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Density-functional study of the metallic state of trans-polyacetylene
Solid State Communications, Printed in Great Britain.
DENSITY-FUNCTIONAL
0038- 1098/93 $6.00 + .OO Pergamon Press Ltd
Vol. 87, No. 5, pp. 487-491, 1993.
STUDY OF THE METALLIC
STATE OF TRANS-POLYACETYLENE
J. Paloheimo Semiconductor
Laboratory,
Technical Research Centre of Finland, SF-021 50 Espoo, Finland and J. von Boehm
Department
of Mathematics, Helsinki University of Technology,
SF-021 50 Espoo, Finland
(Received 23 January 1993 by A.A. Maradudin) The dimerization of trans-polyacetylene is studied as a function of doping, using a self-consistent density-functional calculation in the local-density approximation. The dimerization becomes suppressed at a concentration y = y, g 0.04 (0.03) extra holes (electrons) per CH unit. The results demonstrate the importance of the rr electron states at the Brillouin zone boundaries for lowering the total energy and forming the electron density difference between the single and double bonds, and suggest an undimerized metallic state at higher doping levels y > yc.
1. INTRODUCTION Trans-POLYACETYLENE (trans-PA) is known to undergo a transition to a metallic state by doping. Characteristic to this transition is an abrupt increase in the Pauli susceptibility at a doping level of about 6% (per CH unit) [l, 21. The experimental Pauli spin susceptibility and the electronic specific heat of such highly doped samples indicate a high metallic density of states of g(EF) E 0.08 - 0.12 states/eV-carbon atom at the Fermi energy [l-3]. The conductivities c are large, even exceeding lo5 Scn- t at room temperature [4] and suggesting mean free paths of about 100 A [5]. The temperature dependence is still mostly nonmetallic, the conductivity decreasing with decreasing temperature, but remaining large near zero temperature [4]. However, some samples have shown signs of metallic temperature dependence at high temperatures, suggesting here a contribution from the intrinsic optical phonon-scattering-limited conductivity in the metallic chains [4, 51. On the other hand, the infrared active vibrational (IRAV) modes indicating the presence of charged defects (e.g. solitons or polarons) [6] persist up to the highest doping levels [2, 71. The metallic state has been suggested to be a soliton lattice or an incommensurate Peierls state (the latter corresponding to a dense soliton lattice) [2, 8-l 11, a polaron lattice [12, 131, or an undimerized chain [5, 141. In the last model the dimerization is absent whereas the other models are based on modifications of the dimerized state with the
displacements u, = (-1)“~s of the carbon atoms parallel to the polymer axis (x-axis) (us is the dimerization amplitude). The dependence of the degree of dimerization on the doping level is thus an important factor. The Hartree-Fock (HF) [15, 161 and extended Htickel [17] calculations give the result that the dimerization is quenched by increasing doping. The purpose of the present paper is to study the effect of doping on the dimerization of a single, infinite chain of trans-PA using a self-consistent (SC) method and the density-functional (DF) theory in the local-density approximation (LDA) for exchange and correlation. 2. METHOD We use the SC linear-combination-of-Gaussianorbitals (LCGO) method [18, 191 and a fully general all-electron crystal potential (no pseudopotential or muffin-tin approximation). The exchange-correlation model of Ceperley and Alder parametrized by Perdew and Zunger [20] is used. A restricted basis set including the hydrogen 1s and carbon Is, 2s, 2pX, 2p,, and 2p, orbitals is used. The electron density is divided into two parts, p(r) = p,,(r) + Ap(r). Here p. is the sum of the spherically symmetric pseudoatom electron densities neutralizing the corresponding nuclear charges, and Ap is a relatively small and mostly smooth deformation density, which is expressed in terms of 1l3 = 133 1 plane waves. In the
487
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THE METALLIC
STATE OF Z&INS-POLYACETYLENE
case of a half-filled band, the total charge of p(r) is 14 per unit cell of a dimer C2H2, and Ap(r) is electrically neutral. For other band fillings, the absolute value of the coefficient Ap(G = 0) equals SN/I’s where SN is the number of extra electrons per unit cell of a dimer C2H2, having the volume VO. The extra electron charge has to be balanced by background charges. In normal doping the dopant counter ions reside beside the polymer chains maintaining the charge neutrality of the crystal. In our approach, we approximate the dopant ions by a uniform background charge density lelSN/Vo, which neutralizes the system [and the coefficient Ap(G = O)]. This should be a reasonable approximation at high doping levels because of the strong screening [14] and the overlap of the dopant potentials [21]. In this study ]SN] ranged from 0.0 to 0.3 charges per unit cell, corresponding to a doping level y = ]SN)/2 of O-15% (per CH unit). Both electron (6N > 0) and hole (6N < 0) additions were considered. The C-C-C bond angies were fixed to 120”, the C-H bonds (rC_H = 1.09 A) pointed always perpendicularly to the polymer axis (x-axis), and the unit cell dimensions were fixed to the values a = 2.434A, b = 6.350& c = 4.233 A, The two carbon atoms in the unit cell move approximatively parallel to the Xaxis, the projected displacements being fu and --u. The value of a was obtained using the experimental bond lengths of rc=c = 1.36A and rc_c = 1.45 A [22] and our fixed C-C-C angle. The values of b and c were large enough to make the adjacent chains practically noninteracting. For the present total energy calculations, we used a regular mesh of 21 k’s at k/(r/a) = -1.0, -0.9, -0.8,. . . ,O.O,. . . , +0.8, f0.9, +l.O in the onedimensional Brillouin zone (BZ) (the boundaries k = -r/a and k = +7r/a with a half weight) because this allows us to study the effect of doping with a sufficient uniform accuracy. 3. RESULTS The total energy AE vs u for different doping levels y is shown in Fig. 1. A limited number of points is presented for y = 15% because the numerical accuracy of AE decreases for high y’s due to the larger Ap(r). The dimerization amplitude uo, given by the minimum of AE, becomes exceedingly suppressed with increasing y. The r-r* gap at the BZ boundary [E,(T-n*)] is found to be proportional to u and to decrease with decreasing uo. The transition from the semiconducting state to the undimerized, gapless, metallic state occurs (within our accuracy) at a critical concentration y = y, g 4(3)% per CH unit for
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Fig. 1. Total energy AE (per dimer C2H2) as a function of the dimerization amplitude u. The doping levels for holes (a) and electrons (b), y = 0, 2.5, 5, 10, and 15% per CH unit, are indicated in the figures. The zero levels are arbitrarily chosen, and the lines are only to guide the eye. hole (electron) doping (the AE-curves at low u’s for electrons are less accurate because it is numerically difficult to treat the cancelling x and X* BZ boundary states [19]). The calculated 7r electron density above the polymer plane in the center of the double and single bonds as a function of u for different doping levels y is shown in Fig. 2. The increase of the number of extra holes (electrons) weakens (strengthens) the double (single) bonds thereby suppressing the charge density wave (the difference in the electron density between the double and single bonds). The same conclusions can be drawn from the characteristics of the coefficient Ap[G = (2n/a)u,]. The discontinuity in the charge density at u = 0 for small y’s is due to the discretization of the k mesh [19].
THE METALLIC
Vol. 87, No. 5
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I
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0.01
0.0
STATE OF 7’R/INS-POLYACETYLENE
0.02
u(A)
0.03
r
0.05 m
’ do
s 0.04
0.0: I
0.0
I
1
0.01
1
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Fig. 2. x electron density pli above the polymer plane between the carbon atoms in the double bond at (0.0, 0.0, ~0.458 A) (rism . . g curves) and single bond at (a/2, 0.0, 2 0.458 A) (descending curves) as a function of the dimerization amplitude u: for extra holes (a) and extra electrons (b). The doping levels y (in % per CH unit) are indicated in the figures. a0 is the Bohr radius. The exchange-correlation energy AE,, in the LDA favors dimerization for y = 0% [19]. We found that the shape of AE,,(u) is practically unaffected by extra charges for y 2 15%, although the absolute values are shifted due to the change in the total number of the electrons from the original 14 to 14 + SN per unit cell. 4. DISCUSSION The 7r band states at the BZ boundaries are important for lowering the total energy and forming an electron density difference between the single and double bonds in the dimerization [19, 231. Therefore, extra holes added into the polymer chain are expected
489
to lead to a reduced dimerization, because the important states at the BZ boundary become unoccupied [19, 231. A corresponding effect may be expected for extra electrons, too, because the occupied r* conduction band states at the BZ boundary should cancel the effect of the 7r valence band states at the BZ boundary. In fact, the results of our calculations, shown in Figs. 1 and 2, clearly confirm such a behaviour. The calculated dimerization stabilization energy and amplitude u. decrease with an increasing number of extra holes or electrons as shown in Fig. 1. The transition to a metallic, undimerized state with Eg(xx*) =0 takes place at the critical doping level y, g 4(3)% fo r h o 1es (e 1ec t rons). Since the DF LDA calculations are known to underestimate u. (see [19] and references therein), the actual y, should be somewhat larger. We associate this qualitative change to the experimentally detected sharp increase in the Pauli susceptibility at 2 6% [l, 21. The calculated density of states at the Fermi energy is g(EF) 2 0.12 states/eV-carbon atom above yf, in fair agreement with the experimental values of E 0.08-0.12 states/eVcarbon atom in metallic tram-PA [l-3]. Our results, shown in Fig. 1, agree qualitatively with the HF and extended Htickel results of [15-171. The HF calculation by Tanaka et al. [16] gives the interesting trend dimerized lattice + soliton lattice --+ almost undimerized lattice with increasing number of extra holes. Moreover, their calculated bond lengths become increasingly equal with increasing hole doping. Although we have not been able to study the soliton lattice case because a considerable large unit cell would be needed, we expect that the SC DF LDA calculations would also give the soliton lattice ground state for 0 < y < y,. Next, we consider the different models suggested for the metallic state. The displacements of the CH units in the polaronic lattice at large dopings may be approximated by the expression u, g (-l)n~g,p{ 1 + A sin[(47ry/a)x,]},
(1)
where uo,p is the average dimerization amplitude and A the amplitude of the periodic modulation (0 I A I: 1) with the period a/(2y). The value of A should decrease with increasing doping y. In the limit A + 0, uo,p and the gap between the polaron bands (see, e.g. [13]) would approach u. and E,(T-T*) of a doped uniformly dimerized chain, respectively. Our results (Fig. 1) thus indirectly indicate that the possible polaronic lattice should have a pronounced tendency to change into the undimerized metallic chain with increasing doping. The displacements of the CH units in the soliton
490
THE METALLIC
STATE OF TRANS-POLYACETYLENE
lattice at large dopings may be approximated expression [8] a, g (-1)“~~
sin[(2ry/a)x,],
by the (2)
where uo,s is the dimerization amplitude of the soliton lattice. The band structure has a narrow gap at EF [g(&) = 01,which, in fact, is expected to stabilize the soliton lattice with respect to the undimerized state [23, 241. However, a gap (or a pseudo-gap) is not supported by the experiments but a high density of extended states at EF is observed [l, 4, 51. Although disorder (possibly together with interchain interactions) could induce a finite density of states at EF [2, 8, IO], the experiments show that the high conductivity is based on good order, not on disorder [4, 51. The rapidity of the increase of the Pauli spin susceptibility does not support models based on disorder induced destruction of the incommensurate gap at EF [l]. The results of the Takayama-Lin-Liu-Maki (TLM) [25] model for the energy difference between the undimerized and fully dimerized state (see Fig. 1 in [9]) is qualitatively in agreement with our result, although the soliton lattice has always the lowest energy in the Su-Schrieffer-Heeger (SSH) [26] and TLM models. The SSH and TLM models take the electron-lattice couplings into account, but neglect electron-electron interactions which are known to be important (see e.g. [27] and references therein). In fact, the inclusion of electron-electron interactions within the Hubbard-Peierls model leads to a possibility of a gapless, undimerized metallic state at high doping levels [14]. For reasonable parameter values a transition from the soliton lattice to the metallic state would take place at y = yc z 3% [14]. The doping induced IRAV modes become inactive if the charge density becomes uniform. Therefore, the persistence of the IRAV modes under doping is usually considered to be in agreement with the soliton lattice but in disagreement with the undimerized state and the polaron lattice which should have a nearly uniform electron density and a small oscillator strength at high doping levels [9]. However, although the existence of the IRAV modes does indicate nonuniformities in the electron density, the association to the soliton lattice is indirect [5, 61 (see also [28]). Furthermore, thin barriers between relatively large and perfect metallic regions are needed for explaining the experimental nonmetallic decrease of u with decreasing temperature [4].
an increasing concentration y of extra holes or electrons. An undimerized state is obtained at y = y, 2 4(3)% per CH unit for hole (electron) doping. The results show the importance of the rr band states near the Brillouin zone boundaries for lowering the total energy and forming an electron density difference between the single and double bonds in the dimerization. Our calculations give a simple undimerized metallic state above y, in agreement with most experimental results of the highly doped samples of good quality. Acknowledgements-The authors would like to thank Prof. P. Kuivalainen for the early contributions and fruitful discussions, and Dr H. Stubb for his interest and encouragement during this work. One of us (J.P.) was supported by the Technology Development Centre, Finland (TEKES). REFERENCES 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12.
5. CONCLUSIONS To conclude, our calculations giveOa decrease of the dimerization amplitude from 0.01 A to zero with
Vol. 87, No. 5
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F.Moraes, J. Chen, T.-C. Chung & A. J. Heeger, Synth. Met. 11, 271 (1985); J. Chen & A. J. Heeger, Synth. Met. 24, 311 (1988). X.Q. Yang, D.B. Tanner, M.J. Rice, H.W. Gibson, A. Feldblum & A.J. Epstein, Solid State Commun. 61, 335 (1987). D. Moses, A. Denenstein, A. Pron, A.J. Heeger & A.G. MacDiarmid, Solid State Commun. 36, 219 (1980). A.B. Kaiser & S.C. Graham, Synth. Met. 36,367 (1990); Th. Schimmel, D. Gllser, M. Schwoerer & H. Naarmann, Conjugated Polymers (Edited by J.L. Bredas & R. Silbey), pp. 49- 111. Kluwer, Dordrecht (1991); T. Ishiguro, H. Kaneko, Y. Nogami, H. Ishimoto, H. Nishiyama, J. Tsukamoto, A. Takahashi, M. Yamaura, T. Hagiwara & K. Sato, Phys. Rev. Lett. 69, 660 (1992); M.T.Ahmed, A.B. Kaiser, S. Roth & M.D. Migahed, J. Phys. D: Appl. Phys. 25,79 (1992). S. Kivelson & A.J. Heeger, Synth. Met. 22, 371 (1988); G. Paasch, Synth. Met. 51, 7 (1992). B. Horovitz, Solid State Commun. 41, 729 (1982). D.B.Tanner, G.L. Doll, A.M. Rao, P.C. Eklund. G.A. Arbuckle & A.G. MacDiarmid, Synth. Met. 28, D141 (1989). E.J. Mele & M.J. Rice, Phys. Rev. B23, 5397 (1981). H.-Y. Choi & E.J. Mele, Phys. Rev. B34, 8750 (1986). S. Stafstrom, Phys. Rev. B43, 9158 (1991). E.M. Conwell, H.A. Mizes & S. Jeyadev, Phys. Rev. B40, 1630 (1989). S. Kivelson & A.J. Heeger, Phys. Rev. Lett. 55, 308 (1985); Synth. Met. 17, 183 (1987); J. Voit & H. Biittner, Solid State Commun. 67, 1233 (1988). S. Stafstrijm 8z J.L. Bredas, Phys. Rev. B38, 4180 (1988).
Vol. 87, No. 5 14. 15.
16. 17. 18. 19. 20. 21. 22. 23.
THE METALLIC
STATE OF TRANS-POLYACETYLENE
D. Baeriswyl, J. Carmelo & K. Maki, Synth. Met. 21, 271 (1987). J.L. Brtdas, R.R. Chance & R. Silbey, J. Whys. Chem. 85, 756 (1981); T. Yamabe, K. Tanaka, T. Koike & M. Ueda, Mol. Cryst. Liq. Cryst. 117, 185 (1985). K. Tanaka, M. Okada, T. Koike & T. Yamabe, Synth. Met. 31, 181 (1989). M. Kertesz, F. Vonderviszt & S. Pekker, Chem. Phys. Lett. 90,430 (1982). J. von Boehm, P. Kuivalainen & J.-L. Calais, Phys Rev. B35,8177 (1987). J. Paloheimo & J. von Boehm, Phys. Rev. B46, 4304 (1992); Synth. Met. 55-57, 4443 (1993) and to be published. J.P. Perdew & A. Zunger, Phys. Rev. B23, 5048 (1981). C. Fredriksson & S. Stafstriim, Synth. Met. 4143, 3467 (1991). H. Kahlert, 0. Leitner & G. Leising, Synth. Met. 17, 467 (1987). R.E. Peierls, Quantum Theory of Solids, Clarendon, Oxford (1955).
24.
25. 26. 27.
28.
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Interchain couplings may close the narrow gap [H.A. Mizes & E.M. Conwell, Phys. Rev. B43, 9053 (1991); P. Vogl & D.K. Campbell, Phys. Rev. B41, 12797 (1990); S. Stafstriim, accepted for publication in Phys. Rev. B] thereby weakening the soliton lattice distortion (see
PI).
H. Takayama, Y.R. Lin-Liu & K. Maki, Phys. Rev. B21, 2388 (1980). W.P. Su, J.R. Schrieffer & A.J. Heeger, Phys. Rev. Lett. 42, 1698 (1979); Phys. Rev. B22,2099 (1980). D. Baeriswyl, P. Horsch & K. Maki, Phys. Rev. Lett. 60, 70 (1988); J.T. Gammel & D.K. Campbell, Phys. Rev. Lett. 60, 71 (1988); S. Kivelson, W.-P. Su, J.R. Schrieffer & A.J. Heeger, Phys. Rev. L&t. 60, 72 (1988); J.L. BrCdas & A.J. Heeger, Phys. Rev. Lett. 63,2534 (1989); C. Wu, X. Sun & K. Nasu, Phys. Rev. Lett. 63, 2535 (1989). S.A. Kivelson 8z M.I. Salkola, Synth. Met. 44, 281 (1991).
DENSITY-FUNCTIONAL
0038- 1098/93 $6.00 + .OO Pergamon Press Ltd
Vol. 87, No. 5, pp. 487-491, 1993.
STUDY OF THE METALLIC
STATE OF TRANS-POLYACETYLENE
J. Paloheimo Semiconductor
Laboratory,
Technical Research Centre of Finland, SF-021 50 Espoo, Finland and J. von Boehm
Department
of Mathematics, Helsinki University of Technology,
SF-021 50 Espoo, Finland
(Received 23 January 1993 by A.A. Maradudin) The dimerization of trans-polyacetylene is studied as a function of doping, using a self-consistent density-functional calculation in the local-density approximation. The dimerization becomes suppressed at a concentration y = y, g 0.04 (0.03) extra holes (electrons) per CH unit. The results demonstrate the importance of the rr electron states at the Brillouin zone boundaries for lowering the total energy and forming the electron density difference between the single and double bonds, and suggest an undimerized metallic state at higher doping levels y > yc.
1. INTRODUCTION Trans-POLYACETYLENE (trans-PA) is known to undergo a transition to a metallic state by doping. Characteristic to this transition is an abrupt increase in the Pauli susceptibility at a doping level of about 6% (per CH unit) [l, 21. The experimental Pauli spin susceptibility and the electronic specific heat of such highly doped samples indicate a high metallic density of states of g(EF) E 0.08 - 0.12 states/eV-carbon atom at the Fermi energy [l-3]. The conductivities c are large, even exceeding lo5 Scn- t at room temperature [4] and suggesting mean free paths of about 100 A [5]. The temperature dependence is still mostly nonmetallic, the conductivity decreasing with decreasing temperature, but remaining large near zero temperature [4]. However, some samples have shown signs of metallic temperature dependence at high temperatures, suggesting here a contribution from the intrinsic optical phonon-scattering-limited conductivity in the metallic chains [4, 51. On the other hand, the infrared active vibrational (IRAV) modes indicating the presence of charged defects (e.g. solitons or polarons) [6] persist up to the highest doping levels [2, 71. The metallic state has been suggested to be a soliton lattice or an incommensurate Peierls state (the latter corresponding to a dense soliton lattice) [2, 8-l 11, a polaron lattice [12, 131, or an undimerized chain [5, 141. In the last model the dimerization is absent whereas the other models are based on modifications of the dimerized state with the
displacements u, = (-1)“~s of the carbon atoms parallel to the polymer axis (x-axis) (us is the dimerization amplitude). The dependence of the degree of dimerization on the doping level is thus an important factor. The Hartree-Fock (HF) [15, 161 and extended Htickel [17] calculations give the result that the dimerization is quenched by increasing doping. The purpose of the present paper is to study the effect of doping on the dimerization of a single, infinite chain of trans-PA using a self-consistent (SC) method and the density-functional (DF) theory in the local-density approximation (LDA) for exchange and correlation. 2. METHOD We use the SC linear-combination-of-Gaussianorbitals (LCGO) method [18, 191 and a fully general all-electron crystal potential (no pseudopotential or muffin-tin approximation). The exchange-correlation model of Ceperley and Alder parametrized by Perdew and Zunger [20] is used. A restricted basis set including the hydrogen 1s and carbon Is, 2s, 2pX, 2p,, and 2p, orbitals is used. The electron density is divided into two parts, p(r) = p,,(r) + Ap(r). Here p. is the sum of the spherically symmetric pseudoatom electron densities neutralizing the corresponding nuclear charges, and Ap is a relatively small and mostly smooth deformation density, which is expressed in terms of 1l3 = 133 1 plane waves. In the
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STATE OF Z&INS-POLYACETYLENE
case of a half-filled band, the total charge of p(r) is 14 per unit cell of a dimer C2H2, and Ap(r) is electrically neutral. For other band fillings, the absolute value of the coefficient Ap(G = 0) equals SN/I’s where SN is the number of extra electrons per unit cell of a dimer C2H2, having the volume VO. The extra electron charge has to be balanced by background charges. In normal doping the dopant counter ions reside beside the polymer chains maintaining the charge neutrality of the crystal. In our approach, we approximate the dopant ions by a uniform background charge density lelSN/Vo, which neutralizes the system [and the coefficient Ap(G = O)]. This should be a reasonable approximation at high doping levels because of the strong screening [14] and the overlap of the dopant potentials [21]. In this study ]SN] ranged from 0.0 to 0.3 charges per unit cell, corresponding to a doping level y = ]SN)/2 of O-15% (per CH unit). Both electron (6N > 0) and hole (6N < 0) additions were considered. The C-C-C bond angies were fixed to 120”, the C-H bonds (rC_H = 1.09 A) pointed always perpendicularly to the polymer axis (x-axis), and the unit cell dimensions were fixed to the values a = 2.434A, b = 6.350& c = 4.233 A, The two carbon atoms in the unit cell move approximatively parallel to the Xaxis, the projected displacements being fu and --u. The value of a was obtained using the experimental bond lengths of rc=c = 1.36A and rc_c = 1.45 A [22] and our fixed C-C-C angle. The values of b and c were large enough to make the adjacent chains practically noninteracting. For the present total energy calculations, we used a regular mesh of 21 k’s at k/(r/a) = -1.0, -0.9, -0.8,. . . ,O.O,. . . , +0.8, f0.9, +l.O in the onedimensional Brillouin zone (BZ) (the boundaries k = -r/a and k = +7r/a with a half weight) because this allows us to study the effect of doping with a sufficient uniform accuracy. 3. RESULTS The total energy AE vs u for different doping levels y is shown in Fig. 1. A limited number of points is presented for y = 15% because the numerical accuracy of AE decreases for high y’s due to the larger Ap(r). The dimerization amplitude uo, given by the minimum of AE, becomes exceedingly suppressed with increasing y. The r-r* gap at the BZ boundary [E,(T-n*)] is found to be proportional to u and to decrease with decreasing uo. The transition from the semiconducting state to the undimerized, gapless, metallic state occurs (within our accuracy) at a critical concentration y = y, g 4(3)% per CH unit for
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Fig. 1. Total energy AE (per dimer C2H2) as a function of the dimerization amplitude u. The doping levels for holes (a) and electrons (b), y = 0, 2.5, 5, 10, and 15% per CH unit, are indicated in the figures. The zero levels are arbitrarily chosen, and the lines are only to guide the eye. hole (electron) doping (the AE-curves at low u’s for electrons are less accurate because it is numerically difficult to treat the cancelling x and X* BZ boundary states [19]). The calculated 7r electron density above the polymer plane in the center of the double and single bonds as a function of u for different doping levels y is shown in Fig. 2. The increase of the number of extra holes (electrons) weakens (strengthens) the double (single) bonds thereby suppressing the charge density wave (the difference in the electron density between the double and single bonds). The same conclusions can be drawn from the characteristics of the coefficient Ap[G = (2n/a)u,]. The discontinuity in the charge density at u = 0 for small y’s is due to the discretization of the k mesh [19].
THE METALLIC
Vol. 87, No. 5
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0.01
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STATE OF 7’R/INS-POLYACETYLENE
0.02
u(A)
0.03
r
0.05 m
’ do
s 0.04
0.0: I
0.0
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1
0.01
1
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u(b)
0.03
Fig. 2. x electron density pli above the polymer plane between the carbon atoms in the double bond at (0.0, 0.0, ~0.458 A) (rism . . g curves) and single bond at (a/2, 0.0, 2 0.458 A) (descending curves) as a function of the dimerization amplitude u: for extra holes (a) and extra electrons (b). The doping levels y (in % per CH unit) are indicated in the figures. a0 is the Bohr radius. The exchange-correlation energy AE,, in the LDA favors dimerization for y = 0% [19]. We found that the shape of AE,,(u) is practically unaffected by extra charges for y 2 15%, although the absolute values are shifted due to the change in the total number of the electrons from the original 14 to 14 + SN per unit cell. 4. DISCUSSION The 7r band states at the BZ boundaries are important for lowering the total energy and forming an electron density difference between the single and double bonds in the dimerization [19, 231. Therefore, extra holes added into the polymer chain are expected
489
to lead to a reduced dimerization, because the important states at the BZ boundary become unoccupied [19, 231. A corresponding effect may be expected for extra electrons, too, because the occupied r* conduction band states at the BZ boundary should cancel the effect of the 7r valence band states at the BZ boundary. In fact, the results of our calculations, shown in Figs. 1 and 2, clearly confirm such a behaviour. The calculated dimerization stabilization energy and amplitude u. decrease with an increasing number of extra holes or electrons as shown in Fig. 1. The transition to a metallic, undimerized state with Eg(xx*) =0 takes place at the critical doping level y, g 4(3)% fo r h o 1es (e 1ec t rons). Since the DF LDA calculations are known to underestimate u. (see [19] and references therein), the actual y, should be somewhat larger. We associate this qualitative change to the experimentally detected sharp increase in the Pauli susceptibility at 2 6% [l, 21. The calculated density of states at the Fermi energy is g(EF) 2 0.12 states/eV-carbon atom above yf, in fair agreement with the experimental values of E 0.08-0.12 states/eVcarbon atom in metallic tram-PA [l-3]. Our results, shown in Fig. 1, agree qualitatively with the HF and extended Htickel results of [15-171. The HF calculation by Tanaka et al. [16] gives the interesting trend dimerized lattice + soliton lattice --+ almost undimerized lattice with increasing number of extra holes. Moreover, their calculated bond lengths become increasingly equal with increasing hole doping. Although we have not been able to study the soliton lattice case because a considerable large unit cell would be needed, we expect that the SC DF LDA calculations would also give the soliton lattice ground state for 0 < y < y,. Next, we consider the different models suggested for the metallic state. The displacements of the CH units in the polaronic lattice at large dopings may be approximated by the expression u, g (-l)n~g,p{ 1 + A sin[(47ry/a)x,]},
(1)
where uo,p is the average dimerization amplitude and A the amplitude of the periodic modulation (0 I A I: 1) with the period a/(2y). The value of A should decrease with increasing doping y. In the limit A + 0, uo,p and the gap between the polaron bands (see, e.g. [13]) would approach u. and E,(T-T*) of a doped uniformly dimerized chain, respectively. Our results (Fig. 1) thus indirectly indicate that the possible polaronic lattice should have a pronounced tendency to change into the undimerized metallic chain with increasing doping. The displacements of the CH units in the soliton
490
THE METALLIC
STATE OF TRANS-POLYACETYLENE
lattice at large dopings may be approximated expression [8] a, g (-1)“~~
sin[(2ry/a)x,],
by the (2)
where uo,s is the dimerization amplitude of the soliton lattice. The band structure has a narrow gap at EF [g(&) = 01,which, in fact, is expected to stabilize the soliton lattice with respect to the undimerized state [23, 241. However, a gap (or a pseudo-gap) is not supported by the experiments but a high density of extended states at EF is observed [l, 4, 51. Although disorder (possibly together with interchain interactions) could induce a finite density of states at EF [2, 8, IO], the experiments show that the high conductivity is based on good order, not on disorder [4, 51. The rapidity of the increase of the Pauli spin susceptibility does not support models based on disorder induced destruction of the incommensurate gap at EF [l]. The results of the Takayama-Lin-Liu-Maki (TLM) [25] model for the energy difference between the undimerized and fully dimerized state (see Fig. 1 in [9]) is qualitatively in agreement with our result, although the soliton lattice has always the lowest energy in the Su-Schrieffer-Heeger (SSH) [26] and TLM models. The SSH and TLM models take the electron-lattice couplings into account, but neglect electron-electron interactions which are known to be important (see e.g. [27] and references therein). In fact, the inclusion of electron-electron interactions within the Hubbard-Peierls model leads to a possibility of a gapless, undimerized metallic state at high doping levels [14]. For reasonable parameter values a transition from the soliton lattice to the metallic state would take place at y = yc z 3% [14]. The doping induced IRAV modes become inactive if the charge density becomes uniform. Therefore, the persistence of the IRAV modes under doping is usually considered to be in agreement with the soliton lattice but in disagreement with the undimerized state and the polaron lattice which should have a nearly uniform electron density and a small oscillator strength at high doping levels [9]. However, although the existence of the IRAV modes does indicate nonuniformities in the electron density, the association to the soliton lattice is indirect [5, 61 (see also [28]). Furthermore, thin barriers between relatively large and perfect metallic regions are needed for explaining the experimental nonmetallic decrease of u with decreasing temperature [4].
an increasing concentration y of extra holes or electrons. An undimerized state is obtained at y = y, 2 4(3)% per CH unit for hole (electron) doping. The results show the importance of the rr band states near the Brillouin zone boundaries for lowering the total energy and forming an electron density difference between the single and double bonds in the dimerization. Our calculations give a simple undimerized metallic state above y, in agreement with most experimental results of the highly doped samples of good quality. Acknowledgements-The authors would like to thank Prof. P. Kuivalainen for the early contributions and fruitful discussions, and Dr H. Stubb for his interest and encouragement during this work. One of us (J.P.) was supported by the Technology Development Centre, Finland (TEKES). REFERENCES 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12.
5. CONCLUSIONS To conclude, our calculations giveOa decrease of the dimerization amplitude from 0.01 A to zero with
Vol. 87, No. 5
13.
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