In this report, we have both numerically and analytically calculated the phonon drag thermopower, $S_{pz}$ and, energy loss rate, $F_{pz}$ due to piezoelectric surface acoustic phonons present in the underlying piezoelectric substrate at a distance, $d$ apart from single and bilayer graphene as a function of carrier's temperature, concentration and phonon mean free path. The numerical results are then reduced to analytical results in the Bloch-Gruneisen regime and the corresponding power dependency of the quantities described above are obtained. We have also included the effect of Thomas-Fermi and Random Phase Approximation screening into account in these calculations. The obtained unscreened results are further reduced in much simpler form for a particular case having no distance between the graphene samples and the underlying substrate or at larger values of distances. The temperature dependency in $S_{pz}$ and $F_{pz}$ for both SLG and BLG systems is found to be one order less than the in-plane acoustic phonons dependency, which is directly related to the matrix element of electron-in plane acoustic phonon interaction having a linear dependence on the phonon wave vector, $q.$ Furthermore, a comparison of these calculations is made between the in-plane and piezoelectric surface acoustic phonon modes. One finds a crossover between these phonons modes depicting these are important in providing effective scattering channels for energy relaxation in typical experimental situations at different temperatures.

在此报告中，我们通过数值和分析方式计算了声子阻力热功率， ${\mathrm{小号}}_{pž}$ 而且，能量损失率 $F_{pž}$ 由于下方的压电基板中一定距离处存在压电表面声子， $d$除了单层和双层石墨烯外，还取决于载流子的温度，浓度和声子平均自由程。然后将数值结果简化为Bloch-Gruneisen体制中的分析结果，并获得上述量的相应功率依存关系。在这些计算中，我们还考虑了Thomas-Fermi和随机相位近似筛选的影响。对于在石墨烯样品与下面的基底之间没有距离或距离值较大的特定情况，以更简单的形式进一步减少了获得的未经筛选的结果。温度依赖性${\mathrm{小号}}_{pž}$ 和 $F_{pž}$ 对于SLG和BLG系统，发现其均比面内声子的声子依赖性小一个量级，这与电子-面内声子声子相互作用的矩阵元素直接相关，该线性关系与声子波矢量线性相关， $q。$此外，在平面内和压电表面声子声子模式之间对这些计算进行了比较。人们发现这些声子模式之间的交叉，说明这些声子模式对于在不同温度下典型实验情况下提供有效的散射通道以放松能量具有重要意义。