- Email: [email protected]
Semiclassical angular effect of magnetoresistance and field-induced confinement state in the organic quasi-one-dimensional conductors
Physica B 256±258 (1998) 633±636
Semiclassical angular eect of magnetoresistance and ®eldinduced con®nement state in the organic quasi-onedimensional conductors Toshihito Osada
a,b,*
a
b
Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8666, Japan Research Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan
Abstract We have studied the semiclassical and non-classical behaviors of magnetoresistance (MR) in quasi-one-dimensional (Q1D) conductors. Based on the Boltzmann equation, we numerically investigated possible semiclassical angular eects of MR in the Q1D system, that is, the characteristic patterns of the ®eld-orientation-dependence of MR. The numerical results well reproduced the most of angular dependent MR features observed in an organic conductor (TMTSF)2 ClO4 except a dip structure around B//b0 . As for this non-semiclassical feature, we discuss the possibility of the ®eld-induced electron con®nement onto a single conduction layer. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Q1D organic conductor; Magnetotransport; Angular eect; Field-induced con®nement
1. Introduction In the past 10 years, novel angular eects of magnetoresistance (MR) have been discovered in low-dimensional organic conductors. They are the structures which appear on the angular dependent pattern of MR when the magnetic ®eld orientation is rotated. Particularly, quasi-one-dimensional (Q1D) organic conductors such as (TMTSF)2 X show rich angular eects, the `Lebed resonance' [1], the `Danner±Chaikin oscillations' [2], the `third angular eect' [3], etc. The origins of these angular eects have not necessarily been established yet. In addition, one of these conductors, (TMTSF)2 PF6 , shows anomalous angular dependent features dif-
*
Corresponding author. Fax: 81 3 3478 5472; e-mail: [email protected].
ferent from the others, and the possibility of the `®eld-induced con®nement' enhanced by the electron correlation has been discussed [4±6]. The purpose of this work is to understand those angular dependent behaviors of MR in the Q1D conductors. First, we have numerically surveyed all possible semiclassical angular eects originating from the Fermi surface topology. Then, we discuss the non-classical features due to the electron con®nement by studying the deviation from the semiclassical behaviors. 2. Semiclassical angular eects of magnetoresistance First, based on the semiclassical theory, we carried out the numerical calculation of MR in the Q1D conductors with a pair of the sheetlike Fermi
0921-4526/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 6 7 9 - 6
634
T. Osada / Physica B 256±258 (1998) 633±636
surfaces (FS's). We employ the following tightbinding band model for the Q1D conductor [3]: E k ÿ 2ta cos akx ÿ 2tb cos bky ÿ 2tc cos ckz ÿ EF :
1
Here, we take the x-axis for the 1D axis and the xy-plane for the conduction plane, and choose the parameters so as to simulate (TMTSF)2 ClO4 ; 1000:100:3, a:b:c 3.5:7.0:13.0, and 4ta :4tb :4t pc EF ÿ 2ta (quarter ®lled). The semiclassical electron orbital motion is determined by the equation of motion: hk_ ÿev B;
v 1= h oE k=ok:
2
According to the semiclassical magnetotransport theory, the conductivity is calculated from the electron orbital motion by the kinetic form of the Boltzmann equation: Z0 2e2 X df ÿ vi k; 0 vj k; te 1=sÿixt dt: rij V k dE ÿ1
3 Here, the relaxation time s is assumed as a constant (the relaxation time approximation). We can calculate the DC MR for given conditions by evaluating the above the formulae numerically. Fig. 1 shows the calculated interlayer resistivity qzz as a function of ®eld orientation. The ®eld strength was ®xed to Bs 3.5 ´ 10ÿ10 Ts. In this diagram, the direction and the distance from the origin indicate the ®eld orientation and the resistivity value, respectively. Rich structures, that is, the angular eects are superposed on the moderate background angular dependence. Three conventional angular eects in the Q1D system, the Lebed resonances, the Danner±Chaikin oscillations, and the third angular eect are successfully reproduced on the yz-plane, xz-plane, and xy-plane, respectively. This fact means that these three eects are explained as the semiclassical Fermi surface topological eects in the same way. In Fig. 1, we can see that qzz shows complicated angular dependence near the 1D-axis (x-axis). Fig. 2 shows the detailed angular dependence near the 1D-axis by the density plot, where the brightness indicates the resistivity value. We can ®nd the clear regularity in the apparently compli-
Fig. 1. Dependence of the interlayer MR on the magnetic ®eld orientation in the Q1D conductors. The unit of resistivity is 2pabceB/4e2 .
Fig. 2. Density plot of the interlayer MR for the ®eld orientations near the 1D-axis. The unit of resistivity is 2pabceB/4e2 .
cated feature. The radiating dark lines are the Lebed resonances, where the electron orbit is periodic in the k-space. The blight diamond pat-
T. Osada / Physica B 256±258 (1998) 633±636
terns correspond to the Danner±Chaikin oscillation peaks. The blight horizontal bar in the center is the knife-edge-like resistance peak of the third angular eect, where the closed electron orbit exists. In this way, the three fundamental angular eects are mixed up generating the complicated angular dependence near the 1D-axis. Strictly speaking, additional ®ne patterns are also observed where the o-plane angle is very small (|H| < 2°). In recent experiments, complicated oscillations have been observed near the third angular eect when the magnetic ®eld is rotated with a ®xed o-angle from the conducting plane [7]. This `out-of-plane eect' is well explained by this diagram.
635
Fig. 3. Angular dependence of interlayer MR in a Q1D conductor (TMTSF)2 ClO4 . The dip structure is indicated by a downward arrow.
3. Search for the ®eld-induced con®nement When the magnetic ®eld is applied parallel to the y-axis, the semiclassical width of the z-component of real space pelectron orbits is given by 4tc / vF eB. Here, vF 2ta a=h is the Fermi velocity. If the magnetic ®eld becomes larger than the `con®nement ®eld' Bconf 4tc /vF ec, the semiclassical width becomes less than the interlayer distance c, so that the electron motion is eectively con®ned in a single conducting layer (xy-plane). In this situation, the breakdown of the semiclassical treatment of electron kinetics is expected. Beyond this one-body picture, Strong et al. theoretically discussed that the con®nement is enhanced by the electron correlation through eective reduction of the interlayer transfer tc . According to their theory, in the many-body con®nement state, each conduction layer is decoupled into 2D non-Fermi liquid incoherent with each other, and the resistance is scaled by the magnetic ®eld component normal to the 2D plane: Rxx (B)Rxx 0 Bz 1=2 , Rzz (B)-Rzz 0 Bz 3=2 . Strong et al. applied their theory to an organic Q1D conductor (TMTSF)2 PF6 which shows the anomalous angular dependence of MR. In order to investigate the non-semiclassical features experimentally in real Q1D conductors, we studied an organic Q1D conductor (TMTSF)2 ClO4 , a sister compound of (TMTSF)2 PF6 .
Fig. 3 shows the angular dependence of the interlayer MR when the ®eld is rotated in the plane normal to the 1D axis (yz-plane). Here, in the region labeled as FISDW, the electron system is not in the normal metallic phase. In contrast to (TMTSF)2 PF6 , the angular dependence in the normal phase is almost semiclassical. Only one exception is the dip structure around B//b0 (y-axis) where the interlayer MR should take the maximum value in the semiclassical theory. Since Bconf is estimated as 12.3 T in (TMTSF)2 ClO4 , the one-body con®nement is at least expected to appear around B//b0 in the experimental ®eld range (B < 12 T). The con®nement is one of the plausible explanations for the dip structure around B//b0 . We also studied the possibility of the manybody con®nement by testing the scaling law. In the dip region, the MR seems to obey the Bz 3=2 -law. However, the MR is not necessarily scaled by Bz , but depends on By . So, the many-body con®nement state is not perfectly realized in the present experiment on (TMTSF)2 ClO4 .
Acknowledgements This work was supported by the Torey Science Foundation.
636
T. Osada / Physica B 256±258 (1998) 633±636
References [1] T. Osada et al., Phys. Rev. Lett. 66 (1991) 1525. [2] G.M. Danner et al., Phys. Rev. Lett. 72 (1994) 3714. [3] T. Osada, S. Kagoshima, N. Miura, Phys. Rev. Lett. 77 (1996) 5261.
[4] S.P. Strong, D.G. Clarke, P.W. Anderson, Phys. Rev. Lett. 73 (1994) 1007. [5] G.M. Danner, P.M. Chaikin, Phys. Rev. Lett. 75 (1995) 4690. [6] E.I. Chashechkina, P.M. Chaikin, Phys. Rev. Lett. 80 (1998) 2181. [7] M.J. Naughton et al., Synth. Met. 86 (1997) 1481.
Semiclassical angular eect of magnetoresistance and ®eldinduced con®nement state in the organic quasi-onedimensional conductors Toshihito Osada
a,b,*
a
b
Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8666, Japan Research Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan
Abstract We have studied the semiclassical and non-classical behaviors of magnetoresistance (MR) in quasi-one-dimensional (Q1D) conductors. Based on the Boltzmann equation, we numerically investigated possible semiclassical angular eects of MR in the Q1D system, that is, the characteristic patterns of the ®eld-orientation-dependence of MR. The numerical results well reproduced the most of angular dependent MR features observed in an organic conductor (TMTSF)2 ClO4 except a dip structure around B//b0 . As for this non-semiclassical feature, we discuss the possibility of the ®eld-induced electron con®nement onto a single conduction layer. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Q1D organic conductor; Magnetotransport; Angular eect; Field-induced con®nement
1. Introduction In the past 10 years, novel angular eects of magnetoresistance (MR) have been discovered in low-dimensional organic conductors. They are the structures which appear on the angular dependent pattern of MR when the magnetic ®eld orientation is rotated. Particularly, quasi-one-dimensional (Q1D) organic conductors such as (TMTSF)2 X show rich angular eects, the `Lebed resonance' [1], the `Danner±Chaikin oscillations' [2], the `third angular eect' [3], etc. The origins of these angular eects have not necessarily been established yet. In addition, one of these conductors, (TMTSF)2 PF6 , shows anomalous angular dependent features dif-
*
Corresponding author. Fax: 81 3 3478 5472; e-mail: [email protected].
ferent from the others, and the possibility of the `®eld-induced con®nement' enhanced by the electron correlation has been discussed [4±6]. The purpose of this work is to understand those angular dependent behaviors of MR in the Q1D conductors. First, we have numerically surveyed all possible semiclassical angular eects originating from the Fermi surface topology. Then, we discuss the non-classical features due to the electron con®nement by studying the deviation from the semiclassical behaviors. 2. Semiclassical angular eects of magnetoresistance First, based on the semiclassical theory, we carried out the numerical calculation of MR in the Q1D conductors with a pair of the sheetlike Fermi
0921-4526/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 6 7 9 - 6
634
T. Osada / Physica B 256±258 (1998) 633±636
surfaces (FS's). We employ the following tightbinding band model for the Q1D conductor [3]: E k ÿ 2ta cos akx ÿ 2tb cos bky ÿ 2tc cos ckz ÿ EF :
1
Here, we take the x-axis for the 1D axis and the xy-plane for the conduction plane, and choose the parameters so as to simulate (TMTSF)2 ClO4 ; 1000:100:3, a:b:c 3.5:7.0:13.0, and 4ta :4tb :4t pc EF ÿ 2ta (quarter ®lled). The semiclassical electron orbital motion is determined by the equation of motion: hk_ ÿev B;
v 1= h oE k=ok:
2
According to the semiclassical magnetotransport theory, the conductivity is calculated from the electron orbital motion by the kinetic form of the Boltzmann equation: Z0 2e2 X df ÿ vi k; 0 vj k; te 1=sÿixt dt: rij V k dE ÿ1
3 Here, the relaxation time s is assumed as a constant (the relaxation time approximation). We can calculate the DC MR for given conditions by evaluating the above the formulae numerically. Fig. 1 shows the calculated interlayer resistivity qzz as a function of ®eld orientation. The ®eld strength was ®xed to Bs 3.5 ´ 10ÿ10 Ts. In this diagram, the direction and the distance from the origin indicate the ®eld orientation and the resistivity value, respectively. Rich structures, that is, the angular eects are superposed on the moderate background angular dependence. Three conventional angular eects in the Q1D system, the Lebed resonances, the Danner±Chaikin oscillations, and the third angular eect are successfully reproduced on the yz-plane, xz-plane, and xy-plane, respectively. This fact means that these three eects are explained as the semiclassical Fermi surface topological eects in the same way. In Fig. 1, we can see that qzz shows complicated angular dependence near the 1D-axis (x-axis). Fig. 2 shows the detailed angular dependence near the 1D-axis by the density plot, where the brightness indicates the resistivity value. We can ®nd the clear regularity in the apparently compli-
Fig. 1. Dependence of the interlayer MR on the magnetic ®eld orientation in the Q1D conductors. The unit of resistivity is 2pabceB/4e2 .
Fig. 2. Density plot of the interlayer MR for the ®eld orientations near the 1D-axis. The unit of resistivity is 2pabceB/4e2 .
cated feature. The radiating dark lines are the Lebed resonances, where the electron orbit is periodic in the k-space. The blight diamond pat-
T. Osada / Physica B 256±258 (1998) 633±636
terns correspond to the Danner±Chaikin oscillation peaks. The blight horizontal bar in the center is the knife-edge-like resistance peak of the third angular eect, where the closed electron orbit exists. In this way, the three fundamental angular eects are mixed up generating the complicated angular dependence near the 1D-axis. Strictly speaking, additional ®ne patterns are also observed where the o-plane angle is very small (|H| < 2°). In recent experiments, complicated oscillations have been observed near the third angular eect when the magnetic ®eld is rotated with a ®xed o-angle from the conducting plane [7]. This `out-of-plane eect' is well explained by this diagram.
635
Fig. 3. Angular dependence of interlayer MR in a Q1D conductor (TMTSF)2 ClO4 . The dip structure is indicated by a downward arrow.
3. Search for the ®eld-induced con®nement When the magnetic ®eld is applied parallel to the y-axis, the semiclassical width of the z-component of real space pelectron orbits is given by 4tc / vF eB. Here, vF 2ta a=h is the Fermi velocity. If the magnetic ®eld becomes larger than the `con®nement ®eld' Bconf 4tc /vF ec, the semiclassical width becomes less than the interlayer distance c, so that the electron motion is eectively con®ned in a single conducting layer (xy-plane). In this situation, the breakdown of the semiclassical treatment of electron kinetics is expected. Beyond this one-body picture, Strong et al. theoretically discussed that the con®nement is enhanced by the electron correlation through eective reduction of the interlayer transfer tc . According to their theory, in the many-body con®nement state, each conduction layer is decoupled into 2D non-Fermi liquid incoherent with each other, and the resistance is scaled by the magnetic ®eld component normal to the 2D plane: Rxx (B)Rxx 0 Bz 1=2 , Rzz (B)-Rzz 0 Bz 3=2 . Strong et al. applied their theory to an organic Q1D conductor (TMTSF)2 PF6 which shows the anomalous angular dependence of MR. In order to investigate the non-semiclassical features experimentally in real Q1D conductors, we studied an organic Q1D conductor (TMTSF)2 ClO4 , a sister compound of (TMTSF)2 PF6 .
Fig. 3 shows the angular dependence of the interlayer MR when the ®eld is rotated in the plane normal to the 1D axis (yz-plane). Here, in the region labeled as FISDW, the electron system is not in the normal metallic phase. In contrast to (TMTSF)2 PF6 , the angular dependence in the normal phase is almost semiclassical. Only one exception is the dip structure around B//b0 (y-axis) where the interlayer MR should take the maximum value in the semiclassical theory. Since Bconf is estimated as 12.3 T in (TMTSF)2 ClO4 , the one-body con®nement is at least expected to appear around B//b0 in the experimental ®eld range (B < 12 T). The con®nement is one of the plausible explanations for the dip structure around B//b0 . We also studied the possibility of the manybody con®nement by testing the scaling law. In the dip region, the MR seems to obey the Bz 3=2 -law. However, the MR is not necessarily scaled by Bz , but depends on By . So, the many-body con®nement state is not perfectly realized in the present experiment on (TMTSF)2 ClO4 .
Acknowledgements This work was supported by the Torey Science Foundation.
636
T. Osada / Physica B 256±258 (1998) 633±636
References [1] T. Osada et al., Phys. Rev. Lett. 66 (1991) 1525. [2] G.M. Danner et al., Phys. Rev. Lett. 72 (1994) 3714. [3] T. Osada, S. Kagoshima, N. Miura, Phys. Rev. Lett. 77 (1996) 5261.
[4] S.P. Strong, D.G. Clarke, P.W. Anderson, Phys. Rev. Lett. 73 (1994) 1007. [5] G.M. Danner, P.M. Chaikin, Phys. Rev. Lett. 75 (1995) 4690. [6] E.I. Chashechkina, P.M. Chaikin, Phys. Rev. Lett. 80 (1998) 2181. [7] M.J. Naughton et al., Synth. Met. 86 (1997) 1481.