Quantum tunneling theory of charge-density-wave depinning: the Coulomb-dominated low-temperature regime, microscopic thermal activation, coherence time, and electron-condensate momentum transfer

In a 1986 paper John Bardeen consolidated his theory of charge-density-wave (CDW) depinning by quantum tunneling, a Zener-like model accounting for the nonlinear current-voltage characteristics, the sharp threshold field, photon-assisted-tunneling scaling of the combined dc and ac response, and the current-to-frequency ratio of sliding CDWs in quasi-one-dimensional conductors. Four issues were left open by the model: (1) the form of depinning in the Coulomb-dominated regime of low temperatures and low frequencies, where screening of the long-range Coulomb interaction by thermally excited quasiparticles is incomplete and where the scaling behavior breaks down; (2) a microscopic treatment of the thermal activation observed at low temperature, complementing a classical analysis by Tucker, from which the activated low-frequency response should follow from the basic parameters of the tunneling model; (3) the identification of the coherence-limited relaxation time entering the high-field tunneling conductivity, which fixes its numerical prefactor (the criterion $v_F\tau^\star=\xi_0$, with $v_F$ the Fermi velocity and $\xi_0$ the coherence length, versus $L_d/2$, with $L_d$ a domain length, or some other microscopic criterion); and (4) a microscopic theory of momentum transfer from single-particle electrons to the condensate, replacing the phenomenological electron-phonon relaxation time and effective mass and describing the frequency-dependent partition of spectral weight between tunneling-related transport and additional absorption near the pinning frequency. The question remains open.

Background

A charge density wave (CDW) is a coupled modulation of the conduction-electron density and the crystal lattice that forms below a Peierls transition temperature $T_p$ in quasi-one-dimensional conductors such as NbSe$_3$, orthorhombic TaS$_3$, and the blue bronzes $M_{0.3}$MoO$_3$. The opening of a Peierls gap at the Fermi surface makes single-particle conduction thermally activated, but because the phase of the modulation can translate, the condensate can in principle carry a collective current (Fröhlich conduction). Static impurities pin the phase, and in the classical picture (Fukuyama and Lee; Lee and Rice) the condensate depins and slides only above a sharp threshold electric field $E_T$, producing strongly nonlinear current-voltage curves and narrow-band voltage noise (nearly periodic voltage oscillations generated as the sliding condensate sweeps past the impurity distribution). John Bardeen proposed from 1979 onward a rival quantum picture: depinning occurs by quantum tunneling of the CDW phase through the pinning barrier, in analogy with Josephson tunneling in superconductors. In this model the collective current follows a Zener-like nonlinear form, $I_{CDW}(V)=I_0(V-V_T)\exp(-V_0/V)$ above threshold, approaching an asymptotic high-field CDW conductivity $\sigma_b$; the response to combined dc and ac fields obeys photon-assisted-tunneling scaling (the dc response is modified by the absorption and emission of quanta $\hbar\omega$ of the ac field), consistent with the sliding-conductivity scaling of Sneddon, Cross and Fisher and with interference experiments on orthorhombic TaS$_3$ (Miller, Thorne, Lyons, Tucker and Bardeen). Low-temperature measurements of the temperature dependence of the threshold and of the current-to-frequency ratio were reported by Thorne, Miller, Lyons, Lyding and Tucker, with the tunneling interpretation given by Bardeen. A crucial physical assumption of the model is that the long-range Coulomb interaction of the charge modulation is screened by thermally excited quasiparticles across the Peierls gap, a condition that holds only at temperatures not too far below $T_p$. The 1986 paper by Bardeen distilled this program; its remaining frontiers are the four issues below. First, in the Coulomb-dominated regime of low temperatures and low frequencies, where quasiparticle screening is incomplete, the form of depinning and the breakdown of the scaling behavior required further theoretical development. Second, the thermal activation observed at low temperatures was not derived microscopically; a complementary classical treatment by Tucker (whose companion paper appears in the same proceedings volume) should be extended so that the activated low-frequency response follows from the basic parameters of the tunneling model. Third, the coherence-limited relaxation time $\tau^\star$ entering $\sigma_b$ was fixed only up to order-one ambiguity ($v_F\tau^\star=\xi_0$ versus $L_d/2$, or some other microscopic criterion), leaving the numerical prefactors of $\sigma_b$ and related scales uncertain. Fourth, in materials or alloys with very high pinning frequency $\omega_p$, the partitioning of spectral weight between tunneling-related transport modes and additional absorption near $\omega_p$ was treated only phenomenologically through an electron-phonon relaxation time and an effective mass $M'$, so a microscopic theory of momentum transfer from single-particle electrons to the condensate (the macroscopic phonon) and of its frequency dependence was needed. Subsequent research did not complete this program but bifurcated. Classical finite-size and phase-slip models (Klemm and Robbins; Tucker and collaborators) — phase slips meaning local, thermally activated collapses of the condensate order parameter through which its phase advances by $2\pi$ — reproduced much of the same phenomenology without quantum tunneling, and Grüner's 1988 review consolidated a largely classical mainstream, while Bardeen's own 1989 comparison of classical and quantum models defended the tunneling picture. A separate line made electrostatic charging energy central: the soliton-nucleation Coulomb-blockade model of Miller and co-workers, in which tunneling of microscopic charged solitons (localized kinks of the CDW phase carrying discrete charge), blocked below a threshold set by electrostatic charging energy, replaces tunneling of the phase. Meanwhile low-temperature experiments accumulated evidence compatible with quantum depinning, including temperature-independent thresholds at low temperature, quantum creep, and $h/2e$ Aharonov-Bohm oscillations of a CDW moving through columnar defects. The question posed here asks for the missing microscopic foundation of the original tunneling theory itself.

Problem Statement

The 1986 formulation of CDW depinning by quantum tunneling stands as a phenomenologically successful but microscopically incomplete theory. The open problem is to supply its missing microscopic foundation, specifically: (1) Derive, from a microscopic Hamiltonian for an incommensurate CDW in a quasi-one-dimensional conductor with impurity pinning and the long-range Coulomb interaction, the form of quantum depinning in the Coulomb-dominated regime of low temperatures and low frequencies, where screening by thermally excited quasiparticles is incomplete; establish in this regime whether the Zener-like current-voltage form survives, how it is modified, and where the photon-assisted-tunneling scaling of the combined dc and ac response breaks down. (2) Derive the observed low-temperature thermally activated response microscopically, complementing the classical analysis of Tucker, so that the activated low-frequency conductivity follows from the basic parameters of the tunneling model (Peierls gap, pinning energy, domain structure) rather than from phenomenology. (3) Identify microscopically the coherence-limited relaxation time $\tau^\star$ entering the high-field tunneling conductivity $\sigma_b$ (is $v_F\tau^\star=\xi_0$ with $v_F$ the Fermi velocity and $\xi_0$ the coherence length, $L_d/2$ with $L_d$ a pinning-domain length, or some other microscopic criterion correct?), thereby fixing the numerical prefactors of $\sigma_b$ and of the related current and voltage scales. (4) Construct a microscopic theory of momentum transfer from single-particle electrons to the condensate (the macroscopic phonon), replacing the phenomenological electron-phonon relaxation time and effective mass $M'$ introduced for materials or alloys with very high pinning frequency $\omega_p$, and determine the frequency-dependent partition of spectral weight between tunneling-related transport modes and additional absorption near $\omega_p$. The objective is to supply the missing microscopic foundation of the tunneling model on these issues, with results confrontable with the measured threshold fields, frequency-dependent dielectric response, and current-voltage characteristics of materials such as NbSe$_3$, orthorhombic TaS$_3$, and the blue bronzes. Issues (1)–(4) are independent sub-problems of this single objective.

References

1. John Bardeen, 'Theory of Non-Ohmic Conduction from Charge-Density Waves in NbSe3', Physical Review Letters 42, 1498 (1979). DOI 10.1103/PhysRevLett.42.1498, https://doi.org/10.1103/PhysRevLett.42.1498
2. H. Fukuyama and P. A. Lee, 'Dynamics of the charge-density wave. I. Impurity pinning in a single chain', Physical Review B 17, 535 (1978). DOI 10.1103/PhysRevB.17.535, https://doi.org/10.1103/PhysRevB.17.535
3. P. A. Lee and T. M. Rice, 'Electric field depinning of charge density waves', Physical Review B 19, 3970 (1979). DOI 10.1103/PhysRevB.19.3970, https://doi.org/10.1103/PhysRevB.19.3970
4. Leigh Sneddon, M. C. Cross and Daniel S. Fisher, 'Sliding conductivity of charge-density waves', Physical Review Letters 49, 292 (1982). DOI 10.1103/PhysRevLett.49.292, https://doi.org/10.1103/PhysRevLett.49.292
5. S. E. Barnes and A. Zawadowski, 'Theory of Josephson-type oscillations in a moving charge-density wave', Physical Review Letters 51, 1003 (1983). DOI 10.1103/PhysRevLett.51.1003, https://doi.org/10.1103/PhysRevLett.51.1003
6. J. H. Miller, R. E. Thorne, W. G. Lyons, J. R. Tucker and John Bardeen, 'Dynamics of charge-density waves in orthorhombic TaS3', Physical Review B 31, 5229 (1985). DOI 10.1103/PhysRevB.31.5229, https://doi.org/10.1103/PhysRevB.31.5229
7. R. E. Thorne, J. H. Miller, W. G. Lyons, J. W. Lyding and J. R. Tucker, 'Macroscopic quantum tunneling in quasi one-dimensional metals. I. Experiment', Physical Review Letters 55, 1006 (1985). DOI 10.1103/PhysRevLett.55.1006, https://doi.org/10.1103/PhysRevLett.55.1006
8. John Bardeen, 'Macroscopic quantum tunneling in quasi one-dimensional metals. II. Theory', Physical Review Letters 55, 1010 (1985). DOI 10.1103/PhysRevLett.55.1010, https://doi.org/10.1103/PhysRevLett.55.1010
9. I. V. Krive and A. S. Rozhavsky, 'On the nature of threshold electric field in quasi-one-dimensional commensurate charge-density-waves', Solid State Communications 55, 691 (1985). DOI 10.1016/0038-1098(85)90235-2, https://doi.org/10.1016/0038-1098(85)90235-2
10. D. Reagor and G. Grüner, 'Frequency-dependent response of pinned charge-density-wave condensates: classical versus quantum description', Physical Review Letters 56, 659 (1986). DOI 10.1103/PhysRevLett.56.659, https://doi.org/10.1103/PhysRevLett.56.659
11. John Bardeen, 'Depinning of charge-density waves by quantum tunneling', Physica B+C 143, 14 (1986). DOI 10.1016/0378-4363(86)90043-4, https://doi.org/10.1016/0378-4363(86)90043-4
12. J. R. Tucker, 'Dynamics of sliding charge density waves', Physica B+C 143, 19 (1986). DOI 10.1016/0378-4363(86)90044-6, https://doi.org/10.1016/0378-4363(86)90044-6
13. R. A. Klemm and M. O. Robbins, 'Charge density wave conduction: dynamics and finite size effects', Physica B+C 143, 76 (1986). DOI 10.1016/0378-4363(86)90058-6, https://doi.org/10.1016/0378-4363(86)90058-6
14. T. Saso and Y. Suzumura, 'Quantum effects on impurity pinning and depinning of charge density waves', Physica B+C 143, 143 (1986). DOI 10.1016/0378-4363(86)90076-8, https://doi.org/10.1016/0378-4363(86)90076-8
15. J. R. Tucker, W. G. Lyons, J. H. Miller, R. E. Thorne and J. W. Lyding, 'Origin of the dielectric relaxation frequency in sliding-charge-density-wave systems', Physical Review B 34, 9038 (1986). DOI 10.1103/PhysRevB.34.9038, https://doi.org/10.1103/PhysRevB.34.9038
16. M. O. Robbins and R. A. Klemm, 'Charge-density-wave conduction: dynamics and finite-size effects', Physical Review B 34, 8496 (1986). DOI 10.1103/PhysRevB.34.8496, https://doi.org/10.1103/PhysRevB.34.8496
17. I. V. Krive and A. S. Rozhavsky, 'The effective lagrangian of the charge-density-wave and macroscopic quantum tunneling in random media', Physics Letters A 132, 363 (1988). DOI 10.1016/0375-9601(88)90870-5, https://doi.org/10.1016/0375-9601(88)90870-5
18. J. R. Tucker and W. G. Lyons, 'Low-temperature depinning of sliding charge-density waves', Physical Review B 38, 7854 (1988). DOI 10.1103/PhysRevB.38.7854, https://doi.org/10.1103/PhysRevB.38.7854
19. G. Grüner, 'The dynamics of charge-density waves', Reviews of Modern Physics 60, 1129 (1988). DOI 10.1103/RevModPhys.60.1129, https://doi.org/10.1103/RevModPhys.60.1129
20. John Bardeen, 'Classical versus quantum models of charge-density-wave depinning in quasi-one-dimensional metals', Physical Review B 39, 3528 (1989). DOI 10.1103/PhysRevB.39.3528, https://doi.org/10.1103/PhysRevB.39.3528
21. E. N. Bogachek, I. V. Krive, I. O. Kulik and A. S. Rozhavsky, 'Instanton Aharonov-Bohm effect and macroscopic quantum coherence in charge-density-wave systems', Physical Review B 42, 7614 (1990). DOI 10.1103/PhysRevB.42.7614, https://doi.org/10.1103/PhysRevB.42.7614
22. M. E. Itkis, F. Ya. Nad' and P. Monceau, 'Nonlinear conductivity of quasi-one-dimensional TaS3 at low temperatures', Journal of Physics: Condensed Matter 2, 8327 (1990). DOI 10.1088/0953-8984/2/42/010, https://doi.org/10.1088/0953-8984/2/42/010
23. A. Maiti and J. H. Miller, 'Theory of charge-density-wave tunneling', Physical Review B 43, 12205 (1991). DOI 10.1103/PhysRevB.43.12205, https://doi.org/10.1103/PhysRevB.43.12205
24. Alexander S. Kovalev, Yurij V. Pershin and Alexander S. Rozhavsky, 'Dynamics of conversion of conduction electrons into a collective charge-density-wave current', Physical Review B 53, 16227 (1996). DOI 10.1103/PhysRevB.53.16227, https://doi.org/10.1103/PhysRevB.53.16227
25. Yu. I. Latyshev, O. Laborde, P. Monceau and S. Klaumünzer, 'Aharonov-Bohm effect on charge density wave (CDW) moving through columnar defects in NbSe3', Physical Review Letters 78, 919 (1997). DOI 10.1103/PhysRevLett.78.919, https://doi.org/10.1103/PhysRevLett.78.919
26. J. H. Miller, C. Ordóñez and E. Prodan, 'Time-correlated soliton tunneling in charge and spin density waves', Physical Review Letters 84, 1555 (2000). DOI 10.1103/PhysRevLett.84.1555, https://doi.org/10.1103/PhysRevLett.84.1555
27. Masanori Yumoto, Hidetoshi Fukuyama, Hiroshi Matsukawa and Naoto Nagaosa, 'Impurity effects on quantum depinning of commensurate charge density waves', Journal of the Physical Society of Japan 69, 2953 (2000). DOI 10.1143/JPSJ.69.2953, https://doi.org/10.1143/JPSJ.69.2953
28. Tian Ming-Liang, Mao Zhi-Qiang, Wang Rui-Ping and Zhang Yu-Heng, 'Field dependence of the non-linear conduction of charge-density waves in thallium blue bronze Tl0.3MoO3', Journal of Physics: Condensed Matter 6, 8565 (1994). DOI 10.1088/0953-8984/6/41/018, https://doi.org/10.1088/0953-8984/6/41/018
29. S. G. Zybtsev, V. Ya. Pokrovskii and S. V. Zaitsev-Zotov, 'Quantized states of the charge-density wave in microcrystals of K0.3MoO3', Nature Communications 1, 85 (2010). DOI 10.1038/ncomms1087, https://doi.org/10.1038/ncomms1087
30. Yu. I. Latyshev, A. P. Orlov and P. Monceau, 'Charge density wave transport in NbSe3 at low temperatures under high magnetic field', JETP Letters 93, 99 (2011). DOI 10.1134/S0021364011020081, https://doi.org/10.1134/S0021364011020081
31. A. A. Sinchenko, P. Monceau and T. Crozes, 'Transverse conductivity behavior at threshold electric field for CDW sliding in NbSe3', Physica B: Condensed Matter 407, 1816 (2012). DOI 10.1016/j.physb.2012.01.038, https://doi.org/10.1016/j.physb.2012.01.038
32. John H. Miller, Jr. and Asanga I. Wijesinghe, 'Quantum mechanisms of density wave transport', Physica B: Condensed Matter 407, 1734 (2012). DOI 10.1016/j.physb.2012.01.018, https://doi.org/10.1016/j.physb.2012.01.018
33. J. H. Miller, Jr., A. I. Wijesinghe, Z. Tang and A. M. Guloy, 'Correlated quantum transport of density wave electrons', Physical Review Letters 108, 036404 (2012). DOI 10.1103/PhysRevLett.108.036404, https://doi.org/10.1103/PhysRevLett.108.036404
34. Tian Le, Ruiyang Jiang, Linfeng Tu, Renji Bian, Yiwen Ma, Yunteng Shi, Ke Jia, Zhilin Li, Zhaozheng Lyu, Xuewei Cao, Jie Shen, Guangtong Liu, Youguo Shi, Fucai Liu, Yi Zhou, Li Lu and Fanming Qu, 'Inverse-current quantum electro-oscillations in a charge-density wave insulator', Physical Review B 109, 245123 (2024). DOI 10.1103/PhysRevB.109.245123, https://doi.org/10.1103/PhysRevB.109.245123 (preprint arXiv:2302.08677, https://arxiv.org/abs/2302.08677)
35. I. A. Cohn and S. V. Zaitsev-Zotov, 'Magnetic field induced activated and quantum creep of charge density waves in o-TaS3', Physical Review B 108, 115124 (2023). DOI 10.1103/PhysRevB.108.115124, https://doi.org/10.1103/PhysRevB.108.115124
36. John H. Miller, Jr., Martha Y. Suárez-Villagrán and Johnathan O. Sanderson, 'Quantum transport of charge density wave electrons in layered materials', Materials Today Physics 41, 101326 (2024). DOI 10.1016/j.mtphys.2024.101326, https://doi.org/10.1016/j.mtphys.2024.101326 (preprint arXiv:2310.10512, https://arxiv.org/abs/2310.10512)