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Spin density wave fluctuations and transport properties
Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 765-767 North-Holland
765
Spin density wave fluctuations and transport properties Kazumi Maki Department of Physics, Unicersity of Southern California, Los Angeles, CA 90089·0484, USA
Fluctuation contributions to the electric conductivity and the sound attenuation in the vicinity of the spin density wave transition are considered within single loop approximation. We find that the electric resistance of the extremely clean sample diverges like IT -1;; I-a with a = ~(4- D) where D is the spatial dimension of the fluctuation.
It is well known that a number of quasi-one-dimensional charge-density waves (COW) exhibit the resistive anomaly at the COW transition [1]. For example the electric resistivity diverges with a power law like IT - 1;, I-a. Further there is a nonohmic component in the resistivity; the resistivity decreases with increasing electric field E [2]. The former is usually interpreted in terms of a model proposed by Hom and Guiddoti [3]; the excess resistance due to the scattering by the COW fluctuation. At the more microscopic level the fluctuation contribution to the electric conductivity is calculated from the 5 diagrams given in fig. 1 where the solid lines are the quasi-particle propagator and the wavy lines are the fluctuation propagator. The term considered by Horn and Guiddoti corresponds to the diagram A, which we shall call the anomalous term [4]. Unfortunately, however, since Hom and Guiddoti neglected the vertex renormalization, their result in the. clean limit (l > ~o = v/'IT1;, where I is the electron mean free path and ~o is the intrinsic coherence distance) is no longer valid. Further the diagrams C and C' first considered by Aslamazov and Larkin [5] in their analysis of the superconducting fluctuation was neglected, though they are
small in COW [6). More recently the nonohmic conductivity characteristic to the Frohlich conduction is reported in spin density waves (SOW) in Bechgaard salts like (TMTSFhN03 , (TMTSFhPF6 and (TMTSFhCI04 [7,8]. Therefore it is quite natural to look for the fluctuation induced anomalies in the transport properties of the SOW in quasi-one-dimensional systems. As a model we take an anisotropic Hubbard model as first consider by Yamaji [9]. In the normal state the quasiparticle propagator is given by
(1) where
"'n is the' Matsubara frequency and
H p) =
-2t a cos aPt - 2t b cos bp2 - 2t c cos cP3-/.I
"" ±[v(Pt -PF)-2tbcos bP2-2tccos CP3]
(2) with v = 2t aa sin apF. Here ± signs are for the right going and the left going quasi-particles. The fluctuation propagator (in the clean limit) in the vicinity of T=1;, is given by
where No = ('ITvbc)-" T = In(T/1;,), "'. is the Matsubara frequency and D q 2 is the short-handed expression for
and the anisotropic Fermi velocities are given by
fl'
('
Fig. 1. Lowest order corrections due to SDW fluctuations are shown. Here solid lines are the quasi-particle propagator and wavy lines are the fluctuation propagator. The vertex renorrnalization is not shown for clarity.
(5) and (a, b, c) are the lattice constants. Unlike in a COW, there are four independent fluctuations (3 spin waves and 1 phason). In the pres-
0304-8853/90/S03.50!fJ 1990 - Elsevier Science Publishers B.Y. (North-Holland) and Yamada Science Foundation
K. Maki / Spin density wave fluctuations and transport properties
766
ence of an electric field, T in two of the propagators (amplitude and phase) has to be replaced by (10) 2 -I
T->T+i7~(3)euE(8('ITT») .
(6)
be quite long. In the presence of an electric field E in the chain direction eq. (7) is rewritten as 11 an
-) (E) - _ ~(71'(3» e 2 Tv ( ~
(6) where 110 is the conductivity in the absence of the fluctuation, l1a n
=
-'IT
(7r(3»1/2 eu:~~ ( / :f )(J; + 18)-1
for 30,
ed2u =_4 ( 2f_)(T-8)-lln(T/8) u2 r-r r
2
-
since the electric field induces the oscillation between these two modes. In order to consider the transport properties it is essential to introduce a relaxation mechanism. This is most readily done in terms of impurities. However, the impurity scattering not only gives rise to the quasi-particle life time but also to the vertex renormalization. In particular the fluctuation propagator has to be multiplied by the renormalization factors. Then the electric conductivity in a SOW for T> 'Fe is given as (6)
X
~
V2U)
T + i'
{(iT + 18) -I + Re(iT + ie + ,,18) -I} for 30,
- ;;~2 (/~"f )[(T- 8) -I i ) In(
+Re{(T+i(-8)-lln(T~i()}]
(9) where ( = 7n3)euE(8('ITT)2)-I. Indeed the second term in eq. (9) describes semiquantitatively the observed nonohmic conductivity in 0-TaS) and NbSe). As far as SOWs are concerned no detailed experimental result is available. However, the temperature dependence of the electric resistance looks very similar to the one in a COW. A similar analysis is extended to the elastic anomalies [11]. When the SOW is unpinned we obtain (10)
AC=O
for 20,
(7) where n3) = 1.202 ... , d is the thickness of the sample, D is the dimension of the fluctuations and 8 is the pair breaking parameter given by
for 20,
and
Aa/>.w2[(1->.)c]-1 2
=
~ (7t(3»-1/2(bcT/V2 U3 ) X
{(2f/r+i')(J; +18)-I+J;-I}
for30,
2
and
=
(8) and r l and r 2 are the forward and backward scattering rates due to impurity scattering. Here we neglected the regular term, which is smaller than l1an (i.e. it is less divergent). Compared with the corresponding expressions for a COW, l1an in a SOW is twice as large since in a COW there are only two fluctuation modes. Eq. (7) predicts that the electric resistivity in the extremely with clean system (8 S; 10- 4 ) diverges like I T - 'Fe a = i(4 - D). This exponent is larger by unity from the model by Horn and Guiddoti predicted. For example Richard et al. (2) observed the exponent of 0.5 and 1 for 0-TaS3 and NbSe 3, respectively. They are interpreted in the light of the present theory as due to the 30 and 20 fluctuations, respectively. We believe that the 20 fluctuation in NbSe3 is due to the size effect, since the transverse coherence length in the vicinity of T = 'Fe can
,-a
76 (bc/du2)(2f / r + f ) x(T-o)-lln(T/8)+T- 1 )
for 20,
(11)
where C and a are the sound velocity and attenuation coefficient, respectively. The eq. (10) has been obtained earlier by Nakane (12) for a COW. Acknowledgements I have benefitted from discussion with Pierre Monceau. The present work is supported by National Science Foundation under grant number OMR-86-11829 and OMR-89-15285. References (1) P. Monceau, in: Electronic Properties of Inorganic Quasi-
One-Dimensional Compounds, part II, ed. P. Monceau (Reidel, Dordrecht, 1985) p. 139.
K Maki / Spin density wave fluctuations and transport properties
[2) J. Richard, H. Salva, M.C. Saint-Lager and P. Monceau, J. de Phys 44 (1983) C3-1685. [3) P.M. Hom and D. Guiddoti, Phys. Rev. B 16 (1977) 491. [4) K. Maki, Prog. Theor. Phys. 39 (1968) 397. [5) L.G. Aslamazov and A.L. Larkin, Fiz. Tverdi. Tela 10 (1968) 1104 [Sov. Phys. Solid State 10 (1968) 875]. [6) K. Maki, Phys. Rev. B 41 (1990) 9308. [7) S. Tomic, J.R. Cooper, D. Jerome and K. Bechgaard, Phys. Rev. Lett. 62 (1989) 2466. W. Kang, S. Tomic, J.R. Cooper and D. Jerome, Phys. Rev. B 41 (1990) 4862.
767
[8) T. Sambongi et aI., Solid State Commun. 72 (1989) 817. K. Nomura et al., ibid. 72 (1989) 1123. [9] K. Yamaji, J. Phys. Soc. Jpn. 51 (1982) 2787, 52 (1983) 1361. [10] The result on the nonchimic conductivity in ref. [6) contains an error which is corrected here. [11) K. Maki, Phys. Rev. B 41 (1990) 2657. [12) Y. Nakane, J. Phys. Soc. Jpn. 55 (1986) 2235.
765
Spin density wave fluctuations and transport properties Kazumi Maki Department of Physics, Unicersity of Southern California, Los Angeles, CA 90089·0484, USA
Fluctuation contributions to the electric conductivity and the sound attenuation in the vicinity of the spin density wave transition are considered within single loop approximation. We find that the electric resistance of the extremely clean sample diverges like IT -1;; I-a with a = ~(4- D) where D is the spatial dimension of the fluctuation.
It is well known that a number of quasi-one-dimensional charge-density waves (COW) exhibit the resistive anomaly at the COW transition [1]. For example the electric resistivity diverges with a power law like IT - 1;, I-a. Further there is a nonohmic component in the resistivity; the resistivity decreases with increasing electric field E [2]. The former is usually interpreted in terms of a model proposed by Hom and Guiddoti [3]; the excess resistance due to the scattering by the COW fluctuation. At the more microscopic level the fluctuation contribution to the electric conductivity is calculated from the 5 diagrams given in fig. 1 where the solid lines are the quasi-particle propagator and the wavy lines are the fluctuation propagator. The term considered by Horn and Guiddoti corresponds to the diagram A, which we shall call the anomalous term [4]. Unfortunately, however, since Hom and Guiddoti neglected the vertex renormalization, their result in the. clean limit (l > ~o = v/'IT1;, where I is the electron mean free path and ~o is the intrinsic coherence distance) is no longer valid. Further the diagrams C and C' first considered by Aslamazov and Larkin [5] in their analysis of the superconducting fluctuation was neglected, though they are
small in COW [6). More recently the nonohmic conductivity characteristic to the Frohlich conduction is reported in spin density waves (SOW) in Bechgaard salts like (TMTSFhN03 , (TMTSFhPF6 and (TMTSFhCI04 [7,8]. Therefore it is quite natural to look for the fluctuation induced anomalies in the transport properties of the SOW in quasi-one-dimensional systems. As a model we take an anisotropic Hubbard model as first consider by Yamaji [9]. In the normal state the quasiparticle propagator is given by
(1) where
"'n is the' Matsubara frequency and
H p) =
-2t a cos aPt - 2t b cos bp2 - 2t c cos cP3-/.I
"" ±[v(Pt -PF)-2tbcos bP2-2tccos CP3]
(2) with v = 2t aa sin apF. Here ± signs are for the right going and the left going quasi-particles. The fluctuation propagator (in the clean limit) in the vicinity of T=1;, is given by
where No = ('ITvbc)-" T = In(T/1;,), "'. is the Matsubara frequency and D q 2 is the short-handed expression for
and the anisotropic Fermi velocities are given by
fl'
('
Fig. 1. Lowest order corrections due to SDW fluctuations are shown. Here solid lines are the quasi-particle propagator and wavy lines are the fluctuation propagator. The vertex renorrnalization is not shown for clarity.
(5) and (a, b, c) are the lattice constants. Unlike in a COW, there are four independent fluctuations (3 spin waves and 1 phason). In the pres-
0304-8853/90/S03.50!fJ 1990 - Elsevier Science Publishers B.Y. (North-Holland) and Yamada Science Foundation
K. Maki / Spin density wave fluctuations and transport properties
766
ence of an electric field, T in two of the propagators (amplitude and phase) has to be replaced by (10) 2 -I
T->T+i7~(3)euE(8('ITT») .
(6)
be quite long. In the presence of an electric field E in the chain direction eq. (7) is rewritten as 11 an
-) (E) - _ ~(71'(3» e 2 Tv ( ~
(6) where 110 is the conductivity in the absence of the fluctuation, l1a n
=
-'IT
(7r(3»1/2 eu:~~ ( / :f )(J; + 18)-1
for 30,
ed2u =_4 ( 2f_)(T-8)-lln(T/8) u2 r-r r
2
-
since the electric field induces the oscillation between these two modes. In order to consider the transport properties it is essential to introduce a relaxation mechanism. This is most readily done in terms of impurities. However, the impurity scattering not only gives rise to the quasi-particle life time but also to the vertex renormalization. In particular the fluctuation propagator has to be multiplied by the renormalization factors. Then the electric conductivity in a SOW for T> 'Fe is given as (6)
X
~
V2U)
T + i'
{(iT + 18) -I + Re(iT + ie + ,,18) -I} for 30,
- ;;~2 (/~"f )[(T- 8) -I i ) In(
+Re{(T+i(-8)-lln(T~i()}]
(9) where ( = 7n3)euE(8('ITT)2)-I. Indeed the second term in eq. (9) describes semiquantitatively the observed nonohmic conductivity in 0-TaS) and NbSe). As far as SOWs are concerned no detailed experimental result is available. However, the temperature dependence of the electric resistance looks very similar to the one in a COW. A similar analysis is extended to the elastic anomalies [11]. When the SOW is unpinned we obtain (10)
AC=O
for 20,
(7) where n3) = 1.202 ... , d is the thickness of the sample, D is the dimension of the fluctuations and 8 is the pair breaking parameter given by
for 20,
and
Aa/>.w2[(1->.)c]-1 2
=
~ (7t(3»-1/2(bcT/V2 U3 ) X
{(2f/r+i')(J; +18)-I+J;-I}
for30,
2
and
=
(8) and r l and r 2 are the forward and backward scattering rates due to impurity scattering. Here we neglected the regular term, which is smaller than l1an (i.e. it is less divergent). Compared with the corresponding expressions for a COW, l1an in a SOW is twice as large since in a COW there are only two fluctuation modes. Eq. (7) predicts that the electric resistivity in the extremely with clean system (8 S; 10- 4 ) diverges like I T - 'Fe a = i(4 - D). This exponent is larger by unity from the model by Horn and Guiddoti predicted. For example Richard et al. (2) observed the exponent of 0.5 and 1 for 0-TaS3 and NbSe 3, respectively. They are interpreted in the light of the present theory as due to the 30 and 20 fluctuations, respectively. We believe that the 20 fluctuation in NbSe3 is due to the size effect, since the transverse coherence length in the vicinity of T = 'Fe can
,-a
76 (bc/du2)(2f / r + f ) x(T-o)-lln(T/8)+T- 1 )
for 20,
(11)
where C and a are the sound velocity and attenuation coefficient, respectively. The eq. (10) has been obtained earlier by Nakane (12) for a COW. Acknowledgements I have benefitted from discussion with Pierre Monceau. The present work is supported by National Science Foundation under grant number OMR-86-11829 and OMR-89-15285. References (1) P. Monceau, in: Electronic Properties of Inorganic Quasi-
One-Dimensional Compounds, part II, ed. P. Monceau (Reidel, Dordrecht, 1985) p. 139.
K Maki / Spin density wave fluctuations and transport properties
[2) J. Richard, H. Salva, M.C. Saint-Lager and P. Monceau, J. de Phys 44 (1983) C3-1685. [3) P.M. Hom and D. Guiddoti, Phys. Rev. B 16 (1977) 491. [4) K. Maki, Prog. Theor. Phys. 39 (1968) 397. [5) L.G. Aslamazov and A.L. Larkin, Fiz. Tverdi. Tela 10 (1968) 1104 [Sov. Phys. Solid State 10 (1968) 875]. [6) K. Maki, Phys. Rev. B 41 (1990) 9308. [7) S. Tomic, J.R. Cooper, D. Jerome and K. Bechgaard, Phys. Rev. Lett. 62 (1989) 2466. W. Kang, S. Tomic, J.R. Cooper and D. Jerome, Phys. Rev. B 41 (1990) 4862.
767
[8) T. Sambongi et aI., Solid State Commun. 72 (1989) 817. K. Nomura et al., ibid. 72 (1989) 1123. [9] K. Yamaji, J. Phys. Soc. Jpn. 51 (1982) 2787, 52 (1983) 1361. [10] The result on the nonchimic conductivity in ref. [6) contains an error which is corrected here. [11) K. Maki, Phys. Rev. B 41 (1990) 2657. [12) Y. Nakane, J. Phys. Soc. Jpn. 55 (1986) 2235.