搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

基于声子水动力学方程分析全环绕栅极晶体管的瞬态热输运过程

刘哲 魏浩 崔海航 孙锴 孙博华

引用本文:
Citation:

基于声子水动力学方程分析全环绕栅极晶体管的瞬态热输运过程

刘哲, 魏浩, 崔海航, 孙锴, 孙博华

Analysis of GAAFET’s transient heat transport process based on phonon hydrodynamic equations

Liu Zhe, Wei Hao, Cui Hai-Hang, Sun Kai, Sun Bo-Hua
PDF
HTML
导出引用
  • 相较于经典的傅里叶定律, 声子水动力学模型在描述纳米尺度超快声子热输运中已经展现出显著优势. 全环绕栅极晶体管(GAAFET)通过三维沟道设计极大优化了电学性能, 但其纳米尺度特征也导致自热问题和局部过热的挑战. 基于此, 本文针对纳米尺度GAAFET器件内的声子热传输特性开展理论和数值模拟分析. 首先, 基于声子玻尔兹曼方程严格推导了声子水动力学模型和边界条件, 建立了基于有限元的数值求解手段, 针对新型的GAAFET器件, 分析了表面粗糙度、沟道长度、沟道半径、栅极电介质、界面热阻等因素对其热传输特性的影响规律. 研究结果表明, 本文构建的连续介质框架下基于声子水动力学模型及温度跳跃条件的非傅里叶热分析方法能够精确预测GAAFET内部非傅里叶声子导热过程, 并揭示声子阻尼散射和声子/界面散射的作用机制. 这项工作为进一步优化GAAFET的热可靠性设计, 提高其热稳定性和工作性能提供了重要的理论支持.
    Compared to the classical Fourier’s law, the phonon hydrodynamic model has demonstrated significant advantages in describing ultrafast phonon heat transport at the nanoscale. The gate-all-around field-effect transistor (GAAFET) greatly optimizes its electrical performance through its three-dimensional channel design, but its nanoscale characteristics also lead to challenges such as self-heating and localized overheating. Therefore, it is of great significance to study the internal heat transport mechanism of GAAFET devices to obtain the thermal process and heat distribution characteristics. Based on this, this paper conducts theoretical and numerical simulation analyses on the phonon heat transfer characteristics within nanoscale GAAFET devices. Firstly, based on the phonon Boltzmann equation, the phonon hydrodynamic model and boundary conditions are rigorously derived, establishing a numerical solution method based on finite elements. For the novel GAAFET devices, the effects of factors such as surface roughness, channel length, channel radius, gate dielectric, and interface thermal resistance on their heat transfer characteristics are analyzed. The research results indicate that the larger the surface roughness, the smaller the channel length and the channel radius, the larger the interface thermal resistance leads to the higher hot spot peak temperature. The non-Fourier heat analysis method based on the phonon hydrodynamic model and temperature jump condition within the continuous medium framework constructed in this paper can accurately predict the non-Fourier phonon heat conduction process inside GAAFET and reveal the mechanisms of resistive scattering and phonon/interface scattering. This work provides important theoretical support for further optimizing the thermal reliability design of GAAFET, improving its thermal stability, and operational performance.
      通信作者: 孙锴, sunkai1@ime.ac.cn ; 孙博华, sunbohua@binn.cas.cn
    • 基金项目: 国家自然科学基金(批准号: 62374173)和西安建筑科技大学优秀博士论文培育基金(批准号: 2023XYBPY006)资助的课题.
      Corresponding author: Sun Kai, sunkai1@ime.ac.cn ; Sun Bo-Hua, sunbohua@binn.cas.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 62374173) and the Excellent Doctoral Dissertation Cultivation Fund of Xi’an University of Architecture and Technology, China (Grant No. 2023XYBPY006).
    [1]

    Liu S Q, Li Q H, Yang C, Yang J, Xu L, Xu L Q, Ma J C, Li Y, Fang S B, Wu B C, Dong J C, Yang J B, Lu J 2022 Phys. Rev. Appl. 18 054089Google Scholar

    [2]

    Yoon J S, Rim T, Kim J, Meyyappan M, Baek C K, Jeong Y H 2014 Appl. Phys. Lett. 105 102105Google Scholar

    [3]

    Zhang Q Z, Yin H X, Meng L K, Yao J X, Li J J, Wang G L, Li Y D, Wu Z H, Xiong W J, Yang H, Tu H L, Li J F, Zhao C, Wang W W, Ye T C 2018 IEEE Electron Device Lett. 39 464Google Scholar

    [4]

    Belkhiria M, Echouchene F, Jaba N, Bajahzar A, Belmabrouk H 2021 IEEE Trans. Electron Devices 68 954Google Scholar

    [5]

    Belkhiria M, Alyousef H A, Chehimi H, Aouaini F, Echouchene F 2022 Thin Solid Films 758 139423Google Scholar

    [6]

    Rezgui H, Mukherjee C, Wang Y, Deng M, Kumar A, Müller J, Larrieu G, Maneux C 2023 IEEE Trans. Electron Devices 70 6505Google Scholar

    [7]

    Myeong I, Son D, Kim H, Shin H 2019 IEEE Trans. Electron Devices 66 4631Google Scholar

    [8]

    Chhabria V A, Sapatnekar S S 2019 20th International Symposium on Quality Electronic Design (ISQED). pp235–240

    [9]

    Alvarez P T 2018 Thermal Transport in Semiconductors: First Principles and Phonon Hydrodynamics (1st Ed.) (Switzerland: Springer) pp41–71

    [10]

    Yang N, Zhang G, Li B W 2010 Nano Today 5 85Google Scholar

    [11]

    Guo Y Y, Wang M R 2016 J. Comput. Phys. 315 1Google Scholar

    [12]

    Ran X, Guo Y Y, Wang M R 2018 Int. J. Heat Mass Transfer 123 616Google Scholar

    [13]

    Zhang C, Guo Z L 2021 Int. J. Heat Mass Tranfer 181 121847Google Scholar

    [14]

    Cattaneo C 1948 Atti Sem. Mat. Fis. Univ. Modena 3 83

    [15]

    Vernotte P 1958 Comptes Rendus 246 3154

    [16]

    Tzou D Y 1995 J. Heat Transfer 117 8Google Scholar

    [17]

    Xu M T, Wang L Q 2005 Int. J. Heat Mass Transfer 48 5616Google Scholar

    [18]

    Chen G 2002 J. Heat Transfer 124 320Google Scholar

    [19]

    Cao B Y, Guo Z Y 2007 J. Appl. Phys. 102 053503Google Scholar

    [20]

    Dong Y, Cao B Y, Guo Z Y 2011 J. Appl. Phys. 110 063504Google Scholar

    [21]

    Guyer R A, Krumhansl J A 1966 Phys. Rev. 148 766Google Scholar

    [22]

    Guyer R A, Krumhansl J A 1966 Phys. Rev. 148 778Google Scholar

    [23]

    Alvarez F X, Jou D, Sellitto A 2009 J. Appl. Phys. 105 014317Google Scholar

    [24]

    Hua Y C, Cao B Y 2014 Int. J. Heat Mass Tranfer 78 755Google Scholar

    [25]

    Kaiser J, Feng T L, Maassen J, Wang X F, Ruan X L, Lundstrom M 2017 J. Appl. Phys. 121 044302Google Scholar

    [26]

    Guo Y Y, Wang M R 2018 Phys. Rev. B 97 035421Google Scholar

    [27]

    Beardo A, Hennessy M G, Sendra L, Camacho J, Myers T G, Bafaluy J, Alvarez F X 2020 Phys. Rev. B 101 075303Google Scholar

    [28]

    Rezgui H, Nasri F, Ali A B H, Guizani A A 2020 IEEE Trans. Electron Devices 68 10

    [29]

    Rezgui H, Nasri F, Ali A B H, Guizani A A 2021 Therm. Sci. Eng. Prog. 25 100938Google Scholar

    [30]

    Chen G 2005 Nanoscale Energy Transport and Conversion: a Parallel Treatment of Electrons, Molecules, Phonons, and Photons (New York: Oxford University Press) pp227–275

    [31]

    Peierls R E 1996 Quantum Theory of Solids (Oxford: Clarendon Press) pp45–52

    [32]

    Chen N X, Sun B H 2017 Chin. Phys. Lett. 34 020502Google Scholar

    [33]

    Aissa M F B, Nasri F, Belmabrouk H 2017 IEEE Trans. Electron Devices 64 5236Google Scholar

    [34]

    Sellitto A, Carlomagno I, Jou D 2015 Proc. R. Soc. A 471 20150376Google Scholar

    [35]

    Beardo A, Calvo-Schwarzwälder M, Camacho J, Myers T, Torres P, Sendra L, Alvarez F, Bafaluy J 2019 Phys. Rev. Appl. 11 034003Google Scholar

    [36]

    Zhang Z M 2020 Nano/microscale Heat Transfer (2nd Ed.) (Switzerland: Springer) p235

    [37]

    Mahajan S S, Subbarayan G, Sammakia B G 2011 IEEE Trans. Compon. Packag. Manuf. Technol. 1 1132Google Scholar

    [38]

    Chen G F, Hu B Y, Jiang Z L, Wang Z L, Tang D W 2023 Int. J. Heat Mass Tranfer 202 123676Google Scholar

    [39]

    Lai J H, Su Y L, Bu J H, Li B H, Li B, Zhang G H 2020 IEEE Trans. Electron Devices 67 4060Google Scholar

    [40]

    The Chinese Academy of Sciences 2022 Thermal Management of Electronic Devices p2 (in Chinses) [中国科学院 2022 电子设备热管理 (北京: 科学出版社) 第2页]

    The Chinese Academy of Sciences 2022 Thermal Management of Electronic Devices p2 (in Chinses)

    [41]

    程哲 2021 物理学报 70 236502Google Scholar

    Cheng Z 2021 Acta Phys. Sin. 70 236502Google Scholar

    [42]

    Jeong J, Choi S J, Shim J, Kim E, Kim S K, Kim B H, Kim J P, Suh Y, Beak W J, Geum D, Koh Y, Kim D, Kim S 2023 2023 International Electron Devices Meeting (IEDM) pp1–4

  • 图 1  (a)二维稳态声子导热问题的无量纲温度分布; (b)薄膜法向瞬态导热问题的无量纲温度分布. 符号为文献[11]的LBM结果, 实线为当前声子水动力学方程的模拟结果

    Fig. 1.  (a) Dimensionless temperature distribution of the two-dimensional steady-state phonon thermal conductivity problem; (b) dimensionless temperature distribution of the normal transient thermal conductivity problem of the thin film. The symbols represent the LBM results in Ref.[11], and the solid lines represent the simulation results of current phonon hydrodynamics equation.

    图 2  (a)三维GAAFET的结构示意图; (b)二维轴对称结构

    Fig. 2.  (a) Schematic diagram of three-dimensional GAAFET structure; (b) two-dimensional axisymmetric structure.

    图 3  $ t=30\; {\rm{ps}} $时刻GAAFET器件的表面温度分布 (a) $ p = $$ 0 $; (b) $ p = 0.1 $; (c) $ p = 1 $

    Fig. 3.  Temperature distribution of GAAFET devices at $ t=30\; {\rm{ps}} $: (a) $ p = 0 $; (b) $ p = 0.1 $; (c) $ p = 1 $.

    图 4  $ t=50\; {\rm{ps}} $时刻GAAFET器件的等温线图 (a) $ p = $$ 0.2 $; (b) $ p = 0.4 $; (c) $ p = 0.6 $

    Fig. 4.  2D Isotherm cloud maps of GAAFET devices at $ t=50\; {\rm{ps}} $: (a) $ p = 0.2 $; (b) $ p = 0.4 $; (c) $ p = 0.6 $.

    图 5  不同沟道长度的GAAFET器件的温度和热流密度变化 (a) 硅/氧化物界面中点处温度峰值的时间演变; (b) $ t= $$ 200\; {\rm{ps}} $时刻沿r方向中线上的径向热流密度分布

    Fig. 5.  Temperature and heat flux changes of GAAFET devices with different channel lengths: (a) Time evolution of temperature peak at the midpoint of the silicon/oxide interface; (b) radial heat flux distribution along the centerline in the r-direction at $ t=200\; {\rm{ps}} $.

    图 6  不同沟道半径的GAAFET器件的温度和热流密度变化 (a)硅/氧化物界面中点处温度峰值的时间演变; (b) $ t= $$ 50\; {\rm{ps}} $时刻沿氧化物-半导体界面沟道长度上的热流密度分布

    Fig. 6.  Temperature and heat flux variations of GAAFET devices with different channel radius. (a) Time evolution of peak temperature at the midpoint of the silicon/oxide interface; (b) the heat flux distribution along the channel length of the oxide semiconductor interface at time $ t=50\; {\rm{ps}} $.

    图 7  不同栅极电介质材料的GAAFET器件的温度云图 (a) Si-$ {\rm{SiO}}_2 $; (b) Si-$ {\rm{Al}}_2{\rm{O}}_3 $; (c) Si-$ {\rm{HfO}}_2 $

    Fig. 7.  Temperature distribution of GAAFET devices with different gate dielectric materials: (a) Si-$ {\rm{SiO}}_2 $; (b) Si-$ {\rm{Al}}_2{\rm{O}}_3 $; (c) Si-$ {\rm{HfO}}_2 $.

    图 8  $ t=100\; {\rm{ps}} $时不同热阻GAAFET沿r方向中线上的(a)温升和(b)径向热流密度变化

    Fig. 8.  (a) Temperature rise and (b) radial heat flux variation along the centerline of different thermal resistances GAAFET in the r-direction at $ t=100\; {\rm{ps}} $.

    表 1  Si和介电材料(SiO2, Al2O3, HfO2)的热物性参数[28,37]

    Table 1.  Thermal physical property parameters of Si and dielectric materials (SiO2, Al2O3, HfO2)[28,37]

    相关参数 Si $ {\rm{SiO}}_2 $ $ {\rm{HfO}}_2 $ $ {\rm{Al}}_2{\rm{O}}_3 $
    $ {v_{\rm{g}}}/({\rm{m{\cdot} s^{-1}}}) $ 3000 5900 4800 8800
    $ \kappa /({\rm{W{\cdot} m^{-1}{\cdot} K^{-1}}}) $ 150 1.4 1.0 1.5
    $C/{\rm{(10^6 \;J{\cdot} }}{{\rm{m}}^{\rm{-3}}}{\rm{{\cdot} K^{-1})}} $ 1.50 1.75 2.71 2.65
    $ \varLambda /{\rm{nm}} $ 100 0.4 0.23 0.19
    $ {\tau_{\rm{R}}} = \varLambda /v/{\rm{ps}} $ 33.3 0.068 0.048 0.02
    $ {R_{\rm{TBR}}}/{\rm{({m^2} {\cdot} K{\cdot} GW^{-1})}} $ 0.1 9.03 6.92
    下载: 导出CSV
  • [1]

    Liu S Q, Li Q H, Yang C, Yang J, Xu L, Xu L Q, Ma J C, Li Y, Fang S B, Wu B C, Dong J C, Yang J B, Lu J 2022 Phys. Rev. Appl. 18 054089Google Scholar

    [2]

    Yoon J S, Rim T, Kim J, Meyyappan M, Baek C K, Jeong Y H 2014 Appl. Phys. Lett. 105 102105Google Scholar

    [3]

    Zhang Q Z, Yin H X, Meng L K, Yao J X, Li J J, Wang G L, Li Y D, Wu Z H, Xiong W J, Yang H, Tu H L, Li J F, Zhao C, Wang W W, Ye T C 2018 IEEE Electron Device Lett. 39 464Google Scholar

    [4]

    Belkhiria M, Echouchene F, Jaba N, Bajahzar A, Belmabrouk H 2021 IEEE Trans. Electron Devices 68 954Google Scholar

    [5]

    Belkhiria M, Alyousef H A, Chehimi H, Aouaini F, Echouchene F 2022 Thin Solid Films 758 139423Google Scholar

    [6]

    Rezgui H, Mukherjee C, Wang Y, Deng M, Kumar A, Müller J, Larrieu G, Maneux C 2023 IEEE Trans. Electron Devices 70 6505Google Scholar

    [7]

    Myeong I, Son D, Kim H, Shin H 2019 IEEE Trans. Electron Devices 66 4631Google Scholar

    [8]

    Chhabria V A, Sapatnekar S S 2019 20th International Symposium on Quality Electronic Design (ISQED). pp235–240

    [9]

    Alvarez P T 2018 Thermal Transport in Semiconductors: First Principles and Phonon Hydrodynamics (1st Ed.) (Switzerland: Springer) pp41–71

    [10]

    Yang N, Zhang G, Li B W 2010 Nano Today 5 85Google Scholar

    [11]

    Guo Y Y, Wang M R 2016 J. Comput. Phys. 315 1Google Scholar

    [12]

    Ran X, Guo Y Y, Wang M R 2018 Int. J. Heat Mass Transfer 123 616Google Scholar

    [13]

    Zhang C, Guo Z L 2021 Int. J. Heat Mass Tranfer 181 121847Google Scholar

    [14]

    Cattaneo C 1948 Atti Sem. Mat. Fis. Univ. Modena 3 83

    [15]

    Vernotte P 1958 Comptes Rendus 246 3154

    [16]

    Tzou D Y 1995 J. Heat Transfer 117 8Google Scholar

    [17]

    Xu M T, Wang L Q 2005 Int. J. Heat Mass Transfer 48 5616Google Scholar

    [18]

    Chen G 2002 J. Heat Transfer 124 320Google Scholar

    [19]

    Cao B Y, Guo Z Y 2007 J. Appl. Phys. 102 053503Google Scholar

    [20]

    Dong Y, Cao B Y, Guo Z Y 2011 J. Appl. Phys. 110 063504Google Scholar

    [21]

    Guyer R A, Krumhansl J A 1966 Phys. Rev. 148 766Google Scholar

    [22]

    Guyer R A, Krumhansl J A 1966 Phys. Rev. 148 778Google Scholar

    [23]

    Alvarez F X, Jou D, Sellitto A 2009 J. Appl. Phys. 105 014317Google Scholar

    [24]

    Hua Y C, Cao B Y 2014 Int. J. Heat Mass Tranfer 78 755Google Scholar

    [25]

    Kaiser J, Feng T L, Maassen J, Wang X F, Ruan X L, Lundstrom M 2017 J. Appl. Phys. 121 044302Google Scholar

    [26]

    Guo Y Y, Wang M R 2018 Phys. Rev. B 97 035421Google Scholar

    [27]

    Beardo A, Hennessy M G, Sendra L, Camacho J, Myers T G, Bafaluy J, Alvarez F X 2020 Phys. Rev. B 101 075303Google Scholar

    [28]

    Rezgui H, Nasri F, Ali A B H, Guizani A A 2020 IEEE Trans. Electron Devices 68 10

    [29]

    Rezgui H, Nasri F, Ali A B H, Guizani A A 2021 Therm. Sci. Eng. Prog. 25 100938Google Scholar

    [30]

    Chen G 2005 Nanoscale Energy Transport and Conversion: a Parallel Treatment of Electrons, Molecules, Phonons, and Photons (New York: Oxford University Press) pp227–275

    [31]

    Peierls R E 1996 Quantum Theory of Solids (Oxford: Clarendon Press) pp45–52

    [32]

    Chen N X, Sun B H 2017 Chin. Phys. Lett. 34 020502Google Scholar

    [33]

    Aissa M F B, Nasri F, Belmabrouk H 2017 IEEE Trans. Electron Devices 64 5236Google Scholar

    [34]

    Sellitto A, Carlomagno I, Jou D 2015 Proc. R. Soc. A 471 20150376Google Scholar

    [35]

    Beardo A, Calvo-Schwarzwälder M, Camacho J, Myers T, Torres P, Sendra L, Alvarez F, Bafaluy J 2019 Phys. Rev. Appl. 11 034003Google Scholar

    [36]

    Zhang Z M 2020 Nano/microscale Heat Transfer (2nd Ed.) (Switzerland: Springer) p235

    [37]

    Mahajan S S, Subbarayan G, Sammakia B G 2011 IEEE Trans. Compon. Packag. Manuf. Technol. 1 1132Google Scholar

    [38]

    Chen G F, Hu B Y, Jiang Z L, Wang Z L, Tang D W 2023 Int. J. Heat Mass Tranfer 202 123676Google Scholar

    [39]

    Lai J H, Su Y L, Bu J H, Li B H, Li B, Zhang G H 2020 IEEE Trans. Electron Devices 67 4060Google Scholar

    [40]

    The Chinese Academy of Sciences 2022 Thermal Management of Electronic Devices p2 (in Chinses) [中国科学院 2022 电子设备热管理 (北京: 科学出版社) 第2页]

    The Chinese Academy of Sciences 2022 Thermal Management of Electronic Devices p2 (in Chinses)

    [41]

    程哲 2021 物理学报 70 236502Google Scholar

    Cheng Z 2021 Acta Phys. Sin. 70 236502Google Scholar

    [42]

    Jeong J, Choi S J, Shim J, Kim E, Kim S K, Kim B H, Kim J P, Suh Y, Beak W J, Geum D, Koh Y, Kim D, Kim S 2023 2023 International Electron Devices Meeting (IEDM) pp1–4

  • [1] 桑丽霞, 李志康. Au-TiO2光电极界面声子热输运特性的分子动力学模拟. 物理学报, 2024, 73(10): 103105. doi: 10.7498/aps.73.20240026
    [2] 陈星宇, 周昕, 白星, 余展, 王玉杰, 李欣家, 刘洋, 孙铭泽. 傅里叶鬼成像与正弦鬼成像的等价性分析. 物理学报, 2023, 72(14): 144202. doi: 10.7498/aps.72.20222317
    [3] 王钰豪, 刘建国, 徐亮, 成潇潇, 邓亚颂, 沈先春, 孙永丰, 徐寒杨. 傅里叶红外光谱气体检测限的定性分析. 物理学报, 2022, 71(9): 093201. doi: 10.7498/aps.71.20212366
    [4] 刘乃漳, 姚若河, 耿魁伟. AlGaN/GaN高电子迁移率晶体管的栅极电容模型. 物理学报, 2021, 70(21): 217301. doi: 10.7498/aps.70.20210700
    [5] 张书赫, 邵梦, 张盛昭, 周金华. 傅里叶域中的光线. 物理学报, 2019, 68(21): 214202. doi: 10.7498/aps.68.20190839
    [6] 包立平, 李文彦, 吴立群. 热传导系数跳跃的三维非Fourier温度场分布的奇摄动双参数解. 物理学报, 2019, 68(20): 204401. doi: 10.7498/aps.68.20190144
    [7] 尹灵康, 徐顺, Seongmin Jeong, Yongseok Jho, 王健君, 周昕. 广义等温等压系综-分子动力学模拟全原子水的气液共存形貌. 物理学报, 2017, 66(13): 136102. doi: 10.7498/aps.66.136102
    [8] 李红祺. 随机平衡设计傅里叶振幅敏感性分析方法和拓展傅里叶振幅敏感性分析方法在陆面过程模式敏感性分析中的应用探索. 物理学报, 2015, 64(6): 069201. doi: 10.7498/aps.64.069201
    [9] 于树海, 董磊, 刘欣悦, 凌剑勇. 傅里叶望远镜重构图像虚像分析. 物理学报, 2015, 64(18): 184205. doi: 10.7498/aps.64.184205
    [10] 刘永迪, 李虹, 张波, 郑琼林, 游小杰. 基于双重傅里叶级数的混沌SPWM频谱量化分析. 物理学报, 2014, 63(7): 070503. doi: 10.7498/aps.63.070503
    [11] 苏丽娜, 顾晓峰, 秦华, 闫大为. 单电子晶体管电流解析模型及数值分析. 物理学报, 2013, 62(7): 077301. doi: 10.7498/aps.62.077301
    [12] 强蕾, 姚若河. 非晶硅薄膜晶体管沟道中阈值电压及温度的分布. 物理学报, 2012, 61(8): 087303. doi: 10.7498/aps.61.087303
    [13] 顾江, 王强, 鲁宏. AlGaN/GaN 高速电子迁移率晶体管器件电流坍塌效应与界面热阻和温度的研究. 物理学报, 2011, 60(7): 077107. doi: 10.7498/aps.60.077107
    [14] 黄素娟, 王朔中, 于瀛洁. 共轭对称延拓傅里叶计算全息. 物理学报, 2009, 58(2): 952-958. doi: 10.7498/aps.58.952
    [15] 赵磊, 隋展, 朱启华, 张颖, 左言磊. 分步傅里叶法求解广义非线性薛定谔方程的改进及精度分析. 物理学报, 2009, 58(7): 4731-4737. doi: 10.7498/aps.58.4731
    [16] 段 鹤, 郑毓峰, 张校刚, 孙言飞, 董有忠. 水热法合成FeS2粉晶及其生长热动力学的研究. 物理学报, 2005, 54(4): 1659-1664. doi: 10.7498/aps.54.1659
    [17] 徐云, 张建峡, 杜世培. 动力学系统中非线性项的跳跃随机性. 物理学报, 1991, 40(1): 33-38. doi: 10.7498/aps.40.33
    [18] 邢修三. 晶体中热缺陷的产生动力学. 物理学报, 1988, 37(4): 694-697. doi: 10.7498/aps.37.694
    [19] 蔡履中, 张幼文. 彩虹全息图成象的傅里叶分析. 物理学报, 1982, 31(8): 1020-1029. doi: 10.7498/aps.31.1020
    [20] 成众志. 晶体管变频器分析. 物理学报, 1959, 15(10): 525-534. doi: 10.7498/aps.15.525
计量
  • 文章访问数:  256
  • PDF下载量:  14
  • 被引次数: 0
出版历程
  • 收稿日期:  2024-04-09
  • 修回日期:  2024-05-28
  • 上网日期:  2024-06-18
  • 刊出日期:  2024-07-20

/

返回文章
返回