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Fröhlich conduction in spin density waves and field induced spin density waves
758
Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 758-762 North-Holland
Invited paper
Frohlich conduction in spin density waves and field induced spin density waves Kazumi Maki a b
a
and Attila Virosztek
a.b
Department of Physics, Unicersity of Southern California, Los Angeles, CA 90089-0484, USA Department of Physics, Unicersity of Virginia, Charlottescille, VA 22901, USA
We review the recent theory of phason dynamics in spin density waves in Bechgaard saIts. The theoretical results on the threshold electric field, the nonOhmic conductivity and the microwave conductivity are compared to recent experimental results on spin density waves in Bechgaard salts.
I. Introduction The Frohlich conduction (i.e. the colIective transport associated with sliding motion of the condensate) in quasi one-dimensional charge density waves (COW) is welI established both experimentalIy and theoreticalIy in a number of inorganic and organic compounds [II. A similar conduction in spin density waves (SOW) has been speculated by Lee, Rice and Anderson. [21 even before the discovery of the Frohlich conduction in NbSeJ. Since the discovery of the SOW ground state and the field induced spin density wave (FISOW) ground state in some of Bechgaard salts like (TMTSFhCI04 and (TMTSFhPF6, it is quite natural to look for the possible Frohlich conduction in these systems. Indeed, both the nonOhmic conductivity [3,41 and the microwave response [5,6] of SOWs in (TMTSFhNOJ, (TMTSFh PF6 and (TMTSFhCI04 are reported recently, which indicate the Frohlich conduction in a SOW as well. As a model, we take an anisotropic Hubbard model first introduced by Yamaji [8] in order to study the equilibrium and nonequilibrium properties of Bechgaard salts at low temperatures. The Hamiltonian is given by H= L«p)C/aCpa + VLnqrn_ q ! ,
pa
Within the mean field analysis of eq. (1), Yamaji [7] was able to describe the phase diagram of (TMTSFhPF6 first determined by Jerome et al. [8]. Central to his model is the imperfect nesting. As the pressure increases the three dimensionality by increasing Ib and i., the perfect nesting is increasingly destroyed leading to the decrease of the SOW transition temperature. Later, after the discovery of FISOW by Kwak et al. [9], the same Hamiltonian is used successfulIy to describe the FISOW [10-12]. The spin density wave in the ground state is given by
S(x)
=
2V-]L1(T)"1 cos( Qx + ¢(x»,
(3)
where Ii is a unit vector, ¢(x) is the phase of the order parameter, L1(T) is the amplitude of the order parameter and the nesting vector Q is given by
(4)
Q = (2PF' 'IT/b, 'IT/c).
The SOW possesses two classes of Goldstone bosons; spin wave and phason (sliding SOW) with dispersions [13,14]
and
(1)
(5)
q
respectively, where V = NoV and No = ('lTVbC)-1 is the electron density of states at the Fermi surface per spin and v, V2, VJ are the anisotropic Fermi velocities given by
where
with (v, v 2 , vJ) =
(2/ a a sin
ap «, {flbb, {fleC).
(6)
(2) and J! is the chemical potential and C/a and Cpa are the quasi-particle creation and annihilation operators with momentum p and spin a.
Note that these dispersions are independent of the temperature. However, in the real systems both the spin rotation symmetry and the translational symmetry of SOW are
0304-8853/90/S03.50
K Maki, A. Virosztek
I Spin density \I'aves
759
1.0 I----=-_
broken weakly and these modes acquire small energy gaps . The former is broken by the spin-orbit coupling and magnetic dipole interaction while the latter is broken by crystalline defects or sometimes by commensurability with an underlying crystalline lattice. In the following, we shall focus on phason dynamics.
QS
2. Phase Hamiltonian From the analysis of the fluctuation propagator of the phason, we can write down the pha se Hamiltonian of a SOW [15J
H(9)=JdDX{~No/[(~~f +ii2(~~f 9 +V22(0ay )2+ v32(a9)2] az - enfQ -t} 9£ + Vpin ( 9),
05
TlTe ,',
to
Fig. 1. The dynamic and static condensate densities 10 and II are evalu at ed numerically and shown as funct ions of the reduced temperature TIT,;,
(7)
ii
where = (1 + V)1/2V , I is the condensate density. Further, eq. (7) has to be supplemented by the charge and the current associated with a slow spatial temporal distortion of 9
(8) (9)
applied to eqs . (8) and (9). Conventionally, it was assumed that eq. (8) is associated with It while eq. (9) is with 10 (17). But this is inconsistent with eq. (10)" as well as the ph ason dispersion (5) and therefore should be rejected in favor of eqs. (8) and (9). Finally, the pinning potential Vp in ( 9) is given as [14,15]
Vlmp(9)
=
-(;N
OV2
f .1( T) tanh[ifJ.1(T)]
-x LCos 2(Q'X
j+9(X;),
(12)
i
with Q = 2PF and 11 = Q/'ITbc is the electron density. Equations (8) and (9) satisfy the charge conservation
all c
ajc
at + ax = 0,
(10)
which should be held at all temperatures. The charge conservation associated with the condensate is only broken if catalyzers (e.g. topological defects) like phase vortices are present. A moving phase vortex can convert the charge carried by the condensate to the one carried by the quasi-particle and vice versa [16]. In general, the condensate density I is a complex function of Co> and t = vqt where wand qt are the frequency and wave vector associated with 9. As first noted by Rice et al. [17] f take s different limiting values in the adiabatic limit (i.e. w, t « .10 ) ;
Ior eo > t, for w < t,
(11)
for"'=±t. At T= 0 K, we have 10 = 11 = 1, while in the vicinity of T = 1;, 10 cr. .1(T) and 11 cr. .12(T). The temperature dependences of 10 and II are evaluated numerically and shown in fig. 1. Further the same principle should be
if it is due to the impurities where V2 is the impurity potential. The sum over i extends over the impurity sites and/or is due to the commensurability [2]
where N is the order of the commensurability, eN is a coefficient of order of unity and JV"" 12 t a is the band width. In a Bechgaard salt, N = 4 is most likely. In this case, we show that v;, is one order of magnitude smaller than Vlmp if we have a few ppm impurities. On the other hand, if N = 3, v;, dominates the pinning of the SDW (or in this matter COW as well). In an unlikely case of N ~ 5, we can safely neglect the commensurability completely because v;, is extremely small unless IV is also very small due to some particular circumstances. Phason dynamics is studied by measuring the nonOhmic conductivity, the microwave response and the electromechanical effect (i.e. the change in the elastic constant in the presence of an electric field) . 3. Threshold electric field First, we shall examine the depinning electric field where the SOW starts sliding in response to an electric
K. Ma ki, A. Virosztek / Spin density lI'at'es
760
field E along the chain direction. Following Fukuyarna, Lee and Rice [18] it is important to discrim inate the strong pinning limit and the weak pinning limit though for a SOW in Bechgaard salts the weak pinning limit appears to be more appropriate due to the weak coupling between the SOW and an impurity. In the strong pinning limit, a single impurity pins the local ph ase and the thre shold field is given by [15]
E:(O) = (Qle)(IIJII)( -.rNOV2)2~O'
(14)
and
E:(T)/E:(O) =
(~(T)/~o) tanh[jp.!1(T)]JlI,
(15)
where IIi is the impurity concentration. Equation (14) predicts the threshold field of a few mV Icm for IIJII 10- 6 • Further, E:(T) is almost independent of T for T< tTc and then gradually increases to E:(TdIE:(O) = 1.33. In the weak pinning limit, on the other hand, we obtain;
2)4/4-D E:V(O) = (4 - D/2D)(Qlell) ( !D( -.rNoVo) X
( av
2N,0 )-D/4-D( 1) -1 II; )2/4-D( "0 A )4/4-D ,
(16)
Although the temperature dependence of E 1 in SOWs of (TMTSFhN03 is only available for T< tTc, E l in SOWs in (TMTSF)2PF6 behaves differently depending on whether the resistivity is measured by using conventional contact with silver paint or by clamped contact [3]. In the former case, the temperature dep endence of E, is qualitatively described by the impurity pinning though the observed E, lies in between the 30 and 20 weak pinning limit. On the other hand , in the latter sample the observed E, is consistent with the commensurability pinning except for a rather abrupt jump in E, just at T = Tc [3]. Furthermore the clamped samples exhibit a much sharper jump in the electric resistivity at Tc than the samples with conventional contacts. This rather sharp jump in the resistivity of the clamped samples may imply that they are better samples free of micro cracks. If this is the case, we can also describe the observed E 1 of the samples with the conventional contacts as the sum of two distinct pinning potentials (i.e, impurity and commensurability) though in these samples in the commensurability pinning is reduced roughly by a factor of ~ compared with the ones with clamped contacts .
4. Phason propagator The complex conductivity in a SOW is given by [19]
and
where a = -;r2/ 3, 1) = V2V31v2 is the anisotropy parameter and D is the spatial dimension of the fluctuation. The related Fukuyama-Lee-Rice length L(T) is given by
L(O) = (2av2No/h-IlliHD('lTNoV)2~o)2/4-D, (18) and
L(T)IL(O) = (E:(T)IE:(O»-2/4-D.
(19)
For D = 3 and IIJII - 10- 6 , L(O) - 10 J.1m. This means that only a few (1 or 2) Fukuyama-Lee-Rice domains in specimens of the usual size are present.' Maybe the unusual spectrum of the narrow band noise observed in SOWs of (TMTSFhCI04 (4] is due to large L(T). Further, this implies that the SOW has a stronger size effect than the COW. When the pinning is due to the commensurability, we obtain (l9]
(20) where we assumed N = 4.
+iw/oD- I(w,O) }-I ,
(21)
where r n is the normal (transport) damping constant, and D(w, q) is the phason propagator, 22 22 2 2 ~ D - 1( w,q ) =w2+'IW r.p-vQI-v2q2v3Q3-":' ,
(22) where r p is the phason damping and 2 is the self-energy. When the pinning is due to the commensurability only 2 is given by 2::= w~ = 2v e/I(/O) - 1 E1 (T),
(23)
and wp is the pinning frequency. In particular, the pinning frequency vani shes like (Tc - T)3 / 4 as the temperature approaches the transition temperature. When the pinning is due to impurities, 2 may be obtained within the coherent potential approximation [~8]. For example , in the strong pinning limit we obtain 2::= 2ev/ I ( / o) - I E1(T)(1
+ 'r.£( Ec - Z)
-I ,
(24)
761
K. Maki, A. Virosztek / Spin density waves
where
Within the same approximation, the sliding velocity is determined from [20] (29)
and
If we neglect the 2 scaling rule [21]
un (
£ ) term, we obtain Bardeen's
(25) and £c = 2..1 0 is the cut-off energy. Since e « 1 in most cases the pinning frequency is again approximately give by eq. (23). This suggests that even in the weak pinning limit eq. (23) will hold. Then we expect that w p in the weak pinning limit vanishes like (T - Tc>l/4 as T approaches Tc independently of D. A recent microwave experiment on SDW of (TMTSFhPF6 is analyzed [6] in terms of eq. (21) in the frequency range of co > 10 MHz where the self-energy 2 is negligible. Then eq. (21) predicts that the phason almost exhausts the conductivity sum rule; there is little contribution coming from the quasi-particle excitation W > 2..1(T). Indeed, in the clean limit (Tn/..1 0 « 1) we expect that the quasi-particle contribution is of the order of (Tn / ..1 0 ) 2 [6]. On the contrary, the experiment does reveal a sizeable contribution to o( w) at w = 2..1(T). This seems to imply that the phason mass in SDW is also renormalized in contrast to eq. (7) though m* /m - 2-10 which is much smaller than the one found in a CDW. Here m" and 111 are the phason and the band electron mass, respectively. Although there is no single phonon correction to 111*, there is a two phonon correction which gives 111* /111 =
1+ X
f1)2
2
(
(.;.) A..1 tanh("2..1 )/2
('IT/2 t a wQ r
1
tanh( ~WQ
)/1- 1,
(26)
and wQ is the phonon frequency and A is the dimensionless electron-phonon coupling constant. If the Bechgaard salt has a phonon with wQ - 10 K, eq. (26) will give 111* a few times 111 at low temperatures. We may calculate the self-energy in an unpinned SDW by changing 9(Xi ) in the cosine factor by 90(Xi ) - v,Qt, where v, is the sliding velocity. Then the CPA gives
(27) where (28)
(30) and the nonOhmic conductivity
0(£)
=
oo{1 - II
+ IITn Tp-l( 1 -
i
)0(£ - £t)}, (31)
where 0 0 = e 2n(1IITn ) - I . Here again, we neglected the contribution from the 2 term. Inclusion of the 2 term will change the E dependence near the threshold; E - E, should be replaced by A(£ - £t)3/2 in eq. (30) in the vicinity of the threshold. The present expression (31) appears to describe the observed nonOhmic conductance fairly well.
5. Concluding remarks We have shown that an anisotropic Hubbard model provides a useful starting point for analyzing not only the equilibrium properties but also the transport properties associated with the Frohlich conduction of spin density waves in Bechgaard salts. The experimental data have just started to accumulate. Due to a much longer Fukuyama-Lee-Rice coherence length in a SDW compared to the one in a CDW, the dynamic responses of a SDW can be quite different from those in a CDW. Perhaps, they will provide better controlled data, which make the theoretical comparison more useful. We have benefitted from discussions with G. Gruner and S. Tomic. The present work is supported by the National Science Foundation under grant No. DMR 86-11829 and DMR 89-15285 and by the US Department of Energy under grant No. DEF 605-84-ER45113. References [I] For a review, see P. Monceau, Electronic Properties of Inorganic Quasi One Dimensional Materials, P. Monceau ed. (Reidel, Dordrecht, 1985) p. 139. G. Gruner and A. Zettl, Phys. Rep. II9 (1985) II? G. Gruner, Rev. Mod. Phys. 60 (1988) II29. [2) P.A. Lee, T.M. Rice and P.W. Anderson, Solid State .' Commun. 14 (1974) 703. [3] S. Tornic, J.R. Cooper, D. Jerome and K. Bechgaard, Phys. Rev. Lett. 62 (1989) 462.
762
[4] [5] [6]
17] [8] [9] [10] (11]
K. Maki. A. Virosztek / Spin density waves W. Kang, S. Tornic, J.R. Cooper and D. Jerome, Phys. Rev. B 41 (1990) 4862. T. Sambongi et aI., Solid State Commun. 72 (1989) 817. K. Nomura et aI., Solid State Commun. 72 (1989) 1123. H.H.S. Javadi, S. Sridhar, G. Gruner, L. Chiang and F. Wudl, Phys. Rev. Lett. 55 (1985) 1216. T.W. Kim et al., Phys. Rev. B 42 (1990). K. Yarnaji, J. Phys. Soc. Jpn. 51 (1982) 2787, 52 (1983) 1361. D. Jerome, A. Mazaud, M. Ribault and K. Bechgaard, J. de Phys. Lett. 41 (1980) 195. J.F. Kwak, J.E. Shirber, R.L. Greene and E.M. Engler, Phys. Rev. Lett. 46 (1981) 1296. K. Yamaji, J. Phys. Soc. Jpn. 54 (1985) 1034; Synth. Met. 13 (1986) 29. D. Poilblanc, M. Heritier, G. Montambaux and P. Lederer, J. Phys. e 19 (1986) L321.
[12] A. Virosztek, L. Chen and K. Maki, Phys. Rev. B 34 (1986) 3371. [13] K. Maid and A. Virosztek, Phys. Rev. B 36 (1987) 511. (14] A. Virosztek and K. Maid, Phys. Rev. B 37 (1988) 2028. [15] K. Maki and A. Virosztek, Phys. Rev. B 39 (1989) 9640, B 41 (1990)557. (16] N.P. Ong and K. Maid, Phys. Rev. B 32 (1985) 6582. [17] T.M. Rice, P.A. Lee and M.e. Cross, Phys. Rev. B 20 (1979) 1345. [18] H. Fukuyama and P.A. Lee, Phys. Rev. B 17 (1978) 535. P.A. Lee and T.M. Rice, Phys. Rev. B 19 (1979) 3970. [19] K. Maid and A. Virosztek, in: Proc. Dubrovnik '89, Fizika (Zagreb, 1990); Phys. Rev. B 42 (1990) 655. [20] L. Sneddon, M. Cross and D. Fisher, Phys. Rev. Lett. 49 (1982) 292. [21] J. Bardeen, Physica B 126 (1984) 342, B 143 (1986) 14.
Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 758-762 North-Holland
Invited paper
Frohlich conduction in spin density waves and field induced spin density waves Kazumi Maki a b
a
and Attila Virosztek
a.b
Department of Physics, Unicersity of Southern California, Los Angeles, CA 90089-0484, USA Department of Physics, Unicersity of Virginia, Charlottescille, VA 22901, USA
We review the recent theory of phason dynamics in spin density waves in Bechgaard saIts. The theoretical results on the threshold electric field, the nonOhmic conductivity and the microwave conductivity are compared to recent experimental results on spin density waves in Bechgaard salts.
I. Introduction The Frohlich conduction (i.e. the colIective transport associated with sliding motion of the condensate) in quasi one-dimensional charge density waves (COW) is welI established both experimentalIy and theoreticalIy in a number of inorganic and organic compounds [II. A similar conduction in spin density waves (SOW) has been speculated by Lee, Rice and Anderson. [21 even before the discovery of the Frohlich conduction in NbSeJ. Since the discovery of the SOW ground state and the field induced spin density wave (FISOW) ground state in some of Bechgaard salts like (TMTSFhCI04 and (TMTSFhPF6, it is quite natural to look for the possible Frohlich conduction in these systems. Indeed, both the nonOhmic conductivity [3,41 and the microwave response [5,6] of SOWs in (TMTSFhNOJ, (TMTSFh PF6 and (TMTSFhCI04 are reported recently, which indicate the Frohlich conduction in a SOW as well. As a model, we take an anisotropic Hubbard model first introduced by Yamaji [8] in order to study the equilibrium and nonequilibrium properties of Bechgaard salts at low temperatures. The Hamiltonian is given by H= L«p)C/aCpa + VLnqrn_ q ! ,
pa
Within the mean field analysis of eq. (1), Yamaji [7] was able to describe the phase diagram of (TMTSFhPF6 first determined by Jerome et al. [8]. Central to his model is the imperfect nesting. As the pressure increases the three dimensionality by increasing Ib and i., the perfect nesting is increasingly destroyed leading to the decrease of the SOW transition temperature. Later, after the discovery of FISOW by Kwak et al. [9], the same Hamiltonian is used successfulIy to describe the FISOW [10-12]. The spin density wave in the ground state is given by
S(x)
=
2V-]L1(T)"1 cos( Qx + ¢(x»,
(3)
where Ii is a unit vector, ¢(x) is the phase of the order parameter, L1(T) is the amplitude of the order parameter and the nesting vector Q is given by
(4)
Q = (2PF' 'IT/b, 'IT/c).
The SOW possesses two classes of Goldstone bosons; spin wave and phason (sliding SOW) with dispersions [13,14]
and
(1)
(5)
q
respectively, where V = NoV and No = ('lTVbC)-1 is the electron density of states at the Fermi surface per spin and v, V2, VJ are the anisotropic Fermi velocities given by
where
with (v, v 2 , vJ) =
(2/ a a sin
ap «, {flbb, {fleC).
(6)
(2) and J! is the chemical potential and C/a and Cpa are the quasi-particle creation and annihilation operators with momentum p and spin a.
Note that these dispersions are independent of the temperature. However, in the real systems both the spin rotation symmetry and the translational symmetry of SOW are
0304-8853/90/S03.50
K Maki, A. Virosztek
I Spin density \I'aves
759
1.0 I----=-_
broken weakly and these modes acquire small energy gaps . The former is broken by the spin-orbit coupling and magnetic dipole interaction while the latter is broken by crystalline defects or sometimes by commensurability with an underlying crystalline lattice. In the following, we shall focus on phason dynamics.
QS
2. Phase Hamiltonian From the analysis of the fluctuation propagator of the phason, we can write down the pha se Hamiltonian of a SOW [15J
H(9)=JdDX{~No/[(~~f +ii2(~~f 9 +V22(0ay )2+ v32(a9)2] az - enfQ -t} 9£ + Vpin ( 9),
05
TlTe ,',
to
Fig. 1. The dynamic and static condensate densities 10 and II are evalu at ed numerically and shown as funct ions of the reduced temperature TIT,;,
(7)
ii
where = (1 + V)1/2V , I is the condensate density. Further, eq. (7) has to be supplemented by the charge and the current associated with a slow spatial temporal distortion of 9
(8) (9)
applied to eqs . (8) and (9). Conventionally, it was assumed that eq. (8) is associated with It while eq. (9) is with 10 (17). But this is inconsistent with eq. (10)" as well as the ph ason dispersion (5) and therefore should be rejected in favor of eqs. (8) and (9). Finally, the pinning potential Vp in ( 9) is given as [14,15]
Vlmp(9)
=
-(;N
OV2
f .1( T) tanh[ifJ.1(T)]
-x LCos 2(Q'X
j+9(X;),
(12)
i
with Q = 2PF and 11 = Q/'ITbc is the electron density. Equations (8) and (9) satisfy the charge conservation
all c
ajc
at + ax = 0,
(10)
which should be held at all temperatures. The charge conservation associated with the condensate is only broken if catalyzers (e.g. topological defects) like phase vortices are present. A moving phase vortex can convert the charge carried by the condensate to the one carried by the quasi-particle and vice versa [16]. In general, the condensate density I is a complex function of Co> and t = vqt where wand qt are the frequency and wave vector associated with 9. As first noted by Rice et al. [17] f take s different limiting values in the adiabatic limit (i.e. w, t « .10 ) ;
Ior eo > t, for w < t,
(11)
for"'=±t. At T= 0 K, we have 10 = 11 = 1, while in the vicinity of T = 1;, 10 cr. .1(T) and 11 cr. .12(T). The temperature dependences of 10 and II are evaluated numerically and shown in fig. 1. Further the same principle should be
if it is due to the impurities where V2 is the impurity potential. The sum over i extends over the impurity sites and/or is due to the commensurability [2]
where N is the order of the commensurability, eN is a coefficient of order of unity and JV"" 12 t a is the band width. In a Bechgaard salt, N = 4 is most likely. In this case, we show that v;, is one order of magnitude smaller than Vlmp if we have a few ppm impurities. On the other hand, if N = 3, v;, dominates the pinning of the SDW (or in this matter COW as well). In an unlikely case of N ~ 5, we can safely neglect the commensurability completely because v;, is extremely small unless IV is also very small due to some particular circumstances. Phason dynamics is studied by measuring the nonOhmic conductivity, the microwave response and the electromechanical effect (i.e. the change in the elastic constant in the presence of an electric field) . 3. Threshold electric field First, we shall examine the depinning electric field where the SOW starts sliding in response to an electric
K. Ma ki, A. Virosztek / Spin density lI'at'es
760
field E along the chain direction. Following Fukuyarna, Lee and Rice [18] it is important to discrim inate the strong pinning limit and the weak pinning limit though for a SOW in Bechgaard salts the weak pinning limit appears to be more appropriate due to the weak coupling between the SOW and an impurity. In the strong pinning limit, a single impurity pins the local ph ase and the thre shold field is given by [15]
E:(O) = (Qle)(IIJII)( -.rNOV2)2~O'
(14)
and
E:(T)/E:(O) =
(~(T)/~o) tanh[jp.!1(T)]JlI,
(15)
where IIi is the impurity concentration. Equation (14) predicts the threshold field of a few mV Icm for IIJII 10- 6 • Further, E:(T) is almost independent of T for T< tTc and then gradually increases to E:(TdIE:(O) = 1.33. In the weak pinning limit, on the other hand, we obtain;
2)4/4-D E:V(O) = (4 - D/2D)(Qlell) ( !D( -.rNoVo) X
( av
2N,0 )-D/4-D( 1) -1 II; )2/4-D( "0 A )4/4-D ,
(16)
Although the temperature dependence of E 1 in SOWs of (TMTSFhN03 is only available for T< tTc, E l in SOWs in (TMTSF)2PF6 behaves differently depending on whether the resistivity is measured by using conventional contact with silver paint or by clamped contact [3]. In the former case, the temperature dep endence of E, is qualitatively described by the impurity pinning though the observed E, lies in between the 30 and 20 weak pinning limit. On the other hand , in the latter sample the observed E, is consistent with the commensurability pinning except for a rather abrupt jump in E, just at T = Tc [3]. Furthermore the clamped samples exhibit a much sharper jump in the electric resistivity at Tc than the samples with conventional contacts. This rather sharp jump in the resistivity of the clamped samples may imply that they are better samples free of micro cracks. If this is the case, we can also describe the observed E 1 of the samples with the conventional contacts as the sum of two distinct pinning potentials (i.e, impurity and commensurability) though in these samples in the commensurability pinning is reduced roughly by a factor of ~ compared with the ones with clamped contacts .
4. Phason propagator The complex conductivity in a SOW is given by [19]
and
where a = -;r2/ 3, 1) = V2V31v2 is the anisotropy parameter and D is the spatial dimension of the fluctuation. The related Fukuyama-Lee-Rice length L(T) is given by
L(O) = (2av2No/h-IlliHD('lTNoV)2~o)2/4-D, (18) and
L(T)IL(O) = (E:(T)IE:(O»-2/4-D.
(19)
For D = 3 and IIJII - 10- 6 , L(O) - 10 J.1m. This means that only a few (1 or 2) Fukuyama-Lee-Rice domains in specimens of the usual size are present.' Maybe the unusual spectrum of the narrow band noise observed in SOWs of (TMTSFhCI04 (4] is due to large L(T). Further, this implies that the SOW has a stronger size effect than the COW. When the pinning is due to the commensurability, we obtain (l9]
(20) where we assumed N = 4.
+iw/oD- I(w,O) }-I ,
(21)
where r n is the normal (transport) damping constant, and D(w, q) is the phason propagator, 22 22 2 2 ~ D - 1( w,q ) =w2+'IW r.p-vQI-v2q2v3Q3-":' ,
(22) where r p is the phason damping and 2 is the self-energy. When the pinning is due to the commensurability only 2 is given by 2::= w~ = 2v e/I(/O) - 1 E1 (T),
(23)
and wp is the pinning frequency. In particular, the pinning frequency vani shes like (Tc - T)3 / 4 as the temperature approaches the transition temperature. When the pinning is due to impurities, 2 may be obtained within the coherent potential approximation [~8]. For example , in the strong pinning limit we obtain 2::= 2ev/ I ( / o) - I E1(T)(1
+ 'r.£( Ec - Z)
-I ,
(24)
761
K. Maki, A. Virosztek / Spin density waves
where
Within the same approximation, the sliding velocity is determined from [20] (29)
and
If we neglect the 2 scaling rule [21]
un (
£ ) term, we obtain Bardeen's
(25) and £c = 2..1 0 is the cut-off energy. Since e « 1 in most cases the pinning frequency is again approximately give by eq. (23). This suggests that even in the weak pinning limit eq. (23) will hold. Then we expect that w p in the weak pinning limit vanishes like (T - Tc>l/4 as T approaches Tc independently of D. A recent microwave experiment on SDW of (TMTSFhPF6 is analyzed [6] in terms of eq. (21) in the frequency range of co > 10 MHz where the self-energy 2 is negligible. Then eq. (21) predicts that the phason almost exhausts the conductivity sum rule; there is little contribution coming from the quasi-particle excitation W > 2..1(T). Indeed, in the clean limit (Tn/..1 0 « 1) we expect that the quasi-particle contribution is of the order of (Tn / ..1 0 ) 2 [6]. On the contrary, the experiment does reveal a sizeable contribution to o( w) at w = 2..1(T). This seems to imply that the phason mass in SDW is also renormalized in contrast to eq. (7) though m* /m - 2-10 which is much smaller than the one found in a CDW. Here m" and 111 are the phason and the band electron mass, respectively. Although there is no single phonon correction to 111*, there is a two phonon correction which gives 111* /111 =
1+ X
f1)2
2
(
(.;.) A..1 tanh("2..1 )/2
('IT/2 t a wQ r
1
tanh( ~WQ
)/1- 1,
(26)
and wQ is the phonon frequency and A is the dimensionless electron-phonon coupling constant. If the Bechgaard salt has a phonon with wQ - 10 K, eq. (26) will give 111* a few times 111 at low temperatures. We may calculate the self-energy in an unpinned SDW by changing 9(Xi ) in the cosine factor by 90(Xi ) - v,Qt, where v, is the sliding velocity. Then the CPA gives
(27) where (28)
(30) and the nonOhmic conductivity
0(£)
=
oo{1 - II
+ IITn Tp-l( 1 -
i
)0(£ - £t)}, (31)
where 0 0 = e 2n(1IITn ) - I . Here again, we neglected the contribution from the 2 term. Inclusion of the 2 term will change the E dependence near the threshold; E - E, should be replaced by A(£ - £t)3/2 in eq. (30) in the vicinity of the threshold. The present expression (31) appears to describe the observed nonOhmic conductance fairly well.
5. Concluding remarks We have shown that an anisotropic Hubbard model provides a useful starting point for analyzing not only the equilibrium properties but also the transport properties associated with the Frohlich conduction of spin density waves in Bechgaard salts. The experimental data have just started to accumulate. Due to a much longer Fukuyama-Lee-Rice coherence length in a SDW compared to the one in a CDW, the dynamic responses of a SDW can be quite different from those in a CDW. Perhaps, they will provide better controlled data, which make the theoretical comparison more useful. We have benefitted from discussions with G. Gruner and S. Tomic. The present work is supported by the National Science Foundation under grant No. DMR 86-11829 and DMR 89-15285 and by the US Department of Energy under grant No. DEF 605-84-ER45113. References [I] For a review, see P. Monceau, Electronic Properties of Inorganic Quasi One Dimensional Materials, P. Monceau ed. (Reidel, Dordrecht, 1985) p. 139. G. Gruner and A. Zettl, Phys. Rep. II9 (1985) II? G. Gruner, Rev. Mod. Phys. 60 (1988) II29. [2) P.A. Lee, T.M. Rice and P.W. Anderson, Solid State .' Commun. 14 (1974) 703. [3] S. Tornic, J.R. Cooper, D. Jerome and K. Bechgaard, Phys. Rev. Lett. 62 (1989) 462.
762
[4] [5] [6]
17] [8] [9] [10] (11]
K. Maki. A. Virosztek / Spin density waves W. Kang, S. Tornic, J.R. Cooper and D. Jerome, Phys. Rev. B 41 (1990) 4862. T. Sambongi et aI., Solid State Commun. 72 (1989) 817. K. Nomura et aI., Solid State Commun. 72 (1989) 1123. H.H.S. Javadi, S. Sridhar, G. Gruner, L. Chiang and F. Wudl, Phys. Rev. Lett. 55 (1985) 1216. T.W. Kim et al., Phys. Rev. B 42 (1990). K. Yarnaji, J. Phys. Soc. Jpn. 51 (1982) 2787, 52 (1983) 1361. D. Jerome, A. Mazaud, M. Ribault and K. Bechgaard, J. de Phys. Lett. 41 (1980) 195. J.F. Kwak, J.E. Shirber, R.L. Greene and E.M. Engler, Phys. Rev. Lett. 46 (1981) 1296. K. Yamaji, J. Phys. Soc. Jpn. 54 (1985) 1034; Synth. Met. 13 (1986) 29. D. Poilblanc, M. Heritier, G. Montambaux and P. Lederer, J. Phys. e 19 (1986) L321.
[12] A. Virosztek, L. Chen and K. Maki, Phys. Rev. B 34 (1986) 3371. [13] K. Maid and A. Virosztek, Phys. Rev. B 36 (1987) 511. (14] A. Virosztek and K. Maid, Phys. Rev. B 37 (1988) 2028. [15] K. Maki and A. Virosztek, Phys. Rev. B 39 (1989) 9640, B 41 (1990)557. (16] N.P. Ong and K. Maid, Phys. Rev. B 32 (1985) 6582. [17] T.M. Rice, P.A. Lee and M.e. Cross, Phys. Rev. B 20 (1979) 1345. [18] H. Fukuyama and P.A. Lee, Phys. Rev. B 17 (1978) 535. P.A. Lee and T.M. Rice, Phys. Rev. B 19 (1979) 3970. [19] K. Maid and A. Virosztek, in: Proc. Dubrovnik '89, Fizika (Zagreb, 1990); Phys. Rev. B 42 (1990) 655. [20] L. Sneddon, M. Cross and D. Fisher, Phys. Rev. Lett. 49 (1982) 292. [21] J. Bardeen, Physica B 126 (1984) 342, B 143 (1986) 14.