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Confinement by charge gap in organic conductor Bechgaard salts
Physica B 281&282 (2000) 684}685
Con"nement by charge gap in organic conductor Bechgaard salts M. Tsuchiizu!, Y. Suzumura!,",* !Department of Physics, Nagoya University, Nagoya 464-8602, Japan "CREST, Japan Science and Technology Corporation (JST), Japan
Abstract In Bechgaard salts, which are e!ectively half-"lling due to dimerization, a con"nement has been found for the large charge gap in the optical experiments. In order to understand the crucial role of the charge gap, the irrelevance of interchain hopping corresponding to the con"nement is calculated for two-coupled chains of quarter-"lled Hubbard model with dimerization. It is demonstrated that a transition from decon"nement to con"nement occurs when the charge gap induced by the dimerization becomes larger than the interchain hopping. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Low-dimensional organic Bechgaard salts; Charge gap; Mott insulator; Dimerized Hubbard model
Bechgaard salts, which are quasi-one-dimensional organic conductors, are strongly correlated systems since the repulsive interaction has a magnitude of the order or even larger than the bandwidth. Besides spin density wave state at low temperatures, the normal states of these conductors exhibit unconventional properties found in optical experiments [1]. With increasing interchain transfer integral, a transition from an insulating state to a metallic state occurs indicating a fact that electrons con"ned to the individual chains are decon"ned. In the present paper, such a transition in two-coupled chains with quarter-"lling is examined by extending the previous general calculations, to a study of a model including the dimerization explicitly. We consider a Hamiltonian given by H"! + [t#(!1)jt ](cs c #h.c.) $ jpl j`1pl j,p,l
* Correspondence address. Department of Physics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan. Tel.: 81-52789-2437; fax: 81-52-789-2928. E-mail address: [email protected] (Y. Suzumura)
! 2t + (cs c #h.c.) M jp1 jp2 j,p # ;+ n n , (1) jtl jsl j,l where cs denotes an operator for the electron at the jth jpl site of the lth chain (l"1,2) with spin pn "cs c and jpl jpl jpl t comes from the dimerization. By diagonalizing the $ bilinear terms in Eq. (1), Fermi momentum (velocity) is given by k "p/2aG2t /v (v "J2ta[1!(t /t)2]/ FB M F F $ [1#(t /t)2]1@2) [2]. By applying the bosonization $ method to the lower band, the e!ective Hamiltonian is rewritten as H%&&"H%&&#H%&& [3,4]. The "rst term is 0 */5 given by H%&&"+ (v /4p): dxMK~1(Rh )2#K (Rh )2N 0 l l l l` l l~ (l"o, p, C, S) where v "v (1#;a/pv )1@2, v " o F F p v (1!;a/pv )1@2, K "[1/(1#;a/pv )]1@2, K "[1/(1! F F o F p ;a/pv )]1@2, v "v "v and K "K "1. Phase variF C S F C S ables, h and h (h and h ), express #uctuations o` p` C` S` for the total (transverse) charge density and spin density, respectively. The second term of H%&& is expressed as g lp,l{p{ H%&&" + */5 2p2a2 l,l{,p,p{
P
dx cos J2h cos(J2h !d x), lp l{p{ l{p{
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 2 1 2 - 0
(2)
M. Tsuchiizu, Y. Suzumura / Physica B 281&282 (2000) 684}685
Fig. 1. The l-dependence of t (l) (solid curves) and K (l) (dashed M o curves) is shown for ;"5 (1) and 8 (2) where t "0.1 and M t "0.05. $
where the coupling constants for the forward and backward scatterings are given by !g "g " C`,S~ C~,S` g "!g "!g "!g ";a p`,C` p`,C~ p`,S` p`,S~ with lattice constant a. Quantities g ("0) and C`,S` g ("0) appear through renormalization. We take C~,S~ t"1. The coupling constants for the umklapp scattering are given by g "g "!g "g " o`,C` o`,C~ o`,S` o`,S~ !2;a(t /t)/[1#(t /t)2] [2]. In Eq. (2), d is "nite only $ $ lp for d ("8t /v ) and g "0 for the others. The C` M F lp,l{p{ energy-dependence is calculated by the renormalization group method where the relation between l and energy, u, and/or temperature, ¹, is given by l"ln(=/u)"ln(=/¹) with = being of the order of bandwidth. By taking ="v a~1 and a"2a/p in the F renormalization group equations [3,4], we examine the con"nement}decon"nement transition. In Fig. 1, the l-dependence of t (l) (solid curves) is M shown for both ;"5 (curves (1)) and ;"8 (curves (2)) where t "0.1 and t "0.05. The steep increase of t (l), M d M which is given by t (l)"t el in the absence of interacM M tion, is reduced by the interaction, ;. The case for ;"5 (curves (1)) shows a relevant t (l) corresponding to the M decon"nement, while the case for ;"8 (curves (2)) leads to con"nement due to t (l) decreasing to zero after takM ing a maximum. There is a critical value of ; for # ; where the con"nement (decon"nement) is obtained for ;'; (;(; ). The l-dependence of K (l) decreasing # # o to zero implies a formation of a gap in the total charge excitation since the quarter-"lled band is regarded e!ectively as half-"lled in the presence of the dimerization. The charge gaps for curves (1) and (2) are obtained from * "=e~l* with l "3.46 and 2.27 where * o K (l )"K (0)/2. o * o In Fig. 2, a dimensional crossover temperature, t (l )"1, is shown as a function of ; where the dashed M t curve denotes * . The state of two-coupled chains is o obtained below the solid curve. It is found that t%&& is M suppressed by the intrachain interaction. The arrow de-
685
Fig. 2. A crossover temperature t%&& and charge gap * as M o a function of ; with t "0.1 and t "0.05 where ; K5.87 (the M $ # arrow). The inset shows a phase diagram of con"nement (I) and decon"nement (II) on the plane of t and * for t "0.05 (solid M o $ curve), 0.1 (dashed curve) and 0.2 (dotted curve) where symbols are illustrated in the text.
notes a critical value, ; K5.87, at which * K0.11. In # o the inset of Fig. 2, a phase diagram of con"nement (I) and decon"nement (II) is shown on the plane of t and M * with some choices of t . By noting that the t -dependo $ $ ence of the boundary is small, (e.g., 1.1(* /t (1.4 in o M the interval region of 0.1(t (0.3), the con"neM ment}decon"nement transition is determined essentially by the competition between the charge gap and the interchain hopping energy. Finally, we comment on the metallic states of the Bechgaard salts. In the inset, parameters obtained from the optical experiments [1] and the band calculation [5] are shown for (TMTTF) Br (circle), (TMTSF) PF (tri2 2 6 angle) and (TMTSF) ClO (square), which have 2 4 t /t"0.098, 0.15 and 0.11 for the interchain hopping, M t /t"0.06, 0.09 and 0.08 for the dimerization and * /tK $ o 0.40, 0.11 and 0.10 for the charge gap, respectively. It is found that the TMTTF salts are in the con"nement phase (I) while the TMTSF salts are in the decon"nement phase (II). A good agreement between the present calculation and the optical experiments is obtained. In summary, we have demonstrated explicitly the con"nement}decon"nement transition in the two-coupled Hubbard chains with dimerization to explain the experimental results.
References [1] V. Vescoli et al., Science 281 (1998) 1181. [2] K. Penc, F. Mila, Phys. Rev. B 50 (1994) 11429. [3] Y. Suzumura, M. Tsuchiizu, G. GruK ner, Phys. Rev. B 57 (1998) 15040. [4] M. Tsuchiizu, Y. Suzumura, Phys. Rev. B 59 (1999) 12326. [5] L. Ducasse et al., J. Phys. C 19 (1986) 3805.
Con"nement by charge gap in organic conductor Bechgaard salts M. Tsuchiizu!, Y. Suzumura!,",* !Department of Physics, Nagoya University, Nagoya 464-8602, Japan "CREST, Japan Science and Technology Corporation (JST), Japan
Abstract In Bechgaard salts, which are e!ectively half-"lling due to dimerization, a con"nement has been found for the large charge gap in the optical experiments. In order to understand the crucial role of the charge gap, the irrelevance of interchain hopping corresponding to the con"nement is calculated for two-coupled chains of quarter-"lled Hubbard model with dimerization. It is demonstrated that a transition from decon"nement to con"nement occurs when the charge gap induced by the dimerization becomes larger than the interchain hopping. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Low-dimensional organic Bechgaard salts; Charge gap; Mott insulator; Dimerized Hubbard model
Bechgaard salts, which are quasi-one-dimensional organic conductors, are strongly correlated systems since the repulsive interaction has a magnitude of the order or even larger than the bandwidth. Besides spin density wave state at low temperatures, the normal states of these conductors exhibit unconventional properties found in optical experiments [1]. With increasing interchain transfer integral, a transition from an insulating state to a metallic state occurs indicating a fact that electrons con"ned to the individual chains are decon"ned. In the present paper, such a transition in two-coupled chains with quarter-"lling is examined by extending the previous general calculations, to a study of a model including the dimerization explicitly. We consider a Hamiltonian given by H"! + [t#(!1)jt ](cs c #h.c.) $ jpl j`1pl j,p,l
* Correspondence address. Department of Physics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan. Tel.: 81-52789-2437; fax: 81-52-789-2928. E-mail address: [email protected] (Y. Suzumura)
! 2t + (cs c #h.c.) M jp1 jp2 j,p # ;+ n n , (1) jtl jsl j,l where cs denotes an operator for the electron at the jth jpl site of the lth chain (l"1,2) with spin pn "cs c and jpl jpl jpl t comes from the dimerization. By diagonalizing the $ bilinear terms in Eq. (1), Fermi momentum (velocity) is given by k "p/2aG2t /v (v "J2ta[1!(t /t)2]/ FB M F F $ [1#(t /t)2]1@2) [2]. By applying the bosonization $ method to the lower band, the e!ective Hamiltonian is rewritten as H%&&"H%&&#H%&& [3,4]. The "rst term is 0 */5 given by H%&&"+ (v /4p): dxMK~1(Rh )2#K (Rh )2N 0 l l l l` l l~ (l"o, p, C, S) where v "v (1#;a/pv )1@2, v " o F F p v (1!;a/pv )1@2, K "[1/(1#;a/pv )]1@2, K "[1/(1! F F o F p ;a/pv )]1@2, v "v "v and K "K "1. Phase variF C S F C S ables, h and h (h and h ), express #uctuations o` p` C` S` for the total (transverse) charge density and spin density, respectively. The second term of H%&& is expressed as g lp,l{p{ H%&&" + */5 2p2a2 l,l{,p,p{
P
dx cos J2h cos(J2h !d x), lp l{p{ l{p{
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 2 1 2 - 0
(2)
M. Tsuchiizu, Y. Suzumura / Physica B 281&282 (2000) 684}685
Fig. 1. The l-dependence of t (l) (solid curves) and K (l) (dashed M o curves) is shown for ;"5 (1) and 8 (2) where t "0.1 and M t "0.05. $
where the coupling constants for the forward and backward scatterings are given by !g "g " C`,S~ C~,S` g "!g "!g "!g ";a p`,C` p`,C~ p`,S` p`,S~ with lattice constant a. Quantities g ("0) and C`,S` g ("0) appear through renormalization. We take C~,S~ t"1. The coupling constants for the umklapp scattering are given by g "g "!g "g " o`,C` o`,C~ o`,S` o`,S~ !2;a(t /t)/[1#(t /t)2] [2]. In Eq. (2), d is "nite only $ $ lp for d ("8t /v ) and g "0 for the others. The C` M F lp,l{p{ energy-dependence is calculated by the renormalization group method where the relation between l and energy, u, and/or temperature, ¹, is given by l"ln(=/u)"ln(=/¹) with = being of the order of bandwidth. By taking ="v a~1 and a"2a/p in the F renormalization group equations [3,4], we examine the con"nement}decon"nement transition. In Fig. 1, the l-dependence of t (l) (solid curves) is M shown for both ;"5 (curves (1)) and ;"8 (curves (2)) where t "0.1 and t "0.05. The steep increase of t (l), M d M which is given by t (l)"t el in the absence of interacM M tion, is reduced by the interaction, ;. The case for ;"5 (curves (1)) shows a relevant t (l) corresponding to the M decon"nement, while the case for ;"8 (curves (2)) leads to con"nement due to t (l) decreasing to zero after takM ing a maximum. There is a critical value of ; for # ; where the con"nement (decon"nement) is obtained for ;'; (;(; ). The l-dependence of K (l) decreasing # # o to zero implies a formation of a gap in the total charge excitation since the quarter-"lled band is regarded e!ectively as half-"lled in the presence of the dimerization. The charge gaps for curves (1) and (2) are obtained from * "=e~l* with l "3.46 and 2.27 where * o K (l )"K (0)/2. o * o In Fig. 2, a dimensional crossover temperature, t (l )"1, is shown as a function of ; where the dashed M t curve denotes * . The state of two-coupled chains is o obtained below the solid curve. It is found that t%&& is M suppressed by the intrachain interaction. The arrow de-
685
Fig. 2. A crossover temperature t%&& and charge gap * as M o a function of ; with t "0.1 and t "0.05 where ; K5.87 (the M $ # arrow). The inset shows a phase diagram of con"nement (I) and decon"nement (II) on the plane of t and * for t "0.05 (solid M o $ curve), 0.1 (dashed curve) and 0.2 (dotted curve) where symbols are illustrated in the text.
notes a critical value, ; K5.87, at which * K0.11. In # o the inset of Fig. 2, a phase diagram of con"nement (I) and decon"nement (II) is shown on the plane of t and M * with some choices of t . By noting that the t -dependo $ $ ence of the boundary is small, (e.g., 1.1(* /t (1.4 in o M the interval region of 0.1(t (0.3), the con"neM ment}decon"nement transition is determined essentially by the competition between the charge gap and the interchain hopping energy. Finally, we comment on the metallic states of the Bechgaard salts. In the inset, parameters obtained from the optical experiments [1] and the band calculation [5] are shown for (TMTTF) Br (circle), (TMTSF) PF (tri2 2 6 angle) and (TMTSF) ClO (square), which have 2 4 t /t"0.098, 0.15 and 0.11 for the interchain hopping, M t /t"0.06, 0.09 and 0.08 for the dimerization and * /tK $ o 0.40, 0.11 and 0.10 for the charge gap, respectively. It is found that the TMTTF salts are in the con"nement phase (I) while the TMTSF salts are in the decon"nement phase (II). A good agreement between the present calculation and the optical experiments is obtained. In summary, we have demonstrated explicitly the con"nement}decon"nement transition in the two-coupled Hubbard chains with dimerization to explain the experimental results.
References [1] V. Vescoli et al., Science 281 (1998) 1181. [2] K. Penc, F. Mila, Phys. Rev. B 50 (1994) 11429. [3] Y. Suzumura, M. Tsuchiizu, G. GruK ner, Phys. Rev. B 57 (1998) 15040. [4] M. Tsuchiizu, Y. Suzumura, Phys. Rev. B 59 (1999) 12326. [5] L. Ducasse et al., J. Phys. C 19 (1986) 3805.