JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 12, NO. 4, DECEMBER 2014 415 Intrinsic Limits of Electron Mobility in Modulation-Doped AlGaN/GaN 2D Electron Gas by Phonon Scattering Liang Pang Abstract⎯We theoretically present the intrinsic advantage of modulation-doping technique is that nearly all limits to electron mobility in the modulation-doped the carriers are spatially separated from their donor AlGaN/GaN two-dimensional electron gas (2DEG) due impurities, which strongly suppresses the impurity to effects including acoustic deformation potential (DP) scattering effect. Consequently, the 2DEG carrier mobility scattering, piezoelectric scattering (PE), and polar-optic is further enhanced, making the system a good candidate phonon scattering (POP). We find that DE and PE are for high-frequency devices like the HEMTs. However, the more significant limiting factors at intermediate other scattering effects largely influence the transport temperatures of 40 K to 250 K, while POP becomes properties of the 2DEG carriers. The significance of dominant as room temperature is approached. Detailed different scattering mechanisms under different conditions numerical results are presented for the change of like temperature and electron sheet density, and the electron mobility with respect to temperature and limitations that these mechanisms may have on the device carrier density. We conclude that these three types of performance are crucial from both physical and technical phonon scattering, which are generally determined by points of view[6]. The three phonon scattering types the material properties but not the technical processing, investigated in this paper are acoustic deformation-potential scattering (DP), piezoelectric scattering (PE), and are hard limits to the 2DEG mobility. polar-optical phonon scattering (POP), which are the Index Terms⎯AlGaN/GaN, 2-dimensional electron dominant scattering mechanisms for temperature higher gas, electron mobility, phonon scattering. than 40 K. We study their physical importance, summarize their analytical expressions, and investigate their impacts 1. Introduction on the electron mobility under different temperatures or 2DEG sheet densities. The purpose of this review is to Over the past decade, GaN and related nitrides have facilitate a better physical understanding of these particular been intensively studied for the fabrication of high scattering mechanisms, while emphasizing that it is these temperature, high power, and high frequency electronic three intrinsic material properties that determines the devices, due to the material properties such as high critical fundamental limit of carrier mobility for any given material breakdown field, high saturation electron velocity, good systems. chemical inactivity, and thermal stability. The most prominent example is the high electron mobility transistor (HEMT), in which the two-dimensional electron gas (2DEG) at the AlGaN/GaN interface greatly enhances the electron mobility[1][3]. Because of the large conduction band discontinuity between AlGaN and GaN, and the strong polarization-induced electric field at the interface, the AlGaN/GaN quantum well provides enhanced quantum (a) confinement compared to AlGaAs/GaAs counterpart[4]. Two ways are proposed to dope the structure in order to supply carriers in the 2DEG channel, naming uniform-doping as in Fig. 1 (a) and modulation-doping as in Fig. 1 (b)[5]. The Manuscript received October 2, 2013; revised December 20, 2013. This work was supported in part by the Grainger Center for Electric Machinery and Electromechanics of the University of Illinois. (b) L. Pang is with the Department of Electrical and Computer Engineering, University of Illinois, Urbana 61801, USA. (Corresponding author e-mail: Fig. 1. Conduction band diagram of (a) a uniformly-doped danielangp@ gmail.com). AlGaN/GaN/AlGaN heteostructure and (b) a modulation-doped Digital Object Identifier: 10.3969/j.issn.1674-862X.2014.04.014 AlGaN/GaN/AlGaN heteostructure. 416 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 12, NO. 4, DECEMBER 2014 2. Scattering Mechanisms the solution to the Boltzman equation will yield (1). It should be noted that the solution to the Boltzman equation * The electron mobility, defined by μn=eτ/m , where e is also yields an integral factor term that depends on the angle the electronic charge, τ is the relaxation time, and m* is the between k and k′. However, this term is not included in our effective electron mass, is a critical transport property. calculations since it can be approximated to unity. Although not the focus of this study, it should be noted that 2.2 Piezoelectric Scattering interface-roughness scattering, as well as alloy disorder and For the temperature range used in this study, the dislocation scattering, which are more process-determined [11] factors, are important scattering mechanisms limiting the relaxation rate for PE is given as 2 mobility. Remote modulation-doping scattering from the ⎡⎤2 1913⎛⎞vIzeh 1 ionized impurities in the AlGaN layer is considered =+⎢⎥⎜⎟lt 14 (6) τ ⎢⎥32 32 ⎜⎟vIπτ kE negligible because of the strong screening applied to the PE⎣⎦⎝⎠tl FD DP 2DEG. Consequently, we concern ourselves here only with 1/2 DP, PE, and POP. where kF=(2πns) is the norm of the Fermi wave vector, 2 1/2 2 1/2 It=[(8ħvtkF/3πkBT) +1] , Il=[(8ħvlkF/3πkBT) +1] , z is the 2.1 Acoustic Deformation-Potential Scattering average distance of the electronic wave function from the The relaxation rate for DP is expressed as[7] AlGaN/GaN interface corresponding to the lowest sub-band * 2 (i.e. the width of the quantum well), h14 is the piezoelectric 1 3mEDB kT constant, and vt is the transverse acoustic phonon velocity. = 23 b (1) τρDP 16 vl = The piezoelectric effect is resulted when an electrical where potential is generated in response to a mechanical stress. 13 ⎛⎞33emn2 * This occurs in GaN due to the lack of inversion symmetry b = ⎜⎟s (2) (i.e. GaN has a non-centrosymmetric crystalline structure) 8= 2εε ⎝⎠0 s such that an elastic strain will result in a polarization of the [12] is a variational parameter, vl is the longitudinal acoustic unit cell . Piezoelectric scattering is the additional * phonon velocity, ED is the deformation potential, m is the coupling between the acoustic phonons and the free carriers effective electron mass, kB is the Boltzmann constant, T is caused by this potential. GaN along with other nitrides have the temperature, ρ is the mass density, h is the Plank relatively large piezoelectric constants among III-V constant, e is the electron charge, ns is the conduction semiconductors, so the piezoelectric scattering must be electron concentration, ε0 is the vacuum permittivity, and εs taken into account. Here we will examine (6) to determine is the relative permittivity. The origin of (1) is briefly the dependence of piezoelectric scattering on temperature discussed here. and electron sheet density. The variational parameter b comes from the Fang- [8] Howard wave function 2.3 Polar-Optical Phonon Scattering [14] 1 For POP, the relaxation rate is expressed as ψ ()zb=− (3 /22 )2 z exp( bz / ) (3) FH 1 2emN2 * ()( TGk ) = B 0 (7) which is used for triangular potential wells and in our case * 2 τ POP ε = fyPOP () appropriately describes the confinement profile of electrons −1 −1 in the 2DEG. Based on the typical acoustic where ε=ε0/(ε∞ −εs ) is the effective dielectric constant, [9] ħω is the polar optic phonon energy, k =(2m*ħω /ħ2)1/2 deformation-potential electron scattering rate in 3D, SIII , POP 0 POP [10] is the electron wave vector corresponding to the phonon the scattering rate for 2D electrons, SII, is given by energy, NB(T)=1/[exp(ħωPOP/kBT)−1] is the phonon 2 SIqSdq= () (4) occupation number described by the Bose-Einstein II∫ z III z distribution, and fPOP=1+(1−exp(−y))/y is a function with a 2 * where q=|k−k′ |, qz is the z-component of q, and I(qz) is dimensionless variable y=πħ ns/m kBT. The value of G(k0) [10] given by is determined from the Fang-Howard wave function as[14] 2 22 I()qziqzdzzz= ψ FH ()exp() . (5) bb()89++ kb 3 k ∫ Gk()= 00. (8) 0 8()bk+ 3 By solving the Boltzman transport equation for low 0 electric fields and using the principle of detailed balance, POP is an important mechanism in GaN and other III-V we can apply SII to a 2D linearized version of the Boltzman compounds because of the opposite ionic charges on the equation. Assuming the equi-partition regime, then the two atoms. A distortion in the crystal lattice will result in a phonon occupation number is Nq, which can be macroscopic field that scatters the electrons. The long- approximated by kBT/ħωq. If we also assume that the range potential results in a scattering rate matrix element acoustic phonon scattering is elastic, then ħωq drops out of that is inversely proportional to |k−k′|. This means that the the Dirac delta function inside SII. With these assumptions, scattering prefers small angles and we do not have the PANG: Intrinsic Limits of Electron Mobility in Modulation-Doped AlGaN/GaN 2D Electron Gas by Phonon Scattering 417 1/2 randomized collisions. As a result, one cannot make the the Fermi wave vector is proportional to ns , the result is 1/6 same simplifications for the Boltzman equation as with DP. μPE being proportional to ns . Last but not least, we But we will, as with the other scattering mechanisms, obtained the relationship between POP and ns by analyze its dependence on temperature and electron sheet determining the effect that ns has on fPOP(y) and G(k0). For density. T=300 K and the given parameters, we find that fPOP(y) is a decaying function for the given range of ns. G(k0) also has a 3. Analytical Results dependence on the electron sheet density because it is determined by using the Fang-Howard wave function.