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Figure 7.10 Schematic diagram of a multilayer structure based on piezoelectric material giving the notation used in text.
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of the displacement vector ( >) at the adjacent interfaces are applied as:
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where linear material properties t are assumed, i.e. D{ = eiEi + Pi- Though the piezoelectric polarisation vector can have an arbitrary direction only the component along the growth direction will influence the electronic properties of a quantum well structure and therefore only this component will be considered here. For an arbitrary n-layer structure equation (7.57) gives a system of n 1 linear equations:
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which can be solved provided appropriate boundary conditions across the structure are defined. Though there are no constraints on the choice of the boundary condition, the natural choice are the 'hard wall' boundary conditions, which enforce the total
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te is used to represent the permittivity in this chapter to distinguish it from e which is the commonly accepted symbol for the components of strain.
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potential drop across the structure to be zero. All other possibilities would imply that the quantum well structure acts as a voltage source in the closed circuit. For a periodic structure (e.g. a superlattice) the hard wall and the periodic boundary conditions are equivalent. Knowing the boundary condition the necessary nth equation of the system is defined and reads:
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where Li is the width of the i-th layer. The solution of the system gives a rather simple equation:
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from which the intrinsic electric field can be easily calculated. It is interesting to see that in the quantum wells the electric field exists even in the layers with zero piezoelectric polarisation, this is a consequence of the charge induced at the interfaces with the adjacent piezoelectric layers.
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Figure 7.11 The induced piezoelectric field in a GaN well and AUxGa1-xN barrier layers grown on a GaN substrate as a function of the barrier layer composition x.
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In the case of wurtzite GaN-based quantum wells, the piezoelectric field due to the large strain and piezoelectric constants, can be stronger than 1 MV/cm [156]. Fig. 7.11
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shows such an example, where the piezoelectric field in the well and barrier layers of a single GaN/AlGaN quantum well structure grown on a GaN substrate is calculated as a function of the barrier layer composition. The electric field in an unstrained 20 A wide layer is as large as 5 MV/cm while in a thicker 60 A strained AlGaN layer it is proportionally smaller but still around 1 MV/cm. The piezoelectric field in GaN alloys is an order of magnitude larger than in another widely used material cubic InGaAs which exhibits piezoelectric fields of around ~ 100 kV/cm [157]. 7.7 EFFECT OF PIEZOELECTRIC FIELDS ON QUANTUM WELLS As shown in the previous section, the piezoelectric effect manifests itself as an electric field in the layers of the quantum well structure. To explore the influence of such fields on the band profile and the electronic structure, the simplest case of a single quantum well will be considered. Also, without loss of generality a uniform dielectric constant can be assumed across the structure. Hence, the magnitude of the piezoelectric fields in the quantum well, assuming identical barriers, follows from equation (7.60) as:
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where Pw,b are the polarisation magnitudes in the well and the barrier respectively and e is the constant permittivity. The electric fields in the well and the barrier layer must be of opposite sign as the overall potential drop across the quantum well has to be zero. This can be expressed analytically using equation (7.59) as: Aspx upca encoder on .net
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This implies that the distribution of the electric field across the structure is proportional to the ratio of the layer widths. Consequently, the thinner epitaxial layer will have a higher electric field than that of the thicker. The piezoelectric field alters the quantum well band profile which can be represented as a step-linear potential (with respect to the coordinate system as in Fig. 7.12):
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where lw is the well width and Vb is the barrier height. The previous potential definition is equivalent to a biased quantum well structure and the solutions can be found as a linear combination of Airy's functions as discussed in 2. Another important effect is that the piezoelectric field can be screened by free electrons. The linear potential causes electrons to gather close to the interfaces, therefore inducing space-charges. If the carrier density in the quantum well is comparable to the density of the piezoelectric induced charges the electrostatic field due to the free electrons can suppress the piezoelectric field. However, as the induced charge at the
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