Microscopic origin of the saturated low-temperature electron dephasing rate in indium tin oxide thin films, with the companion spin-orbit and granular weak-localization anomalies of low-carrier-density conducting oxides

Measurements of the electron dephasing time in two-dimensional homogeneous and granular indium tin oxide (ITO) thin films between 0.3 and 90 K (Wu et al., Phys. Rev. B 85, 104204 (2012)) established a now-standard phenomenology: small-energy-transfer electron-electron scattering dominates the dephasing over a wide temperature range, electron-phonon relaxation is negligibly weak because of the low carrier concentration, and below about 1-2 K the dephasing time saturates at a finite value $\tau_\varphi^0$ that scales approximately as the inverse of the diffusion constant and is more than an order of magnitude longer than in polycrystalline metals of the same diffusion constant. The problem is to identify the microscopic origin of this saturated dephasing rate — discriminating among dynamical structural defects (tunneling two-level systems associated with oxygen nonstoichiometry), intrinsic electron-electron-interaction effects, dilute magnetic-impurity scattering, and environmental 1/f noise — and to build a quantitative theory of dephasing in low-carrier-density conducting oxides, including the microscopic explanation of the extraordinarily weak spin-orbit scattering despite heavy In and Sn constituents and of the conditions under which weak localization in granular ITO crosses over from global inter-grain to local single-grain interference. All of these facets remain unresolved in the literature to date.

Background

In a disordered metal at low temperatures, the conductivity receives a quantum-interference correction called weak localization (WL): electron partial waves that traverse closed time-reversed diffusive paths interfere constructively, enhancing backscattering. A small perpendicular magnetic field destroys this interference, and fitting the resulting magnetoresistance to the standard Hikami-Larkin-Nagaoka formula yields the electron dephasing time $\tau_\varphi(T)$, the timescale over which the phase coherence of the diffusive electron states is lost (Hikami, Larkin and Nagaoka, Prog. Theor. Phys. 63, 707 (1980)). The dephasing rate $1/\tau_\varphi$ is set by inelastic processes: in two-dimensional disordered conductors the dominant low-temperature mechanism is the small-energy-transfer (Nyquist) electron-electron process, $1/\tau_\varphi \propto T\ln T$ with a coefficient fixed by the sheet resistance; at higher temperatures the large-energy-transfer electron-electron process ($\propto T^2$) and electron-phonon scattering ($\propto T^2$ in the quasi-ballistic regime) take over. The disorder strength enters through the electron diffusion constant $D$, which is extracted from the measured conductivity, carrier density, and density of states.

Indium tin oxide (ITO) is a Sn-doped In$_2$O$_{3-x}$ transparent conducting oxide with an unusually free-electron-like conduction band (derived from In 5s states) but a carrier concentration $n \approx 2$-$12 \times 10^{20}$ cm$^{-3}$, two to three orders of magnitude below that of ordinary metals, corresponding to a reduced Fermi energy $E_F \lesssim 1$ eV. Wu et al. (Phys. Rev. B 85, 104204 (2012)) measured WL magnetoresistance in two-dimensional homogeneous ITO films and in inhomogeneous (granular) ITO ultrathin films between 0.3 and 90 K. They found the Nyquist electron-electron term dominant from a few kelvin up to several tens of kelvin, a crossover to the large-energy-transfer term at higher temperatures, and a quantitative explanation of why electron-phonon relaxation is negligible: the electron-transverse-phonon rate scales with $n$ and is therefore suppressed in this low-$n$ material, which is what allows WL to survive up to nearly 100 K.

Two of their findings, however, resisted microscopic explanation. First, below about 1-2 K the extracted dephasing time became very weakly temperature-dependent, saturating at a finite value $\tau_\varphi^0$ as $T \to 0$, with $\tau_\varphi^0$ scaling approximately as $1/D$ — the same counter-intuitive disorder dependence previously reported for numerous polycrystalline metals and alloys (Lin and Kao, J. Phys.: Condens. Matter 13, L119 (2001)) — and more than an order of magnitude longer than in polycrystalline metals of the same $D$. Whether a finite low-temperature ('saturated') dephasing rate is intrinsic to disordered conductors has been contested since Mohanty, Jariwala and Webb (Phys. Rev. Lett. 78, 3366 (1997)); the competing explanations include dephasing by environmental 1/f noise (Imry, Fukuyama and Schwab, EPL 47, 608 (1999)), spin-flip scattering from extremely dilute magnetic impurities, whose effect can survive at sub-ppm concentrations and be amplified in Kondo-type scenarios (the resonant enhancement of a magnetic impurity's spin-flip rate below its characteristic Kondo temperature; Pierre and Birge, Phys. Rev. Lett. 89, 206804 (2002); Micklitz, Costi and Rosch, Phys. Rev. B 75, 054406 (2007)), dephasing by tunneling two-level systems (TLS) — dynamical structural defects that tunnel between two configurational states — which under certain conditions yields a temperature-insensitive dephasing time obeying $\tau_\varphi \propto 1/D$ (Aleiner, Altshuler and Galperin, Phys. Rev. B 63, 201401 (2001); Galperin, Kozub and Vinokur, Phys. Rev. B 69, 073102 (2004)), and intrinsic interaction-driven saturation within the theory of Golubev and Zaikin, most recently restated in their 2019 monograph chapters. None of these mechanisms has been universally accepted. Wu et al. argued that the empirical $\tau_\varphi^0 \propto 1/D$ law, reproduced across ITO films from different suppliers and thermal treatments, can hardly arise from random magnetic contamination, and noted that two-level systems associated with oxygen nonstoichiometry plausibly exist in ITO — but they did not, and could not, decide the question.

Second, the spin-orbit scattering in their polycrystalline ITO films was extraordinarily weak: the spin-orbit scattering length exceeded 500 nm (spin-orbit scattering time above roughly 250 ps), even though ITO contains the relatively heavy elements indium and tin, and similarly weak spin-orbit scattering had been reported in polycrystalline In$_{2}$O$_{3-x}$ and SnO$_2$ films. The paper states that this deserves a theoretical explanation.

The inhomogeneous films add a granular-metal question. A granular metal is an array of metallic grains coupled by tunnel junctions; its granularity is characterized by the dimensionless intergrain tunneling conductance $g_T$ and the single-grain level spacing $\delta$. Efetov and Tschersich (Phys. Rev. B 67, 174205 (2003)) and Beloborodov, Lopatin and Vinokur (Phys. Rev. B 70, 205120 (2004)) predicted that WL in the homogeneous formulation is suppressed above a characteristic temperature $T^* \approx g_T\delta/k_B$; see Beloborodov, Lopatin, Vinokur and Efetov (Rev. Mod. Phys. 79, 469 (2007)) for a review. Wu et al. observed WL magnetoresistance in granular ITO persisting to temperatures well above $T^*$ and proposed a smooth crossover to local single-grain interference at a higher temperature $T^*_{WL}$ at which the dephasing length $L_\varphi = \sqrt{D\tau_\varphi}$ becomes comparable to the grain size — a consequence of the unusually weak electron-phonon relaxation keeping Nyquist dephasing dominant — but the microscopic description of WL in the granular regime and the precise crossover conditions were left open.

Problem Statement

Provide a microscopic, quantitative theory of electron phase coherence in low-carrier-density disordered indium tin oxide (and closely related conducting oxides) that explains the dephasing phenomenology measured by Wu et al. (Phys. Rev. B 85, 104204 (2012)) in two-dimensional homogeneous and inhomogeneous ITO films between 0.3 and 90 K. The single overarching objective — identifying the microscopic mechanisms that set the dephasing and spin-relaxation rates in this material class — decomposes into four requirements, all part of the objective but each an independently resolvable facet:

(1) Identify the microscopic origin of the finite, weakly temperature-dependent ('saturated') dephasing rate $1/\tau_\varphi^0$ observed below about 1-2 K, committing to the mechanism or mechanism set responsible. The identification must explain its approximate disorder scaling $\tau_\varphi^0 \propto 1/D$, its magnitude being more than an order of magnitude longer than in polycrystalline metals of the same diffusion constant $D$, and it must decide among the competing proposals — dephasing by dynamical structural defects such as tunneling two-level systems associated with oxygen nonstoichiometry, intrinsic electron-electron-interaction saturation, spin-flip scattering from dilute magnetic impurities, and dephasing by environmental 1/f noise — specifying experimental signatures that would falsify each alternative.

(2) Quantify how ITO's reduced Fermi energy ($E_F \lesssim 1$ eV) and low carrier concentration ($n \approx 2$-$12 \times 10^{20}$ cm$^{-3}$) determine the magnitudes and functional forms of the low-temperature dephasing processes, including the saturated term — extending to the saturated term the level of quantitative agreement already achieved for the Nyquist and large-energy-transfer electron-electron terms and for the suppressed electron-phonon relaxation.

(3) Explain, at the same microscopic level, why spin-orbit scattering is extraordinarily weak in polycrystalline ITO and related oxides containing heavy elements ($L_{so} > 500$ nm, $\tau_{so} > 250$ ps for $D \approx 10$ cm$^2$/s), including why it nevertheless grows with disorder.

(4) Give the microscopic description of weak localization in the granular regime of inhomogeneous ITO films, and the precise conditions under which global (inter-grain) weak localization crosses over to local single-grain interference — clarifying the status of the proposed crossover temperature $T^*_{WL}$, defined by $L_\varphi(T^*_{WL})$ becoming comparable to the grain size, relative to the granular scale $T^* = g_T\delta/k_B$.

The mechanism set must be identified with quantitative, falsifiable consequences that can be compared with the existing ITO magnetotransport data and with purpose-designed discriminating measurements. The problem is a concrete, material-specific instance of the long-standing question of whether the low-temperature saturation of the electron dephasing time in disordered conductors is intrinsic.

References

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