Refinements of the 1984 spin-orbit weak-localization analysis of non-scaling conductance in Si inversion layers: separate scattering times, dephasing temperature dependence, anisotropic spin-orbit, and the crossover to strong localization (Markiewicz, J. Phys. C 17, L23)

R. S. Markiewicz, 'Absence of scaling of two-dimensional conduction in Si inversion layers', J. Phys. C: Solid State Phys. 17 (1984) L23, explained the observed failure of one-parameter conductance scaling in Si MOSFET inversion layers (Davies, Pepper and Kaveh, 1983) by weak spin-orbit scattering within the Hikami-Larkin-Nagaoka (HLN) theory, and closed by listing the assumptions of that analysis that require further investigation: sample-dependence of the fit parameter A, the simplification of equal magnetic-impurity and spin-orbit scattering times, the neglected linear-in-T term of the inelastic rate, the validity of HLN only in the weak-disorder regime (motivating study of the transition to strong localization), and the expected strong anisotropy of spin-orbit scattering in the layer. Markiewicz's specific question — whether his spin-orbit explanation of the 1983 non-scaling data survives the relaxed assumptions — has not been revisited. The problem is therefore open, though of low (largely historiographic) significance.

Background

A silicon MOSFET (metal-oxide-semiconductor field-effect transistor) inversion layer — the two-dimensional electron gas (2DEG) that forms at a Si/SiO2 interface when a gate voltage is applied — was the workhorse experimental system of 1980s localization physics. In a disordered conductor at low temperature, quantum interference between time-reversed diffusive electron paths enhances backscattering and lowers the conductance by a logarithmic amount; this effect is called weak localization, and its suppression by a perpendicular magnetic field yields a calculable magnetoconductance. The theory of Hikami, Larkin and Nagaoka (HLN, 1980) gives this magnetoconductance for a disordered two-dimensional film including spin-orbit scattering (which for strong coupling reverses the sign of the interference correction, an effect called weak anti-localization) and spin-flip scattering by magnetic impurities; each process is characterized by a time — the spin-orbit time $\tau_{so}$, the magnetic-impurity (spin-flip) time $\tau_s$, and an inelastic (dephasing) time $\tau_i$ that cuts off the interference at the thermal diffusion length (the typical distance an electron diffuses between dephasing events). In parallel, the scaling theory of localization of Abrahams, Anderson, Licciardello and Ramakrishnan (1979) predicted that in two dimensions all electron states are localized, with a beta function $\beta = \partial \ln g / \partial \ln L$ (the logarithmic derivative of the conductance $g$ with respect to system size $L$, set by the inelastic diffusion length) that is a universal one-parameter function of $g$ alone. Davies, Pepper and Kaveh (1983) tested this experimentally on Si inversion layers by extracting the temperature derivative $\beta_e = -\partial \ln g / \partial \ln T$ (with $g$ the conductance normalized by $e^2/\hbar$, proportional to the scaling beta function) and found that curves taken at different carrier densities did not collapse onto a single function of $g$ — an apparent failure of one-parameter scaling, superimposed on the logarithmic conductance corrections (interaction and localization) measured earlier in the same system by Davies and Pepper (1982) and by Uren, Davies, Kaveh and Pepper (1981). Markiewicz (1984) proposed that weak spin-orbit scattering accounts for this non-scaling: using the HLN expression

$$\sigma = \sigma_0' + \beta[(1-F)\ln T + \ln(A + T^2)]$$

where $\sigma_0'$ is a temperature-independent conductivity offset, the prefactor $\beta = e^2/(2\pi^2\hbar)$ is Markiewicz's notation for the conductance prefactor (unrelated to the scaling beta function above), $F \approx 0.95$ is an interaction screening factor, and $A$ is a single dimensionless parameter setting the temperature scale of the spin-orbit contribution ($A = 0.36$; the dotted curves in the source's Figures 1 and 2 show a fit with $A = 0.135$), he fitted simultaneously the scaling function $\beta_e(g)$ and the temperature-dependent conductivity $\sigma(T)$ of the Davies-Pepper-Kaveh data, and attributed the implied spin-orbit time $\tau_{so} \approx 6.5 \times 10^{-11}$ s to surface-roughness enhancement via the Meservey-Tedrow (1978) surface relation $(\tau_e/\tau_{so}) \propto Z^4$ between elastic time, spin-orbit time, and atomic number $Z$. The letter's closing paragraph states the simplifying assumptions behind this analysis that require further investigation: the temperature scale parameter $A$ may differ between samples; the equality $\tau_s = \tau_{so}$ was a parameter-reduction device rather than physics; the inelastic rate was taken as pure $\tau_i^{-1} = \bar{A} T^2$ even though Uren et al (1981) had measured $\tau_i^{-1} = \bar{A} T^2 + B T$ with a non-zero linear term; the HLN theory is a perturbative treatment valid only while the logarithmic corrections remain small compared to the Drude conductivity $\sigma_0$, whereas the measured samples often fall outside this regime, so (in the source's words) the understanding gained could be used to explore the transition to strong localization, i.e. the crossover toward truly localized behavior near the mobility edge (the energy separating extended from localized states in the band tail); and, finally, because all carriers occupy the lowest quantum subband for motion perpendicular to the layer (the discrete bound state for motion perpendicular to the Si/SiO2 interface), spin-orbit scattering in a Si inversion layer is expected to be highly anisotropic, $\tau_{so}^x \approx \tau_{so}^y \gg \tau_{so}^z$, so a treatment with a single spin-orbit time may need refinement. Weak localization in Si inversion layers had been analyzed before Markiewicz (e.g. Wheeler, 1981), but his question is specifically about the robustness of his own spin-orbit explanation of the non-scaling.

Problem Statement

The question is the closing list of Markiewicz (1984): to what extent does his HLN-based spin-orbit explanation of the absence of one-parameter scaling in Si MOSFET inversion layers survive when its simplifying assumptions are removed? Concretely: (i) determine whether the dimensionless spin-orbit temperature-scale parameter $A$ must be sample-dependent (the source notes that the dotted curves in its Figures 1 and 2 show a fit with $A = 0.135$, as an alternative to $A = 0.36$); (ii) reanalyze the conductance with the magnetic-impurity (spin-flip) time $\tau_s$ and the spin-orbit time $\tau_{so}$ as independent parameters, instead of the imposed equality $\tau_s = \tau_{so}$ that was introduced only to reduce the number of fitted parameters; (iii) include the empirically measured linear term in the inelastic scattering rate, $\tau_i^{-1} = \bar{A} T^2 + B T$ with $B \neq 0$ (Uren et al 1981, with $\bar{A}$ and $B$ the coefficients of the quadratic and linear terms), in place of the pure $T^2$ form, and determine how the fitted conductance expression $\sigma = \sigma_0' + \beta[(1-F)\ln T + \ln(A + T^2)]$ and the extracted spin-orbit time are modified; (iv) extend the description beyond the weak-disorder regime in which HLN is derived (logarithmic corrections small compared to the Drude conductivity $\sigma_0$), i.e. explore whether these Si inversion layers cross over toward strong localization; and (v) incorporate the expected strong anisotropy of spin-orbit scattering in the inversion layer, $\tau_{so}^x \approx \tau_{so}^y \gg \tau_{so}^z$ (all carriers occupy the lowest subband for motion perpendicular to the layer), in place of a single isotropic spin-orbit time. Items (ii)-(iii) reanalyze the Davies-Pepper-Kaveh (1983) conductance data in order to establish whether the spin-orbit explanation of the non-scaling survives once the imposed equality $\tau_s = \tau_{so}$ is dropped and the measured linear inelastic term is included, and how the fitted conductance expression and the extracted spin-orbit time change; the Davies-Pepper-Kaveh (1983) conductance data and the Uren et al (1981) coefficients $\bar{A}$ and $B$ are not reproduced here and must be obtained (and, where given only graphically, digitized) from those cited sources. Items (i), (iv) and (v) are further assumptions in the same closing list — the sample-dependence of $A$ checked within that fit, the extension beyond the weak-disorder regime toward strong localization, and the anisotropic spin-orbit treatment.

References

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