General Electronic Model of the Interface
In this chapter we introduce a general basic understanding of the situation that electrons, which are moving through a crystal, find when they encounter a material interface. To stress an analogy from the world of theatre, in this chapter we prepare the s
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General Electronic Model of the Interface
In this chapter we introduce a general basic understanding of the situation that electrons, which are moving through a crystal, find when they encounter a material interface. To stress an analogy from the world of theatre, in this chapter we prepare the stage and set up the scenery for the play called "electron transport", which will be performed in the following chapters. The concept developed in this chapter will allow some statements to be made on the charge distribution at the interface. The influence of these charges on the electrostatic potential then will lead to the basic interface conditions for the Poisson equation. We shall content ourselves with a simple, phenomenological model conception that suffices for setting up the models required by device simulation. We will keep mainly with a more macroscopic model, touching the microscopic phenomena only briefly. A comprehensive overview of the microscopic treatment of interfaces and surfaces can be found for instance in the book by Bechstedt and Enderlein [27].
3.1 Energy Band Structures at the Interface In Chapter 2 we based the description of electron transport in the volume on the Boltzmann transport equation. In order to investigate the effect of an abrupt material interface, we have to pay attention to the question what ingredients of the Boltzmann equation change when we cross the interface. Thus, we have to identify the essential properties of the Boltzmann equation that depend on the material species. Looking at (2.4), we find that these are on one hand the scattering mechanisms, and on the other hand the energy band structure W(k), which is implicitly contained in the group velocity v(k) (see Eq. (2.6)) and in the acceleration force. In (2.4), it is assumed that the distribution function and the potential energy vary only smoothly in the position space. Otherwise it would not be possible to use spatial gradients of these quantities when setting up the flux
D. Schroeder, Modelling of Interface Carrier Transport for Device Simulation © Springer-Verlag Wien 1994
3.1 Energy Band Structures at the Interface
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balance depicted in Fig. 2.2. This assumption is violated if the phase space element is adjacent to a material interface. While the scattering terms for such a phase space element remain the same, the flux coming from that side where the interface lies cannot be determined by a gradient, but must be supplied by a respective interface condition. This behaviour will be considered in detail in Chapter 4. Thus we conclude that the scattering terms stay mainly unaffected by the presence of the interface, while the inertia and acceleration terms of (2.4) require special attention because of the change of the band structure. If we go from the first material across the interface to the second material, we find the energy band structure of the first material at the start and the band structure of the second material at the end of this path. At the interface, a transition between the two energy band structures
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