Landau Khalatnikov Research Articles

Multiphoton Amplitudes and Generalized LKF Transformation in Scalar QED Using the Worldline Formalism

We apply the worldline formalism to scalar quantum electrodynamics (QED) to find a Bern–Kosower type master formula for generalized Compton scattering, on-shell and off-shell. Moreover, we use it to study the non-perturbative gauge parameter dependence of amplitudes in scalar QED and, as our main result, find a simple non-perturbative transformation rule under changes of this parameter in x-space in terms of conformal cross ratios. This generalizes the well-known Landau–Khalatnikov–Fradkin transformation (LKFT). We also exemplify how this generilized LKFT works in perturbation theory.

Impact of Parasitic Capacitance and Ferroelectric Parameters on Negative Capacitance FinFET Characteristics

In this letter, we present a compact model and analyze the impact of key parameters on negative capacitance FinFET (NC-FinFET) device operation. The developed model solves FinFET device electrostatics and Landau–Khalatnikov equations self-consistently. An experimental NC-FinFET device is accurately modeled and the experimentally calibrated parameters are used to analyze the NC-FinFETs device performance and its dependence on several key parameters.

Clocked Domain Wall Logic Using Magnetoelectric Effects

A new digital device scheme based on the magnetoelectric coupling and automotion of the magnetic domain wall is proposed. A single device is composed of a ferromagnetic wire and two ferroelectric capacitors served as the input and the output. It is shown that with the initialization in the magnetic states, a single device and the corresponding majority gate can realize the NOT, NAND, and NOR functions. Furthermore, the device has the cascadability, input–output isolation, gain, and nonreciprocity. The device concepts are justified by the numerical calculation, including the Landau–Lifshitz–Gilbert equation for the magnetization dynamics, the Landau–Khalatnikov equation for the electric polarization dynamics, and the electrostatic equations for the open-circuit output voltage and the charge sharing process between devices. The energy dissipation is also quantified and shown to be two to three orders of magnitude less than that in spin-torque-driven devices.

Analysis and Compact Modeling of Negative Capacitance Transistor with High ON-Current and Negative Output Differential Resistance—Part I: Model Description

We present an accurate and computationally efficient physics-based compact model to quantitatively analyze negative capacitance FET (NCFET) for real circuit design applications. Our model is based on the Landau–Khalatnikov equation coupled to the standard BSIM6 MOSFET model and implemented in Verilog-A. It includes transient and temperature effects, and accurately captures different aspects of NCFET. A comprehensive quasi-static analysis of NCFET in its different regions of operation is also performed using a simpler loadline approach. We also analyze the impact of ferroelectric and gate oxide thicknesses on the performance gain of NCFET over MOSFET.

Negative Capacitance Behavior in a Leaky Ferroelectric

We present a simulation study of the negative capacitance effect incorporating leakage through the ferroelectric (FE) negative capacitor. The dynamics of the FE is modeled using the Landau–Khalatnikov equation. When an FE and a dielectric are simply connected in series without a metal contact between them, the stabilization of negative capacitance remains unchanged irrespective of leakage. However, when a metal is used, any finite leakage through the FE makes it impossible to stabilize negative capacitance at the steady state. Nonetheless, when a voltage is applied, the series configuration enters the negative capacitance state and as long as the gate voltage is cycled faster than the time needed by the leakage current to discharge all the capacitors, the transistor shows improved subthreshold swing. These results are expected to provide insight into understanding and analyzing recent experimental results on negative capacitance.

Direct Observation of Negative Capacitance in Polycrystalline Ferroelectric HfO2

To further reduce the power dissipation in nanoscale transistors, the fundamental limit posed by the Boltzmann distribution of electrons has to be overcome. Stabilization of negative capacitance in a ferroelectric gate insulator can be used to achieve this by boosting the transistor gate voltage. Up to now, negative capacitance is only directly observed in polymer and perovskite ferroelectrics, which are incompatible with semiconductor manufacturing. Recently discovered HfO2‐based ferroelectrics, on the other hand, are ideally suited for this application because of their high scalability and semiconductor process compatibility. Here, for the first time, a direct measurement of negative capacitance in polycrystalline HfO2‐based thin films is reported. Decreasing voltage with increasing charge transients are observed in 18 and 27 nm thin Gd:HfO2 capacitors in series with an external resistor. Furthermore, a multigrain Landau–Khalatnikov model is developed to successfully simulate this transient behavior in polycrystalline ferroelectrics with nucleation limited switching dynamics. Structural requirements for negative capacitance in such materials are discussed. These results demonstrate that negative capacitance effects are not limited to epitaxial ferroelectrics, thus significantly extending the range of potential applications.

Switching-Speed Limitations of Ferroelectric Negative-Capacitance FETs

Recently, negative-capacitance FETs (NCFETs) have been proposed to reduce subthreshold slope and help continue supply-voltage scaling alongside channel-length scaling. We investigate the high-frequency switching behavior of NCFETs using the Landau–Khalatnikov equation to model ferroelectric materials. Multidomain interactions in the ferroelectric are considered, resulting in strong agreement with experimental measurements. Operation of NCFETs at gigahertz frequencies is investigated with this experimentally validated multidomain model. We find that the effectiveness of the voltage amplification in NCFETs is strongly dependent on the viscosity coefficient $\rho $ of the ferroelectric, and that a low $\rho $ ( $ ) is required for the operation at the high gigahertz frequencies.

Features of the low-frequency polarization response in the region of the ferroelectric phase transition in multiferroic TbMnO3

The unusual behavior of the low-frequency (10 Hz–1 MHz) permittivity in single crystals of ferroelectric multiferroic TbMnO3 has been experimentally and theoretically studied in detail in the region of the narrow temperature peak of the permittivity, associated with the ferroelectric phase transition (TC ~ 27.4 K). It has been found that the ec′(ω, T) maximum sharply decreases with increasing measured field frequency, while the temperature position of the maximum is independent of frequency. It has been shown that the observed features of the polarization response can be satisfactorily described within the Landau–Khalatnikov polarization relaxation theory.

Effect of Ferroelectric Damping on Dynamic Characteristics of Negative Capacitance Ferroelectric MOSFET

Static Landau theory has been widely used to confirm negative capacitance ferroelectric MOSFET (FeFET) as a low power switch with sub-60-mV/decade subthreshold swing. However, dynamic switching considering damping effect in ferroelectrics is rarely evaluated. Based on Landau-Khalatnikov theory, dynamic FeFET characteristics are obtained. For experimentally extracted Pb(Zr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.2</sub> Ti <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.8</sub> )O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> (PZT) damping constant ξ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FE</sub> = 75 Ω · m, the amplified gate capacitance induced by negative capacitance occurs when frequency is below a few MHz for large ξ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FE</sub> . For transient simulation, I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DS</sub> exhibits a slow response to the rapid change of V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> so that the hysteretic I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DS</sub> -V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> curve is generated. Furthermore, inverter prediction with complementary FeFETs shows that the dynamic short circuit switching power is much higher than the static one. These results are attributed to the large PZT response time in comparison with the operation period. For a guidance on ferroelectric selection and improvement, a wide range of damping constants are used to predict the corresponding maximum operation frequency, within which the dynamic and static responses of FeFETs are the same. It indicates that the ferroelectrics with low damping constant are essential to utilize FeFET advantages in high-frequency circuit applications.

Analytical drain current model for long-channel gate-all-around negative capacitance transistors with a metal–ferroelectric–insulator–semiconductor structure

A carrier-based analytical drain current model was proposed for long-channel gate-all-around negative capacitance transistors with a metal–ferroelectric–insulator–semiconductor structure, which was derived by solving Poisson’s equation and a one-dimensional Landau–Khalatnikov equation. The electrostatic potential, gain of surface potential, and drain current were examined extensively by changing different device parameters, including the ferroelectric film thickness, channel radius, insulator layer thickness, and permittivity of the insulator layer. The device design methodologies are discussed in detail in this paper. A nonhysteretic transfer characteristic with a steep subthreshold swing (<60 mV/decade) was achieved at room temperature by optimizing the device parameters. The developed model is valid for all operation regions without any auxiliary variables or functions.

Physics-Based Circuit-Compatible SPICE Model for Ferroelectric Transistors

We present a SPICE model for ferroelectric transistors (FEFETs) based on time-dependent Landau–Khalatnikov equation solved self-consistently with the transistor equations. The model also considers depolarization fields due to non-ideal contacts. We experimentally characterize FE films to calibrate our model, based on which we analyze the device and circuit implications of FEFETs. We discuss the dependence of the ON current and gate capacitance of FEFETs on the FE thickness and FE material parameters. A ring oscillator analysis shows delay reduction up to 97% at iso-energy for FEFETs compared with MOSFETs at $V_{\mathrm{ DD}} V. FEFET-based SRAMs show 47%–68% larger read stability and 50%–57% lower access time, albeit with an increase in the write time.

Sub-kT/q Switching in Strong Inversion in PbZr0.52Ti0.48O3 Gated Negative Capacitance FETs

Hysteretic switching with a sub- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kT/q</i> steep slope (13 mV/decade at room temperature) is experimentally demonstrated in MOSFETs with PbZr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.52</sub> Ti <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.48</sub> O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> as a ferroelectric (FE) gate insulator, integrated on a silicon channel with a nonperovskite high- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> dielectric (HfO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) as a buffer interlayer. The steep switching is independent of drain bias. For the first time, sub- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kT/q</i> switching due to FE negative capacitance is observed not at low currents, but in strong inversion ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${I}_{d}\sim $ </tex-math></inline-formula> 100 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{A}/\mu \text{m}$ </tex-math></inline-formula> ). Steep switching in strong inversion provides an important point of consistency with the predictions of the Landau–Devonshire theory and the Landau–Khalatnikov equation.

A carrier-based analytical theory for negative capacitance symmetric double-gate field effect transistors and its simulation verification

A carrier-based analytical drain current model for negative capacitance symmetric double-gate field effect transistors (NC-SDG FETs) is proposed by solving the differential equation of the carrier, the Pao-Sah current formulation, and the Landau–Khalatnikov equation. The carrier equation is derived from Poisson’s equation and the Boltzmann distribution law. According to the model, an amplified semiconductor surface potential and a steeper subthreshold slope could be obtained with suitable thicknesses of the ferroelectric film and insulator layer at room temperature. Results predicted by the analytical model agree well with those of the numerical simulation from a 2D simulator without any fitting parameters. The analytical model is valid for all operation regions and captures the transitions between them without any auxiliary variables or functions. This model can be used to explore the operating mechanisms of NC-SDG FETs and to optimize device performance.

Quantitative analysis and prediction of experimental observations on quasi-static hysteretic metal–ferroelectric–metal–insulator–semiconductor FET and its dynamic behaviour based on Landau theory

Due to internal voltage amplification induced by the negative capacitance of ferroelectrics, the metal–ferroelectric–metal–insulator–semiconductor (MFMIS) FET has been widely investigated to explore its potential application in low power devices. Based on Landau theory and stability criterion, a simulation program is implemented and MFMIS structure is quantitatively analyzed. The results show that it can be appropriately designed for both integrated circuits and memory devices by tuning capacitances contributed by MOSFET dielectric stack and ferroelectrics. Our simulation results on electrical characteristics of ferroelectric devices agree well with both quasi-static and dynamic experimental observations. The influence of the ferroelectric/dielectric layer thickness and area as well as temperature on hysteretic polarization-electric field characteristic of a ferroelectric are successfully explained. For a C–V loop sweeping over the gate voltage in MFMIS, possible asymmetry in the accessible negative capacitance region is also interpreted. Moreover, experimentally observed reduction in the equivalent capacitance of the ferroelectric–dielectric bilayer at high frequency is confirmed by Landau–Khalatnikov theory based simulation. Our work provides a more complete and explicit analytical treatment to understand the effect of negative capacitance of a ferroelectric on device performance.

Continuous Theory of Switching in Geometrically Confined Ferroelectrics

A theory of ferroelectric switching in geometrically confined samples like thin films and multilayers with domain structure has been proposed. For that we use Landau–Khalatnikov (LK) equations with free energy functional being dependent on polarization gradients. In this case, the consistent theory can be developed as for thin ferroelectric films and multilayers the domain structure reduces to Fourier series in ferroelectric polarization. The specific calculations are presented for thin film ferroelectric with dead layers and ferro-/paraelectric multilayer. Our theory is generalizable to ferroelectrics and multiferroics with other geometries.

Abnormal dynamic behavior in a ferroelectric bilayer film

Based on the Landau–Khalatnikov equation of motion, the switching dynamic behavior (including switching time, hysteresis loop and remanent polarization) of a ferroelectric bilayer film with surface transition layer within each constituent thin film and an interfacial coupling between two thin films has been first studied. The results show that the serious lag phenomenon of the hysteresis loop of the bilayer film occurs due to the strong strength of the interfacial coupling. It is interesting to note that there is a critical point, which is an equilibrium point of two contrary actions of the surface transition layer and interfacial coupling. The abnormal behavior of the switching time and remanent polarization in the bilayer film is attributed to the competition of the surface transition layer and interfacial coupling.

Influence of electrodes on polarization-reversal characteristics of a ferroelectric thin film

The modified Landau-type free-energy expression and the Landau–Khalatnikov equation of motion are first used to study the influence of electrodes on the properties of polarization reversal in a ferroelectric thin film with surface transition layers. The combined influence of the electrodes and surface transition layers on the dynamic properties, including polarization-reversal distribution, hysteresis loops, and switching current of the ferroelectric thin films, is discussed in detail. The results show that the electrode materials can greatly affect the dynamic properties of a ferroelectric thin film.

Fluctuations Near the Critical Micelle Concentration. II. Ultrasonic Attenuation Spectra and Scaling

Based on a thermodynamic model of amphiphile solutions derived in the first part of the paper, the ultrasonic attenuation of such systems has been considered theoretically, including fluctuations of local concentrations and micelle sizes. At amphiphile concentrations smaller than the critical micelle concentration (cmc), scaling behavior in terms of the concentration distance to the cmc is predicted by theory, in fair agreement with experimental evidence. The scaling function in the sound attenuation below the cmc reveals the unsymmetric broadening in the spectra that clearly emerges from measurements when approaching the cmc. The shape of the scaling function corresponds to the experimental spectra of solutions with comparatively large cmc as well as with the relaxation spectral function of the unifying model of non-critical concentration fluctuations. Above the cmc, an additional relaxation term is predicted in correspondence with the Landau-Khalatnikov term in the sound attenuation of superfluid helium. This term is difficult to verify by measurements because, in the relevant frequency range, other processes may also contribute the ultrasonic attenuation spectra.

Influences of surface transition layer on switching time and coercive field of a ferroelectric thin film

Based on the Landau–Khalatnikov equation of motion, we first study the properties of polarization reversal in a ferroelectric thin film with surface transition layers, which can reflect a more realistic situation compared to the previous treatments of the same problem using the so-called extrapolation length. The influence of the surface transition layer on switching time and coercive field under a step electric field is discussed in detail. The results show that the surface transition layer plays a crucial role in the properties of polarization reversal in a ferroelectric thin film.

Analytical Landau-type model of polarization switching in ferroelectrics

The polarization switching properties of ferroelectric materials have been extensively studied, both experimentally and theoretically owing to their wide applications in industry and hence generating a need to gain a deeper understanding of factors affecting their switching behaviour. In this paper, an analytical Landau theory incorporating the Landau–Khalatnikov equation has been developed to describe the switching properties of second order ferroelectrics. Analytical expressions derived from exact results of Landau theory are used in modelling work carried out in this study for comparison with trends predicted by empirical laws used to describe polarization switching behaviour at high electric fields. As such this work has established the theoretical basis for empirical laws of polarization switching and developed simple numerical tools which can be quickly used to model trends in polarization switching behaviour of ferroelectrics.