摘要拓扑绝缘体这一新物质态是目前凝聚态物理最引人注目的领域之一。和普通绝缘体一样，拓扑绝缘体的内部有能隙，但在其表面由于自旋轨道耦合的作用而形成无能隙并具有满足线性色散的表面电子态。迄今为止，已有4种拓扑绝缘体已被实验确认存在，其中Bi2Se3的能隙最大，并且在第一布里渊区内仅有一个狄拉克点。这些特征非常有利于器件应用，但输运实验却发现，即使在目前最高质量的Bi2Se3晶体中，其费米能级也不处于能隙之中，从而具有体电导。这使得理论预言的拓扑绝缘体的独特性质受到破坏。我们在实验中已成功地使用底门来调控Bi2Se3薄膜的载流子浓度和化学势，从而达到抑制体电导的目的。计算拓扑绝缘体这类材料的化学势对门电压的响应要比传统的半导体体系复杂得多，这主要是因为Bi2Se3除体载流子外，还具有表面上的狄拉克电子。本文从经典的Poisson 方程出发，使用隆格—库塔法三阶、四阶近似对在负偏压下的样品体内的静电势和载流子浓度分布做了数值求解，并与解析解和实验结果做了比较，符合得较好。 计算可以给出输运实验中不能直接测量的表面态电子浓度，从而为理解反弱局域等量子输运性质打下了基础。
关键词：拓扑绝缘体，表面态，狄拉克费米子，静电势，泊松方程
Abstract
A new quantum state called the topological insulator is the most attractive field in the condensed matter physics recently. Comparing with the ordinary insulators, the topological insulators have a gap in the bulk and a gapless surface state originating from the spin orbit coupling satisfy a linear dispersion relation. So far, four kinds of the topological insulators have been confirmed in experiments. Among them the material Bi2Se3 has a largest gap and a single Dirac cone in the first Brillouin zone. These characteristics are benefit to the application in devices. But the transport experiments show that even the high qualified Bi2Se3 crystal, the Fermi level isn’t in the gap, which contribute to the conductivity of the bulk and broke the unique properties predicted in the topological insulators.
In our experiment, we have succeed in using a back gate to modulate the carrier density and the chemical potential in the Bi2Se3 thin film, then, reduce the conductivity of the bulk. It’s more complicate to calculate the effects from the gate than the ordinary semiconductor. Because not the carriers exist in the bulk, but also the dirac fermions at the surface. In this paper, we get a numerical solution of the classical Poisson equation under the negative gate bias using Runge-Kutta three or four order approximation. It turns out to be a good agreement in comparison with the analytical solution and the experiment data. The calculation can give the electron density which can not be measured in the experiment and make a foundation to the understanding of some quantum transporte phenomenon, like the weak anti-localization.
Keywords：Topological Insulator, Surface State, Dirac Fermion, Electrostatic Potential, Poisson Equation
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