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Monte Carlo Simulation of Flash Memory Elements' Electrophysical Parameters
O.G. Zhevnyak1, V.M. Borzdov1, A.V. Borzdov1, A.N. Petlitsky2
1Belarusian State University,
Nezavisimosty Ave., 4, Minsk 220030, Belarus
2JSC "Integral" - "Integral" Holding Management Company,
Kazintsa str., 121A, Minsk 220108, Belarus
Received 18.10.2022
Accepted for publication 22.11.2022
Abstract
Operation of modern flash memory elements is based on electron transport processes in the channel of silicon MOSFETs with floating gate. The aim of this work was calculation of electron mobility and study of the influence of phonon and ionized impurity scattering mechanisms on the mobility, as well as calculation of parasitic tunneling current and channel current in the conductive channel of flash memory element. Numerical simulation during the design stage of flash memory element allows working out guidelines for optimization of device parameters defining its performance and reliability.
In the work such electrophysical parameters, characterizing electron transport, as mobility and average electron energy, as well as tunneling current and current in the channel of the flash memory element are studied via the numerical simulation by means of Monte Carlo method. Influence of phonon and ionized impurity scattering processes on electron mobility in the channel has been analyzed. It is shown that in the vicinity of drain region a sufficient decrease of electron mobility defined by phonon scattering processes occurs and the growth of parasitic tunneling current is observed which have a negative influence on device characteristics.
The developed simulation program may be used in computer-aided design of flash memory elements for the purpose of their structure optimization and improvement of their electrical characteristics.
Keywords: flash memory, electron mobility, electron scattering, tunneling current, Monte Carlo method.
DOI: 10.21122/2220-9506-2022-13-4-276-280
Адрес для переписки: Борздов В.М.
Белорусский государственный университет, пр-т Независимости, 4, г. Минск 220030, Беларусь e-mail: borzdov@bsu.by Для цитирования:
O.G. Zhevnyak, V.M. Borzdov, A.V. Borzdov, A.N. Petlitsky.
Monte Carlo Simulation of Flash Memory Elements'
Electrophysical Parameters.
Приборы и методы измерений.
2022. - Т. 13, № 4. - С. 276-280.
DOI: 10.21122/2220-9506-2022-13-4-276-280
Address for correspondence: Borzdov V.M.
Belarusian State University, Nezavisimosty Ave., 4, Minsk 220030, Belarus e-mail: borzdov@bsu.by
For citation:
O.G. Zhevnyak, V.M. Borzdov, A.V. Borzdov, A.N. Petlitsky.
Monte Carlo Simulation of Flash Memory Elements'
Electrophysical Parameters.
Devices and Methods of Measurements.
2022, vol. 13, no. 4, pp. 276-280.
DOI: 10.21122/2220-9506-2022-13-4-276-280
Моделирование электрофизических параметров элементов флеш-памяти методом Монте-Карло
111 2 О.Г. Жевняк , В.М. Борздов , А.В. Борздов , А.Н. Петлицкий
1 Белорусский государственный университет пр-т Независимости, 4, г. Минск 220030, Беларусь 2ОАО «Интеграл» - управляющая компания холдинга «Интеграл» ул. Казинца, 121А, г. Минск 220108, Беларусь
Поступила 18.10.2022 Принята к печати 22.11.2022
В основе функционирования современных элементов флеш-памяти лежат процессы переноса электронов в проводящем канале кремниевых МОП-транзисторов с плавающим затвором. Целью данной работы являлось проведение вычислительного эксперимента по расчёту подвижности электронов и изучению влияния на подвижность фононного рассеяния и рассеяния на ионизированной примеси, а также расчёт паразитного туннельного тока и тока в проводящем канале элемента флеш-памяти. Проведение вычислительного эксперимента на этапе разработки и проектирования элементов флеш-памяти позволит выработать рекомендации для оптимизации параметров прибора, определяющих быстродействие и надёжность его работы.
Путем численного моделирования электронного переноса в элементе флеш-памяти методом Монте-Карло рассчитаны такие электрофизические параметры, характеризующие перенос, как подвижность, средняя энергия электронов, а также плотность туннельного тока и тока в канале прибора. Изучено влияние процессов рассеяния на фононах и ионизированной примеси на подвижность электронов в канале. Показано, что вблизи области стока происходит существенное снижение подвижности электронов, обусловленное процессами рассеяния на фононах, а также наблюдается рост паразитного туннельного тока, что приводит к ухудшению рабочих характеристик прибора.
Разработанная программа моделирования может быть использована при компьютерном проектировании элементов флеш-памяти с целью оптимизации их конструкции и улучшения электрических характеристик.
Ключевые слова: флеш-память, подвижность электронов, рассеяние электронов, туннельный ток, метод Монте-Карло.
DOI: 10.21122/2220-9506-2022-13-4-276-280
Адрес для переписки: Борздов В.М.
Белорусский государственный университет, пр-т Независимости, 4, г. Минск 220030, Беларусь e-mail: borzdov@bsu.by
Address for correspondence:
Borzdov V.M.
Belarusian State University, Nezavisimosty Ave., 4, Minsk 220030, Belarus e-mail: borzdov@bsu.by
Для цитирования:
O.G. Zhevnyak, V.M. Borzdov, A.V. Borzdov, A.N. Petlitsky.
Monte Carlo Simulation of Flash Memory Elements'
Electrophysical Parameters.
Приборы и методы измерений.
2022. - Т. 13, № 4. - С. 276-280.
DOI: 10.21122/2220-9506-2022-13-4-276-280
For citation:
O.G. Zhevnyak, V.M. Borzdov, A.V. Borzdov, A.N. Petlitsky.
Monte Carlo Simulation of Flash Memory Elements'
Electrophysical Parameters.
Devices and Methods of Measurements.
2022, vol. 13, no. 4, pp. 276-280.
DOI: 10.21122/2220-9506-2022-13-4-276-280
Introduction
In state-of-the-art integrated circuits short channel silicon MOSFETs with floating gate are widely used as the basis of flash memory elements. Modern trends in development of flash memory elements are directed to the miniaturization of their active regions (see, for example, [1, 2]). Further miniaturization of flash memory elements is restricted by growth of parasitic tunneling currents and decrease of performance in data reading mode. High density of parasitic tunneling currents can lead to distortion of data saved in the flash memory. The value of parasitic tunneling current is determined by design features of the MOSFET and conditions causing heating of electron gas in the conductive channel of the device [2-5]. Device performance is directly related to electron mobility in its active region. Electron mobility in the conductive channel of a MOSFET is defined by electron scattering processes. The main electron scattering mechanisms in short channel silicon MOSFETs are phonon scattering and ionized impurity scattering.
The aim of the work was to calculate of electron mobility and study the influence of phonon and ionized impurity scattering mechanisms on the mobility, as well as to calculate of parasitic tunneling current and channel current in the conductive channel of flash memory element.
Simulated device structure
In the Figure 1 schematic cross-section of the simulated silicon MOSFET with floating gate is represented [5, 6]. Dimensions of the regarded transistor are as follows: channel length Lch = 0.2 nm, gate oxide thickness dox = 6 nm, tunnel oxide thickness dtun = 2 nm, floating gate thickness dfloat = 2 nm. Donor impurity concentration in the source and drain regions is Nd = 1026 m-3, acceptor impurity concentration in the substrate is NA = 1024 m3. The depth of the source and drain regions is dj = 100 nm. The gate bias VG and the drain bias VD are supposed equal to 2 V which is a typical value for data reading mode.
Electron mobility and the influence of phonon and ionized impurity scattering mechanisms on the mobility in regarded flash memory element is studied by means of numerical Monte Carlo charge carrier transport simulation. The density of parasitic tunneling current jtun and the density of channel current jch are calculated too. Algorithms and self-consistent procedures may be found in [7].
Figure 1 - Schematic cross-section of the flash memory element based on floating gate MOSFET
During the simulation the dependences of average values of electron concentration Ne, energy Eav and mobility ^ on the position along the conduction channel (X-axis) and into the depth of the channel (Z-axis) are calculated. The self-consistent procedure implies that electron concentration Ne(x, z) at every time step is used to solve Poisson equation and to calculate electric field strength in different points of the conduction channel, and also to calculate ionized impurity scattering rate. Acoustic and intervalley phonon as well as ionized impurity scattering processes are taken into account. Impact ionization process is also incorporated into the simulation algorithm according to Keldysh-type model discussed in [8]. However, calculations proof that its effect on transport properties is negligible for given transistor operation mode.
Acoustic phonon scattering rate is calculated according to the following expression [7, 9, 10]:
Wac = ^^ JW+Oe) (1 + 2aE), nh uj p
(1)
where Dac is the deformation potential for scattering on acoustic phonons; md is electron density-of-states effective mass for corresponding valley; u} is longitudinal sound velocity in silicon; p is silicon mass density; a is nonparabolicity coefficient.
Intervalley phonon scattering rate is defined as follows [7, 9, 10]:
D,
''J™3Z,j
W■■ = 1J Т2прй3
nj yN J +1
(2)
x^E ± ha,j -AEl} (l + 2a (e ± ha,j - AE l})),
x
where D y is the intervalley coupling constant; Zy is the number of possible final equivalent valleys for the transition; y is the phonon frequency and Ny is the number of phonons according to the Bose-Einstein statistics; AEy is the energy difference between the energy minima of initial and final valleys.
Considering transitions between equivalent valleys in silicon for g-type scattering the values of parameters are Z y = 1, AEy = 0, and phonon temperature is Ty = 537 K. For /-type scattering Zy = 4, AEy = 0, and phonon temperatures are T y = 686 K and Ty = 733 K.
Ionized impurity scattering rate is defined according to Brooks-Herring model [7, 9, 10]:
[w h =
N^-Jl e4 (1 + 2aE)
(e0eSi )2 W^d B?
1 + 4
E ^a
V Bt J
(3)
sSi is silicon relative permittivity; lD =
leSie 0 Ea
I Nie2
is Debye screening length for electrons with characteristic concentration N'e and average energy E'av,
Bt = h2/2( )2 md, El = ^0.75/nNA.
Results of the simulation and their discussion
In the Figure 2 simulated dependence of the relation between parasitic tunneling current and drift current in the conductive channel versus the relative position along the device channel is presented. In the Figures 3 and 4 calculated electron average energy and average mobility along the device channel are shown.
Figure 2 - Parasitic tunneling current to drift current ratio dependence versus relative position along the conductive channel
E, eV 0.8 h
0.6
0.4
0.2
0.25 0.5 0.75 x/Lch
Figure 3 - Average electron energy dependence versus the relative position along the conductive channel of the flash memory element
where NA is acceptor doping density at t-th section of the channel; Ea = E(1+aE); s0 is dielectric constant;
Figure 4 - Average electron mobility dependence on the relative position along the conductive channel: 1 - with account of both phonon and ionized impurity scattering; 2 - with account of only phonon scattering; 3 - with account of only ionized impurity scattering
Analysis of the dependences presented in the figures allow to conclude that in the part of the channel close to the drain region a sufficient growth of electron energy is observed. The latter leads to the growth of phonon scattering rates, which, in turn, causes a sufficient decrease of electron mobility. As well, the growth of parasitic tunneling current is observed. The phenomena are quite undesirable for normal operation of flash memory element.
The simulation results also show that the influence of every regarded scattering mechanism on electron mobility varies at different channel regions and is not uniform. As can be seen from the Figure 4, phonon scattering makes a decisive influence on the decrease of electron mobility in the part of the channel close to the drain region, while the influence of ionized impurity scattering is sufficiently reduced in this part of the channel. Such behavior is correlated to a rather steep increase of electron energy in this region.
To analyze variation of electron mobility along the device channel as well as into the depth of the
channel, in Figure 5 the spatial distribution of mobility is presented in the form of a two-dimensional dependence on x and z coordinates.
Figure 5 - Spatial distribution of electron mobility in the simulated flash memory element
As can be seen electron mobility decreases noticeably with the growth of coordinate value.
Conclusion
The dependences of parasitic tunneling current density, average electron energy and mobility versus the position along the channel of the flash memory element have been calculated for data reading mode. The calculation is made by means of Monte Carlo simulation of electron transport in the device.
It is shown that in the vicinity of drain region a sufficient decrease of electron mobility occurs. At the same time, the growth of parasitic tunneling current is observed, which can hinder the data reading process. Analysis of the influence of phonon and ionized impurity scattering processes on electron mobility in the channel of the device has revealed that both of these mechanisms make approximately equal effect on the mobility only in the origin of the channel. In the vicinity of the drain region phonon scattering processes have a determining influence on electron mobility and are responsible for significant decrease of its value and consequent lowering of the device performance.
The simulation program developed in this work may be used in computer-aided design of flash memory elements for the purpose of their structure
optimization and improvement of their electrical characteristics.
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