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Charge density wave depinning: a possible form of macroscopic coulomb blockade
Superlattices
and Microstructures,
Vol. 11, NO. 2, 199.2
215
CHARGE DENSITY WAVE DEPINNING: A POSSIBLE FORM OF MACROSCOPIC COULOMB BLOCKADE J. H. Miller, Jr. Department of Physics and Texas Centerfor Superconducrivity Universityof Houston Houston, Texas 77204-5932 U.SA. (Received 19 May 1991)
A large body of experimental evidence supports the hypothesis that charge density waves (CDW’s) depin by correlated tunneling of macroscopic charge solitons, as proposed by Bardeen, and also refutes the currently popular classical models of CDW depinning, such as the classical deformable model. Important clues to the origin of the threshold voltage, coherent oscillations, and mode-locking observed in CDW’s can be found by examining a striking similarity of these phenomena to the remarkable phenomena of Coulomb blockade and correlated single-electron tunneling observed in small tunnel junctions.
I.
Introduction
The remarkable nonohmic transport properties of inorganic linear chain compounds, due to collective charge density wave (CDW) depinning,lm3 has been the subject of research for more than a decade. A CDW forms in a linear chain compound at temperatures below a characteristic transition temperature, known as the Peierls temperature, and consists of a periodic lattice distortion along the chain direction, which is accompanied by a periodic modulation of the electronic charge density given by &o
1 =
PlCOSWFX
+ du
11,
(1)
where p, and @are, respectively, the amplitude and phase of the CDW. kF is the Fermi wavevector. and x is the position along thi chain direction. An energy gap, known as the Peierls gap, opens up at the Fermi surface and, unlike an ordinarv semiconductor band gap, can actually be displaced in momentum space if the Fermi sea carries a net momentum. As a result, a moving CDW actually transports an electric current, as originally proposed by Fri5hlich.4 The CDW current per spin per chain is given by
ICDW = efd = &
(
$!
>
The CDW drift velocity vd is related to the drift fkquency fd by Vd = &Wfd, where kDW = lT/kF iS the aw wavelength. In a perfect, infinitely long linear chain compound where the CDW wavelength was incommensurate with the lattice, the total energy of the system would be independent
0749-6036/92/020215+03
$02.00/O
of the CDW phase and the CDW could move freely. However, the presence of impurities breaks the “translational” symmetry of the CDW, and results in a pinning potential which is a periodic function of the phase, modulo 21~. It has been found2 that a sharply-defined minimum electric field, or threshold field ET resulting from pinning by impurities, is required in order to depin the CDW and induce a CDW current. Above the threshold field, the CDW current is accompanied by coherent current oscillations when the sample is voltage-biased. An important controversy concerns whether a completely classical description suffices to fully characterize CDW depinning and dynamics,s-10 or whether an intrinsically quantum mechanical description is required,’ 1as suggested by a number of experiments.l2-17 II.
Tunneling
Model
The simplest possible model of a pinned CDW represents the periodic pinning potential as a sine-Gordon potential of the form V(6) = V,,[l
- COS@].
(3)
The potential energy is minimized in Eq. (3) for $ = 0, *2x, Mn, erc. A CDW, being a coupled electron-lattice system, is electrically neutral on average if the phase @ is independent of position. However, a gradient of the phase, or “kink,” carries a net charge per unit length which couples to an applied electric field. A 2x-kink, or chargesoliton, carries a net charge *e per chain per spin, and is topologically stable for an infinitely long sample if, for example, the phase $ approaches zero as x approaches minus infinity, and r$ approaches 2~ as x approaches plus
0 1992 Academic Press Limited
216
SuDerlattices
infinity. A fully quantum mechanical description is obtained by quantizing the sine-Gordon system and utilizing the Fermionic representation to express the CDW Hamiltonian in terms of quantum soliton operators.18 These quantum solitons are essentially dressed electrons which incorporate the strong coupling of the CDW electrons to the lattice distortion and have an energy +[(hc&)2 + E+a]tn, where CO- 10-t+ is a characteristic speed, and where the soliton energy E+ per spin-chain is found to be comparable to the classical soliton energy provided soliton-soliton and soliton-antisoliton interactions are neglected. An applied electric field creates quantum soliton-antisoliton pairs by Zener tunneling across the soliton energy gap, or “pinning gap” EG = 2Ee, resulting in a field-dependent conductivity in the high field limit of the form
can be found by examining a striking similarity of these phenomena to the remarkable phenomenon of correlated single-electron tunneling in small tunnel junctions.19 In a pioneering paper, Fulton and DolanZo confirmed experimentally the existence of charging effects in small circuits of planar tunnel junctions. In linear arrays of small tunnel junctions charge is transferred by mutually repulsive charge solitons.21 resulting in time-correlated tunneling events with frequency I/e. Delsing et al 22 demonstrated this effect by superimposing onto the dc bias an ac signal of frequency f, resulting in partial mode locking at current levels I = if and 2eT. More recently, complete modelocking has been observed in a freouencv-locked “turnstile device” for single electrons by Geerligs et al. 23 An ideal CDW with a complete Peierls gap at low temperatures exhibits a similar scaling of CDW current with frequency of the form ICoW = 2Nefd, where N is the total number of parallel CDW chains, fd represents the drift frequency, which is also equal to the frequency of coherent current or voltage oscillations, and the factor of two counts both spins. When an ac signal of freouencv f is superimposed onto a dc bias, steps of constant -CDW current are observed due to interference or mode locking of the applied frequency f with the internal CDW frequency fd ; both harmonic (&/f = p) and subharmonic (fdlf = p/q ) steps are observed. Moreover, in NbSeg samples of high purity the CDW current scales linearly with E - ~~~~~ which is essentially the ideal behavior expected for Coulomb blockade. It should be noted that both the rigid overdamped oscillator model5Jj and the classical deformable model7-to of CDW depinning predict that the high-current asymptote of the I-V curve should extrapolate through the origin. In order to make the analogy between CDW tunneling and Coulomb blockade more concrete, one must examine the origin of the threshold voltage in each case. In the case of a small tunnel junction, Coulomb blockade results from the fact that an electron tunneling through the junction with a bias voltage V/bias muSt gain enough energy eVbios to overcome the “charging” energy e2/2C, resulting in a threshold voltage VI = e/2C and offset voltage in the current-voltage characteristic V,$-. - ti/2C: For the case of charge density waves, the fact t at pmmng results largely from randomly distributed impurities implies that the CDW phase is correlated only over a Fukuyama-Lee-Rice domain length L,25 which is comparable to the length of a classical soliton. Suppose we consider a CDW “tunnel junction” consisting of N parallel CDW chains in a domain of length L and cross-sectional area A. The capacitance will then be given by:
OCDW
-ENE ,
=ome
for E >> ET ,
(4)
where the characteristic field for Zener tunneling is given by Eo
-
E,2 CI ___
xlleco
One of the main objections to the tunneling hypothesis is the fact that the pinning gap energy EG - 2Ee- 10-4kgT must be substantially smaller than the thermal energy ~BT, in order for the predicted values of Eo to agree with experiment at temperatures T - 100 K at which nonohmic CDW transport is typically observed. It must be emphasized, however, that EC refers to the gap energy per chain. In reasonable quality crystals, the CDW motion is coherent across N - 10’parallel chains, as pointed out by coherence resulting Bardeen. 11 The three-dimensional from electron delocahzation and interchain coupling is, in fact, an important prerequisite for the cccurence of a Peierls transition. If the CDW phase is assumed to be correlated across N parallel chains, then the energy required to thermally excite a collective soliton-antisoliton pair will be NEG. which is substantially larger than ~BT, so that the probability of thermal excitation exp(-NEolkB7) will be exnemely small. On the other hand, the Zener tunneling nrobabilitv will be unaffected bv this rescaline. as can be -&en by e&mining the various q&uities whichgo into (5). If we treat the tunneling events involving N parallel chains (or, in k-space, N transverse wavevectors kl) within a transverse coherence distance as being statistically correlated, then the effective charge e* = Net and energy
c=E(O)A
m = NE(k) will each be scaled up accordingly. The implication of this resealing is that borh the effective energy gap 4 = NEG and slope Ici = disnersion relation in the Iaree k limit N. s By inserting these qua&ties into are found to cancel out, yielding the before. III. Correlated
Tunneling
N UCOof the energy
will be scaled un bv (5) the factors of N same value of Eo as
of CDW Solitons
Important clues to the origin of the sharp threshold field, coherent oscillations, and mode-locking in CDW’s
and Microstructures,
L
Vol. 7 7, No. 2, 1992
’
where c(0) - 107.ca is the zero frequency dielectric constant, which is extremely large because of the small value of the pinning gap per dressed electron. Assuming that each macroscopic soliton has a total charge 2Ne, then the threshold voltage across this region can be estimated by using the analogy to Coulomb blockade: VT
= ETL
2Ne
= 2~
=
2NeL 2E(o)A
,
(7)
Superlattices
and Microstructures,
Vol. 11, No. 2, 1992
217 5.
resulting in the expression @@ET = inen e ,
6. 7.
where n,h = 2NIA is the number of spin-chains per unit area. The universality condition described by Eq. (8) has been observed experimentally in a number of CDW compounds.26,27 ln extremely pure samples, the relevant length L may actually be the distance between contacts. The CDW in the regions under the contacts will be stationarv. so these resons may be considered as CDW electrodes~on the leftand right-hand sides of a “tunnel iunction.” Correlated tunneh~g of quantum solitons betwekn the electrodes could then be treated within a tunneling Hamiltonian formulation,**,*9 where the polarization effects would be included with an additional contribution Q*/2C to the Hamiltonian. The theoretical treatment developed by Averin and Likharev,t9 and others*l could then be applied to describe correlated soliton tunneling. IV.
Conclusion
The possible analogy between correlated tunneling of CDW solitons and correlated single electron tunneling in small tunnel junctions has been suggested as an interesting interpretation of the observed sharp threshold field and coherent oscillations in CDW systems. Several years ago, a number of experimentsl*-‘7 were performed on CDW materials in order to test the predictions of the tunneling model. In particular, the theory of photon-assisted tunneling (PAT)30 was adapted to Bardeen’s model in order to generate predictions of linear and nonlinear rf experiments. The essentially quantitative agreement obiained in these diverse expeiments constitutes-powerful evidence that tunneling is indeed responsible for CDW depinning in an electric%eld. lf the tunneling hypothesis is correct, then CDW transport constitutes an important new cooperative quantum phenomenon. Acknowledgement -- The author gratefully acknowledges support by the State of Texas, the Texas Center for Superconductivity at the University of Houston, DARPA Prime Grant MDA 972-88-G-002, and the A. P. Sloan Foundation.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
24.
25. References P. Monceau, N. P. Ong, A. M. Portis, A. Meerschaut, and J. Rouxel, Physical Review Letters 37, 6902 (1976). R. M. Fleming and C. C. Grimes, Physical Review Letters 42, 1423 (1979). For extensive reviews of CDw’s see Electronic Properties of Quasi-One-DimensionalMaterials,edited by P. Monceau (Reidel, Dordrecht, 1985); Cl. Grtiner, Reviews of Modern Physics 60, 1129 (1988). H. Friihlich, Proceedings of the Royal Sociery of London A223.296 (1954).
26. 27.
28. 29. 30.
Cl. Gruner, A. Zawadowski, P. M. Chaikin, Physical Review Letters 46, 511 (1981). P. Monceau, J. Richard and M. Renard, Physical Review B31, 1396 (1985). L. Sneddon, M. C. Cross, and D. S. Fisher, Physical Review Letters 49, 292 (1982). D. S. Fisher, Physical Review B31, 1396 (1985). R. A. Klemm and J. R. Schrieffer, Physical Review Letters 52, 65 (1984). L. Sneddon, Physical Review L..etters 52,65 (1984). John Bardeen, Physical Review Letters 42, 1498 (1979); ibid 45, 1978 (1980); ibid 55, 1002 (1985). J. H. Miller,Jr., J. Richard, J. R. Tucker, and John Bardeen, Physical Review Letters 51, 1592 (1983). J. H. Miller,Jr., R. E. Thome, W. G. Lyons, J. R. Tucker, and John Bardeen, Physical Review B31, 5229 (1985). R. E. Thome, J. H. Miller, Jr., W. G. Lyons, J. W. Lyding and J. R. Tucker, Physical Review Letters 55, 1006 (1985). J. Richard, R. E. Thorne, W. G. Lyons, J. H. Miller,Jr., and J. R. Tucker, Solid State Communication 52, 183 (1984). R. E. Thome, W. G. Lyons, J. W. Lyding, J. R. Tucker, and John Bardeen, Physical Review B34, 5988 (1986). R. E. Thome, J. R. Tucker, and John Bardeen, Physical Review Letters 58, 828 (1987). A. Maiti and J. H. Miller, Jr., Physical Review B43, 12205 (1991). D. V. Averin and K. K. Likharev, Journal of Low Temperature Physics 62, 345 (1986), and references cited therein. T. A. Fulton and G. J. Dolan, Physical Review Letters 59, 109 (1987). K. K. Likharev, N. S. Bakhvalov, G. S. Kazacha, and S. I. Serdynkova, IEEE Transactions on Magnetics 25, 1436 (1989). P. Delsing, K. K. Likharev, L. S. Kuzmin, and T. Claeson, Physical Review Letters 63, 1861 (1989). L. J. Geerligs, V. F. Anderegg, P. A. M. Holweg, J. E. Mooij, H. Potltier, D. Esteve, C. Urbina, and M. H. Devoret, Physical Review Letters 64, 2691 (1990). R. E. Thome, J. S. Hubacek, W. G. Lyons, J. W. Lyding, and J. R. Tucker, Physical Review B37, R. E. Thome et al., Physical Review 10055 (1988); B35,6348 and 6360 (1987). H. Fukuyama and P. A. Lee, Physical Review B17, 535 (1978); P. A. Lee and J. M. Rice, Physical Review B19, 3870 (1979). Wei-yu Wu, A. Jcinossy, and G. Griiner, Solid State Communication 49, 1013 (1984); G. Grilner, Reviews of Modern Physics 60, 1129 (1988). G. Grtlner, in Charge Density Waves in Solids, edited by Gy Hutiray and J. .%lyom, Lecture Notes in Physics Vol. 217 (Springer-Verlag, Berlin, 1985), pp. 263-278. J. Bardeen, Physical Review Letters 6,57 (1961). M. H. Cohen, L. M. Falicov, and J. C. Phillips, Physical Review Letters 8, 3 16 (1962). J. R. Tucker, IEEE Journal of Quantum Electronics QE-15, 1234 (1979).
and Microstructures,
Vol. 11, NO. 2, 199.2
215
CHARGE DENSITY WAVE DEPINNING: A POSSIBLE FORM OF MACROSCOPIC COULOMB BLOCKADE J. H. Miller, Jr. Department of Physics and Texas Centerfor Superconducrivity Universityof Houston Houston, Texas 77204-5932 U.SA. (Received 19 May 1991)
A large body of experimental evidence supports the hypothesis that charge density waves (CDW’s) depin by correlated tunneling of macroscopic charge solitons, as proposed by Bardeen, and also refutes the currently popular classical models of CDW depinning, such as the classical deformable model. Important clues to the origin of the threshold voltage, coherent oscillations, and mode-locking observed in CDW’s can be found by examining a striking similarity of these phenomena to the remarkable phenomena of Coulomb blockade and correlated single-electron tunneling observed in small tunnel junctions.
I.
Introduction
The remarkable nonohmic transport properties of inorganic linear chain compounds, due to collective charge density wave (CDW) depinning,lm3 has been the subject of research for more than a decade. A CDW forms in a linear chain compound at temperatures below a characteristic transition temperature, known as the Peierls temperature, and consists of a periodic lattice distortion along the chain direction, which is accompanied by a periodic modulation of the electronic charge density given by &o
1 =
PlCOSWFX
+ du
11,
(1)
where p, and @are, respectively, the amplitude and phase of the CDW. kF is the Fermi wavevector. and x is the position along thi chain direction. An energy gap, known as the Peierls gap, opens up at the Fermi surface and, unlike an ordinarv semiconductor band gap, can actually be displaced in momentum space if the Fermi sea carries a net momentum. As a result, a moving CDW actually transports an electric current, as originally proposed by Fri5hlich.4 The CDW current per spin per chain is given by
ICDW = efd = &
(
$!
>
The CDW drift velocity vd is related to the drift fkquency fd by Vd = &Wfd, where kDW = lT/kF iS the aw wavelength. In a perfect, infinitely long linear chain compound where the CDW wavelength was incommensurate with the lattice, the total energy of the system would be independent
0749-6036/92/020215+03
$02.00/O
of the CDW phase and the CDW could move freely. However, the presence of impurities breaks the “translational” symmetry of the CDW, and results in a pinning potential which is a periodic function of the phase, modulo 21~. It has been found2 that a sharply-defined minimum electric field, or threshold field ET resulting from pinning by impurities, is required in order to depin the CDW and induce a CDW current. Above the threshold field, the CDW current is accompanied by coherent current oscillations when the sample is voltage-biased. An important controversy concerns whether a completely classical description suffices to fully characterize CDW depinning and dynamics,s-10 or whether an intrinsically quantum mechanical description is required,’ 1as suggested by a number of experiments.l2-17 II.
Tunneling
Model
The simplest possible model of a pinned CDW represents the periodic pinning potential as a sine-Gordon potential of the form V(6) = V,,[l
- COS@].
(3)
The potential energy is minimized in Eq. (3) for $ = 0, *2x, Mn, erc. A CDW, being a coupled electron-lattice system, is electrically neutral on average if the phase @ is independent of position. However, a gradient of the phase, or “kink,” carries a net charge per unit length which couples to an applied electric field. A 2x-kink, or chargesoliton, carries a net charge *e per chain per spin, and is topologically stable for an infinitely long sample if, for example, the phase $ approaches zero as x approaches minus infinity, and r$ approaches 2~ as x approaches plus
0 1992 Academic Press Limited
216
SuDerlattices
infinity. A fully quantum mechanical description is obtained by quantizing the sine-Gordon system and utilizing the Fermionic representation to express the CDW Hamiltonian in terms of quantum soliton operators.18 These quantum solitons are essentially dressed electrons which incorporate the strong coupling of the CDW electrons to the lattice distortion and have an energy +[(hc&)2 + E+a]tn, where CO- 10-t+ is a characteristic speed, and where the soliton energy E+ per spin-chain is found to be comparable to the classical soliton energy provided soliton-soliton and soliton-antisoliton interactions are neglected. An applied electric field creates quantum soliton-antisoliton pairs by Zener tunneling across the soliton energy gap, or “pinning gap” EG = 2Ee, resulting in a field-dependent conductivity in the high field limit of the form
can be found by examining a striking similarity of these phenomena to the remarkable phenomenon of correlated single-electron tunneling in small tunnel junctions.19 In a pioneering paper, Fulton and DolanZo confirmed experimentally the existence of charging effects in small circuits of planar tunnel junctions. In linear arrays of small tunnel junctions charge is transferred by mutually repulsive charge solitons.21 resulting in time-correlated tunneling events with frequency I/e. Delsing et al 22 demonstrated this effect by superimposing onto the dc bias an ac signal of frequency f, resulting in partial mode locking at current levels I = if and 2eT. More recently, complete modelocking has been observed in a freouencv-locked “turnstile device” for single electrons by Geerligs et al. 23 An ideal CDW with a complete Peierls gap at low temperatures exhibits a similar scaling of CDW current with frequency of the form ICoW = 2Nefd, where N is the total number of parallel CDW chains, fd represents the drift frequency, which is also equal to the frequency of coherent current or voltage oscillations, and the factor of two counts both spins. When an ac signal of freouencv f is superimposed onto a dc bias, steps of constant -CDW current are observed due to interference or mode locking of the applied frequency f with the internal CDW frequency fd ; both harmonic (&/f = p) and subharmonic (fdlf = p/q ) steps are observed. Moreover, in NbSeg samples of high purity the CDW current scales linearly with E - ~~~~~ which is essentially the ideal behavior expected for Coulomb blockade. It should be noted that both the rigid overdamped oscillator model5Jj and the classical deformable model7-to of CDW depinning predict that the high-current asymptote of the I-V curve should extrapolate through the origin. In order to make the analogy between CDW tunneling and Coulomb blockade more concrete, one must examine the origin of the threshold voltage in each case. In the case of a small tunnel junction, Coulomb blockade results from the fact that an electron tunneling through the junction with a bias voltage V/bias muSt gain enough energy eVbios to overcome the “charging” energy e2/2C, resulting in a threshold voltage VI = e/2C and offset voltage in the current-voltage characteristic V,$-. - ti/2C: For the case of charge density waves, the fact t at pmmng results largely from randomly distributed impurities implies that the CDW phase is correlated only over a Fukuyama-Lee-Rice domain length L,25 which is comparable to the length of a classical soliton. Suppose we consider a CDW “tunnel junction” consisting of N parallel CDW chains in a domain of length L and cross-sectional area A. The capacitance will then be given by:
OCDW
-ENE ,
=ome
for E >> ET ,
(4)
where the characteristic field for Zener tunneling is given by Eo
-
E,2 CI ___
xlleco
One of the main objections to the tunneling hypothesis is the fact that the pinning gap energy EG - 2Ee- 10-4kgT must be substantially smaller than the thermal energy ~BT, in order for the predicted values of Eo to agree with experiment at temperatures T - 100 K at which nonohmic CDW transport is typically observed. It must be emphasized, however, that EC refers to the gap energy per chain. In reasonable quality crystals, the CDW motion is coherent across N - 10’parallel chains, as pointed out by coherence resulting Bardeen. 11 The three-dimensional from electron delocahzation and interchain coupling is, in fact, an important prerequisite for the cccurence of a Peierls transition. If the CDW phase is assumed to be correlated across N parallel chains, then the energy required to thermally excite a collective soliton-antisoliton pair will be NEG. which is substantially larger than ~BT, so that the probability of thermal excitation exp(-NEolkB7) will be exnemely small. On the other hand, the Zener tunneling nrobabilitv will be unaffected bv this rescaline. as can be -&en by e&mining the various q&uities whichgo into (5). If we treat the tunneling events involving N parallel chains (or, in k-space, N transverse wavevectors kl) within a transverse coherence distance as being statistically correlated, then the effective charge e* = Net and energy
c=E(O)A
m = NE(k) will each be scaled up accordingly. The implication of this resealing is that borh the effective energy gap 4 = NEG and slope Ici = disnersion relation in the Iaree k limit N. s By inserting these qua&ties into are found to cancel out, yielding the before. III. Correlated
Tunneling
N UCOof the energy
will be scaled un bv (5) the factors of N same value of Eo as
of CDW Solitons
Important clues to the origin of the sharp threshold field, coherent oscillations, and mode-locking in CDW’s
and Microstructures,
L
Vol. 7 7, No. 2, 1992
’
where c(0) - 107.ca is the zero frequency dielectric constant, which is extremely large because of the small value of the pinning gap per dressed electron. Assuming that each macroscopic soliton has a total charge 2Ne, then the threshold voltage across this region can be estimated by using the analogy to Coulomb blockade: VT
= ETL
2Ne
= 2~
=
2NeL 2E(o)A
,
(7)
Superlattices
and Microstructures,
Vol. 11, No. 2, 1992
217 5.
resulting in the expression @@ET = inen e ,
6. 7.
where n,h = 2NIA is the number of spin-chains per unit area. The universality condition described by Eq. (8) has been observed experimentally in a number of CDW compounds.26,27 ln extremely pure samples, the relevant length L may actually be the distance between contacts. The CDW in the regions under the contacts will be stationarv. so these resons may be considered as CDW electrodes~on the leftand right-hand sides of a “tunnel iunction.” Correlated tunneh~g of quantum solitons betwekn the electrodes could then be treated within a tunneling Hamiltonian formulation,**,*9 where the polarization effects would be included with an additional contribution Q*/2C to the Hamiltonian. The theoretical treatment developed by Averin and Likharev,t9 and others*l could then be applied to describe correlated soliton tunneling. IV.
Conclusion
The possible analogy between correlated tunneling of CDW solitons and correlated single electron tunneling in small tunnel junctions has been suggested as an interesting interpretation of the observed sharp threshold field and coherent oscillations in CDW systems. Several years ago, a number of experimentsl*-‘7 were performed on CDW materials in order to test the predictions of the tunneling model. In particular, the theory of photon-assisted tunneling (PAT)30 was adapted to Bardeen’s model in order to generate predictions of linear and nonlinear rf experiments. The essentially quantitative agreement obiained in these diverse expeiments constitutes-powerful evidence that tunneling is indeed responsible for CDW depinning in an electric%eld. lf the tunneling hypothesis is correct, then CDW transport constitutes an important new cooperative quantum phenomenon. Acknowledgement -- The author gratefully acknowledges support by the State of Texas, the Texas Center for Superconductivity at the University of Houston, DARPA Prime Grant MDA 972-88-G-002, and the A. P. Sloan Foundation.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
24.
25. References P. Monceau, N. P. Ong, A. M. Portis, A. Meerschaut, and J. Rouxel, Physical Review Letters 37, 6902 (1976). R. M. Fleming and C. C. Grimes, Physical Review Letters 42, 1423 (1979). For extensive reviews of CDw’s see Electronic Properties of Quasi-One-DimensionalMaterials,edited by P. Monceau (Reidel, Dordrecht, 1985); Cl. Grtiner, Reviews of Modern Physics 60, 1129 (1988). H. Friihlich, Proceedings of the Royal Sociery of London A223.296 (1954).
26. 27.
28. 29. 30.
Cl. Gruner, A. Zawadowski, P. M. Chaikin, Physical Review Letters 46, 511 (1981). P. Monceau, J. Richard and M. Renard, Physical Review B31, 1396 (1985). L. Sneddon, M. C. Cross, and D. S. Fisher, Physical Review Letters 49, 292 (1982). D. S. Fisher, Physical Review B31, 1396 (1985). R. A. Klemm and J. R. Schrieffer, Physical Review Letters 52, 65 (1984). L. Sneddon, Physical Review L..etters 52,65 (1984). John Bardeen, Physical Review Letters 42, 1498 (1979); ibid 45, 1978 (1980); ibid 55, 1002 (1985). J. H. Miller,Jr., J. Richard, J. R. Tucker, and John Bardeen, Physical Review Letters 51, 1592 (1983). J. H. Miller,Jr., R. E. Thome, W. G. Lyons, J. R. Tucker, and John Bardeen, Physical Review B31, 5229 (1985). R. E. Thome, J. H. Miller, Jr., W. G. Lyons, J. W. Lyding and J. R. Tucker, Physical Review Letters 55, 1006 (1985). J. Richard, R. E. Thorne, W. G. Lyons, J. H. Miller,Jr., and J. R. Tucker, Solid State Communication 52, 183 (1984). R. E. Thome, W. G. Lyons, J. W. Lyding, J. R. Tucker, and John Bardeen, Physical Review B34, 5988 (1986). R. E. Thome, J. R. Tucker, and John Bardeen, Physical Review Letters 58, 828 (1987). A. Maiti and J. H. Miller, Jr., Physical Review B43, 12205 (1991). D. V. Averin and K. K. Likharev, Journal of Low Temperature Physics 62, 345 (1986), and references cited therein. T. A. Fulton and G. J. Dolan, Physical Review Letters 59, 109 (1987). K. K. Likharev, N. S. Bakhvalov, G. S. Kazacha, and S. I. Serdynkova, IEEE Transactions on Magnetics 25, 1436 (1989). P. Delsing, K. K. Likharev, L. S. Kuzmin, and T. Claeson, Physical Review Letters 63, 1861 (1989). L. J. Geerligs, V. F. Anderegg, P. A. M. Holweg, J. E. Mooij, H. Potltier, D. Esteve, C. Urbina, and M. H. Devoret, Physical Review Letters 64, 2691 (1990). R. E. Thome, J. S. Hubacek, W. G. Lyons, J. W. Lyding, and J. R. Tucker, Physical Review B37, R. E. Thome et al., Physical Review 10055 (1988); B35,6348 and 6360 (1987). H. Fukuyama and P. A. Lee, Physical Review B17, 535 (1978); P. A. Lee and J. M. Rice, Physical Review B19, 3870 (1979). Wei-yu Wu, A. Jcinossy, and G. Griiner, Solid State Communication 49, 1013 (1984); G. Grilner, Reviews of Modern Physics 60, 1129 (1988). G. Grtlner, in Charge Density Waves in Solids, edited by Gy Hutiray and J. .%lyom, Lecture Notes in Physics Vol. 217 (Springer-Verlag, Berlin, 1985), pp. 263-278. J. Bardeen, Physical Review Letters 6,57 (1961). M. H. Cohen, L. M. Falicov, and J. C. Phillips, Physical Review Letters 8, 3 16 (1962). J. R. Tucker, IEEE Journal of Quantum Electronics QE-15, 1234 (1979).