# Internal electric field

GaN/AlN heterostructures grown in the polar direction sustain large internal electric fields in the layers induced by the difference of the spontaneous and piezoelectric polarization between the well and barrier materials [Bern 97]. The polarization discontinuity gives rise to bound charges of opposite sign at each interface and consequently to an internal electric field of opposite sign in the well and barrier layers. For a thin layer of GaN within an AlN matrix, the internal field in the GaN well can be as large as 10 MV/cm.

As illustrated in Fig. 12.1, the heterostructure potential *U _{H}*(z) in nitride QWs has a saw tooth shape given by:

Fig. 12.1. a) Conduction-band profile and energy levels for a 1.3-nm-thick GaN/AlN QW. b) Calculated energy for *e _{12}* and

*e*ISB transitions for a GaN/AlN QW with 3-nm-thick barriers. c) The same as a), but for a 3-nm-thick QW.

_{13}where *F _{w} (F_{b})* is the electric field in the QW (barrier),

*AE*is the conduction band offset, and the coordinates

^{c}*zi*are defined by the heterostructure geometry.

In the case of nitride superlattices, the internal electric field is often calculated assuming zero potential drop at each period^{[1]} [Bern 98, Lero 98]:

where *L _{w} (L_{b})* is the QW (barrier) thickness. Taking into account the continuity of the perpendicular component of the electric displacement vector D at the heterointerfaces, we obtain:

where *e _{0}* is the vacuum permittivity,

*e*is the relative permittivity of the QW (barrier) material, and

_{w}(e_{b})*P*is the polarization in the QW (barrier). The internal field can be expressed as:

_{w}(P_{b})

These expressions show that the electric field in the QW increases with the polarization discontinuity and with the barrier thickness, and decreases with the well thickness.

The solution of the Schrodinger equation (12.2) for an undoped nitride QW can be expressed analytically in terms of Airy functions. A numerical solution is required for doped heterostructures to account for the subband population.

Figure 12.1b shows the calculated dependence of the ISB transition energy on the well thickness for an undoped GaN/AlN QW with 3-nm-thick AlN barriers. The calculations were performed using effective mass approximation assuming a conduction-band offset of 1.75 eV and a non-parabolicity in GaN given by eq. (12.3). Two regimes can be distinguished for the ISB energy dependence as illustrated in Figs. 12.1a and b on the example of a thin (1.3-nm) and a thick (3-nm) GaN/AlN QW. For small well thicknesses, the *e _{2}* state is confined by the QW interfaces, which means that the transition energy is governed mostly by the well thickness, i.e. by the quantum size effect. When reducing the well thickness, the calculated

*e*transition energy increases, goes through a maximum for ^0.9 nm, and then decreases. Indeed, when the QW thickness is reduced below 0.9 nm, the

_{12}*e*state becomes delocalized in the triangular potential in the right barrier, and is almost insensitive to the QW thickness, whereas the energy of the

_{2}*e*state continues to increase. Experimentally, the ISB absorption vanishes when the well thickness is reduced because of the delocalization of the

_{1}*e*state in the barrier layers and the consequent reduction of the associated oscillator strength. For well thicknesses above 2 nm, the ground and the first excited states are confined by the V-shaped potential in the GaN well. The transition energy is therefore ruled by the magnitude of the internal field.

_{2}The electric field imposes a lowest limit on the ISB energy that can be attained. Indeed, in a wide well both ei and *e _{2}* energy levels are confined by the triangular potential, and the effective QW size is reduced. The lowest ISB energy for GaN/AlN heterostructure is estimated to be ^0.25 eV (A ^5 pm). To reach longer wavelengths, the internal electric field should be reduced, for example, by reducing the Al content in the barriers, as discussed in Section 12.3.4.

The external electric field can be used to efficiently tune the ISB transition energy via the Stark effect. For a 2.5-nm-thick QW a variation of the field by ^0.5 MV/cm shifts the transition energy by more than 50 meV. This effect can be exploited to design electro-optical modulators [Holm 06].

For thick enough QWs, an ISB absorption from the ground state *e*_{i} to the second excited state *e _{3}* can be observed [Hosh 02]. This transition is allowed by the built-in electric field that breaks the potential symmetry. As shown in Fig. 12.1b, the calculated

*e*transition energy is maximal for QW thickness of ~1.5 nm.

_{13}- [1] The validity of periodic conditions will be discussed in Section 12.2.5.