magnitude less than the bandwidth +~1 eV, as estimated for electronic motion along the chain direction. Again, as it was mentioned above, the superconducting transition occurs at T
+~ 1 K. The weak coupling theory of the SDW-transition, based on the "nesting" of the two open Fermi surfaces, explains
many experimental results at low temperatures surprisingly well (see [5]). There are, however,
some facts which contradict trie idea that interactions are so weak: trie magnetic susceptibility
at high enough temperatures (above +~loo K) displays trie considerable temperature change.
This fact cannot be explamed in terms of weakly interacting electrons (see [5d]). We will
address this important issue.
Trie fact that many of the Bechgaard salts may exist in a non-metallic ground state at T
= o,
implies that their low temperature properties are not the properties of a Fermi liquid in the strict sense. On the other hand, the SDW-phase can be removed by an externat pressure, and the same material will now behave metallically down to lowest temperatures. The proximity to an insulating state, hence, casts some reasonable doubts regarding the Fermi liquid description
in the adjacent metallic phase. There is a growing amount of evidences that, at least, some properties of these materials somehow difler from simple expectations of the theory of metals.
Among these properties are: unusual temperature dependence of resistivity, large magnetore- sistance m weak enough fields (+~10 Tesla) and temperatures of order of 10 K, a non-Korringa temperature dependence of the NMR-relaxation rate (see [5d]). Note that in the three cases the new features appear m the kinetic characteristics.
In the next few Sections we show how the aforementioned simple model of two Fermi surface sheets with an approximate "nesting" properties could also explain some of the new findings
in kinetics of the QID-organic conductors at low temperatures. (This model, however, does not rule ont an involvement of diiferent physics [4j). In Section 5, we will make an attempt,
on a speculative level, though, to trace how interactions in QID conductor may renormahze themselves m a manner so that the weak coupling standard treatment of the SDW-phenomena
remams correct. We will stress, however, that there are new facts which cannot be immediately
explamed by the theory, although it seems that the due to understanding of these new findings
lies in a 3D-low temperature physics of these compounds.
2. Model
Before going into more details, it is helpful to recall trie model, some standard notations, and trie main known facts used below. As usual, we approximate trie monoclmic system of trie
Bechgaard salts by an orthorhombic symmetry with a as the lattice period along trie chain
direction, while b and c correspond to periodicity in directions transverse to the chain. For
(TMTSF)2PF6 the values of the three main tunneling integrals along the corresponding axes
are estimated as:
ta tb tc'= o.2 eV 200 K 10 Il (1)
This anisotropy is consistent with data on the anisotropy of conductivity, as obtained at room
temperatures. Such an anisotropy of conducting properties suggests that the "bare" electromc spectrum may be adequately described in terms of a tight binding model with only a few
tunneling (hoppmg) matrix elements involved. (In view of the exceedingly large anisotropy
between the b- and the c-directions, the materials are aise called quasi-two-dimensional (Q2D) conductors). The tight binding spectrum of non-interacting electrons wohld be of a generic
form:
E(p) = vF(+p~ pF) 2tb cospyb 2t~ cospzc 2t[ cos2pyb (2)
The first term describes an electron moving along the chain. It is lmearized in the vicinity
of each of the two open Fermi surfaces, for its right and left sides, respectively. Two other terms, tb and t~, correspond to hopping between nearest-neighboring chains along b and along
c, respectively; as for the t[-term, it measures an effective hopping between the next nearest-
neighboring chains. In absence of the latter term (fi = o) the spectrum in equation (2)
possesses the so-called "nesting" degeneracy:
Ejp + Q)
= -E(P) 13)
Here
Q = Qo + (2pFi ~ ~ (2')
b c
At t[ # o equation (3) takes place only approximately, provided that t[ is small enough compared to tb. (In what follows the tunneling integral t~ introduces no new physics and will be omitted for brevity).
The Bechgaard salts, (TMTSF)2X, are the charge transfer compounds with one electron transferred to each anion, X. Therefore, the salts are hole conductors with one noie per unit cell in trie conducting TMTSF-network. Trie conduction band is half-filled; for a single metalhc chain, 4pF " 27rla.
For simplicity of trie analysis, electron-electron interactions are usually chosen as short range
(Coulomb) interactions. Trie effective screemng in real materials is due to trie fact that ail chains are packed together into the lattice, so that even m absence of interchain hopping charges on neighboring chains may adjust themselves to screen a charge placed on a given
chain. Such a screening will be rather anisotropic.
For electrons confined to one metallic chain ail interactions at the Fermi surface are reduced to the following three interaction constants:
gii the amplitude for the backward scattering of one electron by another;
g21 the forward scattering amplitude (each electron remains on its side of the Fermi surface after scattering);
g31 the Umklapp processes; the total momentum is not conserved.
(For the Hubbard model ail three constants are equal: gi
" g2 " g3 " g"). The 1D analyses
[1, 2j proceeds assuming weak enough interactions, g~ < 1.
Interactions between electrons on a metallic one-dimensional chain may lead to a few insta- bilities which would compete with each other for opening of a gap at the Fermi level [1, 2].
There are three competing channels: superconducting channel (SC) formation of charge den-
sity wave (CDW); the spm density wave channel (SDW). No thermodynamic phase transition is possible in one dimension.
To separate the above channels and to fix one of ihe above mstabilities as the thermody-
namical phase transition, it is necessary to account for three-dimensional features in the real materials. There are only two distinctly diflerent three-dimensional eflects [6j: 1) interactions between electrons on diiferent chains and 2) interchain electron hoppmg. The type of the possi- ble ground state essentially depends on which of these two eifects prevails. However, m presence
of any 3D-features the phase transition would happen at some low but finite temperature, Tc.
The transition tutus ont to be a mean field transitions. The provision that the temperature interval for critical fluctuations is narrow, is given by smallness of ail the interaction constants,
N°12 QID ORGANIC METALS: THEORY AND EXPERIMENT 1701
p+Q
+
à à.H(Q)
P
Fig. l. The generalized magnetic susceptibility for free electrons in its diagrammatic form: two fines correspond to the two Green functions in the matrix element of equation (5).
g~, 1-e-, g~ < 1 (In trie (TMTSF)2X-salts hopping prevails as a 3D eifect; it is assumed below that Tc < tb).
3. SDW-Instability
The nesting instability scenario bas been extensively studied by many authors le-g-, see [5j for review). We shall only outline below some major steps of trie calculations. Thus, in case of trie SDW-instability trie analysis starts with calculation of trie generalized linear magnetic
response to a staggered field, H(uJo, Q)1 Miwo; Q)
= xiwo, Q)Hiwo, Q) 14)
The free electron response function, y(uJo,Q), is proportional to trie matrix element constituted of two Green functions, shown in Figure 1:
xlwo, Q)
+~
TE / dp Gjw~; p)Gjw~ wo; P Q) ~
~ / dpjioJo
+ E(p) Ejp Q)j~~ x jtanhjEjp) /2T) tanhjE(p Q)/2T)j (5)
At Q
= Qo and mû
" o, trie last integral becomes:
v(EF / ~~ tanin ~
+~ 2v(EF) In(É/T) là')
E 2T
where v(EF) is the electronic density of states (per spin), E is a cut-off energy. Note that
when the vector Q is commensurate, the integral over momentum, p, comprises of the two
equal logarithmic contributions arising each from two sides of trie Fermi surface (+ or -),
because 2 Q is trie reciprocal lattice vector.
With the above information concerning trie single "bubble" diagram in Figure 1, consider corrections to x(uJo,Q) due to electron-electron interactions. Some of those corrections are drawn schematically in Figure 2. Adding each new internai block into Figure 2 introduces
a factor to trie matrix element of exactly trie same form as trie integral calculated before,
m equations là, 5'). Trie integral is logarithmically divergent at low temperatures. This is
trie famihar logarithmic problem: at low enough temperatures trie large logarithmic factors may compensate weakness of interactions, and trie whole expression bas to be summed up to take this fact into account. To find trie effective interactions, r2 and r3, one is to solve trie
diagramatic equations for trie two vertices shown m Figure 3. Their form is self-explanatory.
With trie bare interactions, g2 and g3, mdependent on transverse momentum, trie two equa- tions become a system of trie two algebraic equations, and at low temperatures each vertex
+ + +
~2 + ~3 +
fi~
+
+ gi
+...
' '
Fig. 2. Electron-electron interactions (the dashed fines) form the "ladder" corrections ta x(uJo, Q)1
the two Green functions inside each block produce the logarithmic term of the form in equations (5, 5')
+ + ++ + + + + +
r2
1 1 1
~ i
+ -+ + + +
~3
1 1 1
Fig. 3. Equations for the renormalized vertices, r2.r3i their structure reflects the structure of
diagramatic corrections
in Figure 2.
acquires the form:
r*
= g*1
2g* in (j + in ~'
,
r* + r~
= r~ (6)
~
o
(at g2 + g3, denoted as g*). Here To, as defined by the standard relation:
= 2g*In ) (7)
o
determines trie temperature for trie onset of trie SDW-phase. It is easy to verify that the mean field character of the transition is guaranteed at To < tb < EF.
The main contribution into integral (5), which provides the pale m r's, is due to the log-
arithmic contribution from the extended interval To < E < É
+~ EF. Deviations from the exact nesting condition, for instance, due to non-zero t[ m equation (2), only weakly influence the contribution from that main integration interval, and may net destroy formation of a low
temperature SDW-phase.
N°12 QID. ORGANIC METALS: THEORY AND EXPERIMENT 1703
4. Precursor Effects in Kinetics
As it has already been emphasized above, outside a narrow temperature interval around To, AT/To < 1, fluctuations introduce only negligible corrections to ail physical quantities.
Therefore the critical fluctuations cannot account for any considerable eifect in a wide tem-
perature interval (10-40 K) where unexpected features have been observed in the temperature dependence of resistivity (and in the magnetoresistance) for (TMTFT)2X, where X
= PF6 and
Cl04 (see [5d, 8, 9]).
Consider properties of two vertices, r2, r3, in more details. Each of them diverges at T
= To
for Q
= Qo and mû
" o. At T
+~ To and Q
= Qo + q, where q is small:
2tbqyb +~ To (8)
the r's are finite and are of the order of unity. In other words, in the above domain of
the transverse momentum transfer the effective interactions become large compared to the
weak bare interactions, g~ < 1. This interval however, constitutes only a small fraction of the Fermi surface area of order of To/tb < 1. Here lies the main distinction between the 1D and the 2D- and 3D-cases. In case of uncoupled linear chains each chain exercises the
mdependent renormalization of interactions. If a vertex becomes large as the result of this, the
corresponding enhancement takes place along the whole Fermi surface in the Brillouin zone. On the contrary, if three-dimensional eifects cut-off the 1D-renormalization at E
+~ tb, the phase
space becomes restricted by the narrow interval (8), so that corrections due to the proximity
to a phase transition usually do not contribute significantly to the Fermi hquid characteristics.
We show now that in the mortel under discussion, the above interval (8), however, plays an important rote in the kinetics. Thus, for electron relaxation times the above interval m the
momentum space is smgied ouf by the conservation iaws of energy and momentum, which have
to be fulfilled at a real process of electron-electron scattering [10].
To elaborate on the idea, consider in some more details the temperature dependence of re-
sistivity, p(T). When electron-electron scattering is the dominant mechamsm of resistivity, as is probably the case for these rather clean materials m the temperature range under consider-
ation, the dissipation rate for momentum, 1/Tu, is proportional to the matrix element of the
Umklapp scattering, g3. Omitting ail nonessential factors, one can write 1/Tu in the followmg
familiar form:
)
" 1931~j/~lPi~lP2(l~flP3)(l~~lP4)
u
xô(pi + p2 p3 p4 K)à(Ei + E2 E3 E4) dpi dp4 (9)
Here trie Fermi functions, np, together with the à-functions expressmg the conservation laws
of energy and momentum, would lead to the well-known T~-dependence for p(T) (more ex-
actly, in 2D-caà-p(T)
+~
T~lnT). Making use of trie expression (2) for the total electron energy spectrum, as a sum of two components for trie longitudinal and transverse dispersions, respectively:
EIP) = t(P~) + ElPi) (1°)
and choosing trie Umklapp vector as K
= (4pFio;0), trie conservation laws m (9) may be
identically re-written m trie following manner:
