Sääskilahti K, Oksanen J, Tulkki J
Department of Biomedical Engineering and Computational Science, Aalto University, FI-00076 Aalto, Finland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jul;88(1):012128. doi: 10.1103/PhysRevE.88.012128. Epub 2013 Jul 22.
Modeling of thermal transport in practical nanostructures requires making tradeoffs between the size of the system and the completeness of the model. We study quantum heat transfer in a self-consistent thermal bath setup consisting of two lead regions connected by a center region. Atoms both in the leads and in the center region are coupled to quantum Langevin heat baths that mimic the damping and dephasing of phonon waves by anharmonic scattering. This approach treats the leads and the center region on the same footing and thereby allows for a simple and physically transparent thermalization of the system, enabling also perfect acoustic matching between the leads and the center region. Increasing the strength of the coupling reduces the mean-free path of phonons and gradually shifts phonon transport from ballistic regime to diffusive regime. In the center region, the bath temperatures are determined self-consistently from the requirement of zero net energy exchange between the local heat bath and each atom. By solving the stochastic equations of motion in frequency space and averaging over noise using the general fluctuation-dissipation relation derived by Dhar and Roy [J. Stat. Phys. 125, 801 (2006)], we derive the formula for thermal current, which contains the Caroli formula for phonon transmission function and reduces to the Landauer-Büttiker formula in the limit of vanishing coupling to local heat baths. We prove that the bath temperatures measure local kinetic energy and can, therefore, be interpreted as true atomic temperatures. In a setup where phonon reflections are eliminated, the Boltzmann transport equation under gray approximation with full phonon dispersion is shown to be equivalent to the self-consistent heat bath model. We also study thermal transport through two-dimensional constrictions in square lattice and graphene and discuss the differences between the exact solution and linear approximations.
对实际纳米结构中的热输运进行建模需要在系统大小和模型完整性之间进行权衡。我们研究了在一种自洽热浴设置中的量子热传递，该设置由通过中心区域连接的两个引线区域组成。引线区域和中心区域中的原子都与量子朗之万热浴耦合，这些热浴通过非谐散射模拟声子波的阻尼和退相。这种方法对引线区域和中心区域一视同仁，从而实现了系统简单且物理上透明的热化，还能使引线区域和中心区域之间实现完美的声学匹配。增加耦合强度会减小声子的平均自由程，并逐渐将声子输运从弹道 regime 转变为扩散 regime。在中心区域，根据局部热浴与每个原子之间净能量交换为零的要求自洽地确定热浴温度。通过求解频率空间中的随机运动方程，并使用 Dhar 和 Roy [《统计物理杂志》125, 801 (2006)] 推导的一般涨落 - 耗散关系对噪声进行平均，我们得出了热流公式，该公式包含声子传输函数的卡罗利公式，并在与局部热浴耦合消失的极限情况下简化为兰道尔 - 布蒂克尔公式。我们证明热浴温度测量的是局部动能，因此可以解释为真实的原子温度。在消除声子反射的设置中，具有完全声子色散的灰色近似下的玻尔兹曼输运方程被证明与自洽热浴模型等效。我们还研究了通过方形晶格和石墨烯中的二维收缩的热输运，并讨论了精确解与线性近似之间的差异。
