{'fraction_non_alphanumeric': 0.049260415253087254, 'fraction_numerical': 0.021305468856018455, 'mean_word_length': 3.926470588235294, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Electron-electron interaction and tunnel density of states at the Fermi level in high-density two-dimensional electron system

The zero bias anomaly (ZBA) of the tunnel conductivity are investigated in the Al/δ – GaAs tunnel structure with 2D electron concentration in δ-layer to be equal to 3.6*1012 cm−2. We find that the dip (ZBA) in the tunnel density of states near the Fermi level EF reveals logarithmic dependence on the energy e in the range kT < ϵ< ℏ/r. Here ϵ is the energy relative to Ef and r is the elastic relaxation time of 2D electrons. The depth of the ZBA is proportional to ln(T/T0) in the range T = 0.1 – 20 K. These results are in agreement with Aronov-Altshuler theory in which the correction to the 2D density of states is due to the effects of the electron-electron interaction in diffusion channel. The unusual behavior of the tunnel magnetoconductivity is observed.


Introduction
Localization effects in the 2D electron system (2DES) in semiconductor structures are intensively investigated, see for review [1,2]. The main interest in such structures is study of the tunneling density of states (TDOS) near the Fermi level of 2DES. Theory [3] predicts manifestations of inter-electronic interaction in the regime of weak localization of carries when electrons concentration sufficiently high (k F l 1). In this case for the diffusion 2D channel in the TDOS ρ(ε) at the Fermi level E F of 2DES should be observed a considerable increase of the dip ∆ρ/ρ ∼ ln(kT τ/h), when energy |ε| = |E − E F | < kT and L ee = V F (hτ /kT ) 1/2 >> l. Here τ is the elastic relaxation time and l = V F τ , L ee is the phase relaxation length due to electronelectron collisions. Measurements of the TDOS, which confirmed results of [3], were made on the metal-films (see, for example [4,5]), where conditions of the "dirty limit" for metal conductivity is easy to realize. In the semiconductor 2D structures the corrections to the 2DES conductivity had been studied until now, see for example [6,7]. The "dirty limit" in the semiconductor structures with 2DES is well-satisfied in GaAs structures in which 2DES with Fermi energy 100 meV is situated in the self-consistent potential of the δ-layer [8][9][10]. In this work we present studies of the first investigations of TDOS for the semiconducting structure Al/δ-GaAs (k F l 10) in the temperature range 0.03 − 30 K and magnetic field up to 14 Tesla.  The inset shows curves σ (U)/σ (−2mV) for two cases: at B=0 -superconducting gap in Al gate and at B=0.16 T -tunnel zero bias anomaly, T=0.03K.
In our experiments, we were using the MBE grown Al /δ-GaAs tunnel structure. Al gate was grown in situ to achieve high quality of the Al/GaAs interface. δ-layer was grown at the distance of 20 nm from theAl/GaAs interface and the concentration of Si δ-doping was 7 * 10 12 cm −2 . Epitaxial GaAs was slightly unintentionally p-doped at the level of ∼ 10 16 cm −3 . The samples were prepared in the form of the Hall bars with the width of the 2D channel of 0.5 mm and the full length of 1.5 mm. We studied two types of samples. One of them had two tunnel gates of the length of 10 µm close to the ends of the Hall bar (sample "d"), another one (sample "e") had the gate length 1.3 mm. The ρ-contacts were between the gates in the first case and under the gate in the second one. The tunnel characteristics for the sample "d" are shown in Fig.1. It is known, that tunnel conductivity is σ ∝ D * ρ 2D , where D is the tunnel barrier transparency and ρ 2D is 2D density of states in the 2DES electrode. So the steps of ρ 2D (ε=-eU) can be observed in σ(U) dependence, as in Fig.1 [11]. These steps become the dips in tunnel spectrum. We are interested in the ZBA that is seen in Fig.1 as weak feature in σ(U) at U=0. The region of the ZBA, which associated with the TDOS near the Fermi level, is shown on Fig.  2. The inset in Fig.2 shows the superconducting gap in Al gate of Al/δ-GaAs. The gap is observed at T < 1.1 K and magnetic field B=0 that confirms the high quality of our Al/δ-GaAs structures and the possibility to measure effects in density of states at Fermi level in electrodes. Since TDOS near the Fermi level is symmetrical function of energy ε = E − E F , we will used symmetrized tunnel conductivity σ sym (Fig.2) to investigate the ZBA behavior.   Tunnel conductivity σ sym in magnetic field B perpendicular to δlayer plane for sample "e" at 0.03 K. Dashed linesσ bkg for different B.

Results and discussions
Firstly, we analyse the σ sym (U) dependence near the zero bias at different temperatures (see Fig.3). σ sym (U) can be presented as σ sym (U)=(∆σ sym + σ bkg ), where σ bkg is shown in Fig.2 at T=4.2 K.∆ σ sym (U) proportional to the change∆ ρ of TDOS near Fermi level and σ bkg (U ) const for the bias range 0 − 4 mV for the T ≤ 4 K. Consequently the dependencies in Fig.3 demonstrate∆ ρ as a function of U or energy ε. Aronov and Altshuler [3] have shown that∆ ρ/ρ ∼ ln(ετ/h) at kT<| ε |<h/τ and phase relaxation due to electron-electron collision. This dependence is in a good agreement with the data in Fig.3. The low limits of energy region of logarithmic dependence∆ ρ(U) are shown with arrows in Fig.3 for each temperature. The upper limit ish/τ ∼10 meV as follows from mobility value for our samples. The dependence of∆ σ sym (U = 0)/σ bkg ∝ ∆ρ(ε = 0)/ρ as a function of T is presented in Fig.4. One can see that the data correspond to expected dependence∆ ρ(ε = 0)/ρ ∝ ln(kT τ/h) [3]. At the same time the width of the dip near the Fermi level is linear function of T. This dependence is found at the range of temperatures 0.1 − 20 K. Note that the dip of TDOS∆ ρ(ε = 0) is equal to 20% of ρ at temperature 0.03 K in our experiment (Fig.4). Such result points out to possibility of localization of 2D carriers in our δ-layer structure at lower temperatures. Data for another sample with n 2D 10 12 cm −2 (see point curve in Fig.4) show that the strong localization can be achived even at 0.1 K. The investigation of such samples are in progress. Thus our results demonstrate that the phase relaxation time τ ϕ ∼ 1/T , i.e. τ ϕ is associated with electron-electron scattering in diffusion channel of our high density 2DES.
Unexpectedly the tunnel conductivity at low bias strongly depends on magnetic field B perpendicular to δ-layer, as can be seen in Fig.5. The positive magnetoconductivity (for σ bkg ) was observed at low bias voltages and negative one was found at | U |> 10 mV. At the same time position of the dip in TDOS does not depend on magnetic field. Additional experiments are needed for the explanation of this effect.

Conclusion
We performed the first direct spectroscopic measurements of TDOS near the Fermi level of the high density 2DES in GaAs structure. Logarithmic dependencies of TDOS were obtained in wide ranges of energies, ε = 0.02 − 5 meV, and temperatures T, T=0.1 − 20. The data are in a good agreement with the Aronov-Altshuler theory of the electron-electron interaction in disordered 2D conductors at k F l 1. We found also the non-monotonic tunneling magnetoconductivity for the magnetic fields perpendicular to the 2DES plane.

Acknowledgments
The work was supported by RFBR and RAS. We gratefully acknowledge A.S Bugaev and Yu. V. Fedorov for the sample preparation, and M.N. Feiginov for self-consistent calculations of the tunnel spectrum.