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Interaction Effects in Conductivity of a Two-Valley Electron System in High-Mobility Si Inversion Layers

We have measured the conductivity of high-mobility (001) Si metal-oxide-semiconductor field effect transistors (MOSFETs) over wide ranges of electron densities n=(1.8-15)x10^11cm^2, temperatures T=30mK-4.2K, and in-plane magnetic fields B=0-5T. The experimental data have been analyzed using the theory of interaction effects in the conductivity of disordered 2D systems. The parameters essential for comparison with the theory, such as the intervalley scattering time and valley splitting, have been measured or evaluated in independent experiments. The observed behavior of the conductivity, including its quasi-linear increase with decreasing T down to ~0.4K and its downturn at lower temperatures, is in agreement with the theory. The values of the Fermi- liquid parameter obtained from the comparison agree with the corresponding values extracted from the analysis of Shubnikov-de Haas oscillations based on the theory of magnetooscillations in interacting 2D systems.


I. INTRODUCTION
The beginning of the 80's witnessed a triumph of the one-parameter scaling theory of localization 1,2 and the theory of electron-electron interactions in disordered conductors (for a review, see Ref. 3). As a result, the peculiar low-temperature behavior of the conductivity of numerous low-dimensional systems has been successfully attributed to quantum interference effects (see, e.g., Ref. 4). Curiously, application of these ideas to the two-dimensional (2D) electron liquid in Si MOSFETs -one of the most ubiquitous 2D systems -remained a challenge for more than 25 years. In the early experiments 5,6 with low-mobility (µ ∼ 0.1 m 2 /Vs) structures in the regime of relatively high electron densities (n > 10 12 cm −2 ), a decrease of the conductivity with cooling has been observed, apparently in qualitative agreement with the ideas of weak localization 1,7 and electron-electron interactions 3 . However, the quantitative description of this behavior remained a problem 8 . The disagreement with the theory became qualitative with the advent of high-mobility (µ ≥ 2 m 2 /Vs) structures: the low−T conductivity of high-µ Si MOSFETs increased with decreasing temperature 9 , in striking contrast to the behavior of many other 2D systems. This "metallic" behavior of the conductivity is especially pronounced at low electron densities n ∼ (1 − 2) × 10 11 cm −2 , where a five-fold increase of σ(T ) with cooling was observed 10,11,12,13 . Later, the quasi-linear increase of σ(T ) was observed in many high-mobility 2D systems in the "dilute" regime, including GaAs heterostructures (both p-type 14,15,16,17 and n-type 18,19 ), Si/SiGe 20 , and AlAs 21 quantum wells.
The subsequent development of theory and experiment led to significant progress in our understanding of the low-temperature transport in high-mobility systems in the regime of low electron densities. In the mid-80's, the quasi-linear "metallic" dependence observed in the ballistic interaction regime T τ ≫ 1 (τ is the transport mean free time, here and below we set = k B = e = 1) was attributed 22 to weakening of the screening of the scattering potential with increasing T . Observation of a strong negative magnetoconductance induced by the in-plane magnetic field 23 and renormalization of the effective electron mass and g−factor in these structures 24,25 also suggested that the electron-electron interactions play an important role in this phenomenon. More recently, Zala, Narozhny, and Aleiner (ZNA) 26 developed a theory that took into account all interaction contributions to the conductivity, including the exchange ones. This theory offers a unified approach to both ballistic (T τ ≫ 1) and diffusive (T τ ≪ 1) interaction regimes by considering the quantum interference between electron waves scattered off a short-range random potential "dressed" by Friedel oscillations of the electron density. The theory was extended for the case of a long-range scattering potential by Gornyi and Mirlin (GM) 27 . The theories 26,27 naturally incorporate the Altshuler-Aronov results for the interaction corrections to the conductivity in the diffusive regime 3 .
For the diffusive regime, a more general approach to interacting systems based on the nonlinear σ-model has been developed by Finkel'stein 28 . Recently, the renormalization group (RG) equations of this theory 29,30 (obtained in the first order in 1 πσ and in all orders in interaction) have been compared with the conductivity of Si MOSFETs at low electron densities 29,31,32,33 .
The RG equations 28,29,30,34,35 describe the length scale (temperature) evolution of the resistivity and interaction parameters for a 2D electron system in the diffusive regime 28 .
However, at high electron densities, the temperature range corresponding to the diffusive regime shrinks. In contrast, the theory of interaction corrections 26,27 is applicable over a wider T range (that includes both ballistic and diffusive regimes) provided σ ≫ 1 and ∆σ/σ ≪ 1; these assumptions are well justified at high densities.
The theories 26,27 predict that the magnitude and sign of the the interaction correction ∆σ(T, B) is determined by the value of the Fermi-liquid parameter F σ 0 (which can be found by measuring the Shubnikov-de Haas (SdH) oscillations in weak magnetic fields perpendicular to the plane of a 2D structure 24,25 or the magnetoresistance in strong in-plane magnetic fields 36 ). In particular, it is expected that the σ(T ) dependence becomes "metallic" when F σ 0 is negative and its absolute value exceeds a certain threshold (see Sec. II).
The experimental studies of the conductivity in various low-carrier-density 2D systems in the ballistic (high-temperature) regime are in agreement with the ZNA and GM theories.
The "metallicity" in all these systems is enhanced at low n due to an increase of the absolute value of F σ 0 (see, e.g., Refs. 37,38,39). In Si MOSFETs, the interaction effects are especially strong due to the presence of two nearly degenerate valleys in the electron spectrum 29 .
This enhancement, however, diminishes if the temperature T becomes smaller than the valley splitting ∆ V and intervalley scattering rate τ −1 v . As a result, with lowering T , the "metallic" dependence of σ(T ) is expected to become weaker or even to be replaced with an "insulating" one. To the best of our knowledge, this behavior has not been observed for Si MOSFETs prior to our work. This paper aims to study in detail the conductivity of 2D electron liquid in high-mobility (001) Si MOSFETs over a wide temperature range (T = 0.03 − 4.2 K) that includes both the diffusive and ballistic regimes. In particular, we observed for the first time that the "metallic" increase of σ with cooling is followed by the downturn of σ(T ) at lower temperatures. For the purpose of comparison with the ZNA theory 26 , we studied the range of not-too-low densities, n = (1.8 − 15) × 10 11 cm 2 , where the temperature and magnetic field dependences ∆σ(T, B) can still be treated as small corrections to the Drude conductivity σ D . In principle, no fitting parameters are required for comparison with the theory, because we have measured F σ 0 , ∆ V , and τ v in independent experiments. However, below we take a slightly different approach: we obtain the F σ 0 (n) values from fitting the ∆σ(T, B ) dependences with the ZNA theory 26 , and show that these values are consistent with the corresponding values extracted from the analysis of SdH oscillations 25 . We have also revealed shortcomings of earlier analysis of ∆σ(T ), reanalyzed the available data, and compared the extracted values of F σ 0 (n) with corresponding values from other measurements. We conclude that the experimental data are well described by the theory of interaction corrections 26 at intermediate temperatures For a quantitative analysis at ultra-low temperatures (T ≤ 0.3 K), the interaction correction theory should be modified by taking into account finite intervalley scattering rates.
The paper is organized as follows. In Section II we briefly summarize the theoretical results 26 for the interaction corrections to the conductivity of a two-valley system. The experimental data are presented in Section III, along with the data analysis and discussion.
The summary is given in Section IV.

II. INTERACTION CORRECTIONS TO THE CONDUCTIVITY
A. Temperature dependence of the conductivity in zero magnetic field In the ZNA theory 26 , the corrections ∆σ ee to the Drude conductivity σ D = nτ /m b (m b ≈ 0.205m e is the electron band mass in (001) Si MOSFETs; for more detail, see discussion in section III A 3 and references therein) were calculated in both ballistic and diffusive regimes for all orders of the interaction strength and the leading order in 1/(E F τ ) and T /E F . In particular, the theory reproduces the Altshuler-Aronov correction 3 to the conductivity in the diffusive regime. To adapt the theoretical results 26 to the case of (001) Si MOSFETs, one should take into account that the electron spectrum in this system has two almost degenerate valleys 40 . In zero magnetic field, ∆σ ee (T ) for a system with two degenerate valleys in the absence of intervalley scattering can be written as follows 38 : (1) Here δσ C is the so-called "charge" contribution which combines Fock correction and the singlet part of Hartree correction, and δσ T is the "triplet" contribution due to the triplet part of Hartree term. The valley index can be considered as a pseudo-spin in multi-valley systems 29 , and the valley degeneracy determines the number of triplet terms due to the spin exchange processes between electrons in different valleys. For a system with two degenerate valleys, the total number of interaction channels is 4 × 4 = 16, among them 1 singlet and 15 triplet terms (for comparison, there are 1 singlet and 3 triplet terms for a single-valley system).
Below we assume that the scattering potential is short-ranged which is relevant to Si MOSFETs. According to Ref. 26, the charge term does not depend on the details of interactions: whereas the magnitude and sign of the triplet term is controlled by the Fermi-liquid parameter 26 F σ 0 : The functions f and t in Eqs. (2,3) describe the crossover between the diffusive (∆σ ee ∝ ln T ) and ballistic (∆σ ee ∝ T ) regimes; outside the crossover region, they change the value of ∆σ ee by only a few percent. The explicit expressions for these functions can be found in Ref. 26. The diffusive-ballistic crossover is expected over some temperature range near Equations (2,3) describe the quantum corrections in a system with the conductance σ ≫ 1/2π at temperatures well below the Fermi energy T ≪ (1 + F σ 0 ) 2 E F . The sign and the magnitude of ∆σ ee is controlled by the Fermi-liquid parameter F σ 0 . For a rough estimate, deeply in the ballistic regime the ln T terms and the crossover functions t and f in Eqs. (2,3) can be omitted. For example, at T τ = 10, the functions f and t contribute ≈ 4% and ≈ 11%, respectively, to the linear-in-T τ ballistic terms of Eqs. (2,3).
By neglecting these terms, we find that for a single-valley system the linear dependence ∆σ ee (T ) in the ballistic regime becomes "metallic" (dσ/dT < 0) at 3F σ 25, whereas for a system with two degenerate valleys, the "metallic" ∆σ ee (T ) dependences are expected for 15F σ 0 /(1 + F σ 0 ) < −1 or F σ 0 < −0.06. Thus, the valley degeneracy extends the range of F σ 0 (n), and hence, the range of carrier densities n where the conduction exhibits the "metallic" behavior.

B. Magnetoconductivity in the in-plane magnetic field
The in-plane magnetic field, being coupled mostly to electron spins, provides a useful tool for exploring the interaction effects in the low-temperature conductivity of Si MOSFETs 23 .
When the Zeeman energy E Z = g b µ B B (g b = 2 is the bare g-factor, µ B is the Bohr magneton) becomes much greater than T , the number of triplet terms that contribute to ∆σ ee (T ) is reduced from 15 to 7. Similar reduction of triplet terms is expected for a valley splitting ∆ V > T . These two effects have been accounted by the theory of interaction corrections 26,38 ; in the presence of the magnetic field and/or valley splitting the interaction correction to the conductivity can be expressed as follows 38 : where ∆σ ee (T ) is given by Eq.
(1). All the terms ∆σ Z (Z, T ) have a form if the relevant energies Z ≪ E F (Z stands for E Z , ∆ V , and combinations E Z ± ∆ V ). The explicit expressions for the functions K b and K d are given in Ref. 26. In particular, Eq. (6) describes the interaction-driven magnetoconductivity in the magnetic fields which are much weaker than the field of full spin polarization of a system. Below we will neglect the function m(0, T τ ; F σ 0 ) which describes the crossover between the ballistic and diffusive regimes: this function is numerically small and does not modify the value of ∆σ(Z, T ) outside the ballisticdiffusive crossover region by more then one percent.
It is worth mentioning that in the framework of the RG theory, the magnetoconductance can also be described by the Castellani-Di Castro-Lee formula 35,41 which is equivalent to Eq. (5) in the diffusive limit at ∆ V = 0. However, for the analysis of our magnetoconductivity data measured over a wide T -range that includes both diffusive and ballistic regimes, the interaction correction theory 26 is more appropriate than the RG theory 35 .
The interaction correction theory 26 (as well as the RG theory 29,30 ) does not take into account intervalley scattering. This approximation is valid when the intervalley scattering rate τ −1 v is much smaller than T . In the low-temperature limit T ≪ τ −1 v , the electron states in different valleys are completely intermixed at the time scale ∼ T −1 and the correction ∆σ ee (T ) for a two-valley system is expected to coincide with that for a single-valley system.
Since the interaction corrections to the conductivity ∆σ ee (T, B) depend on several parameters such as τ, F σ 0 , ∆ V and τ v , for testing the theoretical results, it is crucial to determine these parameters in independent measurements. This program is realized in the following section.

A. Sample Characterization
We have studied the temperature and magnetic field dependences of the conductivity for high-mobility (001) Si MOSFETs, which demonstrate the "metallic" quasi-linear σ(T ) dependences at intermediate temperatures over a wide range of electron densities n. In this paper we present the data for two (001) Si MOSFETs, Si6-14 and Si1-46, with the gate oxide thickness 190±20 nm and peak mobility µ(0.1K) ≈ 2 m 2 /Vs. The σ(T, B) dependences were measured over the temperature range T = 0.03 − 4.2 K using a standard low-frequency fourterminal technique. The measuring current was chosen sufficiently small (I ∼ 1 − 3 nA) to avoid overheating of electrons within this temperature range 42 . Our experimental setup allowed us to independently control the magnetic field normal to the plane of 2D layer (B ⊥ ≈ −1.5...+1.5 T) and the in-plane magnetic field (B ≈ −8...+8 T); this cross-magnetic field technique has been described in Ref. 43. Unless otherwise stated, an in-plane magnetic field of B = 0.02 T was applied to quench the superconductivity in the current/voltage contact pads and the gate electrode which are made of thin aluminum films.
1. Effective mass m * and g−factor. Interaction corrections to the magnetooscillations.
The electron density n, the electron temperature T , the effective electron mass m * , and g−factor g * have been found from measurements of SdH oscillations (see also Ref. 25). For fitting the oscillations, as the first step of the analysis, we have used Lifshitz-Kosevich (LK) formula 44 which is valid for non-interacting 2D electrons if the amplitude of oscillations is Here is the elastic quantum scattering time. Figure 1 shows that the temperature dependence of the amplitude of the first harmonic, A 1 (T ), is in agreement with Eq. (7) down to the lowest temperatures; this indicates that the electrons are not overheated (with respect to the thermal bath) by the bias current and/or noise.
The F σ 0 (r s ) values can be found from the measurements of the renormalized g-factor: πn is the dimensionless ratio of the Wigner-Seitz radius to the effective Bohr radius, a * B = κ/me 2 , κ = 7.7 is the average dielectric constant of Si and SiO 2 , and m = 0.19m e is the electron band mass in bulk Si 40,46 . In our experiments 25 , g * (n) was obtained from the analysis of SdH oscillations as the ratio of two quantities: the measured renormalized electron spin susceptibility χ * /χ b = g * m * /g b m b and the effective mass m * (χ b is the bare value of spin susceptibility). Observation of the beatings of SdH oscillations in crossed magnetic fields offers a straightforward (model-independent) method of finding g * m * 25,43 . On the other hand, an estimate of m * is based on a model-dependent analysis of the damping factor for the first harmonic of SdH oscillations, A 1 (T, B ⊥ = const).
According to the LK theory, the damping factor can be expressed as Our experiments show that ln A 1 (T, B ⊥ ) varies linearly with T within the experimental range Fig. 3 in Ref. 25); this however does not prove the applicability of the LK theory, which disregards the interaction effects. It was recently shown 45,47 that due to the interference between electron-electron and electron-impurity interactions, the damping factor acquires an additional term in both the diffusive and ballistic regimes where and for a system with two degenerate valleys.
In this case, the T −dependent correction to the Dingle temperature, δT D (T ) T D , is one half of the interaction correction to the conductivity 48 δσ(T ) σ D (this factor 1 2 originates from the difference between the interaction corrections to the momentum relaxation and quantum scattering times 49 ). We note that the empirical procedure used for finding m * in our earlier paper (Ref. 25  /m e = 0.205 ± 0.005, the value which we adopted throughout the paper 58 ; available measurements of the cyclotron resonance 59 do not contradict and do not refine this value. In principle, the aforementioned complete characterization of samples allows us to compare the ∆σ(T, B) dependences with the theory 26 without any fitting parameters (with a caveat that the theory does not take into account the intervalley scattering, see the discussion below). However, throughout this paper we adopt an equivalent, but more convenient procedure: for each electron density, F σ 0 (n) will be considered as a single parameter for fitting ∆σ(T, B), and these values of F σ 0 will be compared with the corresponding values obtained from the SdH oscillations 25 .

B. Temperature dependences of the conductivity at B = 0
The temperature dependences of the conductivity σ(T ) for sample Si6-14 are shown in Fig. 2. In these measurements, we applied a fixed B ⊥ = 0.1 T that is sufficient to suppress the temperature dependence of the WL correction in the studied temperature range. The σ(T ) dependences are non-monotonic for all studied densities [n = (1.8−15)×10 11 cm −2 for Si6-14 and n = (10 − 15) × 10 11 cm −2 for Si1-46]: a quasi-linear increase of σ with cooling, observed down to ∼ 0.5 K, is replaced at lower T with a decrease of σ. Note that in our previous experiments 37 , we observed a trend of σ(T ) saturation at T < 0.4 K rather than the decrease of the conductivity. One of the reasons for this might have been "heating" of electrons by high frequency noise: only after thorough filtering of all leads connected to the sample were we able to decrease the electron temperature down to ∼ 30 mK. Similar downturn of σ(T ), although at much lower temperatures, has been recently observed in high-µ GaAs FETs at low electron densities 60 . We note also that earlier, a downturn of σ(T ) was observed in Si-MOSFETs 61 but at much higher electron densities (n > 30 × 10 11 cm −2 , r s < 1.4) and at much higher temperatures T ∼ 10K (T τ ≈ 1). For such high densities the interaction corrections to σ become negative and the downturn of σ(T ) was related to the crossover between a "metallic" high-temperature σ(T ) dependence (that is due to electron-phonon and intersubband scattering effects) and resulting "insulating" low-temperature σ(T ) dependence (that is due to weak localization and negative interaction corrections contributions to σ(T )).
Below we use the following strategy for analyzing the ∆σ(T ) dependences. First, we find  Fig. 5. The T range available for fitting in this regime "shrinks" rapidly at low n: the growth of |F σ 0 | and decrease of E F with decreasing n, lead to violation of the condition T ≪ (1+F σ 0 ) 2 E F (e.g., at n = 1.8×10 11 cm −2 this occurs at temperatures above 2 K). This might be one of the reasons for the observed deviation of the high-temperature σ(T ) from the linear-in-(T /E F ) theory 26 . Also, the higher-order corrections might become significant at low n when ∆σ(T )/σ D ∼ 1 (see Fig. 2).
After finding the F σ 0 values (which are temperature-independent in the studied temperature range), we proceed with the analysis of the low−T part of the σ(T ) dependences, where the crossover from dσ/dT < 0 to dσ/dT > 0 was observed. We note that the crossover occurs when the temperature becomes smaller than two characteristic temperature scales -∆ V and where x = T τ v . This crossover function provides correct asymptotic limits for N triplet : 3 at Figure 2 shows that using this function, we can reasonably well describe the shape of experimental σ(T ) dependences for all studied electron densities.

D. Magnetoconductivity
To test the theoretical predictions on the magnetoconductivity (MC) induced by in-plane magnetic fields, we also measured the σ(B ) dependences at fixed T . Similar measurements have been performed in the past (see, e.g., Refs. 37,38,63), but no detailed comparison with the theory was carried out at that time. The MC for sample Si6-14 over the field range −4.5 < B < 4.5 T is shown for different densities and temperatures in Fig. 4. In these measurements, special care was taken to reduce the magnetic field component perpendicular to the plane of the structure: even a 1 • misalignment between the sample's plane and the mag- Again, as in the case of fitting the ∆σ(T ) dependences, the only adjustable parameter was the F σ 0 (n) value extracted for each density from fitting the MC at high temperatures (≈ 0.7 K) where the effects of valley splitting or intervalley scattering on ∆σ ee (T, B) can be neglected. Note that all the theoretical curves plotted in Fig. 4 for the same n were calculated for a fixed F σ 0 (n), i.e. neglecting possible dependence F σ 0 (B ). The detailed analysis of the spin susceptibility χ * ∝ g * m * in strong magnetic field, presented in Ref. 64, shows that the product g * m * decreases with an increase of B by as much as ∼ 20%. Our estimate shows that by ignoring the g * (B) dependence, we might reduce the value of |F σ 0 | by ∼ 10% (see below), which is close to the accuracy of extraction of F σ 0 from the data in strong magnetic fields. As B grows and/or n decreases, the data start deviating from the theoretical curves (see Fig. 4 d); this deviation can be attributed to the violation of the condition g b µ B B ≪ E F required for applicability of Eqs. (5,6).
The F σ 0 values obtained from fitting the ∆σ(T ) and ∆σ(B ) dependences with the theory 26 are shown in Fig. 5. For comparison, we have also plotted the F σ 0 values obtained from the analysis of SdH oscillations measured for sample Si6-14 using the theories 44 The F σ 0 values obtained on the basis of the interaction correction theory and plotted in Fig. 5 may be compared also with the values of F σ 0 = −γ 2 /(1 + γ 2 ) predicted by the RG theory and measured experimentally in Refs. 31,32. Extrapolation of our F σ 0 (r s ) data to lower densities, provides the value F σ 0 ≈ −0.5 at n ≈ 1 × 10 11 cm −2 (r s ≈ 8), which is smaller than the value F σ 0 = −0.31 (γ 2 = 0.45) predicted by the one-loop RG theory for the temperature T max corresponding to the ρ(T ) maximum 29,31,32 (e.g., T max ≈ 3 K for n = 1.2 × 10 11 cm −2 ).
The experimental test 31,32 of the RG theory was conducted at temperatures higher than that in the experiments described in the present work. Within the framework of the RG theory, the interaction parameter γ 2 is expected to increases with decreasing T 29,31,32 and, in principle, it can reach at T < 1K the value of ∼ 1 which corresponds to F σ 0 = −0.5. (Note that the factor-of-two increase of γ 2 (from 0.45 to 1) is beyond the range of the applicability of the one-loop RG theory.) Another problem is that the spin susceptibility χ * ∝ g * m * obtained from the SdH data is almost T -independent 64 , in contrast to the expected increase of γ 2 (and, hence, |F σ 0 | and g * ) with cooling. This contradiction can be resolved if the T -dependence of g * is exactly compensated by the opposite T -dependence of m * , so that χ * ∝ g * m * remains almost constant. The reason for this compensation is not clear and requires both experimental and theoretical studies.

IV. CONCLUSION
Our experiments show that the low-T behavior of the conductivity of high-mobility (001) Si MOSFETs is well described by the theory of interaction effects in systems with short-range disorder 26 . Over a wide range of intermediate temperatures (∆ V , τ −1 v , g b µ B B < T ≪ E F ), the interaction effects are strongly enhanced in Si MOSFETs due to the presence of two valleys in the electron spectrum. This factor, in combination with the interactiondriven renormalization of the Fermi-liquid parameter F σ 0 , leads to an increase of σ with decreasing T . At lower temperatures (T < ∆ V , τ −1 v , g b µ B B, E F ), the triplet contribution to ∆σ ee (T ) is significantly reduced due to valley splitting and/or intervalley scattering. As a result, the "metallic" behavior of σ is replaced with a more conventional, "insulating" behavior. The F σ 0 values obtained from fitting the experimental data with the theory 26 agree well with the F σ 0 data obtained from the analysis of SdH oscillations in these samples. However, it remains unclear how to reconcile the F σ 0 values obtained at low n from fitting the σ(T ) and SdH data by using the interaction correction theory with the corresponding values obtained within framework of the RG theory.
We emphasize that for the detailed analysis of the interaction-induced contributions to the conductivity, it is important to measure such parameters as the valley splitting and intervalley scattering rate in independent experiments. Finally, for a quantitative description of the interaction effects to the conductivity ∆σ ee at low temperatures, both the interaction correction theory and RG theory should be extended to the case of a finite intervalley scattering rate.