The Influence of Spontaneous and Piezoelectric Polarization on Novel AlGaN/GaN/InGaN Device Structures
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PsP (C/m 2 ) e 33 (C/mr2 ) e 31 (C/m 2 ) C 1 3 (GPa) C 33 (GPa) a0 (A)
AIN -0.081 1.46 -0.60 108 373 3.112
GaN -0.029 0.73 -0.49 103 405 3.189
InN -0.032 0.97 -0.57 92 224 3.54
Table 1: The constants used to calculate the polarization in III-nitride layers. PSp is the spontaneous polarization. e33 and e31 are piezoelectric constants. C13 and C 33 are elastic deformation constants and ao is the lattice constant.
positive direction is toward the surface by convention.) The polarization in the material can be changed by placing it under strain. This change in polarization is commonly called the piezoelectric polarization and is given by
S2 a - a0 / e3 PPE=2 a0
C13
(1)
C33
where a is the lattice constant under strain, and ao is the lattice constant of the relaxed material. The constants e31 and e33 are piezoelectric constants and C13 and C33 are elastic deformation constants. The total polarization in a given layer is simply the sum of the spontaneous and piezoelectric polarization, i.e., P = PSP + PEP. The constants used in our calculation are from Bernardini et al. [1] and Wright [5] and are shown in Table I. For alloys, the constants are linearly interpolated. At a heterojunction there is usually a change in the polarization on each side. This abrupt change in polarization causes a bound sheet charge. In general, the bound sheet charge is the polarization of the bottom layer minus the polarization of the top layer, a = P (bottom) - P (top). We now examine the particular situation present in ALGaN/GaN field effect transistors. We assume the thin AlGaN layer is pseudomorphically lattice matched to the thick GaN layer below it. Figure 1 shows the situation for both Ga-face and N-face material. In a Ga-faced structure, a positive bound charge is created at the deeper interface which causes the formation of a 2-D electron gas at the lower interface. In N-face material, the positive bound charge is present at the upper interface and the 2-D electron gas will form there. With the proper structure geometry, it may be possible to form a 2-D hole gas at the other interface [6], however, this has yet to be reported experimentally. With a theoretical understanding of polarization we now wish to predict the effect this polarization will have on conduction band diagrams and electron concentrations for a variety of device structures. For this purpose we use a 1-D Schrbdinger-Poisson solver. First the Schrbdinger's equation is solved to find the electron concentration and then Poisson's equation is solved to determine the conduction band profile. The new conduction band profile will modify the solution of the electron concentration, so the process is repeated until there is little change in the solution, i.e., the solution is self-consistent. Our computer program which solves this self-consistent problem, called CBAND, is very similar to other self-consistent solvers [7]. The program however, must be modified to incorporate the effects of the spontaneous and piezoelectric polarization. This is accomplished by adding thin
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