Flexoelectricity is the polar response of an insulator to strain gradients such as bending. While the size dependence makes it weak in bulk systems in comparison to piezoelectricity, it can play a bigger role in nanoscale systems such as thin films and nanotubes (NTs). In this paper we demonstrate using first-principles calculations that the walls of carbon nanotubes (CNTs) and transition metal dichalcogenide nanotubes (TMD NTs) are polarized in the radial direction, the strength of the polarization increasing as the size of the NT decreases. This is reminiscent of a flexoelectric response in bulk insulators, the strain gradient being achieved by bending the 2D monolayers into NTs. For CNTs and TMD NTs with chiral indices (n, m), the radial polarization of the walls P _R diverges below , where C(n, m) is the circumference and a is the monolayer lattice constant. For CNTs, P _R drops to zero above this value but for TMD NTs there is a non-zero polarization, which is ionic rather than electronic. The size dependence of P _R in the TMD NTs is interesting: it increases gradually and reaches a maximum of P _R ∼ 100 C cm⁻² at C(n, m)/a ∼ 15, then decreases until C(n, m)/a ∼ 10 where it starts to diverge. Measurements of the radial strain on the bonds with respect to the monolayers shows that this polarization is the result of a larger strain on the outer bonds than the inner bonds, but did not explain the peculiar size dependence. These results suggest that while the walls of smaller CNTs and TMD NTs are polarized, the walls of larger TMD NTs are also polarized due to a difference in strain on the inner and outer bonds.

柔性电是绝缘体对应变梯度（例如弯曲）的极性响应。虽然与压电相比，尺寸依赖性使其在体系统中较弱，但它可以在薄膜和纳米管 (NT) 等纳米级系统中发挥更大的作用。在本文中，我们使用第一性原理计算证明碳纳米管 (CNT) 和过渡金属二硫属化物纳米管 (TMD NT) 的壁在径向上极化，极化强度随着 NT 尺寸的减小而增加。这让人想起体绝缘体中的挠曲电响应，应变梯度是通过将 2D 单层弯曲成 NTs 来实现的。对于具有手性指数 ( n , m ) 的CNTs 和 TMD NTs，壁的径向极化P _R发散于以下，其中C ( n , m ) 是周长，a是单层晶格常数。对于 CNT，P _R下降到高于此值的零，但对于 TMD NT，则存在非零极化，这是离子极化而不是电子极化。TMD NTs中P _R的大小依赖性很有趣：它逐渐增加并在C ( n , m )/ a ∼ 15处达到P _R ∼ 100 C cm ^-2的最大值，然后减小直到C ( n , m)/ a ∼ 10 开始发散。相对于单层的键上的径向应变的测量表明，这种极化是外键上的应变大于内键的结果，但没有解释特殊的尺寸依赖性。这些结果表明，虽然较小的 CNT 和 TMD NT 的壁被极化，但由于内键和外键的应变差异，较大的 TMD NT 的壁也被极化。