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Gate-tunable electron interaction in high-κ dielectric films

The two-dimensional (2D) logarithmic character of Coulomb interaction between charges and the resulting logarithmic confinement is a remarkable inherent property of high dielectric constant (high-κ) thin films with far reaching implications. Most and foremost, this is the charge Berezinskii-Kosterlitz-Thouless transition with the notable manifestation, low-temperature superinsulating topological phase. Here we show that the range of the confinement can be tuned by the external gate electrode and unravel a variety of electrostatic interactions in high-k films. We find that by reducing the distance from the gate to the film, we decrease the spatial range of the 2D long-range logarithmic interaction, changing it to predominantly dipolar or even to exponential one at lateral distances exceeding the dimension of the film-gate separation. Our findings offer a unique laboratory for the in-depth study of topological phase transitions and related phenomena that range from criticality of quantum metal- and superconductor-insulator transitions to the effects of charge-trapping and Coulomb scalability in memory nanodevices.

The screening length is a major parameter controlling the electric properties of the high-κ films. Thus, their applications require reliable and simple ways of tuning Λ which, at the same time, maintain robustness of the underlying dielectric properties of the system. As we show below, this is achieved by the clever location of the control gate. Adjusting the distance between the high-κ film and the gate, we vary the screening length of the logarithmic interaction and obtain a wealth of the electrostatic behaviors at different spatial scales, enabling to control the scalability and capacitance of the system. In what follows we describe the electrostatic properties of the generic high-κ device with the tunable distance to the control gate.

Model
We consider a point charge, e < 0, located in the middle of a high-κ film of the thickness d, deposited on a dielectric substrate with the dielectric constant, κ b . A metallic gate is separated from the film by a layer of the thickness h with the dielectric constant κ a , see Fig. 1a.
The origin of the cylindrical coordinate system with the z-axis perpendicular to the film's plane, (ρ, θ, z), is placed at the charge location (Fig. 1a). In very thin films, which are the main focus of our study, we disregard the distances smaller than the film thickness and thus consider ρ > d. The relevant physical characteristic scale controlling the electrostatic properties of the system is the screening length Λ . Then the Poisson equations defining the potential distribution created by the charge assumes the form: Here ϕ is the electric potential inside the film, ϕ a and ϕ b are the potentials in the regions above and below the film, respectively, δ 3 (ρ, z) = δ(ρ)δ(z)/2πρ is the 3D Dirac delta-function in the cylindrical coordinates, q = e and q = e/4πε 0 in CGS and SI systems respectively, ε 0 is the vacuum permittivity. The electrostatic boundary conditions are ϕ = ϕ a,b and κ∂ z ϕ = κ a,b ∂ z ϕ a,b at z = ± d/2 at the film surfaces, and ϕ a = 0 at z = h + d/2 at the interface with the electrode. Then, the energy of the interaction with the second identical electron located at the distance ρ (see Fig. 1a) is given by U(ρ) = 2eϕ(ρ). For numerical calculations we use typical values of parameters for a InO film deposited on the SiO 2 substrate: the film dielectric constant, κ  10 4 , the substrate dielectric constant, κ b = 4, and the dielectric constant for the air gap between the film and the gate, κ a = 1, see ref. 2.

Results
Results of the numerical solution to Eq. (1) are shown in Fig. 1b-d. The space coordinates are measured in units Λ defined in the Introduction. Panels (c) and (d) illustrate the cross-section of the configuration of the electric field lines and the color map of the electrostatic potential for two characteristic cases, without and with metallic gate respectively. For illustration purposes we assumed in panels (c) and (d) κ = 100 and symmetric properties of the upper and lower dielectric media, κ a = κ b . It can be immediately seen that introducing the gate localizes potential within the smaller h-dependent screening length Λ * < Λ . Panel (b) presents the ϕ(ρ) plots calculated for the realistic InO/SiO 2 structure and different distances to the gate. One sees how the potential acquires more and more local character as the gate approaches the film surface.
To investigate the ϕ(ρ) dependence inside the film in detail, we find the analytical solution to the system (1). For distances ρ larger than the film thickness d and for κ κ κ  , a b the potential is given by (see Methods): Here J 0 is the zero order Bessel function. Shown in Fig. 2a is the semi-log plot of the potential vs. the distance calculated for the same parameters as in Fig. 1b. We clearly observe the change of behaviour from the logarithmic one to the fast decay at longer distances. The corresponding screening length at which the crossover occurs, Λ *, is evaluated via the abscissa section by the straight line corresponding to ϕ ρ ρ At larger h, the Λ * (h) dependence starts to deviate from the square root behaviour, and, eventually, at sufficiently large h the influence of the gate vanishes and Λ * saturates to Λ . Inspecting more carefully the transition region around h ~ 10 −1 Λ , one observes that the functional dependence of the screened potential changes its character. At these scales the potential is pretty well To gain insight into the observed behaviours of the potential, we undertake the detailed analysis of two asymptotic cases, ρ > h and ρ < h, in which the exact formulae for ϕ(ρ) can be obtained. Considering possible relations between h and other relevant spatial scales, we derive, with the logarithmic accuracy, the asymptotic behaviours of ϕ(ρ) for corresponding sub-cases (see Methods for the details of calculations). Our findings are summarized in Table 1.
(A) At distances less than the film-electrode separation, ρ < h, we assume that  kh coth( ) 1 in Eq. (2) and recover the well-known result for the system without gate 9-11 : is the difference of the zero order Struve and Neumann functions 17 . Making use of the asymptotes for Φ 0 given in Methods we find that at short distances ρ < Λ one obtains logarithmic behavior of Eq. (3), while at large distances the field lines leave the film and one has the 3D Coulomb decay of the potential. 2 . This leads to the different regimes of the potential decay (see Table 1) that are controlled by the new screening lengths, Λ 1,2 = Λ /ξ 1,2 (Λ 1 < Λ 2 ) in the former case and Λ 3 = Λ /|ξ 1 | = Λ /|ξ 2 | in the latter one. In particular, the logarithmic behaviour presented in sections (iii) and (vi) of Table 1, perfectly reproduces the results of computations shown in Fig. 2a At large scales above Λ * , the screened charge potential decays following the power law, ϕ(ρ) ∝ ρ −n , where the exponent varies from n = 1 (3D Coulomb charge interaction) to n = 3 (dipole-like interaction), in accord with the computational results discussed above. Which of the scenarios is realized, depends on the ratio of ρ to Λ 1 , Λ 2 , and Λ 3 , see Table 1. Finally, for the small spacer thickness, the power-law screening transforms into the exponential  , at larger h the noticeable deviation from this dependence is observed and at  Λ h it tends to Λ . (c) The map visualizing the different interaction regimes between charges in the h− ρ coordinates. The gate-dominated regime takes place at ρ < h, i.e. above the dashed diagonal line. Below this line the interaction is only slightly affected by the gate. The regions with the logarithmic interaction, lying at small ρ are highlighted by the blueish colours. This 2D logarithmic interaction becomes screened at distances beyond the screening length. The latter can acquire either of the values Λ , Λ 1 or Λ 3 , depending on the parameters of the system. In the screened regime, the charges interact either as 3D point charges (grayish region, on the right of the separating line Λ 2 ) or as the gate-imaged electric dipoles (yellowish region, on the left of Λ 2 ). At very small gate separation the strong exponential screening takes place (the violet petal). Gray roman numerals indicate the correspondence to analytical formulae in Table 1.

Discussion
The above results describe a wealth of electrostatic regimes in which the high-κ sheets can operate depending on the distance to the control gate. The interrelation between the regimes presented in the Table 1 is conveniently illustrated in Fig. 2c showing the map of the interaction regimes drawn for the InO/SiO 2 heterostructure parameters. Note that the specific structure of the map depends on the particular values of the parameters of the system controlling the ratios between the different screening lengths Λ , Λ 1 , Λ 2 , and Λ 3 . The lines visualizing these lengths mark crossovers between different interaction regimes. The gray roman numerals correspond to the regimes listed in the Table 1. The colors highlight the basic functional forms of interactions between the charges. The bluish area marks the manifestly high-κ regions of the unscreened 2D logarithmic Coulomb interaction. As the distance to the gate becomes less than the separation between the interacting charges, the screening length restricting the logarithmic interaction regimes renormalizes from Λ to either Λ 1 or Λ 3 . The line Λ 2 delimits the large-scale point-like and dipolar-like interaction regimes. At very small h, a petal-shaped region appears in which the potential drops exponentially with the distance at ρ > Λ 3 .
The implications of the tunability of the logarithmic Coulomb interactions are far reaching. The charge logarithmic confinement is the foundation of the charge BKT transition. Thus tuning the range of the confinement offers a perfect laboratory for the study of effects of screening on the BKT transition and related phenomena. Most notably, adjusting the gate spacer, one can can regulate the effects of diverging dielectric constant near the metal-and superconductor-insulator transitions 2 . Addressing the technological applications, we envision a wide use of gate controlled electrostatic screening in the high-κ films-based flash memory circuits. The reduction of the Coulomb repulsion from the 2D long-range logarithmic to the point-or dipolar-and even to the exponential ones will crucially scale down the circuit size, increasing their capacity and reliability.