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PHYSICAL REVIEW B 80, 184417 2009

Anisotropy of the spin density wave onset for (TMTSF)2PF6 in a magnetic field
Ya. A. Gerasimenko, V. A. Prudkoglyad, A. V. Kornilov, and V. M. Pudalov
P. N. Lebedev Physical Institute, Moscow 119991, Russia

V. N. Zverev
Institute for Solid State Physics, Chernogolovka, 142432 Moscow District, Russia

A.-K. Klehe
Clarendon Laboratory, Oxford University, Oxford OX1 3PU, United Kingdom

J. S. Qualls
Sonoma State University, Rohnert Park, California 94928, USA Received 26 June 2009; published 18 November 2009 In order to study the spin density wave transition temperature TSDW in TMTSF 2PF6 as a function of magnetic field, we measured the magnetoresistance Rzz in fields up to 19 T. Measurements were performed for three field orientations B a , b and c at ambient pressure and at P = 5 kbar, that is nearly the critical pressure. For B c orientation we observed quadratic field dependence of TSDW in agreement with theory and with previous experiments. For B b and B a orientations we have found no shift in TSDW within 0.05 K, both at P = 0 and P = 5 kbar. This result is also consistent with theoretical predictions. DOI: 10.1103/PhysRevB.80.184417 PACS number s : 75.30.Fv, 73.43.Nq

I. INTRODUCTION

TMTSF 2PF6 is a layered organic compound that demonstrates a complex phase diagram, containing phases, characteristic of one-dimensional 1D , two-dimensional, and three-dimensional systems. Transport properties of this material are highly anisotropic typical ratio of the conductivity tensor components is xx : yy : zz 105 :103 :1 at T = 100 K Refs. 1 and 2 . At ambient pressure and zero magnetic field the carrier system undergoes a transition to the antiferromagnetically ordered spin density wave SDW state1 with a transition temperature TSDW 12 K. When an external hydrostatic pressure is applied, TSDW gradually decreases and vanishes at the critical pressure of 6 kbar.3 For higher pressures, P 6 kbar, the SDW state is completely suppressed. Application of a sufficiently high magnetic field along the least conducting direction c restores the spin ordering. This occurs via a cascade of the field-induced SDW FISDW states.4 The conventional model for the electronic spectrum is1,5 E0 k =
vF k
x

the system diverges at q = Q0 and the system is unstable against formation of SDW.1 When tb is nonzero, the situation changes drastically: no vector can couple all states on both sides of the Fermi surface, though Q0 still couples a large number of states. The situation called "imperfect nesting" is sketched on Fig. 1 a . Despite the complex behavior of the system, theory6,7 successfully describes the effects of pressure and magnetic field on the SDW transition in terms of the single parameter tb. According to the theory, tb increases with external pressure and conditions for nesting deteriorate. Therefore, under pressure deviations of the system from the ideal 1D model become more prominent and, as a consequence, TSDW decreases. When tb reaches a critical value tb, the SDW transition vanishes. The application of a magnetic field normal to the a direction restricts electron motion in the b-c plane making the system effectively more one dimensional. Theory, Ref. 6, predicts that the transition temperature increases in weak fields B c as T
SDW

k

F

-2tb cos kyb 1

B =T

SDW

B -T

SDW

0 = B2 ,

-2tb cos 2kyb -2tc cos kzc ,

where tb and tc are the nearest-neighbor transfer integrals along b and c directions, respectively, and tb is the transfer integral involving next-to-nearest second-order neighbors. For ideal one-dimensional case, tb = tc = tb = 0, and the Fermi surface consists of two parallel flat sheets. This surface satisfies the so-called ideal nesting condition: there exists a vector Q0 which couples all states across the Fermi surface. In the quasi-one-dimensional case, when tb and tc are nonzero, the Fermi sheets become slightly corrugated. Nevertheless, one can still find a vector, that couples all states across the Fermi surface, therefore the ideal nesting property also holds in this case. It means that the magnetic susceptibility q of
1098-0121/2009/80 18 /184417 6

and further saturates in high fields; here = P is a function of pressure. A number of experiments3,8 ­10 were made to examine the predictions of the theory for the B c case. All these studies confirmed quadratic field dependence of the transition temperature. Nevertheless, the predicted saturation has not been seen until now. Furthermore, Murata et al.11­14 reported an unexpected anisotropy of the TSDW in TMTSF 2PF6 under uniaxial stress, the result seems to disagree with the theory. According to theory,6,7 the only relevant parameter is tb; therefore, one might expect the uniaxial stress along b to affect TSDW stronger than the stress in other directions. Murata et al.,11­14 however, showed experimentally that the
©2009 The American Physical Society

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qy

z

(a)

(b)

FIG. 1. a Schematic view of the Fermi surface in the imperfect nesting model. Dashed and solid lines show Fermi surface with and without tb term in Eq. 1 , respectively tb value is magnified for clarity . Q denotes the nesting vector. b Schematic 3D view of the Fermi surface. Dashed and solid lines are the orbits of an electron, when magnetic field B c and B b , respectively.

uniaxial stress applied along the a direction changed TSDW stronger than the stress in the b direction. The results mentioned above demonstrate that the consistency between the theoretical description and experiment is incomplete. Whereas there is a number of experimental data for the magnetic field B c , for B a and B b only one experiment8 has been done so far at ambient pressure, and none at elevated pressure. Danner et al.8 observed no field dependence for B a and B b at ambient pressure. The absence of a field dependence, however, cannot be considered as a crucial test of the theory because the effect of the magnetic field might be small at ambient pressure. Indeed, according to the theory, elevated pressure enhances any imperfections of nesting, and the effect of magnetic field is expected to become stronger. As a result, the strongest effect should take place at pressures close to the critical value. The aim of the present work, therefore, is to determine experimentally TSDW B dependence for B a and B b near the critical pressure. We report here our measurements of the magnetic field induced shift in TSDW made at P = 0 and 5 kbar for the three orientations B a, B b , and B c . Our main result is that for B a and B b there is no distinct shift of the transition temperature within our measurements' uncertainty 0.05 K at pressure up to 5 kbar and in fields up to 19 T. At the same time, we found quadratic TSDW B dependences for B c both at zero and nonzero pressures, a result in agreement with previous studies by other groups.3,8 ­10 We suggest an explanation of our experimental data, based on the meanfield theory, and show that the latter correctly describes the effect of the magnetic field on TSDW.
II. EXPERIMENT

nique at 10­120 Hz frequencies. The out-of-phase component of the contacts resistance was negligible. The resistance along c axis, Rzz, was measured using two pairs of contacts on top and bottom faces, normal to the c axis. For measurements under pressure the sample and a manganin pressure gauge were inserted into a miniature nonmagnetic spherical pressure cell15 with an outer diameter of 15 mm. The cell was filled with Si-organic pressure transmitting liquid16 PES-1 . The pressure was applied and fixed at room temperature. The pressure values quoted throughout this paper refer to those determined at helium temperature. After pressure was applied, the cell was mounted in a two-axis rotation stage placed in liquid 4He in a bore of a 21 T superconducting magnet at Oxford University. The rotating system enabled rotation of the pressure cell around the main axis by 200° with uncertainty of 0.1° and around the auxiliary axis by 360° with uncertainty of 1° ; this allowed us to set the sample at any desired orientation with respect to the magnetic field direction. Measurements at ambient pressure were performed using more simple rotating system which allowed rotation around only one axis perpendicular to the field direction by 200° with 0.1° uncertainty. This system was mounted in a bore of a 17 T superconducting magnet at ISSP. Samples were cooled very slowly, at the rate of 0.2/0.3 K/min to avoid microcracks. Nevertheless, some samples cooled down at ambient pressure experienced 1­2 microcracks, seen as an irreversible jumps a few percents in the sample resistance. No cracks were observed during cooling of a sample in the pressure cell. During measurements under pressure the temperature of the cell was determined by RuO2 thermometer, and during measurements at ambient pressure--by Cu-Fe-Cu thermocouple and RuO2 thermometer. The temperature was varied slowly in order to ensure that the sample and the thermometer were in thermal equilibrium. The thermal equilibrium condition was verified by the absence of a hysteresis in Rzz T between cooling and heating cycles.
III. RESULTS AND DISCUSSION

Measurements were performed on three samples from the same batch and the results were in qualitative correspondence with each other. Most detailed data taken for two samples are presented in this section.
A. B ё c

Single-crystal samples of TMTSF 2PF6 were grown by conventional electrochemical technique. Measurements were made on three samples from the same batch the typical sample size is 3 0.25 0.1 mm3 along a, b , and c directions, respectively . Eight fine wires 10 m Au wires or 25 m Pt wires were attached to the sample with conductive graphite paste. Two groups of four contacts were made on the two opposite a-b faces of the sample along a axis. All measurements were made by four-probe ac lock-in tech-

Figure 2 shows the temperature dependence of Rzz at ambient pressure and different magnetic fields. Rzz B =0 at ambient pressure is shown in the inset of Fig. 2 a over a large temperature range: as temperature decreases, the resistance decreases monotonically, then exhibits a sharp jump and further increases in a temperature activated manner. The jump at 12 K indicates the transition to the low-temperature spin density wave state. Throughout the paper we define the transition temperature according to the peak in d ln Rzz / d 1 / T , the logarithmic derivative of resistance vs inverse temperature. As the magnetic field applied along c axis grows, TSDW

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FIG. 2. Temperature dependences of Rzz at ambient pressure for six values of magnetic field as shown on panel b aligned with the least conduction direction, B c . a Rzz T for the set of magnetic fields; b logarithmic derivatives of the same data. Inset to panel a demonstrates typical dependence of Rzz versus T. Inset to b shows linear fit for transition temperatures, obtained from the derivative plots, vs B2.

FIG. 3. Temperature dependences of Rzz at an elevated pressure of 5 kbar for a field orientation B c . a Rzz T for the set of magnetic fields. The inset shows TSDW versus B2 for P = 5 kbar filled dots and for P =0 empty circles . b Logarithmic derivatives of the same data. The inset demonstrates that TSDW shift in the magnetic field is much more pronounced at P = 5 kbar than that at ambient pressure.

is shifted progressively to higher temperatures Fig. 2 b . The shift increases quadratically with field, as shown in the inset of Fig. 2 b . Application of pressure P = 5 kbar lowers the zero-field transition temperature down to 6.75 K Figs. 3 a and 3 b . The pressure dependence of TSDW P is known to be strongly nonlinear,17 its slope is small at low pressures and sharply increases in the vicinity of the critical pressure value. Therefore, the factor of 2 decrease in TSDW P from 12 to 6.75 K, compare Figs. 2 and 3 demonstrates that the pressure is close to the critical value. At a pressure of 5 kbar and in the presence of a magnetic field B c , the transition temperature TSDW grows nearly quadratically with field, TSDW B2 see inset of Fig. 3 a . This growth is qualitatively similar to that for zero pressure, however, it is much more pronounced, compared to the former case see Fig. 3 b and the inset of Fig. 3 a . Application of a magnetic field also increases resistance in the SDW state cf. Figs. 2 a and 3 a . In principle, the resistance growth should be related with the increase in TSDW, e.g., due to the increase in the SDW gap in magnetic field.7 However, the data on Fig. 3 as well as previous observations for example, Ref. 9 indicate, that Rzz T cannot be described by a temperature-activated behavior, both in zero and nonzero magnetic fields. Apparently, the observed Rzz T , B dependence is governed by both the increase in the SDW intensity in magnetic field and the magnetoresistance. Therefore, without an adequate model for Rzz T , B the two contributions cannot be separated, even though our data

clearly indicate the correlation of the resistance growth and the increase in TSDW in magnetic field. The observed TSDW B dependence Fig. 3 for our samples in magnetic field along c is qualitatively consistent with theory6,7 and with earlier observations by other groups.3,8 ­10 According to the theory, pressure deteriorates nesting conditions, enhancing the tb term in the energy spectrum Eq. 1 . Therefore, under pressure the number of unnested electrons increases as compared to the ambient pressure case. In contrast to the action of pressure, application of magnetic field B c improves the nesting conditions, both at elevated and at zero pressure, although the number of unnested electrons is larger in the former case. This is predicted to lead to an enhancement of the field dependence of TSDW under pressure. Our data see inset of Fig. 3 a confirms the theoretically predicted enhancement of the TSDW B c dependence at pressures close the critical value. We therefore anticipate that if the TSDW B dependence existed for other field orientations, it would be enhanced at elevated pressures. Correspondingly, we performed an experimental search for this dependence at P close to the critical value of Pc.
B. B ё a,b

When the effect on the or, at least, is illustrate this

magnetic field is applied in the a-b plane, its SDW-transition temperature is either missing much less than for B c . Figures 4 a and 4 b result for one of the orientations, B b . Even

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PHYSICAL REVIEW B 80, 184417 2009

FIG. 4. Temperature dependences of Rzz at ambient pressure in magnetic field 9 T compared for B c and B b orientations. a Panel shows Rzz T and b panel shows logarithmic derivatives of these dependences. Derivative graphs in a larger scale shown on the inset.

though the shape of the Rzz T curves slightly changes with field, the temperature of the transition remains unchanged within our measurement uncertainty of 0.03 K. For comparison, on the same figures we present also the Rzz T data for the B c orientation, demonstrating that the shift of TSDW in the same field of 9 T is an order of magnitude higher for B c. In line with the experimental situation for the B c , one might expect the shift in TSDW if any to be enhanced under pressure. Figures 5 a and 5 b summarize the main result of our paper--the Rzz T dependences across the transition measured for all three field orientations B a , b , and c at P = 5 kbar, close to the critical pressure. At P = 5 kbar and at B = 19 T, the shift TSDW B is as large as 1 K for B c , whereas for B a , b the shift is either missing or vanishingly small, at least a factor of 20 smaller than for B c see Fig. 5 b . Zooming the data in Fig. 5 c on the left panel , one can notice that Rzz T curves for B a , b are slightly shifted from the B = 0 one. However, our measurements uncertainty is comparable with this difference; for this reason, the sources of this uncertainty are analyzed below. There are two possible sources of uncertainties: i the calibration error of the RuO2 thermometer in magnetic fields and ii an uncertainty of the procedure used to determine the transition temperature. The latter contribution was determined by the width of the transition and was estimated to be about 0.02­0.03 K. As for the former one, in all measurements we used RuO2 resistance thermometer whose magnetoresistance was calibrated at 4.2 K. Possible changes in the RuO2 magnetoresistance between 4.2 and 6.7 K are the ma-

FIG. 5. Temperature dependences of Rzz under pressure P = 5 kbar in magnetic field 19 T aligned with B a or B b compared with orientation B c ; a and b panels show Rzz T and their logarithmic derivatives, respectively. These results are corrected due to magnetoresistance of RuO2 thermometer. Panel c zooms in 0.5 K interval near the transition, solid lines are cubic polynomial fits of experimental points.

jor source of our uncertainty and are estimated to be 0.04 K. Thus, we can only quantify the changes TSDW B that are larger than 50 mK. If the transition temperature changes with B a or B b , the changes are to be smaller than the above value.
C. Discussion

i In the changes in the transition temperature result from imperfect nesting. The energy spectrum Eq. 1 contains the only term tb cos kyb responsible for the nesting imperfection. Magnetic field parallel to the c direction eliminates the electron dispersion in the b direction from the system Hamiltonian.18 This effect is somewhat similar to the quasiclassical action of the Lorentz force on the electrons. The force is directed along b axis because the Fermi velocity vF in TMTSF 2PF6 is along the a axis, on average. It makes electrons on the Fermi surface cross the Brillouin

theory,6,7

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PHYSICAL REVIEW B 80, 184417 2009

zone in the ky direction see Fig. 1 a . Such a motion averages the electron's energy over all ky states,18 and all the terms, that contain cos kyb in the electron spectrum Eq. 1 , vanish. Since the tb term, responsible for the nesting imperfection, also vanishes, the magnetic field B c improves nesting conditions; this results in a growth of the transition temperature. This effect is described by the mean-field theory.6,7 In contrast, a magnetic field B b has no effect on tb, therefore, no shift in TSDW should occur in this field orientation. Our result that for B b the shift in TSDW is much less than for B c does not contradict this prediction. ii In principle, the magnetic field B b still can affect the electron dispersion in the c direction. In theory6 such a dispersion is neglected and tb is assumed to be the only term responsible for imperfect nesting. In general, besides tb there are other antinesting terms that can affect TSDW in field B b . Studies of TSDW anisotropy for different field direction may, in principle, provide information on the tb / tc ratio. In what follows, we estimate the tb / tc ratio from our experimental data. In order to do this, we expand the energy spectrum Eq. 1 , E1 k = E0 k -2tbc cos kyb cos kzc -2tc cos 2kzc , 2 where tbc and tc are the next-to-nearest hopping integrals. In the model Eq. 2 and for magnetic field direction B c , the correction to TSDW from the tbc term is considerably smaller, than that from tb, for tb / tc 1. Therefore, the TSDW B c dependence is almost unchanged, when tbc is taken into account. However, for B b the situation is essentially different. The electrons experience now the Lorentz force along c axis, and the corresponding motion along kz averages out all cos kzc terms in the electron spectrum Eq. 2 . Therefore, the contribution of tbc to TSDW B dependence becomes dominant. iii When magnetic field is applied along a axis, it is not expected to alter the electron motion, because the Lorentz force is zero, on average. Correspondingly, there are no terms in the electron spectrum which may be affected by the magnetic field in this orientation and the transition temperature is not expected to depend on the field B a. From the above discussion we conclude that TSDW, in principle, might be affected by the field B b . Bjelis and Maki in Ref. 19 took the tbc term into account and derived an expression for the transition temperature in tilted magnetic field. Based on this result, one can show see Appendix that in high magnetic fields the anisotropy of TSDW B is related with the tb / tc ratio TSDW B c - TSDW 0 TSDW B b - TSDW 0 1t 4t
b c 2 c b 2

der to overcome this problem, we ramped the temperature slowly in a fixed magnetic field 19 T and measured Rzz T . We repeated the procedure for the three field orientations B a , b , c by rotating in situ the pressure cell with the sample with respect to magnetic field direction. The TSDW B data measured this way were used to calculate tb / tc with Eq. 3 . In the calculations we substituted TSDW B a for TSDW 0 because as discussed in iii magnetic field B a does not affect TSDW. Such a procedure enabled us to eliminate the error related with the magnetoresistance of RuO2 thermometer. Yet the difference Tab = TSDW B b - TSDW B a was within the error bar of 0.02 K in the experiment, the upper bound of our estimate Tab = 0.02 K corresponds according to Eq. 3 to the lower bound of tb / tc 7. This estimate agrees with earlier result tb / tc 6 obtained from angle-dependent magnetoresistance studies in the metallic state at 7 kbar.21 The tb / tc estimates indicate that the contribution of the tbc term to TSDW is negligible, a factor of 50 smaller than that of tb.
IV. CONCLUSION

In conclusion, we have measured the magnetic field effect on the transition temperature TSDW to the spin density wave state in TMTSF 2PF6 in fields up to B = 19 T for three orientations B a , b , c , and at pressures up to 5 kbar. Measurements for B c are in qualitative agreement with the meanfield theory6,7 and with results of other groups.3,8 ­10 Our data confirm that the field dependence of TSDW is enhanced as pressure increases and approaches the critical value. The main result of our paper is that the magnetic field dependence of TSDW for B a and for B b is either absent or vanishingly small at least a factor of 20 smaller than for B c even near the critical pressure and at B = 19 T. This shows that the influence of other imperfect nesting terms on TSDW is negligibly small. This result confirms the assumption of the theory that TSDW is determined by the antinesting terms with the biggest contribution from the tb term in the electron spectrum.
ACKNOWLEDGMENTS

,

3

We are grateful to P. D. Grigoriev, A. G. Lebed, and A. Ardavan for their valuable suggestions and discussion of our results. The work was partially supported by the Programs of the Russian Academy of Sciences, RFBR Grants No. 08-0201047 and No. 09-02-12206 , Russian Ministry of Education and Science, the State Program of support of leading scientific schools Grant No. 1160.2008.2 , EPSRC, and the Royal Society.
APPENDIX: DERIVATION OF EQ. (3)

1 is a numeric factor. Since the above relationship where is dominated by tb / tc 2, the shift in TSDW for B b is expected to be considerably weaker than for B c . Figure 5 shows the experimental data for SDW transition in fields B a , b , c . This data enables us to estimate the tb / tc ratio using Eq. 3 .20 However, such a straightforward comparison of TSDW in B = 19 T and B = 0 includes a large uncertainty related with thermometer magnetoresistance. In or-

Bjelis and Maki in Ref. 19 took the tbc term into account and derived an expression for the transition temperature for magnetic field B in b -c plane. General form of this expression involves series of products of Bessel functions. In high magnetic fields the expression can be simplified by saving only the greatest term in the series. Namely, for B c ,

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PHYSICAL REVIEW B 80, 184417 2009

ln

T

SDW

Bc

T

J

2 1

t

b b

J

2 0

t

bc b

SDW

Re

i2 1 + 2 4T

b SDW

-

1 2 A1

. Substitution of the lattice parameters and vF 1.1 105 m / s 9 gives b / B 0.985 K / T . Therefore, in high fields tb / b and tbc / b vanish, leaving only terms with lower-order Bessel functions in the original series. Consequently, we arrive at Eqs. A1 and A2 . One can show by expanding exponents in series, that
b

and for B b , ln T
SDW

Bb

T

J

2 1

t

bc c

TSDW B c - TSDW 0 TSDW B b - TSDW 0 1 i + 2 4T 1 2

J

2 1

t

b b

J t

2 0

t

bc b

,

A3

J

SDW

2 1

bc c

Re

c SDW

-

. A2

1 is a numeric factor. By substituting asymptotic where forms for Bessel functions with small arguments, one can obtain TSDW B c - TSDW 0 TSDW B b - TSDW 0 1t 4t
b c 2 c b 2

Here TSDW = TSDW B =0 , J0,1 are Bessel functions and is a digamma function. When B c , Lorentz force pushes the electrons to cross the Brillouin zone in ky direction with the characteristic frequency of b = evFBb see Fig. 1 b . When B b the frequency is c = evFBc, a factor of two larger than

,

A4

Therefore, by measuring the above differences of the transition temperature one can determine the ratio of the transfer integrals tb / tc.

1

2

3

4

5

6 7 8

9

10

11

12

For review see The Physics of Organic Superconductors and Conductors, edited by A. G. Lebed Springer, Berlin, 2008 ; T. Ishiguro, K. Yamaji, and G. Saito, Organic Superconductors, 2nd ed. Springer, Berlin, 1998 . Through this paper we use the approximation of orthorhombic elementary cell with the basic vectors a, b , and c and the coordinates x , y , z in corresponding directions. J. F. Kwak, J. E. Schirber, P. M. Chaikin, J. M. Williams, H.-H. Wang, and L. Y. Chiang, Phys. Rev. Lett. 56, 972 1986 . J. F. Kwak, J. E. Schirber, R. L. Greene, and E. M. Engler, Phys. Rev. Lett. 46, 1296 1981 . L. P. Gor 'kov and A. G. Lebed, J. Phys. France Lett. 45, 433 1984 . G. Montambaux, Phys. Rev. B 38, 4788 1988 . K. Maki, Phys. Rev. B 47, 11506 1993 . G. M. Danner, P. M. Chaikin, and S. T. Hannahs, Phys. Rev. B 53, 2727 1996 . N. Matsunaga, K. Yamashita, H. Kotani, K. Nomura, T. Sasaki, T. Hanajiri, J. Yamada, S. Nakatsuji, and H. Anzai, Phys. Rev. B 64, 052405 2001 . N. Biskup, S. Tomic, and D. Jerome, Phys. Rev. B 51, 17972 1995 . F. Z. Guo, K. Murata, A. Oda, and H. Yoshino, J. Phys. Soc. Jpn. 69, 2164 2000 . K. Murata, Y. Mizuno, F. Z. Guo, S. Shodai, A. Oda, and H.

13

14

15

16

17

18 19 20

21

Yoshino, Synth. Met. 120, 1071 2001 . K. Murata, K. Iwashita, Y. Mizuno, F. Z. Guo, S. Shodai, H. Yoshino, J. S. Brooks, L. Balicas, D. Graf, K. Storr, I. Rutel, S. Uji, C. Terakura, and Y. Imanaka, J. Phys. Chem. Solids 63, 1263 2002 . K. Murata, K. Iwashita, Y. Mizuno, F. Z. Guo, S. Shodai, H. Yoshino, J. S. Brooks, L. Balicas, D. Graf, K. Storr, I. Rutel, S. Uji, C. Terakura, and Y. Imanaka, Synth. Met. 133-134, 51 2003 . A. V. Kornilov and V. M. Pudalov, Instrum. Exp. Tech. 42, 127 1999 . A. S. Kirichenko, A. V. Kornilov, and V. M. Pudalov, Instrum. Exp. Tech. 48, 813 2005 . W. Kang, S. T. Hannahs, and P. M. Chaikin, Phys. Rev. Lett. 70, 3091 1993 . P. M. Chaikin, Phys. Rev. B 31, 4770 1985 . A. Bjelis and K. Maki, Phys. Rev. B 45, 12887 1992 . Strictly speaking, theoretical models Refs. 6 and 7 for TSDW B do not take finite scattering time into account; however we 1. For the expect Eq. 3 to be valid in the high-field limit, c studied sample we observed the FISDW transitions starting from field B =7 T for T = 2 K and P = 10 kbar . This result provides 1 for our data taken at 19 T and thus the evidence that c ensures the applicability of Eq. 3 . I. J. Lee and M. J. Naughton, Phys. Rev. B 58, R13343 1998 .

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