1.School of Microelectronics, University of Science and Technology of China, Hefei 230022, China 2.State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China 3.Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications, Department of Physics, Hunan Normal University, Changsha 410081, China
Fund Project:Project supported by the Major Instruments Research Program of National Natural Science Foundation of China (Grant No. 61927901), the National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 11925407), and the National Natural Science Foundation of China (Grant No. 11804333)
Received Date:25 April 2021
Accepted Date:09 May 2021
Available Online:07 June 2021
Published Online:20 July 2021
Abstract:Over the past half-century, according to Moore’s law, the sizes of transistors continue shrinking, and the integrated circuits have approached to their physical limits, which puts forward higher requirements for the thermal dissipation capacity of material. Revealing the physical mechanisms of heat conduction in semiconductors is important for thermal managements of devices. Experimentally, it was found that boron arsenide has a very high thermal conductivity compared with diamond, and boron arsenide has lattice constant close to silicon’s lattice constant, which can be heterogeneously integrated into silicon to solve the thermal management problem. However, group III-V boron compounds show abnormal thermal conductivities: the thermal conductivity of boron arsenide is significantly higher than that of boron phosphide and boron antimonide. Here, we use the first-principles calculation and the Boltzmann transport equation to study the thermal conductivity properties of the group III-V boron compounds. Comparison between the IV and III-V semiconductors shows that the high thermal conductivity of boron arsenide is due mainly to the existence of a large frequency gap between the acoustic and the optical branches. The energy sum of two acoustic phonons is less than energy of one optical phonon, which cannot meet the energy conservation requirements of three-phonon scattering, and then seriously restrict the probability of scattering of three phonons. The high thermal conductivity of diamond is due mainly to its great acoustic phonon group velocity. Although the boron phosphide also has a relatively large acoustic phonon group velocity, the frequency gap is relatively small, which cannot effectively suppress the three-phonon scattering, so the thermal conductivity of boron phosphide is less than that of boron arsenide. Although the frequency gap of boron antimonide is similar to that of boron arsenide, the thermal conductivity of boron antimonide is lower than that of boron arsenide due to its smaller acoustic phonon group velocity and larger coupling matrix element. The research provides a new insight into the design of semiconductor materials with high thermal conductivities. Keywords:group III-V semiconductors/ boron arsenide/ thermal conductivity/ anharmonicity
依据上述理论计算方法, 图1(a)中包括了所计算的7种不同闪锌矿结构半导体材料的晶格热导率. 为了检验理论预测的准确性, 给出了部分材料实验测量的热导率数据, 如图中离散点所示. 可以看到对于金刚石(diamond)、BP、BAs, 第一性原理计算结果与实验测得的热导率数值基本一致. 随着温度升高, 所有半导体材料热导率均表现为下降趋势, 其中金刚石具有最高的热导率, 室温下约为2300 W/(m·K), Si, GaP, GaAs热导率较低. 特别需要指出的是, BAs具有与金刚石相比拟的极高热导率[12], 其热导率要高于平均原子质量更小的BP和更大的BSb, 这显然违反了周期表单调变化的性质. 图 1 (a)半导体材料热导率的第一性原理计算(实线), 不同颜色对应不同材料热导率值. 另外离散点来源于已有文献中实验测量的热导率数据, 其中墨绿色的圆圈[29]、正方形[30]和三角形[31]对应BP; 红色的圆圈[14]、正方形[15]和三角形[13]对应BAs; 黑色的圆圈[15]、正方形[32]和三角形[33]对应金刚石材料. (b) 各种不同半导体材料在单位体积下声子群速度的外积(${v_{\lambda} } \otimes {v_{\lambda} }{{/}}{V_0}$)随晶格振动频率的变化. (c)温度为100 K时各种半导体材料声子弛豫时间随晶格振动频率的变化 Figure1. (a) First-principles calculation of thermal conductivity of semiconductor materials (solid line), different colors correspond to different materials. In addition, the discrete points are derived from the thermal conductivity data measured experimentally in existing literature. The olive circles[29], squares[30] and triangles[31] correspond to BP. Red circles[14], squares[15] and triangles[13] correspond to BAs. Black circles[15], squares[32] and triangles[33] correspond to diamond. (b) Outer product of phonon group velocities per unit volume of various semiconductor materials as a function of frequency(${v_{\lambda} } \otimes {v_{\lambda} }{{/}}{V_0}$). (c) Phonon relaxation time of various semiconductor materials at 100 K as a function of frequency.
表1BP, BAs, BSb以及Si的横向光学模式最大振动频率(${\omega _{{\rm{TO}}}}$)、横向声学模式最大振动频率(${\omega _{{\rm{TA}}}}$)、频率间隙($\varDelta = {\omega _{{\rm{TO}}}} - {\omega _{{\rm{TA}}}}$)、组成化合物的元素的原子质量(m, M)、C Table1.Maximum vibration frequency of transverse optical mode, maximum vibration frequency of transverse acoustic mode, frequency gap, atomic mass of elements of compound and elastic force constant of boron phosphide, boron arsenide, boron antimonide and silicon.
图 2 (a)金刚石 (紫色)的晶格振动谱; (b) BAs (红色)和Si (墨绿色)的晶格振动谱; (c) BP (黑色)和BSb (蓝色)的晶格振动谱; (d) BP, BAs, BSb和Si的声子寿命或弛豫时间 Figure2. (a) Lattice vibration spectrum of diamond (violet); (b) lattice vibration spectrum of boron arsenide (red) and silicon (olive); (c) lattice vibration spectra of boron phosphide (black) and boron antimonide (blue); (d) phonon lifetime of boron phosphide, boron arsenide, boron antimony and silicon.
23.3.跃迁矩阵元与声子寿命 -->
3.3.跃迁矩阵元与声子寿命
从表1可以发现BAs和BSb的频率间隙极其接近, 且二者的声学模式均有聚集现象, 但是BAs的热导率却显著高于BSb. 热导率主要由群速度和声子寿命决定. 根据图2(b),(c), BAs的声学支频率略大于BSb, 使得BAs的群速度约为BSb的1.5倍. 对于声子寿命, 除了受频率间隙影响外, 根据(4)式, 还与耦合矩阵元${\varPhi _{\lambda {\lambda ^{'}}{\lambda ^{''}}}}$有关. 图3给出了BP, BAs和BSb的耦合矩阵元随晶格振动频率的变化. 可以看到在低频区BSb的耦合矩阵元大于BAs及BP, 结合玻色分布, 可知BSb的三声子散射比BAs强, 导致BAs的声子寿命约为BSb的1.5倍, 因而耦合矩阵元和群速度的共同作用使得BAs热导率高于BSb. 图 3 BP, BAs和BSb的耦合矩阵元随晶格振动频率的变化 Figure3. Coupling matrix element of BP, BAs and BSb as a function of frequency.
表2各族元素原子p轨道能量(单位为Ry), 其中列出了具体的元素名称 Table2.Atomic p orbital energies of each group of elements (in Ry), and the specific names of the elements are listed.
图 4 (a) BAs和GaAs的晶格振动谱; (b) BP, BAs, BSb, GaP, GaAs的单位体积下群速度外积 Figure4. (a) Lattice vibration spectra of boron arsenide and gallium arsenide; (b) outer product of group velocities per unit volume of boron phosphide, boron arsenide, boron antimony, gallium phosphide and gallium arsenide.
图 1 (a)半导体材料热导率的第一性原理计算(实线), 不同颜色对应不同材料热导率值. 另外离散点来源于已有文献中实验测量的热导率数据, 其中墨绿色的圆圈[29]、正方形[30]和三角形[31]对应BP; 红色的圆圈[14]、正方形[15]和三角形[13]对应BAs; 黑色的圆圈[15]、正方形[32]和三角形[33]对应金刚石材料. (b) 各种不同半导体材料在单位体积下声子群速度的外积(
图 2 (a)金刚石 (紫色)的晶格振动谱; (b) BAs (红色)和Si (墨绿色)的晶格振动谱; (c) BP (黑色)和BSb (蓝色)的晶格振动谱; (d) BP, BAs, BSb和Si的声子寿命或弛豫时间
图 3 BP, BAs和BSb的耦合矩阵元随晶格振动频率的变化
图 4 (a) BAs和GaAs的晶格振动谱; (b) BP, BAs, BSb, GaP, GaAs的单位体积下群速度外积