This section gathers some bibliographical references concerning four families of theoretical models, which have been proposed to account for the cascade of phase transitions induced by the magnetic field in "Bechgaard salts", (TMTSF)₂X. A short summary of the structure of this molecular compounds family can be found in Appendix 1. For an exhaustive review of organic conductors, the reader can refer to the book by T. Ishiguro, K. Yamaji and G. Saito, Organic Superconductors (Springer-Verlag, Heidelberg, 1998), 2nd ed. More details can be found in line in this site, in the publications of our team devoted to this topic. Particularly, see the introduction of the 1993 habilitation dissertation (Calorimetric studies of the multicritical behaviours of the spin-density-waves phases in a molecular compound).

In (b), the "standard model" groups together many variants of the same theoretical description of the Field-Induced Spin Density Wave phases in terms of SDWs quantisation. This model roughly explains the shape of the experimental phase diagram, and accounts for the existence of a quantised Hall effect in a three-dimensional material. Many theoreticians have given their personal touch to this model, coming from all over the world (Russia, France, USA and Japan, in particular).

In (c), the "interaction" model has been proposed by Victor Yakovenko. It does not call on the particular Fermi surface topography, but rather on the distinctive features of the electronic interactions in a quasi-one-dimensional system.

In (d), a series of models attempt to account for finer aspects of the phase diagram, in particular the arborescence that we have experimentally revealed in (TMTSF)₂ClO₄ (see section 5). Going beyond the integer quantisation of the main phase transition series, this kind of models proposes to interpret some phenomena in terms of a fractional quantisation.

Finally, in (e), an independent approach by Andreï Lebed' enables to account for another aspect of the phase diagram that our nanocalorimetry investigations have evidenced: the existence of a tetracritical point (see section 6).

The "standard" model of the Field-Induced Spin Density Wave phases is the first of two families of theoretical models claiming to explain the cascade of phase transitions between quantised magnetic states. This model lies on the assumption that the SDW results from an instability of the quasi-one-dimensional electron gas, because of particular topological properties of the Fermi surface in these materials (see the previous section).

The original idea has been stated by Lev Gor'kov and Andreï Lebed' [1], and independently by Paul Chaikin [2], who made the remark that the magnetic field tends to localise the electronic trajectories along the TMTSF chains. As the field exceeds some threshold, the electron gas effectively becomes one-dimensional, so that the metallic state becomes unstable against the formation of a density wave. It could be a charge density wave (CDW), but the "standard" model assumes, in an ad hoc way, that the interaction berween electrons is rather favourable to a spin density wave (SDW).

The spin density wave instabilities result from the topography of the open Fermi surface (see the previous section), that makes the electronic system extremely unstable to any perturbation of wave vector Q coupling a large number of electronic states from one sheet of the Fermi surface to the other one (due to the "nesting" of the Fermi surface, Fig. 1).

FIG. 1. The Fermi surface of a quasi-1D conductor consists of two sheets located at +kF and -kF (cut view). The Q wave vector allows one sheet of the FS to nest the another, through a simple translation [3].

The instability is related to the divergence of the electronic susceptibility of the 1D electron gas without interaction (the "standard model" assumes the "weak coupling mean field" approximation). The density wave may be considered as resulting from the condensation of the Fermi level states into electron-hole pairs.

Since the nesting is not complete, pockets of non paired carriers remain (hatching strokes of Fig. 1). Quantisation in magnetic field arises from the existence of closed electronic trajectories formed in the pockets, which can thus be quantised in Landau levels. In order to maintain always completely filled an integer number of these levels, the wave vector Q(H) is permanently adjusted to the magnetic field [3]. The consequence is the experimental observation of the Hall plateaus being constant (independently, in Orsay and in the USA), in (TMTSF)₂ClO₄ [5,6] as well as in (TMTSF)₂PF₆ [7,8] (Fig. 2).

FIG. 2. Volume quantised Hall effect in (TMTSF)2PF6 under pressure ([7,8], from the left-hand side to right-hand side).

So each plateau is associated to an order parameter, and is characterised by an integer quantum number: the latter is the number of completely filled Landau levels. According to the "quantised nesting" model, the electron-holes pairs act as "carriers reservoir". The vector Q varies with the field, which costs condensation energy because it distorts the spin density wave. When the energetic cost is too important, the nesting vector jumps to a phase characterised by a new quantum number [3], which yields the phase transitions we have revealed [publis 7, 8, 9], independently of magnetisation experiments in P. Chaikin's team [9].

Other theoretical works have afterwards completed this description [10 à 15; see the book by Ishiguro et al.].

[1] L.P. Gor'kov and A.G. Lebed', On the stability of the quasi-one-dimensional metallic phase in magnetic fields against the spin density wave formation, J. Phys. (Paris) Lett. 45, L-433 (1984).

[2] P.M. Chaikin, Magnetic-field-induced transition in quasi-two-dimensional systems, Phys. Rev. B 31, 4770 (1985).

[3] M. Héritier, G. Montambaux and P. Lederer, Stability of the spin density wave phases in (TMTSF)₂ClO₄: quantized nesting effect, J. Phys. (Paris) Lett. 45, L943 (1984).

[5] M. Ribault, D. Jérome, J. Tuchendler, C. Weyl and K. Bechgaard, Low-field and anomalous high-field Hall effect in (TMTSF)₂ClO₄, J. Phys. (Paris) Lett. 44, L953 (1983).

[6] P.M. Chaikin, Mu-Yong Choi, J.F. Kwak, J.S. Brooks, K.P. Martin, M.J. Naughton, E.M. Engler and R.L. Greene, Tetramethylselenafulfalenium perchlorate, (TMTSF)₂ClO₄, in high magnetic field, Phys. Rev. Lett. 51, 2333 (1983).

[7] J.R. Cooper, W. Kang, P. Auban, G. Montambaux, and D. Jérome, Quantized Hall effect and a new field-induced phase transition in the organic superconductor (TMTSF)₂PF₆, Phys. Rev. Lett. 63, 1984 (1989).

[8] S.T. Hannahs, J.S. Brooks, W. Kang, L.Y. Chiang, and P.M. Chaikin, Quantum Hall effect in a bulk crystal, Phys. Rev. Lett. 63, 1988 (1989).

[9] Naughton M.J., Brooks J.S., Chiang L.Y., Chamberlin R.V., Chaikin P.M., Magnetization study of the field-induced transitions in tetramethyltetraselenafulvalenium perchlorate, (TMTSF)₂ClO₄, Phys. Rev. Lett. 55, 969 (1985).

[10] M. Héritier, G. Montambaux and P. Lederer , Phase diagram of quasi-one-dimensional conductors in strong magnetic field, J. Phys. (Paris) Lett. 46, L-831 (1985).

[11] K. Yamaji, Theory of field-induced spin-density-wave in Bechgaard salts, Synth. Metals 13, 29 (1986).

[12] M.Ya. Azbel, P. Bak and P.M. Chaikin, Spectra and gap amplification for systems with two widely different incommensurate periodicities, Phys. Rev. A 34, 1392 (1986).

[13] A. Virosztek, L. Chen and K. Maki, Thermodynamics of field-induced spin-density-wave states in Bechgaard salts. II, Phys. Rev. B 34, 3371 (1986).

[14] G. Montambaux, M. Héritier and P. Lederer, Spin susceptibility of the two-dimensional electron gas with open Fermi surface under magnetic field, Phys. Rev. Lett. 55, 2078 (1985).

[15] D.Poilblanc, M. Héritier, G. Montambaux and P. Lederer, Quantised density wave ordering induced by a magnetic field in quasi-one-dimensional conductors in the weak-coupling limit, J. Phys. C 19, L321 (1986).

[16] K. Machida and Y. Hori, On the high-field limiting phase in (TMTSF)₂ClO₄, J. Phys. Soc. Jpn. 61, 2216 (1992).

Victor Yakovenko's "in interaction" model has been developped by its author after remarking that the spin density wave induced by the field appears only if the zero-field fundamental state is superconducting [17 to 20]. This a "weak coupling" model, like the "standard model", however it does not assume any ad hoc interaction to a priori justify the stability of the spin density wave with respect to the others. On the very contrary, the model attempts to describe the interaction responsible for the condensation into electron-hole pairs.

Yakovenko describes the electron-hole pairing with the help of Landau wave functions, extended over several neighbouring organic chains (Fig. 3). The nesting of the Fermi surface does not play any role in this model.

FIG. 3. Quantum electron-hole states described by Landau wave functions. Their spatial extension decreases as the magnetic field is increased, which leads to localise the electronic orbits onto the chains.

The quantum states are electron-hole pairs, and the quantum number N of a pair is here the distance (the number of chains) that separate them.

An interesting prediction of the "interaction" model is that it accounts for the reentrance of the metallic state at very high magnetic field (around 27 teslas: Fig. 4 [21]), which the "standard model" does not.

FIG. 4. Experimental phase diagram of (TMTSF)2ClO4 exhibiting at high field the reentrance of the metallic state, obtained by the Princeton group [21]. The temperature axis is vertical, the field axis goes from left to right, and the pressure axis from top to bottom. Under high pressure, the reentrance is repelled towards very high fields (out of range in these experiments).

[17] V.M. Yakovenko, Theory of magnetic-field-induced phase transitions in quasi-one-dimensional conductors, Zh. Eksp. Teor. Fiz. 93, 627 (1987) [Sov. Phys. JETP 66, 355 (1987)].

[18] V.M. Yakovenko, Comment on "Extreme quantum limit in a quasi two-dimensional organic Conductor" (R.V. Chamberlin et al., Phys. Rev. Lett. 60, 1189 (1988)), Phys. Rev. Lett. 61, 2276 (1988).

[19] V.M. Yakovenko, Quasi-one-dimensional conductors in magnetic field: physical consequences of 'non-standard' theoretical approach, Fizika 21, Suppl. 3, 44 (1989).

[20] V.M. Yakovenko, Theory of magnetic-field-induced phase transitions in quasi-one-dimensional conductors, Zh. Eksp. Teor. Fiz. 93, 627 (1987) [Sov. Phys. JETP 66, 355 (1987)].

[21] W. Kang, S.T. Hannahs, and P.M. Chaikin, Giant Lebed magic angle effects and more - Bechgaard salts with pressure and magnetic field, Synth. Metals 56/1, 1936 (1993).

 

In order to explain the departure from the "standard model", such as the negative Hall effect, observed in very well ordered (TMTSF)₂ClO₄ [22] and in (TMTSF)₂PF₆ under pressure [23], Héritier proposed a model implying a high order nesting of the Fermi surface, yielding to a fractional quantisation of the SDW wave vector, and not simply an integer quantisation [24].

The "standard model" accounts for the cascade of FISDW phases by invoking an interference effect between two periodicities appearing along the longitudinal direction:
   -1 - the (cyclotron) magnetic period 2px₀=h/eBb, due to the orbital effect of the magnetic field on the phase shift of the electronic wave functions (where h is the Planck constant, B the magnetic field and b the transverse lattice parameter, i.e., 0.78nm)
   -2- the SDW wave vector: Q=(Q_//;Q_perp).

Because of nesting properties, the relevant period is not 2p/Q_//, but rather 2p/q_// with q_//=Q_//-2k_F. These two periods open a series of band gaps in the quasi particules energy spectrum, the richness of which has not been completely described because a third period of the electronic system has not been incorporated: this is the crystal lattice period along the longitudinal direction, a.

As these three periodicities are explicitely taken into account, the gaps opened by the self-consistent SDW potential are able to stabilise new sub-phases, corresponding to a quantum number n=q_//x₀, which can take rational values and not anymore simply integer, provided that the interaction between electrons is strong enough [24, 25].

In order to take the periodicity along a into account, the dispersion relation of the electronic energy E=E(k) consists of periodically spaced parts, linearised at the Fermi level (Fig. 5). In the case of a half-filled band (which is the (TMTSF)₂ClO₄ case if one considers the weak dimerisation along the a axis), the Fermi wave vector is equal to k_F=p/2a, and the 4k_F vector belongs to the reciprocal lattice.

FIG. 5. Dispersion relation E=E(k), periodical along kx, linearised at the Fermi level, made of periodically spaced arms, with slopes ±vF.

The 2p/4k_F=a period has introduced new commensurate effects with the two periods 2px₀ and 2p/q_//, but siince these latter two are different from the first one by several orders of magnitude, these effects may be neglected: the lattice parameter a of the Bechgaard salts is about 0.7nm, whereas teh wave number x₀ may take values of about300nm, in a field of 3 teslas.

One of the characteristics of this model, unlike the previous two family models, is that it does not assume anymore the weak coupling approximation. On the contrary, it takes into account an intermediate coupling, in a way which is consistent with our nanocalorimetry results (see section 4).

Figure 6 displays the phase diagram as predicted by this model [24, 25]. It exhibits an arborescent structuree that is not without ressembling our experimental phase diagram (voir la section 5).

FIG. 6. Phase diagram predicted by the fractional nesting model [24].

The self-similar structure of this diagram is not without evoking the "Hofstadter butterfly" [26], Fig. 7, which corresponds to the self-similar structure of the energetic spectrum of an electron moving within a crystal lattice and at the same time in a magnetic field [27,26].

FIG. 7. Hofstadter butterfly: it is the energy pectrum of an electron moving on a square crystal lattice, and immersed in a magnetic field [26].

The physics looks much like that of the SDW of fractional quantisation: "something happens" as the field takes rational values, that is to say, as a q number of flux quanta, F₀=h/e (where h is the Planck constant and e the electron charge), pass through p cells of the lattice. The magnetic flux can thus be written as: F/F₀=p/q. The calculation thus shows that q permitted energy sub-bands are expected to open. For example, 2 bands are expected for F/F₀=1/2, but 50000 bands for the extremely close value of F/F₀=24999/50000!!! This is why the spectrum displays such an irregular shape, with a self-similar nature.

In the case of "Bechgaard salts", the situation is even complicated by the presence of another important period: the wave vecteur of the spin density wave (also the nesting vector of the Fermi surface, in the framework of the fractional nesting model).

Independently of Héritier's fractional nesting model, one may cite other theoretical models that attempt to overcome the limitations of the " standard model". All of them interpret the branching of the phase diagram we have observed as the result of the superposition of a large number of integer phases. The mechanism of this superposition is an "Umklapp" effect, a long range coupling of electronic states in the reciprocal space (for example 4k_F) [28-30]. In the next paragraph, we will be more particularily interested in a competing model which exhibits the particularity of having predicted the existence of tetracritical points, even before our experimental discovery was published (see section 6).

[22]M. Ribault, Electronic states below 5 K in (TMTSF)₂ClO₄, Mol. Cryst. Liq. Cryst. 119, 91 (1985).
[23] B. Piveteau, L. Brossard, F. Creuzet, D. Jérome, R.C. Lacoe, A. Moradpour and M. Ribault, Hall effect study of the field-induced instabilities in (TMTSF)₂ClO₄ u nder pressure, J. Phys. C19, 4483 (1986).
[24] M. Héritier, Field-induced quantized magnetic ordering in quasi-one-dimensional conductors, in "Low Dimensional Conductors and Superconductors", edited by D. Jérome and L.G. Caron (NATO-ASI, Plenum Press) 155, 243 (1986).
[25] publi 17
[26] D.R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 34, 2239 (1976).
[27] M. Ya Azbel, Energy spectrum of a conduction electron in a magnetic field, Zh. Eksp. Teor. Fiz. 46, 939 (1964) [Sov. Phys. JETP 19, 634 (1964)].
[28] K. Machida and M. Nakano, Infinite cascades of field-induced spin density wave states in anisotropic two-dimensional conductors, J. Phys. Soc. Jpn. 59, 4223 (1990).
[29] V.M. Yakovenko, Theory of the quantum Hall effect in quasi-one-dimensional conductors, Synth. Metals 43, 3389 (1991).
[30] V.M. Yakovenko, Quantum Hall effect in quasi-one-dimensional conductors, Phys. Rev. B 43, 11353 (1991).

Lebed's model attempts to describe the treelike phase diagram of (TMTSF)₂ClO₄ (section 5) by assuming that the two order parameters of two neighbouring FISDW phases, described by quantum numbers n₀ and n₀+1, are coupled through an "Umklapp" process (at 4k_F) [31]. The diagram proposed by this model (Fig. 8) exhibits a series of tetracritical points (in Griffiths' sense [32]) and "intermediate" phases, corresponding to the superposition of order parameters (here there is no additional quantisation, for instance fractional).

FIG. 8. Phase diagram of FISDW phases, proposed by Lebed' [31]. It describes a series of critical points that are not without evoking the tetracritical point we have experimentally evidenced.

A "pure" phase is described by the integer quantum numbers ±n, whereas an "intermediate" phase between the SDW phases n₀ and n₀-1 is described by four quantum numbers: n₀, -n₀, n₀+1 and -(n₀+1) [31].

This theoretical interpretation is attractive, due to its simplicity, however it imperfectly explains the branching of transition lines. In effect, the model does not describe the arborescence of transition lines that is observed as the temperature is decreased (section 5). Still, it is able to predict how two FISDW sub-phases could exist within an "intermediate" phase, which accounts qualitatively well for our experimental observation of a tetracritical point in the phase diagram of (TMTSF)₂ClO₄ in magnetic field (section 6).

Lebed' proposed in 2002 a new version of its model for the quantised FISDW phases [33]. The reader may find in it bibliographical references later than 1993.

For more information concerning all these models, by the way also concerning the spectacular experimental properties of "Bechgaard salts ", two dissertations may be read on line in this web site:
   -1- François Tsobnang's PhD thesis, in 1991.
   -2- 1993 Habilitation dissertation (Calorimetric studies of the multicritical behaviors of the spin-density-waves phases in a molecular compound).

One also may refer to more general books [34], or to review articles [35,36].

[31] A.G. Lebed', New phases in organic superconductors, JETP Lett. 51, 663 (1990).[Pis'ma v Zh.Eksp.Teor.Fiz. 51, 583 (1990)].
[32] R.B. Griffiths, Proposal of notation at tricritical points, Phys. Rev. B 7, 545 (1973).
[33] A.G. Lebed, Field-Induced Spin-Density-Wave Phases in Quasi-One-Dimensional Conductors: Theory versus Experiments, Phys. Rev. Lett. 88, 177001 (2002).
[34] T. Ishiguro, K. Yamaji and G. Saito, Organic Superconductors (Springer-Verlag, Heidelberg, 1998), 2nd ed.
[35] M. Ribault, in "Low Dimensional Conductors and Superconductors", edited by D. Jérome and L.G. Caron (NATO ASI, Plenum Press) 155, 199 (1986).
[36] J.-P. Pouget, ibid., p.17 (1986).