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Tunneling conduction by phase kink generation in spin density wave conductor (TMTSF)2PF6 below 1 K
Solid State Communications, Printed in Great Britain.
TUNNELING
0038-1098/91 $3.00 -t- .OO Pergamon Press plc
Vol. 80, No. 11, pp. 953-955, 1991.
CONDUCTION BY PHASE KINK GENERATION IN SPIN DENSITY WAVE CONDUCTOR (TMTSF),PF6 BELOW 1 K W. Wonneberger
Department
of Physics, University of Ulm, D-7900 Ulm/Donau,
Germany
(Received 23 August 1991 by B. Miihlschlegel) It is proposed that the tunneling current recently observed by Mihaly et al. in the quasi one-dimensional spin density wave organic conductor (TMTSF),PF, below 1 K is due to phase kink tunneling creation in analogy to the N = 4 commensurate Sine-Gordon chain. A numerical consistent picture is developed explaining the tunneling current and other observed features by involving a commensurate pinning frequency of or/27~ x 300GHz and the relaxation mode due to impurities.
1. INTRODUCTION
to
SPIN DENSITY WAVES (SDW) in organic chain compounds show similar features as charge density waves (CDW). In particular, they possess a collective mode contributing to the linear and nonlinear electromagnetic response [ 1- 131. A new feature of collective transport emerged recently by the work of Mihaly et al. [14] on the Bechgaard salt (TMTSF),PF, (TMTSF = tetramethyltetraselenafulvalene) in its SDW state: At temperatures of 1 K and below, they measured a current electric field law of the form j(E)
= .& exp 1 -&/E),
(1)
which is characteristic for tunneling processes. Further observed features are: 1. The quantities j, and E,, in equation (I) are independent of temperature. 2. The tunneling currentj is only weakly dependent on crystal purity. 3. The pinned mode at wp x 3 x 10ios-’ due to random impurities freezes out with decreasing temperatures. It becomes unobservable at temperatures T 5 1 K when the sliding SDW conduction is superseded by the tunneling conduction (1). 4. The dielectric constant Re E(O = 6 x 104s-‘) drops rapidly from 8 x 10’ at T = 2.3K to lo6 at T = 0.5K. The electric field strength E. in equation (1) is found to be E, = 3 V cm-’ and is too low to explain (1) by Zener tunneling of quasi particles through the SDW gap E* = 26, = 6 x lO’*hs-‘[II, 12Jaccording
2 Eze”e’ 0
=
--
“E8
4liv, e,
g 6 x lO*Vcm-‘.
(2)
In equation (2), vr = 3 x lO’cms_‘wasusedforthe Fermi velocity in chain direction. The observed small value of E, suggests a collective tunneling mechanism involving a lower energy than sg. Bardeen [ 151has suggested that (2) may apply to a CDW weakly pinned by impurities when Ed is replaced by a pinning gap: Ed + 2/7r&liw,. The quantity a is the ratio of Friihlich to band mass. In the case of CDW a is much larger than unity while standard SDW theory requires a = 1 [16, 171. Bardeen’s concept is unlikely to apply here because of feature 2. Furthermore, an additional factor (E/ET - 1) would appear on the r.h.s. of equation (1) where E, is the threshold field due to random pinning. In this paper, we propose phase kink tunneling creation by the electric field in analogy to phase kink generation by the electric field on the N = 4 SineGordon chain as a possible mechanism for the observed tunneling conduction. This concept is not new. It originates from quantum field theory (cf. [18]) and has been developed and applied to CDW conduction by Maki [ 191.A similar concept but involving amplitude rather than phase kinks has been persued by Krive et al. [20]. In the latter case the energy scale in equation (2) is again set by the gap. Currently, it is believed that CDW conduction is dominated by random impurities and requires incommensurability. Here, instead, we postulate that (TMTSF),PF, is commensurate with commensurability N = 4 in the temperature range considered, i.e. below 1 K. There
953
954
TUNNELING
CONDUCTION
are hints that fourth order longitudinal commensurability plays a role in this SDW material. It can be realized by a vanishing dimerization [ 10, 131. 2. THEORY We adopt the standard phase Hamiltonian density of a commensurate SDW [21]:
IN (TMTSF)2 PF,
Vol. 80, No. 11
Following [19] the current density j can then be calculated from the energy balance jE
=
n,2E,r,
(8)
provided the annihilation of phase kinks at impurities proceeds faster than their creation by the field. Thus the current density is given by (9)
with o, = z+ $ v2 9 zlj. Here, 1 denotes the chain direction, n, is the density of chains and or is the oscillation frequency of small phase oscillations around a phase preferred by commensurability. This frequency is different from the pinning frequency c+, due to random impurities which is observed at higher temperatures [I 11. From [21, 221, we estimate wr according to the formula WF
&giK
where C is a constant of order unity and W denotes the electronic bandwidth. At zero temperature, equation (3) describes an insulator for E, = 0, i.e. the linear conductivity vanishes. For E. 9 E, > 0 phase kinks and antikinks are produced with a phase change of + 27clN localized on a length d = vF/oF and carrying charges f 2e,/N per chain. Because of the strong anisotropy, we adopt a one-dimensional description. It is, however, presently not clear what the transversal structure of the phase kinks is. Judging from the anisotropy in (TMTSF)2PF, the kink is practically confined to one chain in the c-direction. In the one-dimensional context the creation rate (per chain and per length) is given by [lg, 191
The electric field strength E, is
(6) which is smaller than the classical field strength E, = hio~/(2Nv,eo) in the rigid particle model for sliding conduction provided N 2 3. The energy drawn from the electric field per creation event and per chain is (7) For simplicity, we use the classical (weak coupling) expression for the kink energy.
This is the required form (1) the square bracket in equation (9) defining the quantity j. in equation (1). The three-dimensional kink structure is, however, expected to decrease the numerical value for j,. The present result also supports feature 2: as long as the crystal is impure enough, the current (9) is independent of impurities. 3. NUMERICAL
EVALUATION
The theoretical concept of Section 2 is now applied to the measurements in [14] on (TMTSF),PF, below 1 K using N = 4. Relevant data are extracted from [ll, 12, 141. Using a = 1, they are: plasma frequency mpr = 2 x 10’5s-’ - ore: frequency of the longitudinal phason. Electron density no = 1.3 x 10” cmp3, Fermi velocity vF z 3 x IO’cm s-‘, Fermi energy 5 x 10’4hs-‘, and normal state conductivity &F x gN = 0(20K) = 9 x 10’6s-‘. With E. = 3Vcm-‘, equation (6) then requires z 2 x 10’2s~’ or a frequency of 300GHz. This OF value should be compatible with equation (4). Using W%sFandCx l,onefindso,x 10’2s-‘whichis of the right order. The energy scale 2E,.is found to be 2E, w 3 x 10”h ss’ = 2kB K. Thus, feature 1 is guaranteed. There are further cross relations between the above quantities. The observed low value of the dielectric constant at the frequency of 10 kHz [14] for T < 1 K must be reconciled with the much larger value at higher temperatures (feature 4). We think that the answer to this lies in the relaxation mode which is expected to be present also in impure commensurate system. The relaxation mode is due to the long range nature of the Coulomb interaction at low temperatures when quasi particles coupled to transversal density wave excitations by inhomogeneous pinning become scarce 123, 241. The spectral weight of the relaxation mode centered around a frequency speak in Im E(O) accounts for the large value of the dielectric constant because of the Kramers-Kronig relation Re E(O < o+) = opposite limit l/rc S d(ln w) Im E(W). In the w ’ Opeak, the measured value of Re E(O) in a com-
TUNNELING
Vol. 88, No. 11
CONDUCTION
(10)
The present numbers give E* x lo6 in good agreement with the measured low temperature value. Some of the quantitative results of a theory of the relaxation mode for weakly [25] and strongly [26] pinned CDW can be transferred to the commensurate SDW. Assuming that the pinning in the partly incommensurate state at 4 K [l I] is weak and described in terms of a Lee-Rice frequency w, x op, wpeakis found from [25] according to Opeak
=
i3 128~’ 2
0°F
State Commun. 46, 21 (1983).
6. 7. 8.
w,.
WLOww
(11)
The dielectric relaxation frequency w, of the quasi particles is estimated in the usual way from the gap dielectric constant E& and the d.c. conductivity a,(T) = 4a,(k, T/A,) exp (- Ao/k, T) [271. It is then found that ~~~~(2.3 K) = 5 x lOas-’ > w = 6 x 104s-’ and 0,,~(1 K) = 5 x 10’s_’ < w. Thus the sudden decrease of Re E(W)near 2 K (feature 4) can be understood as the shifting of the SDW relaxation mode across the measuring frequency w. This effect is probably enhanced by a progressive transition towards commensurability. Finally, feature 3 can also be interpreted as interplay between random impurity pinning and commensurability with commensurability taking over at lower temperatures.
9.
In summary the newly discovered tunneling conduction in (TMTSF),PF, below 1 K can be consistently interpreted in terms of quantum mechanical phase kink generation in the N = 4 commensurate SDW with residual impurities. The essential feature is a high collective mode frequency or corresponding to 300 GHz or 10 cm-’ which should be detectable in the f.i.r. region below the single particle gap at 3Ocm-‘. For a SDW in the clean local limit and with a = 1, the gap is not seen in optics since all the oscillator strength resides in the collective mode. Acknowledgement
-The
author thanks G. Griiner for
helpful discussions. REFERENCES 1.
W.M. Walsh, Jr., F. Wudl, G.A. Thomas, D. Nalewajek, J.J. Hauser, P.A. Lee & T. Poehler, Phys. Rev. Lett. 45, 829 (1980).
L.T. Buravov, V.N. Laukhin & A.G. Khomenko, Zh. Eksp. Teor. Fiz. 88,2185 (1985) (Sov. Phys. JETP 61, 1292 (1985)). H.H.S. Javadi, S. Sridhar, G. Griiner, Long Chiang & F. Wudl, Phys. Rev. Lett. 55, 1216 (1985). S. Tomic, J.R. Cooper, D. Jerome & K. Bechgaard, Phys. Rev. L&t. 62, 462 (1989).
T. Sambongi, N. Nomura, T. Shimizu, K. Ichimura, N. Kinoshita, M. Tokumoto & H. Anzai, Solid State Commun. 72, 817 (1989). N. Nomura, T. Shimizu, K. Ichimura, T. Sambongi, M. Tokumoto, H. Anzai & N. Kinoshita, Solid State Commun. 72, 1123 (1989).
10.
W. Kang, S. Tomic, J.R. Cooper & D. Jerome,
11.
D. Quinlivan, Yong Kim, K. Holcer, G. Griiner & F. Wudl, Phys. Rev. Lett. 65, 1816 (1990). G. Mihaly, Yong Kim & G. Griiner, Phys. Rev. L&t. 66, 2806 (1991). W. Kang, S. Tomic & D. Jerome, Phys. Rev. B43, 1264 (1991). G. Mihaly, Yong Kim & G. Grtiner, Classical
Phys. Rev. B41,4862 (1990).
12. 13. 14.
15. 16.
4. SUMMARY
955
P.M. Chaikin, G. Griiner, E.M. Engler & R.L. Greene, Phys. Rev. Lett. 45,935 (1980). A. Zettl, G. Griiner 8c E.M. Engler, Phys. Rev. B25, 1443 (1982). A. Janossy, M. Hardiman dz G. Grilner, Solid
mensurate SDW system will be &* = u&/w;.
IN (TMTSF)*PF,
17. 18. 19. 20. 21. 22. 23. 24. 25.
Sliding to Tunneling Crossover in Spin Density Wave Transport, preprint, submitted to Phys. Rev. Lett. J. Bardeen, Phys. Rev. Lett. 55, 1010 (1985); Z. Phys. B: Condensed Matter 67,427 (1987). A. Virosztek & K. Maki, Phys. Rev. B37, 2028
(1988). K. Maki & A. Virosztek, Phys. Rev. B41, 557 (1990). M. Stone, Phys. Rev. D14, 3568 (1976). K. Maki, Phys. Rev. Lett. 39, 46 (1978); Phys. Rev. B18, 1641 (1978). I.V. Krive, A.S. Rozhavskii & 1.0. Kulik, Fiz. Nizk. Temp. 12, 1123 (1986) (Sov. J. Low Temp. Phys. 12, 635 (1986)). K. Maki & A. Virosztek, Phys. Rev. B42, 655 (1990). P.A. Lee, T.M. Rice & P.W. Anderson, Solid State Commun. 14, 703 (1974). R.J. Cava, R.M. Fleming, P.B. Littlewood, E.A. Rietman, L.F. Schneemeyer & R.G. Gunn, Phys. Rev. B30, 3228 (1984). P.B. Littlewood, Phys. Rev. B36, 3108 (1987). T. Baier & W. Wonneberger, Z. Phys. B: Condensed Mutter 79,211 (1990); W. Wonneberger, Synthetic
26. 27.
Metals 43, 3793 (1991).
W. Aichmann & W. Wonneberger, Z. Phys. B: Condensed Mutter 84, 375 (1991). U. Eckern, Z. Phys. B: Condensed Matter 74, 419 (1989).
TUNNELING
0038-1098/91 $3.00 -t- .OO Pergamon Press plc
Vol. 80, No. 11, pp. 953-955, 1991.
CONDUCTION BY PHASE KINK GENERATION IN SPIN DENSITY WAVE CONDUCTOR (TMTSF),PF6 BELOW 1 K W. Wonneberger
Department
of Physics, University of Ulm, D-7900 Ulm/Donau,
Germany
(Received 23 August 1991 by B. Miihlschlegel) It is proposed that the tunneling current recently observed by Mihaly et al. in the quasi one-dimensional spin density wave organic conductor (TMTSF),PF, below 1 K is due to phase kink tunneling creation in analogy to the N = 4 commensurate Sine-Gordon chain. A numerical consistent picture is developed explaining the tunneling current and other observed features by involving a commensurate pinning frequency of or/27~ x 300GHz and the relaxation mode due to impurities.
1. INTRODUCTION
to
SPIN DENSITY WAVES (SDW) in organic chain compounds show similar features as charge density waves (CDW). In particular, they possess a collective mode contributing to the linear and nonlinear electromagnetic response [ 1- 131. A new feature of collective transport emerged recently by the work of Mihaly et al. [14] on the Bechgaard salt (TMTSF),PF, (TMTSF = tetramethyltetraselenafulvalene) in its SDW state: At temperatures of 1 K and below, they measured a current electric field law of the form j(E)
= .& exp 1 -&/E),
(1)
which is characteristic for tunneling processes. Further observed features are: 1. The quantities j, and E,, in equation (I) are independent of temperature. 2. The tunneling currentj is only weakly dependent on crystal purity. 3. The pinned mode at wp x 3 x 10ios-’ due to random impurities freezes out with decreasing temperatures. It becomes unobservable at temperatures T 5 1 K when the sliding SDW conduction is superseded by the tunneling conduction (1). 4. The dielectric constant Re E(O = 6 x 104s-‘) drops rapidly from 8 x 10’ at T = 2.3K to lo6 at T = 0.5K. The electric field strength E. in equation (1) is found to be E, = 3 V cm-’ and is too low to explain (1) by Zener tunneling of quasi particles through the SDW gap E* = 26, = 6 x lO’*hs-‘[II, 12Jaccording
2 Eze”e’ 0
=
--
“E8
4liv, e,
g 6 x lO*Vcm-‘.
(2)
In equation (2), vr = 3 x lO’cms_‘wasusedforthe Fermi velocity in chain direction. The observed small value of E, suggests a collective tunneling mechanism involving a lower energy than sg. Bardeen [ 151has suggested that (2) may apply to a CDW weakly pinned by impurities when Ed is replaced by a pinning gap: Ed + 2/7r&liw,. The quantity a is the ratio of Friihlich to band mass. In the case of CDW a is much larger than unity while standard SDW theory requires a = 1 [16, 171. Bardeen’s concept is unlikely to apply here because of feature 2. Furthermore, an additional factor (E/ET - 1) would appear on the r.h.s. of equation (1) where E, is the threshold field due to random pinning. In this paper, we propose phase kink tunneling creation by the electric field in analogy to phase kink generation by the electric field on the N = 4 SineGordon chain as a possible mechanism for the observed tunneling conduction. This concept is not new. It originates from quantum field theory (cf. [18]) and has been developed and applied to CDW conduction by Maki [ 191.A similar concept but involving amplitude rather than phase kinks has been persued by Krive et al. [20]. In the latter case the energy scale in equation (2) is again set by the gap. Currently, it is believed that CDW conduction is dominated by random impurities and requires incommensurability. Here, instead, we postulate that (TMTSF),PF, is commensurate with commensurability N = 4 in the temperature range considered, i.e. below 1 K. There
953
954
TUNNELING
CONDUCTION
are hints that fourth order longitudinal commensurability plays a role in this SDW material. It can be realized by a vanishing dimerization [ 10, 131. 2. THEORY We adopt the standard phase Hamiltonian density of a commensurate SDW [21]:
IN (TMTSF)2 PF,
Vol. 80, No. 11
Following [19] the current density j can then be calculated from the energy balance jE
=
n,2E,r,
(8)
provided the annihilation of phase kinks at impurities proceeds faster than their creation by the field. Thus the current density is given by (9)
with o, = z+ $ v2 9 zlj. Here, 1 denotes the chain direction, n, is the density of chains and or is the oscillation frequency of small phase oscillations around a phase preferred by commensurability. This frequency is different from the pinning frequency c+, due to random impurities which is observed at higher temperatures [I 11. From [21, 221, we estimate wr according to the formula WF
&giK
where C is a constant of order unity and W denotes the electronic bandwidth. At zero temperature, equation (3) describes an insulator for E, = 0, i.e. the linear conductivity vanishes. For E. 9 E, > 0 phase kinks and antikinks are produced with a phase change of + 27clN localized on a length d = vF/oF and carrying charges f 2e,/N per chain. Because of the strong anisotropy, we adopt a one-dimensional description. It is, however, presently not clear what the transversal structure of the phase kinks is. Judging from the anisotropy in (TMTSF)2PF, the kink is practically confined to one chain in the c-direction. In the one-dimensional context the creation rate (per chain and per length) is given by [lg, 191
The electric field strength E, is
(6) which is smaller than the classical field strength E, = hio~/(2Nv,eo) in the rigid particle model for sliding conduction provided N 2 3. The energy drawn from the electric field per creation event and per chain is (7) For simplicity, we use the classical (weak coupling) expression for the kink energy.
This is the required form (1) the square bracket in equation (9) defining the quantity j. in equation (1). The three-dimensional kink structure is, however, expected to decrease the numerical value for j,. The present result also supports feature 2: as long as the crystal is impure enough, the current (9) is independent of impurities. 3. NUMERICAL
EVALUATION
The theoretical concept of Section 2 is now applied to the measurements in [14] on (TMTSF),PF, below 1 K using N = 4. Relevant data are extracted from [ll, 12, 141. Using a = 1, they are: plasma frequency mpr = 2 x 10’5s-’ - ore: frequency of the longitudinal phason. Electron density no = 1.3 x 10” cmp3, Fermi velocity vF z 3 x IO’cm s-‘, Fermi energy 5 x 10’4hs-‘, and normal state conductivity &F x gN = 0(20K) = 9 x 10’6s-‘. With E. = 3Vcm-‘, equation (6) then requires z 2 x 10’2s~’ or a frequency of 300GHz. This OF value should be compatible with equation (4). Using W%sFandCx l,onefindso,x 10’2s-‘whichis of the right order. The energy scale 2E,.is found to be 2E, w 3 x 10”h ss’ = 2kB K. Thus, feature 1 is guaranteed. There are further cross relations between the above quantities. The observed low value of the dielectric constant at the frequency of 10 kHz [14] for T < 1 K must be reconciled with the much larger value at higher temperatures (feature 4). We think that the answer to this lies in the relaxation mode which is expected to be present also in impure commensurate system. The relaxation mode is due to the long range nature of the Coulomb interaction at low temperatures when quasi particles coupled to transversal density wave excitations by inhomogeneous pinning become scarce 123, 241. The spectral weight of the relaxation mode centered around a frequency speak in Im E(O) accounts for the large value of the dielectric constant because of the Kramers-Kronig relation Re E(O < o+) = opposite limit l/rc S d(ln w) Im E(W). In the w ’ Opeak, the measured value of Re E(O) in a com-
TUNNELING
Vol. 88, No. 11
CONDUCTION
(10)
The present numbers give E* x lo6 in good agreement with the measured low temperature value. Some of the quantitative results of a theory of the relaxation mode for weakly [25] and strongly [26] pinned CDW can be transferred to the commensurate SDW. Assuming that the pinning in the partly incommensurate state at 4 K [l I] is weak and described in terms of a Lee-Rice frequency w, x op, wpeakis found from [25] according to Opeak
=
i3 128~’ 2
0°F
State Commun. 46, 21 (1983).
6. 7. 8.
w,.
WLOww
(11)
The dielectric relaxation frequency w, of the quasi particles is estimated in the usual way from the gap dielectric constant E& and the d.c. conductivity a,(T) = 4a,(k, T/A,) exp (- Ao/k, T) [271. It is then found that ~~~~(2.3 K) = 5 x lOas-’ > w = 6 x 104s-’ and 0,,~(1 K) = 5 x 10’s_’ < w. Thus the sudden decrease of Re E(W)near 2 K (feature 4) can be understood as the shifting of the SDW relaxation mode across the measuring frequency w. This effect is probably enhanced by a progressive transition towards commensurability. Finally, feature 3 can also be interpreted as interplay between random impurity pinning and commensurability with commensurability taking over at lower temperatures.
9.
In summary the newly discovered tunneling conduction in (TMTSF),PF, below 1 K can be consistently interpreted in terms of quantum mechanical phase kink generation in the N = 4 commensurate SDW with residual impurities. The essential feature is a high collective mode frequency or corresponding to 300 GHz or 10 cm-’ which should be detectable in the f.i.r. region below the single particle gap at 3Ocm-‘. For a SDW in the clean local limit and with a = 1, the gap is not seen in optics since all the oscillator strength resides in the collective mode. Acknowledgement
-The
author thanks G. Griiner for
helpful discussions. REFERENCES 1.
W.M. Walsh, Jr., F. Wudl, G.A. Thomas, D. Nalewajek, J.J. Hauser, P.A. Lee & T. Poehler, Phys. Rev. Lett. 45, 829 (1980).
L.T. Buravov, V.N. Laukhin & A.G. Khomenko, Zh. Eksp. Teor. Fiz. 88,2185 (1985) (Sov. Phys. JETP 61, 1292 (1985)). H.H.S. Javadi, S. Sridhar, G. Griiner, Long Chiang & F. Wudl, Phys. Rev. Lett. 55, 1216 (1985). S. Tomic, J.R. Cooper, D. Jerome & K. Bechgaard, Phys. Rev. L&t. 62, 462 (1989).
T. Sambongi, N. Nomura, T. Shimizu, K. Ichimura, N. Kinoshita, M. Tokumoto & H. Anzai, Solid State Commun. 72, 817 (1989). N. Nomura, T. Shimizu, K. Ichimura, T. Sambongi, M. Tokumoto, H. Anzai & N. Kinoshita, Solid State Commun. 72, 1123 (1989).
10.
W. Kang, S. Tomic, J.R. Cooper & D. Jerome,
11.
D. Quinlivan, Yong Kim, K. Holcer, G. Griiner & F. Wudl, Phys. Rev. Lett. 65, 1816 (1990). G. Mihaly, Yong Kim & G. Griiner, Phys. Rev. L&t. 66, 2806 (1991). W. Kang, S. Tomic & D. Jerome, Phys. Rev. B43, 1264 (1991). G. Mihaly, Yong Kim & G. Grtiner, Classical
Phys. Rev. B41,4862 (1990).
12. 13. 14.
15. 16.
4. SUMMARY
955
P.M. Chaikin, G. Griiner, E.M. Engler & R.L. Greene, Phys. Rev. Lett. 45,935 (1980). A. Zettl, G. Griiner 8c E.M. Engler, Phys. Rev. B25, 1443 (1982). A. Janossy, M. Hardiman dz G. Grilner, Solid
mensurate SDW system will be &* = u&/w;.
IN (TMTSF)*PF,
17. 18. 19. 20. 21. 22. 23. 24. 25.
Sliding to Tunneling Crossover in Spin Density Wave Transport, preprint, submitted to Phys. Rev. Lett. J. Bardeen, Phys. Rev. Lett. 55, 1010 (1985); Z. Phys. B: Condensed Matter 67,427 (1987). A. Virosztek & K. Maki, Phys. Rev. B37, 2028
(1988). K. Maki & A. Virosztek, Phys. Rev. B41, 557 (1990). M. Stone, Phys. Rev. D14, 3568 (1976). K. Maki, Phys. Rev. Lett. 39, 46 (1978); Phys. Rev. B18, 1641 (1978). I.V. Krive, A.S. Rozhavskii & 1.0. Kulik, Fiz. Nizk. Temp. 12, 1123 (1986) (Sov. J. Low Temp. Phys. 12, 635 (1986)). K. Maki & A. Virosztek, Phys. Rev. B42, 655 (1990). P.A. Lee, T.M. Rice & P.W. Anderson, Solid State Commun. 14, 703 (1974). R.J. Cava, R.M. Fleming, P.B. Littlewood, E.A. Rietman, L.F. Schneemeyer & R.G. Gunn, Phys. Rev. B30, 3228 (1984). P.B. Littlewood, Phys. Rev. B36, 3108 (1987). T. Baier & W. Wonneberger, Z. Phys. B: Condensed Mutter 79,211 (1990); W. Wonneberger, Synthetic
26. 27.
Metals 43, 3793 (1991).
W. Aichmann & W. Wonneberger, Z. Phys. B: Condensed Mutter 84, 375 (1991). U. Eckern, Z. Phys. B: Condensed Matter 74, 419 (1989).