Gate-tunable electron interaction in high-κ dielectric films
Svitlana Kondovych, Igor Luk'yanchuk, Tatyana I. Baturina, Valerii M. Vinokur
GGate-tunable electron interactionin high- κ dielectric films Svitlana Kondovych , Igor Luk’yanchuk , Tatyana I. Baturina , andValerii M. Vinokur University of Picardie, Laboratory of Condensed Matter Physics, Amiens, 80000, France University of Regensburg, Universit ¨atsstraße 31, Regensburg 93053, Germany A. V. Rzhanov Institute of Semiconductor Physics SB RAS, 13 Lavrentjev Avenue, Novosibirsk 630090, Russia Novosibirsk State University, Pirogova str. 2, Novosibirsk 630090, Russia Materials Science Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, Illinois 60637, USA * [email protected] + these authors contributed equally to this work ABSTRACT
The two-dimensional (2D) logarithmic character of Coulomb interaction between charges and the resulting logarithmicconfinement is a remarkable inherent property of high dielectric constant (high- κ ) thin films with far reaching implications. Mostand foremost, this is the charge Berezinskii-Kosterlitz-Thouless transition with the notable manifestation, low-temperaturesuperinsulating topological phase. Here we show that the range of the confinement can be tuned by the external gate electrodeand unravel a variety of electrostatic interactions in high- κ films. Our findings open a unique laboratory for the in-depthstudy of topological phase transitions and a plethora of related phenomena, ranging from criticality of quantum metal- andsuperconductor-insulator transitions to the effects of charge-trapping and Coulomb scalability in memory nanodevices. Introduction
High dielectric constant or high- κ
2D systems enjoy an intense experimental and theoretical attention, see Ref. [1] andreferences therein. The interest is motivated by high technological promise of these systems for fabrication of nanoscalecapacitor 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 ferroelectricsto strongly disordered thin metallic and superconducting films experiencing metal-insulator and superconductor-insulatortransitions, respectively . The profound application of the high- κ sheets is the charge trapping elements for flash memory enabling the storage of the multiple bits in a single memory cell, thus overcoming the scalability limit of a standard flashmemory. 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 chargeremains confined within the film. This ensures the electrostatic integrity and stability with respect to external perturbations andgives rise to the 2D character of the Coulomb interactions between the charges . Namely, the potential produced by thecharge, located inside the high- κ sheet of thickness d , sandwiched between media with κ a and κ b permeabilities, exhibits thelogarithmic distance dependence, ϕ ( ρ ) ∝ ln ( ρ / Λ ) , extending till the fundamental screening length of the potential dimensionalcrossover, Λ = κ d / ( κ a + κ b ) . A striking example of the 2D Coulomb behaviour is the phenomenon of superinsulation instrongly disordered superconducting films . There, in the critical vicinity of the superconductor-insulator transition, thesuperconducting film acquires an anomalously high κ , the Cooper pairs interact according to the logarithmic law, and thesystem experiences the charge Berezinskii-Kosterlitz-Thouless (BKT) transition into a state with the infinite resistance. Anothergeneral consequence of the logarithmic Coulomb interaction, is that the high- κ sheets exhibit the so-called phenomenon of theglobal Coulomb blockade resulting in a logarithmic scaling of characteristic energies of the system with the relevant screeninglength, which is the smallest of either Λ or the lateral system size. In the Cooper pair insulator, this manifests as the logarithmicscaling of the energy controlling the in-plane tunneling conductivity . In the CTM element, this is the logarithmic scalingof its capacitance.The screening length is a major parameter controlling the electric properties of the high- κ films. Thus, their applicationsrequire reliable and simple ways of tuning Λ which, at the same time, maintain robustness of the underlying dielectric propertiesof the system. As we show below, this is achieved by the clever location of the control gate. Adjusting the distance between the a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b igh- κ film and the gate, we vary the screening length of the logarithmic interaction and obtain a wealth of the electrostaticbehaviors at different spatial scales, enabling to control the scalability and capacitance of the system. In what follows wedescribe 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 substratewith the dielectric constant, κ b . A metallic gate is separated from the film by a layer of the thickness h with the dielectricconstant κ a , see Fig.1a. Figure 1.
System geometry and spatial distribution of electrostatic potential. (a) Thin film of thickness d with thedielectric constant κ is deposited on the substrate with the dielectric constant κ b . The metallic gate on top is separated from thefilm by the spacer of thickness h with the dielectric constant κ a . 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 thecharge generating the electric field; the z -axis is perpendicular to the film plane. (b) The electrostatic potential, ϕ , induced bythe charge e < ρ for different distances h between film and electrode. The values of ρ and h are taken in unitsof the characteristic length Λ , the potential ϕ is taken in units q / κ d where q = e / πε and ε is the vacuum permittivity. Thecurves are calculated for κ = , κ a = κ b =
4. (c) and (d) Electric field lines (white) and the color map of the electrostaticpotential induced by charge e < κ = κ a = κ b = z -axis perpendicular to the film’s plane, ( ρ , θ , z ) , is placed at thecharge location (Fig.1a). In very thin films, which are the main focus of our study, we disregard the distances smaller than thefilm thickness and thus consider ρ > d . The relevant physical characteristic scale controlling the electrostatic properties of thesystem is the screening length Λ . Then the Poisson equations defining the potential distribution created by the charge assumes he form:1 ρ ∂ ρ (cid:0) ρ∂ ρ ϕ (cid:1) + ∂ z ϕ = π q κ δ ( ρ , z ) , | z | < d / , (1)1 ρ ∂ ρ (cid:0) ρ∂ ρ ϕ a , b (cid:1) + ∂ z ϕ a , b = , | z | > d / . 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 ( ρ , z ) = δ ( ρ ) δ ( z ) / πρ is the 3D Dirac delta-function in the cylindrical coordinates, q = e and q = e / πε in CGS and SIsystems respectively, ε 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 / ϕ a = z = h + d / ρ (see Fig.1a) is given by U ( ρ ) = e ϕ ( ρ ) . For numerical calculationswe use typical values of parameters for a InO film deposited on the SiO substrate: the film dielectric constant, κ (cid:39) , thesubstrate dielectric constant, κ b =
4, and the dielectric constant for the air gap between the film and the gate, κ a =
1, seeRef. [2].
Results
Results of the numerical solution to Eqs. (1) are shown in Fig. 1b-d. The space coordinates are measured in units Λ defined inthe Introduction. Panels (c) and (d) illustrate the cross-section of the configuration of the electric field lines and the color mapof the electrostatic potential for two characteristic cases, without and with metallic gate respectively. For illustration purposeswe assumed in panels (c) and (d) κ =
100 and symmetric properties of the upper and lower dielectric media, κ a = κ b . It can beimmediately 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 structure and different distances to the gate. One sees how thepotential 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 (cid:29) κ a , κ b the potential is given by (see Methods): ϕ ( ρ ) = q κ d ∞ (cid:90) J ( k ρ ) k + κ a coth ( kh )+ κ b κ d dk . (2)Here J is the zero order Bessel function. Shown in Fig. 2a is the semi-log plot of the potential vs. the distance calculated forthe same parameters as in Fig. 1b. We clearly observe the change of behaviour from the logarithmic one to the fast decay atlonger distances. The corresponding screening length at which the crossover occurs, Λ ∗ , is evaluated via the abscissa sectionby the straight line corresponding to ϕ ( ρ ) ∝ ln ( ρ / Λ ∗ ) at small ρ . Plotting Λ ∗ vs. h in a double-log scale (Fig. 2b) we find Λ ∗ ∝ √ h at h (cid:46) − Λ . At larger h , the Λ ∗ ( h ) dependence starts to deviate from the square root behaviour, and, eventually, atsufficiently large h the influence of the gate vanishes and Λ ∗ saturates to Λ . Inspecting more carefully the transition regionaround h ∼ − Λ , one observes that the functional dependence of the screened potential changes its character. At these scalesthe potential is pretty well described as ϕ ( ρ ) ∝ exp ( − ρ / Λ ∗ ) with the same Λ ∗ ∝ √ h (see Fig. 2a) at h (cid:46) − Λ . At h (cid:38) − Λ the potential decays as a power ϕ ( ρ ) ∝ ρ − n , with n (cid:46) ρ > h and ρ < h , in which the exact formulae for ϕ ( ρ ) can be obtained. Considering possible relations between h and otherrelevant 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 coth ( kh ) (cid:39) : ϕ ( ρ ) = π q κ d Φ (cid:16) ρ Λ (cid:17) , (3)where Φ ( x ) = H ( x ) − N ( x ) is the difference of the zero order Struve and Neumann functions . Making use of the asymptotesfor Φ given in Methods we find that at short distances ρ < Λ one obtains logarithmic behavior of Eq. (3), while at largedistances the field lines leave the film and one has the 3D Coulomb decay of the potential.(B) For ρ > h we find ϕ ( ρ ) = π q κ d ξ − ξ (cid:104) ξ Φ (cid:16) ξ ρ Λ (cid:17) − ξ Φ (cid:16) ξ ρ Λ (cid:17)(cid:105) , where ξ , = ( κ a + κ b ) (cid:20) κ b ± (cid:113) κ b − κ a κ d / h (cid:21) . (4) igure 2. The electrostatic potential in the presence of the gate and the sketch of the regimes of electrostaticinteractions.
The material dielectric parameters are the same as in Fig. 1b. The distances are measured in units of thefundamental screening length Λ and the potential in units q / κ d . (a) Semi-log plots of the electrostatic potential of the pointcharge placed in the middle of the film as functions of the distance for various values of the spacer, h / Λ , increasing from thetop to the bottom. The straight dotted lines are fits to ∝ ln ( ρ / Λ ∗ ) dependencies at small distances from which we determine thescreening lengths Λ ∗ at different h . The dashed lines stand for the ∝ ρ − / exp ( − ρ / Λ ∗ ) dependencies, which provide prettyfair fits for the long-distance behaviour of ϕ ( ρ ) at small h (cid:46) − Λ . (b) The log-log plot of the Λ ∗ on h dependencedetermined from the data given in panel (a). At small separations between the gate and the film, h (cid:46) − Λ , the effectivescreening length follows the law Λ ∗ (cid:39) √ Λ h , at larger h the noticeable deviation from this dependence is observed and at h (cid:38) Λ it tends to Λ . (c) The map visualizing the different interaction regimes between charges in the h − ρ coordinates. Thegate-dominated regime takes place at ρ < h , i.e. above the dashed diagonal line. Below this line the interaction is only slightlyaffected by the gate. The regions with the logarithmic interaction, lying at small ρ are highlighted by the blueish colours. This2D logarithmic interaction becomes screened at distances beyond the screening length. The latter can acquire either of thevalues Λ , Λ or Λ , depending on the parameters of the system. In the screened regime, the charges interact either as 3D pointcharges (grayish region, on the right of the separating line Λ ) or as the gate-imaged electric dipoles (yellowish region, on theleft of Λ ). At very small gate separation the strong exponential screening takes place (the violet petal). Gray roman numeralsindicate the correspondence to analytical formulae in Table I. epending on h , the length-scaling parameters, ξ and ξ can be either the real numbers, if h > d κκ a / κ b , or the complexmutually conjugated numbers, if h < d κκ a / κ b . This leads to the different regimes of the potential decay (see Table 1) thatare controlled by the new screening lengths, Λ , = Λ / ξ , ( Λ < Λ ) in the former case and Λ = Λ / | ξ | = Λ / | ξ | in thelatter one. In particular, the logarithmic behaviour presented in sections (iii) and (vi) of Table 1, perfectly reproduces theresults of computations shown in Fig. 2a. For small h < d κκ a / κ b the empirical screening length Λ ∗ , acquires the form Λ = (cid:112) ( κ / κ a ) dh corresponding to the small- h square-root behaviour inferred from the curve of Fig. 2b. For h > d κκ a / κ b the logarithmic behaviour persists but with Λ ∗ = Λ , which saturates to Λ with growing thickness of the spacer, h , between thefilm and the gate.At large scales above Λ ∗ , the screened charge potential decays following the power law, ϕ ( ρ ) ∝ ρ − n , where the exponentvaries from n = n = ρ to Λ , Λ , and Λ , see Table 1. Finally, for thesmall spacer thickness, the power-law screening transforms into the exponential one, ϕ ( ρ ) ∝ q κ d (cid:113) π Λ ρ e − ρ / Λ , see Methods.This evolution is well seen in the Fig. 2a, as improving fits of the potential curves to the exponential dependencies (shown bydashed lines) upon decreasing h . ρ < h (i) ρ < Λ ϕ ( ρ ) (cid:39) − q κ d ln C ρ Λ (ii) ρ > Λ ϕ ( ρ ) (cid:39) q ( κ a + κ b ) ρ ρ > hh > d κκ a / κ b (iii) ρ < Λ < Λ ϕ (cid:39) − q κ d ln C ρ Λ (iv) Λ < ρ < Λ ϕ (cid:39) ( κ b − κ a κ d / h ) / q ρ (v) Λ < Λ < ρϕ (cid:39) κ b κ a qh ρ h < d κκ a / κ b (vi) ρ < Λ ϕ (cid:39) − q κ d ln C ρ Λ (vii) ρ > Λ ϕ (cid:39) κ b κ a qh ρ Table 1.
Regimes of the interaction.
There are two major regions, short distances, ρ < h , where interaction is only weaklyinfluenced by the gate (upper panel), and large distances, ρ > h , where the gate presence renormalizes the interaction (bottompanel). Logarithmic dependence on ρ appears below the respective screening lengths, Λ , Λ and Λ . Above these lengths thepotential decays according to the power law. The constant C = e γ (cid:39) . ... is the exponent of the Euler constant γ . Discussion
The above results describe a wealth of electrostatic regimes in which the high- κ sheets can operate depending on the distance tothe control gate. The interrelation between the regimes presented in the Table 1 is conveniently illustrated in Fig. 2c showing themap of the interaction regimes drawn for the InO/SiO heterostructure parameters. Note that the specific structure of the mapdepends on the particular values of the parameters of the system controlling the ratios between the different screening lengths Λ , Λ , Λ , and Λ . The lines visualizing these lengths mark crossovers between different interaction regimes. The gray romannumerals correspond to the regimes listed in the Table 1. The colors highlight the basic functional forms of interactions betweenthe charges. The bluish area marks the manifestly high- κ regions of the unscreened 2D logarithmic Coulomb interaction. Asthe distance to the gate becomes less than the separation between the interacting charges, the screening length restricting thelogarithmic interaction regimes renormalizes from Λ to either Λ or Λ . The line Λ delimits the large-scale point-like anddipolar-like interaction regimes. At very small h , a petal-shaped region appears in which the potential drops exponentially withthe distance at ρ > Λ .The implications of the tunability of the logarithmic Coulomb interactions are far reaching. The charge logarithmicconfinement is the foundation of the charge BKT transition. Thus tuning the range of the confinement offers a perfect laboratoryfor 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 . ddressing 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- ordipolar- 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: ϕ a = ∞ (cid:90) A ( k ) e − kz J ( k ρ ) dk + ∞ (cid:90) A ( k ) e kz J ( k ρ ) dk ; (5) ϕ = q κ ∞ (cid:90) e − k | z | J ( k ρ ) dk + ∞ (cid:90) B ( k ) e − kz J ( k ρ ) dk + ∞ (cid:90) B ( k ) e kz J ( k ρ ) dk ; ϕ b = ∞ (cid:90) D ( k ) e kz J ( k ρ ) dk . Making use the specified in the text electrostatic boundary conditions we get a system of linear equations for coefficients A , , B , and D : q κ + B + B e kd = A + A e kd , q κ + B − B e kd = κ a κ A − κ a κ A e kd , (6) q κ + B e kd + B = D , q κ − B e kd + B = κ b κ D , A + A e kh e kd = . In particularly, for B , we obtain: B , = − q κ β , (cid:0) β , + e kd (cid:1) β β − e kd , (7)with β = − κ b / κ + κ b / κ and β = tanh kh − κ a / κ tanh kh + κ a / κ . (8)We are interested in distances, ρ , larger than the film thickness d when the main contribution to integrals (5) is comingfrom k (cid:28) d − . Expanding (7) over the small parameter kd , assuming that κ (cid:29) κ a , κ b in (8) and substituting the resultingcoefficients B , into the integral for ϕ in (5) we obtain the expression (2). Integrals
Integral (2) can be evaluated using the standard table integral ∞ (cid:90) J ( ak ) k + z dk = π Φ ( az ) (9)(here z = x + iy is the complex variable) in two limit cases.i) In the limit ρ < h the main contribution to (2) comes from the high- k values and kh (cid:29)
1. Assuming coth ( kh ) (cid:39) ρ > h the main role is played by the low- k region, kh (cid:28)
1. Then coth ( kh ) (cid:39) / kh and the integral (2) can becalculated by partial fraction decomposition onto two integrals, ϕ ( ρ ) (cid:39) q κ d ξ − ξ ∞ (cid:90) ξ J ( k ρ ) k + ξ Λ − dk − ∞ (cid:90) ξ J ( k ρ ) k + ξ Λ − dk , (10)where ξ and ξ are the given by (4) solutions of the characteristic quadratic equation Λ ξ + γ b ξ + γ a h − =
0. Each of theseintegrals is of the type (9) that permit us to obtain (4). imit expansions
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Acknowledgements
This work was supported by ITN-NOTEDEV FP7 mobility program. The work by V.V. and partly I.L. was supported bythe U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division. The work of T.I.B. wassupported by the Ministry of Education and Science of the Russian Federation and by RSCF (project No 14-22-00143). T.I.B.acknowledge for financial support the Alexander von Humboldt Foundation. uthor contributions statement
T.I.B., I.L., and V.V. conceived the work. S.K., I.L., and V.V. performed calculations. T.I.B. contributed to analysing andpresenting the data. All authors contributed to writing the manuscript.
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