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非晶态材料二氧化铪在微电子器件中具有广泛应用, 理解其微观导热机制对于提升电子器件的性能和可靠性至关重要. 以往的研究大多基于分子动力学和单一准简谐格林-久保方法, 难以准确考虑低频振动模式的导热贡献. 本文基于准简谐格林-久保理论, 结合流体动力学外推法, 对不同有序度的非晶二氧化铪结构的热输运机制进行全面研究. 该方法可有效克服单一准简谐格林-久保方法中的有限尺寸问题. 理论预测表明, 非晶二氧化铪的热导率与微观结构有序度呈现弱相关性. 基于模态分析表明, 中低频振动模式对热导率具有显著贡献, 是单一准简谐格林-久保方法低估非晶二氧化铪热导率的主要原因. 同时, 本文基于非谐动态结构因子分离了传播子和扩散子对非晶二氧化铪导热的贡献, 计算表明扩散子在所有非晶二氧化铪结构导热中均占据主导作用. 然而, 传播子的导热贡献仍不可忽略, 其占比可高达20%以上, 且随着有序度的增大而增大.
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关键词:
- 热导率 /
- 非晶二氧化铪 /
- 准简谐格林-久保理论 /
- 流体力学外推方法
The amorphous hafnia has a wide range of applications in microelectronic devices, and understanding its microscopic thermal transport mechanism is crucial for improving the performance and reliability of electronic devices. Most previous studies are based on molecular dynamics and the single quasi-harmonic Green-Kubo method, which makes it difficult to accurately consider the contribution of low-frequency vibrational modes to thermal conductivity. In this paper, based on the quasi-harmonic Green-Kubo method combined with hydrodynamic extrapolation, the heat transport mechanism of amorphous hafnia structures with different degrees of ordering is comprehensively investigated. The method effectively overcomes the finite size issue inherent in the single quasi-harmonic Green-Kubo method. Theoretical predictions show that the thermal conductivity of amorphous hafnia exhibits a weak correlation with the degree of microstructural ordering. Modal analysis reveals that low- and mid-frequency vibrational modes significantly contribute to thermal conductivity, which is the main reason for the underestimation of the thermal conductivity of amorphous hafnia in the single quasi-harmonic Green-Kubo method. Furthermore, this paper separates the contributions of propagons and diffusons to the thermal conductivity of amorphous hafnia using the anharmonic dynamic structure factor, revealing that diffusons dominate the thermal conductivity of all amorphous hafnia structures. However, the contribution of the propagons to thermal conductivity remains significant, reaching over 20% and increasing with the degree of ordering.-
Keywords:
- thermal conductivity /
- amorphous hafnia /
- quasi-harmonic Green-Kubo theory /
- hydrodynamic extrapolation method
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图 1 生成非晶结构过程中势能和均方位移累计值随模拟时间的变化, 图中灰色虚线之前为熔融过程, 灰色虚线与黑色实线之间为在温度4000 K下的弛豫过程, 红色点划线之后为退火过程
Fig. 1. Variation of potential energy and accumulated mean square displacement with simulation time during the generation of amorphous structures, the melting process is before the gray dashed line, the relaxation process at 4000 K is between the gray dashed line and the black solid line, and the annealing process is after the red dotted line.
图 2 非晶HfO2在不同淬火速度下的(a)径向分布函数和(b)环数分布
Fig. 2. (a) Radial distribution function and (b) ring distribution of amorphous HfO2 at different quenching rates.
图 3 有限尺寸(2592个原子)和流体动力学外推法的非晶HfO2随温度变化的热导率, 图中图例的下标表示淬火速度, 误差棒为热导率标准差
Fig. 3. Thermal conductivity of amorphous HfO2 with temperature for finite size (2592 atoms) and the hydrodynamic extrapolation. The subscripts of the legend denote the quenching rate, error bar is standard deviations of thermal conductivity.
图 4 非晶HfO2热导率随温度变化关系, 图中散点是Panzer等[37]、Lee等[38]、Scott等[4]和Chaubey等[39]报道的非晶HfO2实验数据
Fig. 4. Temperature-dependent thermal conductivity of amorphous HfO2, scattered points in the figure are experimental data for amorphous HfO2 reported by Panzer et al.[37], Lee et al.[38], Scott et al.[4] and Chaubey et al.[39].
图 5 300 K温度下不同淬火速度非晶HfO2 (a) 热容量; (b) 弛豫时间; (c) 模态扩散率; (d) 参与比
Fig. 5. (a) Heat capacity; (b) lifetime; (c) modal diffusivity; (d) participation ratio of amorphous HfO2 with different quenching rates at 300 K.
图 6 (a)—(e) 淬火速度分别为5×1012, 1×1012, 5×1011, 1×1011, 5×1010 K/s的非晶HfO2的纵向非谐动态结构因子; (f)—(j) 为对应淬火速度下的横向非谐动态结构因子; 图中颜色条是根据(8)式计算的非谐动态结构因子强度, 蓝色直线是非谐动态结构因子在低频部分的线性拟合, 红色虚线是选取的截止频率
Fig. 6. (a)–(e) Longitudinal anharmonic dynamic structure factors of amorphous HfO2 with different quenching rates of 5×1012, 1×1012, 5×1011, 1×1011, and 5×1010 K/s; (f)–(j) transverse anharmonic dynamic structure factors at corresponding quenching rates. The colorbar indicates the strength of the anharmonic dynamic structure factor calculated by Eq. (8), the blue straight line is a linear fit of the anharmonic dynamic structure factor in the low frequency part, and the red dashed line is the selected cut-off frequency.
图 7 300 K温度下不同淬火速度非晶HfO2的传播子和扩散子对热导率的贡献, 图中的蓝色文字代表传播子对总热导率的贡献百分比
Fig. 7. Contributions of propagon and diffuson to thermal conductivity of amorphous HfO2 with different quenching rates at 300 K, the blue text in the figure represents the percentage contribution of propagon to the total thermal conductivity.
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