Introduction

High dielectric constant or high-κ 2D systems enjoy an intense experimental and theoretical attention, see ref. 1 and references therein. The interest is motivated by high technological promise of these systems for fabrication of nanoscale capacitor components and for design of the novel memory elements and switching devices of enhanced performance. The high-κ devices comprise unprecedentedly wide spectrum of physical systems ranging from traditional dielectrics and ferroelectrics to strongly disordered thin metallic and superconducting films experiencing metal-insulator and superconductor-insulator transitions, respectively2,3,4,5,6,7. The profound application of the high-κ sheets is the charge trapping elements for flash memory8 enabling the storage of the multiple bits in a single memory cell, thus overcoming the scalability limit of a standard flash memory. The challenging task crucial to applications is establishing the effective tunability of charge-trapping memory (CTM) units allowing for controlling the strength and spatial scale of charge distribution.

The major feature of high-κ systems leading to their unique properties, is that the electric field induced by the trapped charge remains confined within the film. This ensures the electrostatic integrity and stability with respect to external perturbations and gives rise to the 2D character of the Coulomb interactions between the charges9,10,11. Namely, the potential produced by the charge, located inside the high-κ sheet of thickness d, sandwiched between media with κa and κb permeabilities, exhibits the logarithmic distance dependence, φ(ρ) ∝ ln(ρ/Λ), extending till the fundamental screening length of the potential dimensional crossover, Λ = κd/(κa + κb). A striking example of the 2D Coulomb behaviour is the phenomenon of superinsulation in strongly disordered superconducting films2,12,13,14. There, in the critical vicinity of the superconductor-insulator transition, the superconducting film acquires an anomalously high κ, the Cooper pairs interact according to the logarithmic law, and the system experiences the charge Berezinskii-Kosterlitz-Thouless (BKT) transition into a state with the infinite resistance. Another general consequence of the logarithmic Coulomb interaction, is that the high-κ sheets exhibit the so-called phenomenon of the global Coulomb blockade resulting in a logarithmic scaling of characteristic energies of the system with the relevant screening length, which is the smallest of either Λ or the lateral system size. In the Cooper pair insulator, this manifests as the logarithmic scaling of the energy controlling the in-plane tunneling conductivity12,15,16. In the CTM element, this is the logarithmic scaling of its capacitance.

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, κb. A metallic gate is separated from the film by a layer of the thickness h with the dielectric constant κa, see Fig. 1a.

Figure 1: System geometry and spatial distribution of electrostatic potential.
figure 1

(a) Thin film of thickness d with the dielectric constant κ is deposited on the substrate with the dielectric constant κb. The metallic gate on top is separated from the film by the spacer of thickness h with the dielectric constant κa. Interacting charges, e, are located in the middle of the film. The origin of the cylindrical coordinate system, ρ, θ, z, with ρ being the lateral coordinate, is chosen at the location of the charge generating the electric field; the z-axis is perpendicular to the film plane. (b) The electrostatic potential, φ, induced by the charge e < 0 as function of ρ for different distances h between film and electrode. The values of ρ and h are taken in units of the characteristic length Λ, the potential φ is taken in units q/κd where q = e/4πε0 and ε0 is the vacuum permittivity. The curves are calculated for κ = 104, κa = 1, κb = 4. (c) and (d) Electric field lines (white) and the color map of the electrostatic potential induced by charge e < 0 in the cross-sectional plane. Panel (c) displays the field and potential without the gate; panel (d) shows the same in the presence of the gate. In the panels (c) and (d) we take κ = 100, κa = 1, κb = 1.

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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, φa and φb are the potentials in the regions above and below the film, respectively, δ3(ρ, z) = δ(ρ)δ(z)/2πρ is the 3D Dirac delta-function in the cylindrical coordinates, q = e and q = e/4πε0 in CGS and SI systems respectively, ε0 is the vacuum permittivity. The electrostatic boundary conditions are φ = φa,b and κ∂zφ = κa,b∂zφa,b at z = ±d/2 at the film surfaces, and φa = 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 SiO2 substrate: the film dielectric constant, , the substrate dielectric constant, κb = 4, and the dielectric constant for the air gap between the film and the gate, κa = 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, κa = κb. 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/SiO2 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 the potential is given by (see Methods):

Here J0 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 at small ρ. Plotting Λ* vs. h in a double-log scale (Fig. 2b) we find at . 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 ~ 10−1Λ, one observes that the functional dependence of the screened potential changes its character. At these scales the potential is pretty well described as with the same (see Fig. 2a) at . At the potential decays as a power φ(ρ) ∝ ρ−n, with .

Figure 2: The electrostatic potential in the presence of the gate and the sketch of the regimes of electrostatic interactions.
figure 2

The material dielectric parameters are the same as in Fig. 1b. The distances are measured in units of the fundamental screening length Λ and the potential in units q/κd. (a) Semi-log plots of the electrostatic potential of the point charge placed in the middle of the film as functions of the distance for various values of the spacer, h/Λ, increasing from the top to the bottom. The straight dotted lines are fits to dependencies at small distances from which we determine the screening lengths Λ* at different h. The dashed lines stand for the dependencies, which provide pretty fair fits for the long-distance behaviour of φ(ρ) at small . (b) The log-log plot of the Λ* on h dependence determined from the data given in panel (a). At small separations between the gate and the film, , the effective screening length follows the law , at larger h the noticeable deviation from this dependence is observed and at 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, Λ1 or Λ3, 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 Λ2) or as the gate-imaged electric dipoles (yellowish region, on the left of Λ2). 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.

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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.

  1. A

     At distances less than the film-electrode separation, ρ < h, we assume that in Eq. (2) and recover the well-known result for the system without gate9,10,11:

    where Φ0(x) = H0(x) − N0(x) is the difference of the zero order Struve and Neumann functions17. Making use of the asymptotes for Φ0 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. B

    For ρ > h we find

Table 1 Regimes of the interaction.
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Depending on h, the length-scaling parameters, ξ1 and ξ2 can be either the real numbers, if , or the complex mutually conjugated numbers, if . This leads to the different regimes of the potential decay (see Table 1) that are controlled by the new screening lengths, Λ1,2 = Λ/ξ1,2 (Λ1 < Λ2) in the former case and Λ3 = Λ/|ξ1| = Λ/|ξ2| 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. For small the empirical screening length Λ*, acquires the form corresponding to the small-h square-root behaviour inferred from the curve of Fig. 2b. For the logarithmic behaviour persists but with Λ* = Λ1, which saturates to Λ with growing thickness of the spacer, h, between the film and the gate.

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 Λ1, Λ2, and Λ3, see Table 1. Finally, for the small spacer thickness, the power-law screening transforms into the exponential one, , see Methods. This evolution is well seen in the Fig. 2a, as improving fits of the potential curves to the exponential dependencies (shown by dashed lines) upon decreasing h.

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/SiO2 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, Λ1, Λ2, and Λ3. 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 Λ1 or Λ3. The line Λ2 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 > Λ3.

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 transitions2. 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.

Methods

Fourier transformation

We seek the solution of equations (1) in the form:

Making use the specified in the text electrostatic boundary conditions we get the system of linear equations for coefficients , and :

In particular, for B1,2 we obtain:

with

We are interested in distances, ρ, larger than the film thickness d when the main contribution to integrals (5) is coming from . Expanding (7) over the small parameter kd, assuming that in (8) and substituting the resulting coefficients B1,2 into the integral for φ in (5) we obtain the expression (2).

Integrals

Integral (2) can be evaluated using the standard table integral18

(here z = x + iy is the complex variable) in two limit cases.

  1. i

    In the limit ρ < h the main contribution to (2) comes from the high-k values and . Assuming we reduce (2) to (9) and obtain the expression (3).

  2. ii

    In the limit ρ > h the main role is played by the low-k region, . Then and the integral (2) can be calculated by partial fraction decomposition onto two integrals,

where ξ1 and ξ2 are the given by (4) solutions of the characteristic quadratic equation Λξ2 + γbξ + γah−1 = 0. Each of these integrals is of the type (9) that permit us to obtain (4).

Limit expansions

The asymptotic expansions of Φ0 as a function of the complex argument, z = x + iy, are found from the table properties of H0 and N017. When z → 0 the function Φ0 can be approximated as where is the exponent of the Euler constant. At large the Laurent series development is suitable over the whole complex plane except the vicinity of the imaginary axis z = iy, where the real part of this expansion vanishes and the non-analytic contribution prevails. The latter can be accounted for, by presenting ReΦ0(iy) via the Macdonald function K0, that is approximated as at .

Additional Information

How to cite this article: Kondovych, S. et al. Gate-tunable electron interaction in high-κ dielectric films. Sci. Rep. 7, 42770; doi: 10.1038/srep42770 (2017).

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