Phononic thermal conduction and thermal regulation in low-dimensional micro-nano scale systems: Nonequilibrium statistical physics problems from chip heat dissipation

Luo Tian-Lin; Ding Ya-Fei; Wei Bao-Jie; Du Jian-Ying; Shen Xiang-Ying; Zhu Gui-Mei; Li Bao-Wen

doi:10.7498/aps.72.20231546

Abstract

“Heat death”, namely, overheating, which will deteriorate the function of chips and eventually burn the device and has become an obstacle in the roadmap of the semiconductor industry. Therefore, heat dissipation becomes a key issue in further developing semiconductor. Heat conduction in chips encompasses the intricate dynamics of phonon conduction within one-dimensional, two-dimensional materials, as well as the intricate phonon transport through interfaces. In this paper, the research progress of the complexities of phonon transport on a nano and nanoscale in recent three years, especially the size dependent phonon thermal transport and the relationship between anomalous heat conduction and anomalous diffusion are summarized. Further discussed in this paper is the fundamental question within non-equilibrium statistical physics, particularly the necessary and sufficient condition for a given Hamiltonian whose macroscopic transport behavior obeys Fourier’s law. On the other hand, the methods of engineering the thermal conduction, encompassing nanophononic crystals, nanometamaterials, interfacial phenomena, and phonon condensation are also introduced. In order to comprehensively understand the phononic thermal conduction, a succinct overview of phonon heat transport phenomena, spanning from thermal quantization and the phonon Hall effect to the chiral phonons and their intricate interactions with other carriers is presented. Finally, the challenges and opportunities, and the potential application of phonons in quantum information are also discussed.

Keywords:

Authors and contacts

1.
Department of Materials Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
2.
School of Microelectronics, Southern University of Science and Technology, Shenzhen 518055, China
3.
Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
4.
International Quantum Academy, Shenzhen 518017, China

Corresponding author: Shen Xiang-Ying, shenxy@sustech.edu.cn ; Zhu Gui-Mei, zhugm@sustech.edu.cn ; Li Bao-Wen, libw@sustech.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 52250191, 12205138) and the Shenzhen Basic Research Genenal Project, China (Grant No. JCYJ20220530113206015).

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Figure 1. Microprocessor trend data during 1972–2020. Transistor (kilo unites), single-thread performance (SpecINT × 10³) frequency (MHz), typical power (Watts), number of logical cores, density of heat flow (W/cm²)^[4].

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Figure 2. Application scenario of thermal interface materials.

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图 4 一维体系热导率发散的理论研究　(a) 当非线性参数$ \beta =1.5 $时, 一维原子链的热导率随体系尺寸N的变化关系^[13]; (b) 基于不同模型时(FPU-$ \beta $模型, 不同参数的FK模型)$ JN $随原子链长度的变化关系^[15]; (c) 不同的单璧碳纳米管热导率随尺寸的变化关系^[16]; (d) 硅纳米线热导率随尺寸的变化关系^[17]; (e) 单根高分子链热导率随长度发散的理论结果^[18]

Figure 4. Diverged thermal conductivity in 1D systems: (a) Thermal conductivity $ \kappa $ for an FPU lattice with $ \beta =1.5 $ varies with the system size $ N $^[13]; (b) $ JN $ vs. the number of particles $ N $ for different models ($ J $-heat flux) ^[15]; (c) the thermal conductivity vs. tube length L in log-log scale for different tubes at 300 and 800 K^[16]; (d) thermal conductivity of SiNWs (with fixed transverse boundary condition) vs. longitude length $ L_z $^[17]; (e) thermal conductivity of single extended polymer chains of five polymers as a function of chain length ^[18].

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图 5 碳纳米管和氮化硼纳米管热阻随尺寸的变化关系　(a) 归一化的热阻随长度的变化关系, 插图为扫描电镜下悬空热桥法的测试装置^[22]; (b) 碳纳米管的归一化热阻随长度的变化关系(从热阻变化推断出热导率变化为$ \kappa \sim{L}^{0.6} $)^[22]; (c) 氮化硼纳米管的归一化热阻随长度的发散关系(从而得到$ \kappa \sim{L}^{0.5} $)^[22]

Figure 5. Normalized thermal resistance vs . normalized sample length for different samples: (a) The relations between normalized thermal resistance and sample length for carbon nanotubes (CNTs) and boron-nitride nanotubes (BNNTs)^[22]; (b) the relations between normalized thermal resistance and sample length for CNT^[22]; (c) the relations between normalized thermal resistance and sample length for BNNTs^[22].

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图 6 NbSe₃纳米线和Si_0.4Ge_0.6薄膜中的超扩散现象　(a) NbSe₃纳米线材料体系中的超扩散声子热输运实验发现^[24]; (b) Si_0.4Ge_0.6 薄膜中的超扩散热输运^[25]

Figure 6. Experimental evidence of superdiffusive behavior of thermal transport in aligned atomic chains and Si_0.4Ge_0.6 thin films: (a) Observation of superdiffusive phonon transport in NbSe₃ nanowire^[24]; (b) superdiffusive thermal transport in Si_0.4Ge_0.6 thin films^[25]

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图 7 扩散与热传导之间的对应关系^[8]

Figure 7. Connection between heat conduction and anomalous diffusion^[8].

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图 8 扩散方程中的$ \alpha $与热传导方程中$ \beta $之间的关系^[8]

Figure 8. Connection between anomalous diffusion and heat conduction^[8].

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图 9 一些典型二维材料的热导率和平面结构^[38]

Figure 9. Thermal conductivity and layer structure of some typical 2D materials^[38].

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图 10 二维材料反常热输运的理论研究　(a) 基于FPU-β模型的二维晶格热导率随尺寸变化关系^[41]; (b) 基于相互作用势为四次方模型的二维晶格热导率随尺寸变化关系^[41]; (c) 基于不同计算方法得到的二硫化钼热导率随尺寸的变化关系^[42]; (d) 石墨烯热导率随尺寸变化关系; (e) 硅烯热导率随尺寸变化关系^[43]

Figure 10. Anomalous thermal transport in two-dimensional material. (a) κ_GK(N) in the X direction vs. N in N_X ×N_Y lattices. FPU-β lattice. Inset: data plotted in double logarithmic scale. Solid line corresponds to N^0.25[41]. (b) Purely quartic lattices^[41]. (c) The calculated thermal conductivity of MoS₂ at 300 K as a function of sample size^[42]. (d) Length dependence of thermal conductivity of each phonon branch of graphene. (e) Length dependence of thermal conductivity of each phonon branch of silicene^[43].

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图 11 二维材料热导率发散的实验研究　(a) 基于热桥法测量得到的悬空单层石墨烯热导率^[11]; (b) 拉曼光热法得到的不同厚度MoS₂热导率随尺寸的变化关系; (c) 拉曼光热法得到的不同材料二维材料热导率随尺寸的变化关系^[10]

Figure 11. Thermal conductivity of 2D systems: (a) Experimental results on length-dependent thermal conductivity^[11]; (b) thermal conductivity of suspended momolayer/bilayer/trilayer MoS₂ obtained through Raman photothermal method as a function of sample size; (c) thermal conductivity of monolayer WS₂ (red color) and WSe₂ (orange color) obtained through Raman photothermal method as a function of sample size^[10].

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图 12 在室温下观察到三维硅超晶格的超低热导率　(a) 实验设置示意图. 通过一个时间延迟的极紫外探测光束, 在超快红外激光泵浦脉冲激发后, 监测超晶格表面镍光栅的热弛豫^[45]. (b) 超晶格结构的横截面电子显微镜图像. 约500 nm厚的超晶格薄膜由晶体硅组成, 其中穿插有36 nm周期性和约20 nm直径的FCC堆积孔隙, 导致孔隙率为0.385 ± 0.02^[45]. (c) 傅里叶定律(虚线红色)预测值和实验得到的超低热导率数值(灰色). 插图显示了块体硅导热率(黑色)、体积缩减的有效介质理论Eucken & Russell导热率(蓝色)以及明显的超晶格热导率(红色), 后者仅为块体的1%^[45]

Figure 12. Ultra-low thermal conductivity observed in three-dimensional silicon superlattices at room temperature. (a) Schematic diagram of the experimental setup. The thermal relaxation of the nickel grating on the superlattice surface is monitored after ultrafast infrared laser pump pulse excitation using a time-delayed extreme ultraviolet (EUV) probe beam^[45]. (b) cross-sectional electron microscope image of the superlattice structure. The approximately 500-nm-thick superlattice film is composed of crystalline silicon with 36-nm periodic and approximately 20-nm diameter FCC-stacked pores, resulting in a porosity of 0.385 ± 0.02^[45]. (c) Fourier’s law predicted values (dashed red line) and experimentally obtained ultra-low thermal conductivity values (gray). The inset shows the bulk silicon thermal conductivity (black), the volume-reduced Eucken & Russell thermal conductivity (blue), and the distinct superlattice thermal conductivity (red), the latter being only 1% of the bulk value^[45].

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图 13 高度受限纳米体系中声子“黏度”(viscosity)与声子晶体孔隙率(porosity)的统一理论^[45]

Figure 13. Unified theory of phonon “viscosity” and “porosity” in highly constrained nano-systems^[45].

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图 14 (a) 双层旋转石墨烯面内振动, 垂直方向振动对热传导的贡献和总的热导率随旋转角的变化^[71]; (b) 图(a)中(0°—5°)范围内放大的热导率变化^[71]; (c) 不同温度下热导率与旋转角的依赖关系^[71]; (d) 0°—5°转角下不同温度热导率随转角的变化关系^[71]; (e) 双层旋转石墨烯形成的莫尔晶格(Moire)和AA, AB, SP堆积结构的原子位置. 红/蓝颜色的原子对应下层和上层的碳原子^[71]

Figure 14. (a) Total, in-plane, and out-of-plane thermal conductivity of TBG varies as twist angle from 0° to 30° at 300 K^[71]; (b) total thermal conductivity of TBG versus twist angle below 5°^[71]; (c) total thermal conductivity of TBG versus with twist angle at temperatures 300, 400 and 500 K^[71]; (d) normalized thermal conductivity with respect to the value of the untwisted structure as a function of twist angle at 300, 400 and 500 K^[71]; (e) the Moiré lattice formed in TBG and the atomic arrangements of AA, AB, and SP stacks^[71].

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图 15 布拉格反射型声子晶体重声子在声子晶体的运动行为　(a) 处于禁带内的声子穿过周期结构逐渐消失^[72]; (b) 处于禁带外的声子能够穿过周期结构^[72]

Figure 15. Travel behavior of phonons in phononic crystals: (a) Phonons within the bandgap gradually disappear as they pass through the periodic structure^[72]; (b) phonons outside the bandgap are able to pass through the periodic structure^[72].

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图 16 局域共振型声子晶体　(a) 相同材料下具有柱状结构的纳米声子晶体(红色)声子谱、群速度和没有柱状结构的膜(绿色)的声子谱、群速度的对比^[78]; (b) 柱状分子纳米声子晶体的结构^[78]

Figure 16. Phononic crystals with resonant cavities: (a) Comparison of phonon spectra and group velocities between nanophononic crystals with pillars (red) and membranes without pillars (green) made of the same material^[78]; (b) structure of the nanophononic crystal with pillars^[78].

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图 17 不同波长声子传递的信息^[80]

Figure 17. Information transmission by phonons of different wavelengths^[80].

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图 18 (a) 简化后的共振器模型^[81]; (b) 四种不同类型的周期型结构(由上至下分别是打孔结构、翅形结构、柱形结构、材料掺杂结构), 以及四种结构的热导率与结构尺寸的变化关系^[84]; (c) 不同共振器结构GNR的热导率^[87]

Figure 18. (a) Simplified resonator model^[81]; (b) four different types of periodic structures (from top to bottom: holes, wings, pillars, and material-doped structure), and the relationship between thermal conductivity and structural size for the four structures^[84]; (c) thermal conductivity of different resonator structures in GNRs^[87].

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图 19 二维石墨烯声子晶体的热导率随(a)温度, (b)周期长度的变化情况^[88]; (c) 声子晶体在激光加热下的温度变化情况, 符号的大小代表测量误差^[89]

Figure 19. Thermal conductivity variation with temperature (a) and system periodic length (b) ^[88]; (c) temperature changes of the nanophononic crystals under laser heating, where the symbol size represents measurement errors^[89].

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图 20 (a) 三维声子晶体中声子传输方式示意图^[91]; (b) 不同空孔隙率下硅基声子晶体热导率随温度变化情况^[91]

Figure 20. (a) Schematic representation of phonon transport in three-dimensional nanophononic crystals^[91]; (b) temperature dependence of thermal conductivity in silicon-based nanophononic crystals with different porosities^[91].

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图 21 (a) 光滑的VLS Si纳米线的热导率, 阴影区域是具有均方根粗糙度为1—3 Å的理论预测^[93,98]; (b) 粗糙的Si纳米线的热导率(均方根粗糙度为 3—3.25 nm)^[93,99]; (c) 在纳米管(晶格结构)上沉积金属铂的示意图^[95]; (d), (e) 相应的低放大倍率透射电子显微镜图像, C₉H₁₆Pt沉积在电极上前(d)后(e)的碳纳米管(中间的浅灰色线)的扫描电子显微镜图像, 比例尺为5 mm; (f) 热导率测试实验装置示意图^[95]

Figure 21. (a) Thermal conductivity of smooth VLS Si nanowires. The shaded areas represent theoretical predictions result with root mean square roughness of 1–3 Å^[93,98]. (b) Thermal conductivity of rough Si nanowires with root mean square roughness of 3.00–3.25 nm^[93,99]. (c) Schematic description of depositing amorphous C₉H₁₆Pt (black dots) on a nanotube (lattice structure)^[95]. (d), (e) Corresponding low-magnification transmission electron microscopy images of the same carbon nanotube, showing the condition before (d) and after (e) C₉H₁₆Pt deposition. Scanning electron microscopy image of a carbon nanotube (light gray line in the center) with C₉H₁₆Pt deposited on the electrodes, the scale bar is 5 mm. (f) Schematic diagram of experimental device for thermal conductivity test^[95]

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图 22 (a) 周期排布孔硅纳米材料^[102]; (b) 规整排布孔硅纳米材料^[102]; (c) 不同频率的波穿过规整和无规多孔介质的散射情况^[103]; (d) 无规多孔硅热导率在不同孔隙率P = 64％, 71％, 79％, 89％的变化情况^[104]

Figure 22. (a) Periodic porous silicon nanomaterials^[102]; (b) regularly arranged porous silicon nanomaterials^[102]; (c) scattering of waves at different frequencies through periodic porous and amorphous porous media^[103]; (d) variation of thermal conductivity for amorphous porous silicon with different porosities P = 64%, 71%, 79%, 89%^[104].

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图 23 (a) 氦离子辐照后实验实物图和示意图^[106]; (b) 硅纳米线热导率随掺杂浓度的变化^[106]; (c) 单壁碳纳米管热导率随掺杂浓度下降示意图^[16]; (d) 掺杂前后氮化硼热导率随温度的变化^[110]

Figure 23. (a) Experimental images and schematic diagram after helium ion irradiation^[106]; (b) thermal conductivity variation of silicon nanowire with doping concentration^[106]; (c) decrease in thermal conductivity of single-walled carbon nanotubes with doping concentration^[16]; (d) variation of boron nitride thermal conductivity with temperature before and after doping^[110].

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图 24 (a) 由两个不同节段组成的界面的示意图和温度分布图^[2]; (b) z = 0处的理想界面分别延伸到每侧的有限厚度$ {\delta }_{1} $和$ {\delta }_{2} $, 与在界面上的声子反射和折射示意图^[2]

Figure 24. (a) Schematic diagram and temperature distribution of an interface composed of two different segments^[2]; (b) the ideal interface extending to finite thicknesses $ {\delta }_{1} $ and $ {\delta }_{2} $ on each side of $ z=0 $, with phonon reflection and refraction at the interface^[2].

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图 25 计算界面热阻的理论和数值方法比较

Figure 25. Comparison of theoretical and numerical methods for calculating interface thermal resistance.

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图 26 界面热阻的实验测量方法^[55,106,127]

Figure 26. Experimental methods for measuring interface thermal resistance^[55,106,127].

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图 27 界面热阻不对称性的理论计算与实验验证　(a) 理论^[137]; (b) 实验^[126]

Figure 27. Theoretical calculation and experimental verification of interface thermal resistance asymmetry: (a) Theoretical^[137]; (b) experiment^[126].

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图 28 (a) 均匀、突变和质量梯度一维原子链的示意图^[142]; (b) 界面热导与质量梯度中间层的层数关系^[142]; (c) 具有质量梯度Si/Ge界面示意图^[142]; (d) 界面热导与温度的关系^[142]

Figure 28. (a) Schematic diagram of uniform, abrupt, and mass-graded one-dimensional atomic chains^[142]; (b) relationship between interface thermal conductivity and the number of layers in the mass-graded intermediate layer^[142]; (c) schematic diagram of Si/Ge interface with mass gradient^[142]; (d) relationship between interfacial thermal conductance and temperature^[142].

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图 29 (a) 腔光力学系统示意图, 包含光学腔与一维膜阵列的相互作用^[147]; (b) 系统简化模型^[147]; (c), (d) 最低(最高)模式下的声子数^[147]

Figure 29. (a) Schematic diagram of the optomechanical system, including the interaction between the optical cavity and the one-dimensional membrane array^[147]; (b) simplified model of the system^[147]; (c), (d) phonon numbers in the lowest (highest) mode^[147].

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图 30 (a) 具有非线性反馈的机械振子系统^[148]; (b), (c) 最低模式(n=1)的振动能在长时间稳态下占主导地位, 实现最低模式的声子(能量)凝聚^[148]

Figure 30. (a) Mechanical oscillator system with nonlinear feedback^[148]; (b), (c) the vibrational energy of the lowest mode (n=1) dominates in the long-term steady state, achieving phonon (energy) condensation in the lowest mode^[148].

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图 31 稳态下最低模的声子统计. 非线性反馈诱导的 (a) 最低模式的相位图与 (b) 最低模式的噪声功率谱密度^[148]

Figure 31. Depicts the phonon statistics of the lowest mode in the steady state. The phase diagram of the lowest mode induced by nonlinear feedback is shown in Figure (a), while Figure (b) displays the noise power spectral density of the lowest mode^[148].

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图 32 (a) 低温下声学声子热导率的实验值与温度的关系^[152]; 具有(b) catenoidal形量子结构的量子线、(c) 量子点调制的量子结构, 以及(d) 双腔结构调制石墨烯纳米带的量子结构的声子输运和热导率^[155,156]

Figure 32. (a) The relationship between the experimental value of acoustic phonon thermal conductivity and temperature^[152]; phonon transport and thermal conductivity of the quantum structure with (b) catenoidal shaped quantum structure, (c) quantum dot modulated quantum structure, and (d) double cavity structure modulated graphene nanoribbon quantum structure^[155,156].

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图 33 (a) 铁磁器件; (b) 铁磁体/非磁性金属(F/N)界面器件; (c) 磁振子-声子散射器件中的能量交换; (d) 不同外磁场下磁振子非弹性热流随温差的变化^[164]

Figure 33. (a) Ferromagnetic (FM) devices; (b) ferromagnetic/nonmagnetic (F/N) interfaces; (c) the energy exchange in the present magnon-phonon scattering (MPS) devices; (d) the temperature difference and external magnetic field dependence of inelastic heat flow^[164].

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图 34 (a) 声子霍尔效应^[165]; (b) 理论分析, 霍尔热导率$ {K}_{xy}/T $随温度的变化规律^[166]; 基于α-RuCl3实验测得横向热导率$ {K}_{xy} $与(c) 磁场和(d) 温度的关系^[166]

Figure 34. (a) Phonon Hall effect^[165]; (b) three temperature regions for the thermal Hall conductivity^[166]; (c) magnetic field and (d) temperature dependence of the transversal heat conductivity $ {K}_{xy} $of α-RuCl₃^[166].

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图 35 不同温度下(a) 声子霍尔电导率$ {K}_{xy} $^[169]和(b) 一阶导数$ {\mathrm{d}}{K}_{xy}/{\mathrm{d}}h $与磁场 h 的非单调关系, T = 50 (点线), 100 (虚线), 和 300 K (实线) ^[169]

Figure 35. (a) Phonon Hall conductivity $ {K}_{xy} $ vs. magnetic field h for different temperatures^[169]; (b) $ {\mathrm{d}}{K}_{xy}/{\mathrm{d}}h $ as a function of magnetic field at different temperatures: T =50 (dotted line), 100 (dashed line), and 300 K (solid line) ^[169].

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图 36 蜂窝状AB晶格中的谷声子 (a) 蜂窝状AB晶格的声子色散关系以及子格A与B在$ k\left(k'\right) $点的声子振动模式示意图^[172]; (b) 在$ k\left(k'\right) $点处, 子格A与B非局域部分的相位示意图^[172]; (c) 在$ k\left(k'\right) $点处, 1—4各支能带的声子赝角动量^[172]

Figure 36. Valley phonons in a honeycomb AB lattice: (a) Phonon dispersion relation of a honeycomb AB lattice^[172]; (b) phase correlation of the phonon nonlocal part for sublattice A (upper two panels) and sublattice B (lower two panels)^[172]; (c) phonon pseudo-angular momentum (PAM) for bands 1 to 4^[172].

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图 37 (a) 右旋和(b) 左旋的声子谱, 它们显示出相同的色散但相反的手性分布^[177]. 在图(a)中P附近的频率下, (c) 只允许左旋声子从左到右通过螺旋链, (d) 当螺旋的手性发生改变时, 只允许右旋声子从左到右通过螺旋链^[177]

Figure 37. The phonon spectra for (a) right-handed helix and (b) left-handed helix, which show the same dispersion but opposite chirality distribution^[177]; (c) at the frequency around P in figure (a) only left-handed phonons are allowed to pass the helix from left to right, (d) when the chirality of the helix is switched, the situation reverses^[177].

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图 38 (a) 铁磁性材料中自旋塞贝克效应的示意图, 通过向铁磁体施加温度梯度, 在相邻的非磁性层(即Cu)中产生自旋电流^[178]; (b) 手性声子激发的自旋塞贝克效应的示意图, 通过向非磁性手性体系施加温度梯度, 在没有磁化和磁场的情况下, 手性声子通过手性体系在Cu层中产生自旋电流^[178]

Figure 38. (a) Schematic illustration of the spin Seebeck effect in a ferromagnetic material. By applying a temperature gradient to the ferromagnet, a spin current is generated in an adjacent non-magnetic layer (that is, Cu) ^[178]. (b) Schematic illustration of the chiral-phonon-activated spin Seebeck effect. When a temperature difference is applied to a chiral material, a spin current can be produced in the Cu layer due to the propagation of the chiral phonons through the material in the absence of the magnetization and magnetic field^[178].

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