Ann. Physik 3 (1994) 92- 106 Annalen der Physik @ Johann Ambrosius Barth 1994 Metallic solutions of the continuum model for conducting polymers H. W. Streitwoif and H. Puff Max-Planck-ArbeitsgruppeHalbleitertheorie, Hausvogteiplatz 5 - 7, D- 10117 Berlin, Germany Received 8 November 1993, revised version 3 December 1993, accepted 9 December 1993 Abstract. In the continuum approximation of the Su-Schrieffer-Heeger model for degenerate and nondegenerate ground state systems, an exhaustive survey of solutions of the standard gap equations is given. With special emphasis on the electronic energy spectrum, the pertinent results of Takahashi [ 11 are extended to the whole fi, k-plane. Periodic metallic phases and their grand potentials are discussed. Keywords: Su-Schrieffer-Heeger model; periodic lattice distortions; broken-symmetry polaron solution; non-degenerate ground state; metal transition. 1 Introduction Despite considerable effort to explain the insulator-metal transition observed in conducting polymers like polyacetylene or polythiophene its detailed mechanism is still controversial [2 -41. If reducing or oxidizing agents are intercalated between the chains of such polymers charge carriers are added to the chains causing localized conjugation defects. These repel each other, arrange in periodic lattice distortions as soliton or bipolaron lattices, and give rise to electronic states forming soliton or polaron bands in the original Peierls gap. These bands broaden with further doping. At about 5 % doping level in trans-polyacetylene a strong increase of the Pauli susceptibility and a high electrical conductivity is observed [ 5 ] indicating the insulator-metal transition mentioned above. Disorder was quoted [6]to explain the closing of the gap as cause of the metal transition. However, in alkali doped polyacetylene the solitons are pinned to the regularly spaced dopant ions [7] and there seems to be no sufficient disorder of the ions to close the gap. Also for chlorine doping the distance between the ions does not fluctuate very much [8]. Furthermore the solitonic fluctuations around the ions do not appear to be large enough to close the gap [9].StafstrCIm [lo], however, obtained a shrinking of the gap at high doping levels by treating the electronic charge distribution in an extented Su-Schrieffer-Heeger (SSH) model self-consistently. A similar calculation 1111 taking the nearest neighbour interaction of three chains and dopant ions into account and including a full geometry optimization of the bond length dimerization caused a considerable shift of the electronic energy and closed the gap down to the average level spacing of the system. H. W. Streitwolf, H. Puff, Metallic solutions of the continuum model 93 Kivelson and Heeger [2] explained the metal transition by a transition from a soliton to a polaron lattice, where the chemical potential enters the soliton or polaron band. In the SSH-model [12], however, it is well known [3] that even for large doping levels the chemical potential remains in the gap and the insulating charged soliton lattice is the most stable solution. The SSH-model intended to describe the electronic properties of degenerate ground state polymers with conjugated carbon chains is a simple oneband tight-binding model with linear electron-phonon interaction. It reveals the well known Peierls effect: for half-filled band the total energy of the polymer is reduced by a dimerization of the chain which leads to the formation of a gap, leaving the system in an insulating state. In case of non-degenerate ground state materials Brazovskii and Kirova [I31 extended the model by introducing an external bond dimerization. The Peierls effect enhances the gap resulting from this external bond dimerization. Recently Takahashi [ 11, investigating anew the SSH-model in its continuum version as given by Takayama, Lin-Liu and Maki [14], reported a series of solutions besides the normal insulating soliton lattice phase. These solutions represent defect lattices with partially occupied bands, and he stressed the possibility that those which are infinitesimally adjacent to the stable insulating solution could be invoked for the metal transition. Unfortunately, Takahashi chose a parameterization which renders the energy spectra not particularly transparent. Just at this point we depart from Takahashi’s procedure. As in an earlier paper [15], we parameterize the lattice distortion by the band edges of the electronic structure. These are calculated from the gap equations defining stationary points of the grand potential functional, as functions of the chemical potential and a further parameter k (0 5 k i 1) fixing the width of the polaron band. In thermodynamic equilibrium, this parameter is determined by the unique minimum of the grand potential [15]. The set of solutions given is restricted only by the ansatz Eq. (4) but otherwise exhaustive. The results of a preliminary contribution [16] are slightly corrected and extended to finite external bond dimerization. We find a characteristic topological connection between the various solutions, which differs slightly from that given by Takahashi [l], who erroneously claimed the anomalous solution to be infinitesimally connected to the stable phase. The paper is organized as follows. In Section 2 we summarize, for the convenience of the reader, the formal results derived in [ 151: the self-consistent polaron lattice distortion for the Brazovskii-Kirova model, its band structure, and the gap equations which determine the band edges as parameters for the various solutions of the non-linear eigenvalue problem. Section 3 takes up the case of vanishing external bond dimerization. We calculate the phase diagram for all solutions and discuss their connections and the dependence of the band edges of the electronic energy spectrum on the chemical potential. In Section 4 we calculate .the grand potential and the charge carrier density in case of vanishing external bond dimerization. Instead of a detailed stability analysis which deserves considerable effort we investigate the validity of the thermodynamic inequality a20(k,p)/a,u2<0 for the various solutions and draw some conclusions about a possible metal transition into a solitonic phase with the chemical potential in the soliton band. In Section 5 we investigate the various solutions taking into account an external bond dimerization, We discuss the dependence of the band edges on the chemical potential and the topological connections between the various solutions, Finally we 94 Ann. Physik 3 (1994) calculate the grand potential and the charge carrier density also in this case. Investigating the thermodynamic inequality we obtain results very similar to those for vanishing external bond dimerization. 2 The polaron lattice solution In the continuum version [14] of the Su-Schrieffer-Heegermodel [12] generalized by Brazovskii and Kirova [I31 to take into account an external bond dimerization A, in order to model a non-degenerate ground state polymer the electronic eigenvalue equation for an incommensurate phase with nearly half-filled band reads Here f F ) ( x ) are the smoothed Bloch functions at odd and even sites, respectively. 4t0 is the bandwidth, and the function A ( x ) is a continuous generalization of the difference between the distances of neighbouring atoms and their normal phase distance a. It describes the lattice distortion and comprises the external (A,) and the Peierls dimerization. Both the energy spectrum and the lattice distortion define the thermodynamic grand potential per lattice site as a function of the chemical potential p in the limit T+O K by 2a cob)=- 1 (A(x)+A,)~ (Ea-p)+-jdx L a(occ) Lo 2ntoA with I the dimensionless electron-phonon coupling constant. L is the fundamental periodicity length of the system (periodic boundary conditions). The sum runs over occupied states only. The solutions considered in this paper are stationary points of this functional, +4a c 1 3if'd"(x)f',-'*(x) a (occ) =o , (3) subject to the ansatz Eqs. (3) and (4)comprise the feedback of the electronic states to the lattice geometry and thus imply the nonlinearity of the electronic eigenvalue equation (1). In particular, they lead [15] to a nonlinear differential equation for the lattice distortion, 1 1 ( a t o ) 2 A " ( x ) - - A 3 ( ~ ) + r= 0 dx A ( x ) 2 which has a unique everywhere finite solution H. W. Streitwolf. H. Puff, Metallic solutions of the continuum model 2 (E2 -4 9 3 ) A ( x ) s i g n r = -El - E 2 + E 3 + 95 (5) 1-- El -E3 where the modulus k of the Jacobi elliptic function [17] is given by d = ( K ( k ) is the half-period of the incommensurate lattice phase considered, and t = 2 a t o / 1 m . The parameters (integration constants) E l , E2, and E3 are subject to the conditions I rl = El E2E3 and El 2 E22 E32 0. They have an immediate physical significance as the edges of the energy spectrum emerging from Eqs. ( 5 ) and (l), which is confined to the regions 1 E 1 2 El and E22 I E I L E3. Obviously El is the lower conduction (or n *) band edge, -El the upper valence (or n-) band edge, while the two polaron bands are limited by + E , and +E2. Eq. (5) thus describes a polaronic lattice distortion and ( is the width of the polaron. The electronic energy spectrum E ( q ) is given by where n ( p , k ) denotes the complete elliptic integral of the third kind [18]. Obviously q ( + E , ) = 4 ( + E 2 ) = n / 2 d and 4 ( + E 3 )= 0. The constant ya in Eq. (4)is determined by the normalization condition of the Bloch functions [15]: Condition (3) yields the gap equations Because of the linearized band structure in the continuum model the a sums have to be truncated at an energy cut-off Eo%E1which we have chosen to conserve the number of states [15]: 96 Ann. Physik 3 (1994) Using this cut-off, Eqs. (7) and (8) are easily integrated with the result where the abbreviations and ko = k ‘ E t / E 2have been used. From Eq. (10) one concludes immediately 3 Results and discussion for systems with vanishing external bond dimerization We first investigate solutions for A,-*O, i.e. for polymers with a vanishing external bond dimerization. In taking the limit instead of choosing A, = 0 from the very beginning A, serves as a small symmetry breaking potential which gives us the possibility to discuss solutions with spontaneously broken symmetry. Accordingly Eq. (10) has two types of solutions in this limit: the normal soliton-lattice solutions with E3 = 0 and a polaron-lattice solution with E3 # 0 which, therefore, shows a broken symmetry. We now discuss both types of solutions in detail. a) The normal soliton-lattice solutions. For E3 = 0 the polaronic lattice distortion ( 5 ) reduces to the normal soliton lattice With the Landen transformation k , = (1 - k‘)/(l +k‘) and the soliton width = t/(l+k’) we have A(x) s i g n r = (El -E2)sn(x/<,,kl)as first derived by Horovitz [19]. The remaining gap equation (1 1) then simplifies to H. W. Streitwolf, H. Puff, Metallic solutions of the continuum model 97 According to the position of the chemical potential there are three types of normal (i.e. soliton lattice) solutions of Eq.(13). If their conduction band edges are denoted by EY) = E Y ) ( } pI , k ) then Eg) = k'EY' and E f ) = 0. If p is in the gap E, 1 [ p }2 E 2 we find a solution which describes an insulating charged soliton lattice. We denote it by superscript (2). From Eq. (13) we have EL2)(k)= E(:)(k)f1 + k 2 sinh' ( l / L ) and from Eq. (9) This solution exists within the (k-dependent) p-region E ( 2 ) ( k ) rlpl r E i 2 ) ( k )= k'E\''(k) (Fig. 1). At its limits it fits continuously to the other two soliton lattice solutions ((1) and (3)) to be discussed later on. The band structure and charge carrier density per monomer 0.6 0.7 0.8 modulus k 0.9 1 .o Fig. 1 Phase boundaries in a degenerate ground state polymer for the chemical potential of the different solutions of the gap equations. The solitonic metal (1) with ,u in the conduction band exists between Ej2)(k)and p c , the stable charged solitonic insulator (2) between EI2)(k)and Ei2)(k), the solitonic metal (3) with ,u in the soliton band below Ei2)(k),and the anomalous polaron lattice ( k ),u = E,,=. solution ( A ) between , ~ ( ~ ) and 98 Ann. Physik 3 (1994) remain constant over the entire range of the chemical potential. In the limit k-r 1, this solution describes the commensurate phase with Ei2’(1)= n t o / s i n h ( l / A ) ~ E l c, (14) Ei2)(1)= 0 and n(2)= 1. For the first of the soliton lattices adjacent to solution (2) (denoted by (1)) we have Ip I LEl = El’)( Ip I, k ) , i.e. ,u is in the conduction or valence band depending on the type of excess carriers It therefore represents a metallic soliton lattice. We first investigate its k+l-limit, i.e. the commensurate phase with the chemical potential in the conduction band. Then E2 = 0; from Eq. (9) we have E i = E : + ( X ~ ~ ) ~ , and thus Eq. (13) leads to the solution E : = p 2 - ( E l C - lp1)2tanh2(1/A) for E,,? Ipu( zntOexp(-l/A) (15) which yields the charge carrier density per monomer At its upper boundary lp I = Elc, the chemical potential coincides with the conduction band edge. With decreasing \ p i , the conduction band edge also decreases and finally vanishes at the lower boundary J pI = ntoexp (- l / A ) =pc. Simultaneously the charge carrier density increases from n = 1 to its maximum value nc = 1+exp (- U A ) . This solution therefore violates the thermodynamic inequality an @ ) / a p 2 0, indicating that for Ip 1 z E , there cannot exist a physically sensible solution up to k = 1. To obtain solution (1) for k<l, we start at solution (2) where lpl =E12)(k). Similar to the k+l-limit, a solution exists only in the interval E i 2 ) ( k ) r1p1 z,uC (Fig. l), and again E(,‘)(l p 1, k ) decreases with decreasing lp 1. At the lower boundary of the chemical potential the conduction band is maximally filled and El and E2 both vanish (Fig. 2). The solution describes a gapless metal with electron density nc. In the reverse direction, i.e. with increasing Ip 1, a gap and a corresponding fully occupied soliton band develop, and Ip I and El approach each other until at the upper boundary lpl = E l = E(:)(k)a metal-insulator transition to the usual charged soliton lattice (solution (2)) occurs. For the third soliton lattice solution (denoted by superscript (3)) we have lp I IEP), i.e., p in the soliton band. This solution which is confined to the domain IpI sEL2)(k) (Fig. 1) therefore again represents a metal. is a monotonically decreasing function of Ip I (Fig. 2). At its upper limit Ip I = Ei2)(k),we have E r ) (l p l , k ) = Ei2)(k),and solution (3) fits to the solution (2),again corresponding to a metal-insulator transition. For decreasing Ip I the electron density eventually also decreases until 1 p 1 has reached the centre of the soliton band and n = 1. Now the soliton band is half-filled. The thermodynamic inequality for the solutions (1) and (2) will be discussed in Section 5. a3’ ’ In the following we restrict our arguments to n-type doping. Because of the charge conjugation symmetry of the model the results are also valid for p-type doping. H. W. Streitwolf, H. Puff, Metallic solutions of the continuum model 99 Fig. 2 Band edges E , , E2 and (for the anomalous solution) E3 for k = 0.9, k = 0.95, and k = 0.99 in a degenerate ground state polymer (anomalous solution dotted). b) The broken symmetry polaron lattice solution. This solution (denoted by superscript (A)) is characterized by E3 f 0 and exists only in Ip 1 2 E3, provided po = p2. Since ,U is in the polaron the region E2 = -i2 band the solution represents a polaronic metal. The condition po = p2 reads explicitly and the gap equation (11) yields A solution of Eqs. (17), (18) and (9) exists in the range E I c r Ip(2,uA(k)(cf. Fig. 1). At the lower limit I,u1 = pA( k ) , EiA)vanishes (Fig. 2), and the anomalous solution E\A) fits to the third normal solution E y ) . With increasing lp I, both EiA)and EiA)increase and approach each other to coincide finally at the upper limit I , u I = E , c, where the polaronic metal ends in a half-filled Peierls insulator. The band edges were calculated numerically for all solutions. Results for three values of k are plotted in Fig. 2. Apparently the three normal solutions are connected with each other. The lower limit of the chemical potential of solution (1) o( in the conduction band) is independent of the modulus k and, therefore, represents the commensurate limit k-t 1. The upper limit of the anomalous polaron lattice solution again is independent of the modulus k, while its lower limit joins the normal solution (3) with E3 vanishing. The anomalous polaron lattice solution, therefore, is not infinitesimally connected to the stable soliton lattice (2) as was erroneously claimed by Takahashi [l]. The figures are based on the parameter values to= 2.75 eV, E, = 0.9 eV, and I = 0.338, which are appropriate to trans-polyacetylene. 100 Ann. Physik 3 (1994) 4 Thermodynamic potential and charge carrier density From Eq. ( 2 ) we obtain the thermodynamic grand potential per lattice site for degenerate ground state polymers (Ae = 0): for the solitonic lattice distortion and for the anomalous polaronic lattice distortion, with the auxiliary function (for ,u >0) d for E , 2 Ip I 2 E2 (solution (2)) fi(k,,u)= qb) otherwise . In general fi(k,,u)differs from the electron density per monomer Only for the insulating solution (2) and the k-+ I-limit (C-phase) both are equal. Guided by the numerical results we shall now discuss the thermodynamic inequality for the various solutions. Fig. 3 shows the grand potential (+nc,u) and Fig. 4 the excess electron density n (k,,u)- 1 versus the chemical potential for four different values of the modulus k. For each k there is a definite chemical potential for the insulating charged soliton lattice (phase (2), long dashed line) with a minimal grand potential. This represents the well-known stable phase (2) and is depicted by an asterisk in Fig. 3. A discontinuous phase transition from the C-phase to the stable phase (2) is found at p = 0.637 El = 0.573 eV (n- 1 = 3.07%). Infinitesimally adjacent to phase ( 2 ) there are the two solitonic metal solutions with the chemical potential in the H. W. Streitwolf. H. Puff. Metallic solutions of the continuum model 101 0 2-4.705 & P irai J .9 --+ 5 -4.005 (3)/ C 0.0- stable C-phase : -8- 0 n /, U ue-4.015 0.0 v , , , , , I , , , , , , , , , -, 0.2 0.4 0.6 , ,, , ,, 1 .o 0.0 Chemical potential/Elc 1.2 Fig. 3 Grand potential (+n,p) for the various solutions of the gap equations and four values of the modulus (k = 0.8, 0.9, 0.994, 0.999). The stable insulating soliton lattice is indicated by an asterisk *. 0 denotes the point where the chemical potential enters the soliton band. C,is the k+l-limit of phase (1). --- metallic soliton lattice phase (1) and C,;- - insulating soliton lattice phase (2); normal and dimerized commensurate phases, metallic soliton lattice phase (3); * * anomalous polaron lattice phase. - - phase (I>,,- .', .* ,, . ' . # k=0.999 0.0 0.2 0.4 0.6 010 Chemical potentral/Elc 1.0 1.2 1.4 Fig. 4 Excess electron density for the same values of the modulus k as in Fig. 3. The transition from phase (2) to phase (3) for k = 0.8 is denoted by 0 , the transition from phase (2) to phase (1) for k = 0.994 by an asterisk. --- phase (1); - - - phase (2); metallic soliton lattice phase (3); * * * anomalous polaron lattice phase. - 102 Ann. Physik 3 (1994) conduction band (phase (l), short dashed line) or the soliton band (phase (3), solid line). The anomalous metallic polaron lattice is denoted by dotted lines. Obviously, however, it violates the thermodynamic inequality (20) everywhere. For comparison we also plot in Fig.3 the grand potential for the undimerized normal phase and the dimerized commensurate C-phase with the chemical potential in the gap. C1 is the k-o-limit of phase (l), a dimerized lattice with ,u in the conduction band. It also does not satisfy the inequality (20) as already mentioned. Furthermore, calculations of the second functional derivative of the grand potential [20] in this commensurate phase with the chemical potential in the conduction band showed that this phase is unstable since the stationary point of the grand potential turned out to be a maximum instead of a minimum. The grand potential of the undimerized normal phase with no Peierls gap, however, yields a positive compressibility. From Fig. 4 we conclude that phase (3) joins the insulating phase (2) in accord with Eq. (20) if k s 0 . 8 whereas for k>0.8 this thermodynamic inequality is violated in a finite region adjacent to phase (2). For k = 0.8 we therefore find a discontinuous insulator metal transition (denoted by O ) , where the excess electron density jumps from 6.5% to 10.2% and the chemical potential enters the soliton band from above. Despite the fact that the grand potential of this metallic phase is somewhat larger than that of the insulating soliton lattice phase (2) it seems to be the only candidate for the experimentally observed metallic phase showing solitonic infrared spectra provided some further interactions might stabilize it. According to a proposal by Kivelson and Heeger [2] fluctuations or Coulomb interactions might serve as such factors. In a metallic soliton lattice the electrons are no longer bound to solitons but move freely through the lattice which still is subject to a solitonic lattice distortion. The density of states at the chemical potential p 6 E2 is rather large, but fluctuations will round off the singularity in the density of states at the upper soliton band edge [21]. The freely moving electrons will give rise to a large magnetic Pauli susceptibility. As found experimentally the infrared absorption due to the addition of dopant ions persists into the metallic state. It rises linearly with dopant concentration up to very high dopant densities [22] and may be attributed to the solitonic lattice distortion. Finally a metallic soliton lattice has some resemblance with the superpolarons introduced by Takahashi et al. [23]. At the upper edge of phase (2) the metallic phase (1) with the chemical potential in the conduction band satisfies inequality (20) if k50.994. At k = 0.994 a discontinuous metal transition (denoted by an asterisk) occurs where the excess electron density jumps from 4.6% to 6.2%. However, because of the large difference of the grand potential to that of the stable phase (2) (cf. Fig. 3) this transition seems to be a more unlikely candidate for the experimentally observed insulator-metal transition than that discussed above. 5 Solutions for systems with nondegenerate ground state Similar results as for degenerate ground state polymers were obtained for an extension of the Takayama Lin-Liu Maki model as given by Brazovskii and Kirova [13] who introduced an external bond dimerization to describe polymers with non-degenerate ground states like cis-polyacetylene. Also in this case besides the well-known insulating bipolaron lattice there are several metallic solutions of the gap equations (10) and (11) with a polaronic lattice distortion and the chemical potential in the conduction or H. W. Streitwolf, H. Puff, Metallic solutions of the continuum model 103 Fig. 5 Band edges E , , E, and E, for the various solutions of the gap equations in case of a finite external bond dimerization A, = 24 meV for k = 0.9. --- phase (1); - - - phase (2); -phase (3+); * phase (3-). polaron bands (solutions (1) or (3 k)), respectively. There is no solitonic lattice solution. The band edges for a finite external bond dimerization Ae = 24 meV are shown in Fig. 5 for k = 0.9. Comparison with Fig. 2 reveals an interesting topological connection between the various solutions. Actually the anomalous solution of the Ae+O case results from merging of two polaron lattice solutions with the chemical potential in the polaron band. For the solution with the. larger El (Er’)) we have sign r=1 while sign f = - 1 for the lower solution (E{3-)).The (3 -)-solution in its lower branch converges to the solution (3) in the A,+O-limit (Fig. 2). Thus for the normal solution (3) s i g n r = + I for , ~ = p ( ~ ’ ( k ) . In order to discuss the validity of the thermodynamic inequality (20) we have derived the grand potential and the charge carrier density also for the case Ae # 0. We find Af w ( k , p )= - p i ( k , p ) + - - 2ntoA 2nto 104 Ann. Physik 3 (1994) phase (2) o--- 0.2 0.0 0.4 0.6 0.8 , : ' kdJ.8 1.2 1.0 Chsrnlcal potontial/Elc Fig. 6 Excess electron density for the various solutions of the gap equations in case of a finite external bond dimerization Ae = 24 meV for two values of the modulus (k = 0.8 and 0.994). --- phase (1); --- phase (2); -phase (3+); * * phase (3-). - - - , I ,, ,,,, ,. , , ., . ,., , 0.008 u w' 3 s 5 0.004 & phase (3-)!-; 1 z -I 2 0.000 c C c 0 n -0 C E -0.004 (3 -0.008 0.2 0.4 0.6 0.8 Chemical potentlal/Elc 1 .O 1.2 Fig. 7 Grand potential (relative to the gapless N-phase) for the various solutions of the gap equations in case of a finite external bond dimerization Ae = 24 meV and two values of the modulus (k= 0.8 and 0.994). --- phase (1) and dimerized commensurate phase C,; - - - phase (2); -phase (3 +); .* . phase (3 -). Figs. 6 and 7 show the excess electron density per monomer and the grand potential relative to that of the gapless normal phase for two different values of the modulus k. Again the splitting of the former anomalous phase into phases (3 +) and (3 -) is found. Adjacent to phase (2) (long dashed line) there are phase (3+) (solid line) at the lower H. W. Streitwolf, H. Puff, Metallic solutions of the continuum model 105 end (USE$’)) (denoted by 0 ) and phase (1) (short dashed line) at the upper end (USE(:)).As judged from the thermodynamic inequality (20) we again observe in Fig. 6 a discontinuous transition into the metallic phase (3+) with ,usEl2’ in the polaron band for k = 0.8 (denoted by 0 )while for k > 0.8 the solution (3 +) fits to solution (2) with the wrong sign of the derivative. As in the degenerate ground state case (A,+O) this transition to the polaronic lattice phase (3 +) at k = 0.8 is expected to be related to the metal transition observed for nondegenerate ground state polymers like cis-polyacetylene, polyparaphenylene, and polyphenylene-vinylene, provided it is stabilized by further interactions as proposed by Kivelson and Heeger [ 2 ] . Again the grand potential corresponding to solution (1) deviates too much from that of the stable phase ( 2 ) to make phase (1) a likely candidate for an observable metallic phase. The results for finite external bond dimerization are therefore qualitatively very similar to those in case of vanishing Ae. 6 Summary We have solved the gap equations for the continuum version of the Su-SchriefferHeeger-model extended by Brazovskii and Kirova to take into account an external bond dimerization. We have found besides the well known stable insulating lattice solution a number of metallic solutions. Even in case of vanishing external bond dimerization there is one solution representing a polaron lattice distortion, i.e. a solution with a broken symmetry. We parameterized the solutions by the band edges of the electronic energy spectrum and discussed them in terms of the chemical potential. To gain insight into the stability of these solutions we have calculated the corresponding grand potentials and carrier densities and discussed the thermodynamic inequality (20). As the most likely candidate for a realistic metallic phase we thus found a soliton lattice with the chemical potential in the soliton band for degenerate ground state polymers as trans-polyacetylene and a polaron lattice with the chemical potential in the upper polaron band in case of a finite external bond dimerization. Its grand potential is infinitesimally adjacent to that of the usual stable insulating soliton or polaron lattice phase and its compressibility is positive for excess electron densities above 6.5%. Fluctuations or interactions not considered in the model used here (e.g. electron correlation or interchain coupling) might stabilize this metallic phase. Very similar results are obtained for nondegenerate ground state polymers in the Brazovskii-Kirova model. References Takahashi, Prog. Theor. 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