Question

(b) The polarization of GaN-based heterostructures can create a polarization induced band bending at the heterointerface.

1. Explain how this can be give rise to a 2-dimensional electron gas (2DEG) or 2-dimensional hole gas (2DHG)
   20. marks)

Answer 1

Introduction
The AlGaN/GaN system is very attractive for different types of semiconductor devices. Popular applications for AlGaN/GaN heterostructures are, among others, microwave power high electron mobility transistor (HEMTs) (Refs. 1–4) and power switches.5,6 In microwave AlGaN/GaN HEMTs, the layer structure of the active device region (i.e., gated region) frequently consists of a metal gate followed by a GaN cap, an AlGaN barrier and a GaN channel/buffer layer. A characteristic feature of these heterostructures is the formation of a two-dimensional electron gas (2DEG) with high sheet density at the interface between AlGaN barrier and GaN channel layer. Due to strong polarization effects, 2DEGs with sheet densities exceeding 1013 cm2 are easily formed even without intentionally doping the AlGaN barrier.7,8 In GaN HEMTs, the 2DEG serves as the channel and its electron sheet density can be controlled by the gate voltage. It has been predicted by simulations that under certain conditions in GaN/AlGaN/GaN structures a two-dimensional hole gas (2DHG) may form at the upper GaN/AlGaN interface in addition to the 2DEG at the lower AlGaN/GaN interface.9,10 The simultaneous appearance of 2DEGs and 2DHGs in GaN/AlGaN/GaN structures has recently been confirmed by experiments.10–12 The coexistence of electron and hole gases is undesirable for conventional n-channel HEMT operation and should be avoided. For other applications, however, they may be beneficial. Currently there is an intense interest in electron-hole-pair (exciton) condensation effects that may occur in closely spaced 2DEGs and 2DHGs separated by a thin barrier13,14 and an interesting device concept, the bilayer pseudospin field-effect transistor (BiSFET) exploiting these effects has been proposed.15 Another potential application for structures with coexisting 2DEGs and 2DHGs could be varactors with tailored capacitance-voltage characteristics. GaN-based varactor diodes are currently intensively studied.16,17 Finally, 2DHGs alone, i.e., without coexisting 2DEGs, are useful for p-channel GaN-based HEMTs. In Ref. 18, for example, a p-channel GaN-based HEMT with GaN barrier and InGaN channel has been reported. In the present paper, the formation of 2DEGs and 2DHGs and the conditions for the coexistence of two carrier gases in GaN/AlGaN/GaN structures are investigated in detail. Special emphasis is put on the effects of the layer design and the bias conditions on the electron and hole sheet densities. We focus on wurtzite Ga-face structures which are commonly used in GaN-based devices. The paper is organized as follows: In Sec. II the details and results of numerical Schro¨dinger-Poisson simulations are presented. In Sec. III, an analytical model that provides useful insights in the conditions for the formation of coexisting 2DEGs and 2DHGs is developed. Results calculated with the new model are compared with those obtained from computationally more expensive Schro¨dinger-Poisson solutions and an excellent agreement is obtained.

II. NUMERICAL SCHRO¨ DINGER-POISSON SIMULATIONS

A. Simulation framework and basic device structure

To calculate the free carrier distributions and band diagrams we employ the effective-mass approximation. Along the growth direction of GaN/AlGaN/GaN heterostructures (i.e., in the z direction), carriers are confined in deep and narrow quantum wells. This requires a self-consistent solution of the one-dimensional Schro¨dinger and Poisson equations. Since both electrons and holes are considered, two Schro¨- dinger equations – one for electrons and one for holes – have to be solved. In the present work, our in-house Schro¨dingerPoisson solver,19 expanded by the Schro¨dinger equation for holes, is used. The Schro¨dinger equation for electrons reads as

Here, Ei and wi(z) are the energy level and wave function of the ith subband, mk*(z) is position dependent electron effective mass in the z-direction, i.e., perpendicular to the heterointerface, /(z) is the electrostatic potential, and DEC(z) is equal to the conduction band offset at a heterointerface and zero elsewhere in the structure.

To account for the quantization of the hole gas, the Schro¨dinger equation for holes is solved in analogy to the case for the electrons. For simplicity, instead of heavy-, light-, and split-off valence bands we consider only one valence band with an effective mass equal to the density-ofstates effective hole mass.

In general, the Poisson equation is given by

where Ptot(z) is the overall permanent polarization (i.e., the sum of spontaneous and piezoelectric polarization7,8 ), Nþ D ðzÞ and N D ðzÞ are the ionized donor and acceptor concentrations, e(z) is the dielectric constant, and n(z) and p(z) are the position dependent electron and hole concentrations. Since in the present study we assume only n-type doping, N A in Eq. (2) is set to zero.

For the relative dielectric constant er, energy gap EG, conduction and valence band offsets DEC and DEV, spontaneous polarization PSP, and piezoelectric polarization PPZ in GaN and AlGaN, the models from Ref. 8 have been used. The electron effective masses m k (in the z direction) and m ? (normal to the z direction) for GaN and AlN are taken from Ref. 20 and the corresponding masses for AlGaN are obtained by linear interpolation. For the effective hole mass mh, the density-of-states effective hole mass from Ref. 21 is used.

Figure 1 shows the basic design of the heterostructures investigated in the present work. It consists (from top to bottom) of a Schottky gate, an upper GaN layer serving as a cap and having a thickness tcap, an Al0.3Ga0.7N barrier with a thickness tbar, and a thick GaN bulk having a grounded backside contact. The GaN and AlGaN layers are assumed to be unintentionally n-type doped with a homogeneous donor concentration of 1016 cm3 . The surface potential EC0 is defined as the conduction band edge in the cap at the gate contact and is related to the applied gate voltage VG by

where PhiB is the Schottky barrier height and q is the elementary charge.

Table I summarizes important parameters for GaN and Al0.3Ga0.7N used in the simulations. According to Ref. 8, the spontaneous and piezoelectric polarizations result in a bound polarization charge density of 1.39 1013 cm2 at the upper GaN/AlGaN (cap/barrier) interface and 1.39 1013 cm2 at the lower AlGaN/GaN (barrier/bulk) interface.

III. ANALYTICAL CONSIDERATIONS

A. Model

One can gain further insights into the conditions for the hole gas formation by considering the electrostatics of a GaN/AlGaN/GaN heterostructure with the help of simple first-order analytical considerations. Let us first take the simplified electrostatic arrangement of a GaN/AlGaN/GaN structure as shown in Fig. 6, where the GaN cap and the AlGaN barrier are assumed to be fully depleted from mobile carriers and a 2DEG is formed directly at the barrier/bulk interface. In order to model the conditions for the formation of a 2DHG, the potential at the cap/barrier interface, /cap (given in Volt), is considered. It can be modeled as the superposition of two separate contributions. The first contribution is the potential component upol cap caused by the effect of the polarization charge and the second one is the potential component u0 cap resulting from the effect of the applied surface potential,